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--- abstract: 'Guidelines and consistency rules of UML are used to control the degrees of freedom provided by the language to prevent faults. Guidelines are used in specific domains (e.g., avionics) to recommend the proper use of technologies. Consistency rules are used to deal with inconsistencies in models. However, guidelines and consistency rules use informal restrictions on the uses of languages, which makes checking difficult. In this paper, we consider these problems from a language-theoretic view. We propose the formalism of C-Systems, short for “formal language control systems”. A C-System consists of a controlled grammar and a controlling grammar. Guidelines and consistency rules are formalized as controlling grammars that control the uses of UML, i.e. the derivations using the grammar of UML. This approach can be implemented as a parser, which can automatically verify the rules on a UML user model in XMI format. A comparison to related work shows our contribution: a generic top-down and syntax-based approach that checks language level constraints at compile-time.' author: - Zhe Chen - Gilles Motet bibliography: - 'main.bib' title: 'A Language-theoretic View on Guidelines and Consistency Rules of UML' --- Introduction {#Sec:intro} ============ The UML (Unified Modeling Language) is a graphic modeling language developed by OMG (Object Management Group), and defined by the specifications [@UMLi] and [@UMLs]. UML has emerged as the software industry’s dominant modeling language for specifying, designing and documenting the artifacts of systems [@MRR02][@BRJ05]. Evolving descriptions of software artifacts are frequently inconsistent, and tolerating this inconsistency is important [@Bal91][@EC01]. Different developers construct and update these descriptions at different times during development [@NER00], thus resulting in inconsistencies. They develop multiple views on a system providing pieces of information which are redundant or complementary. Constraints exist on these pieces of information whose violation leads to inconsistent models. Inconsistency problems of UML models have attracted great attention from both academic and industrial communities [@UMLcon02][@UMLcon03][@UMLcon04]. A list of 635 consistency rules are identified by [@VMM05a][@VMM05b]. Guidelines, which also contain a set of rules, are often required on models which are specific to a given context. For instance, OOTiA (Object-Oriented Technology in Aviation) demands that “the length of an inheritance should be less than 6” [@FAA]. This context is domain specific. If these constraints are not respected, the presence of faults is not sure but its risk is high. The context can also be technology specific. For instance, “multiple inheritance should be avoided in safety critical, certified systems” (IL \#38 of [@FAA]), if the UML models are implemented by Java code, as this language does not provide the multiple inheritance mechanism. It seems that consistency rules and guidelines are irrelevant at first glance. However, in fact, they have the same origin from a language-theoretical view. We noticed that both of the two types of potential faults in models come from [*the degrees of freedom*]{} offered by languages. These degrees of freedom cannot be eliminated without reducing the language capabilities [@Mot09]. For instance, the multiple diagrams in UML are useful, as they describe various viewpoints on one system, even if they are at the origin of numerous inconsistencies. In the same way, multiple inheritance can be implemented in the C++ language. To prevent these risks of faults, [*the use of languages*]{} must be controlled. To do it, guidelines are old and popular means in industry. However, their expression is informal and their checking is difficult. For instance, 6 months were needed to check 350 consistency rules on an avionics UML model including 116 class diagrams. This paper aims at formalizing [*the acceptable use of languages*]{} and proposing a way to check [*the use correctness*]{}, by considering guidelines and consistency rules from a language-theoretical view. To achieve this goal, acceptable uses of a language are defined as a grammar handling the productions of the grammar of the language. To support this idea, UML must be specified by a formal language, or at least a language with precisely defined syntax, e.g., XMI in this paper. Thus, a graphic model can be serialized. This formalism also provides a deeper view on the origin of inconsistencies in models. This paper is organized as follows. First, we introduce the grammar of UML in XMI in Section \[Sec:G\_UML\]. Then in Section \[Sec:lcs\], we define the [*C-System*]{}, i.e. a formalism containing controlling grammars that restrict the use of the grammar of UML. We illustrate the formalism using examples in Section \[Sec:exmp\]. Related work and implementation of this approach are discussed in Sections \[Sec:rw\] and \[Sec:disc\]. Section \[Sec:conclusion\] concludes the paper. The Grammar of UML in XMI {#Sec:G_UML} ========================= XMI (XML Metadata Interchange) [@UML-XMI] is used to facilitate interchanging UML models between different modeling tools in XML format. Many tools implement the conversion, e.g., Altova UModel$^\circledR$ can export UML models as XMI files. A UML model in XMI is an XMI-compliant XML document that conforms to its XML schema, and is a derivative of the [*XMI document productions*]{} which is defined as a grammar. The XML schema is a derivative of the [*XMI schema productions*]{}. The XMI specification defines both the XMI schema productions and the XMI document productions in [@UML-XMI]. XMI provides a mapping between a UML user model and an XML document, and a mapping between UML (also MOF) and an XML Schema. XMI generates an XML file using the XMI document productions, and generates an XML schema using the XMI schema productions. Each of the two sets of productions composes a context-free grammar in EBNF [@UML-EBNF]. A UML user model can be expressed using an XMI-compliant XML document that conforms to the corresponding XML Schema, and is a derivative of the XMI document grammar. The grammar and its productions for deriving XMI-compliant XML documents of UML models are defined in [@UML-XMI]. The main part of the grammar is given here after. To make our presentation more concise, we omit declaration and version information of XML files (and the related productions whose names start with “1”). To make later reasoning easier, we modified some representations of the productions, but without changing the generative power of the grammar. 1\. The choice operator “$|$” is used to compose several productions with the same left-hand side into a single line in [@UML-XMI]. We decomposed some of these productions into several productions without the choice operator. An original production $n$ having $k$ choices might be divided into a set of productions $\{ n\_ i\}_{1\leq i \leq k}$. For example, the original production 2 with three choices was divided into the productions 2\_1, 2\_2 and 2\_3. 2\. The closure operator “\*” is used to simplify the representation of the grammar in [@UML-XMI], but it also would make the representation of reasoning confusing. Thus, the productions whose names start with “3” were added to replace the productions with closure operators. The **grammar $G$** of UML in XMI includes the following productions (each production is labeled by a name starting with a digit): 3_1: XMIElements ::= 2:XMIElement 3_2: XMIElements ::= 2:XMIElement 3:XMIElements 2_1: XMIElement ::= 2a:XMIObjectElement 2_2: XMIElement ::= 2b:XMIValueElement 2_3: XMIElement ::= 2c:XMIReferenceElement 2a_1: XMIObjectElement ::= "<" 2k:QName 2d:XMIAttributes "/>" 2a_2: XMIObjectElement ::= "<" 2k:QName 2d:XMIAttributes ">" 3:XMIElements "</" 2k:QName ">" 2b_1: XMIValueElement ::= "<" xmiName ">" value "</" xmiName ">" 2b_2: XMIValueElement ::= "<" xmiName "nil=`true'/>" 2c_1: XMIReferenceElement::= "<" xmiName 2l:LinkAttribs "/>" 2c_2: XMIReferenceElement::= "<" xmiName 2g:TypeAttrib 2l:LinkAttribs "/>" 2d_1: XMIAttributes ::= 2g:TypeAttrib 2e:IdentityAttribs 3h:FeatureAttribs 2d_2: XMIAttributes ::= 2e:IdentityAttribs 3h:FeatureAttribs 2e: IdentityAttribs ::= 2f:IdAttribName "=`" id "'" 2f_1: IdAttribName ::= "xmi:id" 2f_2: IdAttribName ::= xmiIdAttribName 2g: TypeAttrib ::= "xmi:type=`" 2k:QName "'" 3h_1: FeatureAttribs ::= 2h:FeatureAttrib 3h_2: FeatureAttribs ::= 2h:FeatureAttrib 3h:FeatureAttribs 2h_1: FeatureAttrib ::= 2i:XMIValueAttribute 2h_2: FeatureAttrib ::= 2j:XMIReferenceAttribute 2i: XMIValueAttribute ::= xmiName "=`" value "'" 2j: XMIReferenceAttribute ::= xmiName "=`" (refId | 2n:URIref)+"'" 2k: QName ::= "uml:" xmiName | xmiName 2l: LinkAttribs ::= "xmi:idref=`" refId "'" | 2m:Link 2m: Link ::= "href=`" 2n:URIref "'" 2n: URIref ::= (2k:QName)? uriReference In the grammar, the symbol “::=” stands for the conventional rewriting symbol “$\rightarrow$” in formal languages theory [@HU79]. Each nonterminal starts with a capital letter, prefixing a label of the related production, e.g., “2:XMIElement” is a nonterminal with possible productions “2\_1, 2\_2, 2\_3”. Each terminal starts with a lowercase letter or is quoted. As an example to illustrate the use of the grammar, Figure \[Fig:ClassDiag\] represents a package [*Root*]{} which includes three classes, where the class [*FaxMachine*]{} is derived from [*Scanner*]{} and [*Printer*]{}. The core part of the exported XMI 2.1 compliant file (using Altova UModel$^\circledR$) is as follows: <uml:Package xmi:id="U00000001-7510-11d9-86f2-000476a22f44" name="Root"> <packagedElement xmi:type="uml:Class" xmi:id="U572b4953-ad35-496f-af6f-f2f048c163b1" name="Scanner" visibility="public"> <ownedAttribute xmi:type="uml:Property" xmi:id="U46ec6e01-5510-43a2-80e9-89d9b780a60b" name="sid" visibility="protected"/> </packagedElement> <packagedElement xmi:type="uml:Class" xmi:id="Ua9bd8252-0742-4b3e-9b4b-07a95f7d242e" name="Printer" visibility="public"> <ownedAttribute xmi:type="uml:Property" xmi:id="U2ce0e4c8-88ee-445b-8169-f4c483ab9160" name="pid" visibility="protected"/> </packagedElement> <packagedElement xmi:type="uml:Class" xmi:id="U6dea1ea0-81d2-4b9c-aab7-a830765169f0" name="FaxMachine" visibility="public"> <generalization xmi:type="uml:Generalization" xmi:id="U3b334927-5573-40cd-a82b-1ee065ada72c" general="U572b4953-ad35-496f-af6f-f2f048c163b1"/> <generalization xmi:type="uml:Generalization" xmi:id="U86a6818b-f7e7-42d9-a21b-c0e639a4f716" general="Ua9bd8252-0742-4b3e-9b4b-07a95f7d242e"/> </packagedElement> </uml:Package> This text is a derivative of the XMI document productions, c.f. the previous grammar $G$. We may use the sequence of productions “2a\_2, 2k(Package), 2d\_2, 2e, 2f\_1, 3h\_1, 2h\_1, 2i” to derive the following sentential form: <uml:Package xmi:id="U00000001-7510-11d9-86f2-000476a22f44" name="Root"> 3:XMIElements "</" 2k:QName ">" Note that the production 2k has a parameter [*xmiName*]{}, i.e. the value of the terminal when apply the production. In a derivation, we specify a value of the parameter as “2k(value)”. For example, “2k(Package)” is a derivation using 2k with [*xmiName*]{} = “Package”. For simplicity, we consider “2k(value)” as a terminal as a whole. We continue to apply productions, and finally derive the XMI file previously presented. Notice that the model of Fig. \[Fig:ClassDiag\] (both in UML and XML) does not conform to the guidelines in OOTiA about multiple inheritance, since it uses multi-inheritance. The model of Fig. \[Fig:ActDiag\] has an inconsistency: “the number of outgoing edges of $ForkNode$ is not the same as the number of incoming edges of $JoinNode$”. In particular, $JoinNode$ joins two $outgoing$ edges from the same $DecicionNode$, This join transition will never be activated, since only one of the two $outgoing$ edges will be fired. We will define a formal model to check the conformance to these rules by controlling the use of the grammar of UML. The C-System: A Formal Language Control System {#Sec:lcs} ============================================== In this section, we propose the formal model for controlling the use of grammars based on classical language theory [@HU79]. Let $G = (N, T, P, S)$ be a grammar, where $N$ is the set of nonterminals, $T$ is the set of terminals, $P$ is the set of productions of the form $l:A \rightarrow \alpha$ where $l$ is the name of the production, $A \in N$, $\alpha \in (N \bigcup T)^*$, and $S$ is the start symbol. A derivation using a specified production $p$ is denoted by $\alpha \overset{p}\Rightarrow \beta$, and multiple derivations are denoted by $\alpha \Rightarrow^* \gamma$. A **controlling grammar** $\hat{G}$ over a **controlled grammar** (or simply grammar) $G = (N, T, P, S)$ is a quadruple $\hat{G}=(\hat{N}, \hat{T}, \hat{P}, \hat{S})$, where $\hat{T}=P$. The language $L(\hat{G})$ is called a **controlling language**. $\Box$ The symbol $\hat{G}$ is read “control $G$” or “$G$ hat”. For making reading easier, we assume that $N\cap\hat{N}=\emptyset$. $\hat{T}=P$ means that the terminals of $\hat{G}$ are exactly the productions of $G$. If we use an automaton $A$ to process the input string, such that $L(A)=L(G)$, then we can also use a **controlling automaton** $\hat{A}$ to represent the controlling language. As we know, each string $w \in L(G)$ has at least one [*leftmost*]{} derivation (denoted by “$lm$”) using a sequence of productions from $P$, e.g. $p_1p_2 ... p_k$. The controlling grammar restricts the derivation in the sense that the sequences of applied productions should be in the language it specifies, i.e., $p_1p_2 ... p_k \in L(\hat{G})$. Formally, we have the following definition. \[Def:LCS\] Given a grammar $G=(N, T, P, S)$, the language of the grammar with a controlling grammar $\hat{G}$ is: $L(G{\overrightarrow{\cdot}}\hat{G}) = \{w | S \underset{lm}{\overset{p_1}\Rightarrow} w_1 \cdots \underset{lm}{\overset{p_k}\Rightarrow} w_k = w$, $p_1,p_2,...,p_k \in P$ and $p_1p_2...p_k \in L(\hat{G}) \}$ We say that $G$ and $\hat{G}$ constitute a **C-System** $C = G{\overrightarrow{\cdot}}\hat{G}$, short for **formal language control system**. The language $L(C) = L(G{\overrightarrow{\cdot}}\hat{G})$ is called a **global system language**.$\Box$ The symbol ${\overrightarrow{\cdot}}$ is called “meta composition”. Its left operand is controlled by the right operand, which is a meta level grammar. If we use automata-based notations, a string $w \in L(A{\overrightarrow{\cdot}}\hat{A})$ if and only if $A$ accepts $w$, and $\hat{A}$ accepts the sequence of the labels of the transitions used. A ***regular C-System*** is a C-System of which the controlled grammar $G$ is a regular grammar (or $A$ is a finite automaton). Some variants of regular C-Systems are proposed for ensuring system safety requirements, e.g. Input/Output C-Systems [@CM09a], Interface C-Systems [@CM09d][@CM09b]. We denote by $\mathcal{C}_R$ the family of regular C-Systems. A ***context-free C-System*** is a C-System of which the controlled grammar $G$ is a context-free grammar (or $A$ is a pushdown automaton). We denote by $\mathcal{C}_{CF}$ the family of context-free C-Systems. Generally, we denote by $\mathcal{C}_X^Y$ the family of C-Systems that consist of $X$-type controlled grammar and $Y$-type controlling grammar, where $X,Y \in \{R, CF\}$. Although $X,Y$ could be also other types in Chomsky hierarchy, e.g. context-sensitive, this is beyond the scope of this paper. Obviously, the set of accepted inputs is a subset of the controlled language, such that the sequence of the applied productions belongs to the controlling language. Consider a simple example as follows. \[Exmp:Gr\_Gr\] Given a regular grammar $G$ and a regular controlling grammar $\hat{G}$: $$G \left\{ \begin{aligned} p_1:\ & S \rightarrow aS\\ p_2:\ & S \rightarrow bS\\ p_3:\ & S \rightarrow \epsilon \end{aligned} \right.~~~~ \hat{G} \left\{ \begin{aligned} \hat{S} &\rightarrow p_1 \hat{S} | p_3 \hat{S} | p_2 A \\ A &\rightarrow p_2A | p_3A | \epsilon \\ \end{aligned} \right.$$ $L(G)$ accepts the language $(a|b)^*$, e.g., $aab$, $abab$. $L(\hat{G})$ accepts the language $(p_1|p_3)^* p_2 (p_2|p_3)^*$. The trivial grammar $G$ is considered to provide a simple illustration of the introduced principles. The grammars $G$ and $\hat{G}$ constitute a regular C-System $C = G {\overrightarrow{\cdot}}\hat{G} \in \mathcal{C}_R^R$. Given the string $aab \in L(G)$, we conclude that $aab \in L(G{\overrightarrow{\cdot}}\hat{G})$, because we have the leftmost derivations $S \overset{p_1}\Rightarrow aS \overset{p_1}\Rightarrow aaS \overset{p_2}\Rightarrow aabS \overset{p_3}\Rightarrow aab$, where $p_1p_1p_2p_3\in L(\hat{G})$ as $\hat{S} \Rightarrow p_1\hat{S} \Rightarrow p_1p_1\hat{S} \Rightarrow p_1p_1p_2A \Rightarrow p_1p_1p_2p_3A \Rightarrow p_1p_1p_2p_3$. On the contrary, we have $abab \not\in L(G{\overrightarrow{\cdot}}\hat{G})$. Although we have the leftmost derivation $S \overset{p_1}\Rightarrow aS \overset{p_2}\Rightarrow abS \overset{p_1}\Rightarrow abaS \overset{p_2}\Rightarrow ababS \overset{p_3}\Rightarrow abab$, $p_1p_2p_1p_2p_3 \not\in L(\hat{G})$. In fact, the language $L(C) = L(G{\overrightarrow{\cdot}}\hat{G})$ is equivalent to the language $a^*b^+$, which is the subset of $(a|b)^*$ satisfying the constraints: “every $a$ should appear before $b$” and “at least one $b$”. $\Box$ We remark here that our model is different from regularly controlled grammars [@GS68][@DPS97], in the sense that we restrict derivations to be [*leftmost*]{} and allow [*context-free controlling grammars*]{}. These differences result in different theoretical results, which are beyond the scope of this paper. Examples {#Sec:exmp} ======== In this section we use some practical examples to illustrate the idea of the previous section. We denote the grammar of UML by $G = (N,T,P,S)$, where $P$ is the set of productions listed in Section \[Sec:G\_UML\], and each production $p \in P$ is labeled by a name starting with a digit. Consider two rules on class diagrams: **Rule 1:** Each class can have at most one generalization. This rule is a guideline, as we mentioned in Section 1 and at the end of Section 2. This rule is also a consistency rule in the context of Java, since Java does not allow multiple inheritance. However we may derive a class from multiple classes in the context of C++. **Rule 2:** Each class can have at most 30 attributes. This rule may be adopted by software authorities as a guideline in avionics, in order to increase the safety of software systems by minimizing the complexity of classes. Note that these rules cannot be explicitly integrated into the grammar of UML, but only recommended as guidelines or consistency rules. We cannot put rule 1 into the standard of UML, since UML models can be implemented with both C++ and Java programming languages. Rule 2 is a restriction for a specific domain, and we should not require all programmers to use limited attributes by specifying the UML language. We aim to specify the rules from the meta-language level, thus control the use of the language. Consider the example of Fig. \[Fig:ClassDiag\], to obtain the associated XMI text, the sequence of applied productions of $G$ in the leftmost derivation is as follow (“...” stands for some omitted productions, to save space): 2a_2, 2k(Package), 2d_2, 2e, 2f_1, 3h_1, 2h_1, 2i, ..., 2k(packagedElement), ..., 2k(Class), ..., 2k(ownedAttribute), ..., 2k(Property), ..., 2k(packagedElement), ..., 2k(packagedElement), ..., 2k(Class), ..., 2k(ownedAttribute), ..., 2k(Property), ..., 2k(packagedElement), ..., 2k(packagedElement), ..., 2k(Class), ..., 2k(generalization), ..., 2k(Generalization), ..., 2k(generalization), ..., 2k(Generalization), ..., 2k(packagedElement), ..., 2k(Package) Let $c,g$ stand for $2k(Class), 2k(Generalization)$, respectively. Note that the occurrence of two $g$ after the third $c$ violates Rule 1. In fact, all the sequences of productions in the pattern “$...c...g...g...$” are not allowed by the rule (there is no $c$ between the two $g$), indicating that the class has two generalizations. Thus, we propose the following controlling grammar $\hat{G}_c$ to restrict the use of the language to satisfy Rule 1: $$\label{eq:G_c} \hat{G}_c \left\{ \begin{aligned} S &\rightarrow c~~Q_c ~|~ D~S ~|~ D \\ Q_c &\rightarrow c~~Q_c ~|~ g~~Q_g ~|~ D~Q_c ~|~ D\\ Q_g &\rightarrow c~~Q_c ~|~ D~Q_g ~|~ D\\ D &\rightarrow \{ p~|~p \in P ~\wedge ~ p \not\in \{c,g\}\} \end{aligned} \right.$$ where $S, Q_c, Q_g, D$ are nonterminals, $D$ includes all productions except $c, g$. $L(\hat{G}_c)$ accepts the sequences of productions satisfying Rule 1. Implicitly, the controlling grammar specifies an automaton $\hat{A}_c$ in Fig. \[Fig:A\_G\_c\], where $\lightning$ is an implicit error state (the dashed circle). Strings of the pattern $D^*cD^*gD^*gD^*$ will lead $\hat{A}_c$ to the error state. ![The Automaton $\hat{A}_c$[]{data-label="Fig:A_G_c"}](fig_01.mps "fig:")\ If the sequence of productions applied to derive a model is accepted by the language $L(\hat{G}_c)$, then the model conforms to Rule 1. In Fig. \[Fig:ClassDiag\], the derivation of the class $FaxMachine$ uses the pattern $D^*cD^*gD^*gD^* \not\in L(\hat{G}_c)$, which leads to $\lightning$ of the automaton, thus it violates Rule 1. On the contrary, the derivations of $Scanner$ and $Printer$ are accepted by $L(\hat{G}_c)$, thus satisfy Rule 1. Now consider Rule 2. Let $c,pr,pe$ stand for $2k(Class)$, $2k(Property)$,\ $2k(PackagedElement)$, respectively. Note that the occurrence of more than 30 $pr$ after a $c$ violates Rule 2. In fact, all the sequences of productions in the pattern “$...c...(pr...)^n, n > 30$” are not allowed by the rule (there is no $c$ between any two $pr$), indicating that the class has more than 30 attributes. To satisfy Rule 2, we propose the following controlling grammar $\hat{G}_p$ to restrict the use of the language: $$\label{eq:G_p} \hat{G}_p \left\{ \begin{aligned} S &\rightarrow pe~~S ~|~ c~~Q_c ~|~ D~S ~|~ D\\ Q_c &\rightarrow pe~S ~|~ c~~Q_c ~|~ pr~~Q_1 ~|~ D~Q_c ~|~ D\\ Q_i &\rightarrow pe~S ~|~ c~~Q_c ~|~ pr~~Q_{i+1} ~|~ D~Q_i ~|~ D ~~(1\leq i < 30) \\ Q_{30} &\rightarrow pe~~S ~|~ c~~Q_c ~|~ D~Q_{30} ~|~ D \\ D &\rightarrow \{ p~|~p \in P ~\wedge ~ p \not\in \{c,pr,pe\}\} \end{aligned} \right.$$ where $S, Q_c, Q_i$ are nonterminals, $D$ includes all productions except $c, pr, pe$. $L(\hat{G}_p)$ accepts the sequences of productions satisfying Rule 2. Implicitly, the controlling grammar specifies an automaton $\hat{A}_p$ in Fig. \[Fig:A\_G\_p\]. Strings of the pattern “$D^*cD^*(pr~D^*)^n, n>30$” will lead $\hat{A}_p$ to the error state. ![The Automaton $\hat{A}_p$[]{data-label="Fig:A_G_p"}](fig_02.mps "fig:")\ If the sequence of productions applied to derive a model is accepted by the language $L(\hat{G}_p)$, then the model conforms to Rule 2. In Fig. \[Fig:ClassDiag\], the derivations of the classes $Scanner$ and $Printer$ use the pattern $D^*cD^*prD^* \in L(\hat{G}_p)$, thus satisfy Rule 2. Thanks to the controlling grammars, when a model violates required rules, the controlling language will reject the model (an implicit error state $\lightning$ will be activated). Some error handling method may be called to process the error, e.g., printing an error message indicating the position and the cause. We can also use controlling grammar to handle a consistency rule concerning activity diagrams. \[Exmp:Act\_Dig\] In an activity diagram, the number of outgoing edges of $ForkNode$ should be the same as the number of incoming edges of its pairwise $JoinNode$. Let $n,f,j,i,o$ stand for $2k(node)$, $2k(ForkNode)$, $2k(JoinNode)$, $2k(incoming)$, $2k(outgoing)$, respectively. We propose the following controlling grammar $\hat{G}_a$ to restrict the use of the language to satisfy the rule: $$\label{eq:G_a} \hat{G}_a \left\{ \begin{aligned} S &\rightarrow N~~F~~I^*~~Q~~O^*~~N ~|~ N~~I^*~~O^*~~N ~|~ D^*\\ Q &\rightarrow O~~Q~~I ~|~ N~~S~~N~~J \\ N &\rightarrow n~~D^*\\ F &\rightarrow f~~D^*\\ J &\rightarrow j~~D^* \\ I &\rightarrow i~~D^* \\ O &\rightarrow o~~D^* \\ D &\rightarrow \{ p~|~p \in P ~\wedge ~ p \not\in \{n,f,j,i,o\}\} \end{aligned} \right.$$ $L(\hat{G}_a)$ accepts all the sequences of productions of the pattern\ $NFI^*O^nN S NJI^nO^*N$, which leads to models respecting the rule. This context-free grammar implicitly specifies a PDA (Pushdown Automaton [@HU79]), which is more complex than the automata in Figures \[Fig:A\_G\_c\] and \[Fig:A\_G\_p\]. Globally, any UML user model $M$ derived from the C-System $C = G {\overrightarrow{\cdot}}\hat{G}_a \in \mathcal{C}_{CF}^{CF}$, i.e. $M \in L(C)$, conforms to the rule in Example \[Exmp:Act\_Dig\]. As a more concrete instance, we consider the model in Fig. \[Fig:ActDiag\]. The XMI-compliant document of the model in Fig. \[Fig:ActDiag\] is the follows: <packagedElement xmi:type="uml:Activity" xmi:id="U937506ed-af64-44c6-9b4c-e735bb6d8cc6" name="Activity1" visibility="public"> <node xmi:type="uml:InitialNode" xmi:id="U16aa15e8-0e5d- 4fd1-930a-725073ece9f0"> <outgoing xmi:idref="Ue9366b93-a45b-43f1-a201-2038b0bd0b30"/> </node> <node xmi:type="uml:ForkNode" xmi:id="U26768518-a40c- 4713-b35e-c267cc660508" name="ForkNode"> <incoming xmi:idref="Ue9366b93-a45b-43f1-a201-2038b0bd0b30"/> <outgoing xmi:idref="Ua800ba9b-e167-4a7c-a9a9-80e6a77edeb7"/> </node> <node xmi:type="uml:DecisionNode" xmi:id="Uc9e4f0de-8da6- 4c98-9b95-b4cde30ccfc0" name="DecisionNode"> <incoming xmi:idref="Ua800ba9b-e167-4a7c-a9a9-80e6a77edeb7"/> <outgoing xmi:idref="Ua4a2b313-13d6-4d69-9617-4803560731ef"/> <outgoing xmi:idref="U6eede33f-98ac-4654-bb17-dbe6aa7e46be"/> </node> <node xmi:type="uml:JoinNode" xmi:id="Ud304ce3c-ebe4- 4b06-b75a-fa2321f8a151" name="JoinNode"> <incoming xmi:idref="Ua4a2b313-13d6-4d69-9617-4803560731ef"/> <incoming xmi:idref="U6eede33f-98ac-4654-bb17-dbe6aa7e46be"/> </node> <edge xmi:type="uml:ControlFlow" xmi:id="Ua4a2b313-13d6-4d69-9617-4803560731ef" source="Uc9e4f0de-8da6-4c98-9b95-b4cde30ccfc0" target="Ud304ce3c-ebe4-4b06-b75a-fa2321f8a151"> <guard xmi:type="uml:LiteralString" xmi:id="U6872f3b3-680c-430e-bdb3-21c0a317d290" visibility="public" value="x>10"/> </edge> <edge xmi:type="uml:ControlFlow" xmi:id="U6eede33f-98ac-4654-bb17-dbe6aa7e46be" source="Uc9e4f0de-8da6-4c98-9b95-b4cde30ccfc0" target="Ud304ce3c-ebe4-4b06-b75a-fa2321f8a151"> <guard xmi:type="uml:LiteralString" xmi:id="Ub853080d-481c-46ff-9f7c-92a31ac24349" visibility="public" value="else"/> </edge> <edge xmi:type="uml:ControlFlow" xmi:id="Ua800ba9b-e167-4a7c-a9a9-80e6a77edeb7" source="U26768518-a40c-4713-b35e-c267cc660508" target="Uc9e4f0de-8da6-4c98-9b95-b4cde30ccfc0"/> <edge xmi:type="uml:ControlFlow" xmi:id="Ue9366b93-a45b-43f1-a201-2038b0bd0b30" source="U16aa15e8-0e5d-4fd1-930a-725073ece9f0" target="U26768518-a40c-4713-b35e-c267cc660508"/> </packagedElement> It is easy to detect that the sequence of applied productions\ “$...nD^*fD^*iD^*oD^*nD^*$ ... $nD^*jD^*iD^*i...$” is not accepted by $L(\hat{G}_a)$ (one $o$ follows $f$, while two $i$ follow $j$), thus there is an inconsistency. We remark here that there are two preconditions of using the controlling grammar concerning the sequences of the model elements in the XML document: 1. $ForkNode$ must appear before its pairwise $JoinNode$; 2. $incoming$ edges must appear before $outcoming$ edges in a node. The two conditions are trivial, since it is easy to control their positions in the exporting XMI documents in implementing such a transformation. Related Work {#Sec:rw} ============ The most popular technique for verifying software correctness is [*model checking*]{} [@CGP00]. In this framework, we have three steps in verifying a system. First, we formalize system behavior as a model (e.g., a transition system, a Kripke model [@HR04]). Second, we specify the properties that we aim at validating using temporal logics. Third, we use a certain checking algorithm to search for a counterexample which is an execution trace violating the specified properties. If the algorithm finds such a counterexample, we have to correct the original design. Most checking tools use specific semantics of UML diagrams. They have the flavor of model checking, e.g., Egyed’s UML/Analyzer [@Egy07a][@Egy07b] and OCL (Object Constraint Language) [@UML-OCL]. At first, developers design UML diagrams as a model. Then, we specify the consistency rules as OCL or similar expressions. Certain algorithms are executed to detect counterexamples that violate the rules [@CPC04]. Note that these techniques do not discriminate the rules on the model level and those concerning the language level features. Unlike these techniques, our framework takes another way of ensuring correctness. It consists of the following steps: 1. Specifying the grammar $G$ of a language. It specifies an [*operational semantics*]{}, which defines what a language is able to model. Developing the grammar is mainly performed by [*language designers*]{}. 2. Modeling correctness rules of the use of languages as a controlling grammar $\hat{G}$. It specifies a [*correctness semantics*]{}, which defines what a language is authorized to derive. This process is the duty of [*safety engineers*]{} whose responsibility is to assure the correct use of the language. 3. The two grammars constitute a consistent language as a whole, that is, any derivations of the global system language is a correct and consistent use of the language. In particular, our work differs from model checking in the following aspects: 1\. Our work has different objectives, and uses different approaches to those of model checking. As we show in Fig. \[Fig:3level\], model checking techniques use a **bottom-up approach** — they verify execution traces $T^*$ at the lower level $L_1$ to prove the correct use of the grammar $G$ at the middle level $L_2$. Whereas our proposal uses a **top-down approach** — we model correctness rules as acceptable sequences of productions ($P^*$) at the higher level $L_3$ to ensure the correct use of $G$. Then any derivatives (at $L_1$) that conform to the C-System $C = G {\overrightarrow{\cdot}}\hat{G}$ are definitely a correct use. So the two techniques are complementary. ![Three Levels of the Framework[]{data-label="Fig:3level"}](fig_03.mps "fig:")\ 2\. Our work and model checking express language-level and model-level constraints, respectively. Language-level constraints are more effective, because they implicitly have reusability. That is, we only need to develop one language-level constraint and apply it to all the models in the language. However, using model checking, we need to replicate model-level constraints for each model. Additionally, model checking can process model-specific constraints. 3\. Our work and model checking use syntax-based and semantics-based approaches (or static and dynamic analysis), respectively. As a result, our approach is generic and metamodel-independent, and concerns little about semantics. So it can be applied to all MOF-compliant languages, not only to UML. However, model checking techniques depends on the semantics of a language, thus specific algorithms should be developed for different models. 4\. Our work and model checking catch errors at compile-time and runtime, respectively. As a result, our approach implements membership checking of context-free languages, which is decidable. That is, it searches in a limited space, which is defined by grammars. Model checking may search in a larger, even infinite space, so we have to limit the space of computing, and introduce the risk of missing solutions. Discussion {#Sec:disc} ========== In this section, we would like to shortly discuss some issues which are beyond the scope of this paper. The first issue concerns the implementation of controlling grammars. The controlled and controlling grammars can be implemented using two parsers separately. The technique for constructing a parser from a context-free grammar is rather mature [@Ear70][@ASU86]. Some tools provide automated generation of parsers from a grammar specification, such as Yacc, Bison. Notice that the inputs of controlling parsers are the sequences of productions applied in the parsing of $L(G)$. So there are communications between the two parsers. Once module $G$ uses a production $p_i$, then the name of the production is sent to $\hat{G}$ as an input. If $L(\hat{G})$ accepts the sequence of productions and $L(G)$ accepts the model, then $L(G {\overrightarrow{\cdot}}\hat{G})$ accepts the model. The second issue deals with multiple rules. If we have multiple guidelines or consistency rules, each rule is formalized using a grammar. We can develop an automated tool that converts the grammars into automata, and then combine these automata to compute an intersection, i.e., an automaton $A'$ [@HU79]. The intersection $A'$ can be used as a controlling automaton, which specifies a controlling language $L(A')$ that includes all the semantics of the rules. The third issue is about the tradeoff between cost and benefits of applying the proposed approach. It seems that writing a controlling grammar is expensive, because it involves formal methods. However, it is probably not the case. As we mentioned, a controlling grammar specify language-level constraints, and can be reused by all the models derived from the controlled grammar. Thus the controlling grammar can be identified and formalized by the organizations who define the language or its authorized usage, e.g., OMG and FAA (Federal Aviation Administration), respectively. Developers and software companies can use the published standard controlling grammar for checking inconsistencies in their models. By contraries, if every user writes their own checking algorithms and codes, e.g., in OCL or other programming languages, the codes will be hard to be reused by other users who have different models to check. Thus the total cost of all the users may be higher. Of course, more empirical results on the tradeoff is a good direction for future work. Conclusion {#Sec:conclusion} ========== We provided a language-theoretic view on guidelines and consistency rules of UML. We proposed the formalism of [C-Systems]{}, short for “formal language control systems”. To the best of our knowledge, none related work proposed similar methodologies. Rules are considered as controlling grammars which control the use of modeling languages. This methodology is generic, syntax-based and metamodel-independent. It provides a top-down approach that checks and reports violations of language level constraints at compile-time. It can be also applied to all MOF-compliant languages, not only to UML, since it does not depend on the specific semantics of languages. Since we focused on the methodological foundation, one of the future work is to develop an automated checking tool implementing the presented principles. We will also examine instant checking techniques of our method. One feature of UML/Analyzer is instant checking, which only verifies the small portion where the model changes, in order to save the cost of checking [@Egy06]. Intuitively, our approach is also easy to be extended to instant checking. We only need to generate the XMI document of the changed part of diagrams (e.g. a class in a class diagram), and verify it. However, this calls for more works in detail.

--- abstract: 'We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.' author: - | Manuel Barros$^1$, José L. Cabrerizo$^2$, Manuel Fernández$^2$\ and Alfonso Romero$^1$\ ${}^1$Departamento de Geometría y Topología, Facultad de Ciencias\ Universidad de Granada, 18071-Granada, Spain.\ E-mail addresses: [[email protected], [email protected]]{}\ ${}^2$Departamento de Geometría y Topología, Facultad de Matematicas\ Universidad de Sevilla, 41012-Sevilla, Spain.\ E-mail addresses: [[email protected], [email protected]]{} title: 'The Gauss-Landau-Hall problem on Riemannian surfaces' --- From a classical picture to a general setting ============================================= Classically, the Landau-Hall problem consists of the motion study of a charged particle in the presence of a static magnetic field, $H$. In this setting, free of any electric field, a particle, of charge $e$ and mass $m$, evolves with velocity $v$ satisfying the Lorentz force law, [@Landau-Lifshitz], $$\frac{dP}{dt}=\frac{e}{c}\; v \times H,$$ where $c$ denotes the light speed, $P=(\epsilon/c^2)\, v$ stands for the momentum of the particle, and $\epsilon = mc^2\left[ 1- (\parallel v \parallel^2/c^2)\right] ^{-1/2}$ is its energy. Since $dP/dt$ is orthogonal to $P$, then $(d/dt) (\| P \|^2)=0$. This implies the constancy of both $\| v\|$ and $\epsilon$. Assume $H$ is stationary, i.e., $H$ is a time-independent vector of the Euclidean space $\R ^3$. With the choice of a suitable orthonormal reference system, we may assume that $H=h \,(0,0,1)$, for some $h \in\mathbb{R}$. In this framework, we have $$\frac{d}{dt}\,v_1(t)=\omega \, v_2(t), \quad \frac{d}{dt}\,v_2(t)=-\omega \, v_1(t),\quad \frac{d}{dt}\,v_3(t)=0,$$ where $\omega=(ehc)/\epsilon$ is constant. Then $$x_1(t)=x_1^0 + r \sin(\omega t + \alpha), \quad x_2(t)=x_2^0 + r \cos(\omega t + \alpha), \quad x_3(t)=x_3^0 + v_3^0t,$$ where $r=\| v\| /\omega$. In particular, if $v_3^0=0$, then the particle describes a circle in the plane $x_3=x_3^0$, with center $(x_1^0,x_2^0,x_3^0)$ and radius $r$. Now, in this plane we consider the $2$-form $F$ defined by $F(X,Y)=\varepsilon<X\times Y,H>$, where $\varepsilon=\pm 1$ is the sign of $h/\omega$. It is clear that $F$ is covariantly constant, and therefore it is a constant multiple of the area element, indeed $F=\varepsilon h\, dx_1\wedge dx_2$. Now, consider the metric $g$ on the plane defined by $g:=\varepsilon(h/\omega)g^0$, where $g^0=< \, ,>$ denotes the Riemannian metric on the plane induced by the usual one of $\mathbb{R}^3$. Define the operator $\Phi$, $g$-equivalent to $F$, by $g(\Phi(X),Y)=F(X,Y)$. Then, the Lorentz force law can be expressed in terms of this form by $$\label{ecuacionenr3} \frac{d}{dt}\,v(t)=\, \Phi(v(t)).$$ This approach to the classical picture can be obviously extended to a more general setting. In fact, it seems natural to define a [*magnetic field*]{} on a $n(\geq 2)$-dimensional Riemannian manifold $(M,g)$, as a closed $2$-form $F$ on $M$. The [*Lorentz force*]{} of a magnetic background $(M,g,F)$ is defined to be the skew-symmetric operator, $\Phi$, given by $$\label{operator} g(\Phi(X),Y)=F(X,Y),$$ for any couple of vector fields $X,Y$ on $M$. Let us remark that $\Phi$ is metrically equivalent to $F$, so no information is lost when $\Phi$ is considered instead $F$. In classical terminology, it is said that $\Phi$ is obtained from $F$ by raising its second index, and $\Phi$ and $F$ are then said to be physically equivalent. On the other hand, there exists another operator $\Phi'$ defined from $F$ via $g$ in a similar way, namely $g(X,\Phi'(Y))=F(X,Y),$ but it is easily seen that $\Phi'=-\Phi$. So, the choice from among $\Phi$ or $\Phi'$ to represent $F,$ using $g$, is not relevant. Along this paper, we will use $\Phi$ to denote the Lorentz force induced from $(M,g,F)$. A (smooth) curve $\gamma$ in $(M,g)$ is called a [*flowline*]{} of the dynamical system associated with the magnetic field $F$ (or simply a flowline of $F,$ or a [*magnetic curve*]{} of $(M,g,F)$), if its velocity vector field, $\gamma^{\prime}$, satisfies the following (Landau-Hall) differential equation, $$\nabla_{\gamma^{\prime}} \gamma^{\prime}=\Phi(\gamma^{\prime}), \eqno{\rm(LH)}$$ where $\nabla$ is the Levi-Civita connection of $g$ \[compare with Eq. (\[ecuacionenr3\])\]. For the trivial magnetic field, $F=0$, the case without the force of a magnetic field, magnetic curves correspond with the geodesics of $(M,g)$. As it is well known, they are nicely characterized as critical points of an energy action and so they represent the trajectories for free fall particles (moving under the influence of only gravity). In the general case, however, magnetic flows are important examples of dynamical systems on Riemannian manifolds whose flowlines, being the trajectories of charged particles in (non trivial) magnetic fields, are not geodesics (Proposition \[homogeneizacion\]) but, as we will see later, they are closely related with the Riemannian structure. Nevertheless, the magnetic curves of $(M,g,F)$ can be also viewed, at least locally, as the solutions of a variational principle. In fact, let $U$ be an open subset of $M$ where $F=d\omega$ for some potential $1$-form $\omega$ (this open subset could be the whole $M$ when $H^{2}(M)=0$). For any two fixed points $p,q\in U$, we consider the space $\Gamma_{pq}$ of smooth curves in $U$ that connect these two points. Now, we choose the action $\mathcal{LH}:\Gamma_{pq}\rightarrow\mathbb{R}$ defined by $$\label{primerfuncional} \mathcal{LH}(\gamma)=\displaystyle \frac{1}{2}\int_{\gamma} g(\gamma',\gamma')dt-\int_{\gamma} \omega(\gamma ')dt.$$ The tangent space of $\Gamma_{pq}$ in $\gamma$ is made up of the smooth vector fields, $V$, along $\gamma$ that vanish at the end points $p,q\in U$. An standard computation involving integration by parts allows one to compute the first variation of this action to be $$\delta(\mathcal{LH})(\gamma)[V]=\displaystyle -\int_{\gamma} g\left( \nabla_{\gamma^{\prime}} \gamma^{\prime}-\Phi(\gamma '),V\right)dt.$$ As a consequence, we get $$\delta(\mathcal{LH})(\gamma)[V]=0, \; \mathrm{for \; any} \; V\in T_{\gamma}\Gamma_{pq} \; \; \mathrm{if \; and \; only \; if \; \gamma \; is \;a \; solution \; of \; (LH)}.$$ This argument shows that the differential equation (LH) is indeed the Euler-Lagrange equation associated with the functional $\mathcal{LH}$. However, it seems natural to realize the old idea of characterizing magnetic curves from a global variational principle. In other words, to obtain the magnetic trajectories of $(M,g,F)$ as solutions of a variational problem that neither it does not involves any local potential nor it does not constraint the topology of $M$. This is, in general, an interesting open problem. One of the main aim of this paper is just to solve it for certain magnetic fields on surfaces. To be precise, we introduce the notion of a Gauss-Landau-Hall magnetic field (in brief, GMF) on an oriented Riemannian surface $(M,g)$. First, we do it in the natural context that surfaces are immersed in Euclidean space $\mathbb{R}^3$ using the Gauss map. However, we notice that the notion of GMF is absolutely intrinsic so it can be considered on surfaces even if they are not regarded in $\mathbb{R}^{3}$. Then, we are able to obtain an amazing result which characterizes the normal flowlines of a GMF as the solutions of a variational principle globally stated. Therefore, those flowlines appear as critical points of an action whose Lagrangian density involves the proper acceleration of particles (relativistic particles with rigidity of order one, in the sense of Plyushchay, [@Plyushchay2],[@Plyushchay]). A priori, these actions describe a massive relativistic boson. However, massless particles with arbitrary helicity are obtained as a limit case, just when the Lorentz force of the GMG increases arbitrarily. Other details on the paper are the following. We first provide in Section 2 an analysis of the existence, uniqueness, extendibility and completeness of the magnetic curves associated with a given $(M,g,F)$. Section 3 deals with uniform magnetic fields on Riemannian surfaces, while the particular case when $(M,g)$ is a revolution surface is studied in Section 4. In Section 5, a one-parameter family $\mathcal{F}_m$ of functionals is considered on an appropriate space of curves $\Lambda$ in the surface. The Euler-Lagrange equation associated to the variational problem is then obtained. In Section 6 we define a Gauss-Landau-Hall magnetic field on a surface, first in $\R^3$, and then in general. In this section, we obtain the main result, Theorem \[importante\], which asserts that the normal flowlines of a GMF coincide with the critical points of the appropriate functional $\mathcal{F}_m$. Stability of the field equation solutions is also studied. In Section 7, we show a characterization theorem for those revolution surfaces whose parallels are all normal magnetic curves associated to a GMF. We close the section studying some particular examples. Completeness of magnetic curves and more ======================================== An early property of the magnetic curves is the following conservation’s law. Particles evolve with constant speed, and so constant energy, along the magnetic trajectories $$\label{velocidadconstante} \frac{d}{dt}\,g(\gamma^{\prime},\gamma^{\prime})=2g(\Phi(\gamma^{\prime}),\gamma^{\prime})=0.$$ In particular, a magnetic curve $\gamma$ is said to be [*normal*]{} if it has unit energy, i.e., $\|\gamma'\| \, ^{2} \equiv 1$. The existence and uniqueness of geodesics, remains true when one considers magnetic curves. Thus, for each $p\in M$ and $v\in T_pM$ there is exactly one inextendible (i.e., maximal) magnetic curve, $\gamma :(-a,a)\longrightarrow M$, of $(M,g,F)$ with $\gamma(0)=p$ and $\gamma^{\prime}(0)=v$, (see for instance [@Sachs-Wu], p. 91). Since the proof of this result does not make use neither the definiteness of $g$ nor the skew-symmetry of $\Phi$, one has a [*present determines the future*]{} type result for an indefinite metric, Lorentzian in particular, and for any smooth operator. Even more, the result also works for solutions of a differential equation that extends that of Landau-Hall in the following terms [@Romero-Sanchez], $$\nabla_{\gamma^{\prime}}\gamma^{\prime}=\Phi(\gamma^{\prime})+X \circ \gamma,$$ where $X$ is a vector field on a semi-Riemannian manifold. This setting includes the important case in Mechanics where $X=-\nabla V$, and $V$ standing for smooth function on $M$, ([@Marsden], Proposition 3.7.4). Nevertheless, the well known homogeneity result for geodesics, works quite different in non trivial magnetic fields. Therefore, if $\gamma$ is the inextendible magnetic curve of $(M,g,F)$ determined from the initial data $(p,v)$, the curve $\beta$, defined by $\beta(t)=\gamma(\lambda t),$ $\lambda \in\mathbb{R}\backslash\{0\}$, is a magnetic trajectory of $(M,g,\lambda F)$ and also, when $\lambda>0$, of $(M,(1/\lambda)\, g,F)$, in both cases determined from initial data $(p,\lambda v)$. Furthermore, the whole families of magnetic curves of $(M,g,F)$ and $(M,\lambda g,\lambda F)$ coincides, for any constant $\lambda>0$. Consequently, we have \[homogeneizacion\] Let $F$ be a non trivial magnetic field on a Riemannian manifold, $(M,g)$. Then, there exists no affine connection on $M$ whose geodesics are the magnetic curves of $(M,g,F)$. A magnetic field $(M,g,F)$ with Lorentz force $\Phi$, provides, in a similar way as in [@O'Neill], Prop. 3.28, with a unique vector field $Q_{\Phi}$ on the tangent bundle $TM$. This is defined to have integral curves being the lifting to $TM$ of the magnetic curves, that is, $t \mapsto (\gamma(t), \gamma^{\prime}(t))$, where $\gamma$ is a magnetic curve of $(M,g,F)$ (compare with [@Lopez]). Certainly this vector field is nothing but the geodesic flow when $F=0$. Once more, neither the definiteness of $g$ nor the skew-symmetry of $\Phi$ is needed to define $Q_{\Phi},$ [@Romero-Sanchez]. On the other hand, the fact that any integral curve of $Q_{\Phi}$ is the velocity of its projection on $M$, allows us to think of $Q_{\Phi}$ as a nice example of the classically so-called [*second order differential equation*]{} on $M$. Because the comment previous to Proposition \[homogeneizacion\], $Q_{\Phi}$ is not an [*spray*]{}, in general. A dynamical system with complete trajectories is often thought in Physics to be persisting eternally. But in many circumstances one has to deal with incompleteness. So, because of its importance, we next give criteria to assert when it holds true. An important tool to study the completeness of the inextendible magnetic curves, i.e., under what assumptions all the inextendible magnetic curves are defined on all $\mathbb{R}$, is the vector field $Q_{\Phi}$. By using Lemma 1.56 in [@O'Neill], it is easily seen the following result. \[extendibilidad\] Let $(M,g)$ be a Riemannian manifold, $F$ a magnetic field on $M$ and $\gamma : [a,b) \longrightarrow M$, $a<b$, a magnetic curve of $F$. The following are equivalent: \(a) $\gamma$ is extendible to $b$ as a magnetic curve. \(b) There exists a sequence $\{t_n\} \longrightarrow b$, $t_n \in [a,b)$ such that the sequence of velocities $\{\gamma '(t_n)\} $ converges in $TM$. Accordingly, a magnetic curve $\gamma : (a,b) \longrightarrow M, \, a,b\in \mathbb{R}, \, a<b$, of $(M,g,F)$ can be extended to some open interval $I, \, (a,b)\subset I,$ if and only if $\gamma (a,b)$ is contained in a compact subset of $M$. Therefore, we get (compare with Theorem 2.1.18 in [@Marsden]) \[nueva\] Let $\gamma$ be an inextendible magnetic curve of $(M,g,F)$ such that $\gamma(a,b)$ lies in a compact subset of $M$, for every finite interval $(a,b)$ in its domain. Then, $\gamma$ must be complete. In particular, if $M$ is assumed to be compact, then we get that any inextendible magnetic curve of $(M,g,F)$ must be complete. This fact can be also obtained as a consequence of Corollary \[completitud\], and it will be stated in Remark \[remarkprimera\] (a), from a different approach. Now, let $\gamma : [a,b) \longrightarrow M$ be a magnetic curve. Its length $L(\gamma)$ satisfies $L(\gamma) \leq (b-a)\sqrt{e}$, where $e$ is the (constant) energy of $\gamma.$ For each $t \in [a,b)$, the distance between $\gamma(a)$ and $\gamma(t)$ satisfies $d(\gamma(a),\gamma(t)) \leq L(\gamma\mid_{[a,t]}) \leq (b-a)\sqrt{e}$, which shows that $\gamma([a,b))$ is contained in the closed metric ball $B$ centered at $\gamma(a)$ and with radius $(b-a)\sqrt{e}$. Therefore, $$\gamma^{\prime}([a,b))\subset \{(p,v)\in TM \, : \, p\in B, \, g(v,v)=e \}\subset TM.$$ Then, we have, \[completitud\] Let $F$ be any magnetic field on a geodesically complete Riemannian manifold $(M,g)$. Then, all the inextendible magnetic curves of $(M,g,F)$ are complete. [**Proof**]{}. If $(M,g)$ is assumed to be geodesically complete, then the Hopf-Rinow theorem implies that $B$ must be compact. Hence $\{(p,v)\in TM \, : \, p\in B, \, g(v,v)=e \}$ is a compact subset of $TM$. Take now a sequence $\{t_n\} \longrightarrow b$, $t_n \in [a,b)$, then $ \{ \gamma '(t_n)\}$ lies in a compact subset of $TM$. So, by passing to a subsequence of $\{t_n\}$, we are under the assumption (b) of Proposition \[extendibilidad\], concluding that $\gamma$ is extendible to $b$ as a magnetic curve. \[remarkprimera\] [(a):]{} If $M$ is assumed to be compact (therefore $(M,g)$ is geodesically complete for any Riemannian metric $g$ on $M$), then we can give an alternative proof of Corollary \[completitud\]. In fact, the previous conservation’s law \[Eq. (\[velocidadconstante\])\] for the length of velocity vectors of magnetic curves, implies that the vector field $Q_{\Phi}$ on $TM$ can be restricted to each spherical tangent bundle $U_eM= \{(p,v)\in TM \, : \, g(v,v)=e \}\subset TM, \, e>0$. But $U_eM$ is compact whenever $M$ is compact, and hence the restriction of $Q_{\Phi}$ to $U_eM$ is a complete vector field. This proves that all the inextendible magnetic curves of $F$ are complete. [(b):]{} Proposition \[homogeneizacion\] has shown a remarkable difference between magnetic curves and geodesics. The following non-connecteness fact complement that result. Let us consider the unit $2$-sphere $\mathbb{S}^{2}(1)$ endowed with its standard round metric $g,$ and let $F$ be the magnetic field $F=\mu\,\Omega _2,$ where $\Omega_2$ is the area $2$-form and $\mu \in \mathbb{R}, \; \mu \neq 0.$ As we will show later (see the comment after Proposition \[proposicionconstante\]), the associated magnetic curves with energy $e$ are circles on $\mathbb{S}^2(1)$ with radius $r=[1+(\mu^2 / e)]^{-1/2}.$ Then, as $r <1,$ any two antipodal points can not be connected by a magnetic curve of $(\mathbb{S}^2,g,F).$ Moreover, for any $p\in \mathbb{S} ^2$, all the inextendible magnetic curves $\gamma$ of $(\mathbb{S} ^2 \setminus \{-p\},g,F)$ such that $\gamma (0)=p$ are complete. [(c):]{} Let $(M,g)$ be a Riemannian manifold where $g$ is an incomplete metric. If $F$ is a magnetic field on $(M,g),$ then there exists a pointwise conformal metric $f^2g$ such that the inextendible magnetic curves of $(M,f^2g,F)$ are complete. In fact, there exists $f\in C^{\infty}(M), \; f>0,$ such that $f^2g$ is geodesically complete, [@Nomizu-Ozeki]. Therefore, Proposition \[completitud\] gives that the magnetic curves of $(M,f^2g,F)$ are complete. [(d):]{} It should be observed that the closedness assumption on the $2$-form $F$ in Proposition \[completitud\] was not used. On the other hand, the skew-symmetry of the tensor field $F$ has played a crucial role \[recall the conservation’s law (\[velocidadconstante\])\]. In fact, consider the tensor field $F=-2x\,dx^2$ on the Euclidean plane $(\mathbb{R}^2,g^0=dx^2+dy^2)$. If $\Phi$ denotes the operator defined from $F$ using Eq. (\[operator\]), then $\Phi(\partial / \partial x)=-2x\, (\partial /\partial x)$ and $\Phi(\partial / \partial y)= 0$. Therefore, $\gamma(t)=(x(t),y(t))$ satisfies Eq. (LH) if and only if $x''(t)+2\,x(t)x'(t)=0$ and $y''(t)=0$. So, $\gamma(t)=(1/t,t)$ is an inextendible incomplete trajectory of $(\mathbb{R}^2,g^0,F).$ [(e):]{} Finally, let us point out that Proposition \[completitud\] cannot be also extended to the indefinite case. In fact, consider $\R^2$ endowed with the Lorentzian metric $g_L=dx^2-dy^2,$ and define the magnetic field $F=-xdx\wedge dy.$ A curve $(x(t),y(t))$ is a magnetic curve of $(\R ^2,g_L,F)$ if and only if it satisfies $x''(t)=x(t)y'(t), \;\, y''(t)=x(t)x'(t).$ Then, $\gamma (t)=(2/t,-2/t)$ is an inextendible magnetic curve which is defined on $(0,\infty).$ Uniform magnetic fields ======================= From now on, $M$ will be an oriented Riemannian surface with standard complex structure $J,$ and area element $\Omega_{2}$ so that $ \Omega_{2}(X,JX)=1$ for any unit vector field $X$ in $M$. Given a curve $\gamma$ in $M$ such that $g(\gamma^{\prime},\gamma^{\prime})=e>0$ is constant, its Frenet apparatus is $\{T=(1 /\sqrt{e})\,\gamma^{\prime},N=JT\}$. If $\kappa$ denotes the curvature function, we have the following well-known Frenet equations $$\nabla_{\gamma^{\prime}} T = \kappa \sqrt{e}\,N, \qquad \nabla_{\gamma^{\prime}} N = -\kappa \sqrt{e}\,T.$$ Obviously, any magnetic field on a surface, $M$, is determined from a smooth function, $f$ (the [*strength*]{}), by $F=f \, \Omega_{2}$. Therefore, the matrix of $\Phi$ in any orthonormal frame, $\{X,JX\}$ is given by $$\left( \begin{array}{rr} 0 & -f \\ f & 0 \end{array} \right).$$ In particular, along a magnetic curve $\gamma$ of $(M,g,F)$, with energy $e$, and relative to its Frenet frame, the Lorentz force is obtained to be $$\left( \begin{array}{rrr} 0 & -\kappa \, \sqrt{e}\\ \kappa \,\sqrt{e} & 0 \end{array} \right).$$ Therefore, we get, \[magneticasdeenergiae\] The curvature of the magnetic curves with energy $e$ is given by $\kappa=f/\sqrt{e}.$ So, the curvature of the normal magnetics curves completely determines the Lorentz force, i.e., $f=\kappa$ along these flowlines. A parallel magnetic field $F$, i.e., a magnetic field with constant strength $f=\mu$, is called a [*uniform magnetic field*]{}. This class of magnetic fields has been extensively considered in the literature from different points of view ([@Lopez],[@Adachi]-[@Wojtkowski], etc.). The geometric partner of the Landau-Hall problem, for uniform magnetic fields, is nothing but the computation of curves with constant curvature. To be precise, we have, \[proposicionconstante\] Let $F=\mu \, \Omega_{2}$ be a uniform magnetic field, with constant strength $\mu$, on a Riemannian surface $(M,g)$. A curve $\gamma$ in $M$, with constant energy $e$, is a magnetic curve of $(M,g,F)$ if and only if it has constant curvature $\kappa=\mu /\sqrt{e}$. On surfaces of constant Gauss curvature, the feature of the normal flowlines of a non-trivial uniform magnetic field $F=\mu \,\Omega_2$ is well-known for any uniform magnetic field. On the Euclidean plane, $\mathbb{R}^{2}$, they are circles with radius $1/|\mu|$. On the $2$-sphere of radius $r$, $\mathbb{S}^{2}(r)$, flowlines with energy $e$ are circles with radius $(r\sqrt{e})/\sqrt{e+r^2\mu ^2}$ ($<r$). In these two backgrounds, the flowlines are always closed. On the other hand, the situation in a hyperbolic plane is quite different. Let $\mathbb{H}^{2}(-G)$ be the upper half-plane (in $\mathbb{R}^{2}$) endowed with the Lobatchevski metric of curvature $-G$, $G>0$, that is, the Poincaré plane. We use Proposition \[proposicionconstante\] joint the basic knowledge of the curves of constant curvature in $\mathbb{H}^{2}(-G)$ (see any basic text of Riemannian geometry) to make trivial the following description of the flowlines which is due to A. Comtet, [@Comtet], and has been mentioned along a large list of references. The behaviour of normal magnetic curves changes according to the ratio between the strength, $\mu$, and the curvature of $\mathbb{H}^{2}(-G)$. Namely, - If $|\mu| / \sqrt{G}>1$, then the trajectories are geodesic circles, and therefore they are closed curves. - If $|\mu| / \sqrt{G} \leq 1$, then the trajectories are non-closed curves which intersect the boundary line, $\partial\mathbb{H}^{2}(-G)$, of the upper half-plane. In particular, they are tangent to this boundary, and so they are horocycles when $|\mu|=\sqrt{G}$. [(a):]{} Let $\gamma$ be a curve with constant geodesic curvature $\kappa \neq 0$ in any of the three previous constant curvature surfaces. Then, for a given uniform magnetic field $F=\mu \,\Omega_2,$ a suitable fitting of the constant speed (and hence, the energy) of $\gamma$ makes this curve to be a magnetic curve of $F.$ [(b):]{} Let $(M,g)$ be again one of the three above space forms and $F=\mu \,\Omega _2$ a uniform magnetic field on $(M,g).$ Then, any magnetic curve $\gamma$ with energy $e$ of $(M,g,F)$ can be then considered as a [*normal*]{} magnetic curve of $\big( M,(1/e)\, g,(1/e)\, F\big)$ (see the comment previous to Proposition \[homogeneizacion\]). The Landau-Hall problem in a surface of revolution ================================================== Let $\alpha(s)=\left(f(s),h(s)\right)$, $a<s<b$, $f(s)>0$, be a parametrization by the arclength of a curve, $\mathrm{C}$, contained in the $\{xz\}$-plane of $\R ^3$. We rotate $\mathrm{C}$ around the $z$-axis to obtain a surface of revolution, say $M_{\alpha}$, with canonical parametrization in $\mathbb{R}^{3}$ $$\label{revolucion} X(s,v)=\left(f(s)\cos v, f(s)\sin v, h(s)\right), \qquad 0\leq v\leq 2\pi.$$ Of course we consider that $M_{\alpha}$ is endowed with the induced metric $g$ of the Euclidean one of $\mathbb{R}^{3}$. Each point of $\mathrm{C}$ describes a parallel, $\gamma_{s}$, which can be parametrized by arclength in the following way $$\gamma_{s}(t)=\left(\, f(s)\cos{\frac{t}{f(s)}}, \; f(s)\sin{\frac{t}{f(s)}}, \; h(s)\,\right),$$ where $0\leq t\leq 2\pi f(s)$. The curvature, $\kappa_{s}$, of $\gamma_{s}$ in $M_{\alpha}$, is computed to be $$\kappa_{s}(t)=\|\nabla_{T_{s}} T_{s}\|=\frac{f^{\prime}(s)}{f(s)},$$ where $T_{s}=\gamma^{\prime}_{s}$ and $\nabla$ is the Levi-Civita connection of $M_{\alpha}$. In particular, $\kappa_{s}$ is constant along $\gamma_{s},$ and so this curve is a good candidate to be a flowline of a suitable uniform magnetic field on $M_{\alpha}$. Let $F=\mu \, \Omega_{2}$ be a uniform magnetic field on $M_{\alpha}$ with constant strength $\mu$. Then $\gamma_{s}$ is a normal magnetic flowline of $(M_{\alpha},g,F=\mu \, \Omega_{2})$ if and only if $\kappa_{s}=\mu$ (Proposition \[proposicionconstante\]). Therefore, the set of magnetic parallels of $(M_{\alpha},g,F=\mu \, \Omega_{2})$ can be identified with the following subset of the interval $(a,b)$ $$\Gamma_{\mu}=\{s\in(a,b) \, : \, f^{\prime}(s)=\mu f(s)\}.$$ To determine those surfaces of revolution whose parallels are all normal magnetic curves of a given uniform magnetic field (that is, those with $\Gamma_{\mu}=(a,b)$) we need to solve the ordinary differential equation $$f^{\prime}(s)=\mu f(s).$$ Obviously, we have two possibilities. The trivial one, corresponding with the case of a trivial magnetic field (the strength vanishes), the flowlines are then geodesics, and the surface of revolution is a right circular cylinder. Otherwise, since the Gauss curvature of a surface of revolution (in the canonical parametrization) is given by $$\label{Gaussrevolution} G(s,t)=-\frac{f^{\prime\prime}(s)}{f(s)},$$ we get that $G(s,t)=-\mu ^{2}$, an hence the surface has constant negative curvature. In particular, we have, \[uniformes\] The parallels of a surface of revolution, $M_{\alpha}$, are all normal magnetic flowlines of a uniform magnetic field, $F=\mu \, \Omega_{2}$, if and only if either: 1. $M_{\alpha}$ is a right circular cylinder (when $\mu=0$), or 2. $M_{\alpha}$ is a bugle surface with Gaussian curvature $-\mu ^{2}$. Let us consider the torus of revolution, $\mathbf{T}(r,R)$, obtained by rotating the circle, $\mathbf{C}$, centered at $(R,0,0)$ and with radius $r$, ($R>r$), around the $z$-axis. The circle can be arclength parametrized by $\alpha(s)=\left(R+r \cos (s/r), \, 0, \, r \sin (s/r)\right)$. Therefore, $f(s)=R+r \cos (s/r)$, with $0\leq s\leq 2\pi r$. Given a uniform magnetic field $F=\mu \, \Omega_{2}$ on $\mathbf{T}(r,R)$, the set of normal magnetic parallels is identified to $$\Gamma_{\mu}=\{s\in [0,2\pi r] \, : \, H_{\mu}(s)=0\},$$ where $$H_{\mu}(s)=R\mu\,+\,r\mu\cos \,(\frac{s}{r}) \,+\,\sin \,(\frac{s}{r}).$$ To study $\Gamma_{\mu}=H_{\mu}^{-1}(0)$, we use elemental calculus. First, we assume that $\mu \neq 0$, otherwise $\Gamma_{0}=H_{0}^{-1}(0)$ is made up of the two parallels that are geodesics in $\mathbf{T}(r,R)$. In this setting, we see that $H_{\mu}$ has exactly two critical points on $\mathbf{C}$ (the maximum and the minimum) which are antipodal. In fact, $H_{\mu}^{\prime}(s)=0$ if and only if $\cot (s/r)=r\mu$ and it happens just in the following two antipodal points $$\begin{aligned} p_{\mu} & = & \left(\,\cos{\frac{s}{r}}, \; \sin{\frac{s}{r}}\,\right)=\left(\frac{r \mu}{\sqrt{1+r^{2}\mu^{2}}}, \, \frac{1}{\sqrt{1+r^{2}\mu^{2}}}\right),\\ q_{\mu} & = & \left(\,\cos{\frac{s}{r}}, \; \sin{\frac{s}{r}}\,\right)=-\left(\frac{r \mu}{\sqrt{1+r^{2}\mu^{2}}}, \, \frac{1}{\sqrt{1+r^{2}\mu^{2}}}\right),\end{aligned}$$ The values of $H_{\mu}$ in $p_{\mu}$ and $q_{\mu}$ are $$\begin{aligned} H_{\mu}(p_{\mu}) & = & R \mu+\sqrt{1+r^{2}\mu^{2}}; \qquad {\rm the \, maximum \, of} \quad H_{\mu}, \\ H_{\mu}(q_{\mu}) & = & R \mu-\sqrt{1+r^{2}\mu^{2}}; \qquad {\rm the \, minimum \, of} \quad H_{\mu}.\end{aligned}$$ We call $D_{\mu}$ the diameter in $\mathbf{C}$ determined by $p_{\mu}$ and $q_{\mu},$ and let $\rho=(R^{2}-r^{2})^{-1/2}.$ We distinguish two cases. \(A) If $\mu>0$, then the point $p_{\mu}$, where $H_{\mu}$ gets its maximum, lies in the first quadrant of the circle $\mathbf{C}$. Since $H_{\mu}(p_{\mu})>0$, then $\Gamma_{\mu}\neq\emptyset$ if and only if $H_{\mu}(q_{\mu})\leq 0$ (the minimum is non positive) that is $\mu \leq \rho$. Certainly if the equality holds, then $H_{\rho}$ vanishes only at $q_{\rho}=\left(-r/R, \, -1/(\rho R)\right)$. Otherwise, $H_{\mu}$ vanishes exactly in two points, say $z_{\mu}$ and $w_{\mu}$, which are separated by $D_{\mu}$ so they lie in different half circles. \(B) If $\mu<0$, then the point $p_{\mu}$, where $H_{\mu}$ gets its maximum, lies in the second quadrant of the circle $\mathbf{C}$. Since the minimum is negative, $H_{\mu}(q_{\mu})<0$, then $\Gamma_{\mu}\neq\emptyset$ if and only if $H_{\mu}(p_{\mu})\geq 0$ and it happens if and only if $\mu \geq -\rho$. It is clear that when the equality holds, then $H_{\rho}$ vanishes just at $p_{\rho}=\left(-r/R, \, 1/(\rho R)\right)$. However, if $\mu>-\rho$, then $H_{\mu}$ has exactly two zeroes, say $z_{\mu}^{\prime}$ and $w_{\mu}^{\prime}$, which are obviously separated by $D_{\mu}$ so they lie in different half circles. All this information can be summarized in the following statement, \[proptoro\] Let $F=\mu \,\Omega_{2}$ be a uniform magnetic field on a torus of revolution, $\mathbf{T}(r,R)$. Then $\left(\mathbf{T}(r,R), \,g, \mu \, \Omega_{2}\right)$ has normal magnetic parallels if and only if $\mu \in \left[-\rho, \, \rho \right]$. Furthermore, 1. If $\mu=-\rho,$ then there is one normal magnetic parallel obtained by rotating the point $\left( R+(r^2/R),r/(\rho R) \right)\in \mathbf{C}$. 2. If $\mu=\rho$, then there is one normal magnetic parallel obtained by rotating the point $\left( R-(r^2/R),-r/(\rho R) \right)\in \mathbf{C}$. 3. If $\mu \in \left(-\rho , \, \rho \right)$, then $H_{\mu}$ has exactly two normal magnetic parallels obtained by rotating two points of $\mathbf{C}$ that are separated by $D_{\mu}$. \[catenoide\] Let $C$ be the cathenoid generated by revolving the cathenary curve $\alpha (t)=(\cosh t,0,t), \; t\in \mathbb{R}$ around the $z$-axis. Let $F$ be a uniform magnetic field with constant strength $\mu \neq 0$, defined on $C$. Then, it can be shown that there exist magnetic parallels if and only if $\mu \in \left[ -1/2,1/2 \right]$. Moreover, we have: 1. If $\mu=1/2$ (resp. $\mu=-1/2$), there exists only a normal magnetic parallel corresponding to $t=\ln (1+\sqrt{2})$ (resp. $t=-\ln (1+\sqrt{2})$). 2. If $\mu \in \left( -1/2,0\right)$ (resp. $\mu \in \left( 0,1/2\right))$, then $C$ has two normal magnetic parallels which are located in the region $t>0$ (resp. $t<0$). \[cicloide\] Let $S$ be the revolution surface obtained by rotating the cycloid curve $\alpha (t)=(a(1-\cos t),0,a(t-\sin t)), \; t\in (0,2\pi)$ around the $z$-axis. It is easy to see: 1. If $\mu >0$, then $S$ has only a normal magnetic parallel corresponding to the parameter value $t_0=2\arccos \, \left[ \left( -1+\sqrt{1+16a^2\mu^2}\right) /(4a\mu)\right].$ 2. If $\mu<0$, then $S$ has also only a normal magnetic parallel corresponding to $t_1=2\pi -t_0$. [Finally, let us consider the cone $M$ generated by the line $\alpha (t)=(at,0,bt), \; a^2+b^2=1, \; t>0$ around the $z$-axis. Then, it can be seen that for any $\mu>0$, there exists a unique normal magnetic parallel given by $t=1/ \mu$.]{} Relativistic particles with rigidity of order one ================================================= The search for Lagrangians describing spinning particles (both massive and massless) has a long history. An interesting and unconventional approach is to provide the necessary extra degrees of freedom by actions whose densities depend on higher order geometrical invariants. In particular, this means that those extra bosonic variables must be encoded in the geometry of the world trajectories. The simplest models are those involving density Lagrangians that depend on the curvature, $\kappa$, of the worldlines ([@Arroyo]-[@Nesterenko1], etc.). In particular, actions that depend linearly from $\kappa$ ([@Plyushchay2],[@Arroyo-Barros-Garay]-[@Plyushchay3], etc.) will be considered in this section. These models describing a massive relativistic boson [@Plyushchay]. Suppose that $\Lambda$ is a suitable space of curves (closed curves or clamped curves, for instance) in a Riemannian surface $(M,g)$. Define a one-parameter family of functionals $\mathcal{F}_{m}:\Lambda\rightarrow\mathbb{R}, \; m\in \R,$ by $$\label{funcional} \mathcal{F}_{m}(\gamma)=\int_{\gamma}(\kappa+m)ds,$$where $s$ stands for the arclength parameter of curves $\gamma\in\Lambda$. In order to obtain the first variation of these actions, we use the following standard machinery (see for instance [@Barros]). For a curve $\gamma:[0,L]\rightarrow M$, we take variations $\Theta=\Theta(t,r):[0,L]\times(-\varepsilon,\varepsilon)\rightarrow M$ with $\Theta(t,0)=\gamma(t)$. Then, we have the variation vector field $W=W(t)=\left( \partial\Theta /\partial r\right)(t,0)$ along the curve $\gamma$. We also put $V=V(t,r)=\left( \partial\Theta /\partial r\right)(t,r)$, $W=W(t,r)$, $v=v(t,r)=\|V(t,r)\|$, $T=T(t,r)$, $N=N(t,r)$, with the obvious meanings. The corresponding reparametrizations will be denoted by $V(s,r)$, $W(s,r)$ etc. The variations of $v$ and $\kappa$ in $\gamma$, in the direction of $W$, can be obtained to be $$\label{v} W(v) = g(\nabla_{T} W,T)\,v,$$ $$\label{curvature} W(\kappa) = g(\nabla_{T}^{2} W,N)-2g(\nabla_{T} W,T)\kappa+G \, g(W,N),$$ here $G$ denotes the Gauss curvature of $(M,g)$ and $\nabla$ its Levi-Civita connection. Now, we use a standard argument that involves the above formulas so as some integrations by parts to obtain the first derivative of $\mathcal{F}_{m}$ $$\label{variacion} \delta\mathcal{F}_{m}(\gamma)[W]=\int_{\gamma} g\left(\Omega(\gamma),W\right) ds+\left[\mathcal{B}(\gamma,W)\right]_{0}^{L},$$where $\Omega(\gamma)$ and $\mathcal{B}(\gamma,W)$ denote the Euler-Lagrange and the boundary operators and they are respectively given by $$\begin{aligned} \Omega(\gamma) & = & (G-m \, \kappa)N, \\ \mathcal{B}(\gamma,W) & = & g(\nabla_{T} W,N)+m \, g(W,T).\end{aligned}$$ \[clamped\] [(Clamped curves)]{} Given points $q_{1}, \, q_{2}\in M$ and unit vectors $x_{1}\in T_{q_{1}}M$ and $x_{2}\in T_{q_{2}}M$, define the space of curves $$\Lambda=\{\gamma:[t_{1},t_{2}]\rightarrow M \, : \, \gamma(t_{i})=q_{i}, \, T(t_{i})=x_{i}, \, N(t_{i})=Jx_{i}, \, 1\leq i\leq 2\}.$$ Then, the critical points of the functional $\mathcal{F}_{m}:\Lambda\rightarrow\mathbb{R}$ are characterized by the following Euler-Lagrange equation $$G\mid_{\gamma}=m \, \kappa,$$ where $G\mid _{\gamma}$ denotes the Gauss curvature of $(M,g)$ along $\gamma$. [**Proof.**]{} Let $\gamma\in\Lambda$ and $W\in T_{\gamma}\Lambda$, then $W$ defines a curve in $\Lambda$ associated with a variation $\Theta$ of $\gamma$. We can perform the following computations along $\Theta$ $$\begin{aligned} W & = & d\Theta(\partial_{r}), \\ \nabla_{T} W & = & f \, T+d\Theta(\partial_{r} T),\end{aligned}$$ where $f=\partial_{r}(\ln \, v)$. We evaluate these formulas along $\gamma$ by making $r=0$ and use that $\Theta$ is a curve in the space $\Lambda$ to obtain $$\begin{aligned} W(t_{i}) & = & 0, \\ \nabla_{T} W(t_{i}) & = & f(t_{i})x_{i}.\end{aligned}$$ As a consequence, $\left[\mathcal{B}(\gamma,W)\right]_{t_{1}}^{t_{2}}=0$. So, using Eq. (\[variacion\]), we have that $\gamma$ is a critical point of $\mathcal{F}_{m}:\Lambda\rightarrow\mathbb{R}$, that is $\delta\mathcal{F}_{m}(\gamma)[W]=0$ for any $W\in T_{\gamma}\Lambda$, if and only if $\Omega(\gamma)=0$ which proves the statement. Similarly, we can obtain the following \[closed\] [(Closed curves)]{} Let $\mathcal{C}$ be the space of immersed closed curves in $(M,g)$. The critical points of the functional $\mathcal{F}_{m}:\mathcal{C}\rightarrow\mathbb{R}$ are those closed curves that are solutions of the following Euler-Lagrange equation $$G\mid _{\gamma}=m \, \kappa.$$ Gaussian magnetic fields ======================== Let $M$ be a surface immersed in the Euclidean three-space, $\mathbb{R}^{3}$, so the metric, $g$ is the induced one. We denote by $N : M \longrightarrow \mathbb{S}^{2}$ its Gauss map and $d\sigma^{2}$ will stand for the area element on the unit round sphere $\mathbb{S}^{2}$. The two form $N^{*}(d\sigma^{2})$ on $M$ can be used, for example, to measure areas of the spherical images or topological total charges of solitons in the $O(3)$ non-linear sigma model (see for instance [@Ody], [@Tsurumaru] and references therein). In this section we will consider magnetic fields of the type $$F=\frac{1}{m} N^{*}(d\sigma^{2}),$$ where $m$ is a non zero constant. We call then [*Gaussian magnetic fields*]{} (GMF). It is well known that $N^{*}(d\sigma^{2})=G \, \Omega_{2}$, $G$ denoting the Gaussian curvature of $(M,g)$ and this, in particular, implies that we can consider these kind of magnetic fields with no mention to the surrounding space. Namely, a GMF is always of the type $$\label{Gaussian} F=\frac{G}{m}\,\Omega_2.$$ The Lorentz force of a GMF is computed to be $\Phi=(G/m) \, J$, where $J$ is the standard complex structure in $M$. In particular, for any unit vector field, $X$, on $M$, the matrix of $\Phi$, in the terminology of Section 3, with respect to an orthonormal frame $\{X,JX\}$ is given by $$\left( \begin{array}{rr} 0 & - \frac{G}{m} \\ \frac{G}{m} & 0 \end{array} \right).$$ In this framework, we can combine Proposition \[magneticasdeenergiae\] and the field equations of the particle models defined from $\mathcal{F}_{m}$, see Eq. (\[funcional\]), to obtain the following amazing relationship between the flow of a GMF and the worldline trajectories of relativistic particles with order one. To be precise, we have, \[importante\] Let $\gamma\in \Lambda$ be a curve [(clamped or closed)]{} in $(M,g)$. Then it is a normal flowline of $\left( M,g,F=(G/m) \, \Omega_{2} \right)$ if and only if it is a critical point [(world line)]{} of the action $\mathcal{F}_{m}:\Lambda\rightarrow\mathbb{R}$ given by $$\mathcal{F}_{m}(\gamma)=\int_{\gamma} (\kappa+m) \, ds.$$ At this point, we can take advantage of the variational approach to study stability of the GMF flowlines. Therefore, we need the second derivative of $\mathcal{F}_{m}$ in a critical point, say $\gamma$ in a suitable space of curves, $\Lambda$ (recall closed or clamped curves). After some long computations (see [@Arroyo] for details) one can obtain the following expression $$\label{segundavariacion} \delta^{2}\mathcal{F}_{m}(\gamma)[W]=\int_{\gamma}g\left(W,\nabla_{W} \Psi\right)|_{\gamma} \, ds,$$ where $\Psi$ denotes the vector field, along a variation of $\gamma$, given by $\Psi=(G-m \, \kappa) \, N$. Now, we choose $W=\Phi \, N$ to obtain $$g\left(W,\nabla_{W} \Psi\right)|_{\gamma}=\Phi^{2} \, N(G-m \, \kappa),$$ where the right hand term is restricted to $\gamma$. However the variation of $\kappa$ was given in Eq. (\[curvature\]); so, in particular, we have $N(\kappa)=\kappa^{2}+G$. Then, one gets from Eq. (\[segundavariacion\]) $$\delta^{2}\mathcal{F}_{m}(\gamma)[W]=\int_{\gamma}\Phi^{2}\left(N(G)-\frac{1}{m}(G^{2}+m^{2}G)\right) \, ds.$$ Hence, we have the following useful test of stability. \[caracterizacionestabilidad\] A critical point, $\gamma\in\Lambda$, of $\mathcal{F}_{m}$ is stable if and only if the function $\; N(G)-\displaystyle ( 1/m)(G^{2}+m^{2}G)$ is signed along $\gamma$. It should be observed that the previous test has the following geometrical meaning. Put $\varphi=G-m \, \kappa$, then $\gamma\subset\varphi^{-1}(0)$, because it is a critical point of $\mathcal{F}_{m}$, and then stability means that $\gamma$ is made up of regular points of $\varphi$. Moreover, observe that this happens if $0$ is a regular value of $\varphi$. Now, let us use all this information in the following elemental setting. We consider $M=\mathbb{S}^{2}(1)$ the unit round sphere. Then any GMF, $F=(G/m) \, \Omega_{2}=(1/m) \, \Omega_{2}$, is uniform. However, when studying uniform magnetic fields on a round sphere, we can not talk about stability of magnetic trajectories. This is not the case of our approach. In this setting, the magnetic curves are, according Theorem \[importante\], the critical points of the functional $$\mathcal{F}_{m}(\gamma)=\int_{\gamma}(\kappa+m) \, ds=\int_{\gamma}\kappa \, ds+m\, L(\gamma),$$ and they are nothing but those curves that satisfy $1=m \, \kappa$, that is geodesic circles with geodesic curvature $\kappa=1/m$. On the other hand, we can use the Gauss-Bonnet formula to see that this variational problem is equivalent to that associated with the action $\mathcal{D}_{m}:\mathbf{D}\longrightarrow\mathbb{R}$ defined by $$\mathcal{D}_{m}(\Delta)=\int_{\Delta} G \, \Omega_{2}+m\int_{\partial\Delta}ds={\rm Area}(\Delta)+m \, L(\partial\Delta),$$ acting on the space $\mathbf(D)$ of simply-connected domains, $\Delta$ in $\mathbb{S}^{2}$, with the same boundary $\gamma=\partial\Delta$. This is nothing but the isoperimetric problem in the round sphere. The solution is a couple of domains $\Delta_{1}$ and $\Delta_{2}$ (the maximum and the minimum) with common boundary a geodesic circle, $\gamma$, of curvature $\kappa=1/m$. Since $N(G)=0$ then, $-(1+m^{2})/m$ has obviously sign for any choice of the coupling constant $m$. Consequently, calling to Proposition \[caracterizacionestabilidad\], the solutions are stable. The case where $m=0$ deserves a few words. First of all GMF with $m=0$ could be considered as a limiting case, however, after our variational approach, it can be identified with the massless Plyushchay model, [@Plyushchay1], which is governed by the Lagrangian $$\mathcal{F}_{0}(\gamma)=\int_{\gamma}\kappa \, ds.$$ This model has been considered with detail in [@Arroyo-Barros-Garay]. For example the sphere does not admit, non only minima (maxima) for this model but also critical points. However, an anchor ring has two critical points corresponding to the two parallels of parabolic points. GMF flowlines on some non-constant Gauss curvature surfaces =========================================================== In this last section we would like to analyze when certain relevant curves on some non-constant Gauss curvature surfaces are in fact magnetic. We recalled the explicit expression of the Gauss curvature, $G(s,v)$, of a surface of revolution, $M_{\alpha}$, see Eq. (\[Gaussrevolution\]). Consequently, we can assert now that a parallel, $\gamma_{s}$, is a normal flowline of the GMF given by $(G/m) \, \Omega_{2}$ on $M_{\alpha}$ if and only if $$\label{parallel} f^{\prime\prime}(s)+m \, f^{\prime}(s)=0.$$ Next, we will obtain the surfaces of revolution whose parallels are all normal flowline of a GMF. In contrast to the case of a uniform magnetic field, where only the bugle surface appeared as a solution (see Proposition \[uniformes\]), now the general solution is made up of a three parameter family of surfaces which includes the bugle surface too. The normal flow of a GMF, $(G/m) \,\Omega_{2}$, in a surface of revolution, $M_{\alpha}$, is invariant under rotations if and only if the profile curve of $M_{\alpha}$ lies in the following three parameter family of arclength parametrized plane curves $$\alpha(s)=\left(f(s), \, \int_{0}^{s}\sqrt{1-f^{\prime}(s)^{2}} \, ds\right),$$ where $$f(s)=\frac{1}{m}\big(a+c \, \exp(-m \, s+b)\big); \qquad a,b,c\in \mathbb{R} \quad {\rm with} \quad a>0.$$ Observe that the general solution of the ordinary differential equation (\[parallel\]) has the form $f(s)=\big(a+c \, \exp(-m \, s+b)\big)/m$, so the proof of the last result becomes obvious. Observe also that the above characterized class of surfaces of revolution includes the bugle surface ($a=0$) and the right circular cylinder ($c=0$) too. It should be noticed the following coupling phenomenon in a surface of revolution, $M_{\alpha}$, between the GMF, $F_{1}=(G/m) \, \Omega_{2}$ and the uniform magnetic field $F_{2}=-m \, \Omega_{2}$, for some values of the coupling constant $m$. Suppose, for example, that $M_{\alpha}=\mathbf{T}(r,R)$ is a torus of revolution and $\rho=(R^2-r^2)^{-1/2}$ (notation as in Section 4). Then, $F_{1}$ always has two parallels being normal magnetic curves, no matter the value of $m$. Now, we use Proposition \[proptoro\] to obtain the following statement, If $-m\in \left( \rho, \, \rho \right)$, then both $F_{1}$ and $F_{2}$ have two normal magnetic parallels coming from points alternatively placed in the profile circle. Moreover they collapse when $-m$ goes to $-\rho$ or $\rho$. [**Proof.**]{} For any value of $m$ in $\mathbb{R}$, $(\mathbf{T}(r,R),g,F_{1})$ has two normal magnetic parallels obtained by rotation of the two antipodal points in $\mathbf{C}$, defined by $\cot \, (s/r)=-r \, m$. These two points are just those determining the diameter $D_{-m}$ that separates the two magnetic parallel of $(\mathbf{T}(r,R),g,F_{2})$ when $-m\in \left(-\rho, \, \rho \right)$. The second part of this statement follows similarly when use points 1 and 2 of Proposition \[proptoro\]. We finish the paper showing several examples. Let $\beta(s)$ be an arclength parametrized curve contained in a plane, $\Pi$ (with unit normal vector $B_{0}$), in $\mathbb{R}^{3}$. We denote by $\{T(s),N(s)\}$ a Frenet frame along $\beta(s)$, so that $T(s)\wedge N(s)=B_{0}$, and $\kappa(s)$ will stand for its curvature function. For a suitable $r>0$, we define a tube of radius $r$, say $\mathrm{T}_{\beta}(r)$, as the surface given by $$X(s,v)=\beta(s)+r\big(\cos(v) \, N(s)+\sin(v) \, B_{0}\big).$$ We denote by $\Lambda_{\beta}=\{\,\gamma_{v}, \, : \, v\in[0,2\pi] \,\}$ the family of curves in the tube obtained when we make $v$ constant. The curvature of these curves in $\mathrm{T}_{\beta}(r)$ can be obtained, from a direct computation, to be $$\kappa_{v}(s)=\frac{\kappa(s) \, \sin(v)}{1-r \, \kappa(s) \, \cos(v)}.$$ Notice that it is not constant unless $\beta(s)$ is chosen to be constant curvature. On the other hand, the Gauss curvature of the tube $\mathrm{T}_{\beta}(r)$ is computed to be $$G(s,v)=-\frac{\kappa(s) \, \cos(v)}{r\big(1-r\kappa(s) \, \cos(v)\big)}.$$ Now, we can apply these formulas together with the Euler-Lagrange equations associated with the GMF, $F=(G/m) \, \Omega_{2}$ (Propositions \[clamped\], \[closed\]), to see that there exist exactly two curves (clamped or closed) in $\Lambda_{\beta}$ that are normal magnetic trajectories. They are obtained for $\cot(v)=-r \, m$ and this is, formally, the same result that we have obtained for a torus of revolution (Proposition \[proptoro\]) which can be regarded as a tube around a circle. Similarly, for a curve, $\beta(s)$, in $\mathbb{R}^{3}$ with Frenet frame $\{T(s),N(s),B(s)\}$, curvature $\kappa(s)$ and torsion $\tau(s)$, one can define the tube $\mathrm{T}_{\beta}(r)$ by $$X(s,v)=\beta(s)+r\big(\cos(v) \, N(s)+\sin(v) \, B(s)\big).$$ In particular, if $\beta(s)$ is a helix ($\kappa$ and $\tau$ are both constant) then the curvature function, $\kappa_{v}(s)$ of the curves in $\Lambda_{\beta}=\{\,\gamma_{v}, \, : \, v\in[0,2\pi]\, \}$ satisfy $$\kappa_{v}^{2}=\frac{\kappa^{2} \, \sin^{2}(v)}{\big(1-r \, \kappa \, \cos(v)\big)^{2}+r^{2}\tau^{2}}.$$ Now, the curves in $\Lambda_{\beta}$ that are normal flowlines of $(G/m) \, \Omega_{2}$ on the helicoidal tube $\mathrm{T}_{\beta}(r)$ correspond with the zeroes of the function $\vartheta:\mathbb{S}^{1}\rightarrow\mathbb{R}$ defined by $$\vartheta(v)=\big(1-r \, \kappa \, \cos(v)\big)^{2}\big(\cos^{2}(v)-r^{2}m^{2}\sin^{2}(v)\big)+r^{2}\tau^{2}\cos^{2}(v).$$ However, we have $$\vartheta(0)=\vartheta(\pi)=(1-r \, \kappa)^{2}+r^{2}\tau^{2}>0 \quad \textrm{and} \quad \vartheta(\frac{\pi}{2})=\vartheta(\frac{3\pi}{2})=-r^{2} m^{2}<0.$$ Therefore, there exist four curves of $\Lambda_{\beta}$ in the flow of $(G/m) \, \Omega_{2}$. [On the cathenoid (Example \[catenoide\]), the GMF given by $(G/m)\,\Omega_2$ has a unique normal magnetic parallel for all $m$. If $m>0$, it is obtained for a $t_0<0$. If $m<0$, then it is obtained for a $t'_0=-t_0>0$. ]{} [On the hyperboloid of revolution obtained from Eq. (\[revolucion\]) by putting $f(t)=\cosh t$ and $h(t)=\sinh t$, the GMF given by $(G/m)\,\Omega_2$ has also a unique normal magnetic parallel for all $m$, analogously to the previous case. ]{} [On the cicloidal surface (Example \[cicloide\]), the GMF given by $(G/m)\,\Omega_2$ has a unique normal magnetic parallel for $m\in\left(-\infty , -1/(4a)\right)\bigcup\left( 1/(4a),\infty\right)$. If $m>0$, $t_0=\arccos \left((1/(4am)\right)$, whereas if $m<0$, then $t'_0=2\pi -t_0$.]{} [**Conclusions**]{} Oriented surfaces, $M$, in $\mathbb{R}^{3}$ admit two natural $2$-forms. First, the area element, $\Omega_{2}$, associated with the induced metric, $g$. Second, the area element, $N^{*}(d\sigma^{2})$, of its spherical image under the Gauss map, $N:M\rightarrow\mathbb{S}^{2}$. It is well known that these $2$-forms are nicely related by $$N^{*}(d\sigma^{2})=G \, \Omega_{2},$$ where $G$ enotes the Gaussian curvature of $g$. In particular, both $2$-forms are intrinsic and then they are defined once we know a Riemannian metric, $g$, on $M$. Associated with these $2$-forms appear two classes of magnetic fields on $(M,g),$ 1. The class made up of the constant multiples of the former one, $\mathcal{C}_{1}=\{\mu \, \Omega_{2} \, : \, \mu\in\mathbb{R}\}$, provides that of uniform magnetic fields, with strength $\mu$, on $(M,g)$. The corresponding Landau-Hall problem has been widely studied along the literature. Even in this paper, we have obtained some new information relative to uniform magnetic field essentially in a surface of revolution. For example, we have characterized right circular cylinders and bugle surfaces as the only surfaces of revolution whose parallels are all normal magnetic flowlines of a uniform magnetic fields. 2. The class of the constant multiples of the later one, $\mathcal{C}_{2}=\{\mu \, N^{*}(d\sigma_{2}) \, : \, \mu\in\mathbb{R}\}$, constitutes a class of magnetic fields that in this paper are introduced under the terminology of Gauss-Landau-Hall magnetic fields (GMF). In this case the strength is given by $\mu \, G$ and obviously both classes coincide when $(M,g)$ has constant curvature. In this paper, we wish to state the importance and nice interest of GMF on surfaces. In fact, the chief result of the paper appears when we study the Landau-Hall problem associated with a GMF (which we call the Gauss-Landau-Hall problem). Then, we are be able to show that this problem is equivalent to the dynamics of a massive relativistic boson. This provides an amazing relationship between two, a priori, quite different physical phenomena. Therefore, we can use two points of view to study each of the two involved problems. On one hand, one can study completeness, homogeneity and so on, in the dynamical study of bosonic worldlines. By the way, we have introduced a section with new results on these topics. But on the other hand, the Gauss-Landau-Hall problem can be regarded as a variational problem globally stated. In this setting, flowlines are critical points of an action which has been used to model relativistic particles with order one rigidity. In particular, we can talk about, and so we study, global stability of normal flowlines of a GMF. Say finally that under this equivalence, the model to describe a massless particle with arbitrary helicity correspond with a limit case obtained when the force of the GMF increases arbitrarily. We believe that this new point of view in the study of GMF is physically remarkable and it could be extended to other classes of magnetic fields. [****]{} Research partially supported by MCYT FEDER Grant BFM 2001-2871-C04 and by Acción Coordinada Grupos de Investigación FQM-324 and FQM-327, Junta de Andalucía. [777]{} L.D. Landau and E. M. Lifschitz, [*Course of Theoretical Physics*]{}, Vol. 1. Mechanics. 3rd edition. (Pergamon Press, Oxford-New York-Toronto, 1976). M. S. Plyushchay, Phys. Lett. B, [**243**]{}, 383 (1990). M.S. Plyushchay, Int. J. Mod. Phys. A, [**4**]{}, 3851 (1989). R. K. Sachs and H. Wu, [*General Relativity for mathematicians*]{}, in Graduate texts in Mathematics, no.48, (Springer-Verlag, New York, 1977). A. Romero and M. Sánchez, “Completeness of the solutions of certain differential equation on semi-Riemannian manifolds: Motion of particles on a force field” (Univ. Granada, umpublished). R. Abraham and J. E. Marsden, [*Foundations of Mechanics*]{}, 2nd. Ed. 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--- abstract: 'We prove that certain acyclic cluster algebras over the complex numbers are the coordinate rings of holomorphic symplectic manifolds. We also show that the corresponding quantum cluster algebras have no non-trivial prime ideals. This allows us to give evidence for a generalization of the conjectured variant of the orbit method for quantized coordinate rings and their classical limits.' author: - Sebastian Zwicknagl title: Poisson and quantum geometry of acyclic cluster algebras --- Introduction ============ In the present paper we investigate the Poisson geometry associated with cluster algebras over the complex numbers defined by acyclic quivers, and relate them to the ideal theory of the corresponding quantum cluster algebras. Our main motivation is the following conjectural analogue of Kirillov’s Orbit method for quantized coordinate rings which has been an open problem for roughly twenty years (see e.g. [@brown-goodearl] or [@Yak-spec Section 4.3] and [@Soi] for the case of compact quantum groups). Let $G$ be a semisimple complex algebraic group and $\CC[g]$ its coordinate ring while $\CC_q[G]$ denote the corresponding quantized coordinate ring. It has been conjectured that there exists a homeomorphism between the space of primitive ideals in $R_q[G]$ and the symplectic leaves of the standard Poisson structure on $G$. For an excellent introduction to this conjecture we refer the reader to Goodearl’s paper [@Goo1]. The conjecture appears extremely difficult to prove and it is only known to be true in the cases of $G=SL_2, SL_3$. The coordinate rings $\CC[G]$ are known to have an upper cluster algebra structure ([@BFZ]) while the quantized coordinate rings are conjectured to have a quantum (upper) cluster algebra structure ([@bz-qclust Conjecture 10.10]). Indeed, it follows from recent results of Geiß, Leclerc and Schröer ([@GLSq Section 12.4]) that $\CC_q[SL_n]$ has a quantum cluster algebra structure. Cluster algebras are nowadays very well-established, hence we do not recall any of the definitions here, and refer the reader to the literature, resp. our Section \[se:Cluster Algebras\]. Most importantly for our purposes, a cluster algebra over $\CC$ is defined by a combinatorial datum in a field of fractions $\CC(x_1,\ldots, x_n)$. We will denote this [*initial seed*]{} by $({\bf x}, B)$ where ${\bf x}=(x_1,\ldots,x_n)$ and $B$ is an integer $m\times n$-matrix with $m\le n$ such that its principal $m\times m$ submatrix is skew-symmetrizable. The cluster variables $x_{m+1},\ldots, x_n$ are the frozen variables which we will call coefficients. A quantum cluster algebra is given by a [*quantum seed*]{} $( {\bf x}, B,\Lambda)$ where $(B,\Lambda)$ where $B$ is as above and $\Lambda$ is a skew-symmetric $n\times n$-matrix such that $(B,\Lambda)$ is a [*compatible pair*]{} (see Section \[se:Cluster Algebras\] for details). The set ${\bf x}=(x_1,\ldots, x_n)$ now lives in the skew-field of fractions $\CC_\Lambda(x_1,\ldots,x_n)$ defined by $\Lambda$. A compatible pair also defines a compatible Poisson structure in the sense of [@GSV] on the cluster algebra given by $({\bf x},B)$. It is well-known that the conjectured quantum cluster algebra structures on the rings $R_q[G]$ and the standard Poisson structure on $\CC[G]$ arise from such a compatible pair. Therefore, we would like to suggest the following conjecture.. \[conj: homeo cluster\] Let $(B,\Lambda)$ be a compatible pair and let $\AA$ and $\AA_q$ be a cluster, resp. quantum cluster algebra defined by $({\bf x},B)$, resp. $({\bf x},B,\Lambda)$. Suppose further that $\AA$ and $\AA_q$ are Noetherian and that $\AA$ is the coordinate ring of the affine variety $X$. Then, there exists a homeomorphism between the space of primitive ideals of $\AA_q$ and the symplectic leaves on $X$ defined by $\Lambda$. In light of Conjecture \[conj: homeo cluster\], we may think of quantum affine space and quantum tori as cluster algebras where all cluster variables are frozen. In this case the corresponding homeomorphism is well known and easy to construct. The other extreme case are cluster algebras without coefficients and here the class that is usually easiest to study are the acyclic cluster algebras. For example, it is known that such a cluster algebra is always Noetherian and the coordinate ring of an affine variety (see [@BFZ] and [@bz-qclust] for the classical and quantum versions). It is our main objective to give evidence for Conjecture \[conj: homeo cluster\] by proving it in this very specific case. It is an immediate consequence of the following two main results. \[th:Classical-intro\] Let $\AA$ be a cluster algebra with initial seed $({\bf x}, B)$ defined by an acyclic quiver where $B$ is invertible satisfying , and suppose that it is the coordinate ring of an affine variety $X$ and that $(B,\Lambda)$ is a compatible pairs. Then $X$ has the structure of a symplectic manifold, whose symplectic form is the corresponding Poisson bivector. \[th:quantum-intro\] Let $\AA_q$ be a quantum cluster algebra with quantum seed $( {\bf x}, B,\Lambda)$ satisfying the assumptions of Theorem \[th:Classical-intro\]. Then $\{0\}$ is the only proper two sided prime ideal in $\AA_q$. Our approach, is similar to that of [@ZW; @tpc], however all the proofs are self-contained and much easier, as our set-up is less general. The main idea is to study the intersection of ideals with the polynomial ring generated by a given cluster–in this case the acyclic seed. We are able to derive rather strong conditions that Poisson prime ideals–resp. two-sided prime ideals in the quantum case– must satisfy and are able to show that no non-trivial ideals satisfying them exist. A straightforward argument, then allows us to conclude that the variety $X$ is a symplectic manifold. We should also remark that we do not know whether any acyclic cluster algebras exist that do not satisfy the assumptions made in Theorem \[th:Classical-intro\]. The paper is organized as follows. We first briefly recall the definitions of cluster algebras and compatible Poisson structures, compatible pairs and quantum cluster algebras in Section \[se:Cluster Algebras\]. Thereafter, we continue with some technical key propositions (Section \[se:key propositions\]) and discuss in Section \[se:Poisson and symp geom\] the symplectic geometry of acyclic cluster algebras. The proof of Conjecture \[conj: homeo cluster\] in our specific case is completed in Section \[se:ideals in qca\] by proving Theorem \[th:quantum-intro\]. Cluster Algebras {#se:Cluster Algebras} ================ Cluster algebras {#se:Cluster Algebras-def} ---------------- In this section, we will review the definitions and some basic results on cluster algebras, or more precisely, on cluster algebras of geometric type over the field of complex numbers $\CC$. Denote by $\mathfrak{F}=\CC(x_1,\ldots, x_n)$ the field of fractions in $n$ indeterminates. Recall that a $m \times m$-integer matrix $B'$ is called skew-symmetrizable if there exists a $m \times m$-diagonal matrix $D$ with positive integer entries such that $B' \cdot D$ is skew-symmetric. Now, let $B$ be a $m\times n$-integer matrix such that its principal $m\times m$-submatrix is skew-symmetrizable. We call the tuple $(x_1,\ldots,x_n, B)$ the [*initial seed*]{} of the cluster algebra and $ (x_1,\ldots x_m)$ a cluster, while ${\bf x}=(x_1,\ldots x_n)$ is called an extended cluster. The cluster variables $x_{m+1},\ldots,x_n$ are called [*coefficients*]{}. We will now construct more clusters, $(y_1,\ldots, y_m)$ and extended clusters ${\bf y}=(y_1,\ldots, y_n)$, which are transcendence bases of $\mathfrak{F}$, and the corresponding seeds $({\bf y}, \tilde B)$ in the following way. Define for each real number $r$ the numbers $r^+={\rm max}(r,0)$ and $r^-={\rm min}(r,0)$. Given a skew-symmetrizable integer $m \times n$-matrix $B$, we define for each $1\le i\le m$ the [*exchange polynomial*]{} $$P_i = \prod_{k=1}^n x_k^{b_{ik}^+}+ \prod_{k=1}^n x_k^{-b_{ik}^-}\ .$$ We can now define the new cluster variable $x_i'\in\mathfrak{F}$ via the equation $$\label{eq:exchange} x_ix_i'=P_i\ .$$ This allows us to refer to the matrix $B$ as the [*exchange matrix*]{} of the cluster $(x_1,\ldots,x_n)$, and to the relations defined by Equation \[eq:exchange\] for $i=1,\ldots,m$ as [*exchange relations*]{}. We obtain that $(x_1,x_2,\ldots, \hat x_i,x_i',x_{i+1},\ldots, x_n)$ is a transcendence basis of $\mathfrak{F}$. We next construct the new exchange matrix $B_i=B'=(b_{ij}')$, associated to the new (extended) cluster $${\bf x}_i=(x_1,x_2,\ldots, \hat x_i,x_i',x_{i+1},\ldots, x_n)$$ via its coefficients $b_{ij}'$ as follows: $\bullet$ $b_{ij}' = -b_{ij}$ if $j \le n$ and $i = k$ or $j = k$, $\bullet$ $b_{ij}' = b_{ij} + \frac{|b_{ik} |b_{kj} + b_{ik} |b_{kj} |}{2}$ if $j \le n$ and $i \ne k$ and $j \ne k$, $\bullet$ $b_{ij}'=b_{ij}$ otherwise. This algorithm is called [*matrix mutation*]{}. Note that $B_i$ is again skew-symmetrizable (see e.g. [@FZI]). The process of obtaining a new seed is called [*cluster mutation*]{}. The set of seeds obtained from a given seed $({\bf x},B)$ is called the mutation equivalence class of $({\bf x},B)$. The cluster algebra $\mathfrak{A}\subset \mathfrak{F}$ corresponding to an initial seed $(x_1,\ldots, x_n,B)$ is the subalgebra of $\mathfrak{F}$, generated by the elements of all the clusters in the mutation equivalence class of $({\bf x},B)$ . We refer to the elements of the clusters as the [*cluster variables*]{}. Notice that the coefficients, resp. frozen variables $x_{m+1},\ldots, x_n$ will never be mutated. Of course, that explains their name. We have the following fact, motivating the definition of cluster algebras in the study of total positivity phenomena and canonical bases. [@FZI Section 3](Laurent phenomenon) Let $\mathfrak{A}$ be a cluster algebra with initial extended cluster $(x_1,\ldots, x_n)$. Any cluster variable $x$ can be expressed uniquely as a Laurent polynomial in the variables $x_1,\ldots, x_n$ with integer coefficients. Moreover, it has been conjectured for all cluster algebras, and proven in many cases (see e.g. [@MSW] and [@FST],[@FT]) that the coefficients of these polynomials are positive. Finally, we recall the definition of the lower bound of a cluster algebra $\AA$ corresponding to a seed $({\bf x}, B)$. Denote by $y_i$ for $1\le i\le m$ the cluster variables obtained from ${\bf x}$ through mutation at $i$; i.e., they satisfy the relation $x_iy_i=P_i$. [@BFZ Definition 1.10] \[def:lower bounds\] Let $\AA$ be a cluster algebra and let $({\bf x}, B)$ be a seed. The lower bound $ \mathfrak{L}_B \subset \AA$ associated with $({\bf x}, B)$ is the algebra generated by the set $\{x_1,\ldots x_n,y_1\ldots, y_m\}$. Upper cluster algebras {#se:upper cluster algebras} ---------------------- Berenstein, Fomin and Zelevinsky introduced the related concept of upper cluster algebras in [@BFZ]. Let $\mathfrak{A} \subset \mathfrak{F}$ be a cluster algebra with initial cluster $(x_1, \ldots, x_n, B)$ and let, as above, $y_1, \ldots, y_m$ be the cluster variables obtained by mutation in the directions $1, \ldots, m$, respectively. \(a) The upper bound $\UU_{{\bf x},B} ( \mathfrak{A})$ is defined as $$\UU_{{\bf x},B} ( \mathfrak{A}) = \bigcap_{j = 1}^m \CC [x_1^{\pm 1}, \ldots x_{j - 1}^{\pm 1}, x_j, y_j, x_{j + 1}^{\pm 1}, \ldots, x_m^{\pm 1}, x_{m+1},\ldots,x_n] \ .$$ \(b) The upper cluster algebra $\UU ( \mathfrak{A})$ is defined as $$\UU ( \mathfrak{A})=\bigcap_{({\bf x'},B')}\UU_{\bf x'} ( \mathfrak{A})\ ,$$ where the intersection is over all seeds $({\bf x}',B')$ in the mutation equivalence class of $({\bf x},B)$. Observe that each cluster algebra is contained in its upper cluster algebra (see [@BFZ]). Poisson structures {#se:poissonstructure} ------------------ Cluster algebras are closely related to Poisson algebras. In this section we recall some of the related notions and results. Let $k$ be a field of charactieristic $0$. A Poisson algebra is a pair $(A,\{\cdot,\cdot\})$ consisting of a commutative $k$-algebra $A$ and a bilinear map $\{\cdot,\cdot\}:A\otimes A\to A$, satisfying for all $a,b,c\in A$: 1. skew-symmetry: $\{a,b\}=-\{b,a\}$ 2. Jacobi identity: $\{a,\{b,c\}\}+\{c,\{a,b\}\}+\{b,\{c,a\}\}=0$, 3. Leibniz rule: $a\{b,c\}=\{a,b\}c+b\{a,c\}$. If there is no room for confusion we will refer to a Poisson algebra $(A,\{\cdot,\cdot\})$ simply as $A$. A [*Poisson Ideal*]{} $\II$ in a Poisson algebra $A$ is an ideal such that $\{\II,A\}\subset \II$, and if $k$ is of characteristic zero, then a Poisson prime ideal is a prime ideal which is also Poisson. Gekhtman, Shapiro and Vainshtein showed in [@GSV] that one can associate Poisson structures to cluster algebras in the following way. Let $\mathfrak{A} \subset \CC[x_1^{\pm 1}, \ldots, x_n^{\pm 1}] \subset \mathfrak{F}$ be a cluster algebra. A Poisson structure $\{\cdot, \cdot\}$ on $\CC [x_1, \ldots, x_n]$ is called log-canonical if $\{ x_i,x_j\}=\lambda_{ij} x_ix_j$ with $\lambda_{ij}\in \CC$ for all $1\le i,j\le n$. The Poisson structure can be naturally extended to $\mathfrak{F}$ by using the identity $0=\{f\cdot f^{-1},g\}$ for all $f,g\in\CC [x_1, \ldots, x_n]$. We thus obtain that $\{f^{-1},g\}=-f^{-2}\{f,g\}$ for all $f,g\in \mathfrak{F}$. We call $\Lambda=\left( \lambda_{ij}\right)_{i,j=1}^n$ the [*coefficient matrix*]{} of the Poisson structure. We say that a Poisson structure on $\mathfrak{F}$ is compatible with $\mathfrak{A}$ if it is log-canonical with respect to each cluster $(y_1,\ldots, y_n)$; i.e., it is log canonical on $\CC[y_1, \ldots, y_n]$. \[re:Class Poisson\] A classification of Poisson structures compatible with cluster algebras was obtained by Gekhtman, Shapiro and Vainshtein in [@GSV Theorem 1.4]. It is easy to see from their description that if $n$ is even, then the cluster algebra has an admissible Poisson structure of maximal rank. We will refer to the cluster algebra $\AA$ defined by the initial seeed $({\bf x},B)$ together with the compatible Poisson structure defined by the coefficient matrix $\Lambda$ with respect to the cluster ${\bf x}$ as the [*Poisson cluster algebra*]{} defined by the [*Poisson seed*]{} $({\bf x},B,\Lambda)$. It is not obvious under which conditions a Poisson seed $({\bf x},B,\Lambda)$ would yield a Poisson bracket $\{\cdot,\cdot\}_\Lambda$ on $\mathfrak{F}$ such that $\{\AA,\AA\}_\Lambda\subset \AA$. We have, however, the following fact. Let $({\bf x},B,\Lambda)$ be a Poisson seed and $\AA$ the corresponding cluster algebra. Then $\Lambda$ defines a Poisson algebra structure on the upper bound $\UU_{{\bf x},B}(\AA)$ and the upper cluster algebra $\UU(\AA)$. Denote as above by $\{\cdot,\cdot\}_\Lambda$ the Poisson bracket on $\mathfrak{F}$ by $\Lambda$. Observe that the algebras $\CC[x_1^{\pm 1},\ldots x_{i-1}^{\pm 1}, x_i,y_i, x_{i+1}^{\pm 1}, \ldots, x_n^{\pm 1}]$ are Poisson subalgebras of the Poisson algebra $\CC[x_1^{\pm 1},\ldots x_n^{\pm 1}]$ for each $1\le i\le m$, as $\{x_i,y_i\}_\Lambda=\{x_i,x_i^{-1}P_i\}_\Lambda\in \CC[x_1,\ldots, x_n]$. If $A$ is a Poisson algebra and $\{B_i\subset A:i\in I\}$ is a family of Poisson subalgebras, then $\bigcap_{i\in I} B_i$ is a Poisson algebra, as well. The assertion follows. Compatible Pairs and Their Mutation {#se:Compatible Pairs and Mut} ----------------------------------- Section \[se:Compatible Pairs and Mut\] is dedicated to compatible pairs and their mutation. Compatible pairs yield important examples of Poisson brackets which are compatible with a given cluster algebra structure, and as we shall see below, they are also integral in defining quantum cluster algebras. Note that our definition is slightly different from the original one in [@bz-qclust]. Let, as above, $m\le n$. Consider a pair consisting of a skew-symmetrizable $m\times n$-integer matrix $B$ with rows labeled by the interval $[1,m]=\{1,\ldots, m\}$ and columns labeled by $[1,n]$ together with a skew-symmetrizable $n\times n$-integer matrix $\Lambda$ with rows and columns labeled by $[1,n]$. \[def:compa pair\] Let $B$ and $\Lambda$ be as above. We say that the pair $(B,\Lambda)$ is compatible if the coefficients $d_{ij}$ of the $m\times n$-matrix $D=B\cdot \Lambda$ satisfy $d_{ij}=d_i\delta_{ij}$ for some positive integers $d_i$ ($i\in [1,m]$). This means that $D=B\cdot \Lambda$ is a $m\times n$ matrix where the only non-zero entries are positive integers on the diagonal of the principal $m\times m$-submatrix. The following fact is obvious. \[le:full rank\] Let $(B,\Lambda)$ be a compatible pair. Then $B\cdot \Lambda$ has full rank. Let $(B,\Lambda)$ be a compatible pair and let $k\in [1,m]$. We define for $\varepsilon\in \{+1,-1\}$ a $n\times n$ matrix $E_{k,\varepsilon}$ via - $(E_{k,\varepsilon})_{ij}=\delta_{ij}$ if $j\ne k$, - $(E_{k,\varepsilon})_{ij}= -1$ if $i=j= k$, - $(E_{k,\varepsilon})_{ij}= max(0,-\varepsilon b_{ik})$ if $i\ne j= k$. Similarly, we define a $m\times m$ matrix $F_{k,\varepsilon}$ via - $(F_{k,\varepsilon})_{ij}=\delta_{ij}$ if $i\ne k$, - $(F_{k,\varepsilon})_{ij}= -1$ if $i=j= k$, - $(F_{k,\varepsilon})_{ij}= max(0,\varepsilon b_{kj})$ if $i= k\ne j$. We define a new pair $(B_k,\Lambda_k)$ as $$\label{eq:mutation matrix and Poisson} B_k= F^T_{k,\varepsilon} B E_{k,\varepsilon}^T \ , \quad \Lambda_k=E_{k,\varepsilon} \Lambda E_{k,\varepsilon}^T\ ,$$ where $X^T$ denotes the transpose of $X$. We chose this rather non-straightforward way of defining $E_{k\varepsilon}$ and $F_{k,\varepsilon}$ in order to show how our definition relates to that of [@bz-qclust]. We will not need it in what follows. The motivation for the definition is the following fact. [@bz-qclust Prop. 3.4] \[pr:comp under mutation\] The pair $(B_k,\Lambda_k)$ is compatible. Moreover, $\Lambda_k$ is independent of the choice of the sign $\varepsilon$. The following fact is clear. Let $\AA$ be a cluster algebra given by an initial seed $({\bf x}, B)$ where $B$ is a $m\times n$-matrix. If $(B,\Lambda)$ is a compatible pair, then $\Lambda$ defines a compatible Poisson bracket on $\mathfrak{F}$ and on $\UU(\AA)$. \[ex:acyclic\] If $m=n$ (i.e. there are no coefficients/frozen variables) and $B$ has full rank, then $(B, \mu B^{-1})$ is a compatible pair for all $\mu\in \ZZ_{> 0}$ such that $\mu B^{-1}$ is an integer matrix. It follows from [@GSV Theorem 1.4] that in this case all compatible Poisson brackets arise in this way. Recall that double Bruhat cells in complex semisimple connected and simply connected algebraic groups have a natural structure of an upper cluster algebra (see [@BFZ]). Berenstein and Zelevinsky showed that the standard Poisson structure is given by compatible pairs relative to this upper cluster algebra structure (see [@bz-qclust Section 8]). Quantum Cluster Algebras {#se: q-cluster algebra spec} ------------------------ In this section we recall the the definition of a quantum cluster algebra, introduced by Berenstein and Zelevinsky in [@bz-qclust]. We define, for each skew-symmetric $n\times n$-integer matrix $\Lambda$, the skew-polynomial ring $\CC_\Lambda^t[x_1,\ldots, x_n]$ to be the $\CC[t^{\pm 1}]$-algebra generated by $x_1,\ldots, x_n$ subject to the relations $$x_ix_j=t^{\lambda_{ij}} x_jx_i\ .$$ Analogously, the quantum torus $H_\Lambda^t=\CC_\Lambda^t[x_1^{\pm 1},\ldots, x_n^{\pm 1} ]$ is defined as the localization of $\CC_\Lambda^t[x_1,\ldots, x_n]$ at the monoid generated by ${x_1,\ldots, x_n}$, which is an Ore set. The quantum torus is clearly contained in the skew-field of fractions $\mathcal{F}_\Lambda$ of $\CC_\Lambda^t[x_1,\ldots, x_n]$, and the Laurent monomials define a lattice $L\subset H_\Lambda^t\subset \mathcal{F}_\Lambda$ isomorphic to $\ZZ^n$. Denote for each ${\bf e}=(e_1,\ldots e_n)\in \ZZ^n$ by $x^{\bf e}$ the monomial $x_1^{e_1}\ldots x_n^{e_n}$. We need the notion of a toric frame in order to define the quantum cluster algebra. A toric frame in $ \mathfrak F$ is a mapping $M:\ZZ^n\to \mathfrak F-\{0\}$ of the form $$M(c)=\phi(X^{\eta(c)})\ ,$$ where $\phi$ is a $\QQ(\frac{1}{2})$-algebra automorphism of $\mathfrak F$ and $\eta: \ZZ^n\to L$ an isomorphism of lattices. Since a toric frame $M$ is determined uniquely by the images of the standard basis vectors $\phi(X^{\eta(e_1)})$,…, $\phi(X^{\eta(e_n)})$ of $\ZZ^n$, we can associate to each toric frame a skew commutative $n\times n$-integer matrix $\Lambda_M$. We can now define the quantized version of a seed. [@bz-qclust Definition 4.5] A quantum seed is a pair $(M,B)$ where - $M$ is a toric frame in $\mathfrak F$. - $B$ is a $n\times m$-integer matrix with rows labeled by $[1,m]$ and columns labeled by $[1,n]$. - The pair $(B,\Lambda_M)$ is compatible. Now we define the seed mutation in direction of an exchangeable index $k\in [1,m]$. For each $\varepsilon\in \{ 1,-1\}$ we define a mapping $M_k: \ZZ^n\to \mathfrak F$ via $$M_k(c)=\sum_{p=0}^{c_k} \binom{c_k}{p}_{q^{d_k}{2}} M(E_\varepsilon c +\varepsilon p b^k)\ , \quad M_k(-c)=M_k(c)^{-1}\ ,$$ where we use the well-known $q$-binomial coefficients (see e.g. [@bz-qclust Equation 4.11]), and the matrix $E_{k,\varepsilon}$ defined in Section \[se:Compatible Pairs and Mut\]. Define $B_k$ to be obtained from $B$ by the standard matrix mutation in direction $k$, as in Section \[se:Cluster Algebras-def\]. One obtains the following fact. [@bz-qclust Prop. 4.7] (a) The map $M_k$ is a toric frame, independent of the choice of sign $\varepsilon$. \(b) The pair $(B_k,\Lambda_{M_k})$ is a quantum seed. Now, given an [*initial quantum seed*]{} $(B, \Lambda_M) $ denote, in a slight abuse of notation, by $X_1=M(e_1),\ldots, X_r=M(e_r)$, which we refer to as the [*cluster variables*]{} associated to the quantum seed $(M,B)$. Here our nomenclature differs slightly from [@bz-qclust], since there one considers the coefficients not to be cluster variables. We now define the seed mutation $$X_k'=M(-e_k+\sum_{b_{ik}>0} b_{ik}e_i)+ M(-e_k-\sum_{b_{ik}<0} b_{ik}e_i)\ .$$ We obtain that $X_k'=M_k(e_k)$ (see [@bz-qclust Prop. 4.9]). We say that two quantum seeds $(M,B)$ and $(M',B')$ are mutation-equivalent if they can be obtained from one another by a sequence of mutations. Since mutations are involutive (see [@bz-qclust Prop 4.10]), the quantum seeds in $\mathfrak{F}$ can be grouped in equivalence classes, defined by the relation of mutation equivalence. The quantum cluster algebra generated by a seed $(M,B)\subset \mathfrak F$ is the $\CC[t^{\pm 1}]$-subalgebra generated by the cluster variables associated to the seeds in an equivalence class. There are definitions of quantum lower bounds, upper bounds and quantum upper cluster algebras (see [@bz-qclust Sections 5 and 7]), analogous to the classical case. Intersections of Ideals with Clusters {#se:key propositions} ===================================== In the present chapter we consider the intersection between Poisson ideals in a cluster algebra and individual clusters. Moreover, we prove quantum analogues of the propositions whenever available. \[pr:TPP super toric\] Let ${\bf x}$ be a cluster, $rank(\Lambda)=n$ and $\II$ be a non-zero Poisson ideal. Then the ideal $\II$ contains a monomial in $x^m\in \CC[{\bf x}]$. Notice first that $\II_{\bf x}=\II\cap \CC[x_1,\ldots, x_n]\ne 0$. Indeed, let $0\ne f\in \II$. We can express $f$ as a Laurent polynomial in the variables $x_1,\ldots, x_n$; i.e., $f=x_{1}^{-c_1}\ldots x_n^{-c_n} g$ where $c_1,\ldots, c_n\in \ZZ_{\ge 0}$ and $0\ne g\in\CC[x_1,\ldots, x_n]$. Clearly, $g= x_{1}^{c_1}\ldots x_n^{c_n}f \in \II_{\bf x}$. We complete the proof by contradiction. Let $f=\sum_{{\bf w}\in \ZZ^n} c_{\bf w} x^{\bf w}\in\II_{\bf x}$ We assume that $f$ has the smallest number of nonzero summands such that no monomial term $c_{\bf w} x^{\bf w}$ with $c_{\bf w}\neq 0$ is contained in $\II$. It must therefore have at least two monomial terms. Assume, as above that $c_{\bf w},c_{{\bf w}'}\ne 0$ and denote by ${\bf v}$ the difference ${\bf v}={\bf w}-{\bf w}'$. Since $\Lambda$ has full rank, there exists $i\in [1,n]$ such that $\{x_i,x^{\bf v}\}\ne 0$. Therefore, $\{x_i, x^{\bf w}\}=cx_ix^{\bf w}\ne dx_ix^{{\bf w}'}\{x_i, x^{{\bf w}'}\}$ for some $c,d\in \CC$. Note that $\{x_i,f\}=\sum_{{\bf w}\in \ZZ^n} c_{\bf w} \lambda_{\bf w}x^{\bf w}x_i$ for certain $ \lambda_{\bf w}\in\ZZ$. Clearly, $cx_if-\{x_i,f\}\in \II$ and $$cx_if-\{x_i,f\}=(c-c)c_{\bf w} x^{\bf w}x_i+(c-d)c_{{\bf w}'} x^{{\bf w}'}x_i+\ldots\ .$$ Hence, $cx_if-\{x_i,f\}\ne 0$ and it has fewer monomial summands than $f$ which contradicts our assumption. Therefore, $\II$ contains a monomial. The proposition is proved. The following fact is an obvious corollary of Proposition \[pr:TPP super toric\]. \[th:intersection\] Let $\AA$ be the cluster algebra defined by a Poisson seed $({\bf x}, B,\Lambda)$ with $rank(\Lambda)=n$. If $\II\subset \AA$ is a non-zero Poisson prime ideal, then $\II$ contains a cluster variable $x_i$. We also have the following quantum version of Proposition \[pr:TPP super toric\] which is proved analogously to the classical case. \[pr:TPP super toric q\] Let $({\bf x},B,\Lambda)$ be a quantum seed, $rank(\Lambda)=n$ and $\II$ a non-zero two-sided ideal. Then the ideal $\II$ contains a monomial in $x^m\in \CC_\Lambda[{\bf x}]$. We do not have a quantum version of Proposition \[th:intersection\] because we do not know whether prime ideals in quantum cluster algebras are completely prime. Poisson ideals in acyclic cluster algebras {#se:Poisson and symp geom} ========================================== In this section we recall results from our previous paper [@ZW; @tpc]. As the proofs are rather short, we shall include them for convenience. Recall e.g. from [@BFZ] that acyclic cluster algebras associated with an acyclic quiver and with trivial coefficients correspond, up to a reordering of the variables of the acyclic seed, to cluster algebras defined by a seed $({\bf x},B)$ where $B$ is a skew-symmetric $n\times n$-matrix with $b_{ij}>0$ if $i<j$. Berenstein, Fomin and Zelevinsky proved in [@BFZ] that such a cluster algebra $\AA$ is equal to both its lower and upper bounds. Thus, it is Noetherian and, if $B$ has full rank, a Poisson algebra with the Poisson brackets given by compatible pairs $(B,\Lambda)$ with $\Lambda=\mu B^{-1}$ for certain $\mu\in \ZZ$ (see Example \[ex:acyclic\]). In order for $B$ to have full rank we have to assume that $n=2k$ is even. Let $P_i=m_i^+ +m_i^-$ where $m_i^+$ and $m_i^-$ denote the monomial terms in the exchange polynomial. Then $\{y_i,x_i\}=\mu_1m_i^+ +\mu_2m_i^-$ for some $\mu_1,\mu_2\in \ZZ$. We, additionally, want to require that $\mu_1\ne \mu_2$. To assure this, we assume that $$\label{eq:poisson gen} \sum_{j=1}^n (b^{-1})_{ij}\left(max(b_{ij},0)+min(b_{ij},0)\right)\ne 0$$ for all $i\in [1,n]$. We have the following result. [@ZW; @tpc] \[th:acyclic\] Let $\AA$ be an acyclic cluster algebra over $\CC$ with trivial coefficients of even rank $n=2k$, given by a seed $(x_1,\ldots, x_{n}, B)$ where $B$ is a skew-symmetric $n\times n$-integer matrix satisfying $b_{ij}>0$ if $i<j$ and suppose that $B$ and $B^{-1}$ satisfy Equation \[eq:poisson gen\] for each $i\in[1,n]$. Then, the Poisson cluster algebra defined by a compatible pair $(B,\Lambda)$ where $\Lambda=\mu B^{-1}$ with $0\ne \mu\in \ZZ$ contains no non-trivial Poisson prime ideals. Suppose that there exists a non-trivial Poisson prime ideal $\II$. Then, $\II\cap {\bf x}$ is nonempty by Proposition \[pr:TPP super toric\], hence $\II\cap {\bf x}=\{x_{i_1},\ldots, x_{i_j}\}$ for some $1\le i_1\le i_2\le \ldots\le i_j\le 2k$. Additionally, observe that $P_{i_1}=m_{i_1}^++m_{i_1}^-$ has to be contained in $\II$, as well as $$\{y_{i_1},x_{i_1}\}=\mu_1m_{i_1}^+ +\mu_2m_{i_1}^-\ .$$ By our assumption, we have $\mu_1\ne \mu_2$, and therefore $m_{i_1}^-\in \II$. Hence, $x_h\in\II$ for some $h\in[1, i_{1}-1]$ or $1\in \II$. But $b_{i,h}<0$ implies that $h<i$ and we obtain the desired contradiction. The theorem is proved. The theorem has the following corollary which was also independently proved by Muller very recently [@Mu; @1], though in more generality. \[co:acyclic smooth\] Let $\AA$ be as in Theorem \[th:acyclic\]. Then, the variety $X$ defined by $\AA=\CC[X]$ is smooth. The singular subset is contained in a Poisson ideal of co-dimension greater or equal to one by a result of Polishchuk ([@Pol]). It is well known that the Poisson ideal must be contained in a proper Poisson prime ideal (see also [@ZW; @tpc]). The assertion follows. The assumption that the cluster algebra has even rank is very important. Indeed, Muller has recently shown that the variety corresponding to the cluster algebra of type $A_3$ has a singularity ([@Mu Section 6.2]). Symplectic Structure -------------------- ### Symplectic geometry of Poisson varieties In this section we will recall some well-known properties of the symplectic structure on Poisson varieties. Our discussion follows along the lines of [@brown-goodearl Part III.5]. If $A$ is a Poisson algebra over a field $k$, then each $a\in A$ defines a derivation $X_a$ on $A$ via $$X_a(b)=\{a,b\}\ .$$ This derivation is called the [*Hamiltonian vectorfield*]{} of $a$ on $A$. Now suppose that $A$ is the coordinate ring of an affine complex variety $Y$. We will associate to the Poisson bracket the [*Poisson bivector*]{} $u\in \Lambda^2 T(Y)$ where $T(Y)$ denotes the tangent bundle of $Y$. Let $p\in Y$ be a point and $\mm_p\subset A$ the corresponding maximal ideal. Let $\alpha,\beta\in \mm_p/\mm_p^2$ be elements of the cotangent space and let $f,g$ be lifts of $\alpha$ and $\beta$, respectively. We define $u_p\in \Lambda^2 T_p(Y)$ $$u_p(\alpha,\beta)=\{f,g\}(p)\ .$$ Note that $u_p$ is a well-defined skew-symmetric form. Indeed, if $I\subset A$ is an ideal and $b\in \II^2$, then for all $a\in A$ $$\{a,b\}\in \II$$ by the Leibniz rule. The form $u_P$ may be degenerate, indeed if it is non-degenerate at every point $p\in Y$, then we call $u_P$ symplectic and, moreover, if $Y$ is connected, then $Y$ is smooth and a (holomorphic) symplectic manifold. Define $$N(p)=\{\alpha \in T^*_p Y:u_p(\alpha,\cdot)=0\}\ ,$$ and $H(p)\subset T_p Y$ its orthogonal complement. The space $H(P)$ is the tangent space of the linear span of the Hamiltonian vectorfields at $p$. Recall that by the Theorem of Frobenius, a Poisson variety $Y$ decomposes as a disjoint union of symplectic leaves, maximal symplectic submanifolds. The tangent space of the symplectic leaf at the point $p$ is $H(p)$. ### The Main Theorem Corollary \[co:acyclic smooth\] implies that $X$ is smooth, hence it has the structure of a complex manifold. We have the following result. \[th:symplectic leaves\] Let $\AA$ be an acyclic cluster algebra as defined above, $X$ an affine variety such that $\AA=\CC[X]$ and let $\Lambda$ define a compatible Poisson bracket. Then, $X$ is a holomorphic symplectic manifold. First, let $p\in X$ be a generic point, by which we mean that $x_i(p)\ne 0$ for all $i=[1,\ldots,n]$. Set $x_i(p)=p_i$. It is easy to see that the Hamiltonian vectorfield $X_{x_i}$ at $p$ evaluates in the local coordinates $(x_1,\ldots, x_n)$ as $$X_{x_i}(p)= (\lambda_{1i} p_i,\ldots, \lambda_{ni} p_i)\ ,$$ where $\Lambda=(\lambda_{ij})_{i,j=1}^n$. Since, $\Lambda$ is non-degenerate, we obtain that the Hamiltonian vectorfields span the tangent space $T_p X$ at $p$. It remains to consider the case when $p_i=0$ for some $i\in[1,n]$. Suppose that $p\in X$ such that $p_i=0$ and $p_j\ne 0$ for all $j<i$. We have to show that the symplectic leaf containing $p$ is not contained in the hyper-surface $x_i=0$. We may assume, employing induction, that if $p_1,\ldots, p_i\ne 0$ then the symplectic leaf at $p$ has full rank. We now claim that $\{x_i,y_i\}(p)\ne 0 $. Indeed, suppose that $\{x_i,y_i\}(p)=(\mu_1 m^+\mu_2 m^-)(p)=0$. Since $p_i=0$ implies that $(m^++m^-)(p)=0$, we would conclude as in the proof of Theorem \[th:acyclic\] that $m^+(p)=m^-(p)=0$, but that is a contradiction to our assumption that $p_j\ne 0$ for all $j<i$. Denote by $u\in \Lambda^2 T(Y)$ the Poisson bivector. We obtain that $u_p(\frac{\delta}{\delta x_i},\cdot)\ne 0$, hence the symplectic leaf containing $p$ is not tangent to the hypersurface $x_i=0$ at $p$. It must contain a point in an analytic neighborhood of $p$ at which $x_i(p)\ne 0$ and $x_j(p)\ne 0$ for all $j<i$ by our assumption. We obtain the desired contradiction and, hence, every symplectic leaf has dimension $n$. But the manifold $X$ is connected and, therefore, cannot be a union of disjoint open submanifolds of equal dimension, hence $X$ contains only one symplectic leaf and the theorem is proved. This result can be easily generalized to acyclic cluster algebras with invertible coefficients (using Remark \[re:Class Poisson\]). This would imply that in the set-up of locally acyclic cluster algebras it should be easy to show that the spectrum of a locally acyclic cluster algebra is a holomorphic symplectic manifold. Ideals in Acyclic Quantum Cluster Algebras {#se:ideals in qca} ========================================== \[th:ideals in qclalg\] Let $\AA_q$ be a quantum cluster algebra with quantum seed $({\bf x}, B,\Lambda)$ satisfying Equation \[eq:poisson gen\]. Then, $\AA_q$ does not contain any non-trivial proper prime ideals. Let $\II$ be a prime ideal in $\AA_q$. We obtain from Proposition \[pr:TPP super toric q\] that $\II$ contains a monomial $x^{\bf v}$ with ${\bf v}\in\ZZ_{\ge 0}^n$. It is easy to observe that we can choose ${\bf v}$ to be minimal with respect to the lexicographic order on $\ZZ^n_{\ge 0}$. Recall that the lexicographic ordering defines ${\bf u}<{\bf w}$ for ${\bf u},{\bf w}\in \ZZ^n$ if and only if there exists $i\in [1,n]$ such that $u_i,w_i$ and $u_j=w_j$ if $j>i$. Recall that this defines a total ordering on $\ZZ^n_{\ge0}$. There exists some $i\in[1,n]$ such that $v_i\ne 0$ and $v_k=0$ for all $k>i$, and we write $x^{\bf v}=x^{\bf v'} x_i^{v_i}$ with ${\bf v'}={\bf v}-v_i{\bf e}_i$. Recall that an ideal $\II$ in a non-commutative ring $R$ is prime if $arb\in \II$ for all $r\in R$ implies that $a\in \II$ or $b\in \II$. Now, since $\AA_q$ is acyclic, we know from [@bz-qclust Theorem 7.3 and 7.5] that it is isomorphic to its lower bound. Hence, employing the notation of Definition \[def:lower bounds\], each element $a\in \AA_q$ can be written as $a=\sum_{p=1}^r c_p x^{\bf w}y_i^h y^{\bf w'}$ where $w_kw'_k=w_i h =0$ and $c_p\in\CC[q^{\pm 1}]$ for all $p\in[1,r]$. Observe that $y_k$ and $x_\ell$ skew-commute if $k\ne \ell$, as they are cluster variables in the cluster ${\bf x}_k$. We now compute that $$x^{\bf v'} ax_i^{j}=\sum_{p=1}^r c_p q^{\lambda_a} x^{\bf w} y_i^h x^{\bf v'} x_i^{v_i} y^{w'}= \sum_{p=1}^r c_p q^{\lambda_p} x^{\bf w} y_i^h x^{\bf v} y^{w'} \in \II\ ,$$ for all $a\in\AA_q$ and certain $\lambda_p\in\ZZ$. Hence, since $\II$ is a prime ideal, $x^{\bf v'}\in \II$ or $x_{i}^{v_i}\in\II$. But ${\bf v'}<{\bf v}$ and $v_i{\bf e}_i\le {\bf v}$, and therefore, $ x_i^{v_i}=x^{\bf v}\in\II$. But we have assumed in Equation \[eq:poisson gen\] that $x_iy_i$ and $y_ix_i$, are not linearly dependent, hence $x_i^{v_i} y_i$ and $y_i x_i^{v_i}$ are not linearly dependent. As in the proof of Theorem \[th:acyclic\], we now argue that $x_i^{v_i-1 }m_i^{+}\in \II\ $, where $P_i=m_i^++m_i^-$ (see Section \[se:Poisson and symp geom\]). But this is a contradiction to our minimality assumption. The theorem is proved. Theorem \[th:ideals in qclalg\] and Theorem \[th:symplectic leaves\] have the following immediate corollary. 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--- abstract: 'In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting from $v$ faults in the circuit have weight greater than $v$. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.' author: - Christopher Chamberland - 'Michael E. Beverland' bibliography: - 'bibtex\_chamberland.bib' title: 'Flag fault-tolerant error correction with arbitrary distance codes' --- Introduction and formalism {#sec:Intro} ========================== Scalable quantum computers are expected to require some form of error correction (EC) to function reliably. Unfortunately, no practical model for a self-correcting quantum memory has been proposed to date, despite considerable effort [@BrownMemory16]. The models that come closest to this goal involve topological protection in the presence of physically imposed symmetries [@KitaevWire01; @KarzigScalable17], but even these are not expected to reduce error rates sufficiently for large computations. Therefore active protocols that require measuring the check operators of an error correcting code are probably necessary to realize scalable quantum computing. There are three general approaches of fault-tolerant error correction (FTEC) applicable to a wide range of stabilizer codes due to Shor [@Shor96], Steane [@Steane97], and Knill [@KL2005]. There are also a number of promising code-specific FTEC schemes, most notably the surface code with a minimum weight matching error correction scheme [@BK98; @DKLP02; @FMMC12]. This approach gives the best fault-tolerant thresholds to date and only requires geometrically local measurements. A high threshold [@Shor96; @AB97; @Preskill98; @KLZ98] implies that relatively imperfect hardware could be used to reliably implement long quantum computations. Despite this, the hardware and overhead requirements for the surface code are sufficiently demanding that it remains extremely challenging to implement in the lab. Fortunately, there are reasons to believe that there could be better alternatives to the surface code. For example, dramatically improved thresholds could be possible using concatenated codes if they enjoyed the same level of optimization as the surface code has in recent years [@Poulin06; @PhysRevA.83.020302]. Another enticing alternative is to find and use efficiently-decodable low density parity check (LDPC) codes with high rate [@Gallager1960; @LDPC13; @TZLDPC14] in a low-overhead FTEC protocol [@Gottesman13LDPC]. For these and other reasons, it is important to have general FTEC schemes applicable to a wide range of codes and to develop new schemes. Shor EC can be applied to any stabilizer code, but typically requires more syndrome measurement repetitions than Steane and Knill EC. Furthermore, all weight-$w$ stabilizer generators are measured sequentially using $w$-qubit verified cat states. On the other hand, Steane EC has higher thresholds than Shor EC and has the advantage that all Clifford gates are applied transversally during the protocol. However, Steane EC is only applicable to CSS [@CS96; @Steane97] codes and uses a verified logical ${|+\rangle}$ state encoded in the same code to simultaneously obtain all $X$-type syndromes, using transversal measurement (similarly for $Z$). Knill EC can also be applied to any stabilizer code but requires two additional ancilla code blocks (encoded in the same code that protects the data) prepared in a logical Bell state. The Bell state teleports the encoded information to one of the ancilla code blocks and the extra information from the transversal Bell measurement gives the error syndrome. Knill EC typically achieves higher thresholds than Shor and Steane EC but often uses more qubits [@Knill05; @Fern08KnillUpperBound]. It is noteworthy that for large LDPC codes, in which low weight generators are required be fault-tolerantly measured, Shor EC is much more favourable than Steane or Knill EC. Many improvements in these schemes have been made. For examples, in [@DA07], ancilla decoding was introduced to correct errors arising during state preparation in Shor and Steane EC rather than simply rejecting all states which fail the verification procedure. In this work, we build on a number of recent papers [@CR17v1; @CR17v2; @Yoder2017surfacecodetwist] that demonstrate flag error correction for particular distance-three and error detecting codes and present a general protocol for arbitrary distance codes. Flag error correction uses extra ancilla qubits to detect potentially problematic high weight errors that arise during the measurement of a stabilizer. We provide a set of requirements for a stabilizer code (along with the circuits used to measure the stabilizers) which, if satisfied, can be used for flag error correction. We are primarily concerned with extending the lifetime of encoded information using fault-tolerant error correction and defer the study of implementing gates fault-tolerantly to future work. Our approach can be applied to a broad class of codes (including but not limited to surface codes, color codes and quantum Reed-Muller codes). Of the three general schemes described above, flag EC has most in common with Shor EC. Further, flag EC does not require verified state preparation, and for all codes considered to date, requires fewer ancilla qubits. Lastly, we note that in order to satisfy the fault-tolerant error correction definition presented in \[subsec:Section0\], our protocol applied to distance-three codes differs from [@CR17v1]. We foresee a number of potential applications of these results. Firstly we believe it is advantageous to have new EC schemes with different properties that can be used in various settings. Secondly, flag EC involves small qubit overhead, hence possibly the schemes presented here and in other flag approaches [@CR17v1; @CR17v2; @Yoder2017surfacecodetwist] will find applications in early qubit-limited experiments. Thirdly, we expect the flag EC protocol presented here could potentially be useful for LDPC codes as described in [@Gottesman13LDPC]. In \[subsec:ReviewChaoReichardt,subsec:Distance5protocol\] we provide important definitions and introduce flag FTEC for distance-three and -five codes. In \[subsec:ApplicationProtocolColorCode\] we apply the protocol to two examples: the [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes, which importantly have a variety of different weight stabilizers. The general flag FTEC protocol for arbitrary distance codes is given in \[subsec:GeneralProtocol\]. A proof that the general protocol satisfies the fault-tolerance criteria is given in \[app:ProtocolGeneralProof\]. In \[subsec:Remarks\] we provide examples of codes that satisfy the conditions that we required for flag FTEC. Flag circuit constructions for measuring stabilizers of the codes in \[subsec:Remarks\] are given \[app:GeneralTflaggedCircuitConstruction\]. We also provide a candidate circuit construction for measuring arbitrary weight stabilizers in \[App:GeneralwFlagCircuitConstruction\]. In \[sec:CircuitLevelNoiseFTEC\], we analyze numerically a number of flag EC schemes and compare with other FTEC schemes under various types of circuit level noise. We find that flag EC schemes, which have large numbers of idle qubit locations, behave best in error models in which idle qubit errors occur with a lower probability than CNOT errors. The remainder of this section is devoted to FTEC and noise model/simulation methods. Fault-tolerant error correction {#subsec:Section0} ------------------------------- Throughout this paper, we assume a simple depolarizing noise model in which idle qubits fail with probability $\tilde{p}$ and all other circuit operations (gates, preparations and measurements) fail with probability $p$, which recovers standard circuit noise when $\tilde{p}=p$. A detailed description is given in \[subsec:NoiseAndNumerics\]. The weight of a Pauli operator $E$ ($\text{wt}(E)$) is the number of qubits on which it has non-trivial support. We first make some definitions, $$\begin{aligned} \mathcal{E}_{t} = \{ E \in \mathcal{P}_{n} | \text{wt}(E) \le t \}, \end{aligned}$$ where $\mathcal{P}_{n}$ is the $n$-qubit Pauli group. \[Def:EpsilontSet\] Given a stabilizer group $\mathcal{S} = \langle g_{1}, \cdots, g_{m} \rangle$, we define the syndrome $s(E)$ to be a bit string, with i’th bit equal to zero if $g_i$ and $E$ commute, and one otherwise. Let $E_{\text{min}}(s)$ be a minimal weight correction $E$ where $s(E)=s$. We say operators $E$ and $E'$ are logically equivalent, written as $E \sim E'$, iff $E' \propto g E$ for $g \in \mathcal{S}$. \[Def:LogEquivDef\] An error correction protocol typically consists of a sequence of basic operations to infer syndrome measurements of a stabilizer code $C$, followed by the application of a Pauli operator (either directly or through Pauli frame tracking [@DA07; @Barbara15; @CIP17]) intended to correct errors in the system. Roughly speaking, a given protocol is fault-tolerant if for sufficiently weak noise, the effective noise on the logical qubits is even weaker. More precisely, we say that an error correction protocol is a $t$-FTEC if the following is satisfied: For $t = \lfloor (d-1)/2\rfloor$, an error correction protocol using a distance-$d$ stabilizer code $C$ is $t$-fault-tolerant if the following two conditions are satisfied: 1. For an input codeword with error of weight $s_{1}$, if $s_{2}$ faults occur during the protocol with $s_{1} + s_{2} \le t$, ideally decoding the output state gives the same codeword as ideally decoding the input state. 2. For $s$ faults during the protocol with $s \le t$, no matter how many errors are present in the input state, the output state differs from a codeword by an error of at most weight $s$. \[Def:FaultTolerantDef\] Here ideally decoding is equivalent to performing fault-free error correction. By codeword, we mean any state ${|\overline{\psi}\rangle} \in C$ such that $g{|\overline{\psi}\rangle} = {|\overline{\psi}\rangle} \thinspace \forall \thinspace g \in \mathcal{S}$ where $\mathcal{S}$ is the stabilizer group for the code $C$. Note that for the second criteria in \[Def:FaultTolerantDef\], the output and input codeword can differ by a logical operator. The first criteria in \[Def:FaultTolerantDef\] ensures that correctable errors don’t spread to uncorrectable errors during the error correction protocol. Note however that the first condition alone isn’t sufficient. For instance, the trivial protocol where no correction is ever applied at the end of the EC round also satisfies the first condition, but clearly is not fault-tolerant. The second condition is not always checked for protocols in the literature, but it is important as it ensures that errors do not accumulate uncontrollably in consecutive rounds of error correction (see [@AGP06] for a rigorous proof and [@CDT09] for an analysis of the role of input errors in an extended rectangle). To give further motivation as to why the second condition is important, consider a scenario with $s$ faults introduced during each round of error correction, and assume that $t/n<s<(2t+1)/3$ for some integer $n$ (see Fig. \[fig:ConditionTwoJustification\]). Consider an error correction protocol in which $r$ input errors and $s$ faults in an EC block leads to an output state with at most $r+s$ errors[^1]. Clearly condition 1 is satisfied. With the above considerations, an input state $E_{1}{|\bar{\psi}\rangle}$ with $\text{wt}(E_{1})\leq s$ is taken to $E_{2}{|\bar{\psi}\rangle}$, with $\text{wt}(E_{2})\leq 2s$ by one error correction round with $s$ faults. After the $j$th round, the state will be $E_{j}{|\bar{\psi}\rangle}$ with the first condition implying $\text{wt}(E_{j})\leq j \cdot s$ provided that $j \leq n$. However, when $j > n$, the requirement of the first condition is no longer satisfied so we cannot use it to upper bound $\text{wt}(E_{j})$. Now consider the same scenario but assuming both conditions hold. The second condition implies that after the first round, the input state $E_{1}{|\bar{\psi}\rangle}$ becomes $E'_{2}{|\bar{\phi}\rangle} = E_{2}{|\bar{\psi}\rangle}$, with $\text{wt}(E_{2}')\leq s$, and where ${|\bar{\phi}\rangle}$ is a codeword. Therefore the codewords are related by: ${|\bar{\phi}\rangle}= (E_{2}^{'\dagger} E_{2}) {|\bar{\psi}\rangle}$, with logical operator $(E_{2}^{'\dagger} E_{2})$ having weight at most $3s$, since $\text{wt}(E_{2})+\text{wt}(E_{2}') \leq 3s$. However, the minimum non-trivial logical operator of the code has weight $(2t+1)>3s$, implying that ${|\bar{\psi}\rangle} = {|\bar{\phi}\rangle}$, and therefore that $\text{wt}(E_{2}) = \text{wt}(E_{2}') \leq s$. Hence, for the $j$th round, $\text{wt}(E_{j}) \leq s$ for all $j$, i.e. the distance from the codeword is not increased by consecutive error correction rounds with $s$ faults, provided $s < (2t+1)/3$. ![An example showing the first fault tolerance condition alone in \[Def:FaultTolerantDef\] is not sufficient to imply a long lifetime. We represent $s$ faults occurring during a round of error correction with a vertical arrow, and a state a distance $r$ from the desired codeword with a horizontal arrow with $r$ above. The first condition alone allows errors to build up over time as in the top figure, which would quickly lead to a failure. However provided $s<(2t+1)/3$, both conditions together ensure that errors in consecutive error correction rounds do not build up, provided each error correction round introduces no more than $s$ faults, which could remain true for a long time.[]{data-label="fig:ConditionTwoJustification"}](ConditionTwoJustification.png){width="45.00000%"} Noise model and pseudo-threshold calculations {#subsec:NoiseAndNumerics} --------------------------------------------- In \[sec:CircuitLevelNoiseFTEC\], we perform a full circuit level noise analysis of various error correction protocols. Unless otherwise stated, we use the following depolarizing noise model: 1. With probability $p$, each two-qubit gate is followed by a two-qubit Pauli error drawn uniformly and independently from $\{I,X,Y,Z\}^{\otimes 2}\setminus \{I\otimes I\}$. 2. With probability $\frac{2p}{3}$, the preparation of the ${|0\rangle}$ state is replaced by ${|1\rangle}=X{|0\rangle}$. Similarly, with probability $\frac{2p}{3}$, the preparation of the ${|+\rangle}$ state is replaced by ${|-\rangle}=Z{|+\rangle}$. 3. With probability $\frac{2p}{3}$, any single qubit measurement has its outcome flipped. 4. Lastly, with probability $\tilde{p}$, each resting qubit location is followed by a Pauli error drawn uniformly and independently from $\{ X,Y,Z \}$. Some error correction schemes that we analyze contain a significant number of idle qubit locations. Consequently, most schemes will be analyzed using three ratios ($\tilde{p} = p$, $\tilde{p} = p/10$ and $\tilde{p} = p/100$) to highlight the impact of idle qubit locations on the logical failure rate. The two-qubit gates we consider are: CNOT, XNOT$ = H_1(\text{CNOT})H_1$, and CZ$ = H_2(\text{CNOT})H_2$. Logical failure rates are estimated using an $N$-run Monte Carlo simulation. During a particular run, errors are added at each location following the noise model described above. Once the error locations are fixed, the errors are propagated through a fault-tolerant error correction circuit and a recovery operation is applied. After performing a correction, the output is ideally decoded to verify if a logical fault occurred. For an error correction protocol implemented using a stabilizer code $C$ and a fixed value of $p$, we define the logical failure rate $$p_{\text{L}}^{(C)}(p) = \lim_{N \to\infty} \frac{N_{\text{fail}}^{(C)}(p)}{N} ,$$ where $N_{\text{fail}}^{(C)}(p)$ is the number of times a logical $X$ *or* logical $Z$ error occurred during the $N$ rounds. In practice we take $N$ sufficiently large to estimate $p_{\text{L}}^{(C)}(p)$, and provide error bars [@AliferisCross07; @CJL16b]. In this paper we are concerned with evaluating the performance of FTEC protocols (i.e. we do not consider performing logical gates fault-tolerantly). We define the pseudo-threshold of an error correction protocol to be the value of $p$ such that $$\begin{aligned} \tilde{p}(p) = p^{(C)}_{L}(p). \label{Def:PseudoThreshDef}\end{aligned}$$ Note that it is important to have $\tilde{p}$ on the left of \[Def:PseudoThreshDef\] instead of $p$ since we want an encoded qubit to have a lower logical failure rate than an unencoded idle qubit. From the above noise model, a resting qubit will fail with probability $\tilde{p}$. Flag error correction for small distance codes {#sec:Section1} ============================================== In this and the next section, we present a $t$-fault-tolerant flag error correction protocol with distance-$(2t+1)$ codes satisfying a certain condition. Our approach extends that introduced by Chao and Reichardt [@CR17v1] for distance three codes, which we first review using our terminology in \[subsec:ReviewChaoReichardt\]. In \[subsec:Distance5protocol\] we present the protocol for distance five CSS codes which contains most of the main ideas of the general case (which is provided in \[app:GeneralFTEC\]). Lastly, in section \[subsec:ApplicationProtocolColorCode\] we provide examples of how the protocol is applied to the [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes. Definitions and Flag $1$-FTEC with distance-3 codes {#subsec:ReviewChaoReichardt} --------------------------------------------------- In what follows, we use the term location to refer to a gate, state preparation, measurement or idle qubit where a fault may occur. Note also that a two-qubit Pauli error $P_{1}\otimes P_{2}$ arising at a two-qubit gate location counts as a single fault. It is well known that with only a single measurement ancilla, a single fault in a blue CNOT of the stabilizer measurement circuit shown in \[fig:StabNonFT\] can result in a multi-weight error on the data block. This could cause a distance-$3$ code to fail, or more generally could cause a distance-$d$ code to fail due to fewer than $(d-1)/2$ total faults. We therefore say the blue CNOTs are *bad* according to the following definition: A circuit location in which a single fault can result in a Pauli error $E$ on the data block with $\mathrm{wt}(E) \ge 2$ will be referred to as a bad location. \[Def:BadErrorDef\] [0.25]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](StabNonFT.png "fig:"){width="\textwidth"} [0.3]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](Weight4Generator.png "fig:"){width="\textwidth"} [0.25]{} ![Circuits for measuring the operator $ZZZZ$ (can be converted to any Pauli by single qubit Cliffords). (a) Non-fault-tolerant circuit. A single fault $IZ$ occurring on the third CNOT (from the left) results in the error $IIZZ$ on the data block. (b) Flag version of \[fig:StabNonFT\]. An ancilla (flag) qubit prepared in ${|+\rangle}$ and two extra CNOT gates signals when a weight two data error is caused by a single fault. Subsequent rounds of error correction may identify which error occurred. Consider an $IZ$ error on the second CNOT, in the non-flag circuit this would result in a weight two error, but in this case, this fault causes the circuit to flag. (c) An alternative flag circuit with lower depth than (b). All bad locations are illustrated in blue.[]{data-label="fig:ErrorPropSteane"}](StabFTwithAncillaV2.png "fig:"){width="\textwidth"} As shown in \[fig:StabFTwithAncilla\], the circuit can be modified by including an additional ancilla (flag) qubit, and two extra CNOT gates. This modification leaves the bad locations and the fault-free action of the circuit unchanged. However, any single fault leading to an error $E$ with $\mathrm{wt}(E) \ge 2$ will also cause the measurement outcome of the flag qubit to flip [@CR17v1]. The following definitions will be useful: Consider a circuit for measuring a stabilizer generator that includes at least one flag ancilla. The ancilla used to infer the stabilizer outcome is referred to as the *measurement qubit*. We say the circuit has flagged if the eigenvalue of a flag qubit is measured as $-1$. If the eigenvalue of a measurement qubit is measured as $-1$, we will say that the measurement qubit flipped. \[Def:GlagMeasureQubitsDef\] The purpose of flag qubits is to signal when high weight data qubit errors result from few fault locations during a stabilizer measurement. Two key definitions are: A circuit[^2] $C(P)$ which, when fault-free, implements a projective measurement of a weight-$w$ Pauli $P$ without flagging is a $t$-flag circuit if the following holds: For any set of $v$ faults at up to $t$ locations in $C(P)$ resulting in an error $E$ with $\text{min}(\text{wt}(E),\text{wt}(E P)) > v$, the circuit flags. \[Def:tFlaggedCircuitDef\] Note that a $t$-flag circuit for measuring a weight-$t$ stabilizer $P$ is also a $k$-flag circuit for any $k>t$. In \[app:GeneralTflaggedCircuitConstruction\] we give constructions for some $t$-flag circuits. Let $\mathcal{E}(g_{i})$ be the set of all errors caused by one fault which caused the circuit $C(g_i)$ to flag. \[Def:FlagErrSetDef1\] Note that the flag error set can contain the identity as well as weight one errors. Suppose all errors in a flag error set $\mathcal{E}(g)$ for a 1-flag circuit $C(g)$ have distinct syndromes. As $C(g)$ is a 1-flag circuit, a single fault that leads to an error of weight greater than one will cause the circuit $C(g)$ to flag. Moreover, when a flag has occurred due to at most one fault, a complete set of fault-free stabilizer measurements will infer the resulting element of the flag error set which has been applied to the data qubits. In fact, one would only require distinct syndromes for errors in the flag error set that are logically inequivalent, as defined in \[Def:LogEquivDef\]. As an example, consider the 1-flag circuit in \[fig:StabFTwithAncilla\]. A single fault at any of the blue CNOT gates can lead to an error $E_{b}$ with $\text{wt}(E_{b}) \le 2$ on the data. The set $\mathcal{E}(Z^{\otimes 4})$ contains all errors $E_{b}$ which resulted from a fault at a blue CNOT gate causing the circuit $C(Z^{\otimes 4})$ of \[fig:StabFTwithAncilla\] to flag, i.e., $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},Y_{q_{3}}Z_{q_{4}} \}$ with qubits $q_1$ to $q_4$. With the above definitions, we can construct a fault-tolerant flag error correction protocol for $d=3$ stabilizer codes satisfying the following condition. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $1$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For every generator $g_{i}$, all pairs of elements $E,E'\in \mathcal{E}(g_{i})$ satisfy $s(E)\neq s(E')$ or $E \sim E'$. \[Def:Flag1FTECcondition\] In other words, we require that any two errors that arise when a circuit $C(g_{i})$ flags due to a single fault must be either distinguishable or logically equivalent. For the following protocol to satisfy the FTEC conditions in \[Def:FaultTolerantDef\], one can assume there is at most 1 fault. If the Flag $1$-FTEC condition is satisfied, the protocol is implemented as follows: ![Tree diagram illustrating the possible paths of the Flag $1$-FTEC Protocol. Numbers enclosed in red circles at the end of the edges indicate which step to implement in the Flag $1$-FTEC Protocol. A dashed line is followed when any of the 1-flag circuits $C(g_{i})$ flags. Solid squares indicate a syndrome measurement using 1-flag circuits whereas rings indicate a decision based on syndrome outcomes. Note that the syndrome measurement is repeated at most three times.[]{data-label="fig:TreeD3Diag"}](TreeD3.png){width="50.00000%"} [0.45]{}  after rotating the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneCodeColor.png "fig:"){width="\textwidth"} [0.45]{}  after rotating the lattice such that the relevant face is on the bottom left. (b) For $g = Z_{q_{1}}Z_{q_{2}}Z_{q_{3}}Z_{q_{4}}$, the flag error set is $\mathcal{E}(g) = \{ I,Z_{q_{3}}Z_{q_{4}},X_{q_{2}}Z_{q_{3}}Z_{q_{4}},Z_{q_{1}}X_{q_{2}},Z_{q_{4}},$ $X_{q_{3}}Z_{q_{4}},X_{q_{3}}Z_{q_{3}}Z_{q_{4}} \}$ which contains all errors arising from a single fault that causes the stabilizer measurement circuit $C(g)$ to flag. Since the Steane code is a CSS code, the $X$ component of an error will be corrected independently allowing us to consider the $Z$-part of the flag error set $\mathcal{E}_Z(g)=\{I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As required, the elements of $\mathcal{E}_Z(g)$ all have distinct syndromes (with satisfied stabilizers represented by a plus). []{data-label="fig:Steane"}](SteaneConditionSatisfied.png "fig:"){width="\textwidth"} A tree diagram for the flag $1$-FTEC Protocol is illustrated in \[fig:TreeD3Diag\]. We now outline the proof that the flag 1-FTEC protocol satisfies the fault-tolerance criteria of \[Def:FaultTolerantDef\] (a more rigorous proof of the general case is presented in \[app:ProtocolGeneralProof\]). To show that Flag $1$-FTEC Protocol satisfies the criteria of \[Def:FaultTolerantDef\], we can assume there is at most one fault during the protocol. If a single fault occurs in either the first or second round leading to a flag, repeating the syndrome measurement will correctly diagnose the error. If there are no flags and a fault occurs which causes the syndromes in the first two rounds to change, then the syndrome during the third round will correctly diagnose the error. There could also be a fault during either the first or second round that goes undetected. But since there were no flags it cannot spread to an error of weight-2. In this case applying a minimum weight correction based on the measured syndrome of the second round will guarantee that the output codeword differs from a valid codeword by an error of weight at most one. Note that the above argument applies irrespective of any errors on the input state, hence the second criteria of \[Def:FaultTolerantDef\] is satisfied. It is worth pointing out that up to three repetitions are required in order to guarantee that the second criteria of \[Def:FaultTolerantDef\] is satisfied (unless the code has the property that all states are at most a weight-one error away from a valid codeword, as in [@CR17v1]). The Steane code is an example which satisfies the Flag $1$-FTEC condition with a simple choice of circuits. To verify this, the representation of the Steane code given in \[fig:SteaneColor1\] is useful. There is an $X$- and a $Z$-type stabilizer generator supported on the four qubits of each of the three faces. First let us specify all six stabilizer measurement circuits. The circuit that measures $Z_{q_1}Z_{q_2}Z_{q_3} Z_{q_4}$ is specified by taking qubits $q_1$, $q_2$, $q_3$, and $q_4$ to be the four data qubits in descending order in the 1-flag circuit in \[fig:StabFTwithAncilla\]. The other two $Z$-stabilizer measurement circuits are obtained by first rotating \[fig:SteaneColor1\] by $120^{\circ}$ and $240^{\circ}$ and then using \[fig:StabFTwithAncilla\]. The $X$-stabilizer circuit for each face is the same as the $Z$-stabilizer circuit for that face, replacing CNOT gates acting on data qubits by XNOT gates. The $Z$ component of the flag error set of the circuit in \[fig:StabFTwithAncilla\] is $\mathcal{E}_Z(Z_{q_1}Z_{q_2}Z_{q_3}Z_{q_4}) = \{ I,Z_{q_1},Z_{q_4},Z_{q_3}Z_{q_4} \}$. As can be seen from \[fig:SteaneColor1\], each of these has a distinct syndrome, thus the measurement circuit for $Z_{q_1}Z_{q_2}Z_{q_3} Z_{q_4}$ satisfies the flag $1$-FTEC condition, as do the remaining five measurement circuits by symmetry. Flag $2$-FTEC with distance-5 codes {#subsec:Distance5protocol} ----------------------------------- Before explicitly describing the conditions and protocol, we discuss some of the complications that arise for codes with $d>3$. For distance-5 codes, we must ensure that if two faults occur during the error correction protocol, the output state will differ from a codeword by an error of at most weight-two. For instance, if two faults occur in a circuit for measuring a stabilizer of weight greater than four, the resulting error $E$ on the data should satisfy $\text{wt}(E) \le 2$ unless there is a flag. In other words, all stabilizer generators should be measured using 2-flag circuits. In another case, two faults could occur during the measurement of *different* stabilizer generators $g_{i}$ and $g_{j}$. If two bad locations fail and are both flagged, and assuming there are no more faults, the measured syndrome will correspond to the product of the error caused in each circuit (which could have weight greater than two). Consequently, one should modify \[Def:FlagErrSetDef1\] of the flag error set to include these types of errors. One then decodes based on the pair of errors that resulted in the measured syndrome, provided logically inequivalent errors have distinct syndromes. Before stating the protocol, we extend some definitions from \[subsec:ReviewChaoReichardt\]. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $t$-flag circuits $C(g_{i})$ for measuring the generator $g_{i}$. Let $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}})$ be the set of all errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which all flagged. \[Def:FlagErrSetDef\] Note that there could be more than one fault in a single circuit $C(g_{i_{k}})$. Examples of flag error sets are given in \[tab:PossibleCorrelatedErrors\] where only contributions from $Z$ errors are included (since the considered code is a CSS code). We also define a general $t$-fault correction set: $$\begin{aligned} \tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s) = \begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty. } \end{cases} \label{eq:GeneralLookupTable}\end{aligned}$$ By $E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$, we are considering the set consisting of products between errors caused by $k$ flags and any error of weight $t-m$. As will be seen below, the correction set will form a critical part of the protocol by specifying the correction applied based on the measured syndrome and flag outcomes over multiple syndrome measurement rounds. In the case where $k$ $t$-flag circuits flagged caused by $k \le m \le t$ faults, the correction applied to the data block will correspond to an element of $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$ if the measured syndrome corresponds to an element in this set (there could also be $t-m$ faults which did not give rise to a flag). However in practice, there could be more than $t$ faults and so the measured syndrome may not be consistent with any element of the set $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m}$. In this case, and for the error correction protocol to satisfy the second criteria of \[Def:FaultTolerantDef\], the correction will correspond to $E_{\text{min}}(s)$. These features are all included in the set $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s)$. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $2$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For any choice of generators $\{ g_{i}, g_{j} \}$: 1. $E,E' \in \mathcal{E}_{2}(g_{i},g_{j}) \Rightarrow s(E)\neq s(E')$ or $E \sim E'$, 2. $E,E' \in \mathcal{E}_{2}(g_{i})\cup (\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1}) \Rightarrow s(E) \neq s(E')$ or $E \sim E'$. \[Def:Flag2FTECcondition\] In order to state the protocol, we define an update rule given a sequence of syndrome measurements using $t$-flag circuits for the counters[^3] $n_{\text{diff}}$ and $n_{\text{same}}$ as follows: For the following protocol to satisfy \[Def:FaultTolerantDef\], one can assume there are at most 2 faults. If the Flag $2$-FTEC condition is satisfied, the protocol is implemented as follows: ![Tree diagram for the Flag $2$-FTEC protocol. Numbers encircled in red at the end of the edges indicate which step to implement in the Flag $2$-FTEC Protocol. A dashed line is followed when any of the 2-flag circuits $C(g_{i})$ flags. Solid squares indicate a syndrome measurement using 2-flag circuits whereas rings indicate a decision based on syndrome outcomes. Edges with different colors indicate the current value of $n_{\text{diff}}$ in the protocol. Note that the protocol is repeated at most 6 times.[]{data-label="fig:TreeDiagramD5"}](TreeD5.png){width="50.00000%"} Note that when computing the update rules, if a flag occurs during the $j$’th round of syndrome measurements, the syndrome is not recorded for that round since all stabilizers must be measured. Thus when computing $n_{\text{diff}}$ and $n_{\text{same}}$ using consecutive syndromes $s_k$ and $s_{k+1}$, we are assuming that no flags occurred during rounds $k$ and $k+1$. In each case of the protocol, the correction sets correspond to those data errors which could arise from up to two faults which are consistent with the conditions of the case. As the elements are logically equivalent (by \[eq:GeneralLookupTable,Def:Flag2FTECcondition\]), which element is applied is unimportant. The general protocol for codes of arbitrary distance is given in \[app:GeneralFTEC\]. Examples of flag 2-FTEC applied to $d=5$ codes {#subsec:ApplicationProtocolColorCode} ---------------------------------------------- [0.22]{} ![Graphical representation of (a) the 19-qubit 2D color code and (b) the 17-qubit 2D color code. The $X$ and $Z$ stabilizers of the code are symmetric, given by the vertices of each plaquette. Both codes have distance-5.[]{data-label="fig:ColorCodeLattices"}](19qubitColorLattice.png "fig:"){width="\textwidth"} [0.25]{} ![Graphical representation of (a) the 19-qubit 2D color code and (b) the 17-qubit 2D color code. The $X$ and $Z$ stabilizers of the code are symmetric, given by the vertices of each plaquette. Both codes have distance-5.[]{data-label="fig:ColorCodeLattices"}](17qubitColorLattice.png "fig:"){width="\textwidth"} In this section we give examples of the flag $2$-FTEC protocol applied to the 2-dimensional [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes, (see \[fig:19qubitLatticeColor,fig:17qubitLatticeColor\]). We first find 2-flag circuits for all generators (weight-4 and -6 for the 19-qubit code and weight-4 and -8 for the 17-qubit code). We also show that the flag 2-FTEC condition is satisfied for both codes. -------------- --------------------- --------------------- ------------------------------------- 1-fault 2-faults 1-fault 2-faults $I$,$Z_{1}$ $I$,$Z_{1}$,$Z_{2}$ $I$,$Z_{1}$,$Z_{6}$ $I$,$Z_{1}$,$Z_{2}$ $Z_{4}$ $Z_{3}$,$Z_{4}$ $Z_{1}Z_{2}$ $Z_{3}$, $Z_{4}$,$Z_{5}$,$Z_{6}$ $Z_{3}Z_{4}$ $Z_{1}Z_{2}$ $Z_{5}Z_{6}$ $Z_{1}Z_{2}$,$Z_{1}Z_{3}$ $Z_{1}Z_{4}$ $Z_{4}Z_{5}Z_{6}$ $Z_{1}Z_{4}$,$Z_{1}Z_{5}$ $Z_{2}Z_{4}$ $Z_{1}Z_{6}$,$Z_{2}Z_{3}$ $Z_{2}Z_{6}$,$Z_{3}Z_{4}$ $Z_{3}Z_{6}$,$Z_{4}Z_{5}$ $Z_{4}Z_{6}$,$Z_{5}Z_{6}$ $Z_{1}Z_{2}Z_{3}$,$Z_{1}Z_{5}Z_{6}$ $Z_{2}Z_{5}Z_{6}$,$Z_{3}Z_{4}Z_{5}$ $Z_{3}Z_{4}Z_{6}$,$Z_{3}Z_{5}Z_{6}$ $Z_{4}Z_{5}Z_{6}$ -------------- --------------------- --------------------- ------------------------------------- : $Z$ part of the flag error set of \[Def:FlagErrSetDef\] for flag circuits used to measure the stabilizers $g_{1} = Z_{1}Z_{2}Z_{3}Z_{4}$ and $g_{3} = Z_{1}Z_{2}Z_{3}Z_{4}Z_{5}Z_{6}$ (we removed errors equivalent up to the stabilizer being measured).[]{data-label="tab:PossibleCorrelatedErrors"} For a 2-flag circuit, two faults leading to an error of weight greater or equal to 3 (up to multiplication by the stabilizer) must always cause at least one of the flag qubits to flag. As shown in \[app:GeneralTflaggedCircuitConstruction\], a 2-flag circuit satisfying these properties can always be constructed using at most four flag qubits. We show 2-flag circuits for measuring weight six and eight generators in \[fig:Flag2CircuitExamples\]. In \[subsec:Remarks\], it will be shown that the family of color codes with a hexagonal lattice satisfy a sufficient condition which guarantees that the flag 2-FTEC condition is satisfied. However, there are codes that do not satisfy the sufficient condition but which nonetheless satisfy the 2-Flag FTEC condition. For the 19-qubit and 17-qubit color codes, we verified that the flag 2-FTEC condition was satisfied by enumerating all errors as one would have to for a generic code. In particular, in the case where the 2-flag circuits $C(g_{i})$ and $C(g_{j})$ flag, the resulting errors belonging to the set $\mathcal{E}_{2}(g_{i},g_{j})$ must be logically equivalent or have distinct syndromes (which we verified to be true). If a single circuit $C(g_{i})$ flags, there could either have been two faults in the circuit or a single fault along with another error that did not cause a flag. If the same syndrome is measured twice in a row after a flag, then errors in the set $\mathcal{E}_{2}(g_{i})\cup (\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1})$ must be logically equivalent or have distinct syndromes (which we verified). If there is a flag but two different syndromes are measured in a row, errors belonging to the set $\mathcal{E}_{1}(g_{i}) \times \mathcal{E}_{1}$ must be logically equivalent or have distinct syndromes (as was already checked). The flag error sets (see \[Def:FlagErrSetDef\]) for the 19-qubit code can be obtained using the Pauli’s shown in \[tab:PossibleCorrelatedErrors\]. [0.32]{} ![ Illustration of 2-flag circuits for measuring (a) $Z^{\otimes 6}$ requiring only two flag qubits and (b) $Z^{\otimes 8}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag2CircuitExamples"}](Weight6Generator.png "fig:"){width="\textwidth"} [0.37]{} ![ Illustration of 2-flag circuits for measuring (a) $Z^{\otimes 6}$ requiring only two flag qubits and (b) $Z^{\otimes 8}$ requiring only three flag qubits. Flag qubits are prepared in the ${|+\rangle}$ state, and measurement qubits in the ${|0\rangle}$ state. []{data-label="fig:Flag2CircuitExamples"}](Weight8Generator.png "fig:"){width="\textwidth"} Given that the flag 2-FTEC condition is satisfied, the flag 2-FTEC protocol can be implemented following the steps of \[subsec:Distance5protocol\] and the tree diagram illustrated in \[fig:TreeDiagramD5\]. Flag error correction protocol for arbitrary distance codes {#app:GeneralFTEC} =========================================================== In this section we first provide the general flag $t$-FTEC protocol in \[subsec:GeneralProtocol\]. In \[subsec:Remarks\] we give a sufficient condition for stabilizer codes that allow us to easily prove that flag FTEC can be applied to a number of infinite code families. We show that the families of surface codes, hexagonal lattice color codes and quantum Reed-Muller codes satisfy the sufficient condition. Lastly, in \[app:GeneralTflaggedCircuitConstruction\], we give general $t$-flag circuit constructions which are applicable to the code families described in \[subsec:Remarks\]. We assume the reader is familiar with all previous definitions. However, to make this section reasonably self contained, we repeat some key definitions below. [6]{} A circuit $C(P)$ which, when fault-free, implements a projective measurement of a weight-$w$ Pauli $P$ without flagging is a $t$-flag circuit if the following holds: For any set of $v$ faults at up to $t$ locations in $C(P)$ resulting in an error $E$ with $\text{min}(\text{wt}(E),\text{wt}(E P)) > v$, the circuit flags. [9]{} Let $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}})$ be the set of all errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which all flagged. We also remind the reader of the correction set $$\begin{aligned} \tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s) = \begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty. } \end{cases} \label{eq:GeneralLookupTableV2}\end{aligned}$$ Conditions and protocol {#subsec:GeneralProtocol} ----------------------- In what follows we generalize the fault-tolerant error correction protocol presented in \[subsec:Distance5protocol\] to stabilizer codes of arbitrary distance. Consider a stabilizer code $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$ and $t$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. For any set of $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$ such that $1 \le m \le t$, every pair of elements $E,E' \in \bigcup_{j=0}^{t-m}\mathcal{E}_{t-j}(g_{i_{1}},\cdots ,g_{i_{m}})\times \mathcal{E}_{j}$ either satisfy $s(E)\neq s(E')$ or $E \sim E'$. \[Def:FlagtFTECcondition\] The above conditions ensure that if there are at most $t=\lfloor (d-1)/2 \rfloor$ faults, the protocol described below will satisfy the fault-tolerance conditions of \[Def:FaultTolerantDef\]. In order to state the protocol, we define an update rule given a sequence of syndrome measurements using $t$-flag circuits for the counters $n_{\text{diff}}$ and $n_{\text{same}}$ as follows (see also \[subsec:Distance5protocol\] and the associated footnote): In each case of the protocol, the correction sets correspond to those data errors which could arise from up to $t$ faults which are consistent with the conditions of the case. As the elements are logically equivalent (by \[eq:GeneralLookupTableV2,Def:FlagtFTECcondition\]), which element is applied is unimportant. For the protocol to satisfy the fault-tolerance criteria, the syndrome measurement needs to be repeated a minimum of $t+1$ times. In the scenario where the most syndrome measurement rounds are performed, $t$ identical syndromes are obtained before a fault causes the $t+1$’th syndrome to change (in which case $n_{\text{diff}}$ would increase by one). Afterwords, one measures the same syndrome $t-1$ times in a row until another fault causes the syndrome to change. This continues until all of the $t$ possible faults have been exhausted. At this stage, $n_{\text{diff}}=t$ so an extra syndrome measurement round will be performed using non-flag circuits. Thus the maximum number of syndrome measurement rounds $n_{\text{max}}$ is given by $$\begin{aligned} n_{\text{max}} = \sum_{j=0}^{t-1}(t-j) + t+1 = \frac{1}{2}(t^{2}+3t+2). \label{Eq:Nmax}\end{aligned}$$ Note that a similar approach by repeating syndrome measurements is used for Shor error correction [@AGP06; @Gottesman2010]. However, our scheme requires fewer syndrome measurement repetitions than is often described for Shor error correction and does not require the preparation and verification of a $w$-qubit cat state when measuring a stabilizer of weight-$w$. [^4] For codes that satisfy the flag $t$-FTEC condition, we also show in \[app:StatePrepAndMeasure\] how to fault-tolerantly prepare and measure logical states using the flag $t$-FTEC protocol. Sufficient condition and satisfying code families {#subsec:Remarks} ------------------------------------------------- The general flag $t$-FTEC condition can be difficult to verify for a given code since it depends on precisely which $t$-flag circuits are used. A sufficient (but not necessary) condition that implies the flag $t$-FTEC condition is as follows: **Sufficient flag $t$-FTEC condition:** Given a stabilizer code with distance $d>1$, and $\mathcal{S} = \langle g_{1},g_{2},\cdots , g_{r} \rangle$, we require that for all $v = 0,1, \dots t$, all choices $Q_{t-v}$ of $2(t-v)$ qubits, and all subsets of $v$ stabilizer generators $\{ g_{i_1},\dots ,g_{i_v} \} \subset \{ g_{1},\cdots , g_{r} \}$, there is no logical operator $l \in N(\mathcal{S}) \setminus \mathcal{S}$ such that $$\begin{aligned} \text{supp}(l) \subset \text{supp}(g_{i_1}) \cup \cdots \cup \text{supp}(g_{i_v}) \cup Q_{t-v},\end{aligned}$$ where $N(\mathcal{S})$ is the normalizer of the stabilizer group. If this condition holds, then the flag $t$-FTEC condition is implied for any choice of $t$-flag circuits $\{ C(g_{1}),C(g_{2}), \cdots , C(g_{r}) \}$. To prove this, we must show that it implies that none of the sets appearing in the $t$-FTEC condition contain elements that differ by a logical operator. Consider the set $\bigcup_{j=0}^{t-m}\mathcal{E}_{t-j}(g_{i_{1}},\cdots ,g_{i_{m}})\times \mathcal{E}_{j}$ for some set of $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$ with $1 \le m \le t$. An error $E$ from this set will have support in the union of the support of the $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$, along with up to $t-m$ other single qubits. Another error $E'$ from this set will have support in the union of support of the same $m$ stabilizer generators $\{ g_{i_{1}},\cdots ,g_{i_{m}} \}$, along with up to $t-m$ other *potentially different* single qubits. If the sufficient condition holds, then $\text{supp}(E E')$ cannot contain a logical operator. The sufficient flag $t$-FTEC condition is straightforward to verify for a number of code families with a lot of structure in their stabilizer generators and logical operators. We briefly provide a few examples. **Surface codes flag $t$-FTEC:** The rotated surface code [@KITAEV97Surface; @TS14; @BK98; @PhysRevLett.90.016803] family [$[\![d^2,1,d]\!]$]{} for all odd $d=2t+1$ (see \[fig:surfacecodeproof\]) satisfies the flag $t$-FTEC condition using any 4-flag circuits. Firstly, by performing an exhaustive search, we verified that the circuit of \[fig:StabFTwithAncilla\] is a 4-flag circuit. As a CSS code, we can restrict our attention to purely $X$-type and $Z$-type logical operators. An $X$ type logical operator must have at least one qubit in each of the $2t+1$ rows of the lattice shown. However, each stabilizer only contains qubits in two different rows. Therefore, with $v$ stabilizer generators, at most $2v$ of the rows could have support. With an additional $2(t-v)$ qubits, at most $2t$ rows can be covered, which is fewer than the number of rows, and therefore no logical $X$ operator is supported on the union of the support of $v$ stabilizers and $2(t-v)$ qubits. An analogous argument holds for $Z$-type logical operators, therefore the sufficient $t$-FTEC condition is satisfied. ![ The $d=5$ rotated surface code. Qubits are represented by white circles, and $X$ and $Z$ stabilizer generators are represented by red and green faces. As in the example, any logical $X$ operator has $X$ operators acting on at least five qubits, with at least one in each row of the lattice, involving an even number in any green face. In this case, no two stabilizer generators can have qubits in five rows, and therefore cannot contain an $X$ type logical operator. The argument is analogous for logical $Z$ operators. []{data-label="fig:surfacecodeproof"}](sufacecodeproof.png){width="30.00000%"} **Color codes flag $t$-FTEC:** Here we show that any distance $d=(2t+1)$ self-dual CSS code with at most weight-6 stabilizer generators satisfies the flag $t$-FTEC condition using any 6-flag circuits (see \[fig:6flagCircuit\] for an example). Examples include the hexagonal color code [@Bombin06TopQuantDist] family [$[\![(3d^2+1)/4,1,d]\!]$]{} (see \[fig:19qubitLatticeColor\]). As a self-dual CSS code, $X$ and $Z$ type stabilizer generators are identically supported and we can consider a pure $X$-type logical operator without loss of generality. Consider an $X$ type logical operator $l$ such that $$\begin{aligned} \text{supp}(l) \subset \text{supp}(g_{i_1}) \cup \cdots \cup \text{supp}(g_{i_v}) \cup Q_{t-v}, \label{eq:SuppRepeated}\end{aligned}$$ for some set of $v$ stabilizer generators $\{ g_{i_1},\dots ,g_{i_v} \} \subset \{ g_{1},\cdots , g_{r} \}$ along with $2(t-v)$ other qubits $Q_{t-v}$. Restricted to the support of any of the $v$ stabilizers $g_i$, $l|_{g_i}$ must have weight 0, 2, 4, or 6 (otherwise it would anti-commute with the corresponding $Z$ type stabilizer). If the restricted weight is 4 or 6, we can produce an equivalent lower weight logical operator $l' = g_i l$, which still satisfies \[eq:SuppRepeated\]. Repeating this procedure until the weight of the logical operator can no longer be reduced yields a logical operator $l_{\text{min}}$ which has weight either 0 or 2 when restricted to the support of any of the $v$ stabilizer generators. The total weight of $l_{\text{min}}$ is then at most $2v+2(t-v) =2t$, which is less than the distance of the code, giving a contradiction which therefore implies that $l$ could not have been a logical operator. An analogous arguments holds for $Z$-type logical operators, therefore the sufficient $t$-FTEC condition is satisfied. This proof can be easily extended to show that any distance $d=(2t+1)$ self-dual CSS code with at most weight-$2 v$ stabilizer generators for some integer $v$ satisfies the flag $t'$-FTEC condition using any $(v-1)$-flag circuits, where $t'= t/\lfloor v/2\rfloor$. **Quantum Reed-Muller codes flag $1$-FTEC:** The [$[\![n=2^m-1,k=1,d=3]\!]$]{} quantum Reed-Muller code family for every integer $m\geq 3$ satisfies the flag 1-FTEC condition using any 1-flag circuits for the standard choice of generators. We use the following facts about the Quantum Reed-Muller code family (see \[app:QRMcodes\] and [@ADP14] for proofs of these facts): (1) The code is CSS, allowing us to restrict to pure $X$ type and pure $Z$ type logical operators, (2) all pure $X$ or $Z$ type logical operators have odd support, (3) every $X$-type stabilizer generator has the same support as some $Z$-type stabilizer generator, and (4) every $Z$-type stabilizer generator is contained within the support of an $X$ type generator. We only need to prove the sufficient condition for $v=0,1$ in this case. For $v=0$, no two qubits can support a logical operator, as any logical operator has weight at least three. For $v=1$, assume the support of an $X$-type stabilizer generator contains a logical operator $l$. That logical operator $l$ cannot be $Z$ type or it would anti-commute with the $X$-stabilizer due to its odd support. However, by fact (3), there is a $Z$ type stabilizer with the same support as the $X$ type stabilizer, therefore implying $l$ cannot be $X$ type either. Therefore, by contradiction we conclude that no logical operator can be contained in the support of an $X$ stabilizer generator. Since every other stabilizer generator is contained within the support of an $X$-type stabilizer generator, a logical operator cannot be contained in the support of any stabilizer generator. Note that the Hamming code family has a stabilizer group which is a proper subgroup of that of the quantum Reed-Muller codes described here. The $X$-type generators of each Hamming code are the same as for a quantum Reed-Muller code, and the Hamming codes are self-dual CSS codes. It is clear that the sufficient condition cannot be applied to the Hamming code since it has even-weight $Z$-type logical operators (which are stabilizers for the quantum Reed-Muller code) supported within the support of some stabilizer generators. **Codes which satisfy flag $t$-FTEC condition but not the sufficient flag $t$-FTEC condition:** [0.3]{} ![(a) A 1-flag circuit for measuring the stabilizer $Z_{8}Z_{9}Z_{10}Z_{11}Z_{12}Z_{13}Z_{14}Z_{15}$ of the [$[\![15,7,3]\!]$]{} Hamming code. However a single fault on the fourth or fifth CNOT can lead to the error $Z_{12}Z_{13}Z_{14}Z_{15}$ on the data which is a logical fault. With the CNOT gates permuted as shown in (b), the [$[\![15,7,3]\!]$]{} satisfies the general flag 1-FTEC condition.](CircuitReichardt1.png "fig:"){width="\textwidth"} [0.3]{} ![(a) A 1-flag circuit for measuring the stabilizer $Z_{8}Z_{9}Z_{10}Z_{11}Z_{12}Z_{13}Z_{14}Z_{15}$ of the [$[\![15,7,3]\!]$]{} Hamming code. However a single fault on the fourth or fifth CNOT can lead to the error $Z_{12}Z_{13}Z_{14}Z_{15}$ on the data which is a logical fault. With the CNOT gates permuted as shown in (b), the [$[\![15,7,3]\!]$]{} satisfies the general flag 1-FTEC condition.](CircuitReichardt2.png "fig:"){width="\textwidth"} Note that there are codes which satisfy the general flag $t$-FTEC condition but not the sufficient condition presented in this section. An example of such a code is the [$[\![5,1,3]\!]$]{} code (see \[tab:StabilizerGeneratorsLists\] for the codes stabilizer generators and logical operators). Another example includes the Hamming codes as was explained in the discussion on quantum Reed-Muller codes. For instance, consider the [$[\![15,7,3]\!]$]{} Hamming code. Using the 1-flag circuit shown in \[fig:CircuitReichardt1\], the [$[\![15,7,3]\!]$]{} will not satisfy the general flag 1-FTEC condition since a single fault can lead to a logical error on the data. As was shown in [@CR17v1], by permuting the CNOT gates resulting in the circuit illustrated in \[fig:CircuitReichardt2\], the flag 1-FTEC condition is satisfied. Circuits {#app:GeneralTflaggedCircuitConstruction} -------- [0.35]{} ![(a) Illustration of a w-flag circuit for measuring the operator $Z^{\otimes w}$ where $w=6$ using the smallest number of flag qubits. (b) Illustration of a 3-flag circuit for measuring $Z^{\otimes 8}$ using the smallest number of flag qubits.[]{data-label="fig:ExamplesOfLargeFlagCircuits"}](6FlagCircuit.png "fig:"){width="\textwidth"} [0.35]{} ![(a) Illustration of a w-flag circuit for measuring the operator $Z^{\otimes w}$ where $w=6$ using the smallest number of flag qubits. (b) Illustration of a 3-flag circuit for measuring $Z^{\otimes 8}$ using the smallest number of flag qubits.[]{data-label="fig:ExamplesOfLargeFlagCircuits"}](3flagWeight8.png "fig:"){width="\textwidth"} In \[subsec:Remarks\] we showed that the family of surface codes, color codes with a hexagonal lattice and quantum Reed-Muller codes satisfied a sufficient condition allowing them to be used in the flag $t$-FTEC protocol. Along with the general 1-flag circuit construction of \[fig:General1FlagCircuitSecCircuit\], the $6$-flag circuit for measuring $Z^{\otimes 6}$ of \[fig:6flagCircuit\] can be used as $t$-flag circuits for all of the codes in \[subsec:Remarks\]. Note that the circuit in \[fig:StabFTwithAncilla\] (which is a special case of \[fig:General1FlagCircuitSecCircuit\] when $w=4$) is a 4-flag circuit which is used for measuring $Z^{\otimes 4}$. Before describing general 1- and 2-flag circuit constructions, we give the following two definitions which we will frequently use: Any CNOT that couples a data qubit to the measurement qubit will be referred to as $\text{CNOT}_{dm}$ and any CNOT coupling a measurement qubit to a flag qubit will be referred to as $\text{CNOT}_{fm}$. In both cases the target qubit will always be the measurement qubit. **1- and 2-flag circuits for weight $w$ stabilizer measurements:** We provide 1- and 2-flag circuit constructions for measuring a weight-$w$ stabilizer. The 1-flag circuit requires a single flag qubit, and the 2-flag circuit requires at most four flag qubits. [0.35]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General1FlagCircuitSecCircuit.png "fig:"){width="\textwidth"} [0.43]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General2FlagCircuitSecCircuit.png "fig:"){width="\textwidth"} [0.43]{} ![(a) General 1-flag circuit for measuring the stabilizer $Z^{\otimes w}$. (b) Example of a 2-flag circuit for measuring $Z^{\otimes 12}$ using our general 2-flag circuit construction. (c) An equivalent circuit using fewer flag qubits by reusing a measured flag qubit and reinitializing it in the ${|+\rangle}$ state for use in another pair of $\text{CNOT}_{\text{fm}}$ gates.[]{data-label="fig:Generall1And2FlagCircuits"}](General2FlagCircuitSecCircuitCompressed.png "fig:"){width="\textwidth"} Without loss of generality, in proving that the circuit constructions described below are 1- and 2-flag circuits, we can assume that all faults occurred on CNOT gates. This is because any set of $v$ faults (including those at idle, preparation or measurement locations) will have the same output Pauli operator and flag measurement results as some set of at most $v$ faults on CNOT gates (since every qubit is involved in at least one CNOT). As was shown in Ref. [@CR17v1], \[fig:General1FlagCircuitSecCircuit\] illustrates a general 1-flag circuit construction for measuring the stabilizer $Z^{\otimes w}$ which requires only two $\text{CNOT}_{\text{fm}}$ gates. To see that the first construction is a 1-flag circuit, note that an $IZ$ error occurring on any CNOT will give rise to a flag unless it occurs on the first or last $\text{CNOT}_{\text{dm}}$ gates or the last $\text{CNOT}_{\text{fm}}$ gate. However, such a fault on any of these three gates can give rise to an error of weight at most one (after multiplying by the stabilizer $Z^{\otimes w}$). One can also verify that if there are no faults, the circuit in \[fig:General1FlagCircuitSecCircuit\] implements a projective measurement of $Z^{\otimes w}$ without flagging. Following the approach in [@LAR11], one simply needs to check that the circuit preserves the stabilizer group generated by $Z^{\otimes w}$ and $X$ on each ancilla prepared in the ${|+\rangle}$ state and $Z$ on each ancilla prepared in the ${|0\rangle}$ state. By using pairs of $\text{CNOT}_{\text{fm}}$ gates, this construction satisfies the requirement. We now give a general 2-flag circuit construction for measuring $Z^{\otimes w}$ for arbitrary $w$ (see \[fig:General2FlagCircuitSecCircuit\] for an example). The circuit consists of pairs of $\text{CNOT}_{\text{fm}}$ gates each connected to a different flag qubit prepared in the ${|+\rangle}$ state and measured in the $X$ basis. The general 2-flag circuit construction involves the following placement of $w/2-1$ pairs of $\text{CNOT}_{\text{fm}}$ gates: 1. Place a $\text{CNOT}_{\text{fm}}$ pair between the first and second last $\text{CNOT}_{\text{dm}}$ gates. 2. Place a $\text{CNOT}_{\text{fm}}$ pair between the second and last $\text{CNOT}_{\text{dm}}$ gates. 3. After the second $\text{CNOT}_{\text{fm}}$ gate, place the first $\text{CNOT}_{\text{fm}}$ gate of the remaining pairs after every two $\text{CNOT}_{\text{dm}}$ gates. The second $\text{CNOT}_{\text{fm}}$ gate of a pair is placed after every three $\text{CNOT}_{\text{dm}}$ gates. As shown in \[fig:General2FlagCircuitSecCircuitCompressed\], it is possible to reuse some flag qubits to measure multiple pairs of $\text{CNOT}_{\text{fm}}$ gates at the cost of introducing extra time steps into the circuit. For this reason, at most four flag qubits will be needed, however, if $w \le 8$, then $w/2-1$ flag qubits are sufficient. We now show that the above construction satisfies the requirements of a 2-flag circuit. If one CNOT gate fails, by an argument analogous to that used for the 1-flag circuit, there will be a flag or an error of at most weight-one on the data. If the first pair of $\text{CNOT}_{\text{fm}}$ gates fail causing no flag qubits to flag, after multiplying the data qubits by $Z^{\otimes w}$, the resulting error $E_{r}$ will have $\text{wt}(E_{r}) \le 2$. For any other pair of $\text{CNOT}_{\text{fm}}$ gates that fail causing an error of weight greater than two on the data, by construction there will always be another $\text{CNOT}_{\text{fm}}$ gate between the two that fail which will propagate a $Z$ error to a flag qubit causing it to flag. Similarly, if pairs of $\text{CNOT}_{\text{dm}}$ gates fail resulting in the data error $E_{r}$ with $\text{wt}(E_{r}) \ge 2$, by construction there will always be an odd number of $Z$ errors propagating to a flag qubit due to the $\text{CNOT}_{\text{fm}}$ gates in between the $\text{CNOT}_{\text{dm}}$ gates that failed causing a flag qubit to flag. The same argument applies if a failure occurs between a $\text{CNOT}_{\text{dm}}$ and $\text{CNOT}_{\text{fm}}$ gate. Lastly, a proposed general $w$-flag circuit construction for arbitrary $w$ is provided in \[App:GeneralwFlagCircuitConstruction\]. **Use of flag information:** As seen in \[fig:6flagCircuit,fig:3flagWeight8,fig:General2FlagCircuitSecCircuit,fig:General2FlagCircuitSecCircuitCompressed\], in general $t$-flag circuits require more than one flag qubit. Apart from their use in satisfying the $t$-flag circuit properties, the extra flag qubits could be used to reduce the size of the flag error sets (defined in \[Def:FlagErrSetDef\]) when verifying the Flag $t$-FTEC condition of \[app:GeneralFTEC\]. To do so, we first define $f$, where $f$ is a bit string of length $u$ (here $u$ is the number of flag qubits) with $f_{i} = 1$ if the i’th flag qubit flagged and 0 otherwise. In this case, the correction set of \[eq:GeneralLookupTableV2\] can be modified to include flag information as follows: $$\begin{aligned} &\tilde{E}_{t}^{m}(g_{i_{1}},\cdots , g_{i_{k}},s,f_{i_{1}},\cdots ,f_{i_{k}}) = \nonumber \\ &\begin{cases} \{ E \in \mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}},f_{i_{1}},\cdots ,f_{i_{k}}) \times \mathcal{E}_{t-m} \\ \text{ such that } s(E) = s \} \\ \{ E_{\text{min}}(s) \} \text{ if above set empty., } \end{cases} \label{eq:CorrectionSetWithFlagInfo}\end{aligned}$$ where $\mathcal{E}_{m}(g_{i_{1}},\cdots , g_{i_{k}},f_{i_{1}},\cdots,f_{i_{k}})$ is the new flag error set containing only errors caused by precisely $m$ faults spread amongst the circuits $C(g_{i_{1}}),C(g_{i_{2}}), \cdots , C(g_{i_{k}})$ which each gave rise to the flag outcomes $f_{i_{1}},\cdots,f_{i_{k}}$. Hence only errors which result from the measured flag outcome would be stored in the correction set. With enough flag qubits, this could potentially broaden the family of codes which satisfy the Flag $t$-FTEC condition. Circuit level noise analysis {#sec:CircuitLevelNoiseFTEC} ============================ The purpose of this section is to demonstrate explicitly the flag 2-FTEC protocol, and to identify parameter regimes in which flag FTEC presented both here and in other works offers advantages over other existing FTEC schemes. In \[subsec:Numerics19\] we analyze the logical failure rates of the [$[\![19,1,5]\!]$]{} color code and compute it’s pseudo-threshold for the three choices of $\tilde{p}$. In \[subsec:CompareFlagecschemes\] we compare logical failure rates of several fault-tolerant error correction schemes applied to distance-three and distance-five stabilizer codes. The stabilizers for all of the studied codes are given in \[tab:StabilizerGeneratorsLists\]. Logical failure rates are computed using the full circuit level noise model and simulation methods described in \[subsec:NoiseAndNumerics\]. Numerical analysis of the [$[\![19,1,5]\!]$]{} color code {#subsec:Numerics19} --------------------------------------------------------- The full circuit-level noise analysis of the flag 2-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code was performed using the stabilizer measurement circuits of \[fig:StabFTwithAncilla,fig:WeightSixGenerators\]. In the weight-six stabilizer measurement circuit of \[fig:WeightSixGenerators\], there are 10 CNOT gates, three measurement and state-preparation locations, and 230 resting qubit locations. When measuring all stabilizer generators using non-flag circuits, there are 42 CNOT and 42 XNOT gates, 18 measurement and state-preparation locations, and 2196 resting qubit locations. Consequently, we expect the error suppression capabilities of the flag EC scheme to depend strongly on the number of idle qubit locations. three-qubit flag EC pseudo-threshold ---------------------------------------------------- -------------------------------------------------------- [$[\![19,1,5]\!]$]{} and $\tilde{p}=p$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ [$[\![19,1,5]\!]$]{} and $\tilde{p}=\frac{p}{10}$ $p_{\mathrm{pseudo}} = (6.70 \pm 0.07) \times 10^{-5}$ [$[\![19,1,5]\!]$]{} and $\tilde{p}=\frac{p}{100}$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ : Table containing pseudo-threshold values for the flag 2-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code for $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$.[]{data-label="tab:PseudoThresholAllThree1915"} ![Logical failure rates of the [$[\![19,1,5]\!]$]{} color code after implementing the flag 2-FTEC protocol presented in \[subsec:Distance5protocol\] for the three noise models described in \[subsec:NoiseAndNumerics\]. The dashed curves represent the lines $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$. The crossing point between $\tilde{p}$ and the curve corresponding to $p^{({\ensuremath{[\![19,1,5]\!]}})}_{L}(\tilde{p})$ in \[Def:PseudoThreshDef\] gives the pseudo-threshold.[]{data-label="fig:PseudoThreshPlots19ColorAllThree"}](CBplotsAllThreeCurves.png){width="50.00000%"} Pseudo-thresholds of the [$[\![19,1,5]\!]$]{} code were obtained using the methods of \[subsec:NoiseAndNumerics\]. Recall that for extending the lifetime of a qubit (when idle qubit locations fail with probability $\tilde{p}$), the probability of failure after implementing an FTEC protocol should be smaller than $\tilde{p}$. We calculated the pseudo-threshold using \[Def:PseudoThreshDef\] for the three cases were idle qubits failed with probability $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$. The results are shown in \[tab:PseudoThresholAllThree1915\]. The logical failure rates for the three noise models are shown in \[fig:PseudoThreshPlots19ColorAllThree\]. It can be seen that when the probability of error on a resting qubit decreases from $p$ to $p/10$, the pseudo-threshold improves by nearly a factor of six showing the strong dependence of the scheme on the probability of failure of idle qubits. Comparison of flag 1- and 2-FTEC with other FTEC schemes {#subsec:CompareFlagecschemes} -------------------------------------------------------- [0.33]{} {width="\textwidth"} {width="\textwidth"} [0.33]{} {width="\textwidth"} {width="\textwidth"} [0.33]{} {width="\textwidth"} {width="\textwidth"} The most promising schemes for testing fault-tolerance in near term quantum devices are those which achieve high pseudo-thresholds while maintaining a low qubit overhead. The flag FTEC protocol presented in this paper uses fewer qubits compared to other well known fault-tolerance schemes but typically has increased circuit depth. In this section we apply the flag FTEC protocol of \[subsec:ReviewChaoReichardt,subsec:Distance5protocol\] to the [$[\![5,1,3]\!]$]{}, [$[\![7,1,3]\!]$]{} and [$[\![19,1,5]\!]$]{} codes. We compare logical failure rates for three values of $\tilde{p}$ with Steane error correction applied to the [$[\![7,1,3]\!]$]{} and [$[\![19,1,5]\!]$]{} codes and with the $d=3$ and $d=5$ rotated surface code. More details on Steane error correction and surface codes are provided in \[app:SurfaceECSection,app:SteaneECSection\]. Note that recent work by Goto has provided optimizations to prepare Steane ancillas [@Goto16]. However, our numerical results for Steane-EC were produced using the methods presented in \[app:SteaneECSection\]. Results of the logical failure rates for $\tilde{p}=p$, $\tilde{p}=p/10$ and $\tilde{p}=p/100$ are shown in \[fig:AllComparisonPlotsCombined\]. Various pseudo-thresholds and required time-steps for the considered fault-tolerant error correction methods are given in \[tab:PseudoThresholdsAllECschemesD3,tab:PseudoThresholdsAllECschemesD5\]. The circuits for measuring the stabilizers of the 5-qubit code were similar to the ones used in \[fig:StabFTwithAncilla\] (for an $X$ Pauli replace the CNOT by an XNOT). For flag-FTEC methods, it can be seen that the [$[\![5,1,3]\!]$]{} code always achieves lower logical failure rates compared to the [$[\![7,1,3]\!]$]{} code. However, when $\tilde{p}=p$, both the $d=3$ surface code as well as Steane-EC achieves lower logical failure rates (with Steane-EC achieving the best performance). For $\tilde{p}=p/10$, flag-EC applied to the [$[\![5,1,3]\!]$]{} code achieves nearly identical logical failure rates compared to the $d=3$ surface code. For $\tilde{p}=p/100$, flag 1-FTEC applied to the [$[\![5,1,3]\!]$]{} code achieves lower logical failure rates than the $d=3$ surface code but still has higher logical failure rates compared to Steane-EC. FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ------------------------------- --------------------- ------------------ ------------------------------------ -------------------------------------------------------- Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (7.09 \pm 0.03) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (3.39 \pm 0.10) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (3.29 \pm 0.16) \times 10^{-4}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (6.29 \pm 0.13) \times 10^{-4}$ Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p/10$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (1.11 \pm 0.02) \times 10^{-4}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (8.68 \pm 0.15) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (1.04 \pm 0.02) \times 10^{-4}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (3.08 \pm 0.01) \times 10^{-4}$ Flag-EC [$[\![5,1,3]\!]$]{} $\tilde{p} = p/100$ $7$ $64 \le T_{\mathrm{time}} \le 88$ $p_{\mathrm{pseudo}} = (2.32 \pm 0.03) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} $9$ $72 \le T_{\mathrm{time}} \le 108$ $p_{\mathrm{pseudo}} = (1.41 \pm 0.05) \times 10^{-5}$ $d=3$ Surface code 17 $\ge 18$ $p_{\mathrm{pseudo}} = (1.37 \pm 0.03) \times 10^{-5}$ Steane-EC [$[\![7,1,3]\!]$]{} $\ge 35$ 15 $p_{\mathrm{pseudo}} = (3.84 \pm 0.01) \times 10^{-5}$ We also note that the pseudo-threshold increases when $\tilde{p}$ goes from $p$ to $p/10$ for both the [$[\![5,1,3]\!]$]{} and [$[\![7,1,3]\!]$]{} codes when implemented using the flag 1-FTEC protocol. This is primarily due to the large circuit depth in flag-EC protocols since idle qubits locations significantly outnumber other gate locations. For the surface code, the opposite behaviour is observed. As was shown in [@FMMC12], CNOT gate failures have the largest impact on the pseudo-threshold of the surface code. Thus, when idle qubits have lower failure probability, lower physical error rates will be required in order to achieve better logical failure rates. For instance, if idle qubits never failed, then performing error correction would be guaranteed to *increase* the probability of failure due to the non-zero failure probability of other types of locations (CNOT, measurements and state-preparation). Lastly, the pseudo-threshold for Steane-EC also decreases with lower idle qubit failure rates, but the change in pseudo-threshold is not as large as the surface code. This is primarily due to the fact that all CNOT gates are applied transversally in Steane-EC, so that the pseudo-threshold is not as sensitive to CNOT errors compared to the surface code. Furthermore, most high-weight errors arising during the state-preparation of the logical ancilla’s will be detected (see \[app:SteaneECSection\]). Hence, idle qubit errors play a larger role than in the surface code, but Steane-EC has fewer idle qubit locations compared to flag-EC (see \[tab:PseudoThresholdsAllECschemesD3\] for the circuit depths of all schemes). FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold -------------------------------- --------------------- ------------------ ------------------------------------- -------------------------------------------------------- Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (9.41 \pm 0.17) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (1.18 \pm 0.02) \times 10^{-3}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/10$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (6.70 \pm 0.07) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (7.38 \pm 0.22) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (4.42 \pm 0.27) \times 10^{-4}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/100$ $22$ $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ $d=5$ Surface code 49 $\ge 18$ $p_{\mathrm{pseudo}} = (2.63 \pm 0.18) \times 10^{-4}$ Steane-EC [$[\![19,1,5]\!]$]{} $\ge 95$ 15 $p_{\mathrm{pseudo}} = (5.60 \pm 0.43) \times 10^{-5}$ Although Steane-EC achieves the lowest logical failure rates compared to the other fault-tolerant error correction schemes, it requires a minimum of 35 qubits (more details are provided in \[app:SteaneECSection\]). In contrast, the $d=3$ surface code requires 17 qubits, and flag 1-FTEC applied to the [$[\![5,1,3]\!]$]{} code requires only 7 qubits. Therefore, if the probability of idle qubit errors is much lower than gate, state preparation and measurement errors, flag-FTEC methods could be good candidates for early fault-tolerant experiments. It is important to keep in mind that for the flag 1-FTEC protocol applied to the distance-three codes considered in this section, the same ancilla qubits are used to measure all stabilizers. A more parallelized version of flag-FTEC applied to the [$[\![7,1,3]\!]$]{} code using four ancilla qubits instead of two is considered in \[app:CompactRepFlagQubit\]. In computing the number of time steps required by the flag $t$-FTEC protocols, a lower bound is given in the case where there are no flags and the same syndrome is repeated $t+1$ times. In \[app:GeneralFTEC\] it was shown that the full syndrome measurement for flag-FTEC is repeated at most $\frac{1}{2}(t^{2} + 3t + 2)$ times where $t = \lfloor (d-1)/2 \rfloor$. An upper bound on the total number of required time steps is thus obtained from a worst case scenario where syndrome measurements are repeated $\frac{1}{2}(t^{2} + 3t + 2)$ times. For distance-five codes, the first thing to notice from \[fig:AllComparisonPlotsCombined\] is that the slopes of the logical failure rate curves of flag-EC applied to the [$[\![19,1,5]\!]$]{} code and $d=5$ surface code are different from the slopes of Steane-EC applied to the [$[\![19,1,5]\!]$]{} code. In particular, $p_{\text{L}} = cp^{3} + \mathcal{O}(p^{4})$ for flag-EC and the surface code whereas $p_{\text{L}} = c_{1}p^{2} + c_{2}p^{3} + \mathcal{O}(p^{4})$ for Steane-EC ($c$, $c_{1}$ and $c_{2}$ are constants that depend on the code and FTEC method). The reason that Steane-EC has non-zero $\mathcal{O}(p^{2})$ contributions to the logical failure rates is that there are instances where errors occurring at two different locations can lead to a logical fault. Consequently, the Steane-EC method that was used is not strictly fault-tolerant according to \[Def:FaultTolerantDef\]. In \[app:SteaneECSection\], more details on the fault tolerant properties of Steane-EC are provided and a fully fault-tolerant implementation of Steane-EC is analyzed (at the cost of using more qubits). For $d=5$, the surface code achieves significantly lower logical failure rates compared to all other distance 5 schemes but uses 49 qubits instead of 22 for the [$[\![19,1,5]\!]$]{} code. Furthermore, due the differences in the slopes of flag-2 FTEC protocol compared with Steane-EC applied to the [$[\![19,1,5]\!]$]{} code, there is a regime where flag-2 FTEC achieves lower logical failure rates compared to Steane-EC. For $\tilde{p}=p/100$, it can be seen in \[fig:AllComparisonPlotsCombined\] that this regime occurs when $p \lesssim 10^{-4}$. We also note that the pseudo-threshold of flag-EC applied to the [$[\![19,1,5]\!]$]{} color code increases for all noise models whereas the pseudo-threshold decreases for the other FTEC schemes. Again, this is due to the fact that flag-EC has a larger circuit depth compared to the other FTEC methods and is thus more sensitive to idle qubit errors. Comparing the flag 2-FTEC protocol (applied to the [$[\![19,1,5]\!]$]{} color code) to all the $d=3$ schemes that were considered in this section, due to the higher distance of the 19-qubit code, there will always be a parameter regime where the 19-qubit color code acheives lower logical failure rates than both the $d=3$ surface code and Steane-EC applied to the [$[\![7,1,3]\!]$]{} code. In the case where $\tilde{p}=p/100$ and with $p \lesssim 1.5 \times10^{-4}$, using flag error correction with only 22 qubits outperforms Steane error correction (which uses a minimum of 35 qubits) and the $d=3$ rotated surface code (which uses 17 qubits). Note the considerable number of time steps involved in a round of flag-EC, particularly in the $d=5$ case (see \[tab:PseudoThresholdsAllECschemesD5\]). For many applications, this is a major drawback, for example for quantum computation when the time of an error correction round dictates the time of a logical gate. However there are some cases in which having a larger number of time-steps in an EC round while holding the logical error rate fixed is advantageous as it corresponds to a longer physical lifetime of the encoded information. Such schemes could be useful for example in demonstrating that encoded logical quantum information can be stored for longer time scales in the lab using repeated rounds of FTEC. Conclusion {#sec:Conclusion} ========== Building on definitions and a new flag FTEC protocol applied to distance-three and -five codes presented in \[sec:Section1\], in \[subsec:GeneralProtocol\] we presented a general flag FTEC protocol, which we called flag $t$-FTEC, and which is applicable to stabilizer codes of distance $d = 2t+1$ that satisfy the flag $t$-FTEC condition. The protocol makes use of flag ancilla qubits which signal when $v$ faults lead to errors of weight greater than $v$ on the data when performing stabilizer measurements. In \[subsec:ApplicationProtocolColorCode,app:GeneralTflaggedCircuitConstruction\] we gave explicit circuit constructions, including those needed for distance 3 and 5 codes measuring stabilizers of weight 4, 6 and 8. In \[subsec:Remarks\] we gave a sufficient condition for codes to satisfy the requirements for flag $t$-FTEC. Quantum Reed-Muller codes, Surface codes and hexagonal lattice color codes were shown to be families of codes that satisfy the sufficient condition. The flag $t$-FTEC protocol could be useful for fault-tolerant experiments performed in near term quantum devices since it tends to use fewer qubits than other FTEC schemes such as Steane, Knill and Shor EC. In \[subsec:CompareFlagecschemes\] we provided numerical evidence that with only 22 qubits, the flag $2$-FTEC protocol applied to the [$[\![19,1,5]\!]$]{} color code can achieve lower logical failure rates than other codes using similar numbers of qubits such as the rotated distance-3 surface code and Steane-EC applied to the Steane code. A clear direction of future work would be to find optimal general constructions of $t$-flag circuits for stabilizers of arbitrary weight that improve upon the general construction given in \[App:GeneralwFlagCircuitConstruction\]. Of particular interest would be circuits using few flag qubits and CNOT gates while minimizing the probability of false-positives (i.e. when the circuit flags without a high-weight error occurring). Finding other families of stabilizer codes which satisfy the sufficient or more general condition for flag $t$-FTEC would also be of great interest. One could also envisage hybrid schemes combining flag EC with other FTEC approaches. Another direction of future research would be to find general circuit constructions for simultaneously measuring multiple stabilizers while minimizing the number of required ancilla qubits. Further, we believe performing a rigorous numerical analysis to understand the impact of more compact circuit constructions on the codes threshold is of great interest. Lastly, the decoding complexity (i.e. generating the flag error set lookup tables) is limited by the decoding complexity of the code. In some cases, for example concatenated codes, it may be possible to exploit some structure to generate the flag error sets more efficiently. In the case of concatenated code, the decoding complexity would be reduced to the decoding complexity of the codes used at every level. Finding other scalable constructions for efficient decoding schemes using flag error correction remains an open problem. Acknowledgements ================ The authors would like to thank Krysta Svore, Tomas Jochym-O’Connor, Nicolas Delfosse and Jeongwan Haah for useful discussions. We also thank Steve Weiss for providing the necessary computational tools that allowed us to complete our work in a timely manner. C. C. would like to acknowledge the support of QEII-GSST and thank Microsoft and the QuArC group for its hospitality where all of this work was completed. Proof that the flag $t$-FTEC protocol satisfies the fault-tolerance criteria of \[Def:FaultTolerantDef\] {#app:ProtocolGeneralProof} ======================================================================================================== Consider the flag $t$-FTEC protocol described in \[subsec:GeneralProtocol\]. If the flag $t$-FTEC condition is satisfied, then both fault-tolerance criteria of \[Def:FaultTolerantDef\] will be satisfied. First note that the protocol always terminates. As was shown in the arguments leading to \[Eq:Nmax\] presented in \[subsec:GeneralProtocol\], the maximum number of syndrome measurement rounds is $\frac{1}{2}(t^{2}+3t+2)$. To prove fault-tolerance, in what follows we assume that there are at most $t$-faults during the protocol. Also, we define a benign fault to be a fault that either leaves all syndrome measurements in the protocol unchanged. By repeating the syndrome measurement using $t$-flag circuits, the following cases exhaust all possible errors for the occurrence of at most $t$ faults. *: The same syndrome is measured $t-n_{\text{diff}}+1$ times in a row and there are no flags.* At any time during the protocol, if there are no flags, there can be at most $t-n_{\text{diff}}$ remaining faults that occur (since it is guaranteed that there were at least $n_{\text{diff}}$ faults). Therefore, if the same syndrome was measured $t-n_{\text{diff}}+1$ times in a row, at least one round (say $r$) had to have been fault-free yielding the correct syndrome corresponding to the data qubit errors present at that time. Applying $E_{\text{min}}(s)$ will remove those errors. Furthermore, since all syndrome measurements are identical and there are no flags, there can be at most $t-n_{\text{diff}}$ errors which are introduced on the data blocks from faults during the $t-n_{\text{diff}}+1$ syndrome measurement rounds (excluding round $r$). Since none of the errors change the syndrome, after applying the correction, the output state can differ from the input codeword by an error of weight at most $t-n_{\text{diff}}$ (if the total number of faults input errors was $t$). For input states afflicted by an error of arbitrary weight, the output state will differ from a valid codeword (but not necessarily the input codeword) by an error of weight at most $t-n_{\text{diff}}$. Thus both conditions of \[Def:FaultTolerantDef\] are satisfied. *: There are no flags and $n_{\text{diff}} = t$.* The only way that $n_{\text{diff}} = t$ is if there were $t$-faults that each changed the syndrome measurement outcome. Further since there were no flags, an error $E$ afflicting the data qubits must satisfy $\text{wt}(E) \le t$. Thus repeating the syndrome measurement using non-flag circuits will correctly identify and remove the error in the case where the number of input errors and faults is $t$ or project the system back to the code space (to a possibly differ codeword) if there were $t$ faults and the input state was afflicted by an error of arbitrary weight . *: A set of $t$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ flagged.* Since $t$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ flagged, then no other faults can occur during the protocol. Hence, when repeating the syndrome measurement using non-flag circuits, the measured syndrome will correspond to an error $E_{r} \in \tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$. Since from the flag $t$-FTEC condition all elements of $\tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$ are logically equivalent, the product of errors resulting from the flag circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{t}}) \}$ will be corrected. Note that for an input error $E_{\text{in}}$ of arbitrary weight and since the final round must be error free, applying a correction a correction from the set $\tilde{E}_{t}^{t}(g_{i_{1}},\cdots g_{i_{t}},s)$ is guaranteed to return the system to the codespace. Thus both conditions of \[Def:FaultTolerantDef\] are satisfied. *: The $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged with $1 \le m < t$, $n_{\text{diff}} = t -m $.* Here we can assume that at any point during the protocol and after the $j$’th flag, the syndrome never repeated more than $t-j-n_{\text{diff}}$ times. Otherwise case 5 of the protocol would already have occurred. As $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ have flagged and $n_{\text{diff}}=t-m$, then there can be no more faults. The final syndrome measurement using non-flag circuits will yield a syndrome corresponding to an error in the set $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots g_{i_{m}},s)$ (and all elements are logically equivalent from the flag $t$-FTEC condition). Applying a recovery operator chosen from this set will thus remove the errors afflicting the data. If the input state differs from a valid codeword by an error of arbitrary weight, by definition of $\tilde{E}_{t}^{m}(g_{i_{1}},\cdots g_{i_{m}},s)$ the output state will be a valid codeword. *: The $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged with $1 \le m < t$, $n_{\text{same}} = t -m - n_{\text{diff}} + 1$.* Given that $m$ circuits $\{ C(g_{i_{1}}), \cdots , C(g_{i_{m}}) \}$ flagged, there are $r$ remaining faults that don’t result in a flag with $n_{\text{diff}} \le r \le t-m$. In this case, after the $m$’th flag, the syndrome measurement was repeated using $t$-flag circuits $t-m- n_{\text{diff}}+1$ times in a row and all syndromes were the same. It is thus guaranteed that at least one of the syndrome measurements $s$ was fault-free and correctly identified the errors arising from the flags and errors causing the syndrome to change giving $n_{\text{diff}}$ (along with some error $E$ which did not cause the circuits to flag with $\text{wt}(E) \le t-m- n_{\text{diff}}$). Consequently, if there are no errors on the input state, the overall error on the data will be $EE_{r}$ with $E_{r} \in \bigcup_{j=0}^{t-m-n_{\text{diff}}}\tilde{E}_{t}^{t-j-n_{\text{diff}}}(g_{i_{1}},\cdots ,g_{i_{m}}, s)$. Since all elements in $\bigcup_{j=0}^{t-m-n_{\text{diff}}}\tilde{E}_{t}^{t-j-n_{\text{diff}}}(g_{i_{1}},\cdots ,g_{i_{m}}, s)$ are logically equivalent from the flag $t$-FTEC condition, by choosing a correction from this set, the output state can differ from the input codeword by an error of at most weight $t-m- n_{\text{diff}}$. If there is an input error of arbitrary weight, then again one of the $t-m- n_{\text{diff}}+1$ rounds will have the correct syndrome $s$. The actual state of the data qubits after the protocol can differ from the state which had the correct syndrome by an error of weight at most $t-m- n_{\text{diff}}$. Therefore applying any correction with syndrome $s$ will return the system to the code space up to an error of weight at most $t-m- n_{\text{diff}}$. Fault-tolerant state preparation and measurement using flag $t$-FTEC {#app:StatePrepAndMeasure} ==================================================================== In this section we show how to fault-tolerantly prepare a logical ${|\overline{0}\rangle}$ state and how to perform fault-tolerant measurements for codes that satisfy the flag $t$-FTEC condition of \[app:GeneralFTEC\]. Note that there are several methods that can be used for doing so. Here we follow a procedure similar to that shown in [@Gottesman2010] when performing Shor EC. However, compared to Shor EC, the flag $t$-FTEC protocol requires fewer qubits. Furthermore, postselection is not necessary. Consider an $n$-qubit stabilizer code $C$ with stabilizer group $\mathcal{S} = \langle g_{1},\cdots , g_{n-k} \rangle$ that can correct up to $t$ errors. Notice that the encoded ${|\overline{0}\rangle}$ state is a $+1$ eigenstate of the logical $\overline{Z}$ operator and all of the codes stabilizer generators. For $k$ encoded qubits, ${|\overline{0}\rangle}$ would be $+1$ eigenstate of $\{ \overline{Z}_{1}, \cdots \overline{Z}_{k} \}$ and all of the codes stabilizers. For notational simplicity, in what follows we assume $k=1$. The state ${|\overline{0}\rangle}$ is a stabilizer state completely specified by the full stabilizer generators of $\mathcal{S}$ and $\overline{Z}$. We can think of $\mathcal{S}' = \langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$ as a stabilizer code with zero encoded qubits and a $2^{0}=1$ dimensional Hilbert space. Thus any state which is a $+1$ eigenstate of all operators in $\mathcal{S}'$ will correspond to the encoded ${|\overline{0}\rangle}$ state. Now, suppose we prepare ${|\overline{0}\rangle}_{\text{in}}$ using a non-fault-tolerant encoding and perform a round of flag $t$-FTEC using the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$. Then by the second criteria of \[Def:FaultTolerantDef\], the output state ${|\overline{0}\rangle}_{\text{out}}$ is guaranteed to be a valid codeword with at most $t$ single-qubit errors. But for the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$ there is only *one* valid codeword which corresponds to the encoded ${|\overline{0}\rangle}$ state. In fact, by the second criteria of \[Def:FaultTolerantDef\], any $n$-qubit input state prepared using non-fault-tolerant circuits is guaranteed to be an encoded ${|\overline{0}\rangle}$ state if there are no more than $t$ faults in the EC round. We point out that the flag $t$-FTEC condition of \[subsec:GeneralProtocol\] is trivially satisfied for $\mathcal{S}'$ since the codes logical operators are now stabilizers. In other words, if two errors belong to the set $\tilde{E}^{m}_{t}(g_{i_{1}}, \cdots, g_{i_{k}},s)$, then their product will always be a stabilizer. Therefore, the flag $t$-FTEC protocol can always be applied for the code $\mathcal{S}'$. To summarize, the encoded ${|\overline{0}\rangle}$ state can be prepared by first preparing any $n$-qubit state using non-fault-tolerant circuits followed by applying a round of flag $t$-FTEC using the extended stabilizers $\langle g_{1}, \cdots g_{n-1}, \overline{Z} \rangle$. This guarantees that the output state will be the encoded ${|\overline{0}\rangle}$ state with at most $t$ single-qubit errors. Now suppose we want to measure the eigenvalue of a logical operator $\overline{P}$ where $P$ is a Pauli. If $C$ is a CSS code and the logical operator being measured is $X$ or $Z$, one could measure the eigenvalue by performing the measurement transversally. So suppose $C$ is not a CSS code. From [@Gottesman2010] we require that performing a measurement with $s$ faults on an input state with $r$ errors ($r+s \le t$) is equivalent to correcting the $r$ errors and performing the measurement perfectly. The protocol for fault-tolerantly measuring the eigenvalue of $\overline{P}$ is described as follows: 1. Perform flag $t$-FTEC. 2. Use a $t$-flag circuit to measure the eigenvalue of $\overline{P}$. 3. Repeat steps 1 and 2 $2t+1$ times and take the majority of the eigenvalue of $\overline{P}$. Step 1 is used to remove input errors to the measurement procedure. However during error correction, a fault can occur which could cause a new error on the data. Thus by repeating the measurement without performing error correction, the wrong state would be measured each time if there were no more faults. But repeating the syndrome $2t+1$ times, it is guaranteed that at least $t+1$ of the syndrome measurements had no faults and that the correct eigenvalue of $\overline{P}$ was measured. Thus taking the majority of the measured eigenvalues will give the correct answer. Note that during the fault-tolerant measurement procedure, if there is a flag either during the error correction round or during the measurement of $\overline{P}$, when error correction is performed one corrects based on the possible set of errors resulting from the flag. Candidate general $w$-flag circuit construction {#App:GeneralwFlagCircuitConstruction} =============================================== {width="105.00000%"} In this section we provide a candidate general $w$-flag circuit construction for measuring the stabilizer $Z^{\otimes w}$. Although we do not provide a rigorous proof that our construction results in a $w$-flag circuit, we give several arguments as evidence that it satisfies all the criteria of a $w$-flag circuit. An illustration of the circuit construction (for $w=12$) is given in \[fig:GeneralWflagFig\] and the description for how the circuit is constructed for arbitrary $w$ is provided in the caption. In what follows, we can restrict our attention to the case in which all $v$ faults occur on CNOT gates in the circuit. The effect on the measurement outcomes and data qubits due to a set of $v$ faults that include faults at idle and measurement locations can always occur due to at most $v$ faults at CNOT locations only (as every qubit is involved in at least one CNOT). Moreover, we can assume that for $\text{CNOT}_{\text{fm}}$ gates, the faults belong to the set $\{ IZ, ZI, ZZ \}$ since $X$ errors would never propagate to the data or affect the measurement outcome of a flag qubit. For $\text{CNOT}_{\text{dm}}$ gates, we can assume that faults belong to the set $\{XZ,XI \}$. We only consider $Z$ errors on the target qubit of a $\text{CNOT}_{\text{dm}}$ for the same reason that was given for $\text{CNOT}_{\text{fm}}$ gates. For the control qubit, an X errors guarantees that the weight of the data qubit error increases even after the application of a satbilizer (since we are measuring $Z^{\otimes w}$). We will use the following useful terminology: we say that a single-qubit Pauli at a time step in the circuit propagates to a qubit at a particular time-step if it would do so in the fault-free circuit. Given a single-qubit Pauli at a time step in the circuit, we say that another qubit is affected by the Pauli if it propagates to that qubit in any time step. We now provide arguments for why the circuit is a $w$-flag circuit. First, note that every $\text{CNOT}_{\text{fm}}$ gate comes as part of a pair with the measurement qubit being the target qubit. This ensures that when the circuit is fault-free, it implements a projective measurement of $Z^{\otimes w}$ without flagging. Next, notice that apart from the last two $\text{CNOT}_{\text{dm}}$ gates, each $\text{CNOT}_{\text{dm}}$ gate is followed by two $\text{CNOT}_{\text{fm}}$ gates, one with its partnering $\text{CNOT}_{\text{fm}}$ located before the first $\text{CNOT}_{\text{dm}}$ and the other partner is located in between the last two $\text{CNOT}_{\text{dm}}$ gates. Thus if there is a single $Z$ error on the measurement qubit which propagates to any of the data qubits, the circuit will flag. In all circuits considered in this section, $s_{0}$ will correspond to the sequence of $\text{CNOT}_{\text{fm}}$ gates that come before the first $\text{CNOT}_{\text{dm}}$ gate. First consider the shorter circuit construction using only the first family of $\text{CNOT}_{\text{fm}}$ gates from the construction in \[fig:GeneralWflagFig\] (see the example in \[fig:ErrorPropLogic\]). We can separate the set of all locations into subsets including two $\text{CNOT}_{\text{fm}}$ gates and one $\text{CNOT}_{\text{dm}}$ gate as shown in \[fig:CNOTsectionAppendix\] (apart from the last $\text{CNOT}_{\text{dm}}$). This circuit segment can increase the weight of the data error by at most one. There are four cases with inputs on the measurement qubit before the first $\text{CNOT}_{\text{fm}}$ and $\text{CNOT}_{\text{dm}}$ being $\{(I,I),(I,Z),(Z,I),(Z,Z)\}$. Note that if the following property held for each segment, then the circuit would be $w$-flag: for all inputs to the segment, if the weight of the data error increases and there are no faults in the segment, the segment flags. Unfortunately, for the input $(Z,Z)$, this is not the case. Both input $Z$ must come from at least two faults. ![Example of five faults that lead to an error of weight six on the data without causing a flag when only the first family of $\text{CNOT}_{\text{fm}}$ gates are used in the construction of \[fig:GeneralWflagFig\] (here $w=10$). Errors arising from faults are shown in blue and the resulting errors after propagating through the CNOT gates are shown in red. []{data-label="fig:ErrorPropLogic"}](ErrorPropLogic.PNG){width="45.00000%"} Note that if $v$ faults results in a data qubit error of weight greater than $v$ without causing the circuit in \[fig:ErrorPropLogic\] to flag, there must be either an $IZ$ fault followed by no fault in a consecutive pair of $\text{CNOT}_{\text{fm}}$ gates belonging to $s_{0}$ or a $ZZ$ fault followed by two $\text{CNOT}_{\text{fm}}$ gates that don’t fail in $s_{0}$. Moreover, a poor choice of ordering of the $\text{CNOT}_{\text{fm}}$ gates in $s_{1},s_{2}$ and $s_{3}$ can result in four faults causing a weight $\frac{w}{2}+1$ error on the data without causing the circuit to flag. Therefore, the ordering of the $\text{CNOT}_{\text{fm}}$ gates in the sets $s_{1},s_{2}$ and $s_{3}$ is chosen such that most $Z$ errors in $s_{0}$ that first propagate to flag qubits connected to gates in $s_{1}$, will then propagate to flag qubits in $s_{3}$ and vice-versa. Typically, if a $Z$ error propagates through multiple $\text{CNOT}_{\text{dm}}$ gates in $s_{1}$, then unless $\text{CNOT}_{\text{fm}}$ gates in $s_{3}$ fail, the flag qubits affected by the $Z$ error would flag. Furthermore, the total number of required failures for gates in $s_{3}$ to cancel the $Z$ errors will typically be equal to the number of times the $Z$ error propagated to the data. There are however cases which don’t flag in which $v$ faults in the circuit construction presented in \[fig:ErrorPropLogic\] lead to more than $v$ errors on the data qubit, such as the example given in the figure. All such problematic cases that we found had a $Z$ error on the target qubit in one of the last few $\text{CNOT}_{\text{fm}}$ gates in $s_0$, followed by a $Z$ error on the target qubit in one of the first few $\text{CNOT}_{\text{dm}}$ gates in $s_1$. Then further $Z$ errors occur throughout the remainder of the circuit which propagate to the data while preventing the flag qubits affected by the previous errors from flagging. Further, a $Z$ error on the control qubit of the second $\text{CNOT}_{\text{fm}}$ in $s_{2}$ cancels the $Z$ which propagates to the flag qubit coupled to that $\text{CNOT}_{\text{fm}}$ gate. ![Illustration of a pair of $\text{CNOT}_{\text{fm}}$ gates as well as a segment of $\text{CNOT}_{\text{dm}}$ followed by $\text{CNOT}_{\text{fm}}$ gate. The first $\text{CNOT}_{\text{fm}}$ gate belongs to the sequence of $\text{CNOT}_{\text{fm}}$ gates that come before the first $\text{CNOT}_{\text{dm}}$ gate (see \[fig:GeneralWflagFig\]). []{data-label="fig:CNOTsectionAppendix"}](CNOTsectionAppendix.PNG){width="40.00000%"} This particular problematic fault pattern would lead to flags if it occurred within the full circuit construction of \[fig:GeneralWflagFig\] (if the additional locations of the larger circuit do not fail). As this was the only type of problematic fault pattern that we found, one would hope that all problematic fault patterns are rendered non problematic provided no additional locations fail. Since the additional $\text{CNOT}_{\text{fm}}$ gates always occur immediately after one of the original $\text{CNOT}_{\text{fm}}$ gates (or after the last $\text{CNOT}_{\text{fm}}$ gate), as far as the flag properties of the original circuit are concerned, no new problematic fault patterns are introduced. We conclude this section by noting that our candidate general $w$-flag circuit construction requires $w-1$ flag qubits and is implemented in $7w-8$ time steps. This is clearly not optimal in general since for example, as shown in \[fig:6flagCircuit\], a $w$-flag circuit was found (for $w=6$) which requires only three flag qubits instead of five and the circuit is implemented in 14 time steps instead of 34. It is thus still an open problem to find optimal $w$-flag circuits for arbitrary $w$. Quantum Reed-Muller codes {#app:QRMcodes} ========================= In this section we first describe how to construct the family of quantum Reed-Muller codes $\text{QRM}(m)$ with code parameters $[\![ 2^{m}-1,k=1,d=3 ]\!]$ following [@ADP14]. We then show that the family of $\text{QRM}(m)$ codes satisfy the sufficient flag 1-FTEC condition of \[subsec:Remarks\]. Reed-Muller codes of order $m$ ($\text{RM}(1,m)$) are defined recursively from the following generator matrices: First, $\text{RM}(1,1)$ has generator matrix $$\begin{aligned} G_{1}=\left( \begin{array}{cc} 1 & 1\\ 0 & 1\\ \end{array} \right), \label{eq:G1mat}\end{aligned}$$ and $\text{RM}(1,m+1)$ has generator matrix $$\begin{aligned} G_{m+1}=\left( \begin{array}{cc} G_{m} & G_{m}\\ 0 & 1\\ \end{array} \right), \label{eq:GMmat}\end{aligned}$$ where 0 and 1 are vectors of zeros and ones in \[eq:GMmat\]. The dual of $\text{RM}(1,m+1)$ is given by the higher order Reed-Muller code $\text{RM}(m-2,m)$. In general, the generator matrices for higher-order Reed-Muller codes $\text{RM}(r,m)$ are given by $$\begin{aligned} H_{r,m+1}=\left( \begin{array}{cc} H_{r,m} & H_{r,m}\\ 0 & H_{r-1,m}\\ \end{array} \right). \label{eq:GRMmat}\end{aligned}$$ with $$\begin{aligned} H_{2,1}=H_{1,1} = \left( \begin{array}{cc} 1 &1\\ 0 & 1\\ \end{array} \right), \label{eq:G21mat}\end{aligned}$$ The $X$ stabilizer generators of $\text{QRM}(m)$ are derived from shortened Reed-Muller codes where the first row and column of $G_{m}$ are deleted. We define the resulting generator matrix as $\overline{G}_{m}$. The $Z$ stabilizer generators are obtained by deleting the first row and column of $H_{m-2,m}$. Similarly, we define the resulting generator matrix as $\overline{H}_{m-2,m}$. As was shown in [@ADP14], $\text{rows}(\overline{G}_{m}) \subset \text{rows}(\overline{H}_{m-2,m})$ and each row has weight $2^{m-1}$. Therefore, all the $X$-type stabilizer generators of $\text{QRM}(m)$ have corresponding $Z$-type stabilizers. By construction, the remaining rows of $\overline{H}_{m-2,m}$ will have weight $2^{m-2}$. Furthermore, every weight $2^{m-2}$ row has support contained within some weight $2^{m-1}$ row of the generator matrix $\overline{H}_{m-2,m}$. Therefore, every $Z$-type stabilizer generator has support within the support of an $X$ generator. Implementation of Steane error correction {#app:SteaneECSection} ========================================= In this section we describe how to implement Steane error correction and discuss its fault-tolerant properties. We also provide a comparison of a version of Steane error correction with flag 2-FTEC protocol described in \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. Steane error correction is a fault-tolerant scheme that applies to the Calderbank-Shor-Steane (CSS) family of stabilizer codes [@Steane97]. In Steane error correction, the idea is to use encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle} = ({|\overline{0}\rangle} + {|\overline{1}\rangle})/\sqrt{2}$ ancilla states to perform the syndrome extraction. The ancilla’s are encoded in the same error correcting code that is used to protect the data. The $X$ stabilizer generators are measured by preparing the encoded ${|\overline{0}\rangle}$ state and performing transversal CNOT gates between the ancilla and the data, with the ancilla acting as the control qubits and the data acting as the target qubits. After applying the transversal CNOT gates, the syndrome is obtained by measuring ${|\overline{0}\rangle}$ transversally in the $X$-basis. The code construction for CSS codes is what guarantees that the correct syndrome is obtained after applying a transversal measurement (see [@Gottesman2010] for more details). [0.25]{} ![(a) Fault-tolerant Steane error correction circuit for distance-three CSS codes. Each line represents an encoded qubit. The circuit uses only two encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (encoded in the same error correcting code which protects the data) to ensure that faults in the preparation circuits of the ancilla’s don’t spread to the data block. (b) Fault-tolerant Steane error correction circuit which can be used for any distance-three CSS stabilizer code encoding the data. There are a total of eight encoded ancilla qubits instead of four. The dark bold lines represent resting qubits. Note that the circuit in \[fig:SteaneCSSforNonPerfect\] could in some cases be used for higher distance CSS codes with appropriately chosen circuits for ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (see [@PR12]).[]{data-label="fig:SteaneECcircuitsAll"}](SteaneECcircuitforPerfectCSS.png "fig:"){width="\textwidth"} [0.25]{} ![(a) Fault-tolerant Steane error correction circuit for distance-three CSS codes. Each line represents an encoded qubit. The circuit uses only two encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (encoded in the same error correcting code which protects the data) to ensure that faults in the preparation circuits of the ancilla’s don’t spread to the data block. (b) Fault-tolerant Steane error correction circuit which can be used for any distance-three CSS stabilizer code encoding the data. There are a total of eight encoded ancilla qubits instead of four. The dark bold lines represent resting qubits. Note that the circuit in \[fig:SteaneCSSforNonPerfect\] could in some cases be used for higher distance CSS codes with appropriately chosen circuits for ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (see [@PR12]).[]{data-label="fig:SteaneECcircuitsAll"}](SteaneECcircuitforPerfectNonCSS.png "fig:"){width="\textwidth"} FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ------------------------------ --------------------- ------------------ ------------------------------------- -------------------------------------------------------- Full Steane-EC $\tilde{p} = p$ $\ge 171$ 15 $p_{\mathrm{pseudo}} = (3.50 \pm 0.14) \times 10^{-3}$ Full Steane-EC $\tilde{p} = p/100$ $\ge 171$ 15 $p_{\mathrm{pseudo}} = (1.05 \pm 0.04) \times 10^{-3}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p$ 22 $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (1.14 \pm 0.02) \times 10^{-5}$ Flag-EC [$[\![19,1,5]\!]$]{} $\tilde{p} = p/100$ 22 $504 \le T_{\mathrm{time}} \le 960$ $p_{\mathrm{pseudo}} = (7.74 \pm 0.16) \times 10^{-5}$ Similarly, the $Z$-stabilizer generators are measured by preparing the encoded ${|\overline{+}\rangle}$, applying CNOT gates transversally between the ancilla and the data with the data acting as the control qubits and the ancilla’s acting as the target qubits. The syndrome is then obtained by measuring ${|\overline{+}\rangle}$ transversally in the $Z$-basis. The above protocol as stated is not sufficient in order to be fault-tolerant. The reason is that in general the circuits for preparing the encoded ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ are not fault-tolerant in the sense that a single error can spread to a multi-weight error which could then spread to the data block when applying the transversal CNOT gates. To make the protocol fault-tolerant, extra ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ ancilla states (which we call verifier qubits) are needed to check for multi-weight errors at the output of the ancilla states. [0.3]{} ![Logical failure rate of the full fault-tolerant Steane error correction approach of \[fig:SteaneCSSforNonPerfect\] and the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. In (a) idle qubits are chosen to fail with a total probability $\tilde{p} = p$ while in (b) idle qubits fail with probability $\tilde{p} = p/100$. The intersection between the dashed curve and solid lines represent the pseudo-threshold of both error correction schemes. []{data-label="fig:ResultFullFaultTolerantSteanevsCB"}](FullFaulTolerantSteanep.png "fig:"){width="\textwidth"} [0.3]{} ![Logical failure rate of the full fault-tolerant Steane error correction approach of \[fig:SteaneCSSforNonPerfect\] and the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} code. In (a) idle qubits are chosen to fail with a total probability $\tilde{p} = p$ while in (b) idle qubits fail with probability $\tilde{p} = p/100$. The intersection between the dashed curve and solid lines represent the pseudo-threshold of both error correction schemes. []{data-label="fig:ResultFullFaultTolerantSteanevsCB"}](FullFaulTolerantSteanep100.png "fig:"){width="\textwidth"} For the ${|\overline{0}\rangle}$ ancilla, multiple $X$ errors can spread to the data if left unchecked. Therefore, another encoded ${|\overline{0}\rangle}$ ancilla is prepared and a transversal CNOT gate is applied between the two states with the ancilla acting as the control and the verifier state acting as target. Anytime $X$ errors are detected the state is rejected and the error correction protocol start over. Further, if the verifier qubit measures a $-1$ eigenvalue of the logical $Z$ operator, the ancilla qubit is also rejected. A similar technique is used for verifying the ${|\overline{+}\rangle}$ state (see \[fig:SteaneCSSforPerfect\]). For the [$[\![7,1,3]\!]$]{} Steane code, an error $E=Z_{i}Z_{j}$ can always be written as $E=\overline{Z}Z_{k}$ where $\overline{Z}$ is the logical $Z$ operator (this is not true for general CSS codes). But ${|\overline{0}\rangle}$ is a $+1$ eigenstate of $\overline{Z}$. Therefore, we don’t need to worry about $Z$ errors of weight greater than one occurring during the preparation of the ${|\overline{0}\rangle}$ state. In [@AGP06] it was shown that unlike for the [$[\![7,1,3]\!]$]{} code, for general CSS codes, the encoded ancilla states need to be verified for both $X$ and $Z$ errors in order for Steane error correction to satisfy the fault-tolerant properties of \[Def:FaultTolerantDef\]. We show the general distance-three fault-tolerant scheme in \[fig:SteaneCSSforNonPerfect\]. Note that the circuit in \[fig:SteaneCSSforPerfect\] will only satisfy the fault-tolerant criteria of \[Def:FaultTolerantDef\] for perfect distance-three CSS codes (see [@AGP06] for more details). In \[subsec:CompareFlagecschemes\] we computed logical failure rates for Steane error correction applied to the [$[\![19,1,5]\!]$]{} code using the circuit of figure \[fig:SteaneCSSforPerfect\] in order to minimize the number of physical qubits. However, since the [$[\![19,1,5]\!]$]{} code is not a perfect CSS code, only the circuit in \[fig:SteaneCSSforNonPerfect\] satisfies all the criteria of \[Def:FaultTolerantDef\]. This explains why the leading order contributions to the logical failure was of the form $p_{\text{L}} = c_{1}p^{2}+c_{2}p^{3} + \mathcal{O}(p^{4})$ instead of $p_{\text{L}} = cp^{3} + \mathcal{O}(p^{4})$ (which would be the case for a distance-5 code). In \[fig:ResultFullFaultTolerantSteanevsCB\] we applied Steane error correction using the circuit of \[fig:SteaneCSSforNonPerfect\] to achieve the full error correcting capabilities of the [$[\![19,1,5]\!]$]{} code. We used methods presented in [@PR12; @CJL16b] in order to obtain the encoded ${|\overline{0}\rangle}$ state (since the [$[\![19,1,5]\!]$]{} code is self-dual, the ${|\overline{+}\rangle}$ state is obtain by interchanging all physical ${|0\rangle}$ and ${|+\rangle}$ states and reversing the direction of the CNOT gates). Note that not all ${|\overline{0}\rangle}$ and ${|\overline{+}\rangle}$ circuits had the same sequence of CNOT gates. This was to ensure that a single fault in two different preparation circuits, i.e. for ${|\overline{0}\rangle}$ and for ${|\overline{+}\rangle}$, would not lead to uncorrectable $X$ or $Z$ errors that would go undetected by the verifier ancillas and at the same time propagate to the data block. The results are compared with the flag 2-FTEC protocol of \[subsec:Distance5protocol\] applied to the [$[\![19,1,5]\!]$]{} for the noise models where idle qubits fail with probability $\tilde{p}=p$ and $\tilde{p}=p/100$. In both cases the logical failure rates have a leading order $p^{3}$ contribution (which is determined from finding the best fit curve to the data). The pseudo-threshold results are given in \[tab:PseudoThreshFullSteane\]. As can be seen, the full Steane-EC protocol using the circuit of \[fig:SteaneCSSforNonPerfect\] achieves significantly lower logical failure rates compared to Steane-EC using the circuit in \[fig:SteaneCSSforPerfect\] at the cost of using a minimum of 171 qubits compared to a minimum of 95 qubits. In contrast, the flag 2-FTEC scheme of \[subsec:Distance5protocol\] has a pseudo-threshold that is one to two orders of magnitude lower than than the full Steane-EC scheme but requires only 22 qubits. Implementation of Surface code error correction {#app:SurfaceECSection} =============================================== [0.2]{} ![(a) The $d=3$ surface code, with data qubits represented by white circles. The $X$ ($Z$) stabilizer generators are measured with measurement ancillas (gray) in red (green) faces (b) For perfectg measurements, the graph $G_{\text{2D}}$ used to correct $X$ type errors (here for $d=5$) consists of a black node for each $Z$-stabilizer, and a black edge for each data qubit in the surface code. White boundary nodes and blue boundary edges are added. Black and blue edges are given weight one and zero respectively. In this example, a two qubit $X$ error has occurred causing three stabilizers to be violated (red nodes). A boundary node is also highlighted and a minimum weight correction (red edges) which terminates on highlighted nodes is found. The algorithm succeeds as the error plus correction is a stabilizer.[]{data-label="fig:Surface17Circuits"}](SurfaceCodeLattice.png "fig:"){width="\textwidth"} [0.2]{} ![(a) The $d=3$ surface code, with data qubits represented by white circles. The $X$ ($Z$) stabilizer generators are measured with measurement ancillas (gray) in red (green) faces (b) For perfectg measurements, the graph $G_{\text{2D}}$ used to correct $X$ type errors (here for $d=5$) consists of a black node for each $Z$-stabilizer, and a black edge for each data qubit in the surface code. White boundary nodes and blue boundary edges are added. Black and blue edges are given weight one and zero respectively. In this example, a two qubit $X$ error has occurred causing three stabilizers to be violated (red nodes). A boundary node is also highlighted and a minimum weight correction (red edges) which terminates on highlighted nodes is found. The algorithm succeeds as the error plus correction is a stabilizer.[]{data-label="fig:Surface17Circuits"}](Graphd5Planar.png "fig:"){width="\textwidth"} We consider the rotated surface code [@KITAEV97Surface; @TS14; @BK98; @DKLP02; @FMMC12; @PhysRevLett.90.016803] as shown in Fig. \[fig:Surface17Lattice\], which has $n=d^2$ data qubits for distance $d$. Although we are concerned with error correction under the circuit level noise model described in \[subsec:NoiseAndNumerics\], it is useful to build intuition by first considering the idealized noise model in which stabilizer measurements are perfect, and single qubit $X$ errors occur with probability $2p/3$ ($Z$ errors can be treated in the same way). An $X$ type error $E$ occurs with probability $\mathcal{O}(p^{\text{wt}(E)})$, and has syndrome $s(E)$. The minimum weight $X$-type correction can be found efficiently for the surface code in terms of the graph $G_{\text{2D}}$ shown in Fig. \[fig:Surfaced5Graph\]. The graph $G_{\text{2D}}$ has a bulk node (black circle) for each $Z$ stabilizer generator, and a bulk edge (black) for each data qubit. A bulk edge coming from a bulk node corresponds to the edge’s data qubit being in the support of the node’s stabilizer. The graph also contains boundary nodes (white boxes) and boundary edges (blue), which do not correspond to stabilizers or data qubits. Each bulk and boundary edge is assigned weight one and zero respectively. The minimum weight decoder is then implemented as follows. After the error $E$ is applied, the nodes corresponding to unsatisfied stabilizers are highlighted. If an odd number of stabilizers was unsatisfied, one of the boundary nodes is also highlighted. Highlighted nodes are then efficiently paired together by the minimum weight connections in the graph, by Edmonds’ algorithm [@Edmonds65; @Kolmogorov09]. The correction $C$ is applied to the edges in the connection. Note that any single $\mathcal{O}(p)$ fault in this noise model corresponds to a weight one edge on the graph. [0.3]{} ![Circuits for measuring (a) $Z$-type, and (b) $X$-type generators. Identity gates (black rectangles) are inserted in the $Z$-type stabilizer measurement circuits to ensure that all measurements are synchronized. Note that unlike in [@FMMC12], to be consistent with the other schemes in this paper, we assume that we can prepare and measure in both the $X$ and $Z$ basis.[]{data-label="fig:Surface17Circuits"}](ZstabMeasSurfaceCode.png "fig:"){width="\textwidth"} [0.3]{} ![Circuits for measuring (a) $Z$-type, and (b) $X$-type generators. Identity gates (black rectangles) are inserted in the $Z$-type stabilizer measurement circuits to ensure that all measurements are synchronized. Note that unlike in [@FMMC12], to be consistent with the other schemes in this paper, we assume that we can prepare and measure in both the $X$ and $Z$ basis.[]{data-label="fig:Surface17Circuits"}](XstabMeasSurfaceCode.png "fig:"){width="\textwidth"} For circuit noise, we introduce a measurement qubit for each stabilizer generator, as represented by gray circles in Fig. \[fig:Surface17Lattice\], and circuits must be specified to implement the measurements, such as those in Fig. \[fig:Surface17Circuits\]. The performance of the code is sensitive to the choice of circuit [@TS14], for example a poor choice could allow a single fault to cause a logical failure for $d=3$ for any choice of decoder. To implement the decoder, first construct a new three dimensional graph $G_{\text{3D}}$ by stacking $d$ copies of the planar graph $G_{\text{2D}}$ that was shown in Fig. \[fig:Surfaced5Graph\], and adding new bulk (boudnary) edges to connect bulk (boudnary) nodes in neighboring layers. We also add additional diagonal edges such that any single $\mathcal{O}(p)$ fault in the measurement circuits corresponds to a weight-one edge in $G_{\text{3D}}$ (see Fig. \[fig:LowerRightAndLowerLeft\]). For simplicity, we do not involve further possible optimizations such as setting edge weights based on precise probabilities and including $X$-$Z$ correlations [@PhysRevA.83.020302]. [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge1.png "fig:"){width="\textwidth"} [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge2.png "fig:"){width="\textwidth"} [0.11]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge3.png "fig:"){width="\textwidth"} [0.15]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge5.png "fig:"){width="\textwidth"} [0.15]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge6.png "fig:"){width="\textwidth"} [0.25]{} ![Examples of a single fault leading to diagonal edges in $G_{\text{3D}}$. Dark arrows represent the CNOT sequence. (a) An $X$ error occurs during the third time step in the CNOT gate acting on the central data qubit. (b) During the fifth time step of this round, the $X$ error is detected by the $Z$ type measurement qubit to the top right. (c) The $X$ error is not detected by the bottom left $Z$ type stabilizer until the following round. (d) An $XX$ error occurs on the third CNOT of an $X$ measurement circuit, which is detected by the $Z$ measurement to the right. (e) Detection by the left $Z$ stabilizer does not occur until the next round. (f) The corresponding edges in $G_{\text{3D}}$, green for (a-c), and blue for (d-e). Here we show two rounds of the graph ignoring boundary edges.[]{data-label="fig:LowerRightAndLowerLeft"}](DiagonalEdge4.png "fig:"){width="\textwidth"} All simulations of the surface code are performed using the circuit noise model in \[subsec:NoiseAndNumerics\], with the graph $G_{\text{3D}}$ described above as follows (to correct $X$ errors): 1. : Stabilizer outcomes are stored over $d$ rounds of noisy error correction, followed by one round of perfect error correction. The net error $E$ applied over all $d$ rounds is recorded. 2. : Nodes in the graph $G_{\text{3D}}$ are highlighted if the corresponding $Z$-type stabilizer outcome changes in two consecutive rounds. [^5] 3. : Find a minimal edge set forming paths that terminate on highlighted nodes. Highlight the edge set. 4. : The highlighted edges in $G_{\text{3D}}$ are mapped edges in the planar graph $G_{\text{2D}}$, and are then added modulo 2. 5. : The $X$-type correction $C_X$ is applied to highlighted edges in $G_{\text{2D}}$. The $Z$ correction $C_Z$ is found analogously. Finally, if the residual Pauli $R = E C_X C_Z$ is a logical operator, we say the protocol succeeded, otherwise we say it failed. FTEC scheme Noise model Number of qubits Time steps ($T_{\mathrm{time}}$) Pseudo-threshold ----------------------------- ----------------- ------------------ ------------------------------------ -------------------------------------------------------- Flag-EC [$[\![7,1,3]\!]$]{} $\tilde{p} = p$ 9 $36 \le T_{\mathrm{time}} \le108$ $p_{\mathrm{pseudo}} = (3.39 \pm 0.10) \times 10^{-5}$ Flag-EC [$[\![7,1,3]\!]$]{} 11 $34 \le T_{\mathrm{time}} \le 104$ $p_{\mathrm{pseudo}} = (2.97 \pm 0.01) \times 10^{-5}$ Compact implementation of flag error correction {#app:CompactRepFlagQubit} =============================================== [$[\![5,1,3]\!]$]{} [$[\![7,1,3]\!]$]{} [$[\![19,1,5]\!]$]{} code [$[\![17,1,5]\!]$]{} code -------------------------------------------------------------------- ------------------------------------------------------------------- --------------------------------------------------------------------- ----------------------------------------------------------------- $X_{1}Z_{2}Z_{3}X_{4}$ $Z_{4}Z_{5}Z_{6}Z_{7}$ $Z_{1}Z_{2}Z_{3}Z_{4}$,  $X_{1}X_{2}X_{3}X_{4}$ $Z_{1}Z_{2}Z_{3}Z_{4}$,  $X_{1}X_{2}X_{3}X_{4}$ $X_{2}Z_{3}Z_{4}X_{5}$ $Z_{2}Z_{3}Z_{6}Z_{7}$ $Z_{1}Z_{3}Z_{5}Z_{7}$,  $X_{1}X_{3}X_{5}X_{7}$ $Z_{1}Z_{3}Z_{5}Z_{6}$,   $X_{1}X_{3}X_{5}X_{6}$ $X_{1}X_{3}Z_{4}Z_{5}$ $Z_{1}Z_{3}Z_{5}Z_{7}$ $Z_{12}Z_{13}Z_{14}Z_{15}$,  $X_{12}X_{13}X_{14}X_{15}$ $Z_{5}Z_{6}Z_{9}Z_{10}$,  $X_{5}X_{6}Z_{9}Z_{10}$ $Z_{1}X_{2}X_{4}Z_{5}$ $X_{4}X_{5}X_{6}X_{7}$ $Z_{1}Z_{2}Z_{5}Z_{6}Z_{8}Z_{9}$,  $X_{1}X_{2}X_{5}X_{6}X_{8}X_{9}$ $Z_{7}Z_{8}Z_{11}Z_{12}$,  $X_{7}X_{8}X_{11}X_{12}$ $X_{2}X_{3}X_{6}X_{7}$ $Z_{6}Z_{9}Z_{16}Z_{19}$,  $X_{6}X_{9}X_{16}X_{19}$ $Z_{9}Z_{10}Z_{13}Z_{14}$,  $X_{9}X_{10}X_{13}X_{14}$ $X_{1}X_{3}X_{5}X_{7}$ $Z_{16}Z_{17}Z_{18}Z_{19}$,  $X_{16}X_{17}X_{18}X_{19}$ $Z_{11}Z_{12}Z_{15}Z_{16}$,  $X_{11}X_{12}X_{15}X_{16}$ $Z_{10}Z_{11}Z_{12}Z_{15}$,   $X_{10}X_{11}X_{12}X_{15}$ $Z_{8}Z_{12}Z_{16}Z_{17}$,  $X_{8}X_{12}X_{16}X_{17}$ $Z_{8}Z_{9}Z_{10}Z_{11}Z_{16}Z_{17}$ $Z_{3}Z_{4}Z_{6}Z_{7}Z_{10}Z_{11}Z_{14}Z_{15}$ $Z_{5}Z_{7}Z_{8}Z_{11}Z_{12}Z_{13}$ $X_{3}X_{4}X_{6}X_{7}X_{10}X_{11}X_{14}X_{15}$ $X_{5}X_{7}X_{8}X_{11}X_{12}X_{13}$ $X_{8}X_{9}X_{10}X_{11}X_{16}X_{17}$ $\overline{X} = X^{\otimes 5}$,$\overline{Z} = Z^{\otimes 5}$ $\overline{X} = X^{\otimes 7}$,$\overline{Z} = Z^{\otimes 7}$ $\overline{X} = X^{\otimes 19}$,$\overline{Z} = Z^{\otimes 19}$ $\overline{X} = X^{\otimes 17}$,$\overline{Z} = Z^{\otimes 17}$ ![Circuit for measuring the $Z$ stabilizer generators of the [$[\![7,1,3]\!]$]{} code using one flag qubit and three measurement qubits. The circuit is constructed such that any single fault at a bad location leading to an error of weight greater than one will cause the circuit to flag. Moreover, any error that occurs when the circuit flags due to a single fault has a unique syndrome.[]{data-label="fig:CompactStabMeasure"}](SteaneCode4Ancilla.png){width="40.00000%"} ![Logical failure rates of the flag 1-FTEC protocols using two and four ancilla qubits applied to the [$[\![7,1,3]\!]$]{} Steane code.[]{data-label="fig:PseudoThreshCompareSteaneDepth"}](CompareCompactFullp.png){width="35.00000%"} In [@CR17v1], it was shown that by using extra ancilla qubits in the flag-EC protocol, it is possible to measure multiple stabilizer generators during one measurement cycle which could reduce the circuit depth. Note that for the Steane code, measuring the $Z$ stabilizers using \[fig:StabFTwithAncilla\] requires only one extra time step. In this section we compare logical failure rates of the [$[\![7,1,3]\!]$]{} code using the flag-EC method of \[subsec:ReviewChaoReichardt\] which requires only two ancilla qubits and a flag-EC method which uses four ancilla qubits but that can measure all $Z$ stabilizer generators in one cycle (see \[fig:CompactStabMeasure\]). All $X$ stabilizers are measured in a separate cycle. Logical failure rates for $\tilde{p}=p$ are shown in \[fig:PseudoThreshCompareSteaneDepth\]. Pseudo-thresholds and the number of time steps required to implement the protocols are given in \[tab:PseudoThreshFullSteane2\]. Note that measuring stabilizers using two ancilla’s requires at most two extra time steps. Furthermore, the extra ancilla’s for measuring multiple stabilizers result in more idle qubit locations compared to using only two ancilla qubits. With the added locations for errors to be introduced, the flag error correction protocol using only two ancilla’s achieves a *higher* pseudo-threshold compared to the protocol using more ancilla’s. Thus assuming that reinitializing qubits can be done without introducing many errors into the system, FTEC using fewer qubits could achieve lower logical failure rates compared to certain schemes using more qubits. Stabilizer generators of various codes. {#app:ListStabGenerator} ======================================= In \[tab:StabilizerGeneratorsLists\] we provide stabilizer generators for the [$[\![5,1,3]\!]$]{} code, [$[\![7,1,3]\!]$]{} Steane code, [$[\![19,1,5]\!]$]{} and [$[\![17,1,5]\!]$]{} color codes. [^1]: This is the case for Shor, Steane and Knill EC with appropriately verified ancilla states. However the surface code does not satisfy this due to hook errors but nonetheless still satisfies condition 1 of \[Def:FaultTolerantDef\]. [^2]: To avoid confusing the notation of $C(P)$ that represents a circuit and $C$ that represents a code space, we always include the measured Pauli in parenthesis unless clear from context. [^3]: $n_{\text{diff}}$ tracks the minimum number of faults that could have caused the observed syndrome outcomes. For example, if the sequence $s_{1},s_{2},s_{1}$ was measured, $n_{\text{diff}}$ would increase by one since a single measurement fault could give rise to the given sequence (for example, this could be caused by a single CNOT failure which resulted in a data qubit and measurement error). However for the sequence $s_{1},s_{2},s_{1},s_{2}$, $n_{\text{diff}}$ would increase by two. [^4]: One could also define update rules analogous to those for $n_{\text{diff}}$ and $n_{\text{same}}$ when implementing Shor-EC which would only require at most $\frac{1}{2}(t^{2}+3t+2)$ syndrome measurement repetitions as in the flag $t$-FTEC protocol. [^5]: For an odd number of highlighted vertices, highlight the boundary vertex.

--- abstract: 'Ablation of Cu and Al targets has been performed with laser pulses in the intensity range of . We compare the measured removal depth with 1D hydrodynamic simulations. The electron-ion temperature decoupling is taken into account using the standard “two-temperature model”. The influence of the early heat transfer by electronic thermal conduction on hydrodynamic material expansion and mechanical behavior is investigated. A good agreement between experimental and numerical matter ablation rates shows the importance of including solid-to-vapor evolution of the metal in the current modeling of the laser matter interaction.' author: - 'J.P. Colombier$^{1,2}$' - 'P. Combis$^{1}$' - 'F. Bonneau$^{1}$' - 'R. Le Harzic$^{2}$' - 'E. Audouard$^{2}$' title: Hydrodynamic simulations of metal ablation by femtosecond laser irradiation --- Introduction ============ Ultrafast phenomena driven by subpicosecond laser pulses have been the subject of thorough investigation for many years in order to explain the ablation of solid materials [@Komashko; @Laville; @Schafer; @Zhigilei]. As opposed to the laser-dielectric interaction where thermal and athermal ablation regimes probably takes place [@Stoian; @Bulgakova; @Feit], the laser-metal interaction is mainly governed by the thermal ablation one [@Gamaly; @Bulgakova]. The laser energy is absorbed by the free electrons first. The pulse duration being shorter than the electron-phonon relaxation time , electrons and ions subsystems must be considered separately. The “two-temperature model” (TTM) describes the thermal transport of energy by the electrons and the energy transfer from the electrons to the lattice [@Anisimov]. Numerous theoretical and experimental previous works have been devoted to the study of the matter ablation with a single laser pulse. Experimental irradiation conditions in current applications are largely investigated to optimize the ablation rate: pulse duration [@Sallé], fluence [@Nolte] and background gas [@Wynne; @Preuss]. However, a complete view including all the relevant physical mechanisms is still lacking. To get a better understanding of the ablation process and to extend the results into situations not covered by the experiments, two kinds of investigations are put at work : (i) a complete identification of the various physical mechanisms responsible for the material removal from the surface, (ii) an evaluation of the impact of these various processes on the amount of ablated matter. In the works previously addressed, few calculations are able to provide a direct comparison with experiments. Most of them are focused on thermal transport and a more detailed description of the physical processes occuring in the material has to be incorporated to really describe the whole ablation process. Among these, works based on hydrodynamic modeling [@Eidmann; @Laville; @Afanasiev; @Komashko] have been recently associated to the TTM to describe ablation process. To overcome the drawbacks of a material fluid treatment, a mechanical extension of the TTM has been proposed to model the ultrafast thermomelasticity behavior of a metal film [@Chen]. Works based on molecular dynamics allow to access to the influence of the ultrafast energy deposition on the thermal and mechanical evolution properties of the material [@Perez; @Meunier]. With a different approach, other authors have performed a microscopic analysis of the mechanisms of ultrashort pulse laser melting by means of a hybrid molecular-dynamic and fluid modelisation [@Schafer; @Zhigilei]. From all these investigations, it appears that none of these effects may be neglegted to reproduce the features of the damage resulting from an ultrashort laser irradiation. Moreover, there is a lack of investigations which combine experimental and theoretical results so that current models are still questionnable. In the present simulations, the TTM provides energy deposition in an expanding material intimately correlated with the processes governing the ablation in the ultrashort pulse case, which is a specificity of our hydrodynamic approach. Simulation results give useful insight into the presented experiment data. Transport properties of electrons are not very well understood in nonequilibrium electron-ion systems. However, the comprehension of these phenomena in the context of ultrafast interaction is essential not only for fundamental purposes but also for micromachining applications. A precise description of the effect of the electronic temperature on the absorption seems to be still unsettled [@Milchberg; @Fisher; @Rethfeld], and it has not been taken into account in the presented calculations. Nevertheless, the model employed in this work uses a large set of current available data. Obviously, numerical calculations are always requiring additional information such as electron-phonon or electron-electron relaxation times, which may be extracted, from experimental data [@Corkum; @Gusev]. Reciprocally, comparison between simulations and experiments allows to validate physical data introduced into the theoretical models. For instance, we shall see in the following that the measure of the pressure variation with time inside the material would be very informative. These data, however, are difficult to measure with a high accuracy. By contrast, experimental measure of laser ablation rate seems to be easier to be obtained and compared with numerical simulations. In this article, numerical and experimental results on ablated matter are reported. For this purpose, the TTM is inserted into a hydrodynamic code in order to describe the material removal. First we detail the physical processes which are taken into account in our computations within the framework of hydrodynamical modeling. Then, the effects caused by relaxation processes on the evolution of shock waves are examined. We next present the experiments which have been performed to obtain ablation depth measurements as a function of the laser fluence. Finally, our discussions are based on results of numerical simulations on Cu and Al samples compared with specific experimental measurements. Theoretical model ================= To represent numerically the interaction between the laser and the metallic target, we used a 1D Lagrangian hydrodynamic code [@Delpor]. Solving the Helmholtz wave equation permitted us to determine the electromagnetic field through the region illuminated by the laser. The deposited energy is then deduced using the Joule-Lenz law. A precise result for the absorption from an arbitrary medium can be obtained from the direct solution of the equation for the electromagnetic field. Let us consider a planar wave propagating along z axis. We write the following Helmholtz equation for the complex amplitude of the electric field $E_{z}$ with frequency $\omega$ : $$\label{three} \Delta E_{z}-\nabla\nabla E_{z}+\left(1+i\frac{4\pi}{\omega}\sigma\left(T,\rho\right)\right)\frac{\omega^{2}}{c^{2}}E_{z}=0$$ where $\sigma(T,\rho)$ is the complex conductivity and $c$ is the light speed. The function $\sigma$ is calculated as a function of the density $\rho(z)$ and the temperature $T(z)$. The relative permittivity of the medium is supposed to be equal to the unity in the case of a metal target. These simulations need accurate data such as transport coefficients in solids, liquids, vapors and dense or diluted plasmas, specially refractive indices [@Palik], electric and thermal conductivities [@Ebeling], and mechanical properties such as material strength. The evolution of the irradiated target is governed by the fluid hydrodynamic and heat diffusion equations connected with a multi-phase equation of state (EOS). Thermodynamic functions that realistically describe characteristics of metal properties in various parts of phase diagram are needed. A such set of different metal EOS is generated by means of a numerical tool developped by the authors of the reference [@EOS]. As an illustration of the EOS used, displays isobars in the phase diagram temperature-specific energy of the copper for a wide range of pressure. Such data reveal the dependence on the thermodynamic properties of the melting and vaporisation processes. The mechanical propagation of shock waves and fracture are also simulated. Elastoplasticity is described by a strain rate independent model (relevant to high strain rate conditions at high pressure, typically beyond ) which accounts for pressure, temperature and strain dependent yield strength and shear modulus [@SCG]. Laser induced stresses are the combination of the hydrostatic pressure and the response to the shearing deformation. In the temperature range of interest here, the effect of radiative energy transport on the hydrodynamic motion is negligible. Hence we ignore energy transport by radiation when solving the hydrodynamic equations. ![Representation of isobars in a phase diagram of the copper EOS in the region of phase transitions.\[EOS\_cu\]](EOS_copper.EPS){width="8.5cm"} Ultrashort laser irradiation and the associated ionic and electronic temperature decoupling require to introduce specific electronic parameters. We assume that free electrons form a thermal distribution during the interaction and use the Fermi-Dirac distribution to determine the electron properties (energy, pressure and heat capacity) as a function of the density and the temperature [@Ashcroft]. The evanescent electromagnetic wave is absorbed by the electrons. In our range of intensity, they are quickly heated at a temperature of few eV. During and after the pulse, the energy diffuses among the electrons and is then transferred to the lattice. Classical heat diffusion plays a significant role as soon as a thermal gradient occurs in the material. Diffusion processes are described by the following equations : $$\begin{aligned} \label{one} \rho \ C_{e} \displaystyle\frac{\partial T_{e}}{\partial t}&=& \displaystyle\nabla \displaystyle\left(K_{e}\displaystyle\nabla T_{e}\displaystyle\right)- \gamma (T_{e}-T_{i}) + S \hspace{1cm} \; \\ \label{two} \rho \ C_{i} \displaystyle\frac{\partial T_{i}}{\partial t}&=& \displaystyle\nabla \displaystyle\left( K_{i} \displaystyle\nabla T_{i}\displaystyle\right)+ \gamma (T_{e}-T_{i})\end{aligned}$$ where $T$, $C$ and $K$ are the temperature, the specific heat and the thermal conductivity respectively. Indices “e” and “i” stand for electron and ion species. $\rho$ is the mass density. $\gamma$ characterizes the rate of energy exchange between the two subsystems and $S$ is the space and time dependent laser source term determined by the Joule-Lenz law. Introduction of the TTM in a hydro-code allows us to take into account the density dependence of both specific heats and conductivities. $C_{e}(T_{e},\rho)$ is calculated with the Fermi model using $\rho$ dependent chemical potentials [@Ashcroft]. The electron thermal conductivity $K_{e}$ is commonly expressed in the form [@Anis-Reth]: $$\label{three}K_{e} = \alpha \frac{(\theta_{e}^{2}+0.16)^{5/4}(\theta_{e}^{2}+0.44)} {(\theta_{e}^{2}+0.092)^{1/2}(\theta_{e}^{2}+\beta \theta_{i})} \theta_{e}$$where $\theta_{e}$ and $\theta_{i}$ are electron and ion temperature normalized to the Fermi temperature ($\theta_{e}=T_{e}/T_{F}$, $\theta_{i}=T_{i}/T_{F}$) and $\alpha$, $\beta$ are material dependent parameters [@Schafer; @Wang]. The linear variation of coupling term with ($T_{e}-T_{i}$) is classic in TTM : we have taken $\gamma=\gamma_{0}$ as for copper [@Corkum], and for aluminum [@Fisher]. It must be noticed that the values of $K_{e}$ and $\gamma$ are subject to considerable uncertainty in literature [@Schafer]. To accurately describe the ultrafast response, we incorporate electronic pressure into the set of hydrodynamic equations. The system of equations for electron and ion subsystems can be written in the Lagrangian form : $$\label{four} \begin{cases} \vspace{0.2cm} \displaystyle\frac{\partial\varepsilon_{e}}{\partial t}= -p_{e}\displaystyle\frac{\partial V}{\partial t}, \hspace{1.9cm} \displaystyle\frac{\partial\varepsilon_{i}}{\partial t}= -p_{i}\displaystyle\frac{\partial V}{\partial t}\\ \displaystyle\frac{\partial u}{\partial t}= -\frac{\partial}{\partial m}(p_{e}+p_{i}),\hspace{1cm} \displaystyle\frac{\partial V}{\partial t}=\frac{\partial u}{\partial m} \end{cases}$$ where $u$ is the fluid velocity, $m$ the mass and $V$ the specific volume. $p_{e}$, $\varepsilon_{e}$, $p_{i}$, $\varepsilon_{i}$ are the pressure and the specific energy of electrons and ions. ![Contour plots of the stress resulting from the irradiation of a copper target by a , laser pulse.\[sptime\]](sptime.eps){width="8.5cm"} Equations (\[one\]) to (\[four\]) connected with a multiphase equation of state (EOS) [@EOS] constitute a self-consistent description of the matter evolution. However, the pressure provided by this EOS is the sum of the electronic and ionic pressure at the equilibrium and has to be replaced by the sum of these two contributions out of equilibrium. The electronic pressure is independently determined by means of the standard fermion gas model. As a consequence, the total pressure used in the above calculations is determined as . Simulations and analysis ======================== To start with, the interaction of a , FWHM gaussian pulse at wavelength with a copper target is investigated. shows the space-time evolution of the induced stress. The metal surface is heated to a maximum of , after the irradiation. At this time, the free surface expands in a liquid state with a velocity of . Due to the electron heating, an electronic compression wave appears at the end of the laser pulse. The electron-ion energy exchange results in a significant increase in the ionic pressure, which propagates inside the metal. Behind the shock, tensile stress occurs associated with very high strain rate around . In order to study the sensitivity of the above results with respect to the physical parameters, we compare in the time variation of the computed stress at depth under standard conditions ($\gamma = \gamma_{0}$ and $K_{e} = K_{e_{o}}$) with those obtained with an increased coupling factor or electronic conductivity. ![Time evolution of the stress in copper at depth resulting from a , laser pulse. Standard conditions :  and (solid line), (dashed line), (dotted line).\[stress\]](stress.eps){width="8.5cm"} In the first hundred picoseconds, the stress growth is directly related to the heating depth. The increased coupling factor leads to a steeper thermal gradient and a lower temperature at compared to the other situations. The resulting stress is therefore lower in this case. The peak of the shock wave, propagating from the surface, reaches the depth at . Increasing the coupling factor accelerates the energy transfer from the electrons to the lattice and results in higher compression. Inversely, an enhanced electronic conductivity spreads the energy spatial profile and yields reduced stresses. It must be noticed that in the three cases, the compression is followed by a high tensile stress greater than the characteristic tensile strength of the material. Nevertheless, the loading in tension is not long enough () to induce a fracture in the medium [@Tuler-Butcher; @strength]. The shock duration and intensity provide a good signature of the balance between the electronic diffusion and the electron-ion coupling. Further improvements will be discussed in the following. To obtain local information on the energy transfer induced by a femtosecond laser irradiation, a high-resolution time measurement of the stress reaching the rear side of a thin sample could be achieved [@Tollier; @Romain]. Such experimental records could be directly compared with our simulations which would led us to refine the physical values accordingly. To validate the computation, we performed ablation measurements and compared them to the current simulations using standard values of for aluminum and copper samples. Experiment details ================== Ultrashort laser pulses are generated by an amplified all solid-state Ti:Saphire laser chain. Low energy pulses are extracted from a mode-locked oscillator (, , , ). The pulses are then injected into an amplifying chain including : an optical pulse stretcher, a regenerative amplifier associated with a two-pass amplifier using a as pumping source, and a pulse compressor. Linearly polarized pulses with wavelength centered around 800 nm, an energy of at repetition rate and typical duration of were delivered. The samples are mounted on a three-motorized-axes system with 0.5 $\mu$m accuracy. Experiments are performed in the image plane of an aperture placed before the objective. The resulting spatial laser profile on sample is “top hat” so that borders and spurious conical drilling effects are reduced. Focusing objectives of or focal lengths to obtain fluences in a range of 0.5 to 35 J/cm$^{2}$ with the same beam size, 16 $\mu$m in diameter, in the image plane. For ablation depth measurements, grooves are machined by moving the sample [@these]. The sample speed is adapted such that each point on the groove axis undergoes 8 consecutive irradiations at each target pass. The number of times the sample passes in front of the fixed beam can be adjusted. shows a scanning electron microscopy (SEM) picture of the machined grooves on copper for 2, 4, 6, 8 and 10 passes. ![SEM picture of machined-groove profiles, from 2 to 10 passes, on a copper sample.\[profilo\]](profilo.eps){width="8.5cm"} The groove depth is measured using an optical profilometer with a depth resolution. The ablation rates for each groove are deduced taking into account the sample speed, the repetition rate of the laser, the beam size and the number of passes. For each energy density, an averaged ablation rate is determined and the number of passes has been chosen to obtain reproducible results. From these experiments on copper and aluminum targets, we evaluate, for different fluences, an ablation depth averaged over a few tens of laser shots. The theoretical ablation depth is deduced from 1D numerical simulations using a criterion discussed hereafter. Results and discussion ====================== At moderate fluence, the propagation of the shock wave induced by the energy deposition on the lattice causes the surface expansion at very high speed and significant non-uniform strain rates. Simultaneously, the surface of the target is melted or vaporized as soon as the conditions of temperature and density required are fulfilled. High strain rates can turn the liquid region into an ensemble of droplets and ablation follows. This process is called the homogeneous nucleation [@nucleation]. Unfortunately, quantitative values on the formation and ejection of liquid droplets are difficult to access because the interfacial solid-liquid microscopic properties of the nucleation centers are not accurately known. Nevertheless, in our simulation we can consider that the liquid layer accelerated outside the target corresponds to the ablated matter. Large values of strain rates () indeed signal that droplet formation may occur and are sufficient to produce ablation. At higher fluence, the surface is strongly vaporized. The gas expands near the free surface and compresses the internal matter. The vaporized part of the target added to the fraction of the above-defined liquid layer constitute the calculated ablated matter. Experimental results on ablation of copper and aluminum are compared with the numerical estimates in . ![Experimental and numerical (solid line) ablation depth as a function of the laser fluence on a copper target obtained with a 170 fs laser pulse. The dashed line shows evolution of the specific removal rate.\[cu\]](copper.eps){width="8.5cm"} ![Experimental and numerical (solid line) ablation depth as a function of the laser fluence on an aluminum target obtained with a 170 fs laser pulse. The dashed line shows evolution of the specific removal rate.\[alu\]](alu.eps){width="8.5cm"} Sharp numerical ablation thresholds are visible at 3 and for Cu and Al targets respectively. At high fluence, the ablation saturates for both materials. This saturation occurs mainly for two reasons. Vaporization and subsequent gas expansion consume energy that does not contribute to the ablation process. Moreover, the liquid layer confinement increases as far as the gas expands and limits the liquid removal. As defined by Feit *et al* [@Feit], a specific removal depth, i.e., depth removed per unit incident fluence could be a relevant parameter to estimate the ablation efficiency at a fixed pulse duration. Calculations of this quantity is carried out as a function of the laser fluence. Dashed curve presented in indicate a maximum value of in the aluminum case. The curve is smoother for copper in and the maximum specific removal depth is reached at a fluence around . This point corresponds to the occurence of a critical behavior which confirms a change in the hydrodynamic behavior. While the thickness of the liquid layer grows with the incident energy, the specific removal depth rises. Afterwards, when the gas expansion starts, a growing part of the laser energy is transformed in kinetic energy and the specific removal depth drops. This suggests that an optimum material removal exists and refers to the situation when the surface vaporisation is limited. It appears that this quantity is relevant for material processing which can be optimized by operating at this optimal fluence. At low fluence, a noticeable discrepancy arises between the experimental and numerical results. The calculated ablated matter for a copper target is below the experimental results, while for an aluminum target, numerical results are above. We suspect that electron transport properties should be further improved. It has been shown that a significant decrease in the electrical conductivity may take place as a result of the electronic temperature increase [@Milchberg], which our model discards. However, the experimental measurements and the theoretical calculations come to a reasonable agreement at higher fluence. As it has been shown by Fisher *et al* [@Fisher], in the vicinity of a 800 nm wavelength, the interband absorption occuring in an aluminum target decreases with increasing temperature. The authors show that, with 50 fs laser pulses, the absorbtion coefficient presents a minimum near ablation threshold, at laser intensity. The evolution of interband absorption with the temperature is not taken into account in our calculations. Consequently, we may overestimate the energy absorbed in this intensity region. For copper, the simulation overestimates the ablation threshold. This can be due to a deficit of physical process comprehension or to the inaccuracy of the parameters introduced in the model. No single value of $\gamma$ or $K_{e}$ can perfectly fit the sets of data shown in Fig. \[cu\] and \[alu\]. As for the discussion of pressure presented above, one can think that a change in $\gamma$ or $K_{e}$ has a similar effect on the threshold fluence $F_{Th}$. Therefore, it is interesting to investigate the fluence threshold dependence on $\gamma$ and $K_{e}$. A parametric analysis has been performed for a copper target. displays the threshold fluence which has been obtained for different parameter couples and . The deposited energy with lower thermal conductivity or higher coupling factor can penetrate deeper into the material. As a consequence, $F_{Th}$ is lowered with respect to that of the reference case and would be comparable with experimental data in the vicinity of the threshold. However, simulations performed in these conditions have shown a higher disagreement with experimental data at higher fluence due to an earlier expanded vapor. On the contrary, for a reduced $\gamma$ or an enhanced $K_{e}$ value, $F_{Th}$ increased and the ablation depth at high fluence is overestimated. Results obtained with one-temperature simulations evidence the importance of the TTM to reproduce the experimental results. The good agreement obtained between experimental data and simulations underlines the need of coupling the TTM model with a hydrodynamic code for ablation simulations in metals. Numerical results presented in this paper give an overall description of processes occuring during ultrashort laser ablation experiments. ![Calculation of the threshold fluence dependence on the coupling parameter $\gamma$ and the electronic conductivity $K_{e}$ for copper. Simulations are performed for different ratio of the standard value of one parameter remaining the second one constant. Consequently, the intersection of curves coincide with the value used in the calculation of ablation rate presented in .\[seuil\]](seuil.eps){width="8.5cm"} Conclusion ========== In this paper, we have reported new results on the interaction of femtosecond laser pulses with metal targets at intensities up to $10^{14}$ W/cm$^{2}$. Numerical computations were carried out by means of a 1D hydrodynamic code describing the laser field absorption and the subsequent phase transitions of matter. Simulations were compared to original ablation experiments performed on aluminum and copper samples. The behavior of the ablation depth as a function of laser fluence confirms the importance of the use of specific two-temperature equation of state and hydrodynamics. An optimum condition for laser fluence has been identified and could provide a precious information for efficient material processing. We have highlighted that ablation process is not only governed by electronic diffusion but also by the high shock formation and propagation. The differences between experimental and numerical results remain, however, more pronounced for low laser fluences. We took a great care to extend the metal properties to the nonequilibrium case. 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--- author: - Olli Taanila title: | Holographic thermalization\ and Oppenheimer-Snyder collapse --- NIKHEF-12\ HIP-2015-22/TH Introduction ============ Holographic methods have been well established in the recent years as a serious tool to understand the dynamics and thermalization of strongly coupled systems. The AdS/CFT duality [@Maldacena:1997re; @Witten:1998qj; @Gubser:1998bc] maps the complex process of thermalization of a strongly coupled $\mathcal{N}=4$ SYM field theory in four dimensions into the relatively simple process of gravitational collapse in 5D gravity. The use of the duality has spawned an abundance of interesting work attempting to gain insight to different aspects of thermalization (e.g. see references 8-37 in [@Keranen:2015fqa]). Work that has attempted to describe states close to those in heavy ion collisions has been performed in e.g. [@Chesler:2010bi; @vanderSchee:2013pia; @Casalderrey-Solana:2013sxa; @Wu:2011yd; @Chesler:2015wra], in the sense that the initial state and the distribution of the energy-momentum in the boundary field theory is close to the one in heavy ion collisions. Although certain simple quantities, such as the aforementioned energy-momentum tensor, are easily computed in these numerical works, the computation of other observables can be very difficult due to the numerical nature of the work. For computing other observables analytical models of the thermalization process are thus favoured. The general theme of these analytical models is then that although their symmetries prohibit them describing the initial state of a heavy ion collision accurately, they are simple enough that the gravitational dynamics can be understood analytically and computations involving different observables can be performed in these backgrounds. The simplest analytical model of gravitational collapse used in holography is the thin-shell model [@Danielsson:1999fa; @Danielsson:1999zt]. It consists of an infinitely thin shell of matter with some equation of state collapsing into a black hole. A variation of this simple model is the Vaidya metric, where the shell is null (see [@Hubeny:2013dea] and references therein). The relative simplicity of the thin-shell model enables the computation of different quantities such as higher correlations functions of the stress tensor [@Lin:2008rw], photon and dilepton production rates [@Baier:2012ax; @Baier:2012tc], the two-point function in the geodesic approximation [@Balasubramanian:2010ce; @Balasubramanian:2011ur], and different kinds of entropies [@Keranen:2014zoa; @Keranen:2015fqa; @Baron:2012fv; @Baron:2013cya] just to name a few. The initial state described by the thin-shell collapse however has a peculiar feature: Due to the translational symmetry of the setup, all the one-point functions in the boundary field theory have their equilibrium value throughout the evolution. This is in stark contrast with the initial state of heavy ion collisions, where the initial state is described by a very inhomogeneous distribution of matter. Thus the thin-shell model cannot describe perhaps the most important observable, the thermalization of the energy-momentum tensor. Thus it is clear that the thin-shell model cannot describe in detail heavy ion collisions or other processes encountered in the laboratory setting. Instead, we hope that by studying these analytical models, we can learn something qualitative about the dynamics of the strongly coupled system. When we compute observables, we hope that their features are generic for thermalization processes, and can be applied to other scenarios as well. It is however not clear which features of the predictions of the thin-shell model are generic and which very specific to that setup. For example, the thin-shell model has an energy-momentum tensor that is zero throughout in the bulk except for a delta function contribution of the shell, and, in fact, certain observables such as the photon and dilepton production rates hves been observed to have oscillatory features [@Baier:2012ax; @Baier:2012tc] for which it is unclear whether they are generic signatures of thermalization or whether they are caused by the rather singular nature of the shell. For the forementioned reasons it is extremely important to employ different analytical models for holographic thermalization. Ideally, we could compute observables in different scenarios and thus find out which predictions seem generic patterns of thermalization and which features are specific to each model. An alternative to the thin-shell model is the Oppenheimer-Snyder (OS) collapse. This describes the gravitational collapse of a homogeneous and isotropic ball of dust initially at rest. This model was first published already in 1939 [@Oppenheimer:1939ue] as a model of star formation in 3+1 dimensional asymptotically Minkowski space-time. In the context of AdS/CFT it was first mentioned in the appendix of [@Giddings:2001ii], but aside from the equations of motion of the collapse it was not developed further there. The advantage of the Oppenheimer-Snyder collapse in the context of holography is that since it describes a ball of dust instead of an infinitely thin shell of matter, the energy-momentum tensor does not have a delta function, but rather is finite everywhere. Thus the resulting space-time is more regular at the edge of the ball than at the location of an infinitely thin shell. The model is still symmetric enough that for example the equations of motion of a spectator field in the bulk can solved at least for some cases. It is the study of the Oppenheimer-Snyder collapse as a model of holographic thermalization that this work is devoted to. We will first introduce the model and solve its gravitational evolution. Then as a test observable, we compute the equal time two-point function in the geodesic approximation. To compare the OS setup with the thin-shell model, we perform the same computations in the thin-shell model. As a by-product of the computation, we discover previously unknown oscillatory solutions of the thin-shell model in global AdS. The paper is organized as follows: In section \[sec:OS\] we introduce the Oppenheimer-Snyder model in asymptotically AdS space-times, and discuss its dynamics. In section \[sec:ts\] we review the thin-shell model and review the dynamics of the collapse. In section \[sec:correlator\] we compute the equal time two-point correlator in the geodesic approximation in both of the models and compare the results. Finally in section \[sec:conclusions\] we conclude and discuss possible developments. Asymptotically Anti-de Sitter Oppenheimer-Snyder collapse {#sec:OS} ========================================================= The Oppenheimer-Snyder collapse is conceptually the simplest model of gravitational collapse: A homogeneous and isotropic ball of pressureless dust undergoing gravitational collapse. Since we are looking for spherically symmetric solutions, we use global coordinates for AdS, where constant-$r$ hypersurfaces have explicit rotational symmetry. Outside the ball the energy-momentum content is zero. The solution with zero $T_{\mu\nu}$, negative cosmological constant and rotational symmetry is given by the AdS-Schwarzschild solution: $$\label{eq:outsidemetric} ds^2 = -f(r) \, dt^2 + \frac{dr^2}{f(r)}+r^2\,d\Omega_n^2 \; , \quad f(r) = 1 + \Lambda\,r^2 -\frac{m}{r^{n-1}} \; ,$$ where $d\Omega_n^2$ is the metric of a $n$-sphere and $\Lambda$ is the cosmological constant. In this coordinate system the edge of the sphere has some time dependent value of the radial coordinate $r_s$. Thus this coordinate patch is valid for $r> r_s$. Inside the ball of dust the simplest coordinate frame to use is the rest frame of the dust. Since the dust is homogeneous and isotropic, we want a solution which has only time-dependent energy-momentum content, negative cosmological constant and rotational symmetry. This metric is of course the hyperbolic Friedmann-Robertson-Walker metric, $$ds^2 = -d\eta^2 + a(\eta)^2 \left[ \frac{d\rho^2}{1+\rho^2} + \rho^2 d\Omega_n^2\right] \; .$$ Imposing the Einstein equation with a negative cosmological constant and a pressureless fluid at rest and then using the energy-momentum continuity for the fluid, we arrive at the Friedmann equation $$\label{eq:friedmann} \left(\frac{da}{d\eta}\right)^2 = 1 + \frac{2}{n(n+1){M_\mathrm{Pl}}^2}\epsilon_0\frac{a_0^{n+1}}{a^{n-1}}-\Lambda a^2 \; ,$$ where $\epsilon_0$ is the energy density of the dust when the scale factor has the value $a_0$. Since this is in the dust’s rest frame, and its total quantity is conserved, the edge of the ball sits at a constant value of the radial coordinate $\rho$, which we shall call $\rho_0$. Thus the validity of this coordinate patch is $0 < \rho < \rho_0$. In order to glue these two coordinate patches together, we relate the two different patches together at the edge of the sphere and then require that the induced metric there is the same in both of the coordinate systems and that the extrinsic curvature is continuous across the edge of the sphere. These conditions are the Israel junction conditions [@Israel:1966rt] with zero energy-momentum content at the hypersurface joining the two patches. Although this might seem like heavy machinery for the problem at hand, we use it since it is applicable later on when we consider the thin-shell case as a comparison to the OS model. The coordinates describing the edge of the sphere are chosen to be the angular coordinates, which are identified to be the same inside and outside, and the proper time at the edge, $\tau$. From the induced metric we can see that we can identify the time coordinate $\eta$ at the edge of the sphere with the proper time at the edge, i.e. $\eta_s = \tau$. From that it follows that $r_s(\tau) = \rho_0 a(\tau)$. Additionally the time and radial coordinates $t_s$ and $r_s$ are related by $$-1 = -f(r_s)\,\dot{t}_s^2 + \dot{r}_s^2/f(r_s) \; .$$ Next we use the Israel junction conditions to relate the curvature on both sides. The junction conditions state that the difference of the curvature[^1] $$K_{ij} = n_\alpha \left( \frac{\partial^2 X^\alpha}{\partial\xi^i\,\partial\xi^j}+\Gamma^\alpha_{\beta\gamma}\frac{\partial X^\beta}{\partial\xi^i}\frac{\partial X^\gamma}{\partial\xi^j} \right)$$ across the joining hypersurface is related to the energy-momentum content at the hypersurface, which in this case is zero. Here $X^\alpha$ are the $n+2$-dimensional coordinates of the $n+1$-dimensional hypersurface parametrized by the $n+1$ coordinates called $\xi^i$, and $n^\alpha$ is the normal vector of the hypersurface. $\gamma_{ij}$ is the induced metric on the surface. Since the energy-momentum content is zero, the extrinsic curvatures inside and outisde are related by $$\left(K_{ij} - K\,\gamma_{ij}\right)_\mathrm{inside} = \left(K_{ij} - K\,\gamma_{ij}\right)_\mathrm{outside} \; .$$ A straightforward computation gives the non-zero components of the above expression in the inside coordinate system to be $$\begin{aligned} K_{\tau\tau} -K\,\gamma_{\tau\tau} &= -n \frac{\sqrt{1+\rho_0^2}}{\rho_0 a(\tau)}\\ K_{\phi_n \phi_n} -K\,\gamma_{\phi_n \phi_n} & = g_{\phi_n \phi_n} (n-1) \frac{\sqrt{1+\rho_0^2}}{\rho_0 a(\tau)} \; ,\end{aligned}$$ while in the outside coordinate system the non-zero components are given by $$\begin{aligned} \label{eq:Koutside1} K_{\tau\tau} -K\,\gamma_{\tau\tau} &= -n\frac{\sqrt{f(r_s)+\dot{r}_s^2}}{r_s}\\ \label{eq:Koutside2} K_{\phi_n \phi_n} -K\,\gamma_{\phi_n \phi_n} & = g_{\phi_n \phi_n} \frac{1}{r_s\,\dot{r}_s}\frac{d}{d\tau} \left( r_s^{n-1}\sqrt{f(r_s)+\dot{r}_s^2}\right)\; .\end{aligned}$$ Equating the expressions of the inside and outside coordinate system, we get the simple identity $$\label{eq:OSeom} f(r_s) + \dot{r}_s^2 = 1 + \rho_0^2 \; .$$ Since $\rho_0$ is a constant, this is an equation of motion for the radial position of the shell in the outside coordinate system, $r_s$. Substituting $r_s = \rho_0 a(\tau)$ into this equation, we get an equation of motion for the scale factor $a$. If one substitutes the explicit functional form of $f$, this can be seen to be exactly the Friedmann equation, already given in equation (\[eq:friedmann\]). In principle one can proceed with the above derivation with also non-zero pressure of the fluid the sphere is composed of, however, one will then see that a solution satisfying both equations (\[eq:friedmann\]) and (\[eq:OSeom\]) can only be found if $p=0$. This is not surprising, since if the pressure of the fluid would be non-zero, there would be a pressure gradient at the edge of the sphere and the fluid would no longer stay in rest in the inside coordinate system. ![A schematic of the coordinate system $(a,\rho)$ inside the sphere. The grey dashed lines are null geodesics. The grey region is the lower half of the manifold, where the sphere is expanding, however, the $a$-coordinate covers only the upper (or lower, depending on the sign choice) region. Note that there are null rays emanating from the center of the sphere which hit the singularity before encountering the edge of the sphere.[]{data-label="fig:inside"}](inside_diagram.pdf) The motion of the Oppenheimer-Snyder sphere ------------------------------------------- Next we analyse the motion and the causal structure of the Oppenheimer-Snyder collapse. To do that, we use inside the scale factor as the time coordinate, that is, we make the coordinate substitution $da = \frac{da}{d\eta}\, d\eta$, so that the metric becomes $$\label{eq:insidemetric} ds^2 = -\frac{da^2}{h(a)} +\frac{a^2}{1+\rho^2} d\rho^2 +a^2\,\rho^2\,d\Omega_n^2 \; ,$$ where the metric function $h(a) = \left(\frac{da}{d\eta}\right)^2$ comes directly from the Friedmann equation (\[eq:friedmann\]). It is convenient to relate this function to $f$ via $$h(a) = 1+\frac{1-f(r_s)}{\rho_0^2} \; .$$ The time evolution of the diameter of the sphere can be found from equation (\[eq:OSeom\]). The first observation is that since $f(r)$ grows without bound, for any $\rho_0$ there is always a turning point ${r_\mathrm{max}}$, where $\dot{r}_s =0$. Furthermore, since $$\frac{d^2r_s}{d\tau^2} = \frac{1}{2}\frac{d}{dr_s} \dot{r}_s^2 = -f'(r_s) \; ,$$ the acceleration at the turning point is always negative, and at the turning point the sphere starts to collapse. This means that there is also a maximum value of the scale factor, ${a_\mathrm{max}}$, which is the value of the scale factor at the turning point. The space-time also has a space-like singularity at $a=0$, which is the “big crunch” of the space-time. Due to the time reversal symmetry of the setup, in principle the evolution of the sphere is such that it emerges from a singularity and expands until it reaches ${r_\mathrm{max}}$, after which it collapses back to a singularity. Thus the coordinate system $(a,\rho)$ covers only one half of the manifold. We consider only the upper half of the space-time which describes the collapse of the shell starting form the turning point, with the view that the system is initially prepared to the state described by the sphere at the turning point. It is also important to note that while there is some value of the “time coordinate” $a=a_h$ when the horizon is formed, there is no qualitative change inside. In fact, although the value of $r_h$ and thus $a_h$ can be changed by tuning the value of $\rho_0$, the Friedmann equation is indifferent to the choice of that parameter. The thin-shell model {#sec:ts} ==================== The thin-shell model has been discussed widely in the literature [@Baron:2012fv; @Lin:2008rw; @Baier:2012ax; @Baier:2012tc; @Keranen:2014zoa; @Keranen:2015fqa] in the context of holographic thermalization. Another variation on the same theme is the Vaidya metric, where the collapsing shell is null (see e.g. [@Hubeny:2013dea]). However, in most of these works the collapse has been studied in the Poincaré patch of the AdS space. In this coordinate system the shell, which is located at constant value of the $r$-coordinate, is actually an infinite flat sheet with translational symmetry. To enable a comparison with the Oppenheimer-Snyder collapse, we need to use the global coordinate system of the AdS-space, where a constant-$r$-hypersurface is an actual spherical shell with rotational symmetry. It turns out that the spherically symmetric case has interesting behaviour not present in the previously studied case in the Poincaré patch. The notation follows closely that of [@Keranen:2015fqa], so we suggest the reader to refer to that one for more details. The metric of the thin-shell model in global AdS is given by $$\label{eq:thinshellmetric} ds^2 = -f(r)\,dt^2 +\frac{dr^2}{f(r)}+r^2\,d\Omega_n^2$$ where the metric function $f$ is now defined by $$\label{eq:thinshellfpm} f = \begin{cases} f_- = 1+\Lambda r^2\,, & r<r_s\\ f_+ = 1+\Lambda r^2 - \frac{m}{r^{n-1}}\,, & r>r_s \end{cases}$$ and the location of the shell $r_s$ is time dependent. The radial coordinate $r$ and the angular coordinates are continuous at the shell, but the time coordinate is discontinuous, and thus there are two different time coordinates, $t_-$ and $t_+$, the time inside and outside of the shell respectively. To parametrize the shell, we use the angular coordinates and the proper time of the shell, $\tau$, and use an overdot to denote a derivative with respect to $\tau$. Unlike in the OS collapse, now both the inside and outside solutions are vacuum solutions, and all the energy-momentum content is located at the shell. This means that the Israel junction conditions are now $$\label{eq:thinshellisrael} \left(K_{ij}- K \,\gamma_{ij} \right)_\mathrm{inside} - \left(K_{ij}- K \,\gamma_{ij} \right)_\mathrm{outside} = -8\pi g_5 S_{ij} \; ,$$ where $S_{ij}$ is now the energy-momentum tensor of the shell in the $\xi$-coordinate system. We use the ideal fluid form, $$S^{ij} = (\rho+p)u^i u^j + p\gamma^{ij} \;, \quad p = c\rho \; ,$$ where $u^i = \delta^i_0$, since the fluid is at rest in the $\xi$-coordinate system. The expressions from equations (\[eq:Koutside1\]) and (\[eq:Koutside2\]) now apply both inside and outside, $f_-$ and $f_+$ substituted for $f$ respectively. Plugging these into the junction conditions of equation (\[eq:thinshellisrael\]), we get two independent equations: $$\begin{aligned} -\frac{n}{r_s} \left( \sqrt{f_-(r_s)+\dot{r}_s^2}-\sqrt{f_+(r_s)+\dot{r}_s^2}\right) & = -8\pi g_5 \rho\\ g_{\phi_n\phi_n}\frac{1}{\dot{r}_s\,r_s}\frac{d}{d\tau} \left[ r_s^{n-1}\left( \sqrt{f_-(r_s)+\dot{r}_s^2}-\sqrt{f_+(r_s)+\dot{r}_s^2}\right) \right] & = -8\pi g_5 g_{\phi_n\phi_n} p\end{aligned}$$ These equations hold for arbitrary relation between $\rho$ and $p$. We however assume a constant equation of state $c$, and thus the above equations are solved by $$\label{eq:thinshelleom} \sqrt{f_-+\dot{r}_s^2}-\sqrt{f_++\dot{r}_s^2} = M\,r_s^{1-n(1+c)}\;,$$ where $M$ is now an integration constant characterizing the energy of the shell. From this we can also solve the equation of motion for the shell, $$\label{eq:rsdot2} \dot{r}_s^2 = \frac{M^2}{4}r_s^{2(1-n(1+c))} - \frac{f_-+f_+}{2}+\frac{r_s^{2(n(1+c)-1)}(f_--f_+)^2}{4M^2}\;.$$ The integration constant $M$ encodes information about the initial conditions of the shell’s motion. Since we want to compare with the motion in the Oppenheimer-Snyder collapse, we assume the shell starts to fall from a turning point $r_0$ at $\tau=0$, where $\dot{r}_s = 0$. We can evaluate equation (\[eq:thinshelleom\]) at the turning point, and thus we get $$M = \frac{\sqrt{f_-(r_0)}-\sqrt{f_+(r_0)}}{r_0^{1-n(1+c)}}\; .$$ This expression with equation (\[eq:rsdot2\]) determines the motion of the thin shell. Trajectories of the thin shell ------------------------------ Next we wish to inspect the possible trajectories of the shell. Instead of solving the position of the shell $r_s$ explicitly as a function of the coordinate time $t_s$, it is sufficient to look at the expression $\dot{r}_s^2$ in equation (\[eq:rsdot2\]). First of all, since $\dot{r}_s^2$ is a square of the physical velocity, it has to be positive. Additionally, since $\ddot{r}_s = \frac{1}{2}\frac{d}{dr_s}\dot{r}_s^2$, the same expression also encodes information of the acceleration for the trajectory. Thus by computing the acceleration at the turning point, $$\left.\frac{d^2r_s}{d\tau^2}\right|_{r_s=r_0}= - \left[ r_0 \Lambda + \frac{m}{r_0^n}\frac{n-1}{4} + \frac{n(1+c)-1}{4}\frac{M^2}{r_0^{n(1+c)}} - \frac{n\,c\,m^2\,r_0^{2cn-1}}{4\,M^2} \right] \; ,$$ we can determine towards which direction the shell starts to move. As long as the expression in the square brackets is positive, the shell starts to fall at the turning point. However, if $$\frac{n\,c\,m^2\,r_0^{2cn-1}}{M^2} \; > \; 4\, r_0 \Lambda + (n-1)\frac{m}{r_0^n} + \left[n(1+c)-1\right]\frac{M^2}{r_0^{n(1+c)}} \; ,$$ instead of collapsing, the shell starts to expand. Thus all trajectories can be divided to two categories: The ones where the shell starts to collapse, and the ones where the shell starts to expand at its turning point $r_0$. Next question then is, what happens to the shell after its initial turning point? Is there another turning point, or will the shell always either collapse to a black hole or expand all the way to the boundary? The EOM cannot be solved analytically for turning points for general values of $n$ and $c$. When investigating the solutions numerically, it turns out that both kinds of behaviour are present. Thus the solutions can be divided into two additional categories: The ones which have only a single turning point, and the ones which have more. Thus we have four general categories of solutions, illustrated in figure \[fig:rsdotfig\]: 1. Shell collapses to a black hole (solid line) 2. Shell starts to collapse, but has a turning point at smaller radius (dotted line) 3. Shell starts to expand, but has a turning point at a greater radius (small dashed line) 4. Shell expands all the way to the boundary (large dashed line) In case 1, the expression in equation (\[eq:rsdot2\]) has only a single root. For case 2, the expression has three roots, however, since the initial condition is given at $r_0$, the smallest root is never reached, and the shell ends up oscillating between the $r_0$ and the middle root. For case 3 the expression has again three roots, and now the shell oscillates between $r_0$ and the largest root. For case 4, the the expression has two roots, but the smaller one is never reached, and instead the shell starts and continues to expand from $r_0$ towards infinity. ![The speed squared of the thin shell for different equations of state. Solid line $c=0.328$, the dotted line $c=0.33$, the small dashed line $c=0.3323$ and the large dashed line $c=0.3334$. The divergence of the velocity at $r_h$ is a sign of a coordinate singularity at the horizon. The other parameters have been chosen to be $m=5$, $\Lambda=1$, $r_0=10$. []{data-label="fig:rsdotfig"}](rsdot2_plot.pdf) The different trajectories in figure \[fig:rsdotfig\] differ only by equation of state. In the case of $c>-1$ and $n=3$, one can look at the powers of the different terms in equation (\[eq:rsdot2\]): At small $r_s$ the first term will dominate no matter what the value of $c$. However, at large $r_s$, if $c<\frac{1}{n}$ the middle term will dominate and the expression will turn negative at sufficiently large $r_s$, and if $c>\frac{1}{n}$ then the last term will dominate and the expression will be positive at large $r_s$. Thus one critical value of $c$ is $\frac{1}{n}$, which separates cases 3 and 4. The critical values of $c$ separating cases 1 and 2, and 2 and 3, respectively, are however dependent on the other choices of parameters and initial conditions, and cannot be solved analytically. Thus the exact numerical values of the choices of $c$ in figure \[fig:rsdotfig\] are not important, but instead it is sufficient to note that if one denotes $c_i$ as the equation of state of the different cases, then $c_1<c_2<c_3<c_4$, $c_3<\frac{1}{n}$ and $c_4>\frac{1}{n}$. The trajectories of the shell encode three qualitatvely different effects: Geodesic motion in AdS, gravitational pull of the shell, and the pressure of the shell. The non-linear interplay of these different effects is the cause of this rich structure for the possible trajectories. The collapsing shell (case 1) and the eternally expanding shell (case 4) are the most familiar solutions, where as the eternally oscillating shell (cases 2 and 3) are more peculiar. In the context of holography it would be tempting to try to interpret these solutions as non-thermalizing states in the boundary field theory. However, when $c\neq0$, the shell is made out of matter that has pressure only in some spatial directions, and not in others. The classical analogy of this would be a sheet of rubber or a network of springs. In order for this shell to have a holographic interpretation in the boundary field theory, there must be a bulk degree of freedom from which the shell is constructed. Whether this is possible for the non-zero values of $c$ is an unanswered question. One familiar case where there is pressure along the shell is an instanton-like solution of e.g. a scalar field in the bulk, but for these solutions the equation of state parameter is usually $c \simeq -1$. As a final note, we must note that for the equation of state $c=0$, which one would imagine to be easiest to construct from the degrees of freedom available in the holographic description, the shell always collapses. Due to the lack of pressure, this is also the configuration we will use when making comparisons with the Oppenheimer-Snyder collapse in the following sections. The equal time two-point correlator {#sec:correlator} =================================== To probe the thermalization described by the Oppenheimer-Snyder collapse and the thin-shell collapse, we compute the equal time two-point correlator of an operator with a large conformal weight using the geodesic approximation. The geodesic approximation (for details and derivation see e.g. [@Banks:1998dd; @Balasubramanian:1999zv]) is essentially the saddle point approximation in the limit of large scaling dimension of the operator whose correlator we are computing. The description for the correlator is then given by $$\label{eq:dictionary} \left\langle \mathcal{O}(\mathbf{x}) \mathcal{O}(\mathbf{x'})\right\rangle \simeq \epsilon^{-2\Delta}e^{-\Delta\sigma(\mathbf{x},\mathbf{x'})}$$ where $\Delta$ is the conformal weight of the bulk particle or equivalently the scaling dimension of the dual operator, $\sigma(\mathbf{x},\mathbf{x'})$ is the length of the geodesic connecting the two boundary points $\mathbf{x}$ and $\mathbf{x'}$, and $\epsilon$ is a cutoff parameter regularizing the length of the geodesic as it approaches the boundary. From this we see that the computation of the equal time two-point correlator is reduced to computing the geodesic length between two points residing in the boundary at equal times. Due to the spherical symmetry of the setup, the geodesic length, and thus the correlator, will only depend on the time coordinate of the points and one angle separating the points in the $n$-sphere which forms the boundary. This angle we shall call $\varphi$. The cutoff $\epsilon$ corresponds to some (large) finite radius $R$ to which the geodesics are computed. Since the divergence of the geodesic length when taking $R \to \infty$ is present even in the case of empty AdS, we define the renormalized geodesic length by $$\tilde{\sigma}(t,\varphi) \equiv \sigma(t,\varphi) - \frac{2}{\sqrt{\Lambda}}\ln 2\sqrt{\Lambda} R \; ,$$ where the last term on the RHS is the divergent part of the length of a spacelike geodesic in empty AdS. The renormalized $\tilde{\sigma}$ is thus finite in the limit $R\to\infty$. The renormalization scheme here is arbitrary, and in fact $\tilde{\sigma}$ will depend on the choice of the scheme. However, we will only compare the lengths of geodesics with matching boundary endpoints, and the choice of the renormalization scheme does not affect the *difference* of the lengths of these geodesics. Similar scheme was used in e.g. [@Hubeny:2013dea]. The derivation of the matching conditions of the geodesics in this section follows the same recipe and computational machinery described in [@Keranen:2015fqa], appendices of which the reader is referred to for more detailed derivations. Calculational strategy ---------------------- We want to know the length of the shortest space-like geodesic connecting two boundary points with the time coordinate $t$ and the angular separation $\varphi$. Although a simple statement, this quantity is not trivial to compute, since if we start from the boundary points we do not know to which direction the geodesic should start to propagate from the boundary – that behaviour encodes the bulk geometry of the space-time. Instead, we approach the problem from a different point of view: A geodesic that connects the two points with the same time coordinate must have a turning point somewhere in the bulk, where both the radial and temporal velocities[^2] are zero. Thus we can start from any point in the bulk and look for geodesics which have a turning point at that point, and then propagate these geodesics all the way boundary. This means that for each set of initial conditions (in the Oppenheimer-Snyder case $(\bar{a},\bar{\rho})$ and in the thin shell case $(\bar{t}_-,\bar{r})$) we can compute the time $t$ when the geodesic hits the boundary, the angular separation $\varphi$, and the length of the geodesic $\sigma$. If we then scan through all possible values of the initial turning point, we should recover a geodesic length for any possible combination of $\varphi$ and $t$. It turns out that one needs to consider also geodesics which penetrate the edge of the sphere or the thin shell *after* the horizon has formed, i.e. geodesics which cross the coordinate singularity at $r=r_h$ in the outside coordinate patch. Although the coordinate system has a singularity at that point, with careful regularization all physical quantities will stay finite and the computation can be carried out in this coordinate system. A more robust approach is to use the Eddington-Finkelstein coordinate system outside, details of which are give in Appendix \[sec:eddingtonfinkelstein\]. Geodesics in the Oppenheimer-Snyder collapse -------------------------------------------- To calculate the length of geodesics connecting two equal time boundary points in the Oppenheimer-Snyder geometry, we need to solve the set of geodesic equations in the two coordinate systems defined by equations (\[eq:outsidemetric\]) and (\[eq:insidemetric\]), and then match the solutions at the edge of the sphere of dust. The geodesic equations in the inside coordinate system are given by $$\begin{aligned} \label{eq:OSingeodesic1} -\frac{h'}{2\,h} \dot{a}^2 + a\,\rho^2\,h\,\dot{\phi}^2 + \frac{a \, h \, \dot{\rho}^2}{1+\rho^2} + \ddot{a} & \;=\; 0 \\ \label{eq:OSingeodesic2} -\rho\,(1+\rho^2)\,\dot{\phi}^2+\frac{2\,\dot{a}\,\dot{\rho}}{a}-\frac{\rho\,\dot{\rho}^2}{1+\rho^2} \ddot{\rho} & \;=\;0\\ \label{eq:OSingeodesic3} \frac{2\,\dot{a}\,\dot{\phi}}{a} + \frac{2\,\dot{\rho}\,\dot{\phi}}{\rho} + \ddot{\phi}& \; =\; 0 \; ,\end{aligned}$$ where $'$ denotes a derivative w.r.t. $a$ and $\dot{\;}$ a derivative w.r.t. the affine parameter of the geodesic, which we choose to be the geodesic length $\sigma$. The geodesic equations (\[eq:OSingeodesic1\]-\[eq:OSingeodesic3\]) can be integrated to give $$\begin{aligned} \label{eq:OSingeoint1} \dot{\phi} & \; = \; \frac{\bar{L}}{a^2\,\rho^2}\\ \label{eq:OSingeoint2} \dot{a}^2 & \; = \; h(a) \left[ \frac{\bar{E}}{a^2}-1\right]\\ \label{eq:OSingeoint3} \dot{\rho}^2 &\;=\; \frac{1+\rho^2}{a^4}\left[ \bar{E}-\frac{\bar{L}^2}{\rho^2}\right] \; ,\end{aligned}$$ where $\bar{E}$ and $\bar{L}$ are integration constants, i.e. constants of motion. Their names are chosen to remind the reader of energy and angular momentum, since they are the analogues of these quantities. The values of the constants of motion are determined from the requirement that the geodesic has a turning point at $(\bar{a},\bar{\rho})$: $$\begin{aligned} \dot{a}^2(a=\bar{a}) = 0 \qquad & \Rightarrow \qquad \bar{E} = \bar{a}^2\\ \dot{\rho}^2(\rho = \bar{\rho}) = 0 \qquad &\Rightarrow \qquad \bar{L} = \pm \bar{a}\bar{\rho}\end{aligned}$$ ![Comparison between the trajectories of the pressureless ($c=0$) thin shell and the OS collapse with equal initial radius $r_0$ as a function of the proper time of the shell. Solid line is the position of the edge of the dust sphere of the OS collapse while the dashed line is the location of the thin shell. The dotted line is the location of the horizon. The difference between the two trajectories gets even smaller if $r_h$ is decreased or $r_0$ increased. Here the other parameters $\Lambda =1$, $m=10$, $n=3$ and $r_0=2$ have been chosen such that the difference is big.[]{data-label="fig:trajectories"}](trajectories.pdf) Now that we know how the geodesic propagates from the turning point to the edge of the sphere, we need to solve how the geodesic propagates outside the sphere, and how the geodesic solution inside is matched to a solution outside. The geodesic equations outside are given by $$\begin{aligned} \frac{f'\,\dot{r}\,\dot{t}}{f} + \ddot{t} & = 0\\ -\frac{f'}{2\,f}\dot{r}^2 + \frac{1}{2}f\,f'\,\dot{t}^2 -r\,f\,\dot{\phi}^2 + \ddot{r} & = 0\\ \frac{2\,\dot{r}\,\dot{\phi}}{r}+\ddot{\phi} & = 0 \; .\end{aligned}$$ These can be integrated to give $$\begin{aligned} \label{eq:outgeoint1} f\,\dot{t} &=E\\ \label{eq:outgeoint2} r^2\dot{\phi} & = L\\ \label{eq:outgeoint3} \dot{r}^2 & = f\left( 1-\frac{L^2}{r^2}\right)+E^2 \; .\end{aligned}$$ To relate the constants of motion outside to the trajectory of the geodesic inside, we construct a coordinate system $(\tau,\lambda)$ which is continuous at the edge of the sphere and its close surroundings and use that to transform the geodesic derivatives in the inside coordinate system to those in the outside coordinate system. This computation is described in more detail in Appendix \[sec:continuous\], but its result is given by $$\begin{aligned} \label{eq:matchingOS1} E &= \sqrt{\frac{\bar{a}^2}{a^2}-1}\sqrt{1+\rho_0^2} + \frac{\bar{a}}{a}\sqrt{1-\frac{\bar{\rho}^2}{\rho^2}}\dot{r}_s\\ \label{eq:matchingOS2} L &= \bar{L} = \pm \bar{a}\bar{\rho} \; .\end{aligned}$$ The procedure to compute the geodesic length for all times and angles is then as follows: Given the initial conditions $\bar{a}$ and $\bar{\rho}$, integrate equations (\[eq:OSingeoint1\] - \[eq:OSingeoint3\]) until the geodesic reaches the edge of the sphere. Then use the matching conditions (\[eq:matchingOS1\]) and (\[eq:matchingOS2\]) to get the constants of motion outside. With the constants of motion, integrate equations (\[eq:outgeoint1\] - \[eq:outgeoint3\]) until the geodesic reaches the boundary, which is represented by some (large but finite) cutoff in the radial coordinate, $R$. Repeat the computation for all possible values of the initial conditions $\bar{a}$ and $\bar{\rho}$. One must discard those cases where either the geodesic does not exit the sphere before $a=0$, or the geodesic does exit the sphere, but turns around and falls back to the sphere and thus never reaches the boundary. Computation in the thin-shell model ----------------------------------- The computation in the thin-shell case is very similar to the Oppenheimer-Snyder case. Since the metric in equations (\[eq:thinshellmetric\]) and (\[eq:thinshellfpm\]) is the same as the metric of equation (\[eq:outsidemetric\]), the geodesic equations both inside and outside are given by the equations (\[eq:outgeoint1\] - \[eq:outgeoint3\]). We shoot the geodesic from its turning point $(\bar{t}_-,\bar{r}_-)$. From the requirement that $\dot{r}=0$ and $\dot{t}=0$ at the turning point, we are able to derive that the constants of motion inside are given by $$E_- = 0 \qquad \text{and} \qquad L_-=\pm \bar{r} \;.$$ We can then integrate the geodesic equations (\[eq:outgeoint1\] - \[eq:outgeoint3\]) until the shell, $r = r_s$, at which point we need to switch to the geodesic equations in the outside coordinate patch. These are of course given by equations (\[eq:outgeoint1\] - \[eq:outgeoint3\]) as well, but now with $f=f_+$ and with different constants of motion. These constants can be derived (for more detail, see e.g. the Appendix B of [@Keranen:2015fqa]) to be $$\begin{aligned} \label{eq:matchingthinshell1} E_+ & = \sqrt{1-\frac{\bar{r}^2}{r_s^2}}\frac{\dot{r}_s}{\sqrt{f_-}}\left( \sqrt{f_-+\dot{r}_s^2}-\sqrt{f_++\dot{r}_s^2} \,\right)\\ \label{eq:matchingthinshell2} L_+ & = \bar{r} \; .\end{aligned}$$ The procedure to compute the geodesic length is after this point very similar to the Oppenheimer-Snyder case: Given the initial radial coordinate $\bar{r}$, integrate the equations for some value of $r_s$. This corresponds to a choice of initial time. Then use the matching conditions (\[eq:matchingthinshell1\]-\[eq:matchingthinshell2\]) to compute the constants of motion outside, after which integrate the geodesic equations (\[eq:outgeoint1\]-\[eq:outgeoint3\]) until the geodesic reaches the ’boundary’, i.e. the radial cutoff $R$. Then rinse and repeat for all values of $\bar{r}$ and $r_s$. Again, one must discard geodesics which never reach the boundary and also take into account geodesics that exit the shell after the horizon has formed. Results ------- We wish to compare the thermalization of Oppenheimer-Snyder to that of a thin-shell collapse. To make the comparison of the two qualitatively different models meaningful, we make the mass of the final black hole and the initial radius of the sphere and shell to be the same for both models. This corresponds to setting the final temperature of the state of the boundary field theory as well as the angular scale of the initial state to be the same for both states. Also, since the sphere of dust is pressureless by construction, we also choose the shell to be pressureless, i.e. $c=0$. The differences in the correlator for the two different models are then due to two distinct sources: different speed of the gravitational collapse and different trajectories of geodesics due to the different geometries. ![Renormalized geodesic length as a function of the angular separation of the boundary points evaluated at different times. Solid line is the equilibrium value, dotted the thin-shell collapse and dashed the Oppenheimer-Snyder collapse. From bottom to upwards the lengths are evaluated at times $t=0$, $0.5$ and $0.75$. The values of other parameters are $n=3$, $m=10$, $r_0=3$ and $\Lambda = 1$.[]{data-label="fig:multilengths"}](multilengths.pdf) ![The unnormalized correlator $e^{-\tilde{\sigma}}$ for boundary angular separation $\phi = 1.76$ as a function of time. The solid line corresponds to the equilibrium case, the dashed to the Oppenheimer-Snyder case and the dotted to the thin-shell collapse. Other parameters are the same as in figure \[fig:multilengths\].[]{data-label="fig:exp1"}](exp1.pdf) ![The unnormalized correlator $e^{-\tilde{\sigma}}$ for boundary angular separation $\phi = 2.94$ as a function of time. The solid line corresponds to the equilibrium case, the dashed to the Oppenheimer-Snyder case and the dotted to the thin-shell collapse. Other parameters are the same as in figure \[fig:multilengths\].[]{data-label="fig:exp2"}](exp2.pdf) The motion of the radius of the dust sphere and the shell are compared in figure \[fig:trajectories\]. Both are released at $t=0$ and start to collapse. The Oppenheimer-Snyder collapse is always faster than the pressureless thin-shell collapse. This can be understood by the distribution of the gravitating matter, since in the OS there is matter in the middle which attracts the outer layers as well. The location of the shell or the edge of the sphere corresponds to the smallest equilibriated scale in the boundary theory, and thus a given scale always thermalizes first in the Oppenheimer-Snyder collapse. It should be noted that the parameters in figure \[fig:trajectories\] have been chosen to make the difference in the evolution of the two different models to be large, and for generic choices of parameters the actual difference between the two trajectories is even smaller. Even if the difference between the two different trajectories is small, it does not imply that the difference in the geodesic length between two boundary points is small as well. In the thin-shell model, all the energy-momentum resides at one radial location, $r_s$, which causes a jump in extrinsic curvature normal to the shell. This jump in curvature causes the geodesic to refract as it crosses the shell. In the Oppenheimer-Snyder model there is no delta function in the energy-momentum tensor and correspondingly no jump in extrinsic curvature which would cause the geodesic to refract. Thus the geodesics in the Oppenheimer-Snyder model are more smooth, and geodesics that penetrate the shell deviate less from the equilibrium then in the thin-shell case. It should be noted that due to the coordinate systems in use, this smoothness is not apparent when numerically computing the trajectories of the geodesics. In figure \[fig:multilengths\] we have plotted the geodesic length between two points as a function of the angular separation of these two points. The solid line is the equilibrium value, the dashed lines the Oppenheimer-Snyder model and the dotted lines the thin-shell mode. The pairs of dashed and dotted lines correspond to different boundary times. First one can observe that the point where the dashed and dotted line meet the solid line corresponds to the largest scale that has equilibriated, which corresponds to the shell motion in figure \[fig:trajectories\]. Furthermore, the Oppenheimer-Snyder geodesics are always closer to equilibrium for all angular scales compared to the thin-shell case. Finally, one can note that the difference between the two models is largest in intermediate times, and very early and very late in the gravitational collapse the difference is smaller. In figures \[fig:exp1\] and \[fig:exp2\] we have plotted the exponential of the geodesic lengths of a fixed angular separation as a function of time. According to equation \[eq:dictionary\], this is directly proportional to the equal time two-point correlator, modulo an overall normalization factor that has not been determined. Again one can see that a given scale thermalizes faster in the Oppenheimer-Snyder model than in the thin-shell case. It should be noted that the thermalization starts with zero velocity, corresponding to the shell or sphere being initially at rest, but reaches the thermal value in *finite* time of the boundary observer. ![The geodesic length between boundary points with angular separation $\phi=2.88$ as a function of time in the OS collapse. For intermediate times there are several geodesics with different lengths connecting the boundary points. The parameters are chosen to be $\Lambda =1$, $m=4$, $n=3$ and $r_0=4$.[]{data-label="fig:discontinuity"}](discontinuity.pdf) When computing the geodesic length, for intermediate times and large angular separation (that is, close to $\varphi = \pi$) we found the geodesic length to be a multivalued function. That is, it appears that for a given angular separation $\varphi$, there are *several* geodesics that connect these two boundary points. To illustrate this we have plotted the lengths of all geodesics connecting boundary points with the angular separation $\phi =2.88$ in figure \[fig:discontinuity\]. One can see that for intermediate times there are *three* different lengths corresponding to this fixed angular separation. If the geodesic approximation is taken at face value, the shortest geodesic is the dominant one, and thus the two-point correlator becomes discontinuous. This feature is present in both the Oppenheimer-Snyder and the thin-shell models. The region where the geodesic length seems to be multivalued is always limited to intermediate times and large angles, i.e. angles close to $\pi$. Similar behaviour was found in the Vaidya model in [@Hubeny:2013dea]. Specifially figure \[fig:discontinuity\] should be compared with figure 7 of [@Hubeny:2013dea]. There the authors argue that the geodesic approximation breaks down, and the multivaluedness of $\sigma$ is not to be taken seriously. In the present work we have followed the authors of [@Hubeny:2013dea] and assumed that the value of the correlator is given by the geodesic length given by the largest values in figure \[fig:discontinuity\] and thus it is continuous. Indeed, when we have produced the plots, we have explicitly selected the points corresponding to the continuous values, and discarded the additional points. Although we can make the correlator thus continuous, its derivative is not: By looking at figure \[fig:discontinuity\] it is clear that if we choose the two curves that extend to small and large times, their intersection (corresponding to $t \simeq 5.5$ in figure \[fig:discontinuity\]) has a kink. Thus for small angles where the geodesic lenght is not multivalued, the correlator is smooth as evident in figure \[fig:exp1\], but for large angles, it is not, as demonstrated by the kink in figure \[fig:exp2\]. The kink is present in both the Oppenheimer-Snyder case (dashed line) and the thin-shell case (dotted line). The discontinuity of the derivative of the correlator has been discussed already in e.g. [@Balasubramanian:2011ur], where it was argued to be physical. It is interesting to note, however, that both in [@Balasubramanian:2011ur] and in [@Hubeny:2013dea] the computation was carried out in the Vaidya metric, which has a null hypersurface separating the inside and outside space-times, i.e. the matter collapses with the speed of light. In this work, however, we have not only looked at time-like shells collapsing, but also the Oppenheimer-Snyder collapse where there is no shell, and thus no discontinuity in the curvature and no delta function inthe energy-momentum tensor. Nevertheless the jump in the time derivative of the correlator is present, demonstrating that the jump does not stem from the null surface in Vaidya nor from the delta-function like distribution of matter. If one assumes that the physical correlator should be smooth, then it seems that the geodesic approximation breaks down not only in the sense that it produces multivaluedness, but also since it does not produce a smooth correlator. Conclusions {#sec:conclusions} =========== The Oppenheimer-Snyder model describes the gravitational collapse of a ball of dust and can be used as an analytical model for holographic thermalization. It serves as an alternative to the previously commonly used thin-shell models. One advantage of the OS model is that the OS metric is smoother than the thin-shell model metric due to the lack of a delta-function in the energy-momentum tensor. More important however is that the OS model offers a *different* analytical model for holographic thermalization: As we are interested in the general features of thermalization in holographic models and not features specific to the thin-shell model, the OS model can function as a new comparison to the thin-shell model, and we can hope that the common features of all these models are generic and can be applied to the physical applications of strongly coupled systems. We investigated the speed of the collapse in the OS model compared to the thin-shell case. We found that the OS model collapses always faster than the corresponding thin-shell model due to the more homogeneous distribution of the energy-momentum. The speed of the collapse is however not an observable, and thus we also computed the equal time two-point function in the geodesic approximation for both cases. With similar initial conditions (same final mass, same initial radius), the OS correlator was found to be at all times closer to equilibrium than the thin-shell correlator. This was due to two separate aspects: First of all the OS model collapses more quickly, but furthermore, since the metric is more smooth at the edge of the sphere than at the location of an infinitely thin-shell, the geodesics are more smooth and refract less, and thus deviate less from the equilibrium geodesics. Thus we concluded that the OS model always thermalizes faster then the thin-shell model. When we computed the evolution of the radius of the thin shell to compare with the OS model, we found that for certain non-zero equations of state the thin shell in global AdS has curious trajectories: In addition to the familiar behaviour of a collapse to a black hole or an infinite expansion all the way to the boundary, we found that there are solutions where the shell ends up oscillating between two turning points, thus never collapsing to a black hole but never dissipating to inifinity either. It is unclear what the interpretation of these states are in the boundary field theory, or if these cases correspond to such equations of state which are unattainable in realistic applications of the duality. When computing the two-point function, we also found that for large times and large angular separations, the geodesic length is multivalued and/or discontinuous. This feature of the geodesic approximation has been previously found in the literature in the Vaidya model [@Hubeny:2013dea], but here we demonstrated for the first time that this discontinuity exists even in the metric which has no delta-functions in the energy-momentum tensor. It seems that the geodesic approximation of the two-point correlator breaks down: not only does the correlator seem multi-valued and discontinuous, but there is also the qualitative issue that the correlator seems to know about physics inside the horizon and thus violating naive understanding of causality. Thus it would be interesting to investigate where exactly the approximation breaks down. If we were investigating the retarded correlator, one could solve a wave equation with certain boundary conditions to obtain the full correlator and then see how this compares with the approximation. However, for the correlator at hand such a simple calculation is not possible. Instead, a procedure described in e.g. [@Keranen:2014lna] is required, which given the non-trivial metric seems a very difficult task. The main interest in this and previous work is to discover which features of the analytical models are generic enough that they can be expected to be present in physically realistic scenarios (e.g. heavy ion collisions). For the equal time two-point correlator we found out that results are qualitatively very similar in both OS and thin-shell models. We also found that the apparent breakdown of the geodesic approximation is present in both models. The next step would be to compute the different observables for the OS model which have already been computed in the thin-shell model in the literature [@Lin:2008rw; @Baier:2012ax; @Baier:2012tc] and see which features of the thermalization are in fact just features of the thin-shell model. For example the photon production rate and dilepton spectrum computed in [@Baier:2012ax; @Baier:2012tc] exhibited oscillations which were speculated to be related to the presence of the delta-function-like shell. Performing the same computation in the OS model would immediately determine the source of the oscillatory behaviour. The author would like to thank Ville Keränen and Javier Mas Solé for useful discussions, and Aleksi Vuorinen for comments on the manuscript. This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Nederlandse organisatie voor Wetenschappelijke Onderzoek (NWO)”. The author would also like to thank the hospitality of the Helsinki Institute of Physics. The continuous coordinate system {#sec:continuous} ================================ To compute the matching conditions, e.g. how a geodesic travers the shell from one coordinate system to another, we construct a coordinate system which is explicitly continuous at the edge of the sphere. This coordinate system uses coordinates $\tau$ and $\lambda$ as its time and radial coordinates. $\lambda$ is defined as the proper distance from the hypersurface which is the edge of the sphere *normal to the hypersurface*. In more practical terms, it is defined as the length of the geodesic that at the hypersurface is normal to the hypersurface, i.e. $dx^\mu/d\lambda = n^\mu$. The time coordinate $\tau$ is then defined to be the proper time of the shell when that that geodesic intercepts the hypersurface. Although this coordinate system has a very complicated form in the bulk, at the hypersurface joining the inside and outside metric it takes the form of a Minkowski metric. We can then transform the geodesic both inside and outside to this coordinate system and propagate the geodesic infinitesimally across the hypersurface. Since the metric is continuous, the geodesic evolves smoothly. The only ingredients we then need for this computation are the partial derivatives of the different coordinates, which we can use to perform this transformation. For the outside (and inside in the thin shell case) metric these are given by $${\doswapr,\tau,\lambda\relax} = \dot{r}_s\;,\; {\doswapr,\lambda,\tau\relax} = \sqrt{f+\dot{r}_s}\;,\; {\doswapt,\tau,\lambda\relax} = \frac{\sqrt{f+\dot{r}_s^2}}{f}\;\;\text{and}\; {\doswapt,\lambda,\tau\relax} = \frac{\dot{r}_s}{f}\;.$$ For the FRW metric inside the sphere in the Oppenheimer-Snyder case, the derivatives are given by $${\doswapa,\tau,\lambda\relax} = -\sqrt{h(a)}\;,\; {\doswap\rho,\tau,\lambda\relax} = 0\;,\; {\doswapa,\lambda,\tau\relax} = 0\;\;\text{and}\; {\doswap\rho,\lambda,\tau\relax} = \frac{\sqrt{1+\rho^2}}{a}\;.$$ Eddington-Finkelstein coordinate system {#sec:eddingtonfinkelstein} ======================================= When computing the geodesics for the equal time two-point correlators, one needs to solve for geodesics that cross the shell or leave the sphere only after the horizon has been formed. Easiest way to avoid the divergences associated with the singularity at the horizon is to use the Eddington-Finkelstein metric for the outside metric, given by $$ds^2 = -f(r) \, dv^2 +2\,dv\,dr +r^2\,d\Omega^2_n \; .$$ The geodesic equations for a space-like geodesic in this coordinate system can be integrated to be $$\begin{aligned} r^2 \dot{\phi} & = L\\ \dot{r}^2 & = f(r)\left[ 1-\frac{L^2}{r^2}\right] + E^2\\ \dot{v} & = \frac{E+\dot{r}}{f(r)} \; .\end{aligned}$$ The location of the shell $(v_s(\tau),r_s(\tau),\ldots)$ obeys $$\dot{r}_s = \frac{1}{2}\left(f\,\dot{v}_s-\frac{1}{\dot{v}_s} \right) \qquad \text{and} \qquad \dot{v}_s = \frac{\dot{r}_s + \sqrt{f+\dot{r}_s^2}}{f} \; .$$ The normal vector of the shell is $$n^r = \dot{v}_s\left( f- \frac{\dot{r}_s}{\dot{v}_s}\right)\qquad \text{and} \qquad n^v = \dot{v}_s \; .$$ The coordinate transformation to the continuous coordinate system $(\tau,\lambda)$ at the shell or the edge of the sphere is then given by $$\begin{aligned} {\doswapr,\lambda,\tau\relax} & = \sqrt{f+\dot{r}_s^2} & {\doswapv,\lambda,\tau\relax} &= \frac{\dot{r}_s+\sqrt{f+\dot{r}_s^2}}{f}\\ {\doswapr,\tau,\lambda\relax} & = \dot{r}_s& {\doswapv,\tau,\lambda\relax} & = \frac{\dot{r}_s+\sqrt{f+\dot{r}_s^2}}{f}\end{aligned}$$ [99]{} J. M. Maldacena, Int. J. Theor. Phys.  [**38**]{} (1999) 1113 \[Adv. Theor. Math. Phys.  [**2**]{} (1998) 231\] \[hep-th/9711200\]. 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[^1]: Note that there is an ambiguity in the literature related to the sign of $K_{ij}$. For example [@Poisson] defines $K_{ij}$ with an opposite sign. In this case the sign ambguity does not matter, but can cause confusion when the energy-momentum of the joining hypersurface is non-zero. [^2]: Here we use the term *velocity* to denote a derivative of the coordinates of the geodesic with respect to the proper length of the geodesic. Although these quantities are not velocities in the physical sense, they appear in the geodesic equations in the same fashion as actual velocities would for time-like geodesics.

--- abstract: 'Motion of many comets is affected by non-gravitational forces caused by outgassing from their surfaces. Outgassing also produces reactive torques resulting in cometary spin evolution. We propose that the two processes are correlated and show that the change of cometary spin rate over its heliocentric orbit scales linearly with the amplitude of its non-gravitational acceleration. The proportionality constant depends on the comet size and orbital elements (semi-major axis and eccentricity) and on the (dimensionless) lever arm parameter $\zeta$ that relates the outgassing-induced torque and acceleration. We determine $\zeta$ for 7 comets for which both non-gravitational acceleration and change of spin period $\Delta P$ were measured and verify this relation. This sample spanning almost 4 decades in $\Delta P$ yields $\log\zeta=-2.21\pm 0.54$, surprisingly small value and spread. We then apply our framework to 209 comets with measured non-gravitational accelerations and determine the objects most likely to exhibit large spin period changes, $\Delta P\gtrsim 20$ min per orbit assuming rotation period of 10 hr and $\zeta$ comparable to our control sample. These objects should be primary targets for future studies of cometary spin variability, further constraining distribution of $\zeta$. Lack of comets with very high expected spin rate changes (which is not equivalent to having the highest non-gravitational acceleration) suggests that (1) cometary fission due to outgassing-driven spin-up must be an important process and (2) the distribution of $\zeta$ has a lower limit $\sim 10^{-3}$.' author: - 'Roman R. Rafikov' bibliography: - 'references.bib' title: 'Non-Gravitational Forces and Spin Evolution of Comets' --- Introduction. {#sect:intro} ============= Motion of many Solar System bodies is known to be affected by non-gravitational forces, which are not accounted for by the combined effect of the Solar gravity and planetary perturbations [@Whipple1950; @Sekanina1981; @Yeomans2004]. These forces exhibit themselves through changes of orbital elements of minor objects, and are naturally explained as the reactive (jet) forces due to outgassing — loss of volatiles from the surface — powered by Solar heating [@Bessel; @Whipple1950; @Marsden1968; @Marsden1969]. Non-gravitational acceleration $\boldsymbol{a}_{\rm ng}$ has been measured for several hundred Solar System objects, both comets [@Krolik; @Szut] and asteroids [@Hui2017]. Loss of volatiles should also affect the spin angular momentum of the outgassing body. Indeed, there is no reason for the mass loss from the surface of any minor object to be perfectly symmetric. Given the complicated shapes of cometary nuclei, it is natural to expect the reactive forces caused by ejection of volatile material from the surface to not point through its center of mass in general. As a result, reactive forces must affect not only the translational motion of an object but also its spin via the associated torques [@Sekanina1981; @Jewitt1997; @Samar2007]. Net torque acting on a body can change the direction of its spin axis (resulting in forced precession) as well as its spin rate. Common underlying origin — outgassing — suggests that non-gravitational acceleration and spin evolution of comets should be correlated in some way. This possibility was explored by @Whipple who showed (using a particular outgassing model) that the rate of forced precession of the cometary spin axis should be proportional to the amplitude of the non-gravitational acceleration $a_{\rm ng}$. In turn, spin precession may cause secular variation of the net non-gravitational linear acceleration, since in many models $\boldsymbol{a}_{\rm ng}$ depends on the geometry of Solar illumination (i.e. incidence angle with respect to local normal) of the cometary surface [@Whipple; @Sekanina1984]. Of course, in practice it is very difficult to determine observationally the direction of cometary spin axis [@Sekanina1981; @Jewitt1997; @Bair], not even mentioning its variation in time caused by forced precession. On the other hand, it is much easier to measure the [*rotation rate*]{} of a minor object and its variation. Indeed, changes of spin period have been measured for several Solar System comets [@Mueller1996; @Knight; @Mottola; @Bode]. And since period variation is just another manifestation of spin angular momentum evolution (complementary to forced precession[^1]), one expects it to also be related to the net non-gravitational acceleration in some way. In this work we use simple heuristic arguments to predict a direct correlation between the changes of the cometary spin period and non-gravitational acceleration acting on them (§\[sect:spin\]). We then use observations of the spin rate variability for a sample of comets to confirm this correlation and to determine its characteristics, namely, the value of the lever arm relating the non-gravitational acceleration to the torque it produces (§\[sect:zeta\]). We then use this correlation to assess spin period variability for a larger sample of comets with measured non-gravitational accelerations (§\[sect:spin\_comets\]). We discuss the implications of our results in §\[sect:disc\]. Connecting non-gravitational acceleration and spin evolution {#sect:spin} ============================================================ We consider a very general picture of outgassing, in which a specific reactive force per unit area ${\bf f}$ acts on every surface element $dS$ of the body. In general, ${\bf f}$ does not have to be normal to the local surface and it could be zero away from the active sites of mass loss. The full non-gravitational force acting on an object is \_[ng]{}=\_S [**f**]{} dS, \[eq:force\] where integration is carried over the full surface. In general, ${\bf F}_{\rm ng}$ varies in time in the inertial frame because of object’s rotation. Only the component along the instantaneous spin direction ${\bf \Omega}$ does not average out to zero and gives rise to the observed anomalous acceleration. At the same time, the net torque acting on an object is =\_S [**R**]{} dS, \[eq:torque\] where ${\bf R}$ is the radius-vector from the body center of mass to a surface element $dS$. Note that in the body frame the torque ${\bf T}$ is invariant with regard to object’s rotation. Also, while the intensity of outgassing (and value of $|{\bf f}|$) definitely vary along the heliocentric orbit, the direction of elementary torque ${\bf R}\times {\bf f}$ likely does not change much. Non-zero torque leads to evolution of the spin angular momentum of the body ${\bf S}$ through $\dot {\bf S}={\bf T}$, resulting in the change of both the direction of ${\bf S}$ (forced precession) as well as its magnitude. The former was explored by @Whipple, and we focus on the latter in this work. For our present purposes, the evolution of the spin rate $\Omega=|{\bf \Omega}|$ can be adequately described as [@Samar] I=T\_, \[eq:spin\_ev\] where $T_\Omega={\bf T}\cdot{\bf \Omega}/\Omega$ is the magnitude of torque ${\bf T}$ along ${\bf \Omega}$ and $I$ is the object’s moment of inertia with respect to the ${\bf \Omega}$ axis (the other components of ${\bf T}$ lead to forced precession of the object’s spin). In general, the integrals in equations (\[eq:force\]) and (\[eq:torque\]) cannot be directly expressed through one another. However, given their form, it is natural to expect the torque $T=|{\bf T}|$ acting on the object to be related to the magnitude of its non-gravitational force $F_{\rm ng}=|{\bf F}_{\rm ng}|$ via some “effective lever arm”. In other words, we should be able to write T\_=D F\_[ng]{}, \[eq:F\] where $\zeta D$ is the effective lever arm (with the dimension of length), $D$ is the characteristic size of the object (e.g. radius for a roughly spherical object), and $\zeta$ is the dimensionless “lever arm” parameter. Coefficient $\zeta$ accounts for the non-central nature of mass loss from the object’s surface and provides a simple connection between $T_\Omega$ and $F_{\rm ng}$. A purely central mass loss (with ${\bf f}$ passing through the center of mass at any point on the surface) would result in $\zeta=0$ — i.e. no torque, but a non-zero force in general. At the other extreme, net force averaging to zero may still give finite $T_\Omega$, meaning $\zeta\to \infty$. However, for a random distribution of multiple outgassing sites over a non-spherical surface one should probably expect $\zeta\lesssim 1$. Obviously, given $T_\Omega$ and $F_{\rm ng}$ one can always define $\zeta$ simply as a dimensionless coefficient relating the two via equation (\[eq:F\]). However, it is also natural to expect that in a large sample of objects $\zeta$ may be distributed in a certain way, which eventually reflects (possibly in a rather complicated fashion) the geometry of outgassing, distribution of object shapes, number of active sites, etc. In the following (§\[sect:zeta\]) we will operate based on this premise and will use observations to determine the gross characteristics of such a distribution of $\zeta$. Equations (\[eq:spin\_ev\]) and (\[eq:F\]) can be combined into =a\_[ng]{}, \[eq:spinev\] where $a_{\rm ng}=|\boldsymbol{a}_{\rm ng}|$ is the magnitude of the non-gravitational acceleration defined through ${\bf F}_{\rm ng}=M\boldsymbol{a}_{\rm ng}$. This relation illustrates the expectation of a direct correlation between the variation of the spin rate (or period) of a minor object and the magnitude of its outgassing-induced linear non-gravitational acceleration $a_{\rm ng}$. Note that, unlike the forced precession rate[^2] derived by @Whipple, $\dot\Omega$ is independent of the spin period of an object. [lcccccccccc]{}\ Name & $a$ & $e$ & $R$ & $(A_r,A_\varphi,A_n)$ & $P$ & $\Delta P$/orbit & $\Delta \Omega$/orbit & $\Delta \Omega_1$ & $\zeta$ & Ref.\ & $\big[$au$\big]$ & & $\big[$km$\big]$ & $\big[10^{-10}$ au d$^{-2}\big]$ & $\big[$hr$\big]$ & $\big[$min$\big]$ & $\big[$s$^{-1}\big]$ & $\big[$s$^{-1}\big]$ & &\ \ \ 2P/Encke & 2.2 & 0.85 & 2.4 & $\left(-0.4^{\pm 0.47},-0.07^{\pm 0.03},0.0\right)$& 11 & 4 & $9.6\times 10^{-7}$ & $3.04\times 10^{-4}$ & 0.031 & 1,2\ 9P/Tempel 1 & 3.15 & 0.51 & 2.8 & $\left(-31^{\pm 14},-0.2^{\pm 8.8}, -4.2^{\pm 2.1}\right)$ & 41 & -14 & $2.4\times 10^{-7}$ & $3.62\times 10^{-4}$ & $6.7\times 10^{-4}$ & 1,2,3\ 10P/Tempel 2 & 3.07 & 0.54 & 6.0 & $\left(2.3^{\pm 0.3},0.1^{\pm 0.02}, 1.8^{\pm 0.3}\right)$ & 9 & 0.27& $9.7\times 10^{-9}$ & $1.93\times 10^{-5}$ & 0.005 & 1,2,4,5\ 19P/Borrelly & 3.6 & 0.62 & 2.5 & $\left(19^{\pm 0.2},-0.78^{\pm 0.07},2.8^{\pm 0.3}\right)$ & 28 & 40 & $1.5\times 10^{-6}$ & $3.0\times 10^{-4}$ & 0.0048 & 2\ 67P/C-G& 3.46 & 0.64 & 1.65 & $\left(11^{\pm 0.2},-0.37^{\pm 0.05},2.5^{\pm 0.16}\right)$ & 12 & -21 & $4.4\times 10^{-6}$ & $3.2\times 10^{-4}$ & 0.014 & 2,6\ 103P/Hartley 2 & 3.47 & 0.7 & 0.58 & $\left(7.6^{\pm 0.5},2.5^{\pm 0.02},0.0\right)$ & 18 & 150 & $1.2\times 10^{-5}$ & $8.3\times 10^{-4}$ & 0.014 & 1,2\ 41P/T-G-K& 3.08 & 0.66 & 0.7 & $\left(170^{\pm 2},43^{\pm 2},14^{\pm 1}\right)$ & 20 & $>1560$ & $4.9\times 10^{-5}$ & $1.56\times 10^{-2}$ & 0.003 & 7\ \ [**Notes**]{}: For every comet we list semi-major axis and eccentricity of its heliocentric orbit, effective physical radius, 3 components of the non-gravitational acceleration at 1 au (assuming @Marsden model (\[eq:gr\])), rotation period, its change over an orbit, change of spin rate over an orbit, $\Delta\Omega_1$ defined by equation (\[eq:dOmmax\]), and lever arm parameter $\zeta$ defined by equation (\[eq:F\]).\ Key to references: (1) @Samar, (2) @Mueller, (3) @Belton2011, (4) @Mueller1996, (5) @Knight, (6) @Mottola, (7) @Bode. Data on $(A_r,A_\varphi,A_n)$ come from JPL Small Body Database (https://ssd.jpl.nasa.gov/?comets) \[table:zeta\] Determination of the lever arm parameter $\zeta$ based on the Solar System data {#sect:zeta} =============================================================================== In this section we verify the correlation (\[eq:spinev\]) between $a_{\rm ng}$ and spin evolution using information about spin period changes of some Solar System objects. This exercise also allows us to estimate the typical value of the dimensionless parameter $\zeta$. There are more than 200 comets, for which the non-gravitational acceleration measurements are reported in the JPL Small Body Database[^3]. However, the changes of the spin period $\Delta P$ (or spin rate $\Delta\Omega$) have been measured for only several comets. Seven of these objects, listed in Table \[table:zeta\], also have measurements of their non-gravitational acceleration available to us, making possible the determination of $\zeta$. There are several other comets for which rotation period variations were reported, e.g. C/2001 K5 (LINEAR) [@Drahus2006] and C/1990 K1 (Levy)[^4] [@Schleicher; @Feldman], however, we were not able to find the non-gravitational acceleration data for them. For simplicity we treat the comets listed in Table \[table:zeta\] as spheres of uniform density with effective radius $D=R$, so that their moment of inertia is $I=(2/5)MR^2$. Then one can use equation (\[eq:spinev\]) to write the change of $\Omega$ accumulated over the full heliocentric orbit of an object as =5R\^[-1]{}\_[r\_p]{}\^[r\_a]{}a\_[ng]{}(r), \[eq:dOm\] where $r_p=a(1-e)$ and $r_a=a(1+e)$ are the perihelion and aphelion distances, correspondingly, and $v_r$ is the heliocentric radial velocity. Clearly, $\Delta\Omega$ depends on the radial profile of the non-gravitational acceleration $a_{\rm ng}(r)$. It is customary to parametrize $\boldsymbol{a}_{\rm ng}(r)$ using a simple model \_[ng]{}(r)=[**A**]{}g(r),     [**A**]{}=(A\_r,A\_,A\_n), \[eq:a\_ng\] where the dimensionless function $g(r)$ describes the radial dependence of the non-gravitational acceleration. It is customary to define $g(r)$ such that $g(1~$au$)=1$. Constant vector ${\bf A}$ sets the amplitude of acceleration vector; it has components $A_r$, $A_\varphi$ in the radial and azimuthal directions, correspondingly, in the plane of the orbit, and $A_n$ normal to it; we depart from the conventional notation ${\bf A}=(A_1,A_2,A_3)$. Many different prescriptions have been proposed for the scaling function $g(r)$ [@Yeomans2004]. The most commonly used expression for $g(r)$, motivated by the expected water sublimation rate powered by Solar heating, is due to @Marsden: g(r)=()\^[-m]{}\^[-k]{}, \[eq:gr\] where $\alpha=0.1113$, $m=2.15$, $n=5.093$, $k=4.6142$, and $r_0=2.808$ au, so that $g(1~$au$)=1$. Note that this expression assumes water vapor production to be symmetric with respect to the perihelion of the orbit, which may not be accurate because of thermal inertia. There is a number of refinements to the prescription (\[eq:gr\]), e.g. accounting for the time lag in water vapor production [@Yeomans1989], secular drift of ${\bf A}$ [@Aksnes], changing orientation of the cometary spin axis [@Whipple; @Sekanina1981], sublimation of other species, i.e. not just water [@Sekanina2015]. Despite these improvements, in this work we employ the conventional prescription (\[eq:gr\]) for computing $\Delta\Omega$. The main reason being that the non-gravitational measurements reported in the Small Body Database assume this model, and we want to be internally consistent in this regard. ![ Values of the dimensionless lever arm parameter $\zeta$ defined by equation (\[eq:F\]) for 7 Solar System comets with simultaneously measured period change (over an orbit) $\Delta P$ and the amplitude of non-gravitational acceleration. Calculation is based on equations (\[eq:dOm1\])-(\[eq:dOmmax\]). Horizontal solid and dashed lines correspond to the mean of $\log\zeta$ and $1\sigma$ standard deviation from it (corresponding values of $\zeta$ are labeled in red). \[fig:zeta\]](Zeta.eps){width="48.00000%"} For an elliptical Keplerian orbit with a semi-major axis $a$ one has v\_r=()\^[1/2]{}r\^[-1]{}. \[eq:vr\] Plugging this expression into equation (\[eq:dOm\]) we find = \_1, \[eq:dOm1\] where \_1 = 5\_[r\_p]{}\^[r\_a]{} \[eq:dOmmax\] is the characteristic value of $\Delta\Omega$ reached when the lever arm parameter $\zeta=1$. Also, we defined $A=\sqrt{A_r^2+A_\varphi^2+A_n^2}$ to account for the fact that all three components of the reactive force ${\bf F}_{\rm ng}$ acting on the comet can lead to its spin evolution. {width="99.00000%"} We can now determine the values of the lever arm parameter $\zeta$ for every comet listed in Table \[table:zeta\]. Using the known orbital elements $a$ and $e$ of these objects, as well as the information on their sizes $R$ and non-gravitational acceleration components ${\bf A}$, we compute $\Delta\Omega_1$ using equation (\[eq:dOmmax\]). We then use equation (\[eq:dOm1\]) and observed change of the spin rate $\Delta\Omega$ to find the value of $\zeta=\Delta\Omega/\Delta\Omega_1$ for each comet. Results of this exercise are listed in Table \[table:zeta\] and are also shown in Figure \[fig:zeta\]. The error bars reflect only the uncertainty in the determination of ${\bf A}$. Uncertain size and shape of many of these comets would introduce additional systematic uncertainty at the level of tens of per cent. At the same time, it is important to note that our determination of $\zeta$ is independent of the cometary bulk density, which is one of the particularly poorly constrained characteristics [@Sosa; @Sosa2011]. One can see that, as expected, $\zeta<1$. Also, the values of $\zeta$ show substantial variation (although the most dramatic outliers also have the biggest errors in the determination of their $a_{\rm ng}$). However, one has to keep in mind that they are computed for objects with $\Delta\Omega$ spanning almost 4 orders of magnitude. Considering these values of $\zeta$ as a realization of an underlying distribution of $\log\zeta$, we find the mean and standard deviation of this distribution to be =-2.21,   \_=0.54. \[eq:stats\] Values of $\zeta$ corresponding to the mean of $\log\zeta$, and deviating from it by $\pm\sigma_{\log\zeta}$ are $\zeta=0.006$, $0.0017$, and $0.021$, correspondingly. In Figure \[fig:correlations\] we plot the observed $\Delta\Omega$ against the “mean theoretically expected” value $\Delta\Omega_{\rm th}=10^{\langle\log\zeta\rangle}\Delta\Omega_1$, computed using equations (\[eq:dOmmax\]) and (\[eq:stats\]). We also do the same for the observed change in period $\Delta P$ and the “mean theoretically expected” value[^5] $\Delta P_{\rm th}=P^2\Delta\Omega_{\rm th}/(2\pi)$. Solid and dotted lines in this Figure reflect the mean and standard deviation of $\log\zeta$ in the relation (\[eq:dOm1\]). Figure \[fig:correlations\] clearly illustrates a strong positive correlation between $\Delta\Omega$ and $\Delta\Omega_{\rm th}$ (or $\Delta\Omega_1$); the Spearman’s rank correlation coefficient is 0.786. This dataset is consistent with the linear dependence predicted by equations (\[eq:dOm1\])-(\[eq:dOmmax\]), thus fully supporting our hypothesis of a direct connection between the non-gravitational acceleration and torques acting on outgassing objects, see equation (\[eq:F\]). This gives us confidence in extending this idea to other minor objects. Expected spin evolution of comets {#sect:spin_comets} ================================= {width="95.00000%"} As mentioned before, determinations of spin period $P$ (or its variation $\Delta P$) are not available for the majority of comets with measured non-gravitational accelerations. This means that we cannot determine the value of the dimensionless lever arm $\zeta$ for these objects. Nevertheless, for each comet we can still use equation (\[eq:dOmmax\]) to compute its $\Delta\Omega_1$ — the change of spin frequency per orbit if the lever arm parameter $\zeta$ were equal to unity. Knowing $\Delta\Omega_1$ and assuming that the determinations of $\zeta$ in §\[sect:zeta\] reflect the true distribution of the lever arm parameter for all comets, we can then [*predict*]{} the expected variation of the spin rate per orbit for all comets with measured $\boldsymbol{a}_{\rm ng}$. In some sense, this procedure is inverse to what has been done in §\[sect:zeta\]. To perform this calculation, we used the Small Body Database to extract a sample of 209 Solar System objects with measured components of ${\bf A}$ in at least one direction. We used their orbital parameters listed in the Database, as well as the information on their sizes, when available, to compute $\Delta\Omega_1$ using equation (\[eq:dOmmax\]). When no information on size was available, we arbitrarily set $R=10$ km, which likely leads to an underestimate of $\Delta\Omega_1$. Note that this calculation does not use information about the rotational periods of these comets, which are not known in most cases. Results of this exercise are presented in Figure \[fig:comets\], where we plot $\Delta\Omega_1$ vs $A=|{\bf A}|$ for each of these objects (using different symbols for objects with and without size determinations). This Figure reveals a general trend of higher $\Delta\Omega_1$ for objects with larger $A$, which is to be expected since $\Delta\Omega_1\propto A$. However, there is a large scatter around a simple linear relation due to the disparity of orbital and physical parameters of individual objects, which also enter equation (\[eq:dOmmax\]). In Figure \[fig:comets\_peri\] we plot $\Delta\Omega_1$ for the same objects but now as a function of their periastron distance $r_p$. There seems to be a general trend of objects with higher $r_p$ to have lower $\Delta\Omega_1$, which is expected because of the steep radial dependence of $g$ on $r$, see equations (\[eq:gr\]) and (\[eq:dOmmax\]). ![ Same as Figure \[fig:comets\], but now with $\Delta\Omega_1$ plotted vs perihelion distance $r_p$. Meaning of the horizontal lines, labels, etc. is the same as in Figure \[fig:comets\]. See text for discussion. \[fig:comets\_peri\]](PeriCometsDOmega.eps){width="50.00000%"} There are two sets of horizontal lines in Figures \[fig:comets\] and \[fig:comets\_peri\]. The upper (red) solid curve (at $\Delta\Omega_1\approx 0.3$ s$^{-1}$) corresponds to spin-up to period $P_{\rm crit}=1$ hr from a non-rotating state, assuming a lever arm parameter $\zeta=10^{\langle\log\zeta\rangle}\approx 0.006$, characteristic of the comets used in §\[sect:zeta\]. The dotted lines mark a standard deviation (in $\log\zeta$) from this value of $\zeta$, see equation (\[eq:stats\]). We do not expect comets to survive at spin periods $\lesssim P_{\rm crit}$ because [*rotational fission*]{} would destroy such rapidly spinning objects [@Davidsson]. Fragmentation of comets is not an uncommon phenomenon [@SekaninaFrag; @Ishiguro; @Steckloff; @Jewitt2016], although its precise causes are still debated. Lower (blue) curves correspond to $10^2$ times lower $\Delta\Omega$ and $\Delta\Omega_1$ (for the same assumptions regarding $\zeta$). This value of $\Delta\Omega$ corresponds to e.g. a period change of $|\Delta P|=|2\pi\zeta P^{-2}\Delta\Omega_1|=1$ hr per orbit for an object initially spinning with $P=10$ hr period (which is rather typical for comets, see Table \[table:zeta\]). Objects above this set of curves should accumulate measurable changes of their spin periods in a single orbit, even if the uncertainty of period measurement is about an hour. One can see that a substantial number of Solar System comets belong to this category. In Table 2 we list all comets for which our calculations give $\Delta\Omega_1>10^{-3}$ s$^{-1}$, translating into $|\Delta P|>20$ min per orbit for initial $P=10$ hr and $\zeta=0.006$. In reading this table it should be remembered that many objects do not have reliable size measurements; we set $R=10$ km for them, which likely results in underestimating the true value of $\Delta P$. Also, measurements of ${\bf A}$ often have substantial uncertainty. Nevertheless, these comets are still expected to be good targets for future measurements of their spin variability. Comets with highest $\Delta\Omega_1$ {#sect:comets} ------------------------------------ Out of the full sample of comets used in this calculation, three objects with the highest $\Delta\Omega_1$ end up lying very close (or above) the breakup line in Figure \[fig:comets\]. Here we look at their properties in some more detail to understand how their existence can be brought in accord with the expectation of their rotational fission. Note that we used $R=10$ km for these objects; if they are smaller, then $\Delta\Omega_1$ would go up making rotational breakup even more of an issue. [**C/2015 D1 (SOHO)**]{} This comet (marked $1$ in Figure \[fig:comets\]) has $a=4.94$ au, $e=0.99427$, corresponding to perihelion distance $r_p=0.0283$ au. Because of the low $r_p$ its $\Delta\Omega_1=2.36$ s$^{-1}$ is the highest among the Solar System comets in our sample. This Sun-skirting comet has, in fact, not survived its perihelion passage in 2015 [@Hui]. It is plausible that rotationally-induced stresses have contributed to its destruction. [**P/1999 J6 (SOHO)**]{} This Jupiter-family comet ($2$, $a=3.1$ au, $e=0.9842$) is also a Sun-skirting object with perihelion at $r_p=0.0491$ au; it has the second-highest $\Delta\Omega_1=0.52$ s$^{-1}$. Despite its low $r_p$ and high $\Delta\Omega_1$, it has been observed over several orbits, which are clearly affected by the non-gravitational effects (its dominant radial acceleration component $A_1$ is determined with $40\%$ accuracy). Most likely explanations for why it has so far survived rotational fission are (1) either a very low value of $\zeta\lesssim 10^{-3}$ characterizing the non-gravitational torque acting on it, or (2) poor characterization of its non-gravitational force behavior (especially near perihelion, where most of $\Delta\Omega_{\rm max}$ gets accumulated) by the equation (\[eq:gr\]), see @Sekanina2015. [**205P/Giacobini-B**]{} Unlike the previous two objects, this Jupiter-family comet ($3$, $a=3.6$ au, $e=0.575$) has rather large perihelion distance $r_p=1.53$ au. Its high $\Delta\Omega_1=0.49$ s$^{-1}$ is caused by large $A$. However, the value of $A$ listed in the Small Body Database has been determined based on only 19 d long orbital arc. This raises a concern about the accuracy of the non-gravitational acceleration measurement for this particular object. Discussion {#sect:disc} ========== Our finding of a clear correlation between the anomalous (non-gravitational) linear acceleration of some minor objects and variation of their rotational period (§\[sect:zeta\]) has important implications for Solar System comets, which we discuss in more detail in §\[sect:disc\_SS\]. The existence of this correlation should not come as a surprise. Both the linear non-gravitational acceleration and the torques affecting spin angular momentum of a comet derive from the same physical process — outgassing from the cometary surface. Thus, it is natural to expect that the two should be linked, which was noted by @Whipple. However, the focus of their work was on forced precession of the cometary spin driven by non-gravitational torques, which is not observable directly. For that reason, @Whipple had to assume a particular model of outgassing, in which $\boldsymbol{a}_{\rm ng}$ itself depends on the comet spin orientation, and to rely on $a_{\rm ng}$ measurements for comet 2P/Encke spanning about two centuries to deduce the evolution of spin orientation for this comet. Obviously, such extensive datasets are available for only for a handful of comets: @Sekanina1984 used the same approach to study a century-long data set on comet P/Kopff. Our work is different from @Whipple in that we focus on the [*period changes*]{} of comets — i.e. changes of the magnitude of the spin vector ${\bf S}$ rather than its orientation — which have been measured for a small number of objects directly. This makes our and @Whipple’s approaches complementary. A full treatment of the non-gravitational spin evolution of comets should consider both the magnitude and the orientation of ${\bf S}$ [@Neishtadt; @Sidor]. However, even with our simple heuristic approach we are able to shed light on the non-gravitational torques acting on comets. In particular, using observational data (see §\[sect:zeta\]) we estimated in a model-independent fashion (i.e. not making any specific assumptions about the shape of the nucleus or geometry of outgassing) the effective dimensionless lever arm parameter $\zeta$, which relates the non-gravitational acceleration and the torque due to outgassing. This calculation assumes $\zeta$ to be constant for a given object and considers changes of only the spin rate, i.e. keeping spin axis fixed. These assumptions are likely somewhat simplistic. First, the geometry of Solar illumination changes as the comet moves along its orbit, likely modifying the distribution of active sites on the comet surface, which would then change both $a_{\rm ng}$ and the non-gravitational torques. As a consequence, parameter $\zeta$ defined by the equation (\[eq:F\]) is expected to vary along the orbit in general. As our analysis focuses on the [*integral*]{} characteristics of spin evolution — spin rate change per orbit and normalization $A$ of $a_{\rm ng}$ with a prescribed form of $g(r)$ (see equations (\[eq:a\_ng\]) and (\[eq:gr\])) — our results and statements regarding $\zeta$ effectively apply only to its orbit-averaged (in some complicated way) value. Second, forced precession of the rotational axis (neglected in our work) may significantly complicate the outgassing-driven spin evolution of comets, possibly making it somewhat stochastic (and better described as a random walk process rather than a simple linear trend in the most extreme cases). Despite these potential issues, based on the strong correlation established in §\[sect:zeta\] between the measured cometary spin period changes and theoretical predictions (Equations (\[eq:dOm1\])-(\[eq:dOmmax\])), we believe that our framework should still adequately describe the gross features of the cometary spin evolution. It is interesting to compare the small values of $\zeta\lesssim 10^{-2}$ inferred in our work with other similar calculations. @Whipple and @Sekanina1984 modeled comet nuclei as axisymmetric spheroids of small oblateness $\alpha\ll 1$, spinning around their (minor) polar axis, with axisymmetric (around spin axis) outgassing. In this particular model the dimensionless lever arm parameter (different from ours as it applies to forced precession) is $\zeta\approx \alpha\sin 2\psi$, where $\psi$ is the latitude of the sub-solar point relative to the spin axis ($0<\psi<\pi/2$). Assuming, very roughly, that $\psi$ (which varies as comet moves along its eccentric orbit) is uniformly distributed, one finds $\zeta\sim (2/\pi)\alpha$. @Whipple found $\alpha\approx 0.03$ for 2P/Encke, which would result in $\zeta\approx 0.02$, close to the upper limit implied by our estimate (\[eq:stats\]), However, substantial rotational brightness modulation of the nucleus of 2P/Encke found by @Luu is at odds with such small a value of $\alpha$. Using the same method, @Sekanina1984 found high oblateness $\alpha\sim 0.15$ for P/Kopff, implying $\zeta\sim 0.1$ for this comet. This is substantially higher than our determinations of $\zeta$, implying that either this comet is an outlier of the distribution of $\zeta$, or that in some cases the model-dependent approach of @Whipple does not fully capture the details of cometary spin evolution. [@Jewitt1997] evaluated outgassing torque and linear acceleration by modeling comet as a rectangular block with a single active site whose intensity is a harmonic function of the Solar illumination angle. Taking his results at face value we find $\zeta=2/\pi$, which is much larger than our findings (\[eq:stats\]). For more realistic comet shapes and multiple active sites [@Jewitt1997] advocated smaller asymmetry of outgassing resulting in $\zeta$ of a few tens, which is still at least an order of magnitude larger than our measurements. Based on observations of spin dynamics of 9P/Tempel 1 @Belton2011 suggested that asymmetry can be even lower (by about an order of magnitude), which would bring $\zeta$ into the range of values found in our work. A number of past studies focused on details of outgassing process, e.g. the dependence of mass loss and solar illumination on the cometary position along the orbit, detailed shape of the nucleus, its thermophysical properties, etc. However, majority of these investigations focused on either the linear non-gravitational acceleration [@Sosa; @Sosa2011; @Maquet; @Hui] or the variation of the rotation period [@Keller; @Mueller] separately, which is insufficient for determining $\zeta$. Our findings will hopefully provide motivation for future studies simultaneously accounting for both effects and their possible complicated interplay [@Whipple; @Samar1995; @KellerProc] that could provide a direct estimate of $\zeta$. The small values of $\zeta$ found in this work are somewhat surprising, as they imply that the effective lever arm for application of non-gravitational force is very small compared to object’s dimensions. In some cases this may be explained by geometric factors. For example, if the shape of the nucleus is close to spherical and outgassing is predominantly normal to the local surface (setup envisaged in [@Whipple] who found rather small effective lever arm for 2P/Encke), then the reactive force ${\bf f}$ should be very close to central at every point, giving rise to very weak torque. If, at the same time, the amplitude of the reactive force $|{\bf f}|$ strongly varies over the surface, then a substantial linear acceleration can be accumulated. This situation would naturally result in small $\zeta$, and could be relevant for some comets in our sample in Table \[table:zeta\]. However, some comets, e.g. 67P/Churyumov-Gerasimenko, have highly non-spherical shapes, so that this argument would not work for them. In such cases smallness of $\zeta$ may argue in favor of a large number of active sites of outgassing randomly distributed over the cometary surface, resulting in a statistical cancellation (upon vector summation and averaging over the orbit) of the torques provided by them. The issue with this explanation may be that cancellation would also substantially reduce the amplitude of the net linear non-gravitational acceleration. However, if some systematic trend (e.g. proximity to sub-solar point) regulates the power of the outgassing vents, then $\boldsymbol{a}_{\rm ng}$ may be suppressed less than ${\bf T}$ upon vector summation over all outgassing sites, resulting in small $\zeta$. It may also be that, starting from an arbitrary rotational state, cometary spin ${\bf S}$ evolves in such a way over multiple approaches to the Sun as to minimize the net torque, something that is worth exploring further [@Neishtadt]. Relatively small spread of $\zeta$ that we found in this work — about an order of magnitude (as $2\sigma_{\log\zeta}\approx 1$) for objects with $\Delta P$ spanning 4 decades — also merits attention. To some degree, it could be a result of the small size of the sample (7 objects) used to derive $\zeta$. On the other hand, there may be some other explanations, e.g. the aforementioned long-term spin evolution converging to some “attractor state” [@Neishtadt]. This discussion makes it clear that measurements of $\zeta$ have a potential to help constrain the physics of cometary outgassing — its geometry, intensity, etc. — at least in a statistical sense. Better characterization of the distribution of $\zeta$ can have substantial impact on our understanding of cometary activity. This provides strong motivation for increasing the number of objects with measured $\Delta P$ and $a_{\rm ng}$. Right now the most basic properties of this distribution represented by equation (\[eq:stats\]) are based on a sample of only 7 objects. Targeted observations of many other comets aimed at measuring their spin variations would allow us to characterize the distribution of $\zeta$ much better. According to Figure \[fig:comets\], even period measurements with uncertainly of tens of minutes may reveal spin variability for dozens of comets with high $\Delta\Omega_1$ (the ones above the blue line in this figure). The objects with highest $\Delta\Omega_1$ listed in the Table \[table:fast\] should have the best chance for revealing such changes, even though only one comet from Table \[table:zeta\] (41P/Tuttle-Giacobini-Kresák) has $\Delta\Omega_1>10^{-3}$ s$^{-1}$ necessary for being listed in Table \[table:fast\]. Although we focused on studying the spin period variability of comets, non-gravitational forced precession [@Whipple; @Sekanina1984] should also be substantial for these objects, likely driving them into excited rotational state. Implications for Solar System comets {#sect:disc_SS} ------------------------------------ Our calculations demonstrate that many Solar System comets (the ones with high $\Delta\Omega_1$) should exhibit rapid evolution of their rotation. Using equations (\[eq:spinev\]) and (\[eq:a\_ng\]) one can estimate the characteristic timescale for the cometary spin evolution $\tau_\Omega=|\Omega/\dot\Omega|$ as \_&=& \^[-1]{} \[eq:tau\_Om\]\ && 670  \^[-1]{}\ && (  )\^[-1]{}. This timescale is comparable to the time comets spend in the inner Solar System (about a year), implying that objects with $A\gtrsim 10^{-8}$ au d$^{-2}$ (quite numerous in Figure \[fig:comets\]) and perihelia within $1$ au should experience substantial changes of their period in a single orbit. Moreover, comets with $A\gtrsim 10^{-7}$ au d$^{-2}$ could be spun up to the limit of rotational breakup in a single orbit even starting from a non-spinning state. Of course, this estimate assumes that the lever arm parameter $\zeta$ of such comets obey the same distribution as for objects studied in §\[sect:zeta\]. In reality, situation is not as dramatic since many comets with high $A$ have perihelia outside 1 au, meaning $g(r)\lesssim 1$ and longer $\tau_\Omega$. Indeed, only three objects discussed in §\[sect:comets\] are close to the “breakup” (red) line in Figure \[fig:comets\] as predicted by Equations (\[eq:dOm1\])-(\[eq:dOmmax\]). Nevertheless, our results do suggest that rotational fission triggered by the non-gravitational torques is an important destruction mechanism for many comets, as advocated by many authors in the past [@Samar1995; @Jewitt2016; @Jewitt2017]. Our adoption of $P_{\rm crit}=1$ hr as a period leading to rotational breakup is rather conservative. Internally weak comets are likely to fission at longer spin periods. On the other hand, our estimates of $\Delta\Omega$ based on Equations (\[eq:dOm1\])-(\[eq:dOmmax\]) assume that non-gravitational torques always spin up comets in the same direction, keeping the rotational axis fixed. As mentioned before, this assumption is likely only a rough approximation; resulting departures from a simple linear trend in $\Omega$ would slow down the spin-up of comets towards rotational fission. Possibility of rotational fission provides an interesting indirect constraint on the values of the lever arm parameter $\zeta$. In Figure \[fig:comets\] there is only one object above the (upper red dotted) line corresponding to spin-up to $P_{\rm crit}=1$ hr with $\zeta=0.0017$ (i.e. $\zeta=10^{\langle\log\zeta\rangle-\sigma_{\log\zeta}}$), and this object is C/2015 D1 (SOHO), which did not survive its perihelion passage, see §\[sect:comets\]. If many comets had substantially smaller values of $\zeta$, then there would be no reason for them to not populate the region $\Delta\Omega_1>1$ s$^{-1}$: given low enough $\zeta$, even such large values of $\Delta\Omega_1$ would still result in low enough $\Delta\Omega$ to avoid rotational fission, see equation (\[eq:dOm1\]). Our interpretation of the lack of comets with $\Delta\Omega_1>1$ s$^{-1}$, most susceptible to rapid rotational evolution, is that there is a [*lower limit*]{} on the values of their lever arm parameter, $\zeta\gtrsim 10^{-3}$, so that all objects with $\Delta\Omega_1$ get spin up to breakup in a single passage through the inner Solar System. In other words, the lack of comets with high values of $\Delta\Omega_1$ is due to a “survival bias” caused by the important role of rotational fission promoted by outgassing. This bias may also explain small values of $\zeta$ found in our work, as objects with large $\zeta$ rapidly spin up and get preferntially destroyed by rotational fission. Summary {#sect:summary} ======= We have investigated the relation between the non-gravitational linear acceleration of comets and their spin evolution driven by outgassing. In our work we focused on spin period changes caused by the non-gravitational torques, as opposed to other studies concentrating on forced precession of the spin axis [@Whipple; @Sekanina1984]. Our main conclusions are listed below. 1. Based on heuristic arguments we proposed a simple linear relation between the variation of cometary spin period (and rate) and the net non-gravitational acceleration. This relation depends on a single parameter — (dimensionless) effective lever arm $\zeta$, linking the non-gravitational torque and acceleration. 2. Using a sample of 7 comets with measured $a_{\rm ng}$ and spin period changes we verified the validity of this relation. We also measured values of $\zeta$ for these objects and found $\log\zeta=-2.21\pm 0.54$ for the whole sample. 3. Using our framework we computed expected changes of spin period (per orbit) for a much larger sample of comets (209 objects) with measured non-gravitational accelerations. Assuming the inferred distribution of $\zeta$ to hold for these comets, we showed that several dozens of them should exhibit spin period changes per orbit of order an hour. A handful of comets may be in danger of rotational fission because of their rapid spin-up due to outgassing torques. This process must be an important channel of cometary destruction. 4. We advocate the use of $\Delta\Omega_1$ — change of the cometary spin rate in a single orbit evaluated for lever arm parameter $\zeta=1$, defined by the equation (\[eq:dOmmax\]) — as a metric for assessing the potential of a particular object to exhibit large changes of spin period. This parameter can be used for guiding target selection for measurements of cometary spin variability, which would help us better constrain the true distribution of $\zeta$. Our results will assist future observations of spin period variability of comets, helping us better understand physics of cometary outgassing. They could also be relevant for spin dynamics of other minor objects, such as asteroids, both in the Solar System [@Jewitt2017] and beyond (R. Rafikov, 2018, in preparation). Financial support for this study has been provided by NSF via grant AST-1409524 and NASA via grant 15-XRP15-2-0139. [lccccccc]{}\ Name & $a$ & $e$ & $R$ & $A_r$ $(A_1)$ & $A_\varphi$ $(A_2)$ & $A_n$ $(A_3)$ & $\Delta \Omega_1$\ & $\big[$au$\big]$ & & $\big[$km$\big]$ & $\big[$au d$^{-2}\big]$ & $\big[$au d$^{-2}\big]$ & $\big[$au d$^{-2}\big]$ & $\big[$s$^{-1}\big]$\ \ \ C/2015 D1 (SOHO)& 4.94 & 0.99427 & - & $1.25\times 10^{-6}$ & 0 & 0 & 2.36\ P/1999 J6 (SOHO) & 3.1 & 0.9842 & - & $3.9\times 10^{-7}$ & $-1.89\times 10^{-9}$ & 0 & 0.518\ 205P/Giacobini-B & 3.60 & 0.575 & - & $7.8\times 10^{-6}$ & $-2.4\times 10^{-7}$ & 0 & 0.486\ C/2017 E4 (Lovejoy) & 477.7 & 0.998967 & - & $1.74\times 10^{-7}$ & $-7.46\times 10^{-8}$ & 0 & 0.0424\ 86P/Wild 3 & 3.62 & 0.369 & 0.86 & $3.1\times 10^{-7}$ & $5.7\times 10^{-8}$ & 0 & 0.0405\ C/2012 V1 (PANSTARRS) & 3785.8 & 0.99945 & - & $2.2\times 10^{-6}$ & $1.1\times 10^{-7}$ & $2.5\times 10^{-7}$ & 0.027\ C/1996 B2 (Hyakutake) & 2272.1 & 0.99989867 & 4.2 & $2.3\times 10^{-8}$ & $3.6\times 10^{-10}$ & 0 & 0.0231\ C/1998 P1 (Williams) & 1698.4 & 0.999325 & - & $2.69\times 10^{-7}$ & $1.46\times 10^{-8}$ & 0 & 0.0227\ 41P/Tuttle-Giacobini-Kresak & 3.085 & 0.661 & 1.4 & $1.74\times 10^{-8}$ & $4.29\times 10^{-9}$ & $1.47\times 10^{-9}$ & 0.0159\ 73P/Schwassmann-Wachmann 3-BT & 3.09 & 0.686 & - & $-1.05\times 10^{-7}$ & $3.11\times 10^{-8}$ & 0 & 0.0148\ 147P/Kushida-Muramatsu & 3.807 & 0.276 & 0.42 & $2.67\times 10^{-7}$ & $-6.9\times 10^{-8}$ & $1.63\times 10^{-7}$ & 0.0140\ C/2002 X5 (Kudo-Fujikawa) & 1208.7 & 0.99984 & - & $2.52\times 10^{-8}$ & $5.83\times 10^{-9}$ & 0 & 0.0127\ 104P/Kowal 2 & 3.263 & 0.639 & 2 & $2.2\times 10^{-8}$ & $-1.06\times 10^{-10}$ & $4.3\times 10^{-9}$ & 0.0117\ C/1987 U3 (McNaught) & 576.5 & 0.99854 & - & $8.23\times 10^{-8}$ & $3.8\times 10^{-9}$ & 0 & 0.0107\ 71P/Clark & 3.137 & 0.494 & 1.36 & $2.12\times 10^{-8}$ & $8.65\times 10^{-9}$ & 0 & 0.0102\ 27P/Crommelin & 9.231 & 0.91898 & - & $-3.66\times 10^{-8}$ & $-1.71\times 10^{-9}$ & $2.08\times 10^{-9}$ & 0.0057\ 87P/Bus & 3.488 & 0.376 & 0.56 & $2.0\times 10^{-8}$ & $-2.33\times 10^{-9}$ & $1.15\times 10^{-9}$ & 0.0057\ 141P/Machholz 2-D & 3.009 & 0.751 & - & $2.27\times 10^{-8}$ & $5.86\times 10^{-9}$ & 0 & 0.00415\ C/2001 A2-A (LINEAR) & 2530.5 & 0.999692 & - & $-2.11\times 10^{-8}$ & $1.92\times 10^{-8}$ & 0 & 0.00404\ 51P/Harrington & 3.714 & 0.542 & 4.8 & $3.87\times 10^{-8}$ & $4.0\times 10^{-9}$ & $-7.13\times 10^{-9}$ & 0.00377\ 75D/Kohoutek & 3.543 & 0.496 & 4.6 & $3.98\times 10^{-8}$ & $8.7\times 10^{-9}$ & 0 & 0.0035\ C/2014 E2 (Jacques) & 688.3 & 0.99904 & - & $2.07\times 10^{-8}$ & $-2.82\times 10^{-9}$ & 0 & 0.00351\ C/2014 Q1 (PANSTARRS) & 1129.4 & 0.99972 & - & $8.21\times 10^{-9}$ & $2.75\times 10^{-9}$ & $-5.56\times 10^{-9}$ & 0.00339\ 51P/Harrington-A & 3.581 & 0.562 & 4.8 & $2.63\times 10^{-8}$ & $3.1\times 10^{-9}$ & 0 & 0.00322\ 5D/Brorsen & 3.101 & 0.81 & - & $1.27\times 10^{-8}$ & $1.34\times 10^{-9}$ & 0 & 0.00278\ 73P/Schwassmann-Wachmann 3-E & 3.062 & 0.694 & - & $1.72\times 10^{-8}$ & $5.81\times 10^{-9}$ & 0 & 0.00256\ P/2007 T2 (Kowalski) & 3.0927 & 0.775 & - & $1.32\times 10^{-8}$ & $2.31\times 10^{-9}$ & 0 & 0.00252\ C/1999 J3 (LINEAR) & 1596.1 & 0.999388 & - & $2.24\times 10^{-8}$ & $-5.42\times 10^{-9}$ & 0 & 0.00246\ C/2002 V1 (NEAT) & 1010.7 & 0.999902 & - & $-2.96\times 10^{-9}$ & $3.79\times 10^{-10}$ & 0 & 0.00238\ 21P/Giacobini-Zinner & 3.499 & 0.71 & 2 & $3.74\times 10^{-9}$ & $-1.04\times 10^{-10}$ & 0 & 0.00232\ C/1985 R1 (Hartley-Good) & 5982.4 & 0.999884 & - & $1.33\times 10^{-8}$ & $-2.16\times 10^{-9}$ & 0 & 0.00217\ 316P/LONEOS-Christensen & 4.328 & 0.166 & - & $7.34\times 10^{-5}$ & 0 & 0 & 0.00212\ 73P/Schwassmann-Wachmann 3-B & 3.062 & 0.693 & - & $1.49\times 10^{-8}$ & $1.96\times 10^{-9}$ & 0 & 0.00212\ C/1993 Y1 (McNaught-Russell) & 134.76 & 0.99356 & - & $1.65\times 10^{-8}$ & $1.22\times 10^{-9}$ & 0 & 0.00207\ C/1999 T1 (McNaught-Hartley) & 8149.7 & 0.999856 & - & $2.44\times 10^{-8}$ & $-9.59\times 10^{-10}$ & 0 & 0.00198\ 101P/Chernykh & 5.785 & 0.594 & 5.6 & $-5.16\times 10^{-8}$ & $1.55\times 10^{-7}$ & 0 & 0.00197\ 168P/Hergenrother & 3.624 & 0.61 & - & $2.66\times 10^{-8}$ & $2.37\times 10^{-9}$ & $-1.25\times 10^{-9}$ & 0.00196\ 76P/West-Kohoutek-Ikemura & 3.471 & 0.539 & 0.66 & $-2.18\times 10^{-9}$ & $-1.85\times 10^{-10}$ & 0 & 0.00186\ 252P/LINEAR & 3.047 & 0.673 & - & $1.05\times 10^{-8}$ & $-7.06\times 10^{-9}$ & 0 & 0.00167\ 88P/Howell & 3.11 & 0.562 & 4.4 & $8.41\times 10^{-9}$ & $-1.6\times 10^{-9}$ & $-5.2\times 10^{-10}$ & 0.00164\ C/2011 F1 (LINEAR) & 2776.2 & 0.999345 & - & $5.67\times 10^{-8}$ & $-2.5\times 10^{-8}$ & $-1.47\times 10^{-9}$ & 0.00157\ 59P/Kearns-Kwee & 4.485 & 0.475 & 1.58 & $3.24\times 10^{-8}$ & $-1.88\times 10^{-9}$ & $2.1\times 10^{-9}$ & 0.00154\ 154P/Brewington & 4.883 & 0.671 & - & $2.9\times 10^{-8}$ & $-2.02\times 10^{-9}$ & $-9.65\times 10^{-9}$ & 0.0015\ C/2012 X1 (LINEAR) & 153.3 & 0.98957 & - & $3.61\times 10^{-8}$ & $2.19\times 10^{-9}$ & $5.33\times 10^{-9}$ & 0.00148\ 240P/NEAT & 3.866 & 0.45 & - & $7.89\times 10^{-8}$ & $3.47\times 10^{-8}$ & $9.91\times 10^{-10}$ & 0.00145\ 205P/Giacobini-A & 3.539 & 0.569 & - & $1.38\times 10^{-8}$ & $-1.72\times 10^{-8}$ & 0 & 0.00138\ 141P/Machholz 2 & 3.019 & 0.749 & - & $7.64\times 10^{-9}$ & $2.21\times 10^{-10}$ & 0 & 0.00134\ 2P/Encke & 2.215 & 0.848 & 4.8 & $-1.34\times 10^{-11}$ & $-2.28\times 10^{-12}$ & $1.8\times 10^{-9}$ & 0.00133\ 42P/Neujmin 3 & 4.876 & 0.584 & 2.2 & $1.27\times 10^{-8}$ & $-1.05\times 10^{-9}$ & 0 & 0.00113\ 255P/Levy & 3.038 & 0.668 & - & $7.33\times 10^{-9}$ & $-4.15\times 10^{-9}$ & 0 & 0.0011\ \ [**Notes**]{}: For every comet we list semi-major axis and eccentricity of its heliocentric orbit, effective physical radius (if available), 3 components of the non-gravitational acceleration at 1 au $(A_r,A_\varphi,A_n)=(A_1,A_2,A_3)$ assuming @Marsden model (\[eq:gr\]), and $\Delta\Omega_1$ — theoretical prediction for the change of spin rate over an orbit for a lever arm parameter $\zeta=1$ defined by equation (\[eq:dOmmax\]). Data on $(A_r,A_\varphi,A_n)$ come from Small Body Database (https://ssd.jpl.nasa.gov/?comets). \[table:fast\] [^1]: @Whipple modeled cometary nucleus as an oblate spheroid with outgassing force pointing through the rotation axis at every point on the surface. This geometry does not result in changes of the spin rate. The fact that cometary spin rate variations are observed (§\[sect:zeta\]) implies that this model is too simplistic. [^2]: Which obeys a relation very similar to (\[eq:spinev\]). [^3]: https://ssd.jpl.nasa.gov/?comets [^4]: Also known as Comet Levy (1990c). [^5]: The small $\Delta\Omega/\Omega$ assumption used in this expression is violated for comet 41P/Tuttle-Giacobini-Kresák, see Table \[table:zeta\] and @Bode.

--- abstract: 'We study the sensitivity of anomalous neutral triple gauge couplings ($aNTGC$) via $pp \rightarrow ZZ$ production in the 4$\ell$ channel at 100 TeV centre of mass energy of future circular hadron collider, `FCC-hh`. The analysis including the realistic detector effects is performed in the mode where both Z bosons decay into same flavor, oppositely charged lepton pairs. The sensitivities to the charge-parity (CP)-conserving $C_{\tilde{B}W} / \Lambda^{4}$ and CP-violating $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ couplings obtained at 95% Confidence Level (C.L.) using the invariant mass distribution of 4$\ell$ system reconstructing the leading and sub-leading Z boson candidates are $[-0.117, \,\, +0.117]$, $[-0.293, \,\, +0.292]$, $[-0.380, \,\, +0.379]$, and $[-0.138, \,\, +0.138]$ in the unit of TeV$^{-4}$, respectively.' author: - 'A. Yilmaz' - 'A. Senol' - 'H. Denizli' - 'I. Turk Cakir' - 'O. Cakir' bibliography: - 'pp2zzPaper.bib' title: 'Sensitivity on Anomalous Neutral Triple Gauge Couplings via $ZZ$ Production at FCC-hh' --- Introduction ============ The studies on the diboson production at colliders play an important role in testing the non-Abelian SU (2)$_{L} \times U (1)_{Y}$ gauge group of the electroweak sector in the Standard Model (SM) and searching for new phenomena at the TeV-energy scale [@Neubauer_2011]. Since there is no triple gauge couplings between the photon and $Z$ boson ($Z\gamma \gamma$ and $Z\gamma Z$) except $WWZ$ and $WW\gamma$ in the SM, pairs of $Z$ bosons cannot be created at a single vertex in the SM. Therefore any deviations from SM predictions on neutral triple gauge couplings (including $ZZ\gamma$, $Z\gamma \gamma$ and $ZZZ$ vertices) can give an indication about new physics beyond the SM. The new physics effects at high energy can be parametrized in the Effective Field Theory (EFT) approach. This theory is general enough to point the most probable places to observe these effects since it is renormalizable, includes the gauge symmetries of the standard model and can be used at both tree level and loop level. There is no concern on violating of the unitary of anomalous couplings in scattering processes at higher energies according to this theory. Anomalous NTG vertices can be added in an effective Lagrangian using EFT approach and parametrized by CP-conserving and CP-violating couplings, while no SM NTGC is present at tree-level [@Green:2016trm]. The production of $ZZ$ dibosons in the 4$\ell$ final state have been studied by various collaborations such as the Large Electron-Positron (LEP) [@Barate:1999jj; @Acciarri:1999ug; @Abdallah:2003dv; @Abbiendi:2003va; @Alcaraz:2006mx; @Schael:2013ita] where the first bounds on anomalous neutral triple gauge couplings (aNTGCs) using e$^+$e$^-$ collider was obtained, the Collider Detector at Fermilab (CDF) [@CDF:2011ab; @PhysRevD.89.112001] and $D\O$ [@Abazov:2007ad; @D0:2013rca] also searched the limits of aNTGC at Tevatron $pp$ collider. Recently, ATLAS [@Aad:2015zqe; @PhysRevD.97.032005] and CMS [@Aaboud:2017rwm; @Sirunyan_2018] collaborations published the improved limits of aNTGCs thanks to the center of mass energy of LHC in the range of 13 TeV at the LHC. This high center of mass energy leads to enhance the cross-section which would widen the range of triple gauge coupling studies. There are also some phenomenological studies for probing the sensitivities of aNTGCs at hadron colliders in the EFT framework [@Senol_2014; @Mangano_2016; @Frye_2016; @DORIGO2018211; @SENOL2018365]. The dimension-eight (dim-8) effective Lagrangian for nTGC in the scope of EFT assuming the local U(1)$_{EM}$ and Lorentz symmetry can be written as [@Degrande_2014] $$\mathcal{L}^{nTGC} = \mathcal{L}_{SM} + \sum_{i} \frac{C_{i}}{\Lambda^{4}} (\mathcal{O}_{i} + \mathcal{O}_{i}^{\dagger}) \label{eqn:lagrangian}$$ where $i$ is the index of equations running over the operators given as $$\begin{aligned} \mathcal{O}_{\widetilde{B}W} & = & i H^{\dagger} \widetilde{B}_{\mu\nu} W^{\nu \rho} \{D_{\rho}, D^{\nu} \} H, \\ \mathcal{O}_{BW} & = & i H^{\dagger} B_{\mu\nu}W^{\nu \rho} \{D_{\rho}, D^{\nu} \} H, \\ \mathcal{O}_{WW} & = & i H^{\dagger} W_{\mu\nu} W^{\nu \rho} \{D_{\rho}, D^{\nu} \} H, \\ \mathcal{O}_{BB} & = & i H^{\dagger} B_{\mu\nu} B^{\nu \rho} \{D_{\rho}, D^{\nu} \} H. \label{eqn:lagrangian2}\end{aligned}$$ where $\widetilde{B}_{\nu\mu}$ is dual $B$ strength tensor. We used the convention given below in the definitions of the operators $$\begin{aligned} B_{\mu\nu} & = & (\partial_{\mu} B_{\nu} - \partial_{\nu} B_{\mu}) \\ W_{\mu\nu} & = & \sigma^{I} (\partial_{\mu} W^{I}_{\nu} - \partial_{\nu} W^{I}_{\mu} + g\epsilon_{IJK} W^{J}_{\mu} W^{K}_{\nu} ) \label{eqn:lagrangian3}\end{aligned}$$ with $\langle \sigma^{I} \sigma^{J} \rangle = \partial^{I \, J} / 2$ and $$D_{\mu} \equiv \partial_{\mu} - i g_{w} W^{i}_{\mu} \sigma^{i} - i \frac{g^{\prime}}{2} B_{\mu} Y \label{eqn:lagrangian4}$$ The coefficients of these four dimension-eight operators describing aNTGC are CP-conserving $C_{\tilde{B}W} / \Lambda^{4}$ and CP-violating $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ couplings. The current limits on $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ couplings of `dim-8` operators converted from the couplings of `dim-6` operators for the process $pp \rightarrow ZZ \rightarrow \ell^+ \ell^- \ell^{\prime +} \ell^{\prime -} $ [@PhysRevD.97.032005] where $\ell = e$ or $\mu$ and $Z\gamma \rightarrow \nu \bar{\nu}\gamma$ [@ATLAS:2018eke] at the center of mass energy $\sqrt{s} = 13$ TeV and integrated luminosity $L_{int} = 36.1$ fb$^{-1} $ from the LHC are given in Table \[tab:limits1\]. In this table, all couplings other than the one under study are set to zero. --------------------------------- -- ----------------------------------------------- ------------------------------------------------------------ Couplings $ (TeV^{-4})$ $ZZ \rightarrow 4 \ell$ [@PhysRevD.97.032005] $Z\gamma \rightarrow \nu \bar{\nu}\gamma$ [@ATLAS:2018eke] $C_{\tilde{B}W} / \Lambda^{4}$ $-5.9, +5.9$ $-1.1, \,\, +1.1$ $C_{WW} / \Lambda^{4}$ $-3.0, +3.0$ $-2.3, \,\, +2.3$ $C_{BW} / \Lambda^{4}$ $-3.3, +3.3$ $-0.65, +0.64$ $C_{BB} / \Lambda^{4}$ $-2.7, +2.8$ $-0.24, +0.24$ --------------------------------- -- ----------------------------------------------- ------------------------------------------------------------ : Observed one dimensional 95% C.L. limits on $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ EFT parameters from LHC.[]{data-label="tab:limits1"} The future circular collider project, FCC [@FCCweb], proposed to have three collider options (FCC-ee, FCC-eh and FCC-hh) working at different center of mass energies. The hadron collider option of FCC (FCC-hh) is planned to reach an integrated luminosity of 20-30 ab$^{-1}$ at 100 TeV center mass energy. FCC-hh, comparing to LHC, has the energy scale by a factor about 7 depending on the process [@Benedikt:2651300]. Exploring the new physics effects in the production of diboson is a challenging task. In the literature ZZ diboson production has been examined in 2 decay channels such as “2$\ell$2$\nu$" and “4$\ell$" channel [@Green:2016trm]. In the first channel $ZZ \rightarrow$ 2$\ell$2$\nu$, one of the Z decays into a neutrino while the other one decaying into a same flavor, oppositely-charged two leptons which leads to increase in the missing transfer energy in the final state. Therefore this channel exposes to a larger background contribution and it is not kinematically reconstructable completely. In the second decay channel $ZZ \rightarrow$ 4$\ell$, not only the first Z boson, but also the other Z boson decays into a same-flavor, oppositely charged two leptons. This process gives rise to include a very low background and kinematically reconstructable in the final state. On the other hand, one needs to take into account the process has small branching fractions results with a low statistics in the final state. This paper will be organized as follows: In section \[sec:simulation\] we will discuss the simulation environment of $ZZ$ diboson production for signal and background at FCC-hh collider. Event selection procedures of our phenomenological study in the 4$\ell$ final state will be given in section \[sec:eventSelect\]. In section \[sec:results\], we will give the collected results for 4$\ell$ final state analysis. Conclusions on the sensitivities of each couplings will be summarized in section \[sec:conclusion\]. Generation of signal and background events {#sec:simulation} ========================================== To obtain the bounds on aNTGC parameters of $ZZ$ diboson production in the framework of the EFT at the FCC-hh. We generated signal and background events for the $pp \rightarrow ZZ $ process by importing the signal `aTGC` implemented through `UFO` model file into `MadGraph5_aMC@NLO v2.6.4` [@Alwall_2014]. The `PYTHIA v8.2` [@SJOSTRAND2015159] package is used for parton showering and hadronization. `LHAPDF v6.1.6` [@Buckley_2015] library and its `NNPDF v2.3` [@BALL2013244] set is used as the default set of parton distribution functions (PDFs) for all simulated MC samples. $3 \times 10^{6}$ events of the signal and the background were generated for each `dim-8` couplings. The detector response is simulated using a detailed description of the FCC-hh detector card implemented in the `Delphes v3.4.1` [@de_Favereau_2014]. All events are analyzed by using the `ExRootAnalysis` [@exRootAnalysis] package with `ROOT v6.16` [@BRUN199781]. The kinematical distributions are normalized to the number of expected events which is defined to be the cross section of each processes including the branchings times integrated luminosity of $\mathcal{L}_{int}$ = 10 ab$^{-1}$. Feynman diagrams that contribute to the signal and its main-background processes are shown in Fig. \[fig:sig\] and Fig. \[fig:bck\], respectively. The red dot represents the aNTGC vertex in the production of $ZZ$. The cross section is calculated with a set of generator level cuts; a lepton is declared to be isolated if the $p_{T}$-sum of all particles within the isolation cone size R$_{iso}$ = 0.3, minimum $p_{T}$ = 10 GeV and $|\eta| <$ 2.5 for the charged leptons. In the calculations, default mass of the $Z$ boson is used as 91.187 GeV. The cross sections of the $ZZ$ process as a function of mentioned four `dim-8` couplings are shown in Fig. \[fig:xSectionPlot\]. In this figure, only one coupling at a time is varied from its SM value and plotted as a function of couplings in the range of limits reported by CMS Collaboration [@Sirunyan_2018]. One can clearly see the deviation from the SM. \[hbt\] ![Cross sections for the process with aNTGCs including CP-conserving and CP-violating terms in the Lagrangian.[]{data-label="fig:xSectionPlot"}](xsectionCouplings.pdf){width="80.00000%"} Event selection {#sec:eventSelect} =============== We consider 4$\ell$ final state in our analysis based on Ref. [@Sirunyan_2018] including three possible options; $e^{+}e^{-}e^{+}e^{-}$, $\mu^+\mu^-\mu^+\mu^-$, and $e^+e^-\mu^+\mu^-$. The preselection for this analysis require the presence of a pair of leptons of the same or different flavors [@Khachatryan_2017]. All permutations of leptons giving a pair of $Z / \gamma^\ast$ candidates are considered within each event. The pairing ambiguity is resolved by ordering the pair of dilepton candidates based on the differences between the reconstructed invariant mass of dilepton canditate ($m_{\ell^+ \ell^-}$) and nominal $Z$ boson mass $m_{Z}$. Therefore, the dilepton candidate with an invariant mass closest to the nominal $Z$ boson mass [@PhysRevD.98.030001], is denoted $leading \, Z$ while the second closest is defined as $subleading \, Z$. In order to see the region where the signal can be enhanced we plotted the transverse momentum of leptons ($p_{T}^{\ell^{1}}$, $p_{T}^{\ell^2}$) versus the reconstructed invariant mass of the $leading$ and $subleading\, Z$ as shown in Fig. \[fig:e\_leadingSubLeadingZmassPTcut0\], Fig. \[fig:mu\_leadingSubLeadingZmassPTcut0\] and Fig. \[fig:eMu\_leadingSubLeadingZmassPTcut0\] for 4$e$, 4$\mu$ and 2$e$2$\mu$ channels, respectively. $p_{T}^{\ell^{1}}$ is labelled as the highest-$p_{T}$ lepton in both $leading$ and $subleading \,Z$. The cut for highest $p_{T}^{\ell^{1}}$ lepton is greater than 20 GeV, and for the subleading lepton is $p_{T}^{\ell^2}>$ 12 GeV (10 GeV) in the $leading \, Z$, while the remaining leptons in the $subleading \, Z$ must have $p_{T}^{\ell^1, \ell^2}>$ 5 GeV for electrons (muons). The pseudo-rapidity cuts of all leptons are applied as $|\eta^{\ell} |<$ 2.5. The distance $\Delta R (\ell^{1} , \ell^{2} )$ between leptons in $\eta$-$\phi$ plane is evaluated by the function $$\Delta R (\ell^{1} , \ell^{2} ) = \sqrt{(\eta^{\ell^{1}} - \eta^{\ell^{2}} )^{2} + (\phi^{\ell^{1}} - \phi^{\ell^{2}})^{2}} \label{eqn:dR}$$ and plotted in Fig. \[fig:deltaR\]. This figure shows $\Delta R$ distributions between two leptons of leading and subleading Z in the first and second column. Each row corresponds to different decay channel aligned for 4$e$, 4$\mu$ and 2$e$2$\mu$, respectively. In order to meet the detector requirement, we applied a cut for all leptons are separated from each others by imposing $\Delta R (\ell^{1} , \ell^{2} ) >$ 0.02. The $pp \rightarrow ZZ$ sensitivity is estimated by using events where a further cut is applied for both invariant mass of $leading \, Z$ and $subleading \, Z$ bosons must be within the range 80 $< m_{leading \, Z} <$ 100 GeV and 60 $< m_{subleading \, Z} <$ 110 GeV, respectively. This ranges were chosen to keep most of the decays in the resonance while removing mostly other processes with 4$\ell$ final states. Decays of the $Z$ bosons to $\tau$ leptons with subsequent decays to electrons and muons are heavily suppressed by requirements on lepton $p_{T}$. The cut flow steps in the analysis for selecting the events are summarized in Table \[tab:tab2\]. After applying the kinematical cuts discussed above, the reconstructed invariant mass of the $leading \, Z$ boson candidates, and a scatter plot showing the correlation between $subleading \, Z$ boson versus $leading \, Z$ boson in simulated events, are shown in Fig. \[fig:all\_channel\_LeadingSubLeadingZmassCut0\]. Channel Signal Background Total ---------- -------- ------------ -------- -- $4e$ 16308 13991 30299 $4\mu$ 32477 26850 59327 $2e2\mu$ 76404 71755 148159 : The number of events yielded for main background and signal (where all couplings equal to zero, except $C_{\widetilde{B}W}=5$) of four-lepton events in the mass region 80 $< m_{4\ell} <$ 100 GeV shown for each final state and combined at FCC-hh with $\mathcal{L}_{int}=$ 10 ab$^{-1}$. \[tab:eventNumbers\] Results {#sec:results} ======= To obtain 95% C.L. limits on the couplings, we apply $\chi^2$ criterion without and with a systematic error. The $\chi^2$ function is defined as follows $$\label{eqn:chi2def} \chi^{2} =\sum_i^{n_{bins}}\left(\frac{N_{i}^{NP}-N_{i}^{B}}{N_{i}^{B}\Delta_i}\right)^{2}$$ where $N_{i}^{NP}$ is the total number of events in the existence of effective couplings, $N_{i}^{B}$ is total number of events of the corresponding SM backgrounds in $i$th bin of the invariant mass of the quartet-leptons distribution, $\Delta_i=\sqrt{\delta_{sys}^2+ 1 / N_i^B}$ is the combined systematic ($\delta_{sys}$) and statistical errors in each bin. The existence of aTGCs will lead to enhance the yield of events at quadruplet-lepton masses. The distribution of the quadruplet-lepton reconstructed mass of events with both leading and subleading $Z$ bosons in the mass range 60–120 GeV for the unified 4$e$, 4$\mu$, and 2$e$2$\mu$ channels are depicted in Fig. \[fig:mZZ4l\]. The limits on probable contributions from aNTGCs are extracted by using this distributions. ![Distributions of the reconstructed four-lepton invariant mass $m_{ZZ}$. In the $m_{ZZ}$ distribution, bin contents are normalized to the bin widths. The lower plot shows the ratio of signal and background int eh bins.[]{data-label="fig:mZZ4l"}](ratioPlot_zzBoson_4l_e_mu_eMu_varBinCutAll.pdf){width="80.00000%"} For the analysis of $ZZ$ production with $quartet-leptons$ in the final state, the number of signal events and one-parameter $\chi^2$ results for each couplings varied with integrated luminosity from 1 ab$^{-1}$ to 30 ab$^{-1}$. In the analysis, only one coupling at a time is varied from its SM value. The results from $\chi^2$ analysis of the couplings describing aTGC interactions of neutral gauge bosons. The coefficients of the operators denoted as $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ are given in Fig. \[fig:sensitivityLimit\]. \[hbt\] ![Estimated sensitivity on aNTG couplings at $95\%$ C.L. as a function of integrated luminosity where there is only one coupling varied at a time from its SM value.[]{data-label="fig:sensitivityLimit"}](chi2sensitivity.pdf "fig:"){width="80.00000%"} We present the results of one-dimensional 95% C.L. confidence intervals at $L_{int}$ = 10 ab$^{-1}$ under the assumption that any excess in signal over background due exclusively to $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ or $C_{BB} / \Lambda^{4}$ are given in Table \[tab:obtainedLimits\]. We also include the effects of systematic errors on the limits. The obtained limits without systematic errosr are one order better than the current limits on these couplings of `dim-8` operators converted from the couplings of `dim-6` operators for the process $pp \rightarrow ZZ \rightarrow \ell^+ \ell^- \ell^{\prime +} \ell^{\prime -} $ [@PhysRevD.97.032005] at the center of mass energy $\sqrt{s} = 13$ TeV and integrated luminosity $L_{int} = 36.1$ fb$^{-1} $ from the LHC. Conclusion and Discussion {#sec:conclusion} ========================= In this paper we present a phenomenological cut based study for probing the limits on the CP-conserving $C_{\tilde{B}W} / \Lambda^{4}$ and CP-violating $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ `dim-8` aNTG couplings via $ZZ \rightarrow 4\ell$ (where $\ell$ = $e$ or $\mu$) production at the FCC-hh. The obtained limits of `dim-8` aNTG couplings at 95% C.L. for $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$, $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ with an $\mathcal{L}_{int}$ = 10 ab$^{-1}$ are one order better than those available prior to this study without systematic error. When we compare these results with the latest search for $\nu \bar{\nu} \gamma$ production [@ATLAS:2018eke] from the LHC, we have better results on $C_{\tilde{B}W} / \Lambda^{4}$, $C_{WW} / \Lambda^{4}$ couplings and improved results on $C_{BW} / \Lambda^{4}$ and $C_{BB} / \Lambda^{4}$ couplings. Even with $5\%$ systematic errors, the obtained bounds for FCC-hh are better than the LHC results on all couplings studied in this paper. The limits of aNTG couplings would benefit from high luminosity and the high energy when the systematic uncertainties are well reduced below $5\%$. Acknowledgement {#sec:acknowledge} =============== This work was partially supported by Turkish Atomic Energy Authority (TAEK) under the project grant no. 2018TAEK(CERN)A5.H6.F2-20. References {#references .unnumbered} ==========

--- abstract: 'We consider active learning of deep neural networks. Most active learning works in this context have focused on studying effective querying mechanisms and assumed that an appropriate network architecture is a priori known for the problem at hand. We challenge this assumption and propose a novel active strategy whereby the learning algorithm searches for effective architectures on the fly, while actively learning. We apply our strategy using three known querying techniques (softmax response, MC-dropout, and coresets) and show that the proposed approach overwhelmingly outperforms active learning using fixed architectures.' author: - | Yonatan Geifman\ Technion – Israel Institute of Technology\ `[email protected]`\ Ran El-Yaniv\ Technion – Israel Institute of Technology\ `[email protected]`\ bibliography: - 'egbib.bib' title: Deep Active Learning with a Neural Architecture Search --- \[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Observation]{} \[theorem\][Assumption]{} \[theorem\][Example]{} Introduction {#sec:introduction} ============ Active learning allows a learning algorithm to control the learning process, by actively selecting the labeled training sample from a large pool of unlabeled instances. Theoretically, active learning has a huge potential, especially in cases where *exponential speedup* in sample complexity can be achieved [@hanneke2011rates; @wiener2015compression; @gelbhart2017relationship]. Active learning becomes particularly important when considering supervised deep neural models, which are hungry for large and costly labeled training samples. For example, when considering supervised learning of medical diagnoses for radiology images, the labeling of images must be performed by professional radiologists whose availability is scarce and consultation time is costly. In this paper, we focus on active learning of image classification with deep neural models. There are only a few works on this topic and, for the most part, they concentrate on one issue: How to select the subsequent instances to be queried. They are also mostly based on the *uncertainty sampling* principle in which querying uncertain instances tends to expedite the learning process. For example, [@gal2017deep] employ a Monte-Carlo dropout (MC-dropout) technique for estimating uncertainty of unlabeled instances. [@wang2016cost] applied the well-known softmax response (SR) to estimate uncertainty. [@sener2018active] and [@geifman2017deep] proposed to use coresets on the neural embedding space and then exploit the coreset loss of unlabeled points as a proxy for their uncertainty. A drawback of most of these works is their heavy use of prior knowledge regarding the neural architecture. That is, they utilize an architecture already known to be useful for the classification problem at hand. When considering active learning of a new learning task, e.g., involving medical images or remote sensing, there is no known off-the-shelf working architecture. Moreover, even if one receives from an oracle the “correct” architecture for the passive learning problem (an architecture that induces the best performance if trained over a very large labeled training sample), it is unlikely that this architecture will be effective in the early stages of an active learning session. The reason is that a large and expressive architecture will tend to overfit when trained over a small sample and, consequently, its generalization performance and the induced querying function (from the overfit model) can be poor (we demonstrate this phenomenon in Section \[sec:experiments\]). To overcome this challenge, we propose to perform a *neural architecture search* (NAS) in every active learning round. We present a new algorithm, the *incremental neural architecture search* (iNAS), which can be integrated together with any active querying strategy. In iNAS, we perform an incremental search for the best architecture from a restricted set of candidate architectures. The motivating intuition is that the capacity of the architectural class should start small, with limited architectural capacity, and should be monotonically non-decreasing along the active learning process. The iNAS algorithm thus only allows for small architectural increments in each active round. We implement iNAS using a flexible architecture family consisting of changeable numbers of stacks, each consisting of a fluid number of Resnet blocks. The resulting active learning algorithm, which we term *active-iNAS*, consistently and significantly improves all known deep active learning algorithms. We demonstrate this advantage of active-iNAS with the above three querying functions over three image classification datasets: CIFAR-10, CIFAR-100, and SVHN. Related Work ============ Active learning has attracted considerable attention since the early days of machine learning. The literature on active learning in the context of classical models such as SVMs is extensive [@CohAtlLad94; @FreundSST93; @tong2001support; @baram2004online; @balcan2013active; @huang2015efficient], and clearly beyond the scope of this paper. Active learning of deep neural models, as we consider here, has hardly been considered to date. Among the prominent related results, we note Gal et al. [@gal2017deep], who presented active learning algorithms for deep models based on a Bayesian Monte-Carlo dropout (MC-dropout) technique for estimating uncertainty. Wang et al. [@wang2016cost] applied the well-known softmax response (SR) idea supplemented with pseudo-labeling (self-labeling of highly confident points) for active learning. Sener and Savarese [@sener2018active] and Geifman and El-Yaniv [@geifman2017deep] proposed using coresets on the neural embedding space and then exploiting the coreset loss of unlabeled points as a proxy for their uncertainty. A major deficiency of most of these results is that the active learning algorithms were applied with a neural architecture that is already known to work well for the learning problem at hand. This hindsight knowledge is, of course, unavailable in a true active learning setting. To mitigate this problematic aspect, in [@geifman2017deep] it was suggested that the active learning be applied only over the “long tail”; namely, to initially utilize a large labeled training sample to optimize the neural architecture, and only then to start the active learning process. This partial remedy suffers from two deficiencies. First, it cannot be implemented in small learning problems where the number of labeled instances is small (e.g., smaller than the “long tail”). Secondly, in Geifman and El-Yaniv’s solution, the architecture is fixed after it has been initially optimized. This means that the final model, which may require a larger architecture, is likely to be sub-optimal. Here, we initiate the discussion of architecture optimization in active learning within the context of deep neural models. Surprisingly, the problem of hyperparameter selection in classical models (such as SVMs) has not been discussed for the most part. One exception is the work of Huang et al. [@huang2015efficient] who briefly considered this problem in the context of linear models and showed that active learning performance curves can be significantly enhanced using a proper choice of (fixed) hyperparameters. Huang et al. however, chose the hyperparameters in hindsight. In contrast, we consider a dynamic optimization of neural architectures during the active learning session. In *neural architecture search* (NAS), the goal is to devise algorithms that automatically optimize the neural architecture for a given problem. Several NAS papers have recently proposed a number of approaches. In [@zoph2016neural], a reinforcement learning algorithm was used to optimize the architecture of a neural network. In [@zoph2017learning], a genetic algorithm is used to optimize the structure of two types of “blocks” (a combination of neural network layers and building components) that have been used for constructing architectures. The number of blocks comprising the full architecture was manually optimized. It was observed that the optimal number of blocks is mostly dependent on the size of the training set. More efficient optimization techniques were proposed in [@liu2017progressive; @pham2018efficient; @real2018regularized; @liu2018darts]. In all these works, the architecture search algorithms were focused on optimizing the structure of one (or two) blocks that were manually connected together to span the full architecture. The algorithm proposed in [@liu2017hierarchical] optimizes both the block structure and the number of blocks simultaneously. Problem Setting {#sec:setting} =============== We first define a standard supervised learning problem. Let ${{\cal X}}$ be a feature space and ${{\mathcal Y}}$ be a label space. Let $P(X,Y)$ be an unknown underlying distribution, where $X \in {{\cal X}}$, $Y \in {{\mathcal Y}}$. Based on a labeled training set $S_m =\{(x_i,y_i)\}$ of $m$ labeled training samples, the goal is to select a prediction function $f\in \cal{F}$, $f:\cal{X}\rightarrow \cal{Y}$, so as to minimize the risk $R_\ell(f)={\mathbf{E}}_{(X,Y)}[\ell(f(x),y)]$, where $\ell(\cdot) \in {\mathbb{R}}^{+}$ is a given loss function. For any labeled set $S$ (training or validation), the empirical risk over $S$ is defined as $\hat{r}_S(f)=\frac{1}{|S|}\sum_{i=1}^{|S|}\ell(f(x_i),y_i).$ In the pool-based active learning setting, we are given a set $U=\{x_1,x_2,...x_u\}$ of unlabeled samples. Typically, the acquisition of unlabeled instances is cheap and, therefore, $U$ can be very large. The task of the active learner is to choose points from $U$ to be labeled by an annotator so as to train an accurate model for a given labeling budget (number of labeled points). The points are selected by a query function denoted by $Q$. Query functions often select points based on information inferred from the current model $f_\theta$, the existing training set $S$, and the current pool $U$. In the mini-batch pool-based active learning setting, the $n$ points to be labeled are queried in bundles that are called mini-batches such that a model is trained after each mini-batch. NAS is formulated as follows. Consider a class ${{\cal A}}$ of architectures, where each architecture $A \in {{\cal A}}$ represents a hypothesis class containing all models $f_\theta \in A$, where $\theta$ represents the parameter vector of the architecture $A$. The objective in NAS is to solve $$\label{eq:nas} A={\operatornamewithlimits{argmin}}_{A\in{{\cal A}}}\min_{f_\theta \in A|S}(R_\ell(f)).$$ Since $R_\ell(f)$ depends on an unknown distribution, it is typically proxied by an empirical quantity such as $\hat{r}_S(f)$ where $S$ is a training or validation set. Deep Active Learning with a Neural Architecture Search {#sec:active_nas} ====================================================== In this section we define a neural architecture search space over which we apply a novel search algorithm. This search space together with the algorithm constitute a new NAS technique that drives our new active algorithm. Modular Architecture Search Space {#sec:search_space} --------------------------------- Modern neural network architectures are often modeled as a composition of one or several basic building *blocks* (sometimes referred to as “cells”) containing several layers [@he2016deep; @iandola2014densenet; @zagoruyko2016wide; @xie2017aggregated; @howard2017mobilenets]. *Stacks* are composed of several blocks connected together. The full architecture is a sequence of stacks, where usually down-sampling and depth-expansion are performed between stacks. For example, consider the Resnet-18 architecture. This network begins with two initial layers and continues with four consecutive stacks, each consisting of two Resnet *basic blocks*, followed by an average pooling and ending with a softmax layer. The Resnet basic block contains two batch normalized $3\times3$ convolutional layers with a ReLU activation and a residual connection. Between every two stacks, the feature maps’ resolution is reduced by a factor of 2 (using a strided convolution layer), and the width (number of feature maps in each layer, denoted as $W$) is doubled, starting from 64 in the first block. This classic architecture has several variants, which differ by the number and type of blocks in each stack. In this work, we consider “homogenous” architectures composed of a single block type and with each stack containing the same number of blocks. We denote such an architecture by $A(B,N_{blocks},N_{stacks})$, where $B$ is the block, $N_{blocks}$ is the number of blocks in each stack, and $N_{stacks}$ is the number of stacks. For example, using this notation, Resnet-18 is $A(B_r,2,4)$ where $B_r$ is the Resnet basic block. Figure \[fig:arch\] depicts the proposed homogeneous architecture.     For a given block $B$, we define a modular architecture search space as ${{\cal A}}=\{A(B,i,j):i\in\{1,2,3,...,N_{blocks}\}, j\in\{1,2,3,...,N_{stacks}\}\}$, which is simply all possible architectures spanned by the grid defined by the two corners $A(B,1,1)$ and $A(B,N_{blocks},N_{stacks})$. Clearly, the space ${{\cal A}}$ is restricted in the sense that it only contains a limited subspace of architectures but nevertheless it contains $N_{blocks}\times N_{stacks}$ architectures with diversity in both numbers of layers and parameters. Search Space as an Acyclic Directed Graph (DAG) {#sec:DAG} ----------------------------------------------- The main idea in our search strategy is to start from the smallest possible architecture (in the modular search space) and iteratively search for an optimal incremental architectural expansion within the modular search space. We define the *depth* of an architecture to be the number of layers in the architecture. We denote the depth of $A(B,i,j)$ by $|A(B,i,j)|=ij\beta + \alpha$, where $\beta$ is the number of layers in the block $B$ and $\alpha$ is the number of layers in the *initial block* (all the layers appearing before the first block) plus the number of layers in the *classification block* (all the layers appearing after the last block). It is convenient to represent the architecture search space as a directed acyclic graph (DAG) $G=(V,E)$, where the vertex set $V=\{A(B,i,j) \}$, $B$ is a fixed neural block (e.g., a Resnet basic block), $i\in\{1,2,\ldots,N_{blocks}\}$ is the number of blocks in each stack, and $j\in\{1,2,3,\ldots,N_{stacks}\}$ is the number of stacks. The edge set $E$ is defined based on two incremental expansion steps. The first step, increases the depth of the network without changing the number of stacks (i.e., without affecting the width), and the second step increases the depth while also increasing the number of stacks (i.e., increasing the width). Both increment steps are defined so as to perform the minimum possible architectural expansion (within the search space). Thus, when expanding $A(B,i,j)$ using the first step, the resulting architecture is $A(B,i+1,j)$. When expanding $A(B,i,j)$ using the second step, we reduce the number of blocks in each stack to perform a minimal expansion resulting in the architecture $A(B,\lfloor \frac{ij}{j+1} \rfloor +1, j+1)$. The parameters of the latter architecture are obtained by rounding up the solution $i'$ of the following problem, $$\begin{array}{l} i'={\operatornamewithlimits{argmin}}_{i'>0}|A(B,i',j+1)| \\ s.t. |A(B,i',j+1)|>|A(B,i,j)| \end{array}.$$ We conclude that each of these steps are indeed depth-expanding. In the first step, the expansion is only made along the depth dimension, while the second step affects the number of stacks and expands the width as well. In both steps, the incremental step is the smallest possible within the modular search space. In Figure \[fig:grid\], we depict the DAG $G$ on a grid whose coordinates are $i$ (blocks) and $j$ (stacks). The modular search space in this example is all the architectures in the range $A(B,1,1)$ to $A(B,5,4)$. The arrows represents all edges in $G$. In this formulation, it is evident that every path starting from any architecture can be expanded up to the largest possible architecture. Moreover, every architecture is reachable when starting from the smallest architecture $A(B,1,1)$. These two properties serve well our search strategy. Incremental Neural Architecture Search -------------------------------------- The proposed *incremental neural architecture search* (iNAS) procedure is described in Algorithm \[alg:evolutionary\_NAS\] and operates as follows. Given a small initial architecture $A(B,i_0,j_0)$, a training set $S$, and an architecture search space ${{\cal A}}$, we first randomly partition the set $S$ into training and validation subsets, $S'$ and $V'$, respectively, $S=S'\cup V'.$ On iteration $t$, a set of candidate architectures is selected based on the edges of the search DAG (see Section \[sec:DAG\]) including the current architecture and the two connected vertices (lines 5-6). This step creates a candidate set, ${{\cal A}}'$, consisting of three models, ${{\cal A}}'=\{A(B,i,j), A(B,\lfloor \frac{ij}{j+1} \rfloor +1, j+1), A(B,i+1, j)\}.$ In line 7, the best candidate in terms of validation performance is selected and denoted $A_t=A(B,i_t,j_t)$. The optimization problem formulated in line 7 is an approximation of the NAS objective formulated in Equation (\[eq:nas\]). The algorithm terminates whenever $A_t=A_{t-1}$, or a predefined maximum number of iterations is reached (in which case $A_t$ is the final output). **iNAS**($S$,$A(B,i_0,j_0)$, ${{\cal A}}$, $T_{iNAS}$) Let $S', V'$ be an train-test random split of $S$ : $i \gets i_{t-1}; j\gets j_{t-1}$ $\begin{array}{ll} {{\cal A}}'= \{&A(B,i,j),\\ &A(B,\lfloor \frac{ij}{j+1} \rfloor +1, j+1),\\ &A(B,i+1, j)\}\\ \end{array}$ ${{\cal A}}' = {{\cal A}}' \cap {{\cal A}}$ $\begin{array}{l} A(B,i_t,j_t) = \\={\operatornamewithlimits{argmin}}_{A\in {{\cal A}}'} \hat{r}_{V'} ({\operatornamewithlimits{argmin}}_{f_\theta \in A}\hat{r}_{S'} (f_\theta)) \end{array}$ break Return $A(B,i_t,j_t)$ \[alg:evolutionary\_NAS\] **[active-iNAS]{}($U$,$A_0$, ${{\cal A}}$, $Q$, $b$, $k$) $t \gets 1$ $S_t \gets$ Sample $k$ points from $U$ at random $U_0 \gets U \backslash S_1$ $A_t \gets$ iNAS($S$, $A_{t-1}$, ${{\cal A}}$) train $f_\theta \in A_t$ using $S$ Return $f_\theta$ $S' \gets Q(f_\theta, S_{t}, U_{t}, b)$ $S_{t+1} \gets S_{t}\cup S'$ $U_{t+1} \gets U_{t} \backslash S'$ $t \gets t+1$** \[alg:active\_nas\] Active Learning with iNAS ------------------------- The *deep active learning with incremental neural architecture search* (active-iNAS) technique is described in Algorithm \[alg:active\_nas\] and works as follows. Given a pool $U$ of unlabeled instances from ${{\cal X}}$, a set of architectures $\cal{A}$ is induced using a composition of basic building blocks $B$ as shown in Section \[sec:search\_space\], an initial (small) architecture $A_0\in\cal{A}$, a query function $Q$, an initial (passively) labeled training set size $k$, and an active learning batch size $b$. We first sample uniformly at random $k$ points from $U$ to constitute the initial training set $S_1$. We then iterate the following three steps. First, we search for an optimal neural architecture using the iNAS algorithm over the search space ${{\cal A}}$ with the current training set $S_t$ (line 6). The initial architecture for iNAS is chosen to be the selected architecture from the previous active round ($A_{t-1}$), assuming that the architecture size is non-decreasing along the active learning process. The resulting architecture at iteration $t$ is denoted $A_t$. Next, we train a model $f_\theta \in A_t$ based on $S_t$ (line 7). Finally, if the querying budget allows, the algorithm requests $b$ new points using $Q(f_\theta,S_t,U_t,b)$ and updates $S_{t+1}$ and $U_{t+1}$ correspondingly. Otherwise the algorithm returns $f_\theta$ (lines 8-14). Theoretical Motivation and Implementation Notes ----------------------------------------------- The iNAS algorithm is designed to exploit the prior knowledge gleaned from samples of increasing size, which is motivated from straightforward statistical learning arguments. iNAS starts with small capacity so as to avoid over-fitting in early stages, and then allows for capacity increments as labeled data accumulates. Recall from statistical learning theory that for a given hypothesis class ${{\cal F}}$ and training set $S_m$, the generalization gap can be bounded as follows with probability at least $1-\delta$, $$R(f)-\hat{r}_{S_m}(f)\leq O(\sqrt{\frac{d\log(m/d)+\log(1/\delta)}{m}}),$$ where $d$ is the VC-dimension of ${{\cal F}}$. Recently, Bartlett et al. [@bartlett2019nearly] showed a nearly tight bound for the VC-dimension of deep neural networks. Let $W$ be the number of parameters in a neural network, let $L$ be the number of layers, and $U$, the number of computation units (neurons/filters), [@bartlett2019nearly] showed that the VC dimension of ReLU-activated regression models is bounded as $VCdim({{\cal F}})\leq O(\bar{L}W\log(U)$, where $\bar{L}{\triangleq}\frac{1}{W}\sum_{i=1}^{L}W_i$ and $W_i$ is the number of parameters from the input to layer $i$. As can be seen, the expansion steps proposed in iNAS are designed to minimally expand the VC-dimension of ${{\cal F}}$. When adding blocks, $W$, $U$ and $\bar{L}$ grow linearly. As a result, the VC-dimension grows linearly. When adding a stack (in the iNAS algorithm), $W$ and $U$ grow sub-exponentially, and $L$ (and $\bar{L}$) also grows. Along the active learning session, $m$ grows linearly in incremental steps, thus, it motivates a linear growth in the VC-dimension (in incremental steps) so as to maintain the generalization gap bound as small as possible. Alternate approaches that are often used, such as a full grid-search on each active round, would not enjoy these benefits and will be prone to overfitting (not to mention that full-grid search could be computationally prohibitive). Turning now to analyze the run time of active-iNAS, when running with small active learning mini-batches, it is evident that the iNAS algorithm will only require one iteration at each round, resulting in only having to train three additional models at each round. In our implementation of iNAS, we apply “premature evaluation” as considered in [@tan2018mnasnet]; our models are evaluated after $T_{SGD}/4$ epochs where $T_{SGD}$ is the total number of epochs in each round. Our final active-iNAS implementation thus only takes $1.75T_{SGD}$ for each active round. For example, in the CIFAR-10 experiment $T_{SGD}=200$ requires less than 2 GPU hours (on average) for an active learning round (Nvidia Titan-Xp GPU). Experiments {#sec:experiments} =========== We first compare active-iNAS to active learning performed with a fixed architecture over three datasets, we apply three querying functions, softmax response, coresets and MC-dropout. Then we analyze the architectures learned by iNAS along the active process. We also empirically motivate the use of iNAS by showing how optimized architecture can improve the query function. Finally, we compare the resulting active learning algorithm obtained with the active-iNAS framework. Experimental Setting -------------------- We used an architecture search space that is based on the Resnet architecture [@he2016deep]. The *initial block* contains a convolutional layer with filter size of $3\times 3$ and depth of 64, followed by a max-pooling layer having a spatial size of $3\times 3$ and strides of $2$. The *basic block* contains two convolutional layers of size $3\times 3$ followed by a ReLU activation. A residual connection is added before the activation of the second convolutional layer, and a batch normalization [@ioffe2015batch] is used after each layer. The classification block contains an average pooling layer that reduces the spatial dimension to $1\times 1$, and a fully connected classification layer followed by softmax. The search space is defined according to the formulation in Section \[sec:search\_space\], and spans all architectures in the range $A(B_r,1,1)$ to $A(B_r,12,5)$. As a baseline, we chose two fixed architectures. The first architecture was the one optimized for the first active round (optimized over the initial seed of labeled points), and which coincidentally happened to be $A(B_r,1,2)$ on all tested datasets. The second architecture was the well-known Resnet-18, denoted as $A(B_r,2,4)$, which is some middle point in our search grid. We trained all models using *stochastic gradient descent* (SGD) with a batch size of 128 and momentum of 0.9 for 200 epochs. We used a learning rate of 0.1, with a learning rate multiplicative decay of 0.1 after epochs 100 and 150. Since we were dealing with different sizes of training sets along the active learning process, the epoch size kept changing. We fixed the size of an epoch to be 50,000 instances (by oversampling), regardless of the current size of the training set $S_t$. A weight decay of $5e$-$4$ was used, and standard data augmentation was applied containing horizontal flips, four pixel shifts and up to 15-degree rotations. The active learning was implemented with an initial labeled training seed ($k$) of 2000 instances. The active mini-batch size ($b$) was initialized to 2000 instances and updated to 5000 after reaching 10000 labeled instances. The maximal budget was set to 50,000 for all datasets[^1]. For time efficiency reasons, the iNAS algorithm was implemented with $T_{iNAS}=1$, and the training of new architectures in iNAS was early-stopped after 50 epochs, similar to what was done in [@tan2018mnasnet]. Active-iNAS vs. Fixed Architecture {#sec:results} ---------------------------------- The results of an active learning algorithm are often depicted by a curve measuring the trade-off between labeled points (or a budget) vs. performance (accuracy in our case). For example, in Figure \[fig:cifar10\](a) we see the results obtained by active-iNAS and two fixed architectures for classifying CIFAR-10 images using the softmax response querying function. In black (solid), we see the curve for the active-iNAS method. The results of $A(B_r,1,2)$ and Resnet-18 ($A(B_r,2,4)$) appear in (dashed) red and (dashed) blue, respectively. The $X$ axis corresponds to the labeled points consumed, starting from $k=2000$ (the initial seed size), and ending with 50,000 . In each active learning curve, the standard error of the mean over three random repetitions is shadowed. We present results for CIFAR-10, CIFAR-100 and SVHN. We first analyze the results for CIFAR-10 (Figure \[fig:cifar10\]). Consider the graphs corresponding to the fixed architectures (red and blue). It is evident that for all query functions, the small architecture (red) outperforms the big one (Resnet-18 in blue) in the early stage of the active process. Later on, we see that the big and expressive Resnet-18 outperforms the small architecture. Active-iNAS, performance consistently and significantly outperforms both fixed architectures almost throughout the entire range. It is most striking that active-iNAS is better than each of the fixed architectures even when all are consuming the entire training budget. Later on we speculate about the reason for this phenomenon as well as the switch between the red and blue curves occurring roughly around 15,000 training points (in Figure \[fig:cifar10\](a)). Turning now to CIFAR-100 (Figure \[fig:cifar100\]), we see qualitatively very similar behaviors and relations between the various active learners. We now see that the learning problem is considerably harder, as indicated by the smaller area under all the curves. Nevertheless, in this problem active-iNAS achieves a substantially greater advantage over the fixed architectures in all three query functions. Finally, in the SVHN digit classification task, which is known to be easier than both the CIFAR tasks, we again see qualitatively similar behaviors that are now much less pronounced, as all active learners are quite effective. On the other hand, in the SVHN task, active-iNAS impressively obtains almost maximal performance after consuming only 20% of the training budget. Analyzing the Learned Architectures ----------------------------------- In addition to standard performance results presented in Section \[sec:results\], it is interesting to inspect the sequence of architectures that have been selected by iNas along the active learning process. In Figure \[fig:architectures\] we depict this dynamics; for example, consider the CIFAR-10 dataset appearing in solid lines, where the blue curve represents the number of parameters in the network and the black shows the number of layers in the architecture. Comparing CIFAR-10 (solid) and CIFAR-100 (dashed), we see that active-iNAS prefers, for CIFAR-100, deeper architectures compared to its choices for CIFAR-10. In contrast, in SVHN (dotted), active-iNAS gravitates to shallower but wider architectures, which result significantly larger numbers of parameters. The iNAS algorithm is relatively stable in the sense that in the vast majority of random repeats of the experiments, similar sequences of architectures have been learned (this result is not shown in the figure). A hypothesis that might explain the latter results is that CIFAR-100 contains a larger number of “concepts” requiring deeper hierarchy of learned CNN layers compared to CIFAR-10. The SVHN is a simpler and less noisy learning problem, and, therefore, larger architectures can play without significant risk of overfitting. Enhanced Querying with Active-iNAS ---------------------------------- In this section we argue and demonstrate that optimized architectures not only improve generalization at each step, but also enhance the query function quality[^2]. In order to isolate the contribution of the query function, we normalize the active performance by the performance of a passive learner obtained with the same model. A common approach for this normalization has already been proposed in [@huang2015efficient; @baram2004online], we define conceptually similar normalization as follows. Let the *relative AUC gain* be the relative reduction of area under the curve (AUC) of the 0-1 loss in the active learning curve, compared to the AUC of the passive learner (trained over the same number of random queries, at each round); namely, $\textnormal{AUC-GAIN}(PA, AC, m)=\frac{AUC_m(PA)-AUC_m(AC)}{AUC_m(PA)},$ where $AC$ is an active learning algorithm, $PA$ is its passive application (with the same architecture), $m$ is a labeling budget, and $AUC_m(\cdot)$ is the area under the learning curve (0-1 loss) of the algorithm with budget $m$. Clearly, high values of AUC-GAIN correspond to high performance and vice versa. In Figure \[fig:AUC-GAIN\], we used the AUC-GAIN to measure the performance of the softmax response querying function on the CIFAR-10 dataset over all training budgets up to the maximal (50,000). We compare the performance of this query function applied over two different architectures: the small architecture ($A(B_r,1,2$), and Resnet-18 ($A(B_r,2,4$). We note that it is unclear how to define AUC-GAIN for active-iNAS because it has a dynamically changing architecture. As can easily be seen, the small architecture dramatically outperforms Resnet-18 in the early stages. Later on, the AUC-GAIN curves switch, and Resnet-18 catches up and outperforms the small architecture. This result supports the intuition that improvements in the generalization tend to improve the effectiveness of the querying function. We hypothesize that the active-iNAS’ outstanding results shown in Section \[sec:results\] have been achieved not only by the improved generalization of every single model, but also by the effect of the optimized architecture on the querying function. Query Function Comparison ------------------------- In Section \[sec:results\] We demonstrated that active-iNAS consistently outperformed direct active applications of three querying functions. Here, we compare the performance of the three active-iNAS methods, applied with those three functions: softmax response, MC-dropout and coreset. In Figure \[fig:comparison\] we compare these three active-iNAS algorithms over the three datasets. In all three datasets, softmax response is among the top performers, whereas one or the other two querying functions is sometimes the worst. Thus, softmax response achieves the best results. For example, on CIFAR-10 and SVHN, the MC-dropout is on par with softmax, but on CIFAR-100 MC-dropout is the worst. The poor performance of MC-dropout over CIFAR-100 may be caused by the large number of classes, as pointed out by [@geifman2017selective] in the context of selective classification. In all cases, coreset is slightly behind the softmax response. This is in sharp contrast to the results presented by [@sener2018active] and [@geifman2017deep]. We conclude this section by emphasizing that our results indicate that the combination of softmax response with active-iNAS is the best active learning method. Concluding Remarks ================== We presented active-iNAS, an algorithm that effectively integrates deep neural architecture optimization with active learning. The active algorithm performs a monotone search for the locally best architecture on the fly. Our experiments indicate that active-iNAS outperforms standard active learners that utilize suitable and commonly used fixed architecture. In terms of absolute performance quality, to the best of our knowledge, the combination of active-iNAS and softmax response is the best active learner over the datasets we considered. Acknowledgments {#acknowledgments .unnumbered} =============== This research was supported by The Israel Science Foundation (grant No. 81/017). [^1]: SVHN contains 73,256 instances and was, therefore, trimmed to 50000. [^2]: We only consider querying functions that are defined in terms of a model (such as all query functions considered here).

--- abstract: | We extend the formula for the Chern classes of blow-ups of algebraic varieties due to Porteous and Lascu–Scott, and of symplectic and complex manifolds due to Geiges and Pasquotto, to the blow–ups of almost complex manifolds. Our approach is based on a concrete partition for the tangent bundle of a blow–up. The use of topological K–theory of vector bundles simplifies the classical approaches. 2000 Mathematical Subject Classification: : 53D35, 57R20 Key words and phrases: : Blow ups; Almost complex manifolds; K–theory, Chern classes. author: - | Haibao Duan[^1]\ Academy of Mathematics and Systems Sciences,\ Chinese Academy of Sciences, Beijing 100190\ [email protected] title: 'A K–theory approach to the tangent invariants of blow–ups' --- Introduction ============ In this paper all manifolds under consideration are in the real and smooth category, which are connected but not necessarily compact and orientable. Let $X\subset M$ be a smooth submanifold whose normal bundle has a complex structure, and let $\widetilde{M}$ be the blow–up of $M$ along $X$. We present a partition for the tangent bundle $\tau _{\widetilde{M}}$ of $\widetilde{M}$ which implies that, if $X\subset M$ is an embedding in the category of *almost complex manifolds*, then the blow-up $\widetilde{M}$ has a canonical almost complex structure, see Theorem 2.3 and Corollary 2.4. The partition on $\tau _{\widetilde{M}}$ is ready to apply to deduce a formula for the tangent bundle $\tau _{\widetilde{M}}$ of the blow–up $\widetilde{M}$ in the $K$–theory $K(\widetilde{M})$ of complex bundles over $\widetilde{M}$, see Theorem 3.2 and Remark 3.3, which in turn yields a formula for the total Chern class $C(\widetilde{M})$ of $\widetilde{M}$, see Theorem 4.6. Historically, the formula for the Chern class of a blow–up of a nonsingular variety was conjectured by J. A. Todd [@[T]] and B. Segre [@[S]], confirmed by I. R. Porteous [@[P]] and by Lascu–Scott [@[LS1]; @[LS2]], generalized to the blow ups of possibly singular varieties along regularly embedded centers by Aluffi [@[A]]. It has also been extended to the blow–ups in the categories of symplectic and complex manifolds by H. Geiges and F. Pasquotto [@[GP]]. While establishing the formula in its natural generality we demonstrate also an approach with deserved simplicity. The author would like to thank P. Aluffi for bring his attention to the topic. Geometry of a blow–up ===================== For an Euclidean vector bundle $\xi $ over a topological space $Y$ write $D(\xi )$ and $S(\xi )$ for the unit disk bundle and sphere bundle of $\xi $, respectively. If $A\subset Y$ is a subspace (resp. if $f:W\rightarrow Y$ is a map) write $\xi \mid A$ (resp. $f^{\ast }\xi $) for the restriction of $\xi $ to $A$ (resp. the induced bundle over $W$). Assume throughout that $i_{X}:X\rightarrow M$ is a smooth embedding that embeds $X$ as a closed subset of $M$, and whose normal bundle $\gamma _{X}$ has a fixed complex structure $J$. Furnish $M$ with an Riemannian metric so that the induced metric on $\gamma _{X}$ is Hermitian in the sense of Milnor [@[MS] p.156]. Let $\pi :E=\mathbb{P}(\gamma _{X})$ $\rightarrow X$ be the complex projective bundle associated with $\gamma _{X}$. Since the tautological line bundle $\lambda _{E}$ on $E$ is a subbundle of the induced bundle $\pi ^{\ast }\gamma _{X}$ we can formulate the composition > $G:D(\lambda _{E})\subset D(\pi ^{\ast }\gamma _{X})\overset{\widehat{\pi }}{\rightarrow }D(\gamma _{X})$, where $\widehat{\pi }$ is the obvious bundle map over $\pi $. Regard $E\subset D(\lambda _{E})$ and $X\subset D(\gamma _{X})$ as the zero sections of the corresponding disk bundles, respectively. **Lemma 2.1.** *The map* $G$* agrees with the projection* $\pi $* on* $E$*,* *and* *restricts to a diffeomorphism* $D(\lambda _{E})\smallsetminus E\rightarrow D(\gamma _{X})\smallsetminus X$*.* **Proof.** Explicitly we have > $D(\lambda _{E})=\{(l,v)\in E\times \pi ^{\ast }\gamma _{X}\mid v\in l\in > E,\left\Vert v\right\Vert ^{2}\leq 1\}$, > > $D(\gamma _{X})=\{(x,v)\in X\times \gamma _{X}\mid v\in \gamma _{X}\mid > x,\left\Vert v\right\Vert ^{2}\leq 1\}$. The inverse of $G$ on $D(\gamma _{X})\smallsetminus X$ is $(x,v)\rightarrow (\left\langle v\right\rangle ,v)$, where $v\in \gamma _{X}\mid x$ with $v\neq 0$ and where $\left\langle v\right\rangle \in E$ denotes the complex line spanned by a non–zero vector $v$.$\square $ It follows from Lemma 2.1 that the map $G$ restricts to a diffeomorphism > $g=$ $G\mid S(\lambda _{E}):S(\lambda _{E})=\partial D(\lambda > _{E})\rightarrow S(\gamma _{X})=\partial D(\gamma _{X})$. With respect to the metric on $M$ identifying $D(\gamma _{X})$ with a tubular neighborhood of $X$ in $M$ we can formulate the adjoint manifold 1. $\widetilde{M}=(M\setminus \overset{\circ }{D(\gamma _{X})})\cup _{g}D(\lambda _{E})$ by gluing $D(\lambda _{E})$ to $(M\setminus \overset{\circ }{D(\gamma _{X})})$ along $S(\lambda _{E})$ using $g$. Moreover, piecing together the identity on $M\setminus \overset{\circ }{D(\gamma _{X})}$ and the map $G$ yields the smooth map 1. $f:\widetilde{M}=(M\setminus \overset{\circ }{D(\gamma _{X})})\cup _{g}D(\lambda _{E})\rightarrow M=(M\setminus \overset{\circ }{D(\gamma _{X})})\cup _{id}D(\gamma _{X})$, which is known as the* blow–up* of $M$ along $X$ with *exceptional divisor* $E$ [@[M]]. Obvious but useful properties of the map $f$ are listed below. **Lemma 2.2.** *Let* $i_{E}:E\rightarrow \widetilde{M}$* (resp.* $i_{X}:X\rightarrow M$*) be the zero section of* $D(\lambda _{E})$* (resp. of* $D(\gamma _{X})$*)* *in view of the decomposition (2.1). Then* > *i) the normal bundle of* $E$* in* $\widetilde{M}$ *is* $\lambda _{E}$*;* > > *ii)* $f^{-1}(X)=E$* with* $f\circ i_{E}=$* *$i_{X}\circ \pi $*;* > > *iii)* $f$* restricts to a diffeomorphism:* $\widetilde{M}\setminus E\rightarrow M\setminus X$*.*$\square $ For a manifold $N$ write $\tau _{N}$ for its tangent bundle. One has then the obvious bundle decompositions $\tau _{D(\lambda _{E})}\mid S(\lambda _{E})=\tau _{S(\lambda _{E})}\oplus \mathbb{R}(\alpha _{1})$; $\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X})=\tau _{S(\gamma _{X})}\oplus \mathbb{R}(\alpha _{2})$, where $\alpha _{1}$ (resp. $\alpha _{2}$) is the outward (resp. inward) unit normal field along the boundary $S(\lambda _{E})=\partial D(\lambda _{E})$ (resp. $S(\gamma _{X})=\partial (M\setminus \overset{\circ }{D(\gamma _{X})})$) with $\mathbb{R}(\alpha _{i})$ the trivial real line bundle spanned by the field $\alpha _{i}$. Moreover, if we let $\tau _{g}$ be the tangent map of the diffeomorphism $g$ then the decomposition (2.1) of $\widetilde{M}$ indicates the partition 1. $\tau _{\widetilde{M}}=\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\dbigcup\limits_{h}\tau _{D(\lambda _{E})}$, where the gluing diffeomorphism > $h:$ $\tau _{D(\lambda _{E})}\mid S(\lambda _{E})\rightarrow \tau > _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X})$ is the bundle map over $g$ with > $h(t\alpha _{1},u)=(t\alpha _{2},\tau _{g}(u))$, $u\in \tau _{S(\lambda > _{E})}$, $t\in \mathbb{R}$. Indeed, the two bundles $\tau _{D(\lambda _{E})}\mid S(\lambda _{E})$ and $\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X})$ admit more subtle decompositions with respect to them the gluing map $h$ in (2.3) admits a useful presentation. Let $p_{E}:$ $\lambda _{E}\rightarrow E$ and $p_{X}:$ $\gamma _{X}\rightarrow X$ be the obvious projections. The same notions will be reserved for their restrictions to the subspaces $S(\lambda _{E})\subset D(\lambda _{E})\subset \lambda _{E}$ and $S(\gamma _{X})\subset D(\gamma _{X})\subset \gamma _{X}$, respectively. For a topological space $Y$ write $1_{\mathbb{C}}$ (resp. $1_{\mathbb{R}}$) for the trivial complex line bundle $Y\times \mathbb{C}$ (resp. the trivial real line bundle $Y\times \mathbb{R}$) over $Y$. For a complex vector bundle $\xi $ write $\xi ^{r}$ for its real reduction. As example the trivialization over* *$S(\lambda _{E})$ 1. $(p_{E}{}^{\ast }\lambda _{E}\mid S(\lambda _{E}))^{r}=\mathbb{R}(\alpha _{1})\oplus \mathbb{R}(J(\alpha _{1}))$ indicates that $p_{E}{}^{\ast }\lambda _{E}\mid S(\lambda _{E})=1_{\mathbb{C}}$, where $J(\alpha _{1})$ is a unit tangent vector field on $S(\lambda _{E})$. Let > $\widehat{g}:g^{\ast }(\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X}))\rightarrow \tau _{M\setminus \overset{\circ }{D(\gamma > _{X})}}\mid S(\gamma _{X})$ be the induced bundle of $g$ over $S(\lambda _{E})$, and let > $\kappa :\tau _{D(\lambda _{E})}\mid S(\lambda _{E})\rightarrow g^{\ast > }(\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X}))$ be the bundle isomorphism over the identity of $S(\lambda _{E})$ so that $h=\widehat{g}\circ \kappa $ [@[MS] Lemma 3.1]. With respect to the Hermitian metric induced from $\gamma _{X}$ one has the orthogonal decomposition $\pi ^{\ast }\gamma _{X}=\lambda _{E}\oplus \lambda _{E}^{\perp }$ in which $\lambda _{E}^{\perp }$ denotes the orthogonal complement of $\lambda _{E}$ in $\pi ^{\ast }\gamma _{X}$. **Theorem 2.3.** *The tangent bundle of the blow up* $\widetilde{M}$* has the partition* > $\tau _{\widetilde{M}}=\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\dbigcup\limits_{\widehat{g}\circ \kappa }\tau _{D(\lambda _{E})}$, *in which* > *i)* $\tau _{D(\lambda _{E})}\mid S(\lambda _{E})=(\pi \circ > p_{E})^{\ast }\tau _{X}\oplus (p_{E}{}^{\ast }\lambda _{E})^{r}\oplus > p_{E}{}^{\ast }Hom(\lambda _{E},\lambda _{E}^{\perp })^{r}$*;* > > *ii)* $g^{\ast }(\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X}))=(\pi \circ p_{E})^{\ast }\tau _{X}\oplus > (p_{E}{}^{\ast }\lambda _{E})^{r}\oplus p_{E}{}^{\ast }(\lambda _{E}^{\perp > })^{r}$*.* *Moreover, with respect to the decompositions i) and ii) the bundle isomorphism* $\kappa $* is given by* > *a)* $\kappa \mid (\pi \circ p_{E})^{\ast }\tau _{X}=id;$ > > *b)* $\kappa \mid (p_{E}{}^{\ast }\lambda _{E})^{r}=id$*;* > > *c)* $\kappa (b)=b(\alpha _{1})$* for* $b\in > Hom(p_{E}{}^{\ast }\lambda _{E},p_{E}{}^{\ast }\lambda _{E}^{\perp })^{r}$*.* **Proof.** It follows from the standard decompositions > $\tau _{E}=\pi ^{\ast }\tau _{X}\oplus Hom(\lambda _{E},\lambda _{E}^{\perp > })^{r}$, $\quad \tau _{D(\lambda _{E})}=(p_{E}{}^{\ast }\lambda > _{E})^{r}\oplus p_{E}{}^{\ast }\tau _{E}$ that 1. $\tau _{D(\lambda _{E})}=(p_{E}{}^{\ast }\lambda _{E})^{r}\oplus (\pi \circ p_{E})^{\ast }\tau _{X}\oplus p_{E}{}^{\ast }Hom(\lambda _{E},\lambda _{E}^{\perp })^{r}$. Similarly, it comes from > $\tau _{D(\gamma _{X})}=p_{X}{}^{\ast }\tau _{X}\oplus p_{X}{}^{\ast }\gamma > _{X}$, $\pi {}^{\ast }\gamma _{X}=\lambda _{E}\oplus \lambda _{E}^{\perp }$, as well as the definition of $f$ that 1. $f^{\ast }\tau _{D(\gamma _{X})}=(p_{E}{}^{\ast }\lambda _{E})^{r}\oplus (\pi \circ p_{E})^{\ast }\tau _{X}\oplus (p_{E}{}^{\ast }\lambda _{E}^{\perp })^{r}$. One obtains the relations i) and ii) by restricting the decomposition (2.5) and (2.6) to the subspace $S(\lambda _{E})\subset D(\lambda _{E})$, respectively. Finally, properties a), b), c) are transparent in view of the relation $h=\widehat{g}\circ \kappa $, together with the description of $g$ indicated in the proof of Lemma 2.1.$\square $ A manifold $M$ is called *almost complex* if its tangent bundle is furnished with a complex structure $J_{M}$ [@[MS] p.151]. Given two almost complex manifolds $(X,J_{X})$ and $(M,J_{M})$ an embedding $i_{X}:$ $X\rightarrow M$ is called *almost complex* if $\tau _{X}$ is a complex subbundle of the restricted bundle $\tau _{M}\mid X$. In this situation the normal bundle $\gamma _{X}$ of $X$ has the canonical complex structure $J$ induced from that on $\tau _{M}\mid X$ and that on $\tau _{X}$, hence the blow–up $\widetilde{M}$ of $M$ along $X$ is defined. Moreover, in view of the decomposition (2.1) we note that > i\) $J_{M}$ restricts to an almost complex structure on $M\setminus \overset{\circ }{D(\gamma _{X})}$; > > ii\) the tubular neighborhood $D(\lambda _{E})$ of $E$ in $\widetilde{M}$ has the canonical almost complex structure so that as a complex vector bundle > > $\qquad \tau _{D(\lambda _{E})}=(\pi \circ p_{E})^{\ast }\tau _{X}\oplus > p_{E}{}^{\ast }\lambda _{E}\oplus Hom(p_{E}{}^{\ast }\lambda > _{E},p_{E}{}^{\ast }\lambda _{E}^{\perp })$ (compare this with (2.5)). Since with respect to the induced complex structures on $\tau _{D(\lambda _{E})}\mid S(\lambda _{E})$ and on $\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X})$ the clutching map $h$ in (2.3) is $\mathbb{C}$–linear by Theorem 2.3, one obtains **Corollary 2.4.** *If* $i_{X}:$* *$X\rightarrow M$* is an embedding in the category of almost complex manifolds, then the blow–up* $\widetilde{M}$* has a canonical almost complex structure that is compatible with that on* $M\setminus \overset{\circ }{D(\gamma _{X})}$* and that on* $D(\lambda _{E})$.$\square $ **Remark 2.5.** The analogue of Corollary 2.4 in the symplectic setting is due to McDuff [@[M] Section 3], which concludes that if $i_{X}:$* *$X\rightarrow M$* *is an embedding of symplectic manifolds, then the blow–up $\widetilde{M}$ admits a symplectic form which coincides with the one on $M\backslash X$ off the exceptional divisor $E$.$\square $ The tangent bundle of a blow–up =============================== For a topological space $Y$ let $K(Y)$ (resp. $\widetilde{K}(Y)$) be the $K$–theory (reduced $K$–theory) of complex vector bundles over $Y$. If $i_{X}: $* *$X\rightarrow M$ is an embedding in the category of almost complex manifolds, then the blow–up $\widetilde{M}$ has a canonical almost complex structure by Corollary 2.4. In particular, the difference $\tau _{\widetilde{M}}-f^{\ast }\tau _{M}$ can be regarded as an element of the ring $\widetilde{K}(\widetilde{M})$. In Theorem 3.2 below we obtain a formula expressing the element $\tau _{\widetilde{M}}-f^{\ast }\tau _{M}\in \widetilde{K}(\widetilde{M})$ in term of the decomposition $p_{E}^{\ast }\gamma _{X}=p_{E}^{\ast }\lambda _{E}\oplus p_{E}^{\ast }\lambda _{E}^{\perp }$. For a relative CW–complex $(Y,A)$ the inclusion $j:(Y,\emptyset )\rightarrow (Y;A)$ induces a homomorphism 1. $j^{\ast }:K(Y;A)\rightarrow \widetilde{K}(Y)$, where $K(Y;A)$ is the relative $K$–group of the pair $(Y;A)$ defined by 1. $K(Y;A)=:\widetilde{K}(Y/A)$. In addition, the group $K(Y;A)$ admits another description useful in the subsequent calculation. **Lemma 3.1** ([@[At] Theorem 2.6.1]). *Any element in the group* $K(Y;A)$* can be represented by a triple* $[\xi ,\eta ;\alpha ]$* in which* $\xi $* and* $\eta $* are vector bundles over* $Y$* and* $\alpha :\xi \mid A\rightarrow $* *$\eta \mid A$* is a bundle isomorphism.* *Moreover, with respect to this representation* *of the group* $K(Y;A)$ *one has* > *i) the triple* $[\xi ,\xi ;id]$ *represents the zero* *for any bundle* $\xi $* over* $Y$*;* > > *ii)* $[\xi ,\eta ;\alpha ]+[\xi _{1},\eta _{1};\alpha _{1}]=[\xi > \oplus \xi _{1},\eta \oplus \xi _{1};\alpha \oplus \alpha _{1}]$*;* > > *iii)* $[\xi ,\eta ;\alpha ]\otimes \gamma =[\xi \otimes \gamma ,\eta > \otimes \gamma ;\alpha \otimes id]$*;* > > *iv)* $j^{\ast }[\xi ,\eta ;\alpha ]=\xi -\eta $*,* *where* $\oplus $* means direct sum of vector bundles (homomorphisms), and where* $\otimes $ *denotes the action* $K(X;A)\otimes K(X)\rightarrow K(X;A)$* defined by the tensor product of vector bundles*.$\square $ For an embedding $i_{X}:X\rightarrow M$ of almost complex manifolds let $f:\widetilde{M}\rightarrow M$ be the blow–up of $M$ along $X$ with exceptional divisor $E$. Consider the composition > $j_{E}:K(D(\lambda _{E}),S(\lambda _{E}))\underset{\cong }{\rightarrow }K(\widetilde{M},\widetilde{M}\smallsetminus \overset{\circ }{D}(\lambda _{E}))\overset{j^{\ast }}{\rightarrow }\widetilde{K}(\widetilde{M})$ in which the first map is the *excision isomorphism*. By Lemma 3.1 the trivialization $\varepsilon :\overline{\lambda }_{E}\mid S(\lambda _{E})\rightarrow 1_{\mathbb{C}}$ indicated by (2.4) defines an element $[p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\in K(D(\lambda _{E}),S(\lambda _{E}))$. **Theorem 3.2.** *In the ring *$\widetilde{K}(\widetilde{M})$* one has* 1. $\tau _{\widetilde{M}}-f^{\ast }\tau _{M}=j_{E}([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}^{\ast }\lambda _{E}^{\perp })$. **Proof.** The partition (2.1) of the blow up $\widetilde{M}$ implies the relation > $\tau _{\widetilde{M}}\mid M\backslash \overset{\circ }{D}(\gamma > _{X})=f^{\ast }\tau _{M}\mid M\backslash \overset{\circ }{D}(\gamma _{X})$ which gives rise to an element $[\tau _{\widetilde{M}},f^{\ast }\tau _{M};id]\in K(\widetilde{M};\widetilde{M}\smallsetminus \overset{\circ }{D(\lambda _{E})})$ by Lemma 3.1. Moreover, with respect to the *excision isomorphism* > $K(\widetilde{M};\widetilde{M}\smallsetminus \overset{\circ }{D(\lambda _{E})})\overset{\cong }{\rightarrow }K(D(\lambda _{E});S(\lambda _{E}))$ we have $[\tau _{\widetilde{M}},f^{\ast }\tau _{M};id]=[\tau _{D(\lambda _{E})},f^{\ast }\tau _{D(\gamma _{_{X}})};\kappa ]$, where > $\kappa :\tau _{D(\lambda _{E})}\mid S(\lambda _{E})\rightarrow g^{\ast > }(\tau _{M\setminus \overset{\circ }{D(\gamma _{X})}}\mid S(\gamma _{X}))$ is the bundle isomorphism specified in Theorem 2.3. Granted with the decompositions of the bundles $\tau _{D(\lambda _{E})}$ and $f^{\ast }\tau _{D(\gamma _{X})}$ in (2.4) and (2.5), as well as the decomposition of bundle map $\kappa $ in Theorem 2.3, we calculate > $\qquad \lbrack \tau _{D(\lambda _{E})},f^{\ast }\tau _{D(\gamma > _{X})};\kappa ]$ > > $=[(\pi \circ p_{E})^{\ast }\tau _{X},(\pi \circ p_{E})^{\ast }\tau > _{X};id]+[p_{E}{}^{\ast }\lambda _{E},p_{E}{}^{\ast }\lambda _{E};id]$ > > $\qquad +[p_{E}{}^{\ast }Hom(\lambda _{E},\lambda _{E}^{\perp > }),p_{E}{}^{\ast }\lambda _{E}^{\perp };\kappa ^{\prime }]$ (by ii) of Lemma 3.1) > > $=[p_{E}{}^{\ast }Hom(\lambda _{E},\lambda _{E}^{\perp }),p_{E}{}^{\ast > }\lambda _{E}^{\perp };\kappa ^{\prime }]$ (by i) of Lemma 3.1) > > $=[p_{E}{}^{\ast }(\overline{\lambda }_{E}\otimes \lambda _{E}^{\perp > }),p_{E}{}^{\ast }\lambda _{E}^{\perp };\kappa ^{\prime }]$ (since $Hom(\lambda _{E},\lambda _{E}^{\perp })=\overline{\lambda }_{E}\otimes > \lambda _{E}^{\perp }$) > > $=[p_{E}{}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon > ]\otimes p_{E}{}^{\ast }\lambda _{E}^{\perp }$ (by iii) of Lemma 3.1) where $\kappa ^{\prime }$ is the restriction of $\kappa $ to the direct summand $p_{E}{}^{\ast }Hom(\lambda _{E},\lambda _{E}^{\perp })$ of $\tau _{D(\lambda _{E})}$ (see c) of Theorem 2.3). Summarizing, in the group $K(D(\lambda _{E}),S(\lambda _{E}))$ we we have the relation > $[\tau _{\widetilde{M}},f^{\ast }\tau _{M};id]=[p_{E}{}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}{}^{\ast }\lambda > _{E}^{\perp }$. Finally, applying the map $j_{E}$ to both sides yields the formula (3.3) by iv) of Lemma 3.1.$\square $ **Remark 3.3.** In the group $K(\widetilde{M})$ formula (3.3) has the concise expression > $\tau _{\widetilde{M}}=f^{\ast }\tau _{M}+i_{E!}(\lambda _{E}^{\perp })$ where $i_{E!}$ is the *Gysin map* in $K$–theory > $i_{E!}:K(E)\overset{\psi _{E}}{\underset{\cong }{\rightarrow }}K(D(\lambda > _{E}),S(\lambda _{E}))\overset{j_{E}}{\rightarrow }\widetilde{K}(\widetilde{M})$, and where $\psi _{E}$ is the *Thom isomorphism* $x\rightarrow $ $\psi _{E}(x)=U_{E}\otimes p_{E}^{\ast }x$, $x\in K(E)$, with $U_{E}\in K(D(\lambda _{E}),S(\lambda _{E}))$ the *Thom class* of $\lambda _{E}$. Indeed, from the general construction [@[At] p.98–99] of Thom classes from the exterior algebras of vector bundles one finds that $U_{E}=[p_{E}^{\ast }\overline{\lambda }_{E},1;\varepsilon ]$.$\square $ The Chern class of a blow up ============================ Based on the formula (3.3) we deduce a formula for the total Chern class $C(\widetilde{M})$ of a blow up $\widetilde{M}$ in the category of almost complex manifolds. In this section the coefficients for cohomologies will be the ring $\mathbb{Z}$ of integers. Preliminaries in Chern classes ------------------------------ Let $BU$ be the classifying space of stable equivalent classes of complex vector bundles, and let $c_{r}\in H^{2r}(BU)$ be the $r^{th}$ Chern class of the universal complex vector bundle over $BU$. Then > $H^{\ast }(BU)=\mathbb{Z}[c_{1},c_{2},\cdots ]$. For a topological space $Y$ let $[Y,BU]$ be the set of homotopy classes of maps from $Y$ to $BU$. In view of the standard identification > $\widetilde{K}(Y)=[Y,BU]$ ([@[Sw] p.210]) we can introduce the total Chern class for elements in $\widetilde{K}(Y)$ as the co–functor $C:\widetilde{K}(Y)\rightarrow H^{\ast }(Y)$ defined > $C(\beta )=1+f^{\ast }c_{1}+f^{\ast }c_{2}+\cdots $, $\beta \in \widetilde{K}(Y),$ where $f:Y\rightarrow BU$ is the classifying map of the element $\beta $. Clearly one has **Lemma 4.1.** *The transformation* $C$*satisfies the next two properties.* *i) If* $\xi _{i}$*,* $i=1,2$*, are two complex vector bundles over* $Y$* with equal dimension and with (the usual) total Chern classes* $C(\xi _{i})$*,* *then* > $C(\xi _{1}-\xi _{2})=C(\xi _{1})C(\xi _{2})^{-1}$. *ii) For a closed subspace* $A\subset Y$* let* $j_{A}:(Y,\emptyset )\rightarrow (Y,A)$* and* $q_{A}:Y\rightarrow Y/A$* be the inclusion and quotient maps, respectively. Then the next diagram commutes:* > $\begin{array}{ccc} > K(Y;A)=\widetilde{K}(Y/A) & \overset{j_{A}^{\ast }}{\rightarrow } & > \widetilde{K}(Y) \\ > C\downarrow & & \downarrow C \\ > H^{\ast }(Y/A) & \overset{q_{A}^{\ast }}{\rightarrow } & H^{\ast }(Y)\end{array}$.$\square $ For two complex vector bundles $\lambda $ and $\xi $ over a space $Y$ with $\dim \lambda =1$, $\dim \xi =m$, and with the total Chern classes > $C(\lambda )=1+t$;$\quad C(\xi )=1+c_{1}(\xi )+c_{2}(\xi )+\cdots +c_{m}(\xi > )$, respectively, assume that the Chern roots of $\xi $ is $s_{1},\cdots ,s_{m}$. That is > $C(\xi )=\tprod_{1\leq i\leq m}(1+s_{i})$ with $c_{r}(\xi )=e_{r}(s_{1},\cdots ,s_{m})$ the $r^{th}$ elementary symmetric function in the roots $s_{1},\cdots ,s_{m}$. The calculation > $C(\lambda \otimes \xi )=\tprod\limits_{1\leq i\leq > m}(1+t+s_{i})=(1+t)^{m}\tprod\limits_{1\leq i\leq m}(1+\frac{s_{i}}{1+t})$ > > $\qquad =(1+t)^{m}[1+\frac{c_{1}(\xi )}{(1+t)}+\frac{c_{2}(\xi )}{(1+t)^{2}}+\cdots +\frac{c_{m}(\xi )}{(1+t)^{m}}]$ shows that 1. $C(\lambda \otimes \xi )=\dsum\limits_{0\leq r\leq m}(1+t)^{m-r}c_{r}(\xi )$. For an Euclidean complex line bundle $\lambda $ over $Y$ with associated disk bundle $p_{\lambda }:D(\lambda )\rightarrow Y$ the Thom space $T(\lambda )$ of $\lambda $ is the quotient space $D(\lambda )/S(\lambda )$. In term of Lemma 3.1 the trivialization $\varepsilon :$ $p_{\lambda }^{\ast }(\overline{\lambda })\mid S(\overline{\lambda })\rightarrow 1_{\mathbb{C}}$ indicated by (2.4) defines the element > $[p_{\lambda }^{\ast }(\overline{\lambda }),1_{\mathbb{C}};\varepsilon ]\in > K(D(\lambda ),S(\lambda ))=\widetilde{K}(T(\lambda ))$. Given a ring $A$ and a set $\{u_{1},\cdots ,u_{k}\}$ of $k$ elements let $A\{u_{1},\cdots ,u_{k}\}$ be the free $A$ module with basis $\{u_{1},\cdots ,u_{k}\}$. In addtion to Theorem 3.2, the next result will play a key role in establishing the formula for Chern class. **Lemma 4.2.** *Let* $e\in H^{2}(Y)$* and* $x\in H^{2}(T(\lambda ))$* be respectively the Euler class and the Thom class of the oriented bundle* $\lambda $*. Then* *i) the integral cohomology ring of* $T(\lambda )$* is determined by the additive presentation* 1. $H^{\ast }(T(\lambda ))=\mathbb{Z}\oplus H^{\ast }(Y)\{x\}$*, * *together with the single relation* $x^{2}+xe=0$*;* *ii) for a complex vector bundle* $\xi $ *over* $Y$*with total Chern class* $C(\xi )=1+c_{1}+\cdots +c_{m}\in H^{\ast }(Y),$ *the total Chern class of the element* $[p_{\lambda }^{\ast }(\overline{\lambda }),1_{\mathbb{C}};\varepsilon ]\otimes p_{\lambda }^{\ast }\xi \in \widetilde{K}(T(\lambda ))$* is* 1. $C([p_{\lambda }^{\ast }(\overline{\lambda }),1_{\mathbb{C}};\varepsilon ]\otimes p_{\lambda }^{\ast }\xi )=(\dsum\limits_{0\leq r\leq m}(1+x)^{m-r}c_{r})C(\xi )^{-1}\in H^{\ast }(T(\lambda )).$ **Proof.** The presentation (4.2) comes immediately from the Thom isomorphism theorem, which states that product with Thom class $x$ yields an additive isomorphism > $H^{r}(Y)\cong H^{r+2}(T(\lambda ))$,$\quad y\rightarrow y\cdot x$,$\quad > y\in H^{r}(X)$ for all $r\geq 0$. For i) it remains to justify the relation $x^{2}+xe=0$. Let $p:S(\lambda \oplus 1_{\mathbb{R}})\rightarrow Y$ be the sphere bundle of the Euclidean bundle $\lambda \oplus 1_{\mathbb{R}}$ and set > $D_{+(-)}(\lambda )=\{(u,t)\in S(\lambda \oplus 1_{\mathbb{R}})\mid t\geq 0$ $(t\leq 0)\}$. It is clear that > a\) $S(\lambda \oplus 1_{\mathbb{R}})=D_{-}(\lambda )\cup D_{+}(\lambda )$, $S(\lambda )=D_{-}(\lambda )\cap D_{+}(\lambda )$, > > b\) both $D_{\pm }(\lambda )$ can be identified with the disk bundle $D(\lambda )$ of $\lambda $. In view of a) we have the canonical map onto the Thom space $T(\lambda )$ > $q:S(\lambda \oplus 1_{\mathbb{R}})\rightarrow T(\lambda )=S(\lambda \oplus > 1_{\mathbb{R}})/D_{-}(\lambda )$. Moreover, letting $u=q^{\ast }x\in H^{2}(S(\lambda \oplus 1_{\mathbb{R}}))$ the ring $H^{\ast }(S(\lambda \oplus 1_{\mathbb{R}}))$ has the presentation (see [@[D] Lemma 4]) > $H^{\ast }(S(\lambda \oplus 1_{\mathbb{R}}))=H^{\ast > }(Y)\{1,u\}/\left\langle u^{2}+ue\right\rangle $. The relation $x^{2}+xe=0$ on $H^{\ast }(T(\lambda ))$ is verified by the relation $u^{2}+ue=0$ on $H^{\ast }(S(\lambda \oplus 1_{\mathbb{R}}))$, together with the fact that the induced ring map $q^{\ast }$ is monomorphic onto the direct summand $\mathbb{Z}\oplus H^{\ast }(Y)\{u\}$ of $H^{\ast }(S(\lambda \oplus 1_{\mathbb{R}})$. For ii) define over $S(\lambda \oplus 1_{\mathbb{R}})$ the complex line bundle $\lambda _{u}$ by > $\lambda _{u}=p^{\ast }\overline{\lambda }\mid D_{+}(\lambda )\cup > _{\varepsilon }1_{\mathbb{C}}\mid D_{-}(\lambda )$ (in view of the partition a)). Then $C(\lambda _{u})=1+u\in H^{\ast }(S(\lambda \oplus 1_{\mathbb{R}})$. Moreover, for a vector bundle $\xi $ over* *$Y$ the element > $[\lambda _{u},1_{\mathbb{C}};\varepsilon ]\otimes p^{\ast }\xi \in > K(S(\lambda \oplus 1_{\mathbb{R}}),D_{-}(\lambda ))$ corresponds to the element $[(p_{\lambda }^{\ast }(\overline{\lambda }),1_{\mathbb{C}};\varepsilon )\otimes p_{\lambda }^{\ast }\xi ]\in K(D(\lambda ),S(\lambda ))$ under the excision isomorphism $K(S(\lambda \oplus 1_{\mathbb{R}}),D_{-}(\lambda ))\cong K(D(\lambda ),S(\lambda ))$ indicated by b), which is also mapped to the element > $\lambda _{u}\otimes p^{\ast }\xi -p^{\ast }\xi \in \widetilde{K}(S(\lambda > \oplus 1_{\mathbb{R}}))$ under the induced homomorphism $j^{\ast }$ of the inclusion $j:(S(\lambda \oplus 1_{\mathbb{R}}),\emptyset )\rightarrow (S(\lambda \oplus 1_{\mathbb{R}}),D_{-}(\lambda ))$ by iv) of Lemma 3.1. It follows from i) of Lemma 4.1 that > $C(j^{\ast }([(\lambda _{u},1_{\mathbb{C}};\varepsilon )\otimes p^{\ast }\xi > ]))=C(\lambda _{u}\otimes p^{\ast }\xi )C(p^{\ast }\xi )^{-1}$ > > $=(\dsum\limits_{0\leq r\leq m}(1+u)^{m-r}c_{r})C(\xi )^{-1}$ (by the formula (4.1)). The commutativity of the diagram in ii) of Lemma 4.1 implies that > $q^{\ast }C([(\lambda _{u},1_{\mathbb{C}};\varepsilon )\otimes p^{\ast }\xi > ])=(\dsum\limits_{0\leq r\leq m}(1+u)^{m-r}c_{r})C(\xi )^{-1}$. One obtains the formula (4.3) from $q^{\ast }(x)=u$ and the injectivity of $q^{\ast }$.$\square $ The integral cohomology ring of a blow–up ----------------------------------------- Let $f:\widetilde{M}\rightarrow M$ be the blow up of $M$ along a submanifold $i_{X}:X\rightarrow M$ whose normal bundle $\gamma _{X}$ has a complex structure and with total Chern class $C(\gamma _{X})=1+c_{1}+\cdots +c_{k}\in H^{\ast }(X),$ $k=\dim _{\mathbb{C}}\gamma _{X}$. Regard $D(\lambda _{E})$ as a normal disk bundle of the exceptional divisor $E$ and consider the quotient map onto the Thom space of $\lambda _{E}$ > $q:\widetilde{M}\rightarrow \widetilde{M}/(M\backslash \overset{\circ }{D}(\gamma _{X})=T(\lambda _{E})$. According to i) of Lemma 4.2 one has **Lemma 4.3.** *Let* $t\in H^{2}(E)$* and* $x\in H^{2}(T(\lambda _{E}))$* be the Euler class and Thom class of the oriented bundle* $\lambda _{E}$*, respectively. Then the ring* $H^{\ast }(T(\lambda _{E}))$ *is determined by the additive presentation* 1. $H^{\ast }(T(\lambda _{E}))=\mathbb{Z}\oplus H^{\ast }(X)[x,tx,\cdots ,t^{k-1}x]$, *together with the relations* > *i)* $x^{2}+tx=0$*; ii)* $t^{k}+c_{1}\cdot t^{k-1}+\cdots > +c_{k-1}\cdot t+c_{k}=0$*.*$\square $ Consider the Gysin maps of the embeddings $i_{X}:X\rightarrow M$ and $i_{E}:E\rightarrow \widetilde{M}$ in cohomology $i_{X!}:H^{\ast }(X)\overset{\psi _{X}}{\underset{\cong }{\rightarrow }}H^{\ast }(D(\gamma _{X}),S(\gamma _{X}))\underset{\cong }{\rightarrow }H^{\ast }(M,M\smallsetminus \overset{\circ }{D}(\gamma _{X}))\overset{j^{\ast }}{\rightarrow }H^{\ast }(M)$ $i_{E!}:H^{\ast }(E)\overset{\psi _{E}}{\underset{\cong }{\rightarrow }}H^{\ast }(D(\lambda _{E}),S(\lambda _{E}))\underset{\cong }{\rightarrow }H^{\ast }(\widetilde{M},\widetilde{M}\smallsetminus \overset{\circ }{D}(\lambda _{E}))\overset{j^{\ast }}{\rightarrow }H^{\ast }(\widetilde{M})$ where $\psi _{X}$ and $\psi _{E}$ are the Thom isomorphisms. We shall set > $\omega _{E}=i_{E!}(1)\in H^{2}(\widetilde{M})$, $\omega _{X}=i_{X!}(1)\in > H^{2k}(M)$. Geometrically, if $M$ is closed and oriented the class $\omega _{X} $ (for instance) is the Poincare dual of the cycle class $i_{X\ast }[X]\in H_{\ast }(M)$. The next result is shown in [@[DL] Theorem 1]. **Theorem 4.4.** *The ring map* $f^{\ast }:H^{\ast }(M)\rightarrow H^{\ast }(\widetilde{M})$* embeds the ring* $H^{\ast }(M)$* as a direct summand of* $H^{\ast }(\widetilde{M})$,*and induces the decomposition* 1. $H^{\ast }(\widetilde{M})=f^{\ast }(H^{\ast }(M))\oplus H^{\ast }(X)\{\omega _{E},\cdots ,\omega _{E}^{k-1}\},$ $2k=\dim _{\mathbb{R}}\gamma _{X}$ *that is subject to the two relations* > *i)* $f^{\ast }(\omega _{X})=\tsum\limits_{1\leq r\leq > k}(-1)^{r-1}c_{k-r}\cdot \omega _{E}^{r}$; > > *ii)* $f^{\ast }(y)\cdot \omega _{E}=i_{X}^{\ast }(y)\cdot \omega > _{E} $*,* $y\in H^{r}(M)$. *Moreover, with respect to the presentations of the rings* $H^{\ast }(T(\lambda _{E}))$ *and* $H^{\ast }(\widetilde{M})$ *in (4.5) and (4.6), the induced ring map* $q^{\ast }$* is determined by* > *iii)* $q^{\ast }(t^{r}x)=(-1)^{r}\omega _{E}^{r+1}$*,* $r\geq > 0$. $\square $ **Remark 4.5.** In [@[GH] p.605] Griffiths and Harris obtained the decomposition (4.6) for blow ups of complex manifolds, while the relations i) and ii) were absent. In the non–algebraic settings partial information on the ring $H^{\ast }(\widetilde{M})$ was also obtained by McDuff in [@[M] Proposition 2.4]. In comparison the structure of $H^{\ast }(\widetilde{M})$ as a ring is completely determined by the additive decomposition (4.6), together with the relations i) and ii). Indeed, granted with the fact that $f^{\ast }(H^{\ast }(M))\subset H^{\ast }(\widetilde{M})$ is a subring, the relations i) and ii) are sufficient to express, respectively, the products of elements in the second summand, and the products between elements in the first and second summands, as elements in the decomposition (4.6). This idea has been applied in [@[DL]] to determine the integral cohomology rings of the varieties of complete conics and complete quadrics in the $3$–space $\mathbb{P}^{3}$, and justify two enumerative problems due to Schubert.$\square $ The Chern class of a blow up ---------------------------- Assume now that $f:\widetilde{M}\rightarrow M$ is a blow up in the category of almost complex manifolds, and that the total Chern class of the normal bundle $\gamma _{X}$ is > $C(\gamma _{X})=1+c_{1}+\cdots +c_{k}$, $\dim _{\mathbb{R}}\gamma _{X}=2k$. Applying the transformation $C:\widetilde{K}(\widetilde{M})\rightarrow H^{\ast }(\widetilde{M})$ to the equality (3.3) and noting the obvious relation > $p_{E}^{\ast }\lambda _{E}^{\perp }=p_{E}^{\ast }\gamma _{X}-p_{E}^{\ast > }\lambda _{E}$ in the $K$–group $K(D(\lambda _{E}))$ one obtains > $C(\widetilde{M})\cdot f^{\ast }C(M)^{-1}=C(j_{E}([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes (p_{E}^{\ast }\gamma > _{X}-p_{E}^{\ast }\lambda _{E}))$ > > $=\frac{C(j_{E}([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}^{\ast }\gamma _{X}))}{C(j_{E}([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}^{\ast > }\lambda _{E}))}$ (by i) of Lemma 4.1) > > $=q^{\ast }\frac{C([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}^{\ast }\gamma _{X})}{C([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes p_{E}^{\ast }\lambda _{E})} > $ (by ii) of Lemma 4.1). Furthermore, from > $C([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes > p_{E}^{\ast }\gamma _{X})=(\dsum\limits_{0\leq r\leq > k}(1+x)^{k-r}c_{r})C(\gamma _{X})^{-1}$, > > $C([p_{E}^{\ast }\overline{\lambda }_{E},1_{\mathbb{C}};\varepsilon ]\otimes > p_{E}^{\ast }\lambda _{E})=(1+x+t)(1+t)^{-1}$ by (4.3) one gets > $C(\widetilde{M})\cdot f^{\ast }C(M)^{-1}=q^{\ast }(\dsum\limits_{0\leq > r\leq k}(1+x)^{k-r}c_{r})(1+t)(1+x+t)^{-1}C(\gamma _{X})^{-1}$ > > $=(\dsum\limits_{0\leq r\leq k}(1+\omega _{E})^{k-r}c_{r})(1-\omega > _{E})C(\gamma _{X})^{-1}$. where the second equality comes from iii) of Theorem 4.4. It implies that > $C(\widetilde{M})-f^{\ast }C(M)=f^{\ast }C(M)\cdot g(\omega _{E})$ with > $g(\omega _{E})=(\dsum\limits_{0\leq r\leq k}(1+\omega > _{E})^{k-r}c_{r})(1-\omega _{E})C(\gamma _{X})^{-1}-1\in H^{\ast }(\widetilde{M})$ a polynomial in $\omega _{E}$ with coefficients in $H^{\ast }(X)$. It is crucial for us to observe from the obvious relation $g(0)=0$ that the polynomial $g(\omega _{E})$ is divisible by $\omega _{E}$. Therefore, the relation ii) in Theorem 4.4 is applicable to yield > $C(\widetilde{M})-f^{\ast }C(M)=f^{\ast }C(M)\cdot g(\omega _{E})$ > > $=i_{X}^{\ast }C(M)g(\omega _{E})$ > > $=C(X)C(\gamma _{X})g(\omega _{E})$ (since $\tau _{M}\mid X=\tau _{X}\oplus > \gamma _{X}$). Summarizing we get **Theorem 4.6.** *With respect to decomposition of the ring* $H^{\ast }(\widetilde{M})$* in (4.6), the total Chern class of the blow up* $\widetilde{M}$* is* > $C(\widetilde{M})=f^{\ast }C(M)+C(X)(\dsum\limits_{0\leq r\leq k}(1+\omega > _{E})^{k-r}c_{r})(1-\omega _{E})-\dsum\limits_{0\leq r\leq k}c_{r})$.$\square $ **Examples 4.7.** The simplest example assuring that Theorem 4.6 is non–trivial in the category of almost complex manifolds is the following one. Recall that the $6$–dimensional sphere $S^{6}$ has a canonical almost complex structure. The blow–up of $S^{6}$ at a point $X\in S^{6}$ is diffeomorphic to the complex projective $3$–space $\mathbb{P}^{3}$, together with an induced almost complex structure $J$ (see Corollary 2.4). By Theorem 4.6 we have > $C(\mathbb{P}^{3},J)=1-2x-4x^{3}$, where $x\in H^{2}(\mathbb{P}^{3})$ is the exceptional divisor. This computation shows that $J$ is different with the canonical complex structure on $\mathbb{P}^{3}$. Similarly, in the recent work [@[Y]] H. Yang classified those $(2n-1)$–connected $4n$–manifolds that admit almost complex structures. In term of the Wall’s invariant $\alpha :H_{2n}(M)\rightarrow \pi _{2n-1}(SO(2n))$ for such a manifold $M$ [@[W]] the set of isotopy classes of smooth embeddings $S^{2n}\rightarrow M$ whose normal bundles have complex structures can be identified with an explicit subset $J(M)$ of the homology group $H_{2n}(M)$, where $S^{2n}$ is the $2n$–dimensional sphere. 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[^1]: The author’s research is supported by 973 Program 2011CB302400 and NSFC 11131008.

--- abstract: 'We report density functional calculations of the electronic structure, Fermi surface, phonon spectrum, magnetism and electron-phonon coupling for the superconducting phase FeSe, as well as the related compounds FeS and FeTe. We find that the Fermi surface structure of these compounds is very similar to that of the Fe-As based superconductors, with cylindrical electron sections at the zone corner, cylindrical hole surface sections, and depending on the compound, other small hole sections at the zone center. As in the Fe-As based materials, these surfaces are separated by a 2D nesting vector at ($\pi$,$\pi$). The density of states, nesting and Fermi surface size increase going from FeSe to FeTe. Both FeSe and FeTe show spin density wave ground states, while FeS is close to an instability. In a scenario where superconductivity is mediated by spin fluctuations at the SDW nesting vector, the strongest superconductor in this series would be doped FeTe.' author: - Alaska Subedi - Lijun Zhang - 'D.J. Singh' - 'M.H. Du' title: 'Density functional study of FeS, FeSe and FeTe: Electronic structure, magnetism, phonons and superconductivity' --- introduction ============ Superconductivity was recently reported in $\alpha$-FeSe$_{1-x}$, with critical temperature $T_c \sim$ 8K. [@hsu] This is of interest both because of the fact that Fe containing superconductors are unusual and because this material shares in common square planar sheets of tetrahedrally coordinated Fe with the Fe-As based high temperature superconductors. [@kamihara; @ren; @ren2; @cwang; @sefat; @kito] $\alpha$-FeSe occurs in the PbO structure. This consists, as mentioned, of Fe square planar sheets, with Se atoms forming distorted tetrahedra around the Fe very similar to the structure of the FeAs planes in LaFeAsO, BaFe$_2$As$_2$ and LiFeAs, which are prototypes of the known families of Fe-As based high-T$_c$ superconductors. [@rotter1; @rotter2; @xcwang] Since the reported $T_c$ = 8K of doped $\alpha$-FeSe is modest, it is important first of all to establish the relationship between this material and the Fe-As based superconductors. We note that LaNiPO is also a superconductor and shares the crystal structure of LaFeAsO, [@watanabe; @tegel] but that it is apparently quite different electronically and can be understood in terms of standard electron-phonon theory,[@subedi] unlike the Fe-As based phases. [@boeri; @mazin] Here we report density functional calculations that show $\alpha$-FeSe and the other known Fe based chalcogenides in this structure to be very similar to that of the Fe-As based superconductors. [@singh-du] In particular the Fermi surface consists of small heavy hole cylinders near the zone center and lighter compensating electron cylinders around the zone corner. We show that the stoichiometric compounds are either very close to a spin density wave instability (FeS) or have an itinerant spin density wave instability without doping (FeSe and FeTe) similar to the Fe-As superconductors. [@cruz] We predict that this itinerant nesting driven magnetic state is strongest in FeTe and in addition that FeTe has the largest Fermi surface of the three compounds. Calculations of the electron-phonon coupling show that doped FeSe is not an electron-phonon superconductor, similar to what was found for the Fe-As phases. Within a spin-fluctuation driven picture of superconductivity the results indicate that FeTe with doping is a likely higher temperature superconductor. first principles methods and structure ====================================== Our calculations of the electronic structure and magnetic properties were performed within the local density approximation with the general potential linearized augmented planewave (LAPW) method, [@singh-book] including local orbitals, [@singh-lo] similar to our previous calculations for the Fe-As based superconductors. [@singh-du; @mazin; @singh-ba] We used LAPW spheres of radius 2.1 $a_0$ for Fe, Se and Te and 1.9 $a_0$ for S. These compounds occur in a simple tetragonal structure with one internal parameter, $z_X$ corresponding to the chalcogen height above the Fe square plane. The experimental lattice parameters [@hsu; @lennie; @finlayson] were employed and we relaxed the chalcogen height via energy minimization. The structural parameters used and some results are presented in Table \[tab-struct\]. The electron-phonon coupling and phonon dispersions were on the other hand obtained using linear response, again with the experimental lattice parameters, with the Quantum Espresso code [@qe; @qenote] within the generalized gradient approximation of Perdew, Burke and Ernzerhof [@pbe] as described for LaFeAsO and LaNiPO. [@subedi; @mazin]   $a$(Å)     $c$(Å)     $z_X$   $N(E_F)$ $m_{SDW}(\mu_B)$ $E_{SDW}$ ------ ------------ ------------ ----------- ---------- ------------------ ----------- FeS 3.6735 5.0328 0.2243 1.35 0.00 0 FeSe 3.765 5.518 0.2343 0.95 0.65 5 FeTe 3.8215 6.2695 0.2496 1.83 1.28 47 : Structural parameters and magnetic properties of PbO-structure Fe$X$. The lattice parameters are from experimental data, while the internal chalcogen structural parameter, $z_X$ is from LDA structure minimization. $m_{SDW}$ is the spin moment within the Fe LAPW sphere (radius 2.1 $a_0$) for the SDW state and $E_{SDW}$ is the energy per Fe of this state relative to the non-spin-polarized state in meV/Fe. $N(E_F)$ is the density of states at the Fermi energy in the non-spin-polarized band structure in eV$^{-1}$ on a per Fe both spins basis.[]{data-label="tab-struct"} electronic structure, phonons and magnetism =========================================== ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-S.ps "fig:"){height="3.4in"} ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-Se.ps "fig:"){height="3.4in"} ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-Te.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-S.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-Se.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-Te.ps "fig:"){height="3.4in"} Our main results for the electronic structure are given in Figs. \[fig-bands\], \[fig-dos\] and \[fig-fermi\], which show the non-spin-polarized band structures, electronic densities of states and Fermi surfaces of FeS, FeSe and FeTe. The calculated values of $N(E_F)$ are given in Table \[tab-struct\]. The phonon dispersions of FeSe are shown in Fig. \[fig-phonon\]. One interesting feature of the phonon dispersion is that they have little dispersion in the $k_z$ and e.g. are very flat along the tetragonal $\Gamma$-$Z$ direction. This presumably reflects anion-anion repulsion, which leads to long bonds between the FeSe layers. the result is that there may be an easy cleavage plane between the Se ions, which may facilitate preparation of clean surfaces for experiments such as photoelectron spectroscopy. The phonon density of states, $G(\omega)$ and electron-phonon spectral function, $\alpha^2 F(\omega)$ are given in Fig. \[fig-a2f\]. The electron-phonon coupling constant for FeSe as obtained in linear response is $\lambda$ = 0.17 with $\omega_{log}$=113 cm$^{-1}$. No superconductivity at any temperature even approaching 1K results with these values within standard electron-phonon theory even if very low values of the Coulomb parameter, e.g. $\mu^*=0.10$ are used in the Allen-Dynes equation. This is similar to what was found previously for LaFeAsO. [@boeri; @mazin] Therefore we conclude that FeSe is not a conventional electron-phonon superconductor. ![(color online) LDA Fermi surface of FeS, FeSe and FeTe from non-spin-polarized calculations with the LDA relaxed $X$ heights. The corners are $\Gamma$ points.[]{data-label="fig-fermi"}](fermi.ps){width="3.4in"} Turning to the electronic structure we find a strong qualitative similarity between these materials and the FeAs-based superconductors. In particular, we find these to be low carrier density metals, with high density of states. This arises from band structures that are closely related to those of the FeAs materials. The chalcogen $p$ states lie well below the Fermi level and are only modestly hybridized with the Fe $d$ states as may be seen from the projected DOS (Fig. \[fig-dos\]). Thus the electronic structure near the Fermi energy derives from metallic Fe$^{2+}$ layers, with direct Fe-Fe interactions. These are embedded inside a largely ionic background which imposes a competing tetrahedral crystal field. As in the Fe-As based materials, there is a pseudogap at an electron count of 6 $d$ electrons per Fe, and $E_F$ lies near the bottom of this pseudogap. We emphasize that this is not the position of a tetrahedral crystal field gap, which would be at 4 electrons, and emphasizes the fact that Fe chalcogen hybridization is not strong compared with the Fe-Fe interactions. This explains the similarity of the electronic structure to that of the FeAs-based materials, which were also found to be substantially ionic in similar calculations. [@singh-du] ![Calculated GGA phonon dispersions of non-spin-polarized FeSe. []{data-label="fig-phonon"}](FeSe_ep0.eps){width="3.4in"} ![Calculated GGA phonon density of state $G(\omega)$ and electron-phonon spectral function $\alpha^2 F(\omega)$ for FeSe. []{data-label="fig-a2f"}](phondos.eps){width="3.4in"} These band structures yield two intersecting elliptical cylindrical electron Fermi surfaces at the zone corner in all three materials. These are compensated by lower velocity hole sections at the zone center – two concentric hole cylinders, and in the case of FeS and FeTe a small closed hole section inside the inner cylinder. This is qualitatively very similar to the FeAs-based materials. It is important to note that cylinders at the zone center and zone corner, if they are the same size, would yield strong nesting peaked at the 2D ($\pi$,$\pi$) point. This will lead in general to enhanced spin fluctuations at the nesting vector, and if sufficiently strong will cause a spin density wave. In fact, an ordered SDW was found both in first principles calculations and in experimental studies for LaFeAsO and many other undoped FeAs-based compounds. [@mazin; @cruz; @dong; @chen; @zhao; @huang; @goldman; @nakai] There is a clear competition between the SDW and the superconducting state in that superconducting samples generally do not show the SDW, while samples with the SDW transition, generally do not show superconductivity. The ground state is an antiferromagnetic cell doubled along the \[11\] in plane direction to yield linear chains of nearest neighbor like spin Fe atoms arranged antiferromagnetically, although the values of the moments are dependent on details, especially the As height above the Fe plane. [@yildirim; @ishibashi; @yin; @mazin-2; @vildosola; @yildirim-2] discussion ========== Turning to the trends, the size of the pseudogap is approximately the same in FeSe and FeS, but is significantly smaller in FeTe. Specifically, there is a greater overlap between the hole and electron bands in the latter compound. This leads to larger Fermi surfaces. The value of $N(E_F)$=1.83 eV$^{-1}$ is also highest in this compound, although it is still lower than the 2.62 eV$^{-1}$ that is obtained for LaFeAsO by the same approach. We used a supercell approach to investigate the SDW with the chalcogen heights fixed to the values calculated by non-spin-polarized energy minimization. We find instabilities for FeSe and FeTe, but not for FeS. The spin moments and the energy of the SDW relative to the non-spin-polarized state are given in Table \[tab-struct\]. As may be seen the SDW is considerably stronger in FeTe than in FeSe, with an energy gain of 47 meV/Fe and a spin-moment of 1.3 $\mu_B$. The corresponding values for LaFeAsO, calculated in the same way are $E_{SDW}$=11 meV/Fe and $m_{SDW}$ = 1.0 $\mu_B$ in a 2.1 $a_0$ Fe sphere. Besides the SDW we find a borderline ferromagnetic tendency in FeTe, without doping when the SDW is not allowed, similar to LaFeAsO. [@singh-du] We do not find ferromagnetic instabilities in either FeSe or FeS, consistent with the lower values of $N(E_F)$ in those materials. The sensitivity of the moment size to the ordering pattern underscores that fact that these are itinerant magnetic systems in the LDA, as opposed to local moment magnets. This means that magnetic ordering is driven by electrons at and near the Fermi surface. On the other hand this is not to say that spin fluctuations are weak in the paramagnetic state above the SDW ordering temperature or the paramagnetic state as realized by doping. In fact, as noted, several authors have found that the As height in the Fe-As compounds is strongly coupled to magnetism and so strong spin fluctuations in the paramagnetic state would help rationalize the underestimated As height in non-spin-polarized LDA calculations. [@yildirim; @ishibashi; @yin; @mazin-2; @yildirim-2] In fact, there is evidence for strong spin fluctuations in the normal state of the FeAs compounds, e.g. from temperature dependent resistivity data indicating strong scattering above the SDW ordering temperature (note a drop in resistivity below the SDW ordering temperature even though the carrier density is strongly reduced), [@mcguire] as well as from spectroscopy. [@mannella] The size of the effects observed implies that these fluctuations should have large amplitudes and therefore unlike the SDW should be rather diffuse in ${\bf q}$-space, which might make them hard to directly observe. [@sf-note] Furthermore, the strong transport signatures especially the enhanced resistivity above the ordering temperature imply substantial coupling between spin flucuations and electrons at the Fermi energy, which is indicative of the itinerant nature of the magnetism. We emphasize also that our results in Table \[tab-struct\] are at the LDA relaxed atomic positions for the non-spin-polarized states. These systems become more magnetic as the chalcogen height is raised. In the case of FeS a recent refinement is available, and gives $z_{\rm S}$=0.2602. [@lennie] This puts the S ions 0.18 Å  higher than in the LDA structure. With this value we find a stable SDW for FeS, with a moment of 1.2 $\mu_B$/Fe. This is the same trend as in the Fe-As based superconducting materials. The result may be taken as an indication that in fact FeS may have a spin density wave as well and at least that there will be strong spin fluctuations in FeS as well as the other PbO-structure Fe chalcogenides. As mentioned, cylindrical Fermi surface sections of equal volume will be nested with nesting vector equal to the separation of the centers of these cylinders. This nesting can be reduced by imperfect matches in shape, three dimensionality, and size mismatch. Size mismatch can arise both from additional Fermi surface sections, such as the extra small hole sections obtained in FeS and FeTe, but not FeSe, or from doping. In particular, electron doping will reduce the size of the hole sections and increase the size of the electron sections consistent with Luttinger’s theorem. The mechanism for superconductivity in the Fe-As based phases has not yet been established. Nonetheless, there are indications that magnetism is associated with superconductivity. These include the modest electron phonon couplings in the materials, the proximity to magnetism and the phase diagrams which show an association between the SDW and superconductivity. We discuss our results within a general spin fluctuation mediated framework. [@moriya; @monthoux; @mazin; @kuroki] In general an itinerant SDW instability arises from a divergence of the real part of the susceptibility $\chi({\bf q})$ at a specific wavevector ${\bf q}$. Superconducting pairing is also associated with the real part of $\chi({\bf q})$ through an integral over the Fermi surface. $\chi({\bf q})$ for ${\bf q}$ connecting different parts of the Fermi surface can contribute to pairing. As a result, when the Fermi surfaces are small and disconnected as in these Fe based materials, the fluctuations associated with nesting will provide substantial interband pairing between the electron and hole sections, but will not provide substantial intraband pairing. [@mazin; @kuroki] As the nesting is reduced, e.g. by doping, $\chi({\bf q})$ will spread out while the peak value will be reduced, consistent with the destruction of the SDW. For circular cylinders of radii differing by $\delta q$, $\chi({\bf q})$ will show a plateau of high $\chi({\bf q})$ around the nesting vector with width $2 \delta q$ (this will persist until the radii differ by a factor of two at which point the center of the plataeu will dip). This means that the two cylinders will still be connected by spin fluctuations associated with their now reduced nesting, and therefore even though the SDW will be suppressed by the reduction in the maximum value of $\chi$ the associated spin fluctuations can still provide superconducting pairing. We emphasize that in this general framework the same parts of the Fermi surface are affected by the SDW and by the superconducting pairing. Thus these two Fermi surface instabilities compete for the same Fermi surface and therefore that there should be at best little coexistence of these two orders. summary and conclusions ======================= We report electronic structures, magnetic properties and electron-phonon calculations for Fe$X$, $X$=S,Se,Te. We find strong similarities to the Fe-As based superconductors, reflecting the ionic nature of the As and chalcogen atoms in these compounds. As in the arsenides, we find that the electron-phonon coupling cannot explain the superconductivity, and furthermore that these compounds display itinerant magnetism. These results imply a similar superconducting nature for the Fe-As phases and FeSe. The trend that we find in going from FeSe to FeTe is interesting in this context. In particular we find quite cylindrical Fermi surfaces and an SDW instability in both compounds. However, the strength of the SDW is substantially higher in FeTe as is the size of the Fermi surface. Within the general framework discussed above, FeTe would be expected to have stronger pairing, and therefore higher $T_c$ than FeSe assuming that the same mechanism applies in both materials, that both materials can be chemically doped to the optimum carrier density and that competing instabilities do not prevent superconductivity in that case. It will be of interest to experimentally probe the similarities of FeSe with those of the Fe-As phases and to search for superconductivity in doped FeTe and in the alloy Fe(Se,Te). We are grateful for helpful discussions with I.I. Mazin, D.G. Mandrus and B.C. Sales. This work was supported by the Department of Energy, Division of Materials Sciences and Engineering. note added ========== Mizuguchi and co-workers [@mizuguchi] recently reported observation of superconductivity at 27K in FeSe under pressure. This is consistent with the conclusion here regarding the relationship between FeSe and the Fe-As based materials. F.C. Hsu, J.Y. Luo, K.W. 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--- abstract: 'Understanding and predicting a material’s performance in response to high-energy radiation damage, as well as designing future materials to be used in intense radiation environments, requires the knowledge of the structure, morphology and amount of radiation-induced structural changes [@dud1; @stoneham; @dud2; @weber; @right]. We report the results of molecular dynamics simulations of high-energy radiation damage in iron in the range 0.2–0.5 MeV. We analyze and quantify the nature of collision cascades both at the global and local scale. We find that the structure of high-energy collision cascades becomes increasingly continuous as opposed to showing sub-cascade branching reported previously. At the local length scale, we find large defect clusters and novel small vacancy and interstitial clusters. These features form the basis for physical models aimed at understanding the effects of high energy radiation damage in structural materials.' address: - 'SEPnet and School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK' - 'London Centre for Nanotechnology, Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, UK' - 'Computational Science and Engineering Department, CCLRC Daresbury Laboratory, Keckwick Lane, Daresbury, Warrington WA44AD, Cheshire, UK' - 'University of Helsinki, P.O. Box 43, FIN-00014 Helsinki, Finland' author: - 'E. Zarkadoula, M. T. Dove and K. Trachenko' - 'S. L. Daraszewicz and D. M. Duffy' - 'M. Seaton and I. T. Todorov' - 'K. Nordlund' title: 'The nature of high-energy radiation damage in iron: Modeling results' --- Introduction ============ Radiation effects are common in nature. Their sources vary from cosmic radiation to decay of isotopes in terrestrial rocks. A large variety of radiation sources are also created and used in science and technology. This includes energy generation in existing nuclear power stations, where kinetic energy of fission products is converted into heat and electricity. When feasible, future fusion reactors will harvest the energy from thermonuclear reactions. In these applications, the energy of emitted particles has a two-fold effect: on one hand, this energy is converted into useful energy, by heating the material; on the other hand, this energy damages the material and degrades the properties important for the operation, including mechanical, thermal, transport and other properties. This is currently a central issue for fusion reactors, where the ability of metal structural components to withstand very high neutron fluxes is intensely discussed [@dud1; @stoneham; @dud2]. Another example is the damage to nuclear reactor materials coming from fission products. In addition, the nuclear industry faces yet another problem, that of radiation damage to materials to be used to encapsulate long-lived radioactive waste [@weber; @synroc2; @gei1]. A heavy energetic particle displaces atoms on its path which, in turn, displace other atoms in the system. A collection of these atoms is often referred to as a “collision cascade” [@Ave98; @ishino; @k6; @k1; @k2; @k3; @k4]. A typical collision cascade created by a heavy 100 keV particle propagates and relaxes on the order of picoseconds and spans nanometers. The resulting structural damage, in the form of amorphous pockets or point defects and their clusters, ultimately defines to what extent materials’ mechanical, thermal and other properties are altered. For example, radiation-induced defects can reduce materials’ thermal conductivity and therefore result in inefficient energy transfer in both fusion and nuclear reactors, heat localization and other unwanted effects. Understanding and predicting these and other effects, as well as designing future materials to be used in intense radiation environments, requires developing physical models. These, in turn, are based on the knowledge of what high-energy radiation damage is in terms of structure and morphology. Molecular dynamics (MD) simulations have been an important method for studying radiation damage in materials because they give access to the small time and length scales of the collision cascades, and give a detailed picture of the damage at the atomistic scale. Previous MD simulations have provided important insights into the radiation damage process [@Ave98; @ishino; @k6; @k1; @k2; @k3; @k4; @stoller2]. However, due to system size limitations in MD simulations, the reported results were limited to energies of about 100 keV. Recently, the number of displaced atoms was reported for 200 keV cascade [@st-rev]. On the other hand, knock-on energies are larger in several important applications. When impacted on 14 MeV neutrons, iron knock-on atoms in fusion reactors reach the energy of up to 1 MeV [@stoneham; @ishino; @st-rev] with an average energy of about 0.5 MeV [@stoller2]. In fission nuclear reactions, the fission product energies are on the order of 50 to 100 MeV, transferring high energy to the surrounding material. The need to simulate realistic energy cascades has been particularly emphasized, with a view that extrapolation of low-energy results may not account for some important features of higher-energy radiation process which can contain novel qualitative features. More generally, the need to simulate length and energy scales that are relevant and appropriate to a particular physical process has been recognized and reiterated [@right]. In this paper, we study the radiation damage process due to high-energy Fe knock-on atoms of 0.2–0.5 MeV energy. We focus on high-energy radiation damage in $\alpha$-iron, the main structural material in fusion and future fission reactors. We analyze and quantify the nature of collision cascades both at the global and local scale. We find that high-energy collision cascades may propagate and relax as increasingly continuous damage structures as opposed to showing sub-cascade branching as assumed previously. At the local length scale, we find large clusters and new defect structures. Methods ======= We have used the DL$\_$POLY program, a general-purpose package designed for large-scale simulations [@dl1; @dl2]. We have simulated systems with 100-500 million of atoms, and run MD simulations on 20000–60000 parallel processors of the HECToR National Supercomputing Service [@hector]. MD simulations were performed at a constant energy and volume ensemble (except for the boundary layer atoms, see below) with an initial temperature set to 300 K, which was preceded by equilibration runs in a constant pressure ensemble at 300 K. Periodic boundary conditions were imposed in all directions. We have implemented several features to handle radiation damage simulations. First, we used a variable time step to account for faster atomic motion at the beginning of the cascade development and its gradual slowing down at later stages. Second, the MD box boundary layer of thickness of about 10 Å was connected to a constant temperature thermostat at 300 K to emulate the effect of energy dissipation into the sample. Third, we have accounted for the electronic energy losses (EEL), particularly important at high energies. EEL is a complicated process that involves a wide range of effects affecting damage production and annealing [@stoneham; @race; @duffy1; @caro]. Taking EEL into account in MD simulations involves models that include slowing-down of atoms due to energy transfer to electron excitation processes as well as feeding this energy back to the system as the phonon energy. The implementation of this dual energy exchange mechanism in MD simulations, based on the two-temperature approach [@ttm; @duffy1; @duffy2], is in progress. In this work, we model electronic energy loss as a friction term added to the equations of motion. The characteristic energy loss relaxation time (taken here as $\tau_{es} = 1.0$ ps), is obtained by relating the stopping strength in the low-velocity regime via Lindhard’s model to the rate of energy loss for a single atom [@lindhard; @duffy2]. Such electronic stopping would only be effective above a certain threshold, where the atoms would have sufficient energy to scatter inelastically. We use a cut-off kinetic energy value (8.6 eV) corresponding to twice of the cohesive energy [@coh2], however a number of other threshold values have been proposed [@coh1; @coh3; @coh4; @coh5]. For , we have used a many-body embedded-atom potential [@mendel], optimized for better reproduction of several important properties of $\alpha$-Fe including the energetics of point defects and their clusters (“M07” from Ref. [@malerba]). At distances shorter than 1 Å, interatomic potentials were joined to short-range repulsive ZBL potentials [@zbl]. To analyze the collision cascade, we have employed two methods. First, an atom is identified as “displaced” if it moves more than distance $d$ from its initial position. The number of displaced atoms, $N_{\mathrm{disp}}$, quantifies the overall amount of introduced damage. Some of this damage recovers back to crystal. To account for this effect, we employed the second method in which an atom is identified as a “defect” if it is further than distance $d$ from any of the crystalline position of the original lattice. Then, the number of defects $N_{\mathrm{def}}$ quantifies both damage production and its recovery. We note that $N_{\mathrm{disp}}$ and $N_{\mathrm{def}}$ depend on $d$ (we use $d=0.75$ Å as a convenient measure). However, the trends discussed in Sec. \[results\], including the two regimes of cascade relaxation as well as dynamics of defects recovery are not sensitive to $d$ provided it is in the sensible range of distances (e.g., too small $d\lesssim 0.1-0.2$ Å will be affected by usual thermal fluctuations whereas may not identify defect atoms). Vacancies or self-interstitial atoms (SIA) are defined to belong to the same defect group (cluster) if within 2 nearest-neighbour distance (plus a 20% perturbation). Second nearest-neighbour (nn) is a common clustering criterion for SIAs [@st-rev; @bjork], however the criteria for vacancy clusters vary significantly (from 1 to 4nn) across the literature [@malerba2]. When identifying cluster size in the the net defect number (the difference between the number of SIAs and the number of vacancies) is reported. Results and discussion {#results} ====================== We discuss the main features of high-energy collision cascades. To account for potentially different collision cascades due to different knock-on directions, we have simulated 4 different directions for each energy, avoiding low-density directions and associated channeling. ![$N_{\mathrm{disp}}$ and $N_{\mathrm{def}}$ from 0.2 MeV (top) and 0.5 MeV (bottom) knock-on atoms for four cascades.[]{data-label="fig1"}](fig1){width="\columnwidth"} $N_{\mathrm{disp}}$ and $N_\mathrm{def}$ are shown in Fig. \[fig1\] for 0.2 and cascades simulated in different knock-on directions. We observe two types of cascade relaxation in Fig. \[fig1\]. The first type is related to the large peak of $N_{\mathrm{disp}}\approx 10^6$ at short times of about 1–2 ps. This peak relaxes during about 10 ps. This peak is often discussed as the “thermal spike” [@spike; @Sam07d] related to melting inside the collision cascade and associated swelling that causes the temporary increase of $N_{\mathrm{disp}}$. At the atomistic level, the increase of $N_{\mathrm{disp}}$ can be understood on the basis of anharmonicity of interatomic interactions: large-scale atomic motion inside the cascade causes the increase of interatomic separations due to anharmonicity. This results in the outward pressure from the cascade on the surrounding lattice and lattice expansion. This elastic deformation lasts several ps, equal to several periods of atomic vibrations during which the energy is dissipated to the lattice, and gives rise to the peak of $N_{\mathrm{disp}}$. Notably, the elastic deformation is reversible irrespective of whether it is followed by the recovery of the structural damage discussed below. In Fig. \[fig1\], $N_{\mathrm{disp}}$, averaged over all knock-on directions at the end of simulation (corresponding to the flat lines in Fig. \[fig1\]) is about 67,000 and 111,000 atoms for 0.2 MeV and 0.5 MeV cascades, respectively. Figures \[fig2\]–\[fig3\] show the snapshots of the collisions cascades at three different stages of cascade development that include the intermediate stage corresponding to the large peak of $N_{\mathrm{disp}}$ in Fig. \[fig1\]. Consistent with , we observe a significantly larger number of atoms involved in large displacements at intermediate times as compared to the final relaxed state. ![Displaced (top) and defect (bottom) atoms in a representative 0.2 MeV collision cascade. The knock-on atom moves from the top left to the bottom right corner. The three frames for each type of atoms correspond to 0.3 ps, 3 ps and 80 ps, respectively. Cascade size (maximal separation between any two atoms in the cascade) is 560 Å. Vacancies (interstitials) are represented in purple (green); we used Atomeye software [@atomeye] to visualize cascade evolution. []{data-label="fig2"}](fig2){width="\columnwidth"} ![(a) and (b) show two representative 0.5 MeV cascades. The knock-on atom moves from the top left to the bottom right corner. In (a) we show both displaced (top) and defect (bottom) atoms at 0.2, 1.5 ps and 100 ps. In (b) we show the defect atoms only at 0.3 ps, 2 ps and 100 ps. Cascade size in (a) and (b) is 950 Å and 1300 Å, respectively. []{data-label="fig3"}](fig3){width="\columnwidth"} The second type of cascade relaxation is related to the dynamics of $N_{\mathrm{def}}$. At short ps times, the large peak of $N_{\mathrm{def}}$ is of the same origin as that seen for $N_{\mathrm{disp}}$. However, dynamics of $N_{\mathrm{def}}$ also reflects the recovery of structural damage. This recovery proceeds by the diffusion and recombination processes during which atoms settle at the newly found crystalline positions. This process lasts up to 20 ps, significantly longer than relaxation time of the first elastic relaxation process (see Fig. \[fig1\]). As a result of this relaxation, $N_{\mathrm{def}}$, averaged over all simulated directions at the end of simulation (corresponding to the flat lines in Fig. \[fig1\]), is about 1,800 and 2,800 atoms, respectively, corresponding to approximately 97% recovery rate as compared to $N_{\mathrm{disp}}$. Such a high recovery rate is in interesting resemblance to some of the resistant ceramic materials, but in contrast to others [@prb; @prb1]. Our simulations provide an important insight about the structure and morphology of high-energy cascades. The existing view of the high-energy cascade is that it branches out to smaller sub-cascades and “pockets” of damage that are well separated from each other. This takes places over a certain energy threshold, even though this threshold was not firmly established [@Nor98; @Hei94; @st-rev]. This picture originated as a result of using binary-collision simulations in combination with MD simulations of low-energy events. Although involving approximations inherent in binary-collision simulations and extrapolations of low-energy MD simulations, this picture would offer a great degree of reduction and simplification: in analyzing the results and consequences of high-energy damage, only small sub-cascades need to be considered. In Figures \[fig2\]–\[fig3\], we observe a qualitatively different picture. Cascades branching is visibly reduced as compared to low-energy events, in that we do not find well-separated sub-cascades. Some cascade branching is seen in the first 0.5 MeV cascade shown in Fig. \[fig3\]a only and, importantly, during an intermediate stage of cascade development only. On the other hand, the final cascade morphology is described by a rather continuous distribution of the damage across about 1000-1300 Å where no distinctly separated sub-cascades can be identified. Common to all collision cascades we have simulated, this picture is particularly visible for defect atoms in the final state of the cascades shown in . Qualitatively, reduced cascade branching and the emergence of a more continuous damage distribution can be understood as follows. For a scattered atom to move far enough from its initial position and form a spatially separated sub-cascade (i.e., branch out) requires a large value of energy transferred to it by the incident atom. In the absence of inelastic losses, the transferred energy, $T$, is $T=\frac{1}{2}T_m(1-\cos(\phi))$, where $T_m$ is the maximal transferred energy and $\phi$ is the scattering angle [@smith]. For large energy of the incident atom, $E$, $\phi$ decreases as $\phi\propto\frac{1}{E}$ [@smith]. We therefore find that for large $E$ and small $\phi$, $T$ decreases as $T\propto\phi^2\propto\frac{1}{E^2}$. Large $E$ and, consequently, small $T$, results in scattered atoms forming the damaged region close to the trajectory of the initial knock-on atom and, therefore, promote a continuous structure of the resulting damage. This is consistent with our current findings. We note that as the incident atom slows down, $T$ increases, leading to sub-cascade branching at the end of the atom trajectory. However, larger $E$ results in the increase of the relative fraction of continuous damage over the fraction of branched cascades. Our finding is important in the context of the long-term evolution of radiation damage. Indeed, a recent kinetic Monte Carlo study [@bjork] has shown that very large defect clusters can have a major effect on the long-term damage development. The discussion has so far concentrated on the large-scale cascade structure and morphology. We now briefly discuss defect structures at the local level. The size and structure of the defect clusters created by the cascades were analyzed and the results are summarized in . The simulations confirm that the normalized fraction of Frenkel pairs (FPs) of 0.3–0.4 is roughly constant for cascades over [@malerba2; @st-rev]. The number of surviving FPs follows the trend from lower cascade energies (Fig. 4). The fraction of surviving interstitials grouped into clusters was found to be 0.58(3) and 0.52(3) for cascade energies of 0.2 and 0.5 MeV, respectively. This is consistent with the results for 50-100 keV cascades [@st-rev; @bjork] hinting at a possibility that the clustering fraction may reach a maximum at . A similar trend was observed for vacancy clustering fractions of 0.33(3) and 0.35(1), respectively. Cascade energy $N_{\rm F}$ NRT fraction of defects Number of isolated vacancies Number of split SIAs Number of vacancy clusters Number of SIA clusters Largest vacancy cluster Largest SIA cluster ---------------- ------------- ------------------------- ------------------------------ ---------------------- ---------------------------- ------------------------ ------------------------- --------------------- 0.2 MeV 900 (200) 0.44 (0.11) 70 (5) 65 (4) 17 (1) 46 (7) 54 89 0.5 MeV 1450 (220) 0.29 (0.04) 150 (14) 170 (15) 36 (7) 84 (13) 47 36 ![ Dependence of the number of surviving FPs on the cascade energy. A comparison between Finnis-Sinclair potential runs started at 100 K (red line [@st-rev]) with the current work (blue). The fits correspond to a single-cascade and sub-cascades (branching) regimes (details in [@st-rev]). []{data-label="fig4"}](fig4){width="\columnwidth"} The majority of interstitial clusters were found to be glissile $<$111$>$ crowdion clusters. The largest defect structure was a composite 89-interstitial cluster, formed from a merger of a set of $<$111$>$ and $<$100$>$ crowdions . Owing to its complex morphology, it will be immobile. This interstitial cluster is quite large, yet consistent with the data reported for lower energies [@soneda; @bjork]. ![ The largest cluster consisting of 89 intersitials. It is mainly composed of a set of $<$100$>$ crowdions (selected region in (a)) and a fraction of normally glissile $<$111$>$ crowdions (region highlighted in (b)). Such cluster morphology blocks the motion of crowdions and results in an overall sessile cluster; similar effect of immobilization of a cluster by another defect was observed in [@gao1]. Interstitials (vacancies) are shown in silver (blue). We used VMD package for visualization of local defect structures [@vmd]. []{data-label="f10"}](fig5){width="\columnwidth"} ![ (a) A (111) projection of a 39-vacancy cluster; (b) a (001) projection of this cluster. The large spheres show the central vacancy for selected constituent ‘vacancy crowdions’, thus emphasizing the dislocation nature of the cluster. []{data-label="f11"}](fig6){width="\columnwidth"} Large vacancy clusters were also observed with the 54-vacancy cluster being the largest one. Several large vacancy clusters formed $<$100$>$ and $<$111$>$ dislocation loop-like configurations \[Fig. \[f11\](a)\]. A cross section of an exemplar vacancy cluster is shown \[Fig. \[f11\](b)\], to emphasize its dislocation-like nature. The smaller vacancy clusters revealed a rich variety of structures, such as hexagonal vacancy clusters with interstitial rings surrounding a central vacancy \[Fig. \[f12\](a)\]. We also observed a wide range of sessile interstitial clusters. Some of these could be clearly identified as being related to the C15 meta-stable phase discussed in [@Marinica], and many were joined to crowdions or crowdion clusters \[Fig. \[f12\](b)\]. Smaller ring-like structures were also observed \[Fig. \[f12\](c)\] in which 6 atoms shared 4 neighbouring lattice sites. Conclusion ========== In summary, we have found novel structural features of radiation damage in iron on both large and local scale which we will need to be included in physical models aimed at understanding and predicting the effects of radiation damage on the mechanical, thermal and transport properties of structural materials. The reported damage structures such as the increased continuous morphology of high-energy collision cascades will form a starting point for long-timescale models in order to understand and predict the effects of radiation damage. 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--- abstract: 'Pre-trained Deep Convolutional Neural Network (CNN) features have popularly been used as full-reference perceptual quality features for CNN based image quality assessment, super-resolution, image restoration and a variety of image-to-image translation problems. In this paper, to get more insight, we link basic human visual perception to characteristics of learned deep CNN representations as a novel and first attempt to interpret them. We characterize the frequency and orientation tuning of channels in trained object detection deep CNNs (e.g., VGG-16) by applying grating stimuli of different spatial frequencies and orientations as input. We observe that the behavior of CNN channels as spatial frequency and orientation selective filters can be used to link basic human visual perception models to their characteristics. Doing so, we develop a theory to get more insight into deep CNN representations as perceptual quality features. We conclude that sensitivity to spatial frequencies that have lower contrast masking thresholds in human visual perception and a definite and strong orientation selectivity are important attributes of deep CNN channels that deliver better perceptual quality features.' author: - | Taimoor Tariq & Munchurl Kim [^1]\ Department of Electrical Engineering\ KAIST\ Daejeon, South Korea\ `{taimoor.tariq,mkimee}@kaist.ac.kr` title: Analysis and Interpretation of Deep CNN Representations as Perceptual Quality Features --- Introduction ============ Quantifying human perception of image quality has been a subject of significant research for quite some time. Full-reference objective metrics such as the PSNR (Peak Signal to Noise Ratio) and SSIM (Structural Similarity Index) ([@6]), being fair metrics of distortion between two images, are not a satisfactory metrics to measure differences in perceptual quality. Considering the recent interest in the applications of deep CNNs in perception-oriented problems such as super-resolution, image-restoration, frame-interpolation and style-transfer etc, research into effective loss metrics that quantify perceptual quality and help train CNNs in delivering better perceptual quality has become paramount. The perceptual loss proposed by [@7] was one of the first to demonstrate how effective the feature representations of pre-trained image classification CNNs could be as features of full-reference perceptual quality, especially when incorporated into loss functions for image restoration. The perceptual loss is now popularly adopted in many image restoration problems such as super-resolution, style transfer, denoising etc. ([@9],[@10],[@11]). [@12] and [@5] further demonstrate how effective deep CNN representations can be as features of perceptual quality, but without any analysis into their characteristics. More recently, [@32] proposed a variation of the perceptual loss called the contextual loss, which still employs deep CNN features as perceptual quality features but uses an approximation of the KL-divergence to quantify distance. The contextual loss has been demonstrated to be quite effective in maintaining natural image statistics during SISR. The recent PIRM Super-Resolution Challenge Report ([@29]) clearly iterates that the perceptual loss and the contextual loss are the most widely used loss functions for CNN based perceptual image Super-Resolution. Nevertheless, like most applications of deep learning, there has been little or no effort to understand and interpret the role of deep CNN representations as effective perceptual quality features. This is quite understandable, as it is difficult to find a direction to approach this problem from. Neural networks are non-linear, which makes a tractable analysis tricky. Furthermore, human perception of quality is also something that is still not understood completely. Most of our basic understanding of human visual perception of quality is in the frequency domain, with models such as the Contrast Sensitivity Function (CSF) ([@33]). To make a connection between deep CNN features and human perception, it is important to realize that deep CNN channels are essentially complex spatial frequency and orientation selective filters. We stimulate pre-trained image classification CNNs with sinusoidal grating stimuli, record the response in the form of mean activation of each channel as function of spatial frequency/orientation of input grating, thus quantifying the frequency and orientation selectivity of different channels. This approach makes it significantly easier to establish a connection between perception models such as the CSF with learned deep feature representations. We hypothesize that two attributes are important for deep CNN channels that are good perceptual quality features. The first attribute is sensitivity to spatial frequencies at which there is minimal contrast masking in human visual perception ([@34]), making the CNN channel sensitive to highly perceivable distortions. The second attribute being a definite and strong orientation selectivity, which helps the channel respond better to image regions with less pattern complexity, where there is less masking for distortions from a perceptual standpoint ([@35]). We verify our hypothesis by designing an Objective Quality Assessment (OQA) experiment ([@24]). OQA experiments correlate the performance of any quality metric with human perception of quality, which is an accepted and standard experimental technique. We group the set of channels in different CNN layers into subsets on the basis of our hypothesis and demonstrate that the group which has channels with our described attributes, delivers a much better as a set of perceptual quality features. We repeat our experiment across multiple layers of many pre-trained image classification networks such as the VGG-16 ([@30]), AlexNet ([@1]), ShuffleNet ([@41]) and SqueezeNet ([@36]). Deep CNN Representations as Perceptual Quality Features ======================================================= The main motivation behind using pre-trained image classification deep CNN representations as perceptual quality features is that instead of a distance measure between two images being a good FR metric, computing distance after non-linear transformation of images into a high dimensional manifold, might result in a better perceptual quality measure. The high dimensional manifold in this case is the manifold of pre-trained CNN features. The general form for the perceptual loss ([@7]) is given by Eq. (1) $$\mathit{l_p}=\frac{1}{M\cdot W\cdot H}\sum\limits_{m=1}^M\Vert\Phi^k_m(I\textsubscript{out})-\Phi^k_m(I\textsubscript{GT})\Vert_2^2$$ Where ’$\Phi^k_m$’ is the feature map corresponding to the ’$\textit{m}^{th}$’ channel in the ’$\textit{k}^{th}$’ layer which as ’$\textit{M}$’ number of total channels with feature map dimensions ’$\textit{H$\cdot$W}$’. As mentioned before, applying pre-trained deep CNN representations as perceptual quality features has proven to be quite effective in FR-IQA methods ([@25]), image restoration ([@10]) and style transfer ([@11]) problems, as iterated by [@29]. However, little else is known of the ability and characteristics of deep CNN representations as perceptual quality features. In this work, using basic human perception models, we aim to get more insight into the role of pre-trained deep representations as perceptual quality features. Problem Formulation =================== Section. 2 iterates the motivation and wide spread use of pre-trained deep CNN representations as features of full-reference perceptual quality. However, there has been no effort to explain and interpret the role of deep representations as perceptual features. We consider a CNN convolution layer as collection channels which deliver perceptual quality features. For example, the *relu3\_2* layer of the VGG-16 has 256 channels. Are all of the channels equally effective in delivering good perceptual quality features? Are some channels better than others and if so, what attributes make them better? The problem in question is important in explaining the role of deep CNN representations as perceptual quality features, but it is somewhat difficult to approach because of the ’black-box’ nature of neural networks. In section. 4.1, we will introduce a methodology to quantify the spatial frequency and orientation tuning of channels in pre-trained CNNs. Using this formulation, we will interpret and explain deep CNN features as perceptual quality features by making use of basic human visual perception models, which rely on spatial frequency and orientation characteristics of input stimuli. In essence, the formulation in Section. 4.1 will act as a bridge to link attributes of deep representations and basic visual perception. \[t!\] {width="\textwidth"} A Psychovisual Approach ======================= Frequency/Orientation Tuning Quantification ------------------------------------------- Our experimental method is inspired by the grating stimulus experiments used by neuro-scientists to study human visual perception characteristics ([@19]). We aim to quantify both the spatial frequency and orientation tuning of different channels in the pre-trained CNN. To quantify the spatial frequency tuning, we generate concentric sinusoidal gratings of a fixed contrast and varying spatial frequencies (cycles per degree), use them to stimulate pre-trained image classification CNNs and record the responses of the feature maps in the form of mean activation versus spatial frequency for each channel. Fig. 1 illustrates the overall scheme of measuring the spatial frequency responses of channels in various convolution layers of the trained VGG-16 network. The reason we are using a concentric pattern is to eliminate the factor of orientation selectivity from this part of the analysis. Some concentric grating stimulus patterns are shown as input to the trained VGG-16 network in Fig. 2.(a). To quantify orientation selectivity at low contrast masking thresholds, we stimulate the pre-trained network with linear pattern sinusoidal gratings with different orientations. The gratings have a fixed spatial frequency, which corresponds the the peak of the Contrast Sensitivity Function (CSF). Some sample grating patterns are shown in Fig. 1. Sample observations of orientation selectivity for channels in different layers of the pre-trained VGG-16 are shown in Fig. 2.(b). Visual Frequency Sensitivity ---------------------------- In this section, we will use the spatial frequency selectivity quantification in section. 4.1 to introduce the concept of visual frequency sensitivity. Human perception of images is largely dependent on attributes of input stimulus. A significant proportion of neuro-science research advocates the role of the early visual cortex as a spatial frequency analyzer ([@14]). Human perception characteristics are dependent on spatial frequency and one of the most significant models that quantifies this characteristic is called the Contrast Sensitivity Function (CSF). The spatial frequencies where the CSF has a higher value, correspond to lower contrast masking thresholds in perception. In essence, this corresponds to a higher probability of perceiving distortions at high CSF valued spatial frequencies. Considering the presented analysis on the spatial frequency selective behavior of deep feature maps. Our hypothesis is that the deep representations that are more sensitive to high CSF valued spatial frequencies, can be better features of perceptual quality. Fig. 3 plots mean activation of two channels versus spatial frequency of the input grating. Feature Map-2 can be seen to have a higher sensitivity compared to Feature Map-1 at high CSF valued spatial frequencies, making Feature Map-2 more sensitive to distortions corresponding to low contrast masking threshold regions in input images. We model this attribute quantitatively as $\mu_1$ defined in Eq. 2 $$\mu_1(k,m)=\sum\nolimits_{f}CSF(f).\abs\Big{\frac{\partial a_m^k}{\partial f}}$$ where ’*k*’ is the index for the convolution layer, ’*m*’ is the feature map index in each convolution layer, ’*CSF*’ is the contrast sensitivity function (CSF), ’*a*’ is the mean activation of the feature map and ’*f*’ is the spatial frequency in cycles per degree. $\mu_1$ quantifies the average sensitivity of a CNN channel weighted by the CSF over different spatial frequencies. The channels having higher $\mu_1$ values should deliver better perceptual features according to our hypothesis, because they can be more sensitive to visually perceivable distortions in input images. \[t!\] ![Two different feature maps may have different sensitivities to important visual frequencies.[]{data-label="fig:Spike_Sorting"}](VFSS_exp.png "fig:"){width="50.00000%"} Orientation Selectivity ----------------------- In addition to the underlying spatial frequency, orientation also plays an important part in human perception of visual stimulus. Neuro-science research indicates that the HVS is highly adapted to extract repeated patterns for visual content representation ([@35]). The complexity of a visual pattern has an effect in its perception. If a pattern is regular, the visual masking for such a pattern is weak, and distortions are easily perceivable. For complex and irregular image patches, the visual system presents a stronger masking effect. We have quantified orientation selectivity of different channels in a pre-trained image classification CNN (VGG-16) in Fig. 2.(b). We observe that a significant proportion of channels show a definite orientation selective tuning, such as the ones represented in Fig. 2(b)(a), Fig. 2(b)(b), Fig. 2(b)(j) etc. There channels should in theory be more sensitive in responding to simple patterns. However, quite a few channels show weaker orientation sensitivity such as the ones represented in Fig. 2(b)(c), Fig. 2(b)(k), Fig. 2(b)(n) and Fig. 2(b)(o) etc. We hypothesize the channels that show strong and definite orientation selective tuning, respond better to regular image patterns, which have lower masking thresholds, making these channels deliver better perceptual quality features. Suppose that within some layer ’*k*’, $a_\theta^m$ be the mean activation of a feature map corresponding to channel ’*m*’ to the input grating of orientation ’$\theta$’. Let the maximum mean activation be $\hat{a^m}=\max_\theta {a_\theta^m}$. We model our orientation selectivity attribute for a channel as $\mu_2$ in Eq. (3). $$\mu_2(k,m) = \sum_\theta{(a_\theta^m - \hat{a^m})^2}$$ Considering our hypothesis, channels with higher $\mu_2$ should deliver relatively better features of perceptual quality. Perceptual Efficacy Score (PE) ------------------------------ Based on our defined attributes, we devise a quantification for the efficacy of channels in pre-trained deep CNNs to deliver good features for perceptual quality, called the Perceptual Efficacy (PE). The perceptual efficacy of a channel with index ’*m*’ in layer ’*k*’ is defined as the product of normalized $\mu_2$ and $\mu_2$. $$PE(k,m) = \frac{\mu_1(k,m)\cdot\mu_2(k,m)}{\sum_m{\mu_1(k,m)}\cdot\sum_m{\mu_2(k,m)}}$$ Experimental Setup ================== We devise an experimental methodology to verify our hypotheses that deep CNN representations that have a higher PE are better perceptual quality features. Let $\sF_k$ be the set of all channels within a layer ’*k*’ of a pre-trained CNN (e.g VGG-16). $$\sF_k = \{\Phi^k_0, \Phi^k_1, \dots, \Phi^k_M \}$$ We constitute subsets of channels from $\sF_k$ based on the quantification of our proposed attributes (PE). For example, if there are 128 channels in the *relu2\_2* layer of the VGG-16, we can group the top 15% (19 channels) of the total 128 with the highest PE as $$\sH\mbox{-}15 = \{\Phi^k_0, \Phi^k_1, \dots, \Phi^k_m \}$$ Similarly, the bottom 15% channels with the lowest PE can be represented as $$\sL\mbox{-}15 = \{\Phi^k_0, \Phi^k_1, \dots, \Phi^k_m \}$$ where $\sH\mbox{-}x$,$\sL\mbox{-}x\subseteq \sF_k$ and $x\in (0,100]$. For $x=100$, the subsets become the complete set of channels $\sF_k$. To validate our hypotheses, it is necessary to demonstrate that subsets containing higher PE valued channels deliver better perceptual quality features compared to subsets with lower PE valued channels. Objective Quality Assessment (OQA) Tests ---------------------------------------- OQA tests correlate the performance of any quality metric, with human subjective assessment and perception of quality ([@24]). Human assessment of perceptual image quality is quantified using the Differential Mean Opinion Score (DMOS) over images with varying levels of distortion. Metrics that have higher correlation with DMOS scores after regression, measured using statistical indicators such as the RMSE (Root Mean Square Error), LCC (Linear Correlation Coefficient) and the SROCC (Spearman Rank Order Correlation Coefficient), are regarded as better quality metrics. In our problem setting, we will use Eq. 1 with the different subsets of channels, as defined in Section 5. We demonstrate that for use with Eq. 1, within different CNN layers, channels having higher PE, give much better correlation with DMOS compared to channels with lower PE. Essentially, we demonstrate that CNN channels with our pre-described attributes are indeed better perceptual quallity features. We use images and DMOS scores from both the LIVE image quality dataset ([@24]) and multiple distortion dataset ([@38]) which collectively include images with Gaussian Blur, JPEG compression, JPEG2000, White Noise as well as images which have been corrupted with multiple types of distortions (such as white noise, Gaussian blur and camera noise) simultaneously. We will repeat our experiment accross multiple layers of several pre-trained image classification CNNs such as AlexNet, ShuffleNet, SqueezeNet and VGG-16. [|l|l|l|l|l|l|]{} Network & Layer & --------- Feature Set --------- : Objective Quality Assessment Test. The correlation of metric scores delivered by Eq. 1 (for different feature subsets) with human subjective assessment of perceptual quality, quantified by DMOS. & RMSE & LCC & SROCC\ & & F & 9.8366 & 0.8146 & 0.8028\ & & H-10 & 8.8286 & 0.8538 & 0.8486\ & & L-10 & 12.3114 & 0.6878 & 0.6806\ & & L-90 & 10.5863 & 0.7813 & 0.7739\ & & F & 9.8149 & 0.8155 & 0.8076\ & & H-2 & 8.8183 & 0.8542 & 0.8476\ & & L-2 & 10.2338 & 0.7874 & 0.7863\ & & L-80 & 9.8485 & 0.8141 & 0.8070\ & & F & 9.7580 & 0.8179 & 0.8155\ & & H-10 & 9.1514 & 0.8419 & 0.8368\ & & L-10 & 12.8110 & 0.6553 & 0.6562\ & & L-70 & 10.3186 & 0.7936 & 0.7931\ & & F & 8.8015 & 0.8548 & 0.8605\ & & H-5 & 8.5467 & 0.8637 & 0.8651\ & & L-5 & 9.8927 & 0.8122 & 0.8197\ & & L-50 & 9.0697 & 0.8450 & 0.8507\ & & F & 11.2791 & 0.7468 & 0.7397\ & & H-10 & 10.8632 & 0.7679 & 0.7625\ & & L-10 & 12.6927 & 0.6632 & 0.6614\ & & L-50 & 11.6555 & 0.7264 & 0.7199\ & & F & 11.4191 & 0.7394 & 0.7314\ & & H-5 & 11.8710 & 0.7142 & 0.7017\ & & L-5 & 12.6857 & 0.6637 & 0.6540\ & & L-50 & 12.0600 & 0.7063 & 0.6988\ & & F & 11.0810 & 0.7570 & 0.7519\ & & H-10 & 9.9055 & 0.8117 & 0.8002\ & & L-10 & 14.2481 & 0.5424 & 0.5583\ & & L-70 & 11.6409 & 0.7272 & 0.7232\ & & F & 9.1354 & 0.8425 & 0.8421\ & & H-10 & 8.8577 & 0.8528 & 0.8477\ & & L-10 & 11.5070 & 0.7346 & 0.7407\ & & L-70 & 9.2306 & 0.8389 & 0.8414\ ------- ------- ------- ------- ------- ------- ------- ------- F H-10 L-10 L-75 F H-2 L-2 L-80 60.23 62.85 56.08 59.83 60.69 62.53 60.10 60.21 F H-5 L-5 L-50 F H-2 L-2 L-75 59.97 60.86 58.28 59.41 64.62 63.38 61.30 62.35 ------- ------- ------- ------- ------- ------- ------- ------- : 2AFC Similarity Test. How well metric decisions conform with human assessment of image triplets . 2AFC Similarity tests --------------------- In the 2AFC test, two distorted images are shown to an observer and he/she is asked to rate which one is closer to the ground truth, in perceptual appearance. This process is repeated for multiple image triplets and observers per-triplet to construct a data-set called the Berkley-Adobpe Perceptual Patch Similarity Data-set (BAPPS) ([@12]). Objective metrics such as the one in Eq. 1 are thereafter evaluated to see how well they conform to the pair-wise human judgment. For example, in an image triplet, let $\mathbf{x_0}$ and $\mathbf{x_1}$ be two distorted versions of the ground truth image $\mathbf{x_g}$ that are shown to 5 human observers, 4 of which judge $\mathbf{x_0}$ to be closer to $\mathbf{x_g}$, as opposed to $\mathbf{x_1}$ being closer to $\mathbf{x_g}$. If an objective metric evaluates $\mathbf{x_0}$ to be closer to the ground truth, it will get an 80% credit which in the opposite case would be 20%. The BAPPS data-set contains images with distortions such as super-resolution, frame-interpolation and deblurring, which do not have subjective DMOS data-sets available online. Therefore, as a secondary experiment, we perform a 2AFC test on super-resolution, frame-interpolation and video-deblurred frame images in the BAPPS data-set with Eq. 1 for different channel subsets defined in Section. 5. In order to verify our hypothesis, we will show that subsets that contain channels with higher PE, deliver better perceptual quality features. Results and Discussions ======================= Table 1 quantifies the correlation of Eq. 1 with DMOS for different subsets of channels, constructed on the basis of our described attributes, as explained in Section 5. Table 1 validates our hypothesis that within a CNN layer, channels which have higher PE (Eq. 4) deliver better perceptual quality features. It can be observed that very small proportions (2%-10% of total) of channels with the highest PE, deliver better perceptual quality features compared to a much higher proportion (50%-90% of total) of channels having lower PE. Furthermore, in a majority of cases, a small proportion of channels that have our described attributes (higher PE), perform even better than the complete set of channels in the layer. This implies that our proposed attributes are indeed important characteristics that make learned deep CNN representations good perceptual quality features. Table. 2 shows the results of our secondary 2AFC similarity test on the super-resolution, frame-interpolation and video-deblurring distorted images in the BAPPS data-set. It can been seen that yet again, similar to the conclusion in the primary QQA experiment, the subsets with channels having higher PE are better perceptual quality features compared to even much larger subsets having channels with lower PE. Future Work =========== We have proposed a model to explain and interpret which channels in pre-trained image classification CNNs deliver better perceptual quality features. The model may be used to improve the use of deep representations as perceptual quality features by helping in feature selection for IQA methods such as ([@25]) and maybe designing channel attentive mechanism to improve the perceptual loss ([@7]). The model may also be reference for learning better perceptual quality feature representations which may benefit a wide variety of applications. Furthermore, the model may be enhanced to include more psychophysical factors such as eccentricity etc. Another possible application may be CNN-based image compression where prior knowledge of the potential efficacy of different channels may help efficient perceptual compression of redundant image data. Conclusions =========== Deep CNN representations of pre-trained image classifications CNNs have been popularly used as perceptual quality features for perception orientated applications such as CNN based quality assessment, image/video super-resolution and many image-to-image translation problems. In this paper, as a novel and first effort, we have linked basic human visual perception models to pre-trained deep CNN representations in order to explain and interpret them as perceptual quality features. 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Squeezenet: Alexnet-level accuracy with 50x fewer parameters and &lt;1mb model size. , abs/1602.07360, 2017. Sebastian Bosse, Dominique Maniry, Klaus-Robert Muller, Thomas Wiegand, and Wojciech Samek. Deep neural networks for no-reference and full-reference image quality assessment. , 27:206–219, 2017. Janus J. Kulikowski, S. Marvcelja, and P. O. Bishop. Theory of spatial position and spatial frequency relations in the receptive fields of simple cells in the visual cortex. , 43:187–198, 1982. Lamberto Maffei and Adriana Fiorentini. The visual cortex as a spatial frequency analyser. , 13 7:1255–67, 1973. Dinesh Jayaraman, Anish Mittal, Anush K. Moorthy, and Alan C. Bovik. Objective quality assessment of multiply distorted images. , pages 1693–1697, 2012. Recommendation itu-r bt.500-11 - methodology for the subjective assessment of the quality of television pictures. 2002. Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott E. Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. , pages 1–9, 2014. Mark Sandler, Andrew F Howard, Menglong Zhu, Andrey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. , pages 4510–4520, 2018. Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. , pages 770–778, 2015. Grating Generation ================== In this section, we present details behind the generation of sinusoidal gratings of different spatial frequencies. The contrast sensitivity function is expressed on the domain of spatial frequency in cycles per degree (cyc/deg). The cycles per degree express the number of sine cycles captured by the observer per unit degree of observation. Obviously, the distance of viewing and dimensions of the screen play an important part in this measurement. We essentially generate gratings in the computer simulation in cycles per pixel. Let the display screen being used in the experiment have a height ’*h*’ inches and resolution ’*r*’ pixels per inch. The optimal viewing distance in psychovisual experiments should satisfy a function called the PVD ([@42]). The PVD is a function that expresses the optimal ratio of viewing distance to the height of the display screen. The optimal viewing distance ’*d*’ for the screen with height ’*h*’ can be calculated using the PVD. The transformation between cycles/degree and cycles/pixel is $$\frac{cycles}{pixel} = \frac{cycles}{degree} \times \frac{degrees}{pixel}$$ Where $$\frac{pixels}{degree} = \frac{180}{\pi \times d \times r}$$ Therefore, $$\frac{cycles}{pixel} = \frac{cycles}{degree} \times \frac{\pi \times d \times r}{180}$$ We have tested with a number of different display systems of SD, 2K and 4K resolutions. Considering that the PVD takes the viewing angle into account, the changes in the resultant spatial frequencies of the gratings are small and insignificant. Therefore, it can be concluded that the choice of display system has a negligible effect on the experiment. For the generation of grating with fixed spatial frequency and varying orientation, the experimental setup is the same. Additional Networks =================== We demonstrate the validity of our hypothesis for other pre-trained image classification CNNs as well. These networks include: - GoogleNet ([@39]) - [MobileNet-v2]{} ([@40]) - [ResNet-18]{} ([@37]) It can be seen in Table 3 that our hypothesis regarding important attributes is valid for these additional CNNs as well. Small proportions of channels (H-(5-10)) with higher PE (Eq. 4) deliver much better perceptual quality features compared to a much higher proportion of channels with lower PE (H-(45-80)) and even the complete set of channels in the layer (F). A scatter plot of some correlations, shown side by side is depicted in Fig. 5. Network Layer Feature Set RMSE LCC SROCC --------- ------- ------------- --------- -------- -------- F 9.2730 0.8370 0.8351 H-5 9.1360 0.8425 0.8364 L-5 12.6595 0.6654 0.6674 L-80 9.6636 0.8218 0.8203 F 10.2264 0.7977 0.8061 H-5 9.8592 0.8137 0.8201 L-5 10.8882 0.7667 0.7750 L-45 10.0326 0.8063 0.8163 F 11.9441 0.7099 0.7017 H-10 11.6059 0.7292 0.7256 L-10 13.7130 0.5884 0.5825 L-70 12.7912 0.6566 0.6505 F 10.1957 0.7991 0.8063 H-10 9.2423 0.8385 0.8459 L-10 13.2810 0.6219 0.6223 L-70 10.7877 0.7716 0.7804 F 10.8622 0.7680 0.7702 H-10 10.0841 0.8040 0.7898 L-10 11.6195 0.7284 0.7339 L-75 11.2807 0.7467 0.7549 F 9.1073 0.8436 0.8611 H-5 9.2559 0.8379 0.8509 L-5 10.1132 0.8028 0.8072 L-75 9.3484 0.8344 0.8518 : Objective Quality Assessment Test. The correlation of metric scores delivered by Eq. 1 (for different feature subsets) with human subjective assessment of perceptual quality, quantified by DMOS. [^1]: Use footnote for providing further information about author (webpage, alternative address)—*not* for acknowledging funding agencies. Funding acknowledgements go at the end of the paper.

--- abstract: 'We calculate the covariant one-loop quantum gravitational effective action for a scalar field model inspired by the recently proposed nonminimal natural inflation model. Our calculation is perturbative, in the sense that the effective action is evaluated in orders of background field, around a Minkowski background. The effective potential has been evaluated taking into account the finite corrections. An order-of-magnitude estimate of the one-loop corrections reveals that gravitational and non-gravitational corrections have same or comparable magnitudes.' author: - Sandeep Aashish - Sukanta Panda bibliography: - 'ref.bib' - 'references.bib' title: Covariant quantum corrections to a scalar field model inspired by nonminimal natural inflation --- \[intro\]Introduction ===================== A fully consistent quantum theory of gravity has remained elusive despite longstanding efforts to construct a gravity theory that is valid at the planck scale (see Ref. [@woodard2009] for a comprehensive review). A well known problem with quantizing gravity perturbatively is the inability to consistently absorb the divergences, giving rise to non-renormalizability. However, in the past two decades or so, a more modern view has developed where general relativity is studied as a quantum effective field theory at low energies [@donoghue1994]. This treatment allows separation of quantum effects from known low energy physics from those that depend on the ultimate high energy completion of the theory of gravity (see Ref. [@donoghue2015] for a review by Donoghue and Holstein). More recently, with the availability of high precision data from experiments probing the early universe, especially inflation era, it has become important to consider quantum gravitational corrections in early universe cosmology [@fabris2012; @krauss2014; @woodard2014]. This has motivated several studies of aspects of quantum gravitational corrections in inflationary universe, see for example Refs. [@klemm2004; @cognola2005; @cognola2012; @hebecker2017; @herranen2017; @bounakis2018; @markkanen2018; @ruf2018; @heisenberg2019]. One of the well-known methods employed in such studies is to compute the effective action, which is known to be the generator of 1PI diagrams [@buchbinder1992; @vilkovisky1992]. An advantage of this technique is that one can directly obtain divergence structure at a given loop order, without going through the hassle of summing over individual feynman diagram contributions. Other applications include the calculation of effective potential [@coleman1973]. The computation of effective action is most commonly carried out using the background field method, where small fluctuations about a classical background field are quantized, not the total field. In general, it turns out that the results consequently depend on the choice of background field [@falkenberg1998; @labus2016; @ohta2016]. In case of gravity, which is treated as a gauge theory, it is therefore important to ensure that there are no fictitious dependence of conclusions on the choice of gauge and background. Hence, in this work, we use DeWitt-Vilkovisky’s covariant effective action approach that systematically yields gauge and background independent effective action [@dewitt1967b; @*dewitt1967c; @parker2009]. We consider a recently proposed modification of the natural inflation (NI) model [@freese1990], wherein a periodic nonminimal coupling term similar to NI potential is added along with a new parameter, that eventually leads to a better fit with Planck results [@ferreira2018]. These phenomenological implications are in no way the only motivation for considering this model in the present work. Rather, it serves as a toy model to achieve our mainly three objectives, which are as follows. First, to set up the computation using symbolic manipulation packages to evaluate one-loop covariant effective action upto quartic order terms in the background field. As a starting point, we work in the Minkowski background. Second, we aim to recover and establish past results. And third, we wish to estimate the magnitudes of quantum gravitational corrections from the finite contributions at least for the effective potential, since there are typically several thousands of terms one has to deal with. The organization of this paper is as follows. In Sec. \[sec2\], we introduce and briefly review the nonminimal natural inflation model. Sec. \[sec3\] covers a review of covariant effective action formalism, notations, and the methodology of our calculations. Sec. \[sec4\] constitutes a major part of this paper, detailing the calculations of each contributing term mentioned in Sec. \[sec3\], along with the divergent part, loop integrals, and renormalization. Some past results and their extensions have also been presented. Finally, in Sec. \[sec5\], we derive the effective potential including the finite corrections from the loop integrals, and perform an order-of-magnitude estimation of quantum corrections. \[sec2\]Periodic nonminimal natural inflation model =================================================== Natural inflation was first introduced by Freese [*et al*.]{}[@freese1990] as an approach where inflation arises dynamically (or *naturally*) from particle physics models. In natural inflation models, a flat potential is effected using pseudo Nambu-Goldstone bosons arising from breaking the continuous shift symmetry of Nambu-Goldstone modes into a discrete shift symmetry. As a result, the inflation potential in a Natural inflation model takes the form, $$\begin{aligned} \label{eq01} V(\phi) = \Lambda^4 \left(1 + \cos(\phi/f)\right);\end{aligned}$$ where the magnitude of parameter $\Lambda^4$ and periodicity scale $f$ are model dependent. However, majority of natural inflation models are in tension with recent Planck 2018 results [@planck2018x]. This work concerns a recently proposed extension of the original natural inflation model introducing a new periodic non-minimal coupling to gravity [@ferreira2018]. The authors in [@ferreira2018] showed that the new model leads to a better fit with observation data thanks to the introduction of a new parameter in the nonminimal coupling term, with $n_{s}$ and $r$ values well within $95\%$ C.L. region from combined Planck 2018+BAO+BK14 data. An important feature of this model is that $f$ becomes sub-planckian, contrary to a super-planckian $f$ in the original natural inflation model [@freese1990], and thus addresses issues related to gravitational instanton corrections [@banks2003; @rudelius2015a; @rudelius2015b; @montero2015; @hebecker2017]. Our objective here is to study one-loop quantum gravitational corrections to the natural inflation model with non-minimal coupling, using Vilkovisky-DeWitt’s covariant effective action approach [@parker2009]. Unfortunately, calculating the covariant effective action exactly is highly nontrivial, though non-covariant effective actions can in principle be evaluated using proper time methods. Hence, we take a different route by employing a perturbative calculation of one-loop effective action, in orders of the background scalar field. This requires us to apply a couple of approximations. First, we work in the regime where potential is flat, i.e. $\phi \ll f$, which is generally true during slow-rolling inflation. Second, the background metric is set to be Minkowski. This choice is debatable, since it does not accurately represent an inflationary scenario, but has been used before [@saltas2017; @bounakis2018] as a first step towards studying quantum corrections. The action for the nonminimal natural inflation in the Einstein frame is given by, $$\begin{aligned} \label{eq02} S = \int d^4 x \sqrt{-g}\left(-\dfrac{2 R}{\kappa^2} + \dfrac{1}{2}K(\phi)\phi {}_{;a} \phi {}^{;a} + \dfrac{V(\phi)}{(\gamma(\phi))^4}\right)\end{aligned}$$ where, $$\label{eq03} \gamma(\phi)^2 = 1 + \alpha\left(1+\cos\left(\dfrac{\phi}{f}\right)\right),$$ and, $$\label{eq04} K(\phi) = \dfrac{1 + 24\gamma'(\phi)^2/\kappa^2}{\gamma(\phi)^2}.$$ $V(\phi)$ is as in Eq. (\[eq01\]). Here, $\phi_{;a} \equiv \nabla_{a}\phi$ denotes the covariant derivative. In the region where potential is flat, $\phi/f \ll 1$, and we expand all periodic functions in Eq. (\[eq02\]) upto quartic order in $\phi$ followed by rescaling $\sqrt{k_0}\phi \to \phi$: $$\label{action} S \approx \int d^4 x \sqrt{-g} \left(- \frac{2 R}{\kappa^2} + \tfrac{1}{2} \frac{m^2}{k_0} \phi^2 + \tfrac{1}{24} \frac{\lambda}{k_0^2} \phi^4 + \tfrac{1}{2} (1 + \frac{k_1}{k_0^2} \phi^2) \phi {}_{;a} \phi {}^{;a}\right) + \mathcal{O}(\phi^5)$$ where parameters $m, \lambda,k_0$ and $k_1$ have been defined out of $\alpha, f$ and $\Lambda^4$ in from Eq. (\[eq02\]): $$\begin{aligned} \label{param} m^2 &=& \dfrac{\Lambda^4 (2\alpha - 1)}{(1 + 2\alpha)^3 f^2};\nonumber \\ \lambda &=& \dfrac{\Lambda^4 (8\alpha^2 - 12\alpha + 1)}{(1 + 2\alpha)^4 f^4}; \nonumber \\ k_0 &=& \dfrac{1}{1 + 2\alpha}; \nonumber \\ k_1 &=& \dfrac{\alpha(\kappa^2 f^2 + 96\alpha^2 + 48\alpha)}{2\kappa^2 f^4 (1 + 2\alpha)^2}. \nonumber \\\end{aligned}$$ We have also omitted a constant term appearing in ($\ref{action}$) because such terms are negligibly small in early universe. The action (\[action\]) is in effect a $\phi^4$ scalar theory with derivative coupling. \[sec3\]Effective action formalism ================================== A standard procedure while calculating loop corrections in quantum field theory, is to use the well known background field method, according to which a field is split into a classical background and a quantum part that is much smaller in magnitude (and hence treated perturbatively) [@falkenberg1998; @labus2016; @ohta2016]. A by-product of this procedure is the background and gauge dependence of quantum corrections. We briefly review here the covariant effective action formalism, that yields gauge-invariant and background field independent results, employed in this work. Interested reader is advised to see Ref. [@parker2009] (chap. 7) for a detailed review. Quantization of a theory $S[\varphi]$ with fields $\varphi^{i}$ is performed about a classical background $\bar{\varphi}^{i}$: $\varphi^{i} = \bar{\varphi}^{i} + \zeta^{i}$, where $\zeta^{i}$ is the quantum part. Here, $\varphi^i$ is the local coordinate of a point in the ‘field space’ and represents any scalar or vector or tensor field(s) in the coordinate space. The index $i$ in field space corresponds to all gauge indices and coordinate dependence of fields. This way of writing the field-space equivalent of a coordinate space quantity (such as a vector field) is called condensed notation [@dewitt1964]. In our case, $\varphi^{i}=\{g_{\mu\nu}(x),\phi(x)\}$; $\bar{\varphi}^{i}=\{\eta_{\mu\nu},\bar{\phi}(x)\}$ where $\eta_{\mu\nu}$ is the Minkowski metric; and, $\zeta^{i}=\{\kappa h_{\mu\nu}(x),\delta\phi(x)\}$. The fluctuations $\zeta^{i}$ are assumed to be small enough for a perturbative treatment to be valid, viz. $|\kappa h_{\mu\nu}|\ll 1; |\delta\phi|\ll |\phi|$. In this limit, the infinitesimal general coordinate transformations can be treated as gauge transformations associated with $h_{\mu\nu}$ [@donoghue1994; @donoghue2017]. In fact, for any metric $g_{\mu\nu}(x)$, this infinitesimal transformation takes the form, $$\begin{aligned} \label{aeaf0} \delta g_{\mu\nu} = -\delta\epsilon^\lambda g_{\mu\nu,\lambda}-\delta\epsilon^\lambda \,_{,\mu}g_{\lambda\nu}-\delta\epsilon^\lambda \,_{,\nu}g_{\lambda\mu}.\end{aligned}$$ In the condensed notation, an infinitesimal gauge transformation of any field $\varphi^i$ is given by, $$\begin{aligned} \label{aeaf1} \delta\varphi^{i} = K^{i}_{\alpha}[\varphi]\delta\epsilon^{\alpha},\end{aligned}$$ where $K^{i}_{\alpha}$ is identified as the generator of gauge transformations, while $\delta\epsilon^{\alpha}$ are the gauge parameters. As with einstein notations, repeated (or contracted) indices in the condensed notation represent a sum over all the associated associated gauge or tensor indices and integral over all coordinate indices. The gauge fixing condition is given by fixing a functional $\chi_{\alpha}[\bar{\varphi}]$ so that it intersects each gauge orbit in field space only once. Including the gauge-fixing condition(s) and corresponding ghost determinant(s), the covariant one-loop effective action is given by [@huggins1987; @toms2007] $$\begin{aligned} \label{aeaf2} \Gamma = -\ln\int[d\zeta]\exp\left[\dfrac{1}{2}\left(-\zeta^{i}\zeta^{j}\Big(S_{,ij}[\bar{\varphi}] - \Gamma^{k}_{ij}S_{,k}[\bar{\varphi}]\Big) - \frac{1}{2\alpha}f_{\alpha\beta}\chi^{\alpha}\chi^{\beta}\right)\right]-\ln\det Q_{\alpha\beta}[\bar{\varphi}],\end{aligned}$$ as $\alpha\longrightarrow 0$ (Landau gauge). Here, $[d\zeta]\equiv \prod_{i} d\zeta$. A few comments on Eq. (\[aeaf2\]) are in order. The first term inside the exponential is the covariant derivative of the action functional with respect to $\zeta^{i}$ in field space. $\Gamma^{k}_{ij}$ are the field-space connections defined with respect to the field-space metric $G_{ij}$, and are responsible for general covariance of Eq. (\[aeaf2\]). In general, the field-space connections have complicated, non-local structure especially in presence of a gauge symmetry. However, they reduce to the standard Christoffel connections, in terms of $G_{ij}$, when $\chi_{\alpha}$ is chosen to be the Landau-DeWitt gauge i.e. $\chi_{\alpha} = K_{\alpha i}[\bar{\varphi}]\zeta^{i} = 0$, along with $\alpha\to 0$ [@mackay2010; @bounakis2018]. $f_{\alpha\beta}$ is any symmetric, positive definite operator and makes no non-trivial contribution to effective action [@parker2009]. Note also that the contributions from connection terms, and hence the question of covariance, is relevant for off-shell analyses, since $S_{,i} = 0$ on-shell. $\det Q_{\alpha\beta}$ is the ghost determinant term that appears during quantization. This term is absorbed into the exponential by introducing Faddeev-Popov ghosts, $c^{\alpha}$ and $\bar{c}^{\alpha}$, so that [@parker2009], $$\begin{aligned} \label{aeaf3} \ln\det Q_{\alpha\beta} = \ln\int [d \bar{c}^{\alpha}] [d c^{\beta}] \exp \left(- \bar{c}^{\alpha} Q_{\alpha\beta} c^{\beta} \right).\end{aligned}$$ As a result, $$\begin{aligned} \label{aeaf4} \Gamma[\bar{\varphi}] = -\ln\int [d\zeta] [d \bar{c}^{\alpha}] [d c^{\beta}] \exp\left[-\dfrac{\zeta^{i}\zeta^{j}}{2}\Big(S_{,ij}[\bar{\varphi}] - \Gamma^{k}_{ij}S_{,k}[\bar{\varphi}]\Big) - \frac{1}{4\alpha}f_{\alpha\beta}\chi^{\alpha}\chi^{\beta} - \bar{c}^{\alpha} Q_{\alpha\beta} c^{\beta}\right].\end{aligned}$$ The computation of Eq. (\[aeaf4\]) traditionally has involved the use of proper time method, such as employing the heat kernel technique. For Laplace type operators (coefficients of $\zeta^{i}\zeta^{j}$ in the exponential), of the form $g^{\mu\nu}\nabla_{\mu}\nabla_{\nu} + Q$ (where $Q$ does not contain derivatives), the heat kernel coefficients are known and are quite useful because they are independent of dimensionality [@dewitt1964]. However, these operators in general are not Laplace type, as in the present case. A class of nonminimal operators such as the one in Eq. (\[aeaf4\]) can be transformed to minimal (Laplace) form using the generalised Schwinger-DeWitt technique [@barvinsky1985], but in practice the implementation is quite complicated and specific to a given Lagrangian. Examples of such an implementation can be found in Refs. [@alvarez2015; @steinwachs2011]. We take a different approach here, calculating the one-loop effective action perturbatively in orders of the background field. While one does not obtain exact results in a perturbative approach, unlike the heat kernel approach, it is possible to obtain accurate results upto a certain order in background fields which is of relevance for a theory in, say, the early universe. Some past examples are Refs. [@saltas2017; @bounakis2018]. Moreover, our implementation of this method using xAct packages for Mathematica [@xact; @xpert] is fairly general in terms of its applicability to not only scalars coupled with gravity, but also vector and tensor fields (see, for instance, Ref. [@aashish2019b]). A caveat at this time, is that the perturbative expansions are performed about the Minkowski background and not a general metric background. However, a generalization to include FRW background is part our future plans. For convenience, we write the exponential in the first term of $\Gamma$ as, $$\begin{aligned} \label{aeaf5} \exp[\cdots] &=& \exp\left\{- \left(\tilde{S}[\bar{\varphi}^{0}]+\tilde{S}[\bar{\varphi}^{1}]+\tilde{S}[\bar{\varphi}^{2}]+\tilde{S}[\bar{\varphi}^{3}]+\tilde{S}[\bar{\varphi}^{4}]\right)\right\}\nonumber \\ &\equiv & \exp\left\{- (\tilde{S}_{0}+\tilde{S}_{1}+\tilde{S}_{2}+\tilde{S}_{3}+\tilde{S}_{4})\right\}\end{aligned}$$ $\tilde{S}_0$ yields the propagator for each of the fields $\zeta^{i}$. The rest of the terms are contributions from interaction terms, which we assume to be small. Treating $\tilde{S}_{1},...,\tilde{S}_{4}$ as perturbative, and expanding Eq. (\[aeaf5\]), $\Gamma[\bar{\varphi}]$ can be written as, $$\begin{aligned} \label{aeaf6} \Gamma[\bar{\varphi}] &=& -\ln\int [d\zeta] [d \bar{c}^{\alpha}] [d c^{\beta}] e^{-S_0}(1-\delta S + \dfrac{\delta S^2}{2} + \cdots) ; \nonumber \\ &=& - \ln (1 - \langle\delta S\rangle + \dfrac{1}{2}\langle\delta S^2 \rangle + \cdots); \end{aligned}$$ where $\delta S = \sum_{i=1}^4 \tilde{S}_{i}$ and $\langle\cdot\rangle$ represents the expectation value in the path integral formulation. Finally, we use $\ln(1 + x)\approx x$ to find the contributions to $\Gamma$ at each order of background field. We only use the leading term in the logarithmic expansion, since all higher order terms will yield contributions from disconnected diagrams viz-a-viz $\langle\delta S\rangle^{2},$ etc which we ignore throughout our calculation. Moreover, since we are interested in terms upto quartic order in background field, we truncate the Taylor series in Eq. (\[aeaf6\]) upto $\delta S^{4}$. With these considerations, the final contributions to $\Gamma$ at each order of $\bar{\varphi}$ is: $$\begin{aligned} \label{aeaf7} \mathcal{O}(\bar{\varphi}) &:& \langle\tilde{S}_{1}\rangle;\nonumber \\ \mathcal{O}(\bar{\varphi}^2) &:& \langle\tilde{S}_{2}\rangle - \dfrac{1}{2}\langle\tilde{S}_{1}^{2}\rangle; \nonumber \\ \mathcal{O}(\bar{\varphi}^3) &:& \langle\tilde{S}_{3}\rangle - \langle\tilde{S}_{1}\tilde{S}_{2}\rangle + \dfrac{1}{6} \langle\tilde{S}_{1}^{3}\rangle;\nonumber \\ \mathcal{O}(\bar{\varphi}^4) &:& \langle\tilde{S}_{4}\rangle - \langle\tilde{S}_{1}\tilde{S}_{3}\rangle + \dfrac{1}{2}\langle\tilde{S}_{1}^{2}\tilde{S}_{2}\rangle -\dfrac{1}{2}\langle\tilde{S}_{2}^{2}\rangle - \dfrac{1}{24}\langle\tilde{S}_{1}^{4}\rangle .\end{aligned}$$ Also, we recall that the metric fluctuations have a factor of $\kappa$. Accordingly, the terms in Eq. (\[aeaf7\]) will also contain powers of $\kappa$. It turns out, as will be shown below, that all contributions are at most of the order $\kappa^4$. Expecting $\mathcal{O}(\kappa^4)$ terms to be significantly suppressed, we only take into account the corrections upto $\mathcal{O}(\kappa^2)$. In what follows, we will detail the evaluation of terms in Eq. (\[aeaf7\]). \[sec4\]Covariant one-loop corrections ====================================== Setup ----- The first step towards writing $\Gamma[\bar{\varphi}]$ in Eq. (\[aeaf4\]) is to identify the field space metric, given in terms of the field-space line element, $$\begin{aligned} ds^{2} &= G_{ij}d\varphi^{i} d\varphi^{j} \\ \label{acqc0} &= \int d^{4}x d^{4}x' \left(G_{g_{\mu\nu}(x)g_{\rho\sigma}(x')}dg_{\mu\nu}(x) dg_{\rho\sigma}(x') + G_{\phi(x)\phi(x')} d\phi(x) d\phi(x')\right).\end{aligned}$$ A prescription for identifying field space metric is to read off the components of $G_{ij}$ from the coefficients of highest derivative terms in classical action functional [@vilkovisky1984b]. For the scalar field $\phi(x)$, the field-space metric is chosen to be, $$\begin{aligned} \label{acqc1} G_{\phi(x)\phi(x')}=\sqrt{g(x)}\delta(x,x');\end{aligned}$$ For the metric $g_{\mu\nu}(x)$, a standard choice for field-space metric is [@parker2009; @mackay2010] $$\begin{aligned} \label{acqc2} G_{g_{\mu\nu}(x)g_{\rho\sigma}(x')}=\dfrac{\sqrt{g(x)}}{\kappa^2}\left\{ g^{\mu(\rho}(x)g^{\sigma)\nu}(x)- \frac{1}{2}g^{\mu\nu}(x)g^{\rho\sigma}(x) \right\}\delta(x,x'),\end{aligned}$$ where the brackets around tensor indices in the first term indicate symmetrization. As a convention, we choose to include $\kappa^2$ factor in Eq. (\[acqc2\]) to account for dimensionality of the length element in Eq. (\[acqc0\]), although choosing otherwise is also equally valid as long as dimensionality is taken care of. The inverse metric can be derived from the identity $G_{ij}G^{jk}=\delta^{k}_{i}$: $$\begin{aligned} \label{acqc3} G^{g_{\mu\nu}(x)g_{\rho\sigma}(x')}&=&\kappa^2\left\{ g_{\mu(\rho}(x)g_{\sigma)\nu}(x)- \frac{1}{2}g_{\mu\nu}(x)g_{\rho\sigma}(x) \right\}\delta(x,x'); \\ \label{acqc4} G^{\phi(x)\phi(x')}&=&\delta(x,x').\end{aligned}$$ Next, using Eqs. (\[acqc1\])-(\[acqc4\]), one can find the Vilkovisky-DeWitt connections $\Gamma^{k}_{ij}$ which has an identical definition to the Christoffel connections thanks to the Landau-DeWitt gauge choice. Out of a total of six possibilities there are three non-zero connections obtained as follows: $$\begin{aligned} \label{acqc5} \Gamma^{g_{\lambda\tau}(x)}_{g_{\mu\nu}(x')g_{\rho\sigma}(x'')} &= \delta(x'',x')\delta(x'',x)\left[-\delta^{(\mu}_{(\lambda}g^{\nu)(\rho}(x)\delta^{\sigma)}_{\tau)} + \dfrac{1}{4}g^{\mu\nu}(x)\delta^{\rho}_{(\lambda}\delta^{\sigma}_{\tau)} + \dfrac{1}{4}g^{\rho\sigma}(x)\delta^{\mu}_{(\lambda}\delta^{\nu}_{\tau)} \right. \nonumber \\ & \left. + \dfrac{1}{4}g_{\lambda\tau}(x)g^{\mu(\rho}(x)g^{\sigma)\nu}(x) - \dfrac{1}{8} g_{\lambda\tau}(x)g^{\mu\nu}(x)g^{\rho\sigma}(x) \right] \\ \Gamma^{g_{\lambda\tau}(x)}_{\phi(x')\phi(x'')} &= \dfrac{\kappa^2}{4}\delta(x'',x')\delta(x'',x) g_{\lambda\tau}(x) \\ \Gamma^{\phi(x)}_{\phi(x') g_{\lambda\tau}(x'')} &= \dfrac{1}{4}\delta(x'',x')\delta(x'',x) g^{\lambda\tau}(x) = \Gamma^{\phi(x)}_{g_{\lambda\tau}(x') \phi(x'')}.\end{aligned}$$ Note that upon substituting into Eq. (\[aeaf4\]), all calculations here are evaluated at the background field(s) which in our case is the Minkowski metric and a scalar field $\bar{\phi}(x)$. We also recall that this rather unrestricted choice of background is allowed because of the DeWitt connections that ensure gauge and background independence. As alluded to earlier, the Landau-DeWitt gauge condition, $K_{\alpha i}[\bar{\varphi}]\zeta^{i} = 0$, is given in terms of the gauge generators $K_{\alpha i}$. Since there is only one set of transformations vi-a-viz general coordinate transformation, there exists one gauge parameter which we call $\xi^{\lambda}(x)$. In the condensed notation, this corresponds to $\delta\epsilon^{\alpha}$ where $\alpha\to (\lambda,x)$. Gauge generator on the gravity side $K^{g_{\mu\nu}}_{\lambda}(x,x')$ is read off from Eq. (\[aeaf0\]), while $K^{\phi}_{\lambda}(x,x')$ is read off from the transformation of $\phi$: $$\begin{aligned} \label{acqc6} \delta_{\xi}\phi = -\partial_{\mu}\phi \xi^{\lambda} .\end{aligned}$$ Substituting in the definition of $\chi_{\alpha}[\bar{\varphi}]$ in coordinate space, we obtain $$\begin{aligned} \label{acqc7} \chi_{\lambda}[\bar{\phi}] &=& \int d^{4}x' \left( K_{g_{\mu\nu} \lambda}(x,x') \kappa h_{\mu\nu}(x') + K_{\phi \lambda}(x,x')\delta\phi(x') \right) \nonumber \\ &=& \dfrac{2}{\kappa}\left(\partial^{\mu}h_{\mu\lambda} - \dfrac{1}{2}\partial_{\lambda}h\right) - \omega\partial_{\lambda}\bar{\phi}\delta\phi .\end{aligned}$$ where $\omega$ is a bookkeeping parameter, which we adopt from Ref. [@mackay2010]; a second such parameter $\nu$ appears with all Vilkovisky-DeWitt connection terms to keep track of gauge (non-)invariance. That is, we write $S_{;ij}=S_{,ij} - \nu \Gamma^{k}_{ij}S_{,k}$. As shown later, playing with these parameters reproduces past non-gauge-invariant results. Here and throughout, the indices of field-space quantities like the gauge generator are raised and lowered using field-space metric defined in Eqs. (\[acqc1\]) - (\[acqc4\]). Lastly, we choose $f^{\alpha\beta} = \kappa^{2}\eta^{\lambda \lambda'}\delta(x,x')$ in Eq. (\[aeaf4\]) to determine the gauge fixing term. One last piece needed before background-field-order expansions, the ghost term $Q_{\alpha\beta}$. We use the definition [@parker2009], $Q_{\alpha\beta} \equiv \chi_{\alpha, i}K^{i}_{\beta}$, to obtain $$\begin{aligned} \label{acqc8} Q_{\mu\nu} = \left(-\dfrac{2}{\kappa} \eta_{\mu\nu}\partial_{\alpha}\partial^{\alpha} + \omega\partial_{\mu}\bar{\phi}\partial_{\nu}\bar{\phi}\right)\delta(x,x').\end{aligned}$$ \[loopint\]Loop integrals and divergent parts --------------------------------------------- Substituting the connections, the gauge fixing term and the ghost term along with action (\[action\]) in Eq. (\[aeaf4\]), and employing the notations in Eq. (\[aeaf5\]), we obtain: $$\begin{aligned} \label{s0} \tilde{S}_{0} &=& {\int d^4{x} } \Big[\frac{m^2 (\delta \phi)^2}{2 k_0} + \tfrac{1}{2} \delta \phi{}_{,a} \delta \phi{}^{,a} - 2 h^{ab} h^{c}{}_{c}{}_{,a}{}_{,b} - \frac{2 \bar{c}^{a} c_{a}{}^{,b}{}_{,b}}{\kappa} + h^{ab} h^{}{}_{a}{}^{c}{}_{,b}{}_{,c} - \frac{h^{ab} h^{}{}_{a}{}^{c}{}_{,b}{}_{,c}}{\alpha} \nonumber \\ && + h^{a}{}_{a} h^{bc}{}_{,b}{}_{,c} + \frac{h^{a}{}_{a} h^{bc}{}_{,b}{}_{,c}}{\alpha} - \tfrac{1}{2} h^{ab} h^{}{}_{ab}{}^{,c}{}_{,c} + \tfrac{1}{2} h^{a}{}_{a} h^{b}{}_{b}{}^{,c}{}_{,c} - \frac{h^{a}{}_{a} h^{b}{}_{b}{}^{,c}{}_{,c}}{4 \alpha} \Big] \\ \label{s1} \tilde{S}_{1} &=& {\int d^4{x} } \Bigl[\frac{m^2 \kappa \delta \phi h^{a}{}_{a} \bar{\phi}}{2 k_0} - \frac{m^2 \kappa \nu \delta \phi h^{a}{}_{a} \bar{\phi}}{4 k_0} - \tfrac{1}{2} \kappa \delta \phi h^{b}{}_{b} \bar{\phi} {}^{,a}{}_{,a} + \tfrac{1}{4} \kappa \nu \delta \phi h^{b}{}_{b} \bar{\phi} {}^{,a}{}_{,a} - \tfrac{1}{2} \kappa \delta \phi h^{b}{}_{b}{}_{,a} \bar{\phi} {}^{,a} \nonumber \\ && + \frac{\kappa \omega \delta \phi h^{b}{}_{b}{}_{,a} \bar{\phi} {}^{,a}}{2 \alpha} + \kappa \delta \phi \bar{\phi} {}^{,a} h^{}{}_{a}{}^{b}{}_{,b} - \frac{\kappa \omega \delta \phi \bar{\phi} {}^{,a} h^{}{}_{a}{}^{b}{}_{,b}}{\alpha} + \kappa \delta \phi h^{}{}_{ab} \bar{\phi} {}^{,a}{}^{,b}\Bigr] \\ \label{s2} \tilde{S}_{2} &=& {\int d^4{x} } \Bigl[ -\frac{m^2 \kappa^2 h^{}{}_{ab} h^{ab} \bar{\phi}^2}{8 k_0} + \frac{m^2 \kappa^2 h^{a}{}_{a} h^{b}{}_{b} \bar{\phi}^2}{16 k_0} + \frac{\lambda \bar{\phi}^2 (\delta \phi)^2}{4 k_0^2} - \frac{m^2 \kappa^2 \nu \bar{\phi}^2 (\delta \phi)^2}{8 k_0} \nonumber \\ && + \frac{k_1 \bar{\phi}^2 \delta \phi{}_{,a} \delta \phi{}^{,a}}{2 k_0^2} + \frac{2 k_1 \delta \phi \bar{\phi} \bar{\phi} {}_{,a} \delta \phi{}^{,a}}{k_0^2} - \tfrac{1}{8} \kappa^2 h^{}{}_{bc} h^{bc} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} + \tfrac{1}{16} \kappa^2 \nu h^{}{}_{bc} h^{bc} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \nonumber \\ && + \tfrac{1}{16} \kappa^2 h^{b}{}_{b} h^{c}{}_{c} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} - \tfrac{1}{32} \kappa^2 \nu h^{b}{}_{b} h^{c}{}_{c} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} + \frac{k_1 (\delta \phi)^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{2 k_0^2} - \tfrac{1}{16} \kappa^2 \nu (\delta \phi)^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \nonumber \\ && + \frac{\kappa^2 \omega^2 (\delta \phi)^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{4 \alpha} + \omega c^{a} \bar{c}^{b} \bar{\phi} {}_{,a} \bar{\phi} {}_{,b} + \tfrac{1}{2} \kappa^2 h^{}{}_{a}{}^{c} h^{}{}_{bc} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b} - \tfrac{1}{4} \kappa^2 \nu h^{}{}_{a}{}^{c} h^{}{}_{bc} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b} \nonumber \\ && - \tfrac{1}{4} \kappa^2 h^{}{}_{ab} h^{c}{}_{c} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b} + \tfrac{1}{8} \kappa^2 \nu h^{}{}_{ab} h^{c}{}_{c} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}\Bigr] \\ \label{s3} \tilde{S}_{3} &=& {\int d^4{x} } \Bigl[\frac{\kappa \lambda \delta \phi h^{a}{}_{a} \bar{\phi}^3}{12 k_0^2} - \frac{\kappa \lambda \nu \delta \phi h^{a}{}_{a} \bar{\phi}^3}{24 k_0^2} + \frac{k_1 \kappa \nu \delta \phi h^{b}{}_{b} \bar{\phi}^2 \bar{\phi}{}^{,a}{}_{,a}}{4 k_0^2} + \frac{k_1 \kappa h^{b}{}_{b} \bar{\phi}^2 \bar{\phi}{}_{,a} \delta \phi{}^{,a}}{2 k_0^2} \nonumber \\ && + \frac{k_1 \kappa \delta \phi h^{b}{}_{b} \bar{\phi} \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{2 k_0^2} + \frac{k_1 \kappa \nu \delta \phi h^{b}{}_{b} \bar{\phi} \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{4 k_0^2} - \frac{k_1 \kappa h^{}{}_{ab} \bar{\phi}^2 \delta \phi{}^{,a} \bar{\phi}{}^{,b}}{k_0^2} - \frac{k_1 \kappa \delta \phi h^{}{}_{ab} \bar{\phi} \bar{\phi}{}^{,a} \bar{\phi}{}^{,b}}{k_0^2}\Bigr] \end{aligned}$$ $$\begin{aligned} \label{s4} \tilde{S}_{4} &=& {\int d^4{x} } \Biggl[- \frac{\kappa^2 \lambda h^{}{}_{ab} h^{ab} \bar{\phi}^4}{96 k_0^2} + \frac{\kappa^2 \lambda h^{a}{}_{a} h^{b}{}_{b} \bar{\phi}^4}{192 k_0^2} - \frac{\kappa^2 \lambda \nu \bar{\phi}^4 (\delta \phi)^2}{96 k_0^2} - \frac{k_1 \kappa^2 h^{}{}_{bc} h^{bc} \bar{\phi}^2 \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{8 k_0^2} \nonumber \\ && + \frac{k_1 \kappa^2 \nu h^{}{}_{bc} h^{bc} \bar{\phi}^2 \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{16 k_0^2} + \frac{k_1 \kappa^2 h^{b}{}_{b} h^{c}{}_{c} \bar{\phi}^2 \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{16 k_0^2} - \frac{k_1 \kappa^2 \nu h^{b}{}_{b} h^{c}{}_{c} \bar{\phi}^2 \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{32 k_0^2} \nonumber \\ && - \frac{k_1 \kappa^2 \nu \bar{\phi}^2 (\delta \phi)^2 \bar{\phi}{}_{,a} \bar{\phi}{}^{,a}}{16 k_0^2} + \frac{k_1 \kappa^2 h^{}{}_{a}{}^{c} h^{}{}_{bc} \bar{\phi}^2 \bar{\phi}{}^{,a} \bar{\phi}{}^{,b}}{2 k_0^2} - \frac{k_1 \kappa^2 \nu h^{}{}_{a}{}^{c} h^{}{}_{bc} \bar{\phi}^2 \bar{\phi}{}^{,a} \bar{\phi}{}^{,b}}{4 k_0^2} \nonumber \\ && - \frac{k_1 \kappa^2 h^{}{}_{ab} h^{c}{}_{c} \bar{\phi}^2 \bar{\phi}{}^{,a} \bar{\phi}{}^{,b}}{4 k_0^2} + \frac{k_1 \kappa^2 \nu h^{}{}_{ab} h^{c}{}_{c} \bar{\phi}^2 \bar{\phi}{}^{,a} \bar{\phi}{}^{,b}}{8 k_0^2}\Biggr]\end{aligned}$$ Here, the indices $(a,b,c,\dots)$ and $(\mu,\nu,\rho,\dots)$ are used interchangeably to denote the gauge indices. $\tilde{S}_0$ leads to the well known free theory propagators for gravity and massive scalar field and the ghost field respectively, $$\begin{aligned} \label{prop} D(x,x') &=&{\int\frac{d^4{k}}{(2\pi)^4}}e^{ik\cdot(x-x')}D(k) = \langle\delta\phi(x)\delta\phi(x')\rangle; \nonumber \\ D_{\alpha\beta\mu\nu}(x,x') & =& {\int\frac{d^4{k}}{(2\pi)^4}} e^{ik\cdot(x-x')}D_{\alpha\beta\mu\nu}(k) = \langle h_{\alpha\beta}(x)h_{\mu\nu}(x')\rangle ; \\ D^{G}_{\mu\nu}(x,x') &=& {\int\frac{d^4{k}}{(2\pi)^4}}e^{ik\cdot(x-x')}D^{G}_{\mu\nu}(k) = \langle\bar{c}_{\mu}(x)c_{\nu}(x')\rangle \nonumber ;\end{aligned}$$ where, $$\begin{aligned} \label{props} D(k) &=& \frac{1}{k^2+\frac{m^2}{k_0}}; \\ \label{propg} D_{\alpha\beta\mu\nu}(k) &=&\frac{\delta_{\alpha\mu}\delta_{\beta\nu}+\delta_{\alpha\nu}\delta_{\beta\mu} -\delta_{\alpha\beta}\delta_{\mu\nu}}{2 k^2} + (\alpha-1)\frac{\delta_{\alpha\mu}k_\beta k_\nu+\delta_{\alpha\nu}k_\beta k_\mu+\delta_{\beta\mu}k_\alpha k_\nu+\delta_{\beta\nu}k_\alpha k_\mu}{2k^4}; \\ \label{propgh} D^{G}_{\mu\nu}(k) &=& \dfrac{1}{k^2}\delta_{\mu\nu}.\end{aligned}$$ Looking at the structure of rest of the terms $\tilde{S}_i$, it is straightforward to conclude that all terms with odd combinations of $h_{\mu\nu}(x)$ and $\delta\phi(x)$ appearing in Eqs. (\[aeaf7\]) will not contribute to the effective action, since $\langle h_{\alpha\beta}(x)\delta\phi(x')\rangle = 0$. Therefore, $\langle \tilde{S}_{1}\rangle = 0$ and there is no contribution at $\mathcal{O}(\bar{\phi})$ to the one-loop effective action. Similarly, $\langle \tilde{S}_{3}\rangle = \langle \tilde{S}_{1}\tilde{S}_{2}\rangle = \langle \tilde{S}_{1}^3\rangle = 0$, and hence at $\mathcal{O}(\bar{\phi}^{3})$ too, there is no contribution to the effective action. Hence, the only non-zero contributions in Eq. (\[aeaf7\]) come at $\mathcal{O}(\bar{\phi}^{2})$ and $\mathcal{O}(\bar{\phi}^{4})$. In the latter, we ignore $\langle S_{1}^{4}\rangle$ terms since they are relevant at $\mathcal{O}(\kappa^4)$ and above while we are interested in terms upto $\kappa^2$ order. Expectation value of $\tilde{S}_{i}$ consists of local terms, and thus describes contributions from tadpole diagrams. The ghost term appears only in $\tilde{S}_{2}$. However, it can be shown that at $\mathcal{O}(\bar{\phi}^{2})$ it yields no nontrivial contributions, and as a result, has usually been ignored in past literature where only quadratic order corrections were considered [@mackay2010; @saltas2017; @bounakis2018]. Consider the ghost propagator (\[propgh\]). Because there is no physical scale involved, the term containing ghost in (\[s2\]) yields, $$\begin{aligned} \label{loop00} \left\langle {\int d^4{x} }\omega c^{a} \bar{c}^{b} \bar{\phi} {}_{,a} \bar{\phi} {}_{,b} \right\rangle &=& {\int d^4{x} }\omega\bar{\phi} {}_{,a} \bar{\phi} {}_{,b} \langle c^{a} \bar{c}^{b} \rangle \nonumber \\ &=& {\int d^4{x} }\omega\bar{\phi} {}_{,a} \bar{\phi} {}_{,b} {\int\frac{d^4{k}}{(2\pi)^4}}\delta^{ab}\dfrac{1}{k^2},\end{aligned}$$ which in four dimensions gives no physical result. The only nontrivial ghost contribution comes at quartic order in background field. Eventually, finding the one-loop corrections then boils down to evaluating upto $\kappa^{2}$ order, the quadratic and quartic order corrections from the following: $$\label{loop01} \Gamma = \langle\tilde{S}_{2}\rangle - \dfrac{1}{2}\langle\tilde{S}_{1}^{2}\rangle + \langle\tilde{S}_{4}\rangle - \langle\tilde{S}_{1}\tilde{S}_{3}\rangle + \dfrac{1}{2}\langle\tilde{S}_{1}^{2}\tilde{S}_{2}\rangle -\dfrac{1}{2}\langle\tilde{S}_{2}^{2}\rangle .$$ In principle, solving Eq. (\[loop01\]) broadly consists of two steps: (i) writing each term in terms of the Fourier space integral(s) of Green’s functions found in Eqs. (\[props\]) - (\[propgh\]); and (ii) solving the resulting loop integrals. In this section, we restrict ourselves to writing just the divergent part of effective action, since there are already several thousand terms to deal with and writing their finite parts would introduce unnecessary complexity. We do consider finite part in the subsequent section, where we evaluate the effective potential after assuming all derivatives of background fields to be zero. ### Calculating $\langle \tilde{S}_{i} \rangle$ We first deal with $\langle\tilde{S}_{2}\rangle$ and $\langle\tilde{S}_{4}\rangle$. For convenience, we do not explicitly write the tensor indices of correlators, fields and their coefficients. First, the derivatives of field fluctuations are transformed to momentum space: $$\begin{aligned} \label{loop02} \int d^{4}x A(x) \langle\partial^{m}\delta(x) \ \partial^{n}\delta(x')\rangle \longrightarrow \int d^{4}x \dfrac{d^{4}p}{(2\pi^{4})} A(x) (-ip)^{m} (ip)^{n} \langle \delta_{p}(x)\delta_{p}(x')\rangle,\end{aligned}$$ where, $\delta (x)$ and $A(x)$ represent the field fluctuations and coefficients respectively. $\delta(x)$ here represents any of the fields ($\delta\phi(x),h_{\mu\nu}(x),c_{\mu}(x),\bar{c}_{\mu}(x)$), and is not to be confused with the dirac delta function $\delta(x,x')$. $\langle \delta_{p}\delta_{p}\rangle$ represents the propagator(s) in momentum space. Then, $\langle \delta_{p}\delta_{p}\rangle$ is replaced with values of Green’s function to obtain the loop integrals. For solving integrals here, we primarily use the results in Ref. [@bardin1999] to evaluate the divergent terms in dimensional regularization, except for some higher rank two-point integrals that appear below, which we solve by hand using well known prescriptions [@romao2019; @george1975]. There are three types of loop integrals coming from Eqs. (\[s2\]) and (\[s4\]): $$\begin{aligned} \label{loop03} \int d^{4}x \dfrac{d^{4}p}{(2\pi^{4})} A(x)\dfrac{1}{p^{2}} ; \quad \int d^{4}x \dfrac{d^{4}p}{(2\pi^{4})} A(x)\dfrac{p^{\mu}p^{\nu}}{p^{4}}; \quad \int d^{4}x \dfrac{d^{4}p}{(2\pi^{4})} A(x)\dfrac{1}{p^{2}+\frac{m^2}{k_{0}}}.\end{aligned}$$ The first two integrals are poleless, and vanish due to the lack of a physical scale [@bardin1999]. The third integral is straightforward and contributes to the divergent part. See appendix \[a1\] for values of all integrals appearing here, including finite parts for some integrals used in the next section. ### Calculating $\langle \tilde{S}_{i}\tilde{S}_{j} \rangle$ $\langle \tilde{S}_{1}\tilde{S}_{1} \rangle$ and $\langle \tilde{S}_{1}\tilde{S}_{3} \rangle$ contain terms of the form, $$\begin{aligned} \label{loop04} & {\int d^4{x} }{\int d^4{x'} } A(x)B(x')\langle\delta\phi(x)\delta\phi(x')\rangle \langle h(x)h(x')\rangle \\ =&{\int d^4{x} }{\int\frac{d^4{k}}{(2\pi)^4}}\frac{d^4 k'}{2\pi^4}\frac{d^4 k''}{2\pi^4} A(x)\tilde{B}(k'') D_{\phi\phi}(k)D_{hh}(k') e^{-i(k+k')\cdot x}\delta^{(4)}(k+k'-k'') \nonumber \\ \label{loop05} =& {\int d^4{x} }A(x){\int\frac{d^4{k}}{(2\pi)^4}}e^{-ik\cdot x}\tilde{B}(k) {\int\frac{d^4{k'}}{(2\pi)^4}} D_{\phi\phi}(k-k')D_{hh}(k')\end{aligned}$$ where $A(x), B(x')$ are classical coefficients, and $D_{\phi\phi}, D_{hh}$ are scalar and gravity propagators respectively; $\tilde{B}(k)$ is the Fourier transform of $B(x')$. There are also the derivatives of Eq. (\[loop04\]) present, and are dealt with in a way similar to Eq. (\[loop02\]), leading to factors of $k'^{\mu}$ in the loop integrals. Consequently, we encounter three types of loop integrals: $$\begin{aligned} \label{loop06} {\int\frac{d^4{k'}}{(2\pi)^4}} \dfrac{k'^{a}\dots k'^{b}}{(k'-k)^2 + \frac{m^2}{k_0}}; \quad {\int\frac{d^4{k'}}{(2\pi)^4}} \dfrac{k'^{a}\dots k'^{b}}{k'^{2} ((k'-k)^2 + \frac{m^2}{k_0})}; \quad {\int\frac{d^4{k'}}{(2\pi)^4}} \dfrac{k'^{a}\dots k'^{b}}{k'^{4} ((k'-k)^2 + \frac{m^2}{k_0})};\end{aligned}$$ which constitute standard one-, two- and three-point $n$-rank integrals ($n=0,1,2$). Likewise, $\langle \tilde{S}_{2}\tilde{S}_{2} \rangle$ yields $4-$point correlators given by, $$\begin{aligned} \label{loop07} \langle\delta\phi(x)\delta\phi(x)\delta\phi(x')\delta\phi(x')\rangle; \ \langle\delta\phi(x)\delta\phi(x)h(x')h(x')\rangle; \ \langle\delta\phi(x)\delta\phi(x)\bar{c}(x') c(x')\rangle; \nonumber \\ \langle h(x) h(x)\bar{c}(x') c(x')\rangle; \ \langle \bar{c}(x) c(x) \bar{c}(x') c(x')\rangle; \ \langle h(x) h(x) h(x') h(x')\rangle\end{aligned}$$ The second, third and fourth terms in (\[loop07\]) are of the form $\langle \delta(x)\delta(x)\delta'(x')\delta'(x')\rangle$ (again, $\delta(x),\delta(x')$ denote the fields), thereby corresponding to disconnected tadpoles and hence do not give any meaningful contribution. The rest of $4-$point correlators in Eq. (\[loop07\]) are resolved into $2-$point functions using Wick theorem [@peskin1995; @schwartz2013]. Fortunately, the last term involving only graviton propagators can be ignored since it only contains $\mathcal{O}(\kappa^4)$ terms. Moreover, $\langle \bar{c}(x)\bar{c}(x')\rangle = \langle c(x)c(x')\rangle = 0$. Therefore, after applying Wick theorem, the final contribution in Eq. (\[loop07\]) comes from, $$\begin{aligned} \label{loop08a} \langle\delta\phi(x)\delta\phi(x)\delta\phi(x')\delta\phi(x')\rangle &=& \langle\delta\phi(x)\delta\phi(x')\rangle\langle\delta\phi(x)\delta\phi(x')\rangle + \langle\delta\phi(x)\delta\phi(x')\rangle\langle\delta\phi(x)\delta\phi(x')\rangle; \\ \label{loop08b} \langle \bar{c}(x) c(x) \bar{c}(x') c(x')\rangle &=& \langle \bar{c}(x) c(x')\rangle \langle \bar{c}(x) c(x')\rangle.\end{aligned}$$ Using Eq. (\[loop08a\]) and (\[loop08b\]) in $\langle \tilde{S}_{2}\tilde{S}_{2} \rangle$ along with Eqs. (\[props\])-(\[propgh\]), and Fourier transforming according to Eq. (\[loop02\]) gives rise to upto rank-4 two-point integrals: $$\begin{aligned} \label{loop09} {\int\frac{d^4{k'}}{(2\pi)^4}} \dfrac{k'^{a}\dots k'^{b}}{(k'^{2}+\frac{m^2}{k_0}) ((k'-k)^2 + \frac{m^2}{k_0})};\end{aligned}$$ ### Calculating $\langle \tilde{S}_{i}\tilde{S}_{j}\tilde{S}_{k} \rangle$ The last term to be evaluated is $\langle \tilde{S}_{1}\tilde{S}_{1}\tilde{S}_{2} \rangle$. It consists of six-point correlators given by, $$\begin{aligned} \label{loop10} &\langle h(x)h(x'')\delta\phi(x)\delta\phi(x'')\bar{c}(x') c(x') \rangle ; \langle h(x)h(x'')\delta\phi(x)\delta\phi(x'')\delta\phi(x')\delta\phi(x') \rangle ; & \nonumber \\ &\langle h(x)h(x'')\delta\phi(x)\delta\phi(x'') h(x')h(x') \rangle &\end{aligned}$$ Again, the last term can be ignored since it has no terms upto $\mathcal{O}(\kappa^2)$. And the first term can be written as $\langle h(x)h(x'')\delta\phi(x)\delta\phi(x'')\rangle\langle\bar{c}(x') c(x') \rangle$, which implies disconnected diagrams and thus can also be ignored. So, ghost terms only end up in $\langle \tilde{S}_{2}\tilde{S}_{2} \rangle$. Hence, only the second term needs to be evaluated, which after applying Wick theorem similar to Eq. (\[loop08a\]) turns out to be, $$\begin{aligned} \label{loop11} \langle h(x)h(x'')\delta\phi(x)\delta\phi(x'')\delta\phi(x')\delta\phi(x') \rangle = & \langle h(x)h(x'')\rangle\langle\delta\phi(x)\delta\phi(x')\rangle\langle\delta\phi(x')\delta\phi(x'') \rangle \nonumber \\ & + \langle h(x)h(x'')\rangle\langle\delta\phi(x)\delta\phi(x')\rangle\langle\delta\phi(x')\delta\phi(x'') \rangle ,\end{aligned}$$ A typical scalar integral in $\langle \tilde{S}_{1}\tilde{S}_{1}\tilde{S}_{2} \rangle$ takes the form, $$\begin{aligned} \label{loop12} &&{\int d^4{x} }{\int d^4{x'} }{\int d^4{x''} }{\int\frac{d^4{k}}{(2\pi)^4}}{\int\frac{d^4{k'}}{(2\pi)^4}}{\int\frac{d^4{k''}}{(2\pi)^4}} A(x)B(x')C(x'')\times \nonumber \\ && e^{-ik\cdot(x'-x)}e^{-ik''\cdot(x-x'')}e^{-ik'\cdot (x''-x')}D_{\phi\phi}(k)D_{\phi\phi}(k')D_{hh}(k'') \nonumber\\ & =& {\int d^4{x} }{\int\frac{d^4{p}}{(2\pi)^4}}{\int\frac{d^4{k}}{(2\pi)^4}} A(x)\tilde{B}(p)e^{-ip\cdot x} \tilde{C}(k)e^{-ik\cdot x}{\int\frac{d^4{k'}}{(2\pi)^4}} \times \nonumber \\ && D_{\phi\phi}(k'-p-k) D_{\phi\phi}(k'-k)D_{hh}(k'),\end{aligned}$$ resulting in scalar and tensor two-, three- and four-point integrals: $$\begin{aligned} \label{s112} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{a}\dots k'^{b}}{d_{0}d_{1}d_{2}d_{3}}; {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{a}\dots k'^{b}}{d_{0}d_{1}d_{2}}; {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{a}\dots k'^{b}}{d_{0}d_{1}}.\end{aligned}$$ where, $$\begin{aligned} \label{loop13} d_{0} = (k'-k)^2 + \frac{m^2}{k_0}; \ d_{1} = (k'-k-p)^{2} + \frac{m^2}{k_0}; \ d_{2} = d_{3} = k'^2.\end{aligned}$$ There are upto rank-3 four-point integrals in $\langle \tilde{S}_{1}\tilde{S}_{1}\tilde{S}_{2} \rangle$, and hence have no divergent part [@bardin1999]. ### Divergent part In total, there are several thousand terms that eventually add up to give the divergent part of Eq. (\[loop01\]). After solving all the above integrals and extracting their divergent parts using dimensional regularization, we end up with Fourier transforms $\tilde{B}(k)$ (and $\tilde{C}(p)$ in case of six-point functions) with or without factors of $k^{a}$ and/or $p^{a}$. These expressions are transformed back to coordinate space as follows: $$\begin{aligned} \label{divp0} {\int d^4{x} }{\int\frac{d^4{p}}{(2\pi)^4}}{\int\frac{d^4{k}}{(2\pi)^4}}A(x)\tilde{B}(p)\tilde{C}(k)e^{-ip\cdot x}e^{-ik\cdot x} k^{a}\dots k^{b} p^{\mu}\dots p^{\nu} \to \nonumber \\ {\int d^4{x} }(i\partial^{\mu})\dots(i\partial^{\nu})B(x) (i\partial^{a})\dots(i\partial^{b})C(x)\end{aligned}$$ and likewise for other cases including $\langle \tilde{S}_{i}\tilde{S}_{j} \rangle$ and $\langle \tilde{S}_{i}\rangle$. Substituting these results for the divergent part in Eq. (\[loop01\]), we get, $$\begin{aligned} \label{divp1} divp(\Gamma) &=& {\int d^4{x} } L \Bigg[\frac{ k_{1} m^4 \bar{\phi}^2}{2 k_{0}^4} + \frac{3 m^4 \kappa^2 \bar{\phi}^2}{4 k_{0}^2} - \frac{ m^2 \lambda \bar{\phi}^2}{4 k_{0}^3} - \frac{5 m^4 \kappa^2 \nu \bar{\phi}^2}{8 k_{0}^2} \nonumber \\ && + \frac{3 m^4 \kappa^2 \nu^2 \bar{\phi}^2}{16 k_{0}^2} + \frac{ k_{1} m^2 \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{2 k_{0}^3} - \frac{3 m^2 \kappa^2 \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{4 k_{0}} + \frac{17 m^2 \kappa^2 \nu \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{16 k_{0}} \nonumber \\ && - \frac{3 m^2 \kappa^2 \nu^2 \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{8 k_{0}} + \frac{ m^2 \kappa^2 \omega \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{4 k_{0}} + \frac{ m^2 \kappa^2 \nu \omega \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}}{8 k_{0}} - \tfrac{3}{8} \kappa^2 \nu \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}{}^{,b}{}_{,b} \nonumber \\ && + \tfrac{3}{16} \kappa^2 \nu^2 \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}{}^{,b}{}_{,b} - \tfrac{1}{4} \kappa^2 \omega \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}{}^{,b}{}_{,b} - \tfrac{1}{8} \kappa^2 \nu \omega \bar{\phi} \bar{\phi}{}^{,a}{}_{,a}{}^{,b}{}_{,b} \nonumber \\ && - \frac{3 k_{1}^2 m^4 \bar{\phi}^4}{256 k_{0}^6 \pi^2} - \frac{3 k_{1} m^4 \kappa^2 \bar{\phi}^4}{32 k_{0}^4 \pi^2} + \frac{ k_{1} m^2 \lambda \bar{\phi}^4}{16 k_{0}^5 \pi^2} + \frac{ m^2 \kappa^2 \lambda \bar{\phi}^4}{32 k_{0}^3 \pi^2} \nonumber \\ && - \frac{ \lambda^2 \bar{\phi}^4}{128 k_{0}^4 \pi^2} + \frac{ k_{1} m^4 \kappa^2 \nu \bar{\phi}^4}{16 k_{0}^4 \pi^2} - \frac{17 m^2 \kappa^2 \lambda \nu \bar{\phi}^4}{768 k_{0}^3 \pi^2} - \frac{3 k_{1} m^4 \kappa^2 \nu^2 \bar{\phi}^4}{128 k_{0}^4 \pi^2} \nonumber \\ && + \frac{ m^2 \kappa^2 \lambda \nu^2 \bar{\phi}^4}{128 k_{0}^3 \pi^2} - \frac{19 k_{1}^2 m^2 \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{128 k_{0}^5 \pi^2} + \frac{ k_{1} \lambda \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{64 k_{0}^4 \pi^2} - \frac{ \kappa^2 \lambda \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{64 k_{0}^2 \pi^2} \nonumber \\ && - \frac{ k_{1} m^2 \kappa^2 \nu \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{128 k_{0}^3 \pi^2} + \frac{3 \kappa^2 \lambda \nu \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{128 k_{0}^2 \pi^2} - \frac{ \kappa^2 \lambda \nu^2 \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{128 k_{0}^2 \pi^2} + \frac{ k_{1} m^2 \kappa^2 \omega \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{64 k_{0}^3 \pi^2} \nonumber \\ && - \frac{ \kappa^2 \lambda \omega \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{192 k_{0}^2 \pi^2} - \frac{ k_{1} m^2 \kappa^2 \nu \omega \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{128 k_{0}^3 \pi^2} + \frac{ \kappa^2 \lambda \nu \omega \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}}{384 k_{0}^2 \pi^2} - \frac{11 k_{1}^2 m^2 \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{128 k_{0}^5 \pi^2} \nonumber \\ && - \frac{3 k_{1} m^2 \kappa^2 \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{32 k_{0}^3 \pi^2} - \frac{3 k_{1} \lambda \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{64 k_{0}^4 \pi^2} + \frac{5 k_{1} m^2 \kappa^2 \nu \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{32 k_{0}^3 \pi^2} + \frac{ \kappa^2 \lambda \nu \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{256 k_{0}^2 \pi^2} \nonumber \\ && - \frac{3 k_{1} m^2 \kappa^2 \nu^2 \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{64 k_{0}^3 \pi^2} + \frac{3 k_{1} m^2 \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{64 k_{0}^3 \pi^2} - \frac{ \kappa^2 \lambda \omega \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{32 k_{0}^2 \pi^2} + \frac{ k_{1} m^2 \kappa^2 \nu \omega \bar{\phi}^2 \bar{\phi} {}_{,a} \bar{\phi} {}^{,a}}{128 k_{0}^3 \pi^2} \nonumber \\ && + \frac{7 k_{1}^2 \bar{\phi}^2 \bar{\phi} {}^{,b}{}_{,b}{}_{,a} \bar{\phi} {}^{,a}}{128 k_{0}^4 \pi^2} + \frac{5 k_{1} \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}^{,b}{}_{,b}{}_{,a} \bar{\phi} {}^{,a}}{384 k_{0}^2 \pi^2} - \frac{ k_{1} \kappa^2 \nu \omega \bar{\phi}^2 \bar{\phi} {}^{,b}{}_{,b}{}_{,a} \bar{\phi} {}^{,a}}{256 k_{0}^2 \pi^2} - \frac{ k_{1}^2 \bar{\phi}^2 \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,a}{}_{,b}}{24 k_{0}^4 \pi^2} \nonumber \\ && - \frac{3 k_{1} \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,a}{}_{,b}}{128 k_{0}^2 \pi^2} - \frac{ k_{1}^2 \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{256 k_{0}^4 \pi^2} + \frac{ k_{1} \kappa^2 \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{32 k_{0}^2 \pi^2} - \frac{3 k_{1} \kappa^2 \nu \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{64 k_{0}^2 \pi^2} \nonumber \\ && + \frac{3 k_{1} \kappa^2 \nu^2 \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{128 k_{0}^2 \pi^2} + \frac{ k_{1} \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{64 k_{0}^2 \pi^2} - \frac{ k_{1} \kappa^2 \nu \omega \bar{\phi}^2 \bar{\phi} {}^{,a}{}_{,a} \bar{\phi} {}^{,b}{}_{,b}}{128 k_{0}^2 \pi^2} + \frac{ k_{1}^2 \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{16 k_{0}^4 \pi^2} \nonumber \\ && + \frac{ k_{1} \kappa^2 \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{16 k_{0}^2 \pi^2} - \frac{25 k_{1} \kappa^2 \nu \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{256 k_{0}^2 \pi^2} + \frac{3 k_{1} \kappa^2 \nu^2 \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{64 k_{0}^2 \pi^2} + \frac{3 k_{1} \kappa^2 \omega \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{32 k_{0}^2 \pi^2} \nonumber \\ && - \frac{ k_{1} \kappa^2 \nu \omega \bar{\phi} \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}^{,b}{}_{,b}}{64 k_{0}^2 \pi^2} + \frac{ k_{1}^2 \bar{\phi}^2 \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}^{,b}{}_{,b}}{384 k_{0}^4 \pi^2} - \frac{ k_{1} \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}^{,b}{}_{,b}}{192 k_{0}^2 \pi^2} - \frac{ k_{1} \kappa^2 \nu \omega \bar{\phi}^2 \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}^{,b}{}_{,b}}{256 k_{0}^2 \pi^2} \nonumber \\ && - \frac{ k_{1}^2 \bar{\phi}^3 \bar{\phi} {}^{,a}{}_{,a}{}^{,b}{}_{,b}}{256 k_{0}^4 \pi^2} + \frac{ k_{1} \kappa^2 \nu \bar{\phi} {}_{,a} \bar{\phi} {}^{,a} \bar{\phi} {}_{,b} \bar{\phi} {}^{,b}}{256 k_{0}^2 \pi^2} - \frac{ k_{1} \kappa^2 \bar{\phi} \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,b}}{16 k_{0}^2 \pi^2} + \frac{ k_{1} \kappa^2 \nu \bar{\phi} \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,b}}{128 k_{0}^2 \pi^2} \nonumber \\ && - \frac{ k_{1} \kappa^2 \omega \bar{\phi} \bar{\phi} {}^{,a} \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,b}}{16 k_{0}^2 \pi^2} - \frac{ k_{1}^2 \bar{\phi}^2 \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,a}{}^{,b}}{128 k_{0}^4 \pi^2} - \frac{ k_{1} \kappa^2 \bar{\phi}^2 \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,a}{}^{,b}}{32 k_{0}^2 \pi^2} - \frac{ k_{1} \kappa^2 \omega \bar{\phi}^2 \bar{\phi} {}_{,a}{}_{,b} \bar{\phi} {}^{,a}{}^{,b}}{32 k_{0}^2 \pi^2} \Bigg]\end{aligned}$$ where, $L=-1/8\pi^{2}\epsilon$ ($\epsilon=n-4$) as the dimensionality $n\to 4$. As expected, there are no $\alpha$ dependent terms. Although not explicitly shown here, factors of $1/\alpha$ appear in individual pieces in Eq. (\[loop01\]). However, when all contributions are added to evaluate $\Gamma$, these terms cancel so that the final result is gauge-invariant. Final result for divergent part of $\Gamma$ after removing bookkeeping parameters ($\omega\to 1$, $\nu \to 1$) in the Landau gauge ($\alpha \to 0$) leads to the covariant corrections: $$\begin{aligned} \label{finalEA} divp(\Gamma) &=& \int d^4 x\Big[\frac{ k_{1} m^4 \bar{\phi}^2}{2 k_{0}^4} + \frac{5 m^4 \kappa^2 \bar{\phi}^2}{16 k_{0}^2} - \frac{ m^2 \lambda \bar{\phi}^2}{4 k_{0}^3} + \frac{ k_{1} m^2 \bar{\phi} \bar{\phi} {}^{,a}{}_{,a}}{2 k_{0}^3} + \frac{5 m^2 \kappa^2 \bar{\phi} \bar{\phi} {}^{,a}{}_{,a}}{16 k_{0}} \nonumber \\ && - \tfrac{9}{16} \kappa^2 \phi \phi {}^{,a}{}_{,a}{}^{,b}{}_{,b} - \frac{3 k_{1}^2 m^4 \bar{ \phi}^4}{32 k_{0}^6} - \frac{7 k_{1} m^4 \kappa^2 \bar{ \phi}^4}{16 k_{0}^4} + \frac{ k_{1} m^2 \lambda \bar{ \phi}^4}{2 k_{0}^5} + \frac{13 m^2 \kappa^2 \lambda \bar{ \phi}^4}{96 k_{0}^3} \nonumber \\ && - \frac{ \lambda^2 \bar{ \phi}^4}{16 k_{0}^4} - \frac{19 k_{1}^2 m^2 \bar{ \phi}^3 \bar{ \phi}{}^{,a}{}_{,a}}{16 k_{0}^5} + \frac{ k_{1} \lambda \bar{ \phi}^3 \bar{ \phi}{}^{,a}{}_{,a}}{8 k_{0}^4} - \frac{ \kappa^2 \lambda \bar{ \phi}^3 \bar{ \phi}{}^{,a}{}_{,a}}{48 k_{0}^2} \nonumber \\ && - \frac{11 k_{1}^2 m^2 \bar{ \phi}^2 \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a}}{16 k_{0}^5} + \frac{9 k_{1} m^2 \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a}}{16 k_{0}^3} - \frac{3 k_{1} \lambda \bar{ \phi}^2 \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a}}{8 k_{0}^4} - \frac{7 \kappa^2 \lambda \bar{ \phi}^2 \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a}}{32 k_{0}^2} \nonumber \\ && + \frac{7 k_{1}^2 \bar{ \phi}^2 \bar{ \phi}{}^{,b}{}_{,b}{}_{,a} \bar{ \phi}{}^{,a}}{16 k_{0}^4} + \frac{7 k_{1} \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}^{,b}{}_{,b}{}_{,a} \bar{ \phi}{}^{,a}}{96 k_{0}^2} - \frac{ k_{1}^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a} \bar{ \phi}{}^{,b}{}_{,a}{}_{,b}}{3 k_{0}^4} - \frac{3 k_{1} \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a} \bar{ \phi}{}^{,b}{}_{,a}{}_{,b}}{16 k_{0}^2} \nonumber \\ && - \frac{ k_{1}^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a}{}_{,a} \bar{ \phi}{}^{,b}{}_{,b}}{32 k_{0}^4} + \frac{ k_{1} \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a}{}_{,a} \bar{ \phi}{}^{,b}{}_{,b}}{8 k_{0}^2} + \frac{ k_{1}^2 \bar{ \phi} \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a} \bar{ \phi}{}^{,b}{}_{,b}}{2 k_{0}^4} + \frac{23 k_{1} \kappa^2 \bar{ \phi} \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a} \bar{ \phi}{}^{,b}{}_{,b}}{32 k_{0}^2} \nonumber \\ && + \frac{ k_{1}^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a} \bar{ \phi}{}_{,a}{}^{,b}{}_{,b}}{48 k_{0}^4} - \frac{7 k_{1} \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}^{,a} \bar{ \phi}{}_{,a}{}^{,b}{}_{,b}}{96 k_{0}^2} - \frac{ k_{1}^2 \bar{ \phi}^3 \bar{ \phi}{}^{,a}{}_{,a}{}^{,b}{}_{,b}}{32 k_{0}^4} + \frac{ k_{1} \kappa^2 \bar{ \phi}{}_{,a} \bar{ \phi}{}^{,a} \bar{ \phi}{}_{,b} \bar{ \phi}{}^{,b}}{32 k_{0}^2} \nonumber \\ && - \frac{15 k_{1} \kappa^2 \bar{ \phi} \bar{ \phi}{}^{,a} \bar{ \phi}{}_{,a}{}_{,b} \bar{ \phi}{}^{,b}}{16 k_{0}^2} - \frac{ k_{1}^2 \bar{ \phi}^2 \bar{ \phi}{}_{,a}{}_{,b} \bar{ \phi}{}^{,a}{}^{,b}}{16 k_{0}^4} - \frac{ k_{1} \kappa^2 \bar{ \phi}^2 \bar{ \phi}{}_{,a}{}_{,b} \bar{ \phi}{}^{,a}{}^{,b}}{2 k_{0}^2}\Big]\end{aligned}$$ If instead we turn off the DV connections by setting $\nu = 0$ and choose $\alpha = 1, \omega = 1$, we recover gauge-dependent results obtained in the past by Steinwachs and Kamenshchik [@steinwachs2011], where they calculated the one-loop divergences for a general scalar-tensor theory that in the single field limit (with the identifications $U=1, G=K$, and $V=V/\gamma^4$ in their notations) encompasses the model (\[action\]). Similarly, in the case $k_1 = 0, k_0 = 1$ we recover the gauge-invariant calculations of Mackay and Toms [@mackay2010] (excluding cosmological constant and nonminimal coupling to gravity). Renormalization and Comparisons ------------------------------- Eq. (\[finalEA\]) is not really renormalizable, because of the presence of higher derivative terms, particularly the fourth order derivatives of $\bar{\phi}(x)$, that has no counterterms. However, we need not worry about these UV divergences since the current framework is an effective theory approach, and we assume that such non-renormalizable terms are resolved by some high energy theory. For now, we still can construct counterterms from the classical action functional to absorb corresponding divergent parts, which will inturn induce 1-loop corrections to the parameters $\frac{m^2}{k_0}, \frac{k_1}{k_0^2}, \frac{\lambda}{k_0^2}$ of the theory (\[action\]). We start by re-writing Eq. (\[finalEA\]) in the form, $$\begin{aligned} \label{ren00} divp(\Gamma) = L {\int d^4{x} } (A \bar{\phi}\Box\bar{\phi} + B\bar{\phi}^2 + C \bar{\phi}^4 + D \bar{\phi}^2 \partial_{\mu}\bar{\phi}\partial^{\mu}\bar{\phi})\end{aligned}$$ where we have ignored the terms not present in the classical background action. We note that the terms of the form $\bar{\phi}^{3}\Box\bar{\phi}$ in Eq. (\[finalEA\]) are transformed to $-3\bar{\phi}^2\partial_{\mu}\bar{\phi}\partial^{\mu}\bar{\phi}$ after by-parts integration. The coefficients $A,B,C,D$ are read off from Eq. (\[finalEA\]): $$\begin{aligned} \label{ren01} A &=& \dfrac{5m^{2}\kappa^{2}}{16 k_{0}} + \dfrac{k_{1} m^{2}}{2 k_{0}^{3}}; \nonumber \\ B &=& \dfrac{k_{1}m^{4}}{2 k_{0}^{4}} + \dfrac{5 m^{4}\kappa^{2}}{16 k_{0}^{2}} - \dfrac{m^{2}\lambda}{4 k_{0}^{3}}; \nonumber \\ C &=& - \dfrac{3 k_{1}^{2}m^{4}}{32 k_{0}^{6}} - \dfrac{7 k_{1}m^{4}\kappa^{2}}{16 k_{0}^{4}} + \dfrac{k_{1}m^{2}\lambda^2}{k_{0}^{5}} + \dfrac{13 m^2 \kappa^2 \lambda}{96 k_0^3} - \dfrac{\lambda^2}{16 k_0^4}; \nonumber \\ D &=& \dfrac{23 k_1^2 m^2}{8 k_0^5} + \dfrac{9 k_1 m^2 \kappa^2}{16 k_0^3} - \dfrac{3 k_1 \lambda}{4 k_0^4} - \dfrac{5\kappa^2 \lambda}{32 k_0^2}.\end{aligned}$$ Taking into account the field Renormalization $\bar{\phi}\to Z^{1/2}\bar{\phi}$, the classical background Lagrangian reads, $$\begin{aligned} \label{ren02} \mathcal{L}_{Z} = -\dfrac{1}{2} Z \bar{\phi}\Box\bar{\phi} + \frac{1}{2}\dfrac{m^2}{k_0} Z \bar{\phi}^2 + \frac{\lambda}{24 k_0^2} Z^2 \bar{\phi}^4 + \frac{1}{2}\frac{k_1}{k_0^2} Z^2 \bar{\phi}^2 \partial_{\mu}\bar{\phi}\partial^{\mu}\bar{\phi}\end{aligned}$$ Suppose, the renormalized Lagrangian is given in terms of renormalized parameters as follows, $$\begin{aligned} \label{ren03} \mathcal{L}_{r} = -\dfrac{1}{2} \bar{\phi}\Box\bar{\phi} + \frac{1}{2}\left(\dfrac{m^2}{k_0}\right)_{r} \bar{\phi}^2 + \frac{1}{24}\left(\frac{\lambda}{k_0^2}\right)_{r} \bar{\phi}^4 + \frac{1}{2}\left(\frac{k_1}{k_0^2}\right)_{r} \bar{\phi}^2 \partial_{\mu}\bar{\phi}\partial^{\mu}\bar{\phi}\end{aligned}$$ where $(\cdot)_{r}$ represents the renormalized parameter. The counterterm Lagrangian is then defined as $\delta\mathcal{L} = \mathcal{L}_{r} - \mathcal{L}_{Z}$. Accordingly, the counterterms for field and other parameters are as follows: $$\begin{aligned} \label{ren04} \delta_{Z} = Z-1; & \quad \delta\left(\dfrac{m^2}{k_0}\right) = \dfrac{m^2}{k_0}Z - \left(\dfrac{m^2}{k_0}\right)_{r} ; \nonumber \\ \delta\left(\dfrac{\lambda}{k_0^2}\right) = \dfrac{\lambda}{k_0^2} Z^2 - \left(\dfrac{\lambda}{k_0^2}\right)_{r}; & \quad \delta\left(\dfrac{k_1}{k_0^2}\right) = \dfrac{k_1}{k_0^2} Z^2 - \left(\dfrac{k_1}{k_0^2}\right)_{r}.\end{aligned}$$ These counterterms are fixed by demanding that $divp(\Gamma) = -{\int d^4{x} }\delta\mathcal{L}$. With some algebraic manipulations, the counterterms read, $$\begin{aligned} \label{ren05} \delta_{Z} = -\dfrac{A}{4\pi^2 \epsilon} ; & \quad \delta\left(\dfrac{m^2}{k_0}\right) = \dfrac{B}{4\pi^2 \epsilon} ; \nonumber \\ \delta\left(\dfrac{\lambda}{k_0^2}\right) = \dfrac{3 C}{\pi^2 \epsilon}; & \quad \delta\left(\dfrac{k_1}{k_0^2}\right) = \dfrac{D}{4\pi^2\epsilon}. \end{aligned}$$ Using Eq. (\[ren05\]) in (\[ren04\]), we find the one-loop corrections to coupling parameters in terms of the coefficients $A,B,C,D$, $$\begin{aligned} \label{ren06} \Delta \left(\dfrac{m^2}{k_0}\right) = \dfrac{m^2 A}{4\pi^2 k_0 \epsilon} + \dfrac{B}{4\pi^2 \epsilon}; \nonumber \\ \Delta\left(\dfrac{\lambda}{k_0^2}\right) = \dfrac{3 C}{\pi^2 \epsilon} + \dfrac{\lambda A}{2\pi^2 k_0^2 \epsilon}; \\ \Delta \left(\dfrac{k_1}{k_0^2}\right) = \dfrac{k_1 A}{2\pi^2 k_0^2\epsilon} + \dfrac{D}{4\pi^2\epsilon}. \nonumber \end{aligned}$$ For the sake of comparisons, and also as a crosscheck, we point out that upon choosing $\nu=0,\alpha=1,\omega=0$ in the case $k_1 = 0, k_0 = 1$, the gauge-dependent one-loop quantum gravitational correction to $\phi^4$ theory first calculated by Rodigast and Schuster [@rodigast2010a] is recovered: $\Delta\lambda = \frac{\kappa^2}{4\pi^2\epsilon}(m^2\lambda - 3\lambda^2/4\kappa^2)$. Note that, all gravitational corrections in Eq. (\[ren06\]) appear with a factor of $\kappa$, while the ones without it are nongravitational corrections that could in principle be obtained from flat space quantum field theory. Also, in the gauge covariant version of the same case (viz. $\nu=1,\alpha=0,\omega=1$ with $k_1 = 0, k_0 = 1$), our results match that of Pietrykowski [@artur2013]. In a similar spirit, we would like to shed some light on the extensions of the work of Ref. [@mackay2010]. There, a self-interacting scalar field with nonminimal coupling to gravity (of the form $\xi R \phi^2/2$) was considered and the corresponding field and mass renormalizations were studied. The action in Ref. [@mackay2010] matches ours if we put $k_1 = 0, k_0 = 1$ and add $\xi R \phi^2/2$. However, corrections to quartic coupling including contributions from the nonminimal coupling have not been calculated so far. Without going into the details, partly because the process is more or less unchanged, we present here the covariant one-loop corrections to quartic coupling $\lambda$ so as to complete the analysis of Ref. [@mackay2010], $$\begin{aligned} \label{ren07} \delta \lambda = \dfrac{3\lambda^2}{16\pi^2\epsilon} + \dfrac{\kappa^2}{\pi^2\epsilon}\left(\dfrac{9}{16}m^2\lambda + \dfrac{21}{8}m^2\lambda\xi^2 - \dfrac{3}{2}m^2\lambda\xi\right). \end{aligned}$$ \[sec5\]Effective potential =========================== It is evident from the analysis so far that extracting any more information, in the form of finite corrections for example, is a cumbersome task. A resolution to this problem lies in making a reasonable compromise, wherein the derivatives of background fields are ignored basis the assumption that either the background field is constant due to a symmetry or it is slowly varying. The resulting effective action is known as effective potential. One of the first instances of this workaround is the well known Coleman Weinberg potential [@coleman1973; @weinberg1973]. This approximation holds up especially during inflation, where the slow-rolling condition requires fields to be slowly varying. In this section, we evaluate the effective potential of the theory (\[action\]) including finite terms and infer cosmological implications. We begin by substituting $\partial^{\mu}\bar{\phi}=0$ in Eqs. (\[s1\])-(\[s4\]), resulting in, $$\begin{aligned} \label{es1} \tilde{S}_{1} &=& {\int d^4{x} } \Bigl[\frac{m^2 \kappa \delta \phi h^{a}{}_{a} \bar{\phi}}{2 k_0} - \frac{m^2 \kappa \nu \delta \phi h^{a}{}_{a} \bar{\phi}}{4 k_0} \Bigr]; \\ \label{es2} \tilde{S}_{2} &=& {\int d^4{x} } \Bigl[ -\frac{m^2 \kappa^2 h^{}{}_{ab} h^{ab} \bar{\phi}^2}{8 k_0} + \frac{m^2 \kappa^2 h^{a}{}_{a} h^{b}{}_{b} \bar{\phi}^2}{16 k_0} + \frac{\lambda \bar{\phi}^2 (\delta \phi)^2}{4 k_0^2} - \frac{m^2 \kappa^2 \nu \bar{\phi}^2 (\delta \phi)^2}{8 k_0} \nonumber \\ && + \frac{k_1 \bar{\phi}^2 \delta \phi{}_{,a} \delta \phi{}^{,a}}{2 k_0^2} \Bigr]; \\ \label{es3} \tilde{S}_{3} &=& {\int d^4{x} } \Bigl[\frac{\kappa \lambda \delta \phi h^{a}{}_{a} \bar{\phi}^3}{12 k_0^2} - \frac{\kappa \lambda \nu \delta \phi h^{a}{}_{a} \bar{\phi}^3}{24 k_0^2} \Bigr]; \\ \label{es4} \tilde{S}_{4} &=& {\int d^4{x} } \Bigl[- \frac{\kappa^2 \lambda h^{}{}_{ab} h^{ab} \bar{\phi}^4}{96 k_0^2} + \frac{\kappa^2 \lambda h^{a}{}_{a} h^{b}{}_{b} \bar{\phi}^4}{192 k_0^2} - \frac{\kappa^2 \lambda \nu \bar{\phi}^4 (\delta \phi)^2}{96 k_0^2} \Bigr].\end{aligned}$$ Using the above expressions in Eq. (\[loop01\]) and following the steps outlined in the Sec. \[loopint\], we obtain the covariant effective potential, $$\begin{aligned} \label{efpot0} \Gamma_{eff}[\bar{\phi}] &=& \dfrac{1}{8\pi^2}{\int d^4{x} } [A_{1}\frac{1}{\epsilon}\bar{\phi}^2 + A_{2}\bar{\phi}^2 + B_{1}\frac{1}{\epsilon}\bar{\phi}^4 + B_{2}\bar{\phi}^4]\end{aligned}$$ where, $A_1$ and $B_1$ are the same as $B$ and $C$ from Eq. (\[ren01\]) respectively, and, $$\begin{aligned} \label{efpot1} A_{2} &=& (\gamma + \log(\pi)) (- \frac{ k_1 m^4}{4 k_0^4} - \frac{5 m^4 \kappa^2}{32 k_0^2} + \frac{ m^2 \lambda}{8 k_0^3}) + \frac{3 k_1 m^4}{8 k_0^4} + \frac{ m^4 \kappa^2}{4 k_0^2} - \frac{ m^2 \lambda}{8 k_0^3} \nonumber \\ && + (- \frac{ k_1 m^4}{4 k_0^4} - \frac{5 m^4 \kappa^2}{32 k_0^2} + \frac{ m^2 \lambda}{8 k_0^3}) \log(\frac{m^2}{ k_0 \mu^2}) \nonumber \\ && - \frac{1}{\bar{\phi}} {\int\frac{d^4{k}}{(2\pi)^4}} e^{-ik\cdot x} \tilde{\bar{\phi}} \left( \frac{3 m^4 \kappa^2 \log\bigl(1 + \frac{ k_0 k^2}{m^2}\bigr)}{32 k_0^2} - \frac{3 m^6 \kappa^2 \log\bigl(1 + \frac{ k_0 k^2}{m^2}\bigr)}{32 k_0^3 k^2}\right); \\ \label{b2} B_{2} &=& - \frac{9 k_1^2 m^4}{128 k_0^6} - \frac{5 k_1 m^4 \kappa^2}{16 k_0^4} + \frac{ k_1 m^2 \lambda}{4 k_0^5} + \frac{25 m^2 \kappa^2 \lambda}{192 k_0^3} - \frac{ \lambda^2}{16 k_0^4} \nonumber \\ && + (\gamma + \log(\pi)) (\frac{3 k_1^2 m^4}{64 k_0^6} + \frac{7 k_1 m^4 \kappa^2}{32 k_0^4} - \frac{ k_1 m^2 \lambda}{4 k_0^5} - \frac{13 m^2 \kappa^2 \lambda}{192 k_0^3} + \frac{ \lambda^2}{32 k_0^4}) \nonumber \\ && + (\frac{3 k_1^2 m^4}{64 k_0^6} + \frac{5 k_1 m^4 \kappa^2}{32 k_0^4} - \frac{ k_1 m^2 \lambda}{8 k_0^5} - \frac{13 m^2 \kappa^2 \lambda}{192 k_0^3} + \frac{ \lambda^2}{32 k_0^4}) \log(\frac{m^2}{ k_0 \mu^2}) \nonumber \\ && - \frac{1}{\bar{\phi}^3}{\int\frac{d^4{k}}{(2\pi)^4}} e^{-ik\cdot x} \tilde{\bar{\phi}^3} \Bigg( \frac{ m^2 \kappa^2 \lambda \log\bigl(1 + \frac{ k_0 k^2}{m^2}\bigr)}{32 k_0^3}\Bigg) \nonumber \\ && + \frac{1}{\bar{\phi}^2}{\int\frac{d^4{k}}{(2\pi)^4}} e^{-ik\cdot x} \tilde{\bar{\phi}^2} \Bigg( -\frac{ m^2 \kappa^2 \lambda \log\Bigl(\frac{1 + \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}{-1 + \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}\Bigr) \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}{32 k_0^3} + \nonumber \\ && \frac{ \lambda^2 \log\Bigl(\frac{1 + \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}{-1 + \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}\Bigr) \bigl(1 + \frac{4 m^2}{ k_0 k^2}\bigr)^{1/2}}{32 k_0^4} - \frac{ m^4 \kappa^2 \lambda \log\bigl(1 + \frac{ k_0 k^2}{m^2}\bigr)}{32 k_0^4 k^2} \nonumber \\ && + \frac{ k_1^2 \arctan\Bigl(\frac{ k_0^{1/2} k}{\bigl(4 m^2 - k_0 k^2\bigr)^{1/2}}\Bigr) k^{3} \bigl(4 m^2 - k_0 k^2\bigr)^{1/2}}{64 k_0^{9/2}} \Bigg) \nonumber \\ && + \frac{1}{\bar{\phi}}{\int\frac{d^4{p}}{(2\pi)^4}} e^{-ip\cdot x} \tilde{\bar{\phi}} \frac{3 k_1 m^4 \kappa^2 \log\Bigl(\frac{1 + \bigl(1 + \frac{4 m^2}{ k_0 p^2}\bigr)^{1/2}}{-1 + \bigl(1 + \frac{4 m^2}{ k_0 p^2}\bigr)^{1/2}}\Bigr) \bigl(1 + \frac{4 m^2}{ k_0 p^2}\bigr)^{1/2}}{32 k_0^4} .\end{aligned}$$ The logarithmic terms appearing in expressions above are dealt with as follows. In the context of the present problem and the effective theory treatment, we restrict ourselves to the condition $k\ll 10^{-6}M_p$ so that $\frac{k_0 k^2}{m^2}\ll 1$ (more on this later) using the order-of-magnitude estimates of parameters in Eq. (\[param\]) from the results of [@ferreira2018]. Hence, logs involving this fraction can be expanded in a Taylor series. On the other hand, $\sqrt{1 + \frac{m^2}{k_0 k^2}}\approx \sqrt{\frac{m^2}{k_0 k^2}}$. For the $\arctan(\cdots)$ term, we use $\arctan(x)\approx x$ for small $x$. After these expansions, all terms with factors of $k$ will vanish since we assume the derivatives of $\bar{\phi}$ to be zero. Hence, all the integrands of momenta integrals in Eqs. (\[efpot1\],\[b2\]) reduce to c-numbers times Fourier transforms of $\bar{\phi}^{n}$. Using these simplifications, the coefficients $A_{2}$ and $B_{2}$ are obtained as, $$\begin{aligned} \label{efpot2} A_{2} &=& (\gamma + \log(\pi)) (- \frac{ k_1 m^4}{4 k_0^4} - \frac{5 m^4 \kappa^2}{32 k_0^2} + \frac{ m^2 \lambda}{8 k_0^3}) + \frac{3 k_1 m^4}{8 k_0^4} + \frac{ m^4 \kappa^2}{4 k_0^2} - \frac{ m^2 \lambda}{8 k_0^3} \nonumber \\ && + \dfrac{3 m^4 \kappa^2}{32 k_0^2} + (- \frac{ k_1 m^4}{4 k_0^4} - \frac{5 m^4 \kappa^2}{32 k_0^2} + \frac{ m^2 \lambda}{8 k_0^3}) \log(\frac{m^2}{ k_0 \mu^2}); \nonumber \\ B_{2} &=& - \frac{9 k_1^2 m^4}{128 k_0^6} - \frac{5 k_1 m^4 \kappa^2}{16 k_0^4} + \frac{ k_1 m^2 \lambda}{4 k_0^5} + \frac{25 m^2 \kappa^2 \lambda}{192 k_0^3} - \frac{ \lambda^2}{16 k_0^4} \nonumber \\ && -\dfrac{m^2\kappa^2\lambda}{32 k_0^3} + \dfrac{\lambda^2}{32 k_0^4} - \dfrac{m^2 \kappa^2 \lambda }{32 k_0^3} + \dfrac{3 k_1 m^4 \kappa^2}{32 k_0^4} \nonumber \\ && + (\gamma + \log(\pi)) (\frac{3 k_1^2 m^4}{64 k_0^6} + \frac{7 k_1 m^4 \kappa^2}{32 k_0^4} - \frac{ k_1 m^2 \lambda}{4 k_0^5} - \frac{13 m^2 \kappa^2 \lambda}{192 k_0^3} + \frac{ \lambda^2}{32 k_0^4}) \nonumber \\ && + (\frac{3 k_1^2 m^4}{64 k_0^6} + \frac{5 k_1 m^4 \kappa^2}{32 k_0^4} - \frac{ k_1 m^2 \lambda}{8 k_0^5} - \frac{13 m^2 \kappa^2 \lambda}{192 k_0^3} + \frac{ \lambda^2}{32 k_0^4}) \log(\frac{m^2}{ k_0 \mu^2}) \end{aligned}$$ The counterterms for quadratic and quartic terms have a similar form to Eq. (\[ren05\]), so that the effective potential can be written in terms of renormalized parameters which can be calculated from Eq. (\[ren06\]) with $A=0$. The effective action takes the form, $$\begin{aligned} \label{efpot3} V_{eff} = \dfrac{1}{2}\dfrac{m^2}{k_0} \bar{\phi}^2 + \dfrac{1}{4!}\dfrac{\lambda}{k_0^2}\bar{\phi}^4 + A_{2}\bar{\phi}^2 + B_{2}\bar{\phi}^4.\end{aligned}$$ Estimating the magnitude of corrections --------------------------------------- Making a definitive statement about cosmological implications of quantum corrected potential requires an analysis in the FRW background, which unfortunately is out of scope of the present work. However, we can get an order-of-magnitude estimate of the quantum corrections to the effective potential using the values of parameters $k_0, k_1, m^2, \lambda$ from the results of Ref. [@ferreira2018]. From the action (\[action\]), the Einstein equations are given by, $$\begin{aligned} \label{om0} 3 H^2 &=& \dfrac{\kappa^2}{8}\left(-3\dot{\bar{\phi}}^{2} - 3 \dfrac{k_1}{k_0^2}\bar{\phi}^2\dot{\bar{\phi}}^2 + \dfrac{m^2}{k_0}\bar{\phi}^2 + \dfrac{\lambda}{12 k_0^2}\bar{\phi}^4\right); \nonumber \\ 2\dot{H} + 3 H^2 &=& \dfrac{\kappa^2}{8}\left(-\dot{\bar{\phi}}^{2} - \dfrac{k_1}{k_0^2}\bar{\phi}^2\dot{\bar{\phi}}^2 + \dfrac{m^2}{k_0}\bar{\phi}^2 + \dfrac{\lambda}{12 k_0^2}\bar{\phi}^4\right),\end{aligned}$$ from which we obtain in the de-Sitter limit ($\dot{H}\sim\dot{\phi}\sim 0$), $$\begin{aligned} \label{om1} 3H^2 = \dfrac{\kappa^2}{8}\left( \dfrac{m^2}{k_0}\bar{\phi}^2 + \dfrac{\lambda}{12 k_0^2}\bar{\phi}^4 \right). \end{aligned}$$ The field equation for $\bar{\phi}$ reads, $$\begin{aligned} \label{om2} (a + \frac{k_1}{k_0^2}\bar{\phi}^2)\ddot{\bar{\phi}} + \dfrac{k_1}{k_0^2}\bar{\phi}\dot{\bar{\phi}}^2 + (3aH + \frac{2k_1}{k_0^2}H\bar{\phi}^2)\dot{\bar{\phi}} - \dfrac{m^2}{k_0}\bar{\phi} - \dfrac{\lambda}{6k_0^2}\bar{\phi}^3 = 0.\end{aligned}$$ Applying the de-Sitter conditions, Eq. (\[om2\]) yields the de-Sitter value of $\bar{\phi}$, $$\begin{aligned} \label{om3} \bar{\phi}_{0}^2 = - \dfrac{6 k_0 m^2}{\lambda}.\end{aligned}$$ Using Eq. (\[om3\]) in (\[om1\]), we find the de-Sitter value of Hubble parameter $H_{0}$: $$\begin{aligned} \label{om4} H_0^2 = - \dfrac{\kappa^2 m^4}{8\lambda}.\end{aligned}$$ Clearly, the condition for existence of de-Sitter solutions is $\lambda < 0$. Demanding this condition in Eq. (\[param\]), along with $m^2>0$ and $\bar{\phi}<f$, leads to a constraint on the parameter $\alpha$ of the original theory (\[eq02\]): $0.5 < \alpha \lesssim 1$. Following the results of [@ferreira2018], we choose $0.5\lesssim\alpha\lesssim 0.6 \sim \mathcal{O}(1)$. Near this value of $\alpha$, $f\sim M_{p}=1/\kappa$ and $\Lambda \sim 10^{16} GeV$. Substituting these in Eqs. (\[param\]), we find $m^2 \sim \Lambda^{4}/f^{2} \sim 10^{-12}M_{p}^{2}$; $\lambda \sim \Lambda^{4}/f^{4} \sim 10^{-12}$. Similarly, $k_0 \sim 1$ while $k_{1}\sim M_{p}^{-2}$. This also implies that in the low energy limit where momenta $k\ll 10^{13}$ GeV $\ll M_p$, $k_{1}k^2/k_{0}^2 \ll \lambda/k_{0}^2$, i.e. the derivative coupling term is suppressed. From the above, we can estimate the order of magnitude contributions of terms in $A_{2}$ and $B_{2}$ at $\mathcal{O}(\bar{\phi}^2)$ and $\mathcal{O}(\bar{\phi}^4)$ respectively. We estimate the magnitude of each type of term present at both orders. At quadratic order in background field, we find, $$\begin{aligned} \label{om5} \dfrac{\kappa^2 m^4}{k_{0}^2} \sim \dfrac{\lambda m^2}{k_0^3} \sim \dfrac{k_{1}m^4}{k_0^4} \sim 10^{14} GeV^2 .\end{aligned}$$ Similarly, at quartic order in background field, $$\begin{aligned} \label{om6} \dfrac{\kappa^2 m^4 k_1}{k_0^4} \sim \dfrac{\kappa^2 m^2 \lambda}{k_0^3} \sim \dfrac{\lambda}{k_0^4} \sim \dfrac{k_1 m^2 \lambda}{k_0^5} \sim 10^{-24} .\end{aligned}$$ Quite an interesting observation here is that the magnitudes of gravitational and non-gravitational corrections turn out to be exactly the same for both quadratic and quartic order contributions. However, the corresponding quantum corrections are smaller by an order of $10^{-12}$ compared to $m^2$ and $\lambda$. \[end\]Conclusion ================= The nonminimal natural inflation model in consideration here is approaximately described by a massive scalar field model with quartic self interaction and a derivative coupling in the region where $\phi/f<1$. We study one-loop corrections to this theory, about a Minkowski background, using a covariant effective action approach developed by DeWitt-Vilkovisky. The one-loop divergences and corresponding counterterms have been obtained. Along the way, we also recover several past results, both gauge-invariant non-gauge-invariant, for similar theories. In one such exercise, we obtain the $\phi^4$ coupling correction in a theory with nonminimal coupling of scalar field to gravity, originally considered in Ref. [@mackay2010] and thereby extend their result. Finite corrections have been taken into account for the calculation of effective potential, where we assume that the background field changes sufficiently slowly so that all derivatives of background field(s) can be ignored. Although cosmologically relevant inferences are not feasible as long as the metric background is Minkowski and not FRW, we can still estimate approximately the magnitudes of quantum corrections. Using the range of parameters applicable to our model, we find that the gravitational and non-gravitational corrections are of same order of magnitudes, while still being expectedly small compared to $m^2$ and $\lambda$. This is quite an interesting observation, since one would naively assume that gravitational corrections are $\kappa^2$ suppressed and thus would necessarily be small. There is thus enough motivation to go a step further, and calculate gravitational corrections in the FRW background so that cosmologically relevant inferences can be derived. This work was partially funded by DST (Govt. of India), Grant No. SERB/PHY/2017041. \[a1\]Loop Integrals ==================== Most of the loop integrals are calculated using the well known PV reduction method [@bardin1999]. Some integrals, namely (\[int4\],\[int5\]) are calculated the general method outlined in Ref. [@romao2019]. Finite parts have been calculated for integrals needed for evaluating the effective potential. Integrals in $\langle\tilde{S}_{2}\rangle$,$\langle\tilde{S}_{4}\rangle$: $$\begin{aligned} {\int\frac{d^4{k}}{(2\pi)^4}}\dfrac{k_{\mu}k_{\nu}}{k^2+\frac{m^2}{k_0}} &=& \dfrac{g_{\mu\nu}}{16\pi^2}\Big(\frac{m^4}{8 k_0^2} - \frac{m^4 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{4 k_0^2}\Big) \\ {\int\frac{d^4{k}}{(2\pi)^4}}\dfrac{k_{\mu}}{k^2+\frac{m^2}{k_0}} &=& 0 \\ {\int\frac{d^4{k}}{(2\pi)^4}}\dfrac{1}{k^2+\frac{m^2}{k_0}} &=& \dfrac{1}{16\pi^2}\frac{m^2 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{k_0}\end{aligned}$$ Integrals in $\langle\tilde{S}_{1}\tilde{S}_{1}\rangle$, $\langle\tilde{S}_{1}\tilde{S}_{3}\rangle$: $$\begin{aligned} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu} k'^{\nu}}{k'^4 ((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \biggl(\tfrac{1}{4} g^{\mu\nu} \bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) \nonumber \\ && + \frac{k^{\mu} k^{\nu} \Bigl(\tfrac{1}{2} \bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) + \tfrac{1}{2} \bigl(- \frac{2}{\epsilon} + \gamma + \log(\pi)\bigr)\Bigr)}{2 k^2}\biggr) \\ {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu} k'^{\nu}}{k'^2 ((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \tfrac{1}{3} k^{\mu} k^{\nu} \bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) \nonumber \\ && - \tfrac{1}{4} g^{\mu\nu} \bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) \bigl(\frac{m^2}{k_0} + \tfrac{1}{3} k^2\bigr)\end{aligned}$$ $$\begin{aligned} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu}}{k'^2 ((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \dfrac{k^{\mu}}{2}(\frac{2}{\epsilon}-\gamma - \log(\pi)) \\ {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{2}}{((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \Bigg[4 \Bigl(\frac{m^4}{8 k_0^2} - \frac{m^4 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{4 k_0^2}\Bigr) \nonumber \\ && + \frac{m^2 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr) (k^2)}{k_0}\Bigg] \\ {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{1}{k'^2 ((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \Big(2 + \frac{2}{\epsilon} - \gamma - \log(\pi) - \log(\frac{m^2}{k_0 \mu^2}) \nonumber \\ && - \log\bigl(1 + \frac{k_0 (k^2)}{m^2}\bigr) \bigl(1 + \frac{m^2}{k_0 (k^2)}\bigr)\Big)\\ {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{1}{((k'-k)^2+\frac{m^2}{k_0})} &=& \dfrac{1}{16\pi^2} \frac{m^2 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{k_0}\end{aligned}$$ Integrals in $\langle\tilde{S}_{2}\tilde{S}_{2}\rangle$: $$\begin{aligned} \label{int4} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{4}}{(k'^2 + \frac{m^2}{k_0}) ((k'-k)^2+\frac{m^2}{k_0})} = \dfrac{1}{16\pi^2} \Bigg(\frac{9 m^4}{16 k_0^2} + \frac{3 m^4}{4 \epsilon k_0^2} - \frac{3 m^4 \gamma}{8 k_0^2} - \frac{3 m^4 \log(\pi)}{8 k_0^2} \nonumber \\ - \frac{3 m^4 \log(\frac{m^2}{k_0 \mu^2})}{8 k_0^2} + \frac{7 m^2 (k^2)}{8 k_0} + \frac{7 m^2 (k^2)}{4 \epsilon k_0} \nonumber \\ - \frac{7 m^2 \gamma (k^2)}{8 k_0} - \frac{7 m^2 \log(\pi) (k^2)}{8 k_0} - \frac{7 m^2 \log(\frac{m^2}{k_0 \mu^2}) (k^2)}{8 k_0} + \tfrac{1}{8} (k^2)^2 \nonumber \\ + \frac{(k^2)^2}{8 \epsilon} - \tfrac{1}{16} \gamma (k^2)^2 - \tfrac{1}{16} \log(\pi) (k^2)^2 - \tfrac{1}{16} \log(\frac{m^2}{k_0 \mu^2}) (k^2)^2 \nonumber \\ - \dfrac{\arctan\Bigl(\frac{k_0^{1/2} (k^2)^{1/2}}{\bigl(4 m^2 - k_0 (k^2)\bigr)^{1/2}}\Bigr) (k^2)^{3/2} \bigl(4 m^2 - k_0 (k^2)\bigr)^{1/2}}{8 k_0^{1/2}}\Bigg)\\ \label{int5} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{2}k'^{\mu}}{(k'^2 + \frac{m^2}{k_0}) ((k'-k)^2+\frac{m^2}{k_0})} = -\dfrac{1}{16\pi^2} (\frac{2}{\epsilon}-\gamma - \log(\pi))\dfrac{3m^2}{2 k_0}k^{\mu}\end{aligned}$$ $$\begin{aligned} \label{int1} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu}k'^{\nu}}{(k'^2 + \frac{m^2}{k_0}) ((k'-k)^2+\frac{m^2}{k_0})} = \dfrac{1}{16\pi^2} \nonumber \\ g^{\mu\nu} \biggl(\frac{m^2 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{6 k_0} + \tfrac{1}{18} \bigl(- \frac{6 m^2}{k_0} - (k^2)\bigr) \nonumber \\ - \frac{\bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) \Bigl(\frac{2 m^4}{k_0^2} + (k^2)^2 - 2 \bigl(\frac{m^4}{k_0^2} - \frac{2 m^2 (k^2)}{k_0}\bigr)\Bigr)}{12 (k^2)}\biggr) \nonumber \\ + k^{x1} k^{x2} \biggl(\frac{m^2 \bigl(-1 - \frac{2}{\epsilon} + \gamma + \log(\pi) + \log(\frac{m^2}{k_0 \mu^2})\bigr)}{3 k_0 (k^2)} + \frac{\frac{6 m^2}{k_0} + (k^2)}{18 (k^2)} \nonumber \\ + \frac{\bigl(\frac{2}{\epsilon} - \gamma - \log(\pi)\bigr) \Bigl(\frac{2 m^4}{k_0^2} - \frac{3 m^2 (k^2)}{k_0} + (k^2)^2 - 2 \bigl(\frac{m^4}{k_0^2} - \frac{2 m^2 (k^2)}{k_0}\bigr)\Bigr)}{3 (k^2)^2}\biggr)\\ \label{int2} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu}}{(k'^2 + \frac{m^2}{k_0}) ((k'-k)^2+\frac{m^2}{k_0})} = -\dfrac{1}{16\pi^2} k^{\mu} \frac{1}{2}\Big(-\frac{2}{\epsilon} + \gamma + \log(\pi)\Big)\\ \label{int3} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{1}{(k'^2 + \frac{m^2}{k_0}) ((k'-k)^2+\frac{m^2}{k_0})} = \dfrac{1}{16\pi^2} \Big(2 + \frac{2}{\epsilon} - \gamma - \log(\pi) - \log(\frac{m^2}{k_0 \mu^2}) \nonumber \\ - \log\Bigl(\frac{1 + \bigl(1 + \frac{4 m^2}{k_0 (k^2)}\bigr)^{1/2}}{-1 + \bigl(1 + \frac{4 m^2}{k_0 (k^2)}\bigr)^{1/2}}\Bigr) \bigl(1 + \frac{4 m^2}{k_0 (k^2)}\bigr)^{1/2}\Big)\end{aligned}$$ Integrals of type (\[int1\],\[int2\],\[int3\]) are also present in $\langle\tilde{S}_{1}\tilde{S}_{1}\tilde{S}_{2}\rangle$. The rest of the integrals are, $$\begin{aligned} {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu}k'^{\nu}k'^{\rho}}{d_0 d_1 d_2} &=& \dfrac{1}{16 \pi^2} \dfrac{1}{12} \Big(\frac{2}{\epsilon} - \gamma - \log(\pi)\Big) \nonumber \\ && \times(g^{\nu \rho} (2 k^{\mu} + p^{\mu}) + g^{\rho\mu } (2 k^{\nu } + p^{\nu }) + g^{\mu \nu } (2 k^{\rho} + p^{\rho}) \\ {\int\frac{d^4{k'}}{(2\pi)^4}}\dfrac{k'^{\mu}k'^{\nu}}{d_0 d_1 d_2} &=& \dfrac{1}{16 \pi^2} g^{\mu\nu} \Big(\frac{2}{\epsilon} - \gamma - \log(\pi)\Big)\end{aligned}$$

--- abstract: 'We investigate the stellar population of star-forming galaxies at *z* $\sim$ 4 by focusing on their slope of rest-frame ultraviolet (UV) continuum, $\beta$ where $f_{\lambda} \propto \lambda^{\beta}$. We investigate the sample of bright Lyman Break Galaxies (LBGs) with *i’* $\leq$ 26.0 in the Subaru/XMM-Newton Deep Survey field by using the SED fitting analysis. We find that the apparently redder ($\beta_{\mathrm{obs}} > -1.73$) LBGs tend to be dusty ($\mathrm{Av} > 1.0$), have the young stellar population ($\beta_{\mathrm{int}} < -2.42$), and the intrinsically active star-forming galaxies (SFR $\gtrsim$ a few $\times\ 10^{2}\,\mathrm{M_{\solar}}\>\mathrm{yr^{-1}}$). It means that a significant fraction of the UV–selected LBGs at *z* $\sim$ 4 is on-going active and dust obscured star-forming galaxies. We compare the IR to UV luminosity ratio assuming the dust attenuation laws with the sub-millimeter observations from previous works. The result suggests that the Calzetti-like dust attenuation law is preferable for the active and dusty star-forming LBGs at $z = 4$. We also find that an extrapolation of the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation toward the fainter magnitude range below our sample magnitude limit intersects the $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation previously obtained in the deeper narrow-area observations at $M_{\mathrm{UV}} = -18.9$ and $\beta = -1.94$, which coincides with the break point of $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation observed so far. The coincidence suggest that we see the almost dust-free population at $M_{\mathrm{UV,obs}} \gtrsim -18.9$.' author: - 'Satoshi <span style="font-variant:small-caps;">Yamanaka</span>' - 'Toru <span style="font-variant:small-caps;">Yamada</span>,' title: 'The UV spectral slope beta and stellar population of most active star-forming galaxies at z$\sim$4' --- Introduction {#S1Intr} ============ The ultraviolet (UV) continuum spectrum of star-forming galaxies is characterized by the spectral index $\beta$ in the form of $f_{\lambda} \propto \lambda^{\beta}$ [@Calz94]. The $\beta$ values are related with physical quantities such as age, metallicity, and dust extinction of the galaxies. In the case of less dust attenuation, younger age, and lower metallicity, the galaxy has the larger negative $\beta$ value (e.g., [@Bouw10]; [@Stan16]). Since it is relatively simple to measure $\beta$ values even if objects are at high redshift, the $\beta$ index is a useful tool to probe their physical quantities. The typical value of $\beta$ found in the previous works is $\sim -1.7$ at $z \sim 4$ for $\sim L_{*}$ galaxies with $M_{\mathrm{UV}} \sim -21.0$ ([@Bouw12]; [@Fink12]), and it is bluer at higher redshift up to $z \sim 7$ ($\beta$–*z* relation: e.g., [@Wil11]). At the fainter magnitude (e.g. $M_{\mathrm{UV}} \sim -19.0$), the observed $\beta$ value has still uncertainties and the relation between $\beta$ and UV absolute magnitude ($\beta$–$M_{\mathrm{UV}}$ relation) has been a subject of debate for the several years (e.g., [@Bouw12]; [@Bouw14]; [@Dunl12]; [@Dunl13]; [@Fink12]; [@Rog14]). and report that bright galaxies have redder $\beta$ values and faint galaxies have bluer $\beta$ values, while and report that $\beta$ values are constant over the observed magnitude range. This inconsistency in the $\beta$–$M_{\mathrm{UV}}$ relation can be caused by both large photometric errors for faint galaxies and selection bias. In order to reveal the true $\beta$–$M_{\mathrm{UV}}$ relation, a further large sample of objects with small photometric uncertainties is needed and/or it is necessary to assess the incompleteness of the observed $\beta$ distribution. Recently, @Dunc15 and @Bouw14 show that the $\beta$ value decreases with the $M_{\mathrm{UV}}$ value. finds the trend by combining the results of the previous literature ([@Bouw14];[@Dunc14]; [@Dunl12]; [@Dunl13]; [@Fink12]; [@Rog14]; [@Wil11]) and discusses the trend by assessing the observational bias (incompleteness) of the observed $\beta$ distribution in the faint magnitude range. The $\beta$–$M_{\mathrm{UV}}$ relation is understood as another aspect of the mass-metallicity relation seen in star-forming galaxies as the $\beta$ value depends on the dust extinction ([@Bouw09], , ). Moreover, it is suggested that there is a “knee” in the $\beta$–$M_{\mathrm{UV}}$ relation at $M_{\mathrm{UV}} \sim -19.0$, and the dependence of $\beta$ on $M_{\mathrm{UV}}$ becomes weaker at $M_{\mathrm{UV}} \gtrsim -19.0$ than at $M_{\mathrm{UV}} < -19.0$ [@Bouw14]. The change of the dependence is interpreted as the change of the dependence of the dust extinction on the UV luminosity or the stellar mass (e.g., [@Pann09]; [@Redd10]). The semi-analytic model predicts the sudden change of the dust-to-gas mass ratio at the critical metallicity and the dust mass rapidly increase at the metallicity level larger than the critical metallicity (e.g., [@Hira11]). The change of the dependence of $\beta$ on $M_{\mathrm{UV}}$ perhaps indicates the existence of the critical metallicity. The redshift dependence of $\beta$ is interpreted as the dust attenuation history or the dust production history in star-forming galaxies. Interestingly, the dust attenuation history at $z \gtrsim 3$ estimated from the redshift dependence of $\beta$ is smoothly connected with the dust attenuation history at $z \lesssim 3$ estimated from the direct measurements of both IR and UV luminosity [@Burg13]. The dust attenuation history is also used for revealing the history of the true (dust-corrected) cosmic Star Formation Rate (SFR) density in the high-z universe because it is still difficult to obtain the IR luminosity for high-z star-forming galaxies (e.g., [@Bouw09], ; [@Mada14]). Currently, the $\beta$–*z* relation has been used for considering the source of cosmic reionization, assuming that the $\beta$ value represents the production rate of hydrogen ionizing photons since the production rate is susceptible to the stellar population (e.g., [@Dunc15]; [@Bouw15], ). We note that the samples in most of the literature are overlapped and not independent since the set of GOODS-South/HST or HUDF/HST data is mostly used so far in their works. Due to the small observed area, the number of bright objects at $z = 4$ in the field is limited (except for [@Rog14]) and these previous studies focus on relatively faint galaxies at high redshift. The $\beta$ distribution of luminous objects is, however, also important since such population provides important clues to understand early star-formation history in the universe (e.g., [@Cucc12]; [@Hatf18]). In fact, by using stacked images, @LeeKS11 investigate the $\beta$ value of ultra-luminous star-forming galaxies ($19.46 \leq I < 24.96$) at *z* $\sim$ 4 and they find that the $\beta$ value of the stacked star-forming galaxies tends to be redder toward the brighter magnitude range. When taking all the previous works into consideration, the investigation for individual and ”normal” luminous galaxies ($L \sim L_{*}$) is very comprehensive. Recent Atacama Large Millimeter/submillimeter Array (ALMA) observations have revealed the dust properties of high redshift star-forming galaxies through the relation between the ratio of IR to UV (so called IRX) and the UV slope $\beta$ (IRX–$\beta$ relation; e.g., [@Cpk15]; [@Bouw16]; [@McLu18]). In order to interpret these results, it is very important to investigate the detailed relation between the UV slope $\beta$ and the stellar population which is hidden by dust extinction. On the other hand, there is difficulty in studying the intrinsic values of the $\beta$, $M_{\mathrm{UV}}$, and stellar population for the luminous and massive galaxies, since the effect of dust extinction on their color degenerates with the age and metallicity of their stellar population: the more massive systems are on average older and metal rich. It is essential to resolve these degeneracy and understand the intrinsic properties of star formation and effects of dust extinction. In this paper, we present the results of the UV slope $\beta$ and stellar population for the relatively bright galaxies with $M_{\mathrm{UV}} \lesssim -20$ at $z \sim 4$ in relatively wide-area and deep Subaru/XMM-Newton Deep Survey (SXDS) field. Wide area coverage is essential to sample rare luminous galaxies. In section 2, we explain the data and sample selection. In section 3, we describe our method to evaluate the $\beta$ value. In section 4, we show the result of the observed $\beta$–$M_{UV}$ relation, and we assess the incompleteness of our sample selection. In section 5, we discuss the intrinsic $\beta$–$M_{UV}$ relation, most active star-forming galaxies, and dust attenuation law for $z \sim 4$ galaxies. In section 6, we give our conclusions to summarize this work. In regard to the cosmological parameters, we assume $\Omega_{m,0} = 0.3$, $\Omega_{\Lambda,0} = 0.7$, $H_{\mathrm{0}} = 70\>\mathrm{km\>s^{-1}\>Mpc^{-1}}$. Finally throughout this work we apply the AB magnitude system ([@OkeGunn]; [@Fkgt96]). Data and sample selection {#S2Dtss} =========================== Data {#S2s1da} ------ In our analysis, we select the Lyman Break Galaxies (LBGs) at $z \sim 4$ in the SXDS field which is partially covered by other surveys (i.e., UDS-UKIDSS/UKIRT, UDS-CANDELS/HST, and SEDS/Spitzer). Figure \[fig1\] shows the field map of each survey and indicates that UDS-CANDELS and SEDS survey do not cover the entire field of SXDS or UDS-UKIDSS. Our catalog includes *B*, *V*, *R*, *i’*, *z’*, and updated-*z’* from Subaru/Suprime-Cam, *J*, *H*, and *K* from UKIRT/WFCAM, F125W and F160W from HST/WFC3, and 3.6$\>\micron$ and 4.5$\>\micron$ from Spitzer/IRAC. The *B*, *V*, *R*, *i’*, *z’* band images are taken from the archived SXDS data [@Furu08]. The CCD of Subaru/Suprime-Cam is replaced with the new CCD of Hamamatsu Photonics in 2008, and the total response function improve especially at the longer wavelength. The Subaru/Suprime-Cam *z’* band images are taken after the replacement again, and we call the images the updated-*z’* band. The limiting magnitude of the updated-*z’* images is $\sim 0.5\>\mathrm{mag}$ deeper than the archived SXDS data [@Furu16]. The *J*, *H*, and *K* band images are taken from the UKIRT Deep Sky Survey (UKIDSS: [@Law07]) DR10, and the F125W and F160W band images are taken from the Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS: [@Grog11]; [@Koek11]). For the data from *B* to *K* band, we use 2”-diameter aperture magnitude measured by using the SExtractor[^1] ver.2.5.0 [@BeAr96]. The smaller PSF images are convolved with Gaussian kernels to be matched in FWHM of the stars (1”.0) with the original updated-*z’* band image. On the other hand the 3.6$\>\micron$ and 4.5$\>\micron$ photometry are taken from the Spitzer Extended Deep Survey (SEDS) catalog [@Ash13] and we apply aperture correction. Finally, we pick up the objects which are in the overlapped region covered by both the SXDS and UDS-UKIDSS fields (see figure \[fig1\]: the overlapped region is filled by yellow slanting lines) because we need both optical and NIR photometry for estimating the UV slope $\beta$ value of *z* $\sim$ 4 galaxies. The information of the imaging data is summarized in table \[tab1\]. ![ Field map of imaging data used in our analysis. The blue solid line show the sky coverage of the SXDS field; the green dot-dash line represent the sky coverage observed with Subaru/updated-*z’*; The red dash line represents the sky coverage of the UDS-UKIDSS field; the black dotted line represents the sky coverage of the UDS-CANDELS field; and the brown two-dot-dash line represents the sky coverage of the SEDS field. Our catalog consists of the objects within the area covered by all of the SXDS, updated-*z’*, and UDS field since we use the *i’*, *z’*, updated-*z’*, and *J* band photometry for estimating the UV slope $\beta$ value. This area is filled by the yellow slanting lines. []{data-label="fig1"}](Pic.Region_SXDS_new1812.eps){width="80mm"} \[tab1\] (1)@Furu08; (2)@Furu16; (3); (4)@Grog11; (5)@Koek11; (6)@Ash13;\ The values show the total magnitude where the completeness of the source detection is 50%. Sample selection and SED fitting {#S2s2ss} ---------------------------------- From the photometric sample, we select the objects satisfying all the following criteria. - $i' \leq 26.0$ - Subaru/*z’* or Subaru/updated-*z’* $\geq$ 2$\,\sigma$ - UKIRT/*J* or HST/F125W $\geq$ 2$\,\sigma$ - $B - R > 1.2$, $R - i' < 0.7$, and $B - R > 1.6(R - i') + 1.9$ - $3.5 \leq z_{phot} < 4.5$ with reduced $\chi^2 \leq 2$ The criteria (1) is applied so as to select the galaxies bright enough to have small photometric errors, and the magnitude threshold corresponds to $S / N \gtrsim 11.5$. The criteria (2) and (3) are required to estimate the $\beta$ value accurately. Due to the stellar spikes and/or saturated pixels, some objects are not detected in the deeper Subaru/updated-*z’* or HST/F125W imaging but detected in the shallower Subaru/*z’* or UKIRT/*J* imaging. Therefore, we use such the criteria (2) and (3). The criteria (4) is the *BRi’*–LBG selection investigated by @Ouch04 for the Subaru/Suprime-Cam filter set, and this criteria is intended to pick up star-forming galaxies at *z* $\sim$ 4. In @Ouch04, the detectability of *z* $\sim$ 4 galaxies and the contamination rate from low-*z* galaxies are discussed, and leaders should refer to the reference for more details. After the criteria (1), (2), (3), and (4), the total number of objects is $\sim$ 2100. The criteria (5) is applied so as to select the reliable galaxies at *z* $\sim$ 4. The reduced $\chi^{2}$ value is calculated for each galaxy as $\chi^2 / \mathrm{d.o.f}$, in which d.o.f $=$ (number of observed broad-band filters for each galaxy) $-$ (number of free parameters in the fitting). In the selection procedure, our concern is only the photometric redshift, and thus we adopt the number of free parameters $=$ 1. As a result, our catalog contains $\sim$ 1800 objects which are visually checked in order to avoid stellar spikes and/or saturated pixels. We write down the detail of $\sim$ 300 objects, which are excluded by the criteria (5). Among $\sim$ 300 objects, (i) $\sim$ 130 objects are the reduced $\chi^2 > 2$ objects at $3.5 \leq z_{phot} < 4.5$, (ii) $\sim$ 90 objects are the low-z interlopers at $z_{phot} < 3.0$, and (iii) $\sim$ 80 objects are the slightly lower/higher redshift objects at $3.0 \leq z_{phot} < 3.5$ or $4.5 \leq z_{phot} < 5.0$. First of all, it is reasonable that we exclude the objects in the sub-sample (ii) because they are the possible contamination in our study. On the other hand, the objects in the sub-sample (i) and (iii) are the potential LBGs at $3.5 \leq z_{phot} < 4.5$. We check the influence of the sub-sample (i) and (iii) on the UV slope $\beta$, the UV magnitude, and the other quantities, and consequently we confirm that the $\sim$ 300 objects excluded by the criteria (5) does not change our results and the criteria (5) is reasonable. \[tab2\] For the investigation of photometric redshift and stellar population, we use the Hyperz[^2] photometric redshift code ver.1.1 [@Bolz00] with the @Bruz03 templates[^3] (hereafter BC03) and the STARBURST99[^4] [@Leit99] templates (hereafter SB99). The BC03 templates are chosen as ”typical galaxy” models and constructed from five different star formation histories (SFHs), thirty age values, and three metallicity values with the Chabrier Initial Mass Function (IMF). The SB99 templates are adopted as ”young star-forming galaxy” models and constructed from two SFHs, thirty age values, four metallicity values, and two extreme nebular continuum cases with the Kroupa IMF. In the run of the Hyperz, the dust attenuation value, Av, is ranged from 0.0 to 3.0 with $\Delta \mathrm{Av} = 0.1$ assuming the @Calz00 attenuation law for dust extinction curve. The details of the parameters are summarized in table \[tab2\]. The motivation to use the BC03 and SB99 model templates simultaneously is that the BC03 template is not enough to describe young star-forming galaxies: A time step, which is critical to make spectra of young-age galaxies, in the SB99 computation is much smaller than that in the BC03 computation. Although the ages of the SB99 template set is very young, we consider that our template sets are optimal for fitting the spectrum of young star-forming galaxies. We note that we compare the photometry and the UV slope $\beta$ value used in this work with those of the published catalog. In the UDS-CANDELS field (the small area enclosed with the black dotted line in figure \[fig1\]), a multi-wavelength photometry catalog has been published by the CANDELS collaborators [@Gala13], and the catalog has the total flux of Subaru/*BVRi’z’*, UKIRT/*JHK*, HST/F125WF160W, and Spitzer/3.6$\>\micron$4.5$\>\micron$. Their photometry of the Subaru and UKIRT bands are consistent with ours if we take account of the difference in the measurement method. The UV slope $\beta$ value of our catalog is also comparable to that of the published catalog when applying the same measurement method for UV slope $\beta$ to both catalogs. However, the difference of the HST and Spitzer bands is slightly larger than expected. We consider that for the HST data the difference is attributed to our very large PSF correction factor applied to be matched with the Subaru images. For the Spitzer data the difference is due to the uncertainty in the aperture correction. We remark that our catalog tends to have the slightly bluer *K* $-$ \[3.6\] and *K* $-$ \[4.5\] colors than those of the published one. Example results of SED fitting {#S2s3exs} -------------------------------- We show two examples in figures \[fig2\] and \[fig3\] in order to show the validity of our SED fitting analysis. The first figure is the red LBG observed with Spitzer ($\beta_{\mathrm{obs}} = -1.27$ and $M_{\mathrm{UV,obs}} = -20.38$), and the second figure is the blue LBG observed without Spitzer ($\beta_{\mathrm{obs}} = -2.39$ and $M_{\mathrm{UV,obs}} = -20.32$). In the top nine panels of each figure, we show the stamps of the imaging data from Subaru/*B* to UKIRT/*K*. The center of the images represent the detected position, and the green two straight lines in each image with 1” length are placed at 1” from the detected position. In the bottom left panel, we show the observed photometry and best-fit SED. The blue plus points with the error bar represent the observed photometry and its uncertainty, and the red solid line represents the best-fit SED. The black open circle with the arrow represents the upper limit of the photometry at 2$\sigma$ level. In the bottom right three panels, we show the $\chi^{2}$ map of our SED fitting analysis on one-dimensional space. The vertical axis represents the $\chi^{2}$ value and the horizontal axis represents the photometric redshift, dust attenuation, and age. The horizontal black dashed line in each panel represents the minimum $\chi^{2}$ value. We emphasize that the two examples we show here are categorized as faintest objects, and most of the other objects are fitted better than the examples. Our SED fitting procedure performs well due to the deep imaging data of UKIRT/*HK* which covers the wavelength of Balmer break for LBGs at *z* $\sim$ 4. {width="140mm"} {width="140mm"} Measurement method of UV slope $\beta$ {#S3Mmuvb} ======================================== According to the original definition of @Calz94, the UV slope $\beta$ value should be estimated from the Spectral Energy Distribution (SED) from 1250Å to 2600Å through the 10 fitting windows. However, spectroscopic observation generally requires a long exposure time for measuring the continuum flux from high-z galaxies due to their faint continuum. Furthermore, the number of objects which can be observed at a time is limited depending on slit configuration. Therefore, it is impractical to accurately measure the continuum flux from spectroscopic data for all the $\sim$ 1800 targets in our sample. Instead, we apply the simple power-law fitting to the broad-band photometry with the following functional form, $$M(\lambda_{x}) = -2.5 (\beta + 2) \log \lambda_{x} + Const \label{eq1}$$ where $\lambda_{x}$ is the effective wavelength of $x$th broad-band filter, $M$($\lambda_{x}$) is the measured magnitude of $x$th broad-band filter, and [*Const*]{} is a constant value. This method is suitable since the bias in the $\beta$ estimation is small ([@Fink12]; [@Rog13]). For the fitting, we conduct the least square fitting to the *i’*, *z’*, updated-*z’*, and *J* band filters which cover the rest-frame wavelength from $\sim$ 1500Å to $\sim$ 2500Å for the objects at *z* $=$ 3.5–4.5. In our analysis, an uncertainty of the $\beta$ value represents the uncertainty of the least square fitting when taking account of the photometric error of the broad-band filters. Although using the larger number of the photometry data points results in more accurate determination of the UV slope $\beta$ value, we need to select the optimal broad-band filters for the fitting so as to avoid strong spectral features. In the rest-frame UV and optical wavelength range, the redshifted Ly$\alpha$ ($\lambda$ $=$ 1216Å) line (or Lyman break) and Balmer break ($\lambda$ $\sim$ 3600Å) can be a contamination affecting the broad-band photometry. Figure \[fig4\] illustrates the position of the Lyman and Balmer breaks in the observed-frame, and filter profiles of the optical and NIR broad-band. For the sake of clarity, we show three model spectra with clear Lyman and Balmer breaks at *z* $=$ 3, 4, and 5, and we omit the filter profiles of Subaru/updated-*z’*, HST/F125W, F160W, Spitzer/3.6$\>\micron$, and 4.5$\>\micron$ bands. This figure indicates that the *R* ($\lambda$ $\sim$ 6000–7000Å) and *H* ($\lambda$ $\sim$ 15000-17000Å) band filters are probably affected by Lyman or Blamer breaks, and the wavelength coverage from *i’* to *J* band (black horizontal line in upper panel of figure \[fig4\]) is suitable for calculating the UV slope $\beta$ value of *z* $\sim$ 4 LBGs. Consequently, we use the Subaru/*i’*, *z’*, updated-*z’*, UKIRT/*J*, and HST/F125W band filters. Another critical point is that the robustness of the $\beta$ measurement for the faint and blue galaxies is strongly influenced by the depth of the imaging data at longer wavelength in our analysis. As mentioned above, we select the $i' \leq 26.0$ LBGs, and we use the Subaru/*i’*, *z’*, updated-*z’*, UKIRT/*J*, and HST/F125W band filters for the $\beta$ measurement. Under the condition, for instance, the galaxies with $i' = 26.0$ and $\beta < -2.0$ have the larger photometric uncertainty in the *z’*, updated-*z’*, *J*, and F125W band filters compared with the galaxies with $i' = 26.0$ and $\beta \geq -2.0$, and hence the bluer galaxies have the larger uncertainty in $\beta$. Moreover, the sample completeness of the extremely blue galaxies such as $\beta \sim -3.0$ is also influenced by the depth of the imaging data although the extremely blue objects are rare. From equation \[eq1\], the case of $\beta = -2.0$ is derived from $M(\lambda_{x}) = Const$, namely, all the broad-band photometry is same. For the appropriate $\beta$ measurement with the wide $\beta$ range, it is required that the magnitude threshold of *i’* band is brighter than the $\sim$2–3$\sigma$ limiting magnitude of the *z’*, updated-*z’*, *J*, and F125W bands. Our selection criteria described in section \[S2s2ss\] is applied by taking the aspect into consideration. We use the conservative criteria and consider that our selection does not cause the strong bias in the $\beta$ distribution except for the extremely faint and blue objects. In order to quantify and discuss the influence, we estimate the recovery fraction in section \[S4s2RF\]. ![ Model spectra and important broad-band filters. In the top panel, the green, blue, and red solid lines represent the model spectrum at *z* $=$ 3, 4, and 5, respectively, which clearly show the Lyman and Balmer breaks. The length of the black horizontal line denotes the wavelength coverage of Subaru/*i’*, Subaru/*z’*, and UKIRT/*J* band filters which are used for calculating the UV slope $\beta$. In the bottom panel, we show the Subaru/*BVRi’z’* and UKIRT/*JHK* band filters from left to right. For the sake of clarity, the filter responses of UKIRT/*JHK* are multiplied by an arbitrary constant. []{data-label="fig4"}](Pic.speccomp5-multi.eps){width="80mm"} Results {#S4Rslt} ======= ![ Observed UV slope $\beta$ vs. UV absolute magnitude at rest-frame 1500Å ($\beta$–$M_{\mathrm{UV}}$ relation). The UV absolute magnitude is calculated by integrating the flux of the best-fit SED model template from rest-frame 1450Å to 1550Å. The green filled circles represent the individual objects in the area of the SEDS (Spitzer field) and the blue filled squares represent the individual objects out of the area. The magenta open circles with the error bar represent the mean UV slope $\beta$ value, standard deviation (left side), and mean error (right side) of the UV slope $\beta$ for each magnitude bin, and they are summarized in table \[tab3\]. []{data-label="fig5"}](Pic2-Official.SEDSBetaMuv+1sig-v1cal.v1BobsMobs.eps){width="80mm"} Relation between $\beta$ and $M_{UV}$ {#S4S1BM} ------------------------------------- Figure \[fig5\] shows the obtained distribution of the UV slope $\beta$ as a function of the UV absolute magnitude at rest-frame 1500Å ($\beta$–$M_{\mathrm{UV}}$ relation). The UV absolute magnitude for each object is calculated from the best-fit SED model template by integrating the flux from rest-frame 1450Å to 1550Å. The green filled circles show the objects in the area observed with Spitzer and the blue ones show the objects out of the area. There seems to be no notable systematic difference in the distribution of the sample with and without the Spitzer data, and the lack of the information about 3.6$\>\micron$ and 4.5$\>\micron$ does not influence the selection for *z* $\sim$ 4 star-forming galaxies very much. In fact, we conduct the Kolmogorov–Smirnov test (K–S test) for the whole sample and some magnitude sub-samples. The null hypothesis of the K–S test is that the samples with (green) and without (blue) SEDS/Spitzer data are derived from the same distribution. The p-values of all the test are $>> 5\,\%$, and the null hypothesis is not rejected at 5$\%$ significance level. Therefore, there is no evidence that the information about 3.6$\>\micron$ and 4.5$\>\micron$ influences the $\beta$–$M_{\mathrm{UV}}$ relation. The magenta open circles with the error bar indicate the mean $\beta$ value and the mean $M_{\mathrm{UV}}$ value for each magnitude bin. The standard deviation of the $\beta$ distribution is indicated by the thick marks toward the left side, and the typical uncertainty in the $\beta$ value for the individual objects is shown by the thick marks toward the right side. For the mean values, we just apply the simple geometric mean without taking account of the individual uncertainty in $\beta$ for the individual objects. This is because the mean $\beta$ value can be biased toward positive values if we weight the $\beta$ values by the individual uncertainty of the objects, i.e., the uncertainty is not symmetric and becomes smaller toward positive $\beta$ values than the opposite. The mean $\beta$ value, standard deviation, typical uncertainty, and mean $M_{\mathrm{UV}}$ value for each bin are summarized in table \[tab3\]. In addition, the other useful information such as the median values is also summarized. \[tab3\] The standard deviation is clearly larger than the typical uncertainty in the $\beta$ value except for the two of the most faint magnitude bin. Therefore, in the magnitude range of $M_{\mathrm{UV}} \lesssim -20.5$, the observed $\beta$ distribution is more scattered than the typical uncertainty and the observed scatter represents the variation of the stellar population and dust extinction among the sample. In the magnitude range of $M_{\mathrm{UV}} \gtrsim -20.5$, the typical uncertainty in $\beta$ becomes as large as the standard deviation, and particularly the objects with $M_{\mathrm{UV}} \gtrsim -20.0$ are not uniformly distributed. Although we does not add any criteria to our sample since our purpose is to investigate the bright LBGs with *i’* $\leq$ 26.0, we check the sample completeness in section \[S4s2RF\] and discuss the results with some $M_{\mathrm{UV}}$ sub-samples. The mean $\beta$ value seems not to decrease with the mean UV magnitude. In order to quantify the $\beta$–$M_{\mathrm{UV}}$ trend, we conduct the least-square linear fitting for our mean $\beta$ and $M_{\mathrm{UV}}$ values described in table \[tab3\]. We use the four data points in $-22.0 \leq M_{\mathrm{UV}} \leq -20.0$, and the slope of the fitted linear equation becomes $-0.02\,\pm\,0.02$, which is nearly zero compared with the value in the previous works for the similar redshift, $-0.13\,\pm\,0.02$ [@Bouw14] and $-0.10\,\pm\,0.03$ [@Kurc14]. We note that our target is the relatively bright LBGs and the dynamic range of $M_{\mathrm{UV}}$ in our study is smaller than that in the previous works, $-22.0 \leq M_{\mathrm{UV}} \leq -15.5$ [@Bouw14] and $-21.0 \leq M_{\mathrm{UV}} \leq -15.0$ [@Kurc14]. When calculating the slope of the $\beta$–$M_{\mathrm{UV}}$ relation, we use *standard deviation of the mean* as an uncertainty of each mean $\beta$ value. The uncertainty is calculated from the standard deviation divided by the number of galaxies in each bin ($= \sigma_{\beta} / N_{\mathrm{obj}}$) described in table \[tab3\], and thus the uncertainty of the slope of the $\beta$–$M_{\mathrm{UV}}$ relation is much smaller than the mean error of $\beta$. In figure \[fig6\], we show the direct comparison of our result to the results from *z* $\sim$ 4 super luminous stacked LBGs ([@LeeKS11], red diamonds) and *z* $\sim$ 4 faint LBGs ([@Bouw14], green triangles). The red data points are calculated by us from the photometry described in @LeeKS11 since all the $\beta$ values and its uncertainties are not described in their paper. The error bars denotes the typical uncertainty in the $\beta$ value. The error bars of @Bouw14 show the sum of the random and systematic error described in their paper. The blue circles with the error bars show our result which is same as the magenta points in figure \[fig5\] but the error bars are the typical uncertainty. In the magnitude range of $-22.0 \leq M_{\mathrm{UV}} \leq -20.0$, our results consistent with the previous works. Although the dynamic range of $M_{\mathrm{UV}}$ is smaller than the previous works, the luminous star-forming galaxies at *z* $\sim$ 4, which are selected from the ground-based wide field images, seem to show the weaker $\beta$–$M_{\mathrm{UV}}$ relation over the magnitude range of $-22.0 \leq M_{\mathrm{UV}} \leq -20.0$. ![ Comparison of this work with the previous studies at the similar redshift. For the sake of clarity, the vertical scale is changed from figure \[fig5\]. The red open diamonds and green open triangle with the error bars show the results from @LeeKS11 and @Bouw14, respectively. The blue open circles with the error bars show the result from our sample which is same as the magenta points in figure \[fig5\], but the error bars are the typical uncertainty. []{data-label="fig6"}](Pic.liter-comp-new1812.v1.eps){width="80mm"} It seems that the distribution of the objects in figure \[fig5\] is truncated and the shape of the distribution looks like a “triangle”. This must be a result from either some physical constraint or some sample selection bias, or both. For example, a galaxy at *z* $=$ 4.5 with the *i’*-band magnitude *i’* $= 26.0$ has $M_{UV} \lesssim -20.1$, and therefore the number of objects, which are selected from our selection criteria, decrease in the region around $M_{UV} \sim -20$ and $\beta \sim -2.5$. By using only our three data points in $-22.0 \leq M_{\mathrm{UV}} \leq -20.5$, the slope of the relation indeed becomes $-0.09\,\pm\,0.04$ which is quite similar to the value of the previous works. However, the number of the objects in the brightest (most left) bin is much less than the other bins (see column 8 in table \[tab3\]), and the slope is almost estimated from only the two data points. We need to assess the incompleteness of the observed $\beta$ distribution in order to take account of our selection bias, which is discussed in the next section. Recovery Fraction {#S4s2RF} ----------------- As shown in figure \[fig5\], the observed $\beta$ distribution is restricted in the “triangle” zone. It seems that there are three truncations, namely, (a) at the top left side, (b) at the bottom left side, and (c) at the bottom right side. In order to discuss the reason of those truncation and evaluate the validity of our results, we calculate the recovery fraction which is the number ratio of recovered objects to input objects by using Monte Carlo method. At first, we make a uniform input distribution on $\beta$–$M_{\mathrm{UV}}$ space for the quantitative discussion. For this purpose we consider the $8 \times 13 = 104$ grids with $\Delta \beta = 0.5$ and $\Delta M_{UV} = 0.25$, and we generate 300 mock galaxy spectra whose $\beta$ and $M_{UV}$ values place in each the small grid (so the total spectra are $104 \times 300 = 31,200$). The mock spectra are constructed from the BC03 or SB99 model templates which are the similar template sets as described in section \[S2s2ss\]. All of the parameters such as SFH, dust attenuation value, age, metallicity, and source redshift are determined at random. We note that the range of the dust attenuation value and source redshift for the mock galaxies are different from the range described in section \[S2s2ss\], and they are $0.0 \leq {\rm Av} \leq 1.5$ and $3.5 \leq z_{s} \leq 4.5$. The number of the age step is also changed from 30 into 15. If a resultant spectrum does not place in the designated small grid, we repeat the procedure until the desired $\beta$ and $M_{\mathrm{UV}}$ values. Second, we calculate the apparent magnitude of the broad-band filters for each mock spectrum and we put the artificial galaxies on the real observed images from Subaru/*B* to UKIRT/*K* by using the IRAF mkobjects task. Since we check that the impact of Spitzer/3.6$\>\micron$ and 4.5$\>\micron$ is negligible for the $\beta$–$M_{\mathrm{UV}}$ relation in section \[S4S1BM\], we omit both the information in our simulation. The size and shape of the mock galaxies are also determined at random so that the size distribution of our simulated objects reproduces the observed size distribution. Finally these embedded mock galaxies are re-detected, re-measured, and re-compiled by the same manners described in section \[S2s2ss\]. In the SED fitting procedure, however, we change the number of the age step from 30 to 15 in order to save the computer resource. After the compilation, we count the number of final recovered objects for each small grid and calculate the number ratio of recovered to input objects. The final result includes the impact from the image quality, the magnitude criteria, the LBG selection, and the photo-*z* selection. Note that the prepared objects are restricted by only the rest-frame [*UV*]{} information and thus the rest-frame [*optical*]{} information such as Balmer break is purely determined at random. ![ Recovery fraction which is the number ratio of recovered to input objects on $\beta$–$M_{\mathrm{UV}}$ space. The black grid lines represent the area where we prepare the input objects uniformly throughout the $\beta$–$M_{\mathrm{UV}}$ space and a total of 31,200 mock galaxies is distributed. The colored area represents the detected area where we can find recovered objects. The white colored area represents the non-detection area where we cannot find any recovered objects. []{data-label="fig7"}](Pic-Official.BC5.BetaRF.v3.eps){width="80mm"} Figure \[fig7\] shows the final recovery fraction map by the color-coded area. The vertical axis is the UV slope $\beta$ and the horizontal axis is the absolute magnitude at rest-frame 1500Å. Although the UV absolute magnitude for input objects is given as total magnitude, the UV absolute magnitude for recovered objects is calculated from 2”-diameter aperture photometry. Therefore we convert the total magnitude of input objects to the 2” aperture magnitude by the aperture correction: $M_{\mathrm{UV,aperture}} = M_{\mathrm{UV,total}} + 0.352$. The black lattice lines indicate each area where $\sim$ 300 mock galaxies (or input objects) are distributed except for both the faintest and brightest magnitude bins where $\sim$ 150 mock galaxies are distributed. The white area represents the non-detection area which means that there are no recovered objects. We find that the relative value of the recovery fraction is roughly homogeneous over the area where the observed objects are distributed ($-2.5 \lesssim \beta \lesssim -0.5$ and $-22.0 \lesssim M_{\mathrm{UV}} \lesssim -20.0$) and does not depend on the UV magnitude except for the area around the truncation (c). The rough homogeneity of the relative value indicates that the observed $\beta$–$M_{\mathrm{UV}}$ distribution is not strongly biased at least over the area of $-2.5 \leq \beta \leq -0.5$ and $-22.0 \leq M_{\mathrm{UV}} \leq -20.0$, and our measurement for the $\beta$–$M_{\mathrm{UV}}$ relation described in section \[S4S1BM\] is reasonable. In other words, we find no evidence that the truncation (a) and (b) are artificial, and they must be caused by some other reasons. On the other hand, this figure also shows that the truncation (c) is artificially caused by our sample selection. Our simulation indicates that the truncation (c) is attributed to the selection criteria of the detection in Subaru/*z* or Subaru/updated-*z’*, and UKIRT/*J* or HST/F125W. The figure also indicates that the recovery fraction locally peaks around $\beta \sim -2.0$ and there are some fluctuated peaks at $\beta \sim -0.25$. Qualitatively, prominent spectral features can be easily identified by the SED fitting procedure, and then the recovery fraction may become higher than the other $\beta$ values. Since the Lyman Break technique prefers to select blue galaxies such as $\beta \sim -2.0$, our simulation indeed reflects our sample selection rather than the assumption about input objects. In addition, red galaxies such as $\beta \sim -0.25$ have clear Balmer Break if their red color is due to the aged stellar population. In our simulation the rest-frame optical information is determined at random, and hence the too many input galaxies, which have the prominent Balmer Break, are probably generated to have $\beta \sim -0.25$. In conclusion, the inhomogeneity of the recovery fraction seen in figure \[fig7\] is due to our sample selection criteria adopting the Lyman Break technique and photo-*z* estimation. The truncation (a) and (b) can be interpreted as follows: The observed number of LBGs decreases toward the brighter UV magnitude and the average $\beta$ value converges in $\beta \sim -1.7$. The decrease of LBGs along with the UV magnitude is explained by the drop of UV Luminosity Function since the characteristic luminosity of *z* $\sim$ 4 LBGs is $M_{\mathrm{UV}}^{*} = -21.14$ [@Yoshi06]. However, it is not clear why the average UV slope $\beta$ value converges in $\beta \sim -1.7$. Qualitatively the galaxies with $\beta \gtrsim -1.7$ should contain a large amount of dust and their UV magnitude becomes fainter due to the dust obscuration. Therefore the red and bright galaxies are a rare or almost impossible population, and it causes the truncation (a). On the other hand, the galaxies with $\beta \lesssim -1.7$ contain a less amount of dust and the galaxies can remain bright UV magnitude. As we cannot find the blue and bright galaxies from figure \[fig5\], such the objects are indeed a rare population in the observational data. In summary, we conclude that the truncation (a) and (b) are not only caused by our sample selection and are most likely caused by some physical requirements, and the truncation (c) is clearly caused by our sample selection. In order to understand what make the blue and bright galaxies rare and to reveal the reason of the truncation (b), we discuss the underlying stellar population of LBGs for our sample in section \[S5Dis\]. Discussion {#S5Dis} ========== Relation between Intrinsic $\beta$ and Intrinsic $M_{\mathrm{UV}}$ {#S5s1IBM} ------------------------------------------------------------------ We here estimate the dust-corrected $\beta$ (hereafter we call it [*intrinsic*]{} UV slope, $\beta_{\mathrm{int}}$) and the dust-corrected $M_{\mathrm{UV}}$ (hereafter we call it [*intrinsic*]{} UV absolute magnitude, $M_{\mathrm{UV,int}}$). In section \[S4s2RF\], our simulation indicates that the observed distribution on $\beta$–$M_{\mathrm{UV}}$ space is caused by some physical reasons. Both observed $\beta$ and $M_{\mathrm{UV}}$ value strongly depend on the dust attenuation value, and hence it is helpful to investigate the $\beta$–$M_{\mathrm{UV}}$ distribution before the dust reddening. In our discussion, we assume that the reasonable best-fit physical quantities are estimated from the SED fitting analysis in which the observed photometry covers the wavelength range between rest-frame $\sim$ 900Å and $\sim$ 4400Å (or $\sim$ 9000Å in part) for *z* $\sim$ 4 objects. {width="140mm"} We calculate the intrinsic UV slope by equation \[eq1\] for the intrinsic magnitude of Subaru/*i’*, Subaru/updated-*z’*, and UKIRT/*J* band filters. For estimating the intrinsic magnitude, we convolve the intrinsic SED, which is reproduced with the best-fit physical quantities without any dust extinction, with the three broad-band filters. We note that the intrinsic UV slope depends on the prepared model templates in the SED fitting (i.e., SFH, age, and metallicity) and has discrete values in out discussion. Figure \[fig8\] shows the conversion from observed to intrinsic value for $\beta$ and $M_{\mathrm{UV}}$. The left panel (a) of figure \[fig8\] shows the *observed* $\beta$ as a function of the *observed* $M_{\mathrm{UV}}$ ($\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation, same as figure \[fig4\]). The right panel (b) of figure \[fig8\] shows the *intrinsic* $\beta$ as a function of the *intrinsic* $M_{\mathrm{UV}}$ ($\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation). The blue filled circles, green filled triangles, and red filled squares represent individual objects with the best-fit dust attenuation value of Av $<$ 0.5, 0.5 $\leq$ Av $<$ 1.0, and Av $\geq$ 1.0, respectively. In the panel (a), we confirm that the objects with the higher dust attenuation value are distributed at the upper area where the $\beta_{\mathrm{obs}}$ value becomes redder. This trend is natural and is not inconsistent with the previous studies reported as the relation between the $\beta_{\mathrm{obs}}$ and dust attenuation value (IRX–$\beta$ relation: e.g., [@Calz94]; [@Meur99]; [@Take12]). In the panel (b), due to the large dust correction, the objects with the higher dust attenuation value and the redder $\beta_{\mathrm{obs}}$ value tend to be distributed at the bottom left area where the $\beta_{\mathrm{int}}$ and $M_{\mathrm{UV,int}}$ value becomes bluer and brighter. Moreover, the trend of the distribution is different from the panel (a), namely, the slope of the $\beta$–$M_{\mathrm{UV}}$ relation is nearly constant or positive. We discuss this distribution for different sub-samples in the following. Both two panels in figure \[fig910\] shows the same $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ distribution but the color-coding represent the different sub-samples classified according to with and without SEDS/Spitzer (left) and $M_{\mathrm{UV,obs}}$ (right). In the left panel, the red filled squares and blue filled circles indicate the objects with and without SEDS/Spitzer data, respectively. In the right panel, the blue filled circles, green filled triangles, and red filed squares indicate the objects with $M_{\mathrm{UV,obs}} > -20.5$, $-20.5 \geq M_{\mathrm{UV,obs}} > -21.0$, and $M_{\mathrm{UV,obs}} < -21.0$, respectively. The large open circle, triangle, and square with the error bars in each panel represent the median value and the median uncertainty for each sub-sample. We calculate the dust attenuation value at $\chi^{2}_{min} + 1$ for the individual objects as the uncertainty of the dust attenuation, and then the uncertainty in Av is converted into the uncertainty in $\beta_{\mathrm{int}}$ and $M_{\mathrm{UV,int}}$. Therefore, the error bars in figure \[fig910\] denote the uncertainty in Av. We also show the histogram of $\beta_{\mathrm{int}}$ and $M_{\mathrm{UV,int}}$ for each sub-sample. {width="70mm"} {width="70mm"} From figure \[fig910\], there seems not to be systematic difference in the $\beta_{\mathrm{int}}$ distribution. From the left panel, we find that the information of the Spitzer data does not influence the estimation of the $\beta_{\mathrm{int}}$ value although the uncertainty in $\beta_{\mathrm{int}}$ and $M_{\mathrm{UV,int}}$ for the sub-sample with Spitzer tends to be smaller than that for the sub-sample without Spitzer. From the right panel, we find that the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ distribution of each $M_{\mathrm{UV,obs}}$ sub-sample is almost parallel to each other. When calculating the slope of the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation for each sub-sample by the same manner described in section \[S4S1BM\], we obtain the value of $0.12 \pm 0.01$ ($M_{\mathrm{UV,obs}} > -20.5$), $0.14 \pm 0.01$ ($-20.5 \geq M_{\mathrm{UV,obs}} > -21.0$), and $0.16 \pm 0.02$ ($M_{\mathrm{UV,obs}} < -21.0$). It means that the variation of the $M_{\mathrm{UV,obs}}$ value does not significantly affect the shape of the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ distribution. Although the faint objects such as the objects with $M_{\mathrm{UV,obs}} > -20.5$ have the $\beta_{\mathrm{obs}}$ value with the large uncertainty, the $\beta_{\mathrm{int}}$ values are reasonably well evaluated after the SED fitting using the all photometric data points. {width="140mm"} Figure \[fig11\] shows the same $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ distribution as figure \[fig910\] but the color-coding represent the sub-samples classified according to Av and $\beta_{\mathrm{obs}}$. In the left panel, the blue filled circles, green filled triangles, and red filled squares indicate the objects with Av $<$ 0.5, 0.5 $\leq$ Av $<$ 1.0, and Av $\geq$ 1.0, respectively. In the right panel, the blue filled circles and red filled squares indicate the objects with $\beta_{\mathrm{obs}} \leq -1.73$ and $\beta_{\mathrm{obs}} > -1.73$, respectively. $\beta_{\mathrm{obs}} = -1.73$ represents the median $\beta_{\mathrm{obs}}$ value for our whole sample. The open circle, triangle, and square with the error bars in each panel represent the median value and the median uncertainty for each sub-sample. As mentioned above, the uncertainty in $\beta_{\mathrm{int}}$ and $M_{\mathrm{UV,int}}$ is estimated from the uncertainty in Av. Figure \[fig11\], interestingly, shows that the objects which are dusty and redder in the observed $\beta$ tend to be bluer in the intrinsic $\beta$ and brighter in the intrinsic $M_{\mathrm{UV}}$. In addition, the intrinsic $\beta$ value slightly increases with the intrinsic $M_{\mathrm{UV}}$ value and the trend is opposite of that of the $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation. Our result can be interpreted as follows. The more intense ongoing star-forming galaxies, whose [*intrinsic*]{} $\beta$ and $M_{\mathrm{UV}}$ value are bluer and brighter, generate and/or contain a large amount of dust, and the [*observed*]{} $\beta$ and $M_{\mathrm{UV}}$ value result in a redder and fainter value due to the dust attenuation. Then, the nearly constant $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ distribution observed in our analysis is formed by the galaxies which have a blue $\beta_{\mathrm{int}}$ and bright $M_{\mathrm{UV,int}}$ value because they are distributed at the area of a red $\beta_{\mathrm{obs}}$ and faint $M_{\mathrm{UV,obs}}$ value. According to our SED fitting analysis, a young-age stellar population is responsible for the bluest $\beta_{\mathrm{int}}$ value. In other words, there are some young-age galaxies with the bluest $\beta_{\mathrm{int}}$ value in the brightest $M_{\mathrm{UV,int}}$ range, but there are no intermediate-age and old-age galaxies with the bluest $\beta_{\mathrm{int}}$ value in the brightest $M_{\mathrm{UV,int}}$ range. This is not surprising because the intrinsic UV luminosity is expected to be sensitive to the age of the stellar population. It is hard to sustain a very high star formation rate with the intermediate and long time duration due to rapid gas depletion. The UV luminosity is dominated by the stars at ”turn-off point” on Hertzsprung–Russell Diagram which is an age indicator of the young stellar population. We, however, emphasize that other parameters such as metallicity and/or IMF can also explain the reason of the bluest $\beta_{\mathrm{int}}$ value. Indeed some literature argue that dusty star-forming galaxies have a ”top-heavy” IMF although the discussion still continues ([@Bau05], [@Tacc08], [@Bas10]). Under the top-heavy IMF environment, hot and massive stars can be formed more and more, and the bluer $\beta_{\mathrm{int}}$ value is easily produced. Otherwise, among the galaxies with the bluest $\beta_{\mathrm{int}}$ value, there may be a post-primordial starburst which is dominated by extremely metal-poor (or PopIII) stars. In order to investigate the star formation activity, we plot the Star Formation Rate (SFR) of the individual objects as a function of their stellar mass in figure \[fig12\]. For the estimation of SFR, we convolve the best-fit template with the GALEX/FUV filter response curve and use the calibration for FUV luminosity ([@Hao11]; [@Kenn12]). For estimating the stellar mass, we multiply the best-fit normalization factor to the output from the BC03 model template. Figure \[fig12\] shows the dust-corrected SFR as a function of the stellar mass, and the blue circle, green triangle, and red squares represent the individual objects with the dust attenuation value of Av $<$ 0.5, 0.5 $\leq$ Av $<$ 1.0, and Av $\geq$ 1.0, respectively. The large open circle, triangle, and square with the error bars show the median value and median uncertainty of each sub-sample. The uncertainty in SFR is estimated from the uncertainty in Av, and the uncertainty in stellar mass is estimated from the photometric uncertainty of *K* band since the estimation of the stellar mass is almost determined by the *K* band photometry. We also plot the previous results of the SFR–$\mathrm{M_{*}}$ relation called as main-sequence of star forming galaxies at *z* $\sim$ 4 ([@Spe14], [@Ste14], [@Cap17]). ![ Dust-corrected SFR vs. stellar mass estimated from the BC03 model templates. The SFR is estimated from the luminosity at the *GALEX*/FUV filter by using the @Hao11 calibration. The stellar mass is calculated multiplying the normalization factor to the output of the BC03 model. The color-coding represents the best-fit dust attenuation value. We also draw the SFR–$\mathrm{M_{*}}$ relation at *z* $\sim$ 4 reported by the previous works ([@Spe14], [@Ste14], [@Cap17]). []{data-label="fig12"}](Pic.SFRMs-new1812.v2AvSingle.eps){width="80mm"} The figure shows that the most intense star-forming galaxies in our sample have SFR $\gtrsim$ a few $\times\ 10^{2}\,\mathrm{M_{\solar}}\>\mathrm{yr^{-1}}$, and they are the objects with Av $\geq$ 1.0. Since most of the objects with Av $\geq$ 1.0 have $\beta_{\mathrm{obs}} > -1.73$ and $\beta_{\mathrm{int}} \leq -2.42$ ($\beta_{\mathrm{int}} = -2.42$ is the median $\beta_{\mathrm{int}}$ value for our whole sample) from figure \[fig11\], our analysis indicates that the highly dust-attenuated and intense star-forming galaxies at *z* $\sim$ 4 tend to have $\beta_{\mathrm{obs}} > -1.73$ and $\beta_{\mathrm{int}} \leq -2.42$. When comparing our result with the previous works, our median value of Av $<$ 0.5 and 0.5 $\leq$ Av $<$ 1.0 sub-sample is consistent with the relation from the previous works although the distribution of our sample is significantly scattered. The sub-sample with Av $\geq$ 1.0 tends to be distributed above the relation from the previous works, and the deviation of the median value from the relation is larger than the uncertainty in SFR. Because the galaxies distributed above the star formation main sequence are classified as starburst phase (e.g., [@Cap17]; [@Bisi18]), we consider that the objects with Av $\geq$ 1.0 are indeed in the starburst phase and our results is not inconsistent with the previous works. In conclusion, we find some highly dust-attenuated (Av $\geq$ 1.0) and intense star-forming (SFR $\gtrsim$ a few $\times\ 10^{2}\,\mathrm{M_{\solar}}\>\mathrm{yr^{-1}}$) galaxies at *z* $\sim$ 4 which have $\beta_{\mathrm{obs}} > -1.73$ and $\beta_{\mathrm{int}} \leq -2.42$. Finally, we consider a simple case in which the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ trend monotonically continues in the fainter magnitude range. According to @Bouw14, the $\beta_{\mathrm{obs}}$ value becomes bluer when the $M_{\mathrm{UV,obs}}$ value becomes fainter, but the slope of the $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation becomes flatter in $M_{\mathrm{UV,obs}} \gtrsim -19.0$. In order to establish both the observed and intrinsic $\beta$–$M_{\mathrm{UV}}$ relation without contradiction, it is expected that the $\beta_{\mathrm{int}}$ value becomes redder and converges to the certain $\beta$ value toward the fainter magnitude range. When we extrapolate the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation faintward below our sample magnitude limit, we will find the intersection point of the observed and intrinsic $\beta$–$M_{\mathrm{UV}}$ relation. Since the dust attenuation value becomes smaller toward the fainter magnitude range along the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation, the intersection point (or convergence point) will represent the position of the appearance of nearly dust-free population. Our $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation shows $\beta_{\mathrm{int}} = 0.61 + 0.14 M_{\mathrm{UV,int}}$ by the same manner described in section \[S4S1BM\], and the $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation from @Bouw14 shows $\beta_{\mathrm{obs}} = -4.39 - 0.13 M_{\mathrm{UV,obs}}$ in $M_{\mathrm{UV,obs}} \leq -18.8$. As a result, both relations intersect at $M_{\mathrm{UV}} = -18.9$ and $\beta = -1.94$ and its point corresponds to the break point of $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation at $M_{\mathrm{UV}} = -18.8$ and $\beta = -1.95$ reported by @Bouw14. Therefore the transition of the $\beta_{\mathrm{obs}}$-$M_{\mathrm{UV,obs}}$ relation around $M_{\mathrm{UV}} \sim -18.8$ indicates that we really see the almost dust-free population in $M_{\mathrm{UV}} > -18.8$, and the apparently bluest star-forming galaxies at *z* $\sim$ 4 distribute around $\beta \sim -2.0$. Case of fixed star formation history and SMC attenuation law {#S5s2fhsmc} -------------------------------------------------------------- In this section and the following sections (section \[S5s3zjkd\] and \[S5s4iasfg\]), we verify our results by using different and somewhat independent ways. We emphasize that these verification is intended not only to check the results from our SED fitting analysis but also to strengthen our suggestion, i.e., we find the dusty and on-going active star-forming galaxies at *z* $\sim$ 4. First of all, we repeat the SED fitting analysis (1) by fixing SFH parameter and (2) by using SMC attenuation law for dust extinction curve from @Pre84 and @Bouch85. Figures \[fig13\] and \[fig14\] show the result of the case (1) and (2), respectively. The figures show the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation, and the fixed SFH parameter or dust extinction curve used in the SED fitting is labeled on the top of each panel. In figure \[fig13\], the first and second rows show the results of the BC03 model template and the third row shows the results of the SB99 model template. In all the panels except for the case of the SMC attenuation law, the blue, green, and red points represent the individual objects with the best-fit dust attenuation value of Av $<$ 0.5, 0.5 $\leq$ Av $<$ 1.0, and Av $\geq$ 1.0, respectively. In the case of the SMC attenuation law, the blue, green, and red points represent the objects with Av $<$ 0.3, 0.3 $\leq$ Av $<$ 0.6, and Av $\geq$ 0.6, respectively. In figure \[fig14\], the large diamonds with the error bars represent the median value and the median uncertainty for each sub-sample. Although the error bars is quite large for the objects with Av $\geq$ 0.6 in the right panel, it is caused from the small number of objects in the sub-sample. {width="140mm"} From figure \[fig13\], we find that the global trend of the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation does not significantly change among SFH parameters, which supports our interpretation described in section \[S5s1IBM\]. We note that the $\beta_{\mathrm{int}}$ value has discrete values and makes discrete sequences, especially in the panel (c). It is attributed to the age step of the prepared model template in the SED fitting, and the more large number of the age step will dilute the discrete sequences. However, it is not critical when taking account of the moderate uncertainty in photometry. In brief, the effect of dust attenuation significantly distorts the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation, which is probably positive, and then the $\beta$–$M_{\mathrm{UV}}$ relation results in the negative $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation reported by the previous works. In $-22 \lesssim M_{\mathrm{UV,obs}} \lesssim -20$, however, the $\beta_{\mathrm{obs}}$ value seems to be constant to the $M_{\mathrm{UV,obs}}$ value (constant $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation) due to the existence of dusty active star-forming population. {width="130mm"} Figure \[fig14\] shows that the best-fit Av value from the SMC attenuation law becomes much smaller than that from the attenuation law because the slope of the dust extinction curve of the SMC is much steeper than that of the . Consequently, we cannot identify the intrinsically active star-forming galaxies which show the high dust attenuation (Av $>$ 1.0), blue $\beta_{\mathrm{int}}$ value ($\beta_{\mathrm{int}} < -2.42$), and red $\beta_{\mathrm{obs}}$ value ($\beta_{\mathrm{obs}} > -1.7$), although we can again find that the intrinsic $\beta$ value slightly increases with the intrinsic $M_{\mathrm{UV}}$ value. Actually, recent works of Atacama Large Millimeter/submillimeter Array (ALMA) observations report that the SMC dust attenuation law is appropriate for normal star forming galaxies at high redshift (e.g., [@Cpk15]; [@Bouw16]). On the other hand, as discussed in section \[S5s4iasfg\], the -like attenuation law is partly required to reproduce the results of the Submillimeter Common User Bolometer Array 2 (SCUBA2) from @Copp15 and @Kopr18. *zJK*-diagram {#S5s3zjkd} --------------- In this section, we compare the observed color of the *z’JK* band photometry with the predicted color which is estimated from the model simulation. Since our sample tends to have a larger photometric error in the broad-band filters at longer wavelength owing to the depth of the imaging data, the weight of the broad-band filters at longer wavelength becomes smaller than the opposite in the SED fitting analysis. It is possible that the photometry of the *z’JHK* band filters does not have a considerable constraint on the best-fit SED. We therefore focus on the observed color of the *z’JK* band photometry, and more directly compare the observed value with the predicted value in the color–color space. For the model simulation, we calculate the color of the two SFH model templates with some condition: One is the BC03 Instantaneous Burst model (hereafter IB), and the other is the BC03 Continuous Constant star formation model (hereafter CSF). We consider that the IB and CSF SFH model is most opposite case in star formation activity, and the models are helpful to interpret the observed results. For the sake of simplicity, we fix the metallicity value with $Z = 0.2 Z_{\solar}$ and the redshift value with *z* $=$ 3.5 and 4.5. In order to clarify the variation of the colors depending on the dust and age, we calculate the colors of the IB and CSF model templates with (a) the fixed age but the variable dust ranging from 0.0 to 3.0, and (b) the variable age ranging from 10$\>\mathrm{Myr}$ to 15.0$\>\mathrm{Gyr}$ but the fixed dust. {width="140mm"} Figure \[fig15\] shows the $z - J$ vs. $J - K$ color–color diagram. The vertical axis is the $z - J$ color and the horizontal axis is the $J - K$ color. The $J - K$ color can trace the Balmer break of galaxies at *z* $\sim$ 4 and the $z - J$ color represents the observed UV slope $\beta$. The blue filled triangles and the red filled circles denote the objects with $\beta_{\mathrm{obs}} \leq -1.73$ and $\beta_{\mathrm{obs}} > -1.73$ in our sample, respectively. In this figure, we only use the objects satisfying all the following condition, $z' > 3\,\sigma$, $J > 3\,\sigma$, and $K > 3\,\sigma$, so as to calculate the reliable colors. The blue and red large circles with the error bars denote the median value and median uncertainty in the observed colors. In the left panel (a), the green lines represent the CSF model template with age $=$ 10$\>\mathrm{Myr}$, and the purple lines represent the IB model template with age $=$ 100$\>\mathrm{Myr}$. The solid and dashed lines represent the case of *z* $=$ 3.5 and 4.5, respectively, and the space between the lines is filled with the shaded area. The solid circles on each line indicate the dust attenuation value of Av $=$ 0.0, 1.0, 2.0, and 3.0 from bottom left to top right, and the given value is labeled beside the circles. In the right panel (b), the green lines represent the CSF model template with Av $=$ 1.0, and the purple lines represent the IB model template with Av $=$ 0.0. The solid and dashed lines represent the case of *z* $=$ 3.5 and 4.5, respectively, and the space between the lines is filled with the shaded area. The solid circles on each line indicate the age value of 10$\>\mathrm{Myr}$, 100$\>\mathrm{Myr}$, 500$\>\mathrm{Myr}$, and 1$\>\mathrm{Gyr}$ from bottom left to top right, and the given value is labeled beside the circles. We note that we omit the label of 100$\>\mathrm{Myr}$ for the green line in the panel (b) since the corresponding point is placed under the median value and cannot be seen. The figure indicates that the observed distribution of the $\beta_{\mathrm{obs}} > -1.73$ sub-sample tends to be reproduced by the star-forming, dusty, and very young-age, that is bluer $\beta_{\mathrm{int}}$, population. Although we only show the extreme and slightly arbitrary cases in the figure, we can deduce the other possibilities from the examples such as star-forming, less dusty, and middle-age population. However, when we take the other possibilities into consideration, the above interpretation is not changed because the direction of the increase in age and dust is different. We consider that the observed $J - K$ color of the $\beta_{\mathrm{obs}} > -1.73$ sub-sample is not sufficiently red, and thus the middle-age and old-age population is not preferred in the SED fitting analysis. The observed distribution of the $\beta_{\mathrm{obs}} \leq -1.73$ sub-sample tends to be reproduced by the less star-forming, less dusty, and young-age population, although the sub-sample can be also reproduced by the star-forming, less dusty, and middle-age population. We note that there are some outliers in our sample, but most of them have a lower signal to noise ratio ($S/N \sim 3$–$5$) in *J* and/or *K* band than the other objects. In summary, the interpretation from the *z’JK* color–color diagram is consistent with the interpretation from our SED fitting analysis, and therefore a part of star-forming galaxies at *z* $\sim$ 4 in our sample is indeed classified as dusty star-forming population. Expected features of most active star-forming galaxies at *z* $\sim$ 4 {#S5s4iasfg} ------------------------------------------------------------------------ Last of this paper, we show two estimation for the IR features of the active star-forming galaxies at *z* $\sim$ 4: One is the luminosity ratio of IR to UV so called IRX, and the other is the flux density at observed-frame 850$\>\micron$, $S_{850}$. Our sample does not have the rest-frame IR information for the individual objects and therefore we use the approximate conversion. For estimating the IRX value, we apply the empirical conversion between ${\rm IRX_{TIR-FUV}}$ and ${\rm A_{FUV}}$ for low-*z* galaxies reported by @Burg05: ${\rm A_{FUV}} = -0.028[{\rm log_{10}}L_{\rm TIR}/L_{\rm FUV}]^3 + 0.392[{\rm log_{10}}L_{\rm TIR}/L_{\rm FUV}]^2 + 1.094[{\rm log_{10}}L_{\rm TIR}/L_{\rm FUV}] + 0.546$. For estimating the $S_{\mathrm{850}}$ value, we first calculate the total (bolometric) IR luminosity by utilizing the not dust-corrected FUV luminosity and the IRX value, and then we convert the total IR Luminosity into the flux density at observed-frame 850$\>\micron$. In the conversion, we use the modified blackbody $+$ power-law formula as the dust thermal emission and the total IR luminosity is estimated by integrating the modeled spectrum from 8$\>\micron$ to 1000$\>\micron$ in the rest-frame. The formula is, $$S (\nu, T_{d}) \propto \left\{ \begin{array}{ll} \frac{\nu^{\beta}\nu^{3}}{e^{h\nu/kT_{d}}-1} & (\nu \leq \nu_{c}); \\ \nu^{-\alpha} & (\nu > \nu_{c}), \end{array} \label{eq2} \right.$$ where $S(\nu, T_{d})$ is the flux density at $\nu$ for a dust temperature $T_{d}$ in the units of Jy and $\beta$ is a dust emissivity index. The connecting frequency, $\nu_{c}$, is calculated from, $$\Biggl. \frac{dS}{d\nu} \Biggr|_{\nu=\nu_{c}} = -\alpha. \label{eq3}$$ For the sake of simplicity, we fix all the above parameters and the source redshift referring to @Copp15: the dust temperature of $T_{d} = 38\>$K, the dust emissivity index of $\beta_{\mathrm{dust}} = 1.5$, the power-law index of $\alpha = 1.7$, and the source redshift of $z = 3.87$. We emphasize that the cautious treatment is required for the comparison between our result and the previous results presented in this paper. {width="52mm"} {width="52mm"} {width="52mm"} Figure \[fig161718\] shows the IRX–$\beta$ relation obtained from the attenuation law (left) and SMC attenuation law (middle). The vertical axis is the predicted IRX value, and the horizontal axis is the observed UV slope $\beta$. The small dots represent each object, and the color-coding is same as the figure \[fig14\]. The large blue, green, and red square with the error bars represent the median value and median uncertainty for each sub-sample. In the right panel, we show the median values of our result and the previous works from (2016: AM16, magenta square), (2017: F17, red triangle), and (2016: B16, orange circle). The sample of AM16 is LBGs at *z* $\sim$ 3 in the COSMOS field and the IR luminosity is obtained from the stacked image of the Herschel and AzTEC. The sample of F17 is massive star-forming galaxies at *z* $\sim$ 3.2 in the COSMOS field, which are distributed on the main-sequence of star formation, and the IR luminosity is obtained from the stacked image of the ALMA. We note that both the samples consist of the relatively more massive ($\mathrm{M_{*} \gtrsim 10^{10} M_{\solar}}$) and lower redshift LBGs compared with our sample. The sample of B16 is LBGs at *z* $=$ 4–10 in the Hubble Ultra Deep Field, and the IR Luminosity is obtained from the stacked image of the ALMA. For B16, the data points in this panel represent the 2$\,\sigma$ upper limit of the formal uncertainty for the $\mathrm{M_{*} < 10^{9.75} M_{\solar}}$ sample described in table 13 of their paper, and thus their sample is relatively less massive (and possibly higher redshift) LBGs compared with our sample. The black solid and dashed lines show the relation based on the attenuation law from @Meur99 and @Take12, respectively. The black dot-dashed line shows the relation based on the SMC attenuation law from @Bouw16. In the case of the attenuation law (left panel), our sample shows the systematically bluer UV slope $\beta$ and the systematic offset becomes larger at the larger IRX value. According to previous works for lower redshift star-forming galaxies (e.g., [@Redd06]; [@Hein13]; [@Oteo13]; [@Alva16]), normal star-forming galaxies are distributed along the IRX–$\beta$ relation, and IR luminous galaxies such as Luminous InfraRed Galaxies (LIRGs; $L_{\mathrm{TIR}} > 10^{11}\,L_{\solar}$) or Ultra Luminous InfraRed Galaxies (ULIRGs; $L_{\mathrm{TIR}} > 10^{12}\,L_{\solar}$) are distributed above the IRX–$\beta$ relation. The offset of our red points implies the presence of IR excess galaxies at at *z* $\sim$ 4 such as local LIRG/ULIRGs although the systematic shift can be attributed to the uncertainty of IRX which comes from the conversion from $A_{\mathrm{FUV}}$ to $\mathrm{IRX_{TIR−-FUV}}$ and/or the failure of the SED fitting analysis. In the case of the SMC attenuation law (middle panel), our sample also shows the systematically bluer UV slope $\beta$ especially at the larger IRX value. Most of our sample, however, show the moderate IRX value (IRX $\leq$ 10), and we find a few IR excess galaxies in our sample. In conclusion, our sample indicates the presence of the IR excess galaxies at *z* $\sim$ 4. When comparing the previous works (right panel in figure \[fig161718\]), our sample from attenuation law tends to have the bluer UV slope $\beta$ at the larger IRX value than all the previous works while our sample from SMC attenuation law is comparable to the those of AM16 and F17. Our results from both attenuation law are not consistent with the result of B16. We note that the difference in the stellar mass of the sample is critical for the IRX–$\beta$ relation since both the IRX and $\beta$ values depend on the stellar mass (e.g., [@Alva16]; [@Bouw16]; [@Fink12]; [@Fuda17]), and we consider that the inconsistency between our results and B16 is attributed to the difference in the stellar mass. The red data point from F17 at $\beta \sim -1.7$ and IRX $\sim$ 10 (most left side point) is comparable to our result from attenuation law although the other data point from F17 is comparable to that from SMC. The authors mention that the most left side point is uncertain because of the small sample size of the bin. Therefore, our result from SMC attenuation law is not inconsistent with the previous works. {width="60mm"} {width="60mm"} For the further verification of our result, figure \[fig1920\] shows the prediction of $S_{\mathrm{850}}$ for the case of the attenuation law (Left) and the SMC attenuation law (Right). The vertical axis is the predicted $S_{850}$ value and the horizontal axis is the predicted IRX value. The blue open diamonds represents the individual objects detected at $> 5\,\sigma$ level in *K* band photometry in our sample. The green filled circle with the orange error bars denotes the median value and median uncertainty of the whole sample. The uncertainty is also estimated from the uncertainty in Av. The horizontal magenta solid line denotes the flux density for the stacked LBG at *z* $\sim$ 4 measured in @Copp15 whose sample is quite similar to our sample. Since the sample in @Copp15 consists of the *K* band detected objects, we only show the $K > 5\,\sigma$ objects in the figure. According to @Copp15, the flux density measured for the stacked image is $S_{\mathrm{850}} = 0.411 \pm 0.064\>$mJy. We note that the result of @Copp15 is derived from the SED template library constructed by @Swin14 and the modified blackbody $+$ power-law formula is used just for checking the validity of their SED fitting analysis. The figure shows that the predicted $S_{\mathrm{850}}$ flux from the SMC attenuation law is insufficient to reproduce the stacking result of @Copp15, but the result from the attenuation law is consistent with the stacking result. This comparison indicates that a part of *z* $\sim$ 4 LBGs are indeed significantly dust attenuated and there must be IR luminous star-forming galaxies in our sample. Alternatively, at least the SMC attenuation law is unsuitable for high-*z* and *K*-detected LBGs. However, the difference between @Copp15 result and ours can be due to the fact that the stacking result is the average weighted by luminosities while our median values are not. Since the red LBGs in our sample can be easily detected and measured by using the ALMA, future ALMA observations for individual detection will potentially solve the discrepancy. We consider the possible interpretation of our optical/NIR-based IRX–$\beta$ relation. The IRX–$\beta$ relation is expressed as $\log_{10} \mathrm{IRX} = \log_{10}(10^{0.4*\mathrm{c1}*(\beta - \beta_{0})} - 1.0) + \mathrm{c2}$ where c1, c2, and $\beta_{0}$ are a constant value. The c1 value is the slope of the relation between dust attenuation Av and the observed UV slope $\beta$, $d \mathrm{Av}/d \beta$, which is specified by the dust extinction curve. The c2 value represents the bolometric correction because the observed UV and IR Luminosity is not the representative value and we need the correction factor for the observed values. The $\beta_{0}$ value is the intrinsic UV slope $\beta$ as investigated in this paper. In short, the IRX–$\beta$ relation assumes that the extinction curve and the stellar population hidden by dust does not vary significantly with the physical quantities of the star-forming galaxies. In the previous works for the IR-based IRX–$\beta$ relation, by using the fixed $\beta_{0}$ value ($\sim -2.2$), the authors discuss the suitable extinction curve for reproducing the IRX–$\beta$ relation seen in high redshift galaxies (e.g., [@Cpk15]; [@Alva16]; [@Bouw16]). @Redd18 explain the the IRX–$\beta$ relation of the *z* $\sim$ 2 galaxies by using the SMC attenuation law and the more bluer $\beta_{0}$ value ($\sim -2.6$), which is derived from the recent stellar population synthesis model. Moreover, @LeeKS12 and @Redd12 discuss the variation of the extinction curve according to the observed UV magnitude and the age of star-forming galaxies. From our analysis, assuming a certain dust extinction curve, the observed properties are not represented by the IRX–$\beta$ relation with the fixed $\beta_{0}$ value, and it is required that there is the variation of the intrinsic $\beta$ value or the variation of the extinction curve, or both, depending on the physical quantities of the star-forming galaxies. The prediction of the $S_{850}$ flux indicates that our sample is expected to include the highly dust attenuated and IR luminous galaxies which are explained by the attenuation law. Therefore, while the less dusty galaxies can be characterized by either attenuation law of and SMC, the highly dust attenuated galaxies are most likely characterized by the attenuation law and the bluer intrinsic $\beta$ value. Although it is difficult to confirm the variation from our results, there seems to be the variation of the intrinsic $\beta$ value or the extinction curve according to the physical quantities of the star-forming galaxies. Conclusion {#S6Conc} ========== In this work, we investigate the UV slope $\beta$ and stellar population of bright star-forming galaxies at *z* $\sim$ 4 in the SXDS field which is the wide-area and deep survey field. We use the imaging data of Subaru/*BVRi’z’*updated-*z’*, UKIRT/*JHK*, HST/F125WF160W, and Spitzer/3.6$\>\micron$ 4.5$\>\micron$, and we construct the sample of star-forming galaxies at *z* $\sim$ 4 by both Lyman Break technique and photometric redshift selection. The UV slope $\beta$ is calculated by the simple power-law fit, and the stellar population is estimated from the optical and NIR photometry thorough the SED fitting analysis. Consequently, we find a sign that some star-forming galaxies, which experience on-going active star formation and suffer heavy dust attenuation, really exist in the *z* $\sim$ 4 universe. We list our main results below. - There seems to be little dependence of the observed UV slope $\beta$ on the observed UV absolute magnitude $M_{\mathrm{UV}}$ in the range of $-22.0 \lesssim M_{\mathrm{UV}} \lesssim -20.0$ although the dynamic range of $M_{\mathrm{UV}}$ is limited. The slope of the $\beta$–$M_{\mathrm{UV}}$ relation is $-$0.02 $\pm$ 0.02, and it is more shallower than the previous studies for similar redshift but fainter LBGs ($-$0.13 $\pm$ 0.02 from [@Bouw14] and $-$0.10 $\pm$ 0.03 from [@Kurc14]). - For investigating the dependence of the UV slope $\beta$ on the dust attenuation, age, metallicity, and SFH, we calculate the *intrinsic* (dust-corrected) UV slope, $\beta_{\mathrm{int}}$, and *intrinsic* UV absolute magnitude, $M_{\mathrm{UV,int}}$, by using the results of the SED fitting analysis. The star-forming galaxies with the bluest $\beta_{\mathrm{int}}$ and brightest $M_{\mathrm{UV,int}}$ value are the dusty star-forming population which is observed with the red $\beta_{\mathrm{obs}}$ value. The dusty star-forming population has $\beta_{\mathrm{obs}} > -1.73$, $\mathrm{Av} \geq 1.0$, $\beta_{\mathrm{int}} \leq -2.42$, and SFR $\gtrsim$ a few $\times\ 10^{2}\,\mathrm{M_{\solar}}\>\mathrm{yr^{-1}}$, and we see the flat $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ distribution due to such population. - We find the intersection point of the $\beta_{\mathrm{int}}$–$M_{\mathrm{UV,int}}$ relation and the $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation by extrapolating our relation toward the fainter magnitude range. The intersection point represents the position of the appearance of nearly dust-free population, and it is at $\beta = -1.94$ and $M_{UV} = -18.88$ which is close to the break point of $\beta_{\mathrm{obs}}$–$M_{\mathrm{UV,obs}}$ relation reported by @Bouw14. - Our result does not depend on the SFHs used in the SED fitting analysis. However, our result depends on the assumption of the attenuation law. The best-fit dust attenuation value assuming the SMC attenuation law is found to be smaller than that obtained with the attenuation law. The trend that the intrinsic $\beta$ value increases with the intrinsic $M_{\mathrm{UV,int}}$ value appears for both the cases. - We compare the observed color of the *zJK* broad-band filters with the expected colors. Since the *z-J* color traces the UV slope $\beta$ and the *J-K* color traces the Balmer break of *z* $\sim$ 4 LBGs, we can also infer the stellar population by the observed quantities. The observed color of the $\beta_{\mathrm{obs}} > -1.73$ sub-sample of the *z* $~$ 4 star-forming galaxies is well reproduced by star-forming, dusty, and young-age (blue $\beta_{\mathrm{int}}$) population. - We estimate the IRX ($= L_{\mathrm{TIR}}/L_{\mathrm{FUV}}$) value and the flux density at observed-frame 850$\>\micron$, $S_{\mathrm{850}}$, from only the optical and NIR imaging data. The optical/NIR-based IRX–$\beta$ relation indicates the variation of the intrinsic $\beta$ value or the variation of the dust attenuation law, or both, according to the physical quantities of the star-forming galaxies. The $S_{\mathrm{850}}$ value estimated from the SMC attenuation law is not consistent with the stacking results of @Copp15, and thus the attenuation law is preferable to the *z* $\sim$ 4 intrinsically luminous LBGs. - Our analysis indicates that a significant fraction of *z* $\sim$ 4 LBGs are the highly dust attenuated and IR luminous population such as ULIRGs/LIRGs. This population has not been recognized very well in the previous analysis but is important in understanding early phase of galaxy formation possibly linking the typical blue LBGs and the further very red sub-mm selected galaxies. We appreciate M. Kajisawa, A. Inoue, K. mawatari, and T. Hashimoto for helpful comments and discussions. This work is mainly based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. The UKIDSS project is defined in @Law07. UKIDSS uses the UKIRT Wide Field Camera (WFCAM; [@Cas07]). The photometric system is described in @Hew06, and the calibration is described in @Hodg09. The pipeline processing and science archive are described in Irwin et al (2009, in prep) and @Hamb08. We used UKIDSS data release 10. This work is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Data analysis was in part carried out on the open use data analysis computer system at the Astronomy Data Center, ADC, of the National Astronomical Observatory of Japan. We used the interactive analysis servers (anam\[01-16\]), the batch processing servers (bapm\[01-06\]), the terminal workstations (new-r\[01-13\]), and the disk space (home and mfst). This work was supported by JSPS KAKENHI Grant Number JP26400217. lvarez-M[á]{}rquez, J., Burgarella, D., Heinis, S., et al. 2016, , 587, A122 Ashby, M. L. N., Willner, S. P., Fazio, G. G., et al. 2013, , 769, 80 Baugh, C. 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--- abstract: 'In the paper, we have discussed the problem of the existence of the metallic hydrogen in the superconducting state inside the cold giant planet. We have shown that the [*cold*]{} planet represents the final state of the thermal evolution of free-floating giant planet.' author: - '*R. Szcz[ȩ]{}[s]{}niak$^{1}$, M.W. Jarosik$^{1}$, M. Szcz[ȩ]{}[s]{}niak$^{2}$, A. Durajski$^{1}$*' title: 'The Final State of the Thermal Evolution of Free-Floating Giant Planet' --- [ $^{1}$Institute of Physics, Cz[ȩ]{}stochowa University of Technology, Al. Armii Krajowej 19, 42-200 Cz[ȩ]{}stochowa, Poland\ $^{2}$Ul. S[ł]{}oneczna 44, 34-700 Rabka-Zdr[ó]{}j, Poland\ [email protected]]{} Introduction {#introduction .unnumbered} ============ In the year of $1935$ Wigner and Huntington predicted that under the influence of high pressure ($p$) the molecular phase of solid hydrogen passes into a metallic phase [@Wigner]. Recently, the theoretical calculations fixed the metallization pressure for the solid hydrogen at $400$ GPa [@Stadele]. In the case of the liquid hydrogen, Nellis [*et al.*]{} inform about the observation of the insulator-metal transition for pressure of $140$ GPa and the temperature $3000$ K [@Weir]. This result confirms the fact that the liquid hydrogen in the metallic state exists inside Jupiter and Saturn [@Guillot]. We notice that the hydrogen phase diagram from the planetary science aspect is in detail discussed by Fortney and Hubbard in [@Fortney1]. In the year of 1968 Ashcroft suggested the possibility of occurrence of additional phase transition in the solid hydrogen; from the metallic to the superconducting state [@Ashcroft]. The obtained results concerning superconductivity in the solid metallic hydrogen can shed light on the internal structure of the hypothetical cold giant planets. We define the [*cold*]{} planets as a class of planets whose mass is close to the mass of Saturn or exceeds it. Additionally, they have the characteristics: (i) The [*cold*]{} planets are primarily composed of the hydrogen. (ii) Some part of the solid metallic hydrogen is in the superconducting state. The liquid metallic hydrogen inside the planet like Saturn exists between 200 GPa and 1000 GPa [@Stevenson]. So, only the low-pressure superconducting state one can consider in the solid hydrogen for low values of the temperature. In the model calculations the best choice represents the planet that possessess the Jupiter’s mass, where the liquid metallic hydrogen is induced between 200 GPa and 4000 GPa [@Stevenson]. In the paper we will calculate the maximum surface temperature of the [*cold*]{} Jupiter. We will consider the high- and low-pressure superconducting state inside the planet ($p=2000$ GPa or $450$ GPa). We notice that these states are characterized by the maximum values of the critical temperature ($T_{C}$). Additionally, we will show that [*cold*]{} planet represents the final state of the thermal evolution of free-floating giant planet. The critical temperature for the metallic hydrogen: $p=2000$ GPa {#the-critical-temperature-for-the-metallic-hydrogen-p2000-gpa .unnumbered} ================================================================ The superconducting properties of the metallic hydrogen for $p=2000$ GPa are determined by the Eliashberg equations [@eliashberg]: $$\label{r1} \Delta_lZ_{l}=\frac{\pi}{\beta} \sum_{m=1}^{M} \frac{\left[K^{+}\left(l,m\right)-2\mu\left(m\right)\right]} {\sqrt{\omega_m^2+\Delta_m^2}} \Delta_{m},$$ $$\label{r2} Z_l=1+\frac {\pi}{\beta }\sum_{m=1}^{M} \frac{K^{-}\left(l,m\right)} {\sqrt{\omega_m^2+\Delta_m^2}}\frac{\omega_m}{\omega _l},$$ where $\Delta_{l}\equiv\Delta\left(i\omega_{l}\right)$ and $Z_{l}\equiv Z\left(i\omega_{l}\right)$ are the superconducting order parameter and the wave function renormalization factor respectively. The symbol $\omega_{l}\equiv \left(\pi / \beta\right)\left(2l-1\right)$ denotes the Matsubara frequencies; $\beta\equiv\left(k_{B}T\right)^{-1}$, where $k_{B}$ is the Boltzmann constant. The functions $K^{\pm}\left(l,m\right)$ are defined by the formula: $K^{\pm}\left(l,m\right)\equiv K\left(l-m\right)\pm K\left(l+m-1\right)$, where $K\left(l-m\right)$ is the pairing kernel: $$\label{r3} K\left(l-m\right)\equiv\int_0^{\Omega_{\rm{max}}}d\Omega\frac{\alpha^{2}F\left(\Omega\right) 2\Omega}{\left(\omega_l-\omega_m\right) ^2+\Omega ^2}.$$ The Eliashberg function ($\alpha^{2}F\left(\Omega\right)$) for $p=2000$ GPa was determined in [@Maksimov2]; the maximum phonon frequency ($\Omega_{\rm{max}}$) is equal to $570$ meV. The influence of in-between electron Coulomb interaction on the superconducting state is modeled by the function: $\mu\left(m\right)\equiv \mu^{*}\Theta\left(\omega_{\rm{c}}-\left|\omega_{m}\right|\right)$, where $\mu^{*}$ is the Coulomb pseudopotential. The symbol $\Theta$ denotes the Heaviside unit function and $\omega_{c}$ is the phonon cut-off frequency ($\omega_{c}=3\Omega_{\rm{max}}$). By using the Eliashberg equations we have shown in the paper [@Szczesniak1] that the dependency of the critical temperature on the Coulomb pseudopotential can be precisely parameterized by the [*modified*]{} Allen-Dynes formula [@allen]: $$\label{r4} k_{B}T_{C}=f_{1}f_{2}\frac{\omega_{{\rm ln}}}{1.2}\exp\left[\frac{-1.04\left(1+\lambda\right)} {\lambda-\mu^{*}\left(1+0.62\lambda\right)}\right],$$ where the electron-phonon coupling constant ($\lambda$) and the logarithmic phonon frequency ($\omega_{{\rm ln}}$) are defined by: $$\label{r5} \lambda\equiv 2\int^{\Omega_{\rm{max}}}_{0}d\Omega\frac{\alpha^{2}F\left(\Omega\right)}{\Omega},$$ $$\label{r6} \omega_{{\rm ln}}\equiv \exp\left[\frac{2}{\lambda} \int^{\Omega_{\rm{max}}}_{0}d\Omega\frac{\alpha^{2}F\left(\Omega\right)} {\Omega}\ln\left(\Omega\right)\right].$$ In the considered case $\lambda=7.32$ and $\omega_{{\rm ln}}=89.2$ meV. The functions $f_{1}$, $f_{2}$ have the form [@allen]: $$\label{r7} f_{1}\equiv\left[1+\left(\frac{\lambda}{\Lambda_{1}}\right)^{\frac{3}{2}}\right]^{\frac{1}{3}}, \qquad f_{2}\equiv 1+\frac {\left(\frac{\sqrt{\omega_{2}}}{\omega_{\rm{ln}}}-1\right)\lambda^{2}} {\lambda^{2}+\Lambda^{2}_{2}},$$ where: $$\label{r8} \omega_{2}\equiv \frac{2}{\lambda} \int^{\Omega_{\rm{max}}}_{0}d\Omega\alpha^{2}F\left(\Omega\right)\Omega$$ and $\omega_{2}=25.942$ eV. Finally: $$\label{r9} \Lambda_{1}\equiv 2.41\left(1-1.285\mu^{*}\right), \qquad \Lambda_{2}\equiv 3.15\left(1+5.513\mu^{*}\right)\left(\frac{\sqrt{\omega_{2}}}{\omega_{\rm{ln}}}\right).$$ The value of $T_{C}$ can be easily calculated on the basis of the formula \[r4\]. In particular, for $\mu^{*}\in\left(0.1, 0.5\right)$ the critical temperature changes the value from $631$ K to $413$ K. The characteristic of the [*cold*]{} Jupiter {#the-characteristic-of-the-cold-jupiter .unnumbered} ============================================ Below, we calculate the maximum surface temperature of the [*cold*]{} Jupiter $\left[T_{s}\right]_{\rm{max}}$ (the temperature at the pressure $p_{\rm{s}}$=100 kPa). In the first step, we determine the relationship between the surface temperature and the internal temperature $T_{\rm{in}}$ (the temperature at the pressure $p_{\rm{in}}> p_{\rm{s}}$). By using the adiabatic model we obtain [@Stevenson]: $$\label{r11} T_{\rm{s}}=\left(\frac{p_{\rm{s}}}{p_{\rm{in}}}\right)^{\Gamma} T_{\rm{in}},$$ where the adiabatic index $\Gamma$ equals the Jupiter’s index; $\Gamma =0.275$. The parameter $\Gamma$ has been calculated under the assumption that the surface temperature of the Jupiter equals $170$ K and $T_{\rm{in}}=21000$ K for $p_{\rm{in}}=4000$ GPa [@Stevenson]. Next, for the high-pressure superconducting state we take: $p_{\rm{in}}=2000$ GPa and $T_{C}\in\left(413, 631\right)$. To describe the low-pressure case we use: $p_{\rm{in}}=450$ GPa and $T_{C}=242$ K [@Cudazzo]. We notice that this state is characterized by the maximum value of $T_{C}$ and its properties are obtained in the framework of the recently introduced density functional theory of superconductivity. The dependence of the internal temperature on the surface temperature is shown in the figure \[f1\]. The horizontal lines correspond to the values of the critical temperature. On the basis of the presented results, we conclude that the maximum surface temperature of the [*cold*]{} Jupiter is situated between 4 K and 6.2 K. ![ The dependence of the internal temperature on the surface temperature for considered values of $p_{\rm{in}}$. The horizontal lines denote the values of the critical temperature.[]{data-label="f1"}](durajski1.eps) ![The dependence of the surface temperature on the time.[]{data-label="f2"}](durajski2.eps) The superconducting state inside the Jupiter can not exist because the value of the Jupiter’s surface temperature is too high. This fact is connected with Jupiter’s young age ($4.7$ Gyr) [@Fortney2]. In the last step, we calculate the surface temperature of the old free-floating giant planet. We assume that the planet’s age is equal to $12.7$ Gyr (the age of the oldest known planet) [@Thorsett] and the maximum temperature rises during the formation to $10^{5}$ K [@Stevenson]. The evolutionary model of the giant planet was first presented by Hubbard [@Hubbard]. The thermal history calculations for non-rotating planet of the Jupiter’s radius $R_{\rm{J}}$ and the Jupiter’s mass $M_{\rm{J}}$ can be done by using the equation: $$\label{r12} 4\pi R_{\rm{J}}^{2}\sigma T^{4}_{\rm{eff}}=L\left(M_{\rm{J}}, t, X\right),$$ where $\sigma$ is the Stefan-Boltzmann constant, $T_{\rm{eff}}$ is the planet’s effective temperature, $L$ is the planet’s luminosity and $t$ is the planet’s age. The symbol $X$ denotes the specified composition of the various mass fractions. In particular, we assume that the planet has the core of the mass $10M_{\rm{Earth}}$ and its thermal evolution is independent of the separation of the helium from the metallic hydrogen. The figure \[f2\] shows our result. We see that in the Universe the [*cold*]{} Jupiter can not yet exist because the surface temperature for $t=12.7$ Gyr is too high ($113$ K). We obtain that the superconducting state will be induced inside the planet for $t\simeq 10^{5}$ Gyr. So, the [*cold*]{} planet represents the final state of the thermal evolution of the free-floating giant planet. Conclusions {#conclusions .unnumbered} =========== We have calculated the maximum surface temperature of the [*cold*]{} Jupiter whose internal structure is close to structure of Jupiter. We have shown that $\left[T_{s}\right]_{\rm{max}}$ is situated between $4$ K and $6.2$ K. We have proved that the [*cold*]{} planet represents the final state of the thermal evolution of the free-floating giant planet. Acknowledgement {#acknowledgement .unnumbered} =============== Authors would like to thank Dr Bogdan Wszo[ł]{}ek for given organizational aid. [3]{} [ Wigner E., Huntington H.B. Chem J. Phys., V. 3, pp. 764-765 (1935) St[ä]{}dele M., Martin R.M. Phys. Rev. Lett., V. 84, pp. 6070-6119 (2000) Weir S.T., Mitchell A.C., Nellis W.J. Phys. Rev. Lett., V. 76, pp. 1860-1897 (1996) Guillot T. Physics Today, V. 57, pp. 63-65 (2004) Fortney J.J., Hubbard W.B. Astroph. J., V. 608, pp. 1039-1079 (2004) Ashcroft N.W. Phys. Rev. Lett., V. 21, pp. 1748-1789 (1968) Stevenson D.J. ’Planetary Structure and Evolution’, California Institute of Technology, (private materials) For discussion of the Eliashberg equations \[originally formulated by Eliashberg G.M., Soviet Phys. JETP, V. 11, pp. 696-750 (1960)\] we refer to:  \ (a) Allen P.B., Mitrovi[ć]{} B. ’Solid State Physics: Advances in Research and Applications’, Academic, (1982)  \ (b) J.P. Carbotte, Marsiglio F. ’The Physics of Superconductors’, Springer(2003) Maksimov E.G., Savrasov D.Yu. Solid State Commun, V. 119, pp. 569-580 (2001) Szcz[ȩ]{}[ś]{}niak R., Jarosik M.W. send to: Solid State Commun Allen P.B., Dynes R.C. Phys. Rev., V. 12, pp. 905-912 (1975) Cudazzo P., Profeta G., Sanna A., at al. Phys. Rev. Lett., V. 100, pp. 257001-257011 (2008) Fortney J.J., Hubbard W.B. arXiv:astro-ph/0312386v1. Contribution to: ’The Search for Other Worlds’, University of Maryland (2003) (a) Thorsett S.E., Arzoumanian Z., Taylor J.H. Astroph. J. Lett., V. 412, L33 (1993)\ (b) Sigurdsson S., Richer H.B., Hausen B.M., et al. Science V. 301, pp. 193 (2003) Hubbard W.B. Icarus, V. 30, pp. 305-313 (1977) ]{}

--- abstract: 'We present a theory of spin waves in a non-condensed gas of spin-1 bosons: providing both analytic calculations of the linear theory, and full numerical simulations of the nonlinear response. We highlight the role of spin-dependent contact interactions in the dynamics of a thermal gas. Although these interactions are small compared to the thermal energy, they set the scale for low energy long wavelength spin waves. In particular, we find that the polar state of $^{87}$Rb is unstable to collisional mixing of magnetic sublevels even in the normal state. We augment our analytic calculations by providing full numerical simulations of a trapped gas, explicitly demonstrating this instability. Further we show that for strong enough anti-ferromagnetic interactions, the polar gas is unstable. Finally we explore coherent population dynamics in a collisionless transversely polarized gas.' author: - 'Stefan S. Natu' - 'Erich J. Mueller' title: 'Spin waves in a spin-$1$ normal Bose gas' --- Introduction ============ To date, all spin wave experiments in non-condensed systems have been on spin-$\frac{1}{2}$ or pseudo-spin-$\frac{1}{2}$ systems. Here we ask the following questions: what is the nature of spin waves in a thermal spin-$1$ gas? How do these differ from the well understood spin-$\frac{1}{2}$ case? These questions are of experimental interest, with several groups possessing the technology to study them using ultra-cold gases [@mukund; @chapman; @sengstock]. In the context of cold gases, spin waves were first discussed by Bashkin [@bashkin], and independently by Lhuillier and Laloë [@laloe]. The key finding was that spin exchange collisions can give rise to weakly damped spin waves even in a non-degenerate gas. Ultra cold gases have provided an exciting setting for observing these spin phenomena. In particular, experiments on pseudo spin-$\frac{1}{2}$ Bose and Fermi systems by the JILA [@cornell] and Duke [@du] groups have observed coherent collective oscillations in an otherwise classical gas. One expects the physics of a spin-$1$ gas to be far richer than a pseudo-spin-$\frac{1}{2}$ system. This is amply demonstrated by experiments on condensed spin-$1$ gases [@mukund2; @sengstock2; @chapman3; @chapman4]. One of the most dramatic observations was that of a dynamical instability in Bose condensed $^{87}$Rb, studied by Sadler *et al.* [@sadler]. Beginning with a gas pumped into the $m_{F} = 0$ state, they observed the spontaneous formation of transverse ferromagnetic domains. Will a similar instability be observed in the normal state? We find that there is an exponentially growing mode in the unmagnetized gas, even in the normal state, with a wavelength comparable to typical cloud sizes. From a theoretical perspective, the source of novel physics in a dilute spin-$1$ Bose gas (such as $^{87}$Rb, $^{23}$Na) is the structure of the interactions, described by *two* coupling constants: $c_{0}$ and $c_{2}$, representing spin-independent and spin-dependent contact interactions. The interaction Hamiltonian density takes the form $H_{\rm int} = c_0 n^2/2 + c_2 \langle \vec{\textbf{S}} \rangle^2/2$, where $n$ and $\vec{\bf S}$ are the local density and spin density respectively [@jason]. The coefficient $c_{2}$ has no corresponding analog in the spin-$\frac{1}{2}$ case. This interaction gives rise to spin mixing collisions, where two atoms in the $m_{F} = 0$ hyperfine sub-level can combine to form atoms in the $m_{F} = \pm 1$ states [@ketterle]. Another important consideration is the quadratic Zeeman effect, which arises from the hyperfine interaction and the difference in the coupling between the electronic and nuclear spins. The linear Zeeman effect can be neglected in the Hamiltonian as the total spin of the atoms is a conserved quantity. We begin our analysis by setting up the problem, and reviewing spin waves in a spin-$\frac{1}{2}$ gas. This allows us to highlight the differences with the spin-$1$ case. Next, we turn to the spin-$1$ gas. Starting from a microscopic Hamiltonian, we obtain a linearized Boltzmann equation about the ferromagnetic ($m_{F} = 1$) and polar ($m_{F} = 0$) states, and calculate the spin wave dispersion in each case. We find that a polar gas with ferromagnetic interactions is dynamically unstable towards spin-mixing collisions for small enough Zeeman fields, analogous to the condensed case. By explicit calculation of the dispersion relation, we show that for strong enough anti-ferromagnetic interactions, an instability occurs in the polar state [@mueller2]. Following these analytic calculations, we perform numerical simulations of a trapped gas using an effective $1$D Boltzmann equation. We explore the evolution of transversely polarized spins, and investigate dynamical instabilities. Our work complements prior work on the kinetics of a normal spin-$1$ Bose gas by Endo and Nikuni [@nikuni]. While their focus is on the effect of collisions in the damping of collective modes of a trapped gas, with particular emphasis on dipole modes, we focus here on the collisionless, Knudsen regime. Basic Setup =========== Kinetic Equations ----------------- As described by Ho [@jason] and Ohmi and Machida [@Ohmi], the second-quantized Hamiltonian for a spin-$1$ gas, expressed in a frame where each spin component is rotating at its Larmor frequency is $$\begin{aligned} \label{eq:1} {\cal H} = \int d\vec{\textbf{r}}~ \psi_{a}^{\dagger}\left(-\frac{1}{2m} + U( \vec{r}, t) + q \textbf{S}_{z} \cdotp \textbf{S}_{z}\right)\psi_{a} + \frac{c_{0}}{2}\psi^{\dagger}_{a}\psi_{a^{'}}^{\dagger}\psi_{a^{'}}\psi_{a} + \frac{c_{2}}{2}\psi^{\dagger}_{a}\psi^{\dagger}_{a^{'}}\vec{\textbf{S}}_{ab}\cdotp \vec{\textbf{S}}_{a^{'}b^{'}}\psi_{b^{'}}\psi_{b}\end{aligned}$$ where $a = (-1, 0, 1)$, is the quantum number for the z-component of the spin, $\psi_{\sigma}(\textbf{r})$ is the field annihilation operator obeying bosonic commutation relations, and $U (\vec{r}, t)$ is the trapping potential. Here $\vec{\textbf{S}}$ denotes the dimensionless vector spin operator. Throughout, we set $\hbar = 1$. Boldface is used to denote matrices, and arrows denote vectors. The interaction strengths, expressed in terms of the scattering lengths in the spin-$2$ and $0$ channel ($a_{2}, a_{0}$) are $c_{0} = 4\pi(a_{0} + 2a_{2})/3m$ and $c_{2} = 4\pi(a_{2} - a_{0})/3m$. A negative $c_{2}$ favors a ferromagnetic state with $\langle \vec{\textbf{S}} \rangle = 1$, while for positive $c_{2}$, the equilibrium state is one with $\langle \vec{\textbf{S}} \rangle = 0$, where all the atoms are in the $m_{F} = 0$ state, or simply an incoherent mixture. Additionally, one considers the quadratic Zeeman effect $q \propto B^{2} $, which favors a state with $m_{F} = 0$ ($\langle \vec{\textbf{S}}\cdotp \vec{\textbf{S}} \rangle = 0$). Thus for a gas with negative $c_{2}$ such as $^{87}$Rb, the spin dependent contact interaction competes with the quadratic Zeeman field, giving rise to interesting dynamics [@sadler]. In the condensed gas, dipolar interactions may also be important [@mukund; @mukund2]. At the lower densities found in a normal gas these interactions, which fall as $1/r^{3}$, may be neglected. Following standard arguments [@kadanoff] we obtain the equations of motion for the Wigner density matrix $\textbf{F}_{ab}(\vec{p}, \vec{R}, t)$, whose elements are $f_{ab}(\vec{p}, \vec{R}, t)= \int d \vec{r} e^{-i\vec{p}\cdotp \vec{r}} \langle \psi^{\dagger}_{a}(\vec{R} - \frac{\vec{r}}{2}, t) \psi_{b}(\vec{R} + \frac{\vec{r}}{2}, t) \rangle$. The diagonal components of the spin density matrix, when integrated in momentum, give the densities of each of the spin species. The off diagonal terms, often referred to as *coherences*, are responsible for spin dynamics. Here $\vec{p}$ represents momentum, $\vec{R}$ and $\vec{r}$ denote the center of mass and relative coordinates. The Wigner function is the quantum analog of the classical distribution function. By taking moments of the Wigner function, we obtain physical observables such as the density $\textbf{n}(\vec{R}, t) = \int \frac{d\vec{p}}{(2 \pi)^{3}}~\textbf{F}(\vec{p}, \vec{R}, t)$ and spin current $\vec{\textbf{j}}(\vec{R}, t) = \int \frac{d\vec{p}}{(2 \pi)^{3}}~\vec{p}~\textbf{F}(\vec{p}, \vec{R}, t)$. The equation of motion takes the standard form [@bashkin]: $$\label{eq:2} \frac{\partial}{\partial t}\textbf{F} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}\textbf{F}= i\Big[\textbf{V}, \textbf{F}\Big] + \frac{1}{2}\Big\{\vec{\nabla}_{R}\textbf{V},\vec{\nabla}_{p}\textbf{F}\Big\} + \textbf{I}_{c}$$ where $\textbf I_{c}$ is the collision integral and $\textbf V$ is the interaction potential. This form of the Boltzmann equation (\[eq:2\]) is completely general, and holds for any non-condensed spinor gas. The role of spin enters in determining the dimension of the density matrix, and the exact form of the interaction potential $\textbf{V}$. The first two terms on the right hand side of the Boltzmann equation (\[eq:2\]) arise from forward and backward scattering collisions. While the former type merely alter the mean field seen by the atoms, backward scattering collisions allow the colliding atoms to exchange momentum. The last term in the Boltzmann equation is the collision integral, responsible for energy relaxation. While a detailed derivation of the collision integral is non trivial [@nikuni], the qualitative properties are well described within a simple relaxation time approximation $I_{c} = -(f - f_{0})/\tau$. The relaxation time ($\tau$) is proportional to the elastic scattering rate $\tau^{-1}_{el} \sim 8\pi a^{2}_{0}v_{T}n$, where $v_{T} = \sqrt{\frac{2 k_{B}T}{m}}$ is the thermal velocity and $n$ is the density. The precise expression for $\bf{V}$ determined by including the interactions within the Hartree-Fock description is$$\label{eq:3} \textbf{V} = \left(U + c_{0}\text{Tr}(\textbf n)\right)\textbf 1 + c_{0}\textbf n + c_{2}\vec{\textbf{S}}\textbf n \cdotp \vec{\textbf{S}} + c_{2} \vec{M}\cdotp\vec{\textbf{S}}$$ where $\textbf{1}$ is the identity matrix, Tr denotes the trace operation, and $\vec{\textbf{M}} = \text{Tr}(\vec{\textbf{S}}~\textbf{n})$ is the magnetization. Our form for the interaction energy is equivalent to that of Endo and Nikuni [@nikuni]. One may explicitly check that this interaction potential is rotationally invariant in spin space. In experiments the external trapping potential $U$ is often spin-dependent, an effect which is readily incorporated. The spin independent interaction gives rise to self and cross interaction terms. The latter give rise to coherences and are encoded in the second term in (\[eq:3\]). In addition, the mean-field potential alters the trapping potential seen by all the atoms by an amount $c_{0}n_{tot}$, where the total density $n_{tot} = \text{Tr}(\textbf{n})$. The contribution of the spin-dependent interaction is more subtle and can be understood as follows: The first term accounts for spin dynamics such as spin-relaxation collisions. For ferromagnetic interactions ($c_{2} < 0$), the last term increases the density of regions where the atomic spins are aligned with respect to one another ($ |M| = 1$). For a fully polarized gas in the $m_{F} = +1$ sublevel, this is $c_{2}\textbf{S}_{z}$, while for an unmagnetized ($m_{F} = 0$) gas it is zero. Finally, we note that while the self- and cross- interactions between the three sublevels produce diagonal and off-diagonal contributions to the interaction potential, the spin-relaxation or population exchange interactions only gives rise to coherences, and are absent for a single component gas. Qualitative Features -------------------- Here we elaborate on our argument for why one expects richer spin physics in a spin-$1$ system versus a spin-$\frac{1}{2}$ one. We assume a uniform, collisionless gas with $\nabla_{R}c_{0}n/m v_{T} \ll c_{0}n$, such that all the physics is governed by the commutator in (\[eq:2\]). The pseudo-spin $\frac{1}{2}$ case may be understood starting from the fact that that a $2 \times 2$ matrix ($A$) can be decomposed into $\textbf{A} = A_{0}\textbf{I} + A_{\mu}\sigma_{\mu}$, where $A_{0} = \text{Tr}(\textbf{A})$, and $\vec{\sigma}$ are the Pauli matrices. Expressing the density matrix and interaction potential in this way, the equations of motion are $\frac{\partial}{\partial t}F_{0} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{0} = 0$ and $\frac{\partial}{\partial t}{F_{\mu}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{\mu} = - F_{\rho}V_{\nu}\epsilon_{\rho\nu\mu}$, where $\epsilon_{\rho\nu\mu}$ is the completely antisymmetric unit tensor. The second equation, which is responsible for much of the spin wave physics in a spin-$\frac{1}{2}$ system simply says that the interactions act as an effective magnetic field about which the spins precess. To analyze the spin-$1$ case, we follow Ohmi and Machida [@Ohmi] and use a Cartesian basis ($\psi =\{ \psi_{x}, \psi_{y}, \psi_{z}\}$). This representation is related to the spherical ($\{1, 0, -1\}$) basis as follows [@mueller]: $\psi_{x} = \frac{1}{\sqrt{2}}(\psi_{1} - \psi_{-1})$,$\psi_{x} = \frac{i}{\sqrt{2}}(\psi_{1} + \psi_{-1})$ and $\psi_{z} = \psi_{0}$. In the Cartesian basis the irreducible decomposition of a spin-$1$ system is $\textbf{A} = A_{0}\textbf{I} + i \epsilon_{abc}A^{(1)}_{c} + A^{(s)}$, where the scalar $A_{0} = \frac{\text{Tr}(A)}{3}$, $\textbf{A}^{(a)} = \epsilon_{abc}A^{(1)}_{c}$ is a completely antisymmetric matrix proportional to the vector spin $\langle\textbf{S}\rangle$ order, and $\textbf{A}^{(s)}$ is a symmetric traceless tensor which is related to the spin fluctuations $\langle \textbf{S}_{a}\textbf{S}_{b} + \textbf{S}_{b}\textbf{S}_{a} \rangle$, and is a nematic degree of freedom. Writing the density matrix ($\textbf{F}$) and interaction ($\textbf{V}$) in terms of their respective irreducible decompositions we obtain three equations of motion: $\frac{\partial}{\partial t}F_{0} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F_{0} = 0$, $\frac{\partial}{\partial t}{F^{(a)}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F^{(a)} = i[\textbf{V}^{(a)}, \textbf{F}^{(a)}] + i[\textbf{V}^{(s)}, \textbf{F}^{(s)}]$, and $\frac{\partial}{\partial t}{F^{(s)}} + \frac{\vec{p}}{m}\cdotp\vec{\nabla}_{R}F^{(s)} = i[\textbf{V}^{(a)}, \textbf{F}^{(s)}] + i[\textbf{V}^{(s)}, \textbf{F}^{(a)}]$. Here the second equation describes the evolution of $\langle \textbf{S} \rangle$ and the spin currents, while the last equation describes dynamics of the nematicity ($\textbf{A}^{(s)}$). The new feature is that the spin and nematic degrees of freedom are coupled by the interaction matrix. Using the Levi-Civita identities, it is easy to see that $i[\textbf{V}^{(a)}, \textbf{F}^{(a)}] $ is the spin-$1$ analog of the corresponding spin-$\frac{1}{2}$ term which acts as an effective magnetic field for the spins. But the last term $i[\textbf{V}^{(s)}, \textbf{F}^{(s)}]$ shows that fluctuations of the nematicity can change the spin dynamics. Spin waves in a two-component gas --------------------------------- As a starting point for understanding the spin-$1$ gas, we review the spin-$\frac{1}{2}$ case. As explained in [@oktel], spin dynamics in the cold collision regime are governed by momentum exchange collisions with a characteristic timescale $\tau^{-1}_{ex} \sim \frac{4\pi a_{12}n}{m}$, where $a_{12}$ is the two-body scattering length. When $k_{B}T \ll \frac{1}{m a^{2}_{12}}$, or alternatively, the thermal deBroglie wavelength is large compared to the scattering length $\Lambda_{T} = \sqrt{\frac{2\pi}{m k_{B}T}} \ge a_{12}$, $\tau_{ex} \ll \tau_{el}$, and for times shorter than $\tau_{el}$, particles exchange momentum several times without significantly altering the energy distribution. Therefore excitations with wavelength longer than $v_{T}\tau_{ex}$ are a collective effect. For typical densities ($\sim 10^{14}$cm$^{-3}$) and scattering lengths, $n^{1/3}a \sim 0.01$ the condensation temperature ($T_{c}$) for the interacting system may be approximated to that of an ideal Bose gas [@baym]. Thus in order to see collective spin phenomena, we require $T_{c} \sim \langle \omega \rangle N^{\frac{1}{3}} < k_{B}T < \frac{1}{m a^{2}_{12}}$, where $\langle \omega \rangle = (\omega_{x}\omega_{y}\omega_{z})^{1/3}$ is the average trap frequency [@pethick]. Although this is typically a wide temperature range $10^{-7} < T < 10^{-3}$K, diffusive relaxation damps out spin waves at higher temperatures. A two-component gas has a longitudinal and a transverse spin mode. The longitudinal mode has a linear dispersion, but is strongly Landau damped [@bashkin]. By contrast, the transverse mode is weakly damped and propagates with dispersion $\omega = \frac{k^{2}v^{2}_{T}\tau_{ex}}{2}\left(1 - i \frac{\tau_{ex}}{\tau_{D}}\right)$, where $\frac{\tau_{ex}}{\tau_{D}} \sim \frac{a_{12}}{\Lambda_{T}} \le 1$ [@levy]. These weakly damped spin waves have been observed in the context of spin polarized Hydrogen by Johnson *et al.* [@cornell; @du; @lee]. Spin waves in the spin-$1$ gas ============================== In this section we linearize Eq.(\[eq:2\]) about stationary states. In Sec.IV we consider more general dynamics. Excitations about the $m_{F}=1$ state ------------------------------------- We begin by linearizing about the $m_{F} = 1$ state. We start with a homogenous gas of particles with a Maxwellian velocity distribution and initial density $n_{0}$. The self-interaction between two particles is proportional to $a_{2}$, and so we define $\Omega_{int} = (c_{0} + c_{2})n_{0}$. We begin by considering the collisionless limit, $1 \ll \Omega_{int}\tau_{D}$. (150, 160)(10, 10) (-38, 160)[(a)]{} (92, 160)[(b)]{} (-38, 87)[(c)]{} (92, 83)[(d)]{} (-25, 95)[ (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q = 4|c_{2}|n_{0}$ (green, dotted), $q = 4.5|c_{2}|n_{0}$ (red, thin). (b) Gapless, linear dispersion for small $k$ at $q = 4|c_{2}|n_{0}$. The solid black curve is the small $k$ expansion (\[eq:7\]). (c) $q = 2|c_{2}|n_{0}$ dispersion in detail. The horizontal black line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig1.eps "fig:")]{} (95, 95)[ (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q = 4|c_{2}|n_{0}$ (green, dotted), $q = 4.5|c_{2}|n_{0}$ (red, thin). (b) Gapless, linear dispersion for small $k$ at $q = 4|c_{2}|n_{0}$. The solid black curve is the small $k$ expansion (\[eq:7\]). (c) $q = 2|c_{2}|n_{0}$ dispersion in detail. The horizontal black line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig2.eps "fig:")]{} (-25, 15)[ (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q = 4|c_{2}|n_{0}$ (green, dotted), $q = 4.5|c_{2}|n_{0}$ (red, thin). (b) Gapless, linear dispersion for small $k$ at $q = 4|c_{2}|n_{0}$. The solid black curve is the small $k$ expansion (\[eq:7\]). (c) $q = 2|c_{2}|n_{0}$ dispersion in detail. The horizontal black line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig3.eps "fig:")]{} (95, 15)[ (a) Dispersion relation $\omega^2(k)$ for various values of $q$ using $^{87}$Rb parameters ($c_{2} < 0$) at $T= 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$: q=0 (solid, black), $q = 2|c_{2}|n_{0}$ (blue, dashed), $q = 4|c_{2}|n_{0}$ (green, dotted), $q = 4.5|c_{2}|n_{0}$ (red, thin). (b) Gapless, linear dispersion for small $k$ at $q = 4|c_{2}|n_{0}$. The solid black curve is the small $k$ expansion (\[eq:7\]). (c) $q = 2|c_{2}|n_{0}$ dispersion in detail. The horizontal black line is $\omega^{2} = q(q - 4|c_{2}|n_{0})$, which is the gap predicted by the small $k$ expansion (\[eq:7\]). (d) $q=0$ dispersion in detail. Horizontal dashed line at $\omega^{2} = -4c^{2}_{2}n^{2}_{0}$ indicates the most unstable mode. Vertical dotted line is at $k\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$.](fig4.eps "fig:")]{} (-28, 110) **$(\omega/c_{0}n_{0})^{2}$** (95, 115) **$\omega/c_{0}n_{0}$** (-30, 35) **$(\omega/c_{0}n_{0})^{2}$** (92, 30) **$(\omega/c_{0}n_{0})^{2}$** (155, 87)[**$k \Lambda_{T}$**]{} (35, 87)[**$k \Lambda_{T}$**]{} (155, 7)[**$k \Lambda_{T}$**]{} (35, 7)[**$k \Lambda_{T}$**]{} Linearizing about this state, and dropping the collision term, the Boltzmann equation can be written in Fourier space: $$\begin{aligned} \label{eq:4} (-\omega + \frac{\vec{k}\cdotp\vec{p}}{m})\tilde{\delta\textbf{F}} = \Big[\textbf{V}_{0}, \tilde{\delta\textbf{F}}\Big] + \Big[\tilde{\delta\textbf{V}}, \textbf{F}_{0}\Big] + \\\nonumber \frac{1}{2}\Big\{\vec{k}~\tilde{\delta\textbf{V}},\vec{\nabla}_{p}\textbf{F}_{0}\Big\} \end{aligned}$$ where $\textbf{F}_{0}$ is the initial distribution and $\textbf{V}_{0}$ is the initial interaction potential. The quantities $\tilde{\delta\textbf{F}}$ and $\tilde{\delta\textbf{V}}$ are Fourier transforms of the change in the distribution and interaction, defined in the usual way $\delta f(\vec{k}, p, \omega) = \int d \vec{r}dt e^{i(\vec{k}\cdotp\vec{r} - \omega t)} \delta f(p,r, t)$. Inserting the form for the interaction potential (\[eq:3\]), we obtain the equations of motion for the density and spin fluctuations. We find that four long wavelength ($|k|v_{T} \ll \Omega_{int}$), low frequency ($\omega \ll \Omega_{int}$), spin modes propagate in this system. These modes have a quadratic dispersion, characteristic of ferromagnetic systems, but have a $k=0$ offset. Two of the modes are transverse spin waves, and the other two are quadrupolar modes where the longitudinal magnetization fluctuates along with the transverse spin fluctuations ($\langle S_\mu S_\nu\rangle -\langle S_\nu S_\mu\rangle$). The spin modes are given by the equation $1 = -(c_{0} + c_{2})\int\frac{d\vec{p}}{(2\pi)^{3}}\frac{f_{0} + \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0}}{\omega +\Omega_{int} + q - \frac{\vec{k}\cdotp\vec{p}}{m}}$ [@landau]. The transverse spin modes have dispersion $\omega(k) = \frac{k^{2}v^{2}_{T}}{2 \Omega_{int}} - q$. The fact that this has a negative energy at $k=0$ signifies that in the presence of a quadratic Zeeman shift, the initial state is not the thermodynamic ground state. One can decrease the energy by rotating the spins to be transverse to the magnetic field. However the frequency stays real, meaning that the initial state is metastable, and transverse spin will not be spontaneously generated. Meanwhile, the quadrupolar modes have dispersion $\omega(k) = -4c_{2}n_{0} + \frac{k^{2}v^{2}_{T}}{2 \Omega_{int}}$. In a gas with positive $c_{2}$, the frequency vanishes at a finite wave-vector, indicating a thermodynamic instability in the anti-ferromagnetic gas. Conversely for negative $c_{2}$ the spin-dependent contact interaction leads to a gap in the quadrupolar mode spectrum. As pointed out by Leggett [@leggett], one can also derive these equations by writing the macroscopic equations for the density and spin current: $$\begin{aligned} \label{eq:5} \partial_{t}\textbf{n} + \frac{\vec{\nabla}_{\textbf{R}}\cdotp\vec{\textbf{j}}}{m} = i\Big[\textbf{V}, \textbf{n} \Big] \\\nonumber \partial_{t}\vec{\textbf{j}} + \frac{\vec{\nabla}_{\textbf{R}}\textbf{Q}}{m} = i\Big[\textbf{V}, \vec{\textbf{j}} \Big] - \frac{1}{2}\Big\{\vec{\nabla}_{\textbf{R}}\textbf {V},\textbf{n} \Big\} - \frac{\vec{\textbf{j}}}{\tau_{D}}\end{aligned}$$ where the energy density $\textbf{Q} = \int d{\vec{p}}~(\vec{p}\cdotp\vec{p})~\textbf{F}$. In order to obtain a closed set of equations, we approximate $\textbf{Q} \approx \int d{\vec{p}}~\frac{1}{3}\langle\vec{p}\cdotp\vec{p}\rangle~\textbf{F} = \frac{p_{T}}{2}\textbf{n}$, where $p_{T} = mv_{T}$. In a collisionless gas, sufficiently close to equilibrium, these approaches are equivalent. An advantage of the Leggett approach is that it provides access to the dispersion at large $k$ more readily than the former approach. Solving (\[eq:5\]), we get the standard relations $\omega(k) = q - \frac{k^{2}v^{2}_{T}\tau_{D}/2}{(1 + (\Omega_{int}\tau_{D})^{2})} (\Omega_{int}\tau_{D} - i)$, and $\omega(k) = -4c_{2}n_{0} + \frac{k^{2}v^{2}_{T}\tau_{D}/2}{(1 + (\Omega_{int}\tau_{D})^{2})} (\Omega_{int}\tau_{D} - i)$. Setting $\tau_{D}$ to infinity, we once again recover the relationships derived above. Excitations about the $m_{F} = 0$ state --------------------------------------- Next we consider the case of a gas in the $m_{F} = 0$ state. As before, low energy dynamics are driven by a combination of the quadratic Zeeman energy, and the spin-dependent contact interaction. As we now show, in the case where $c_{2}$ is negative, these energies compete, driving a [*dynamic*]{} instability. Linearizing equation (\[eq:4\]) yields the relation: $$\label{eq:6} 1 = -\frac{4c^{2}_{2}\chi_{1}\chi_{2}}{(1+ (c_{0}+c_{2})\chi_{2})(1-(c_{0}+c_{2})\chi_{1})}$$ where the response functions are $\chi_{1} = \int \frac{d\vec{p}}{(2\pi)^{3}}\frac{(f_{0} + \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0})}{-\omega +\Omega_{d}+ \frac{\vec{k}\cdotp\vec{p}}{m}}$ and $\chi_{2} = \int \frac{d\vec{p}}{(2\pi)^{3}}\frac{(f_{0} - \frac{1}{2}\vec{k}\cdotp\vec{\nabla}_{p}f_{0})}{-\omega -\Omega_{d}+ \frac{\vec{k}\cdotp\vec{p}}{m}}$ and $\Omega_{d} = (c_{0}-c_{2})n_{0}-q$. For $|k|v_{T} \ll c_{0}n_{0}$ and $\omega \ll c_{0}n_{0}$. The resulting dispersion is: $$\label{eq:7} \omega^{2}(k) = q(q + 4c_{2}n_{0}) + (q + 2c_{2}n_{0})\frac{k^{2}v^{2}_{T}}{c_{0}n_{0}}$$ (50, 100) (-60, 3)[![\[fig:-2\] Microwave-field induced instability in a polar gas for $q<0$: The modes of a polar gas with ferromagnetic interactions ($c_{2} < 0 $) are plotted for $q = 4c_{2}n$ (top) and $q = 2c_{2}n$ (bottom) (hence $q < 0$). The gaps are given by $q(q + 2c_{2}n_{0})$ in each case. Note that for negative $q$, a finite wave-vector instability sets in. Increasing $|q|$ pushes the instability to larger wave-vectors. The dashed line is the most unstable frequency for $q=0$ (see Fig. \[fig:-1\](d)) given by $\omega^{2} = - (2c_{2}n_{0})^{2}$. The timescale for the onset of the instability is independent of $q$. We set $T = 1\mu$K and $n_{0} = 10^{14}$cm$^{-3}$.](pmicro.eps "fig:")]{} (-70, 35) **$(\omega/c_{0}n_{0})^{2}$** (90, -6)[**$k \Lambda_{T}$**]{} The spin and quadrupole modes in the $m_{F} = 0$ state are degenerate. For $c_{2} > 0$, or for large enough $q$, one obtains a gapped quadratic mode. For $c_{2} <0$ however, the dispersion goes from quadratic to linear at $q=4|c_{2}|n_{0}$. Upon lowering $q$ further, the $m_{F} = 0$ state undergoes a [*dynamic*]{} instability where the excitation frequency is complex. At $q=2|c_{2}|n_{0}$, the leading order term in the small $k$ expansion vanishes. The structure of the modes should be contrasted with those in the $m_{F} = 1$ state. Even though either case may not be a thermodynamically stable starting point for anti-/ferromagnetic interactions (respectively), only in the antiferromagnetic case do small fluctuations grow exponentially. Taking the $\tau_{D} \rightarrow \infty$ limit and solving (\[eq:5\]), the dispersion relation is determined by the equation: $$\begin{aligned} \label{eq:8} \Big((\omega - \Omega_{s})(\omega + \Omega_{d}) - \frac{v^{2}_{T}k^{2}}{2}\Big)\Big((\omega + \Omega_{s})(\omega - \Omega_{d}) - \frac{v^{2}_{T}k^{2}}{2}\Big) = -4(c_{2}n_{0})^{2}(\omega - \Omega_{d})(\omega + \Omega_{d})\end{aligned}$$ where $\Omega_{s} = q + 2c_{2}n_{0}$. Expanding (\[eq:8\]) to lowest order in $k$ returns Eq.(\[eq:7\]). The roots of (\[eq:8\]) for negative $c_{2}$ are plotted in Fig. \[fig:-1\](a) for various values of the quadratic Zeeman energy $q$. The temperature is set to $1\mu$K, density to $10^{14}$ cm$^{-3}$, and we use the scattering lengths for $^{87}$Rb, $a_{0}(a_{2}) = 5.39(5.31)$nm. For the homogenous gas in consideration here, this is well above the temperature for Bose-Einstein condensation. As shown in Fig. \[fig:-1\](b), at $q=0$, $\omega^{2}(k)$ is negative for small $k$, indicative of an instability. The timescale for the onset of this instability ($t_{ins}$) is determined by the most unstable mode, which from (\[eq:8\]) occurs at $\omega^{2} = - (2 |c_{2}|n_{0})^{2}$, giving $t_{ins} \sim \frac{1}{2|c_{2}|n_{0}}$.The corresponding wave-vector is $k_{ins}\Lambda_{T} = \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$. Solving for where the mode frequency vanishes, we find that all wave-vectors $k$ smaller than a critical wave-vector $k_{c} = \sqrt{2}k_{ins}$ lead to unstable modes. For achievable experimental densities ($\sim 10^{14}$cm$^{-3}$) and temperatures ($\sim 1\mu$K), this wave-length of the instability $\lambda_{c} \sim 100\Lambda_{T} = 10\mu$m, which is comparable to typical cloud sizes. The time for the onset of this instability $\sim 0.25$s. Although this timescale is easily achievable in experiment, small collisional timescales ($\sim 50$ms) at these temperatures may make the instability undetectable in experiment. *Microwave induced instability in a ferromagnetic polar gas*:$-$ [^1] Gerbier *et al.* [@gerbier] have shown that a weak, microwave driving field, off resonant with the $F=1 \rightarrow F = 2$ transition, may be used to tune the energy levels of the $F=1$ hyperfine sublevels. In this sense, it plays the same role as the quadratic Zeeman shift. By changing the detuning, one can change the effective $q$ from positive ($E_{m_{F} = 0} < E_{m_{F} = \pm1}$) to negative ($E_{m_{F} = 0} > E_{m_{F} = \pm1}$), where $E$ denotes the energy of the hyperfine sublevels. It is therefore of experimental and theoretical interest to consider what happens to the $m_{F} = 1, 0$ states at negative $q$. For the $m_{F} =1$ state, the spin mode becomes gapped by the quadratic Zeeman energy, and the thermodynamic instability disappears. This is to be expected as a negative $q$ favors a ground state with $\langle \textbf{S}\cdotp\textbf{S} \rangle = 1$. The polar state however is more interesting. From (\[eq:7\]), it is clear that for ferromagnetic interactions ($c_{2} < 0$), the first term on the right hand side is always positive, and spectrum becomes gapped. The second term however is negative, signaling a finite wave-vector instability. Analyzing (\[eq:8\]) (see Fig.\[fig:-2\]), we find that increasing $|q|$ pushes the instability to larger wave-vectors, but frequency of the most unstable mode remains largely unchanged. The experimental consequence is that $q<0$, one should observe a reduction in the size of ferromagnetic domains, while leaving the timescale for the onset of the instability unaffected. (150, 90)(10, 10) (-165, 100)[(a)]{} (-35, 100)[(b)]{} (95, 100)[(c)]{} (220, 100)[(d)]{} (-160, 15)[]{} (-35, 15)[]{} (95, 15)[]{} (220, 15)[]{} (-45, 42) **$\omega/c_{0}n_{0}$** (84, 42) **$\omega/c_{0}n_{0}$** (213, 42) **$\omega/c_{0}n_{0}$** (-170, 42) **$\omega/c_{0}n_{0}$** (200, 7)[**$k \Lambda_{T}$**]{} (70, 7)[**$k \Lambda_{T}$**]{} (-60, 7)[**$k \Lambda_{T}$**]{} (325, 7)[**$k \Lambda_{T}$**]{} *Large anti-ferromagnetic interactions*: The modes of the polar gas, when $c_{2} > 0$ can be obtained from (\[eq:8\]). For small $c_{2}/c_{0}$, the polar state is stable toward spin and nematic fluctuations. However, when $c_{2}$ and $c_{0}$ are commensurate, an instability can set in. This latter limit, which has been largely unexplored, is primarily of academic interest as all current spinor gas experiments have $|c_{2}| \ll c_{0}$. For simplicity we restrict our analysis to the case of no magnetic field ($q=0$). As shown in Fig. \[fig:-3\](a), for $c_{2} = 0$ we find a gapped and ungapped quadratic mode with dispersions $\omega(k) = \sqrt{\frac{1}{2}({\Omega^{2}_{d} \pm \sqrt{ \Omega^{4}_{d} - (v_{T}k)^{4}}})}$, where $\Omega_{d}$ reduces to $c_{0}n_{0}$ in this limit. As $c_{2}$ is increased, for some $k_{c}$, the gapped an ungapped modes meet, and for $k > k_{c}$, imaginary frequencies appear (Fig. \[fig:-3\](b)). While the exact expression for $k_{c}$ is complicated, and depends on $T$ and density, we estimate the appearance of imaginary wave-vectors when $\Omega_{s} \approx \Omega_{d}$ i.e. $c_{2} \approx c_{0}/3$. The case of $c_{2} = c_{0}$ is special, and there is a single linear mode with frequency $\omega(k) = \frac{1}{\sqrt{2}}v_{T}k$ (Fig. \[fig:-3\](c)). However as $c_{2} > c_{0}$ (Fig. \[fig:-3\](d)), imaginary frequencies appear once again for a bounded set of $k$ values: $0 < k < k_{max}(c_{2}/c_{0})$. Summary of Results ------------------ In Fig. \[fig:-4\] we summarize the results of our stability analysis in terms of the interaction parameters $c_{0}$ and $c_{2}$ for zero $q$. The following general features are clear: (50, 100)(10, 10) (-80, 13)[![\[fig:-4\] Stability phase portrait in terms of the interaction strengths $c_{0}n$ and $c_{2}n$ at zero Zeeman field. The interaction strengths are measured in nK to facilitate comparison with typical experimental values. The four distinct regions are demarcated by solid black lines. I: For $c_{2} < 0$, but $c_{2} > -c_{0}$, the ferromagnetic state is the thermodynamic ground state. II: For $c_{2} > 0$, $c_{2} \le c_{0}/3$, the polar state is the ground state. III: Here $a_{2} (= c_{0} +c_{2}) < 0$, and the gas is dynamically unstable towards collapse. IV: When $c_{2} \ge c_{0}/3$, the polar state becomes metastable. In the presence of finite populations of $\pm1$, phase separation is expected to occur. In this region, the ferromagnetic state is thermodynamically unstable. Finally, the open and filled circles are typical interaction scales in $^{23}$Na and $^{87}$Rb for $n \sim 10^{14}$cm$^{-3}$.](phaseplot "fig:")]{} (30,7) (55, 8) (-80, 47) (-81, 70) 1. For $c_{0} > 0$ and $c_{2}< 0$, the ground state is a ferromagnet. 2. For $c_{0} > 0$ and $c_{2} > 0$, but $c_{2} \le c_{0}/3$, the ground state is polar. 3. In region III the scattering length in the spin-$2$ channel ($a_{2}$) is negative, and the gas becomes mechanically unstable. 4. In region IV, the polar state is metastable, in the presence of finite populations of $\pm1$ components, the system will phase separate [@mueller2]. The ferromagnet is thermodynamically unstable here. 5. The regime of current day experiments on spin-$1$ gases is that of $|c_{2}| \ll c_{0}$, indicated by the open and filled circles in the figure. For $^{87}$Rb, at $n \sim 10^{14}$cm$^{-3}$, $c_{0}n \sim 50$nK and $|c_{2}|n \sim 0.25$nK (filled). For $^{23}$Na, at the same density $c_{0}n \sim 85$nK and $c_{2}n \sim 4.25$nK (open). Numerical Investigation ======================= In an experiment, the atoms are typically confined to a harmonic trap. Calculating the modes in a trap is much more difficult than in free space. Example calculations [@nikuni] do not always yield simple physical pictures. Moreover experiments are typically performed in a limit where linear response is not applicable [@du]. For these reasons, we perform a numerical study of the spin dynamics for a trapped gas. Furthermore while the calculations in part I pertained to the collisionless regime, using a simple relaxation time approximation, we model collisions in the gas. We address two questions here: 1. What happens if the initial state is not a stationary state with respect to the external magnetic field ? 2. Can the instability in the polar state be observed experimentally? To address question $1$, we consider a ferromagnet in the $x$ direction, in the presence of a field along $z$. We find that of coherent population dynamics in a collisionless gas. By controlling the rate of decay of spin current, one may be able to experimentally observe coherent oscillations even in a classical spinor gas. With regards to $2$, taking into account realistic experimental parameters, the polar state instability should not be visible in $^{87}$Rb. The collision time is set by $c_{0}$, while the time for spin dynamics by $c_{2}$. Owing to the small ratio of $|c_{2}|/c_{0} \sim 0.005$, collisions lead to relaxation before any coherences can develop. Numerical setup --------------- We consider parameters such that the all the motion takes place in one dimension producing an effective 1D Boltzmann equation. Assuming that the distribution function in the transverse directions is frozen into a Boltzmann form, we can integrate them out [@natu], yielding renormalized scattering lengths ($a_{0}/a_{2}$). In what follows, we normalize all lengths by the oscillator length $\sqrt{1/(m\omega_{z})}$ and momenta by $\sqrt{m k_{B} T}$. Using a phase-space conserving split-step approach [@teuk], we numerically integrate (\[eq:2\]). We have verified that the tempero-spatial grid is fine enough that our results are independent of step-size. We use a radial trapping frequency $\omega_{r} = 2\pi \times 250$Hz, axial trapping frequency of $\omega_{z} = 2\pi \times 50$Hz, roughly $5 \times 10^{5}$ atoms, and set the temperature to $1\mu$K. The initial distribution function in position and momentum is given by the equilibrium distribution $e^{-\beta(\frac{p^{2}}{2m} + \frac{1}{2}m\omega^{2}_{z}z^{2})}$. Ferromagnetic gas in the x-direction ------------------------------------ (100, 80)(10, 10) (-48, 87)[(a)]{} (60, 87)[(b)]{} (60, 15)[ Evolution of central density of $m_{F} = 0$ component (normalized to initial density $n_{0} = n_{00} + n_{11} + n_{-1-1} (t = 0)$) as a function of time for negative $c_{2}$ ($^{87}$Rb parameters (thick line)) and positive $c_{2}$ ($^{23}$Na parameters (dashed curve)) for $q \sim 15$Hz. ($B \sim 0.1$G). The line at $n_{00}/n_{0} = 0.5$ is a guide to the eye. (b) Thick curve: evolution of $m_{F} = 0$ population in $^{23}$Na for larger $q \sim 60$Hz ($B \sim 0.3$G). Dashed curve: suppression of oscillations due to finite relaxation rate ($\tau_{el} \sim 80$ms) in $^{23}$Na within a relaxation time approximation. Parameters used in the simulations: $\omega_{r} = 2\pi \times 250$Hz, $\omega_{z} = 2\pi \times 50$Hz and $N =5 \times 10^{5}$.](fig8.eps "fig:")]{} (-57, 15)[ Evolution of central density of $m_{F} = 0$ component (normalized to initial density $n_{0} = n_{00} + n_{11} + n_{-1-1} (t = 0)$) as a function of time for negative $c_{2}$ ($^{87}$Rb parameters (thick line)) and positive $c_{2}$ ($^{23}$Na parameters (dashed curve)) for $q \sim 15$Hz. ($B \sim 0.1$G). The line at $n_{00}/n_{0} = 0.5$ is a guide to the eye. (b) Thick curve: evolution of $m_{F} = 0$ population in $^{23}$Na for larger $q \sim 60$Hz ($B \sim 0.3$G). Dashed curve: suppression of oscillations due to finite relaxation rate ($\tau_{el} \sim 80$ms) in $^{23}$Na within a relaxation time approximation. Parameters used in the simulations: $\omega_{r} = 2\pi \times 250$Hz, $\omega_{z} = 2\pi \times 50$Hz and $N =5 \times 10^{5}$.](fig9.eps "fig:")]{} (50, 40) **$n_{00}/n_{0}$** (-68, 40) **$n_{00}/n_{0}$** (160, 08)[**$t(s)$**]{} (40, 08)[**$t(s)$**]{} The spin density matrix for a ferromagnet pointing in the $x-$ direction is: $\textbf{F} = \psi^{*}\psi$, where $\psi = \{\frac{1}{2}, \frac{1}{\sqrt{2}}, \frac{1}{2}\}$. Numerically integrating (\[eq:5\]), we find that for a magnetic field of $0.1$G ($q \sim 15$Hz), in the absence of collisional relaxation, the populations of the three hyperfine sublevels coherently oscillate about their equilibrium values, as shown in Fig. \[fig:-5\](a). The amplitude of the oscillations depends on the magnitude of the spin dependent contact interaction. The evolution is not simply a precession, rather it involves an oscillation between ferromagnetism and nematicity. For $c_{2} < 0 (> 0)$ the population of the $m_{F} = 0$ sublevel oscillates between $0.5$ and a larger (smaller) value. The frequency and amplitude of the oscillations also depend strongly on the magnitude of the quadratic Zeeman energy. Increasing $q$ increases the frequency of oscillations, but decreases the amplitude (Fig. \[fig:-5\](b)). Similar results have been obtained by Chang *et al.* [@chapman2] in condensed $^{87}$Rb, although they considered a different initial spin configuration. The coherences oscillate at a frequency set by the external perturbation, in this case the quadratic Zeeman energy. The resulting phase factors can be interpreted as rotations in spin space. The spin-dependent contact interaction couples spin and density, and oscillations in the densities of the three components are seen. Increasing $q$, causes the spin vector to rotate faster, leading to a rapid averaging to its initial value over the timescale of the spin-dependent contact interaction, reducing the size of the effect. *Collisions*:$-$ Experiments will also involve collisions between the atoms. A detailed derivation of collisional relaxation rates can be found in [@nikuni]. We are concerned here with the experimental feasibility, hence we use a simple relaxation time approximation. The relaxation rate is proportional $(\frac{a_{0}+ 2a_{2}}{3})^{2} \sim a^{2}_{0}$. The amplitude of the oscillations in the populations of the three sub-levels is directly proportional to the size of $c_{2}$. From the trap dimensions, particle number, and the scattering lengths for $^{87}$Rb ($|c_{2}|/c_{0} \sim 0.005$), we estimate $\tau_{el} \sim 20$ms. On this timescale, virtually no oscillations are seen. The story is different for $^{23}$Na where the smaller scattering length implies relaxation times almost 4 times longer than in Rb, and the ratio of $|c_{2}|/c_{0}$ is $10$ times larger. As shown in Fig. \[fig:-5\](a,b) an experiment with sodium atoms should be able to detect population dynamics. Polar state instability ----------------------- (100, 150)(10, 10) (-55, 170)[(a)]{} (68, 170)[(b)]{} (-55, 78)[(c)]{} (68, 78)[(d)]{} (68, 5)[ Collisionless evolution of the $m_{F} = 0~(+1)$ densities (solid/dotted curves) for $c_{2} = -c_{0}$, for an initial state $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin fluctuation), where $\epsilon = 0.01$. The dashed curve shows the $m_{F} = 0$ populations when $\psi_{initial} = \{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$ (nematic fluctuation). The densities are normalized to the total initial density $n_{0} = n_{00} + n_{11} + n_{-1-1}(t = 0)$. (b) A gapped and ungapped (solid blue and green) exist for $c_{2} = 0$. When $c_{2} \rightarrow -c_{0}$, (dashed blue and green curves) the energy of the gapped mode increases (green dashed) while the other mode (blue dashed) becomes unstable. The dashed orange curve is the imaginary part of this mode. (c) Collisionless evolution of central density of $m_{F} = 0, +1$ components (dotted/dashed respectively) for negative $c_{2}$ ($c_{2}/c_{0} \sim 0.05$ and $c_{2} \sim 2$Hz) for $q \ll |c_{2}|n$ ($B \sim 1$mG). Conservation of $S_{z}$ implies that the $m_{F} = -1$ component evolves in the same way as $m_{F} = 1$. (d) Collisionless evolution of coherences - top curve is the $n_{0-1}$ component and bottom is the $n_{1-1}$ component normalized to the initial total density. The $n_{1-1}$ component tends to -0.5, coherence is maintained, and the final state is a polar state along $x$.](fig13.eps "fig:")]{} (-50, 5)[ Collisionless evolution of the $m_{F} = 0~(+1)$ densities (solid/dotted curves) for $c_{2} = -c_{0}$, for an initial state $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin fluctuation), where $\epsilon = 0.01$. The dashed curve shows the $m_{F} = 0$ populations when $\psi_{initial} = \{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$ (nematic fluctuation). The densities are normalized to the total initial density $n_{0} = n_{00} + n_{11} + n_{-1-1}(t = 0)$. (b) A gapped and ungapped (solid blue and green) exist for $c_{2} = 0$. When $c_{2} \rightarrow -c_{0}$, (dashed blue and green curves) the energy of the gapped mode increases (green dashed) while the other mode (blue dashed) becomes unstable. The dashed orange curve is the imaginary part of this mode. (c) Collisionless evolution of central density of $m_{F} = 0, +1$ components (dotted/dashed respectively) for negative $c_{2}$ ($c_{2}/c_{0} \sim 0.05$ and $c_{2} \sim 2$Hz) for $q \ll |c_{2}|n$ ($B \sim 1$mG). Conservation of $S_{z}$ implies that the $m_{F} = -1$ component evolves in the same way as $m_{F} = 1$. (d) Collisionless evolution of coherences - top curve is the $n_{0-1}$ component and bottom is the $n_{1-1}$ component normalized to the initial total density. The $n_{1-1}$ component tends to -0.5, coherence is maintained, and the final state is a polar state along $x$.](fig12.eps "fig:")]{} (68, 95)[ Collisionless evolution of the $m_{F} = 0~(+1)$ densities (solid/dotted curves) for $c_{2} = -c_{0}$, for an initial state $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin fluctuation), where $\epsilon = 0.01$. The dashed curve shows the $m_{F} = 0$ populations when $\psi_{initial} = \{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$ (nematic fluctuation). The densities are normalized to the total initial density $n_{0} = n_{00} + n_{11} + n_{-1-1}(t = 0)$. (b) A gapped and ungapped (solid blue and green) exist for $c_{2} = 0$. When $c_{2} \rightarrow -c_{0}$, (dashed blue and green curves) the energy of the gapped mode increases (green dashed) while the other mode (blue dashed) becomes unstable. The dashed orange curve is the imaginary part of this mode. (c) Collisionless evolution of central density of $m_{F} = 0, +1$ components (dotted/dashed respectively) for negative $c_{2}$ ($c_{2}/c_{0} \sim 0.05$ and $c_{2} \sim 2$Hz) for $q \ll |c_{2}|n$ ($B \sim 1$mG). Conservation of $S_{z}$ implies that the $m_{F} = -1$ component evolves in the same way as $m_{F} = 1$. (d) Collisionless evolution of coherences - top curve is the $n_{0-1}$ component and bottom is the $n_{1-1}$ component normalized to the initial total density. The $n_{1-1}$ component tends to -0.5, coherence is maintained, and the final state is a polar state along $x$.](Fig21.eps "fig:")]{} (-55, 95)[ Collisionless evolution of the $m_{F} = 0~(+1)$ densities (solid/dotted curves) for $c_{2} = -c_{0}$, for an initial state $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin fluctuation), where $\epsilon = 0.01$. The dashed curve shows the $m_{F} = 0$ populations when $\psi_{initial} = \{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$ (nematic fluctuation). The densities are normalized to the total initial density $n_{0} = n_{00} + n_{11} + n_{-1-1}(t = 0)$. (b) A gapped and ungapped (solid blue and green) exist for $c_{2} = 0$. When $c_{2} \rightarrow -c_{0}$, (dashed blue and green curves) the energy of the gapped mode increases (green dashed) while the other mode (blue dashed) becomes unstable. The dashed orange curve is the imaginary part of this mode. (c) Collisionless evolution of central density of $m_{F} = 0, +1$ components (dotted/dashed respectively) for negative $c_{2}$ ($c_{2}/c_{0} \sim 0.05$ and $c_{2} \sim 2$Hz) for $q \ll |c_{2}|n$ ($B \sim 1$mG). Conservation of $S_{z}$ implies that the $m_{F} = -1$ component evolves in the same way as $m_{F} = 1$. (d) Collisionless evolution of coherences - top curve is the $n_{0-1}$ component and bottom is the $n_{1-1}$ component normalized to the initial total density. The $n_{1-1}$ component tends to -0.5, coherence is maintained, and the final state is a polar state along $x$.](fig.20.eps "fig:")]{} (62, 135) **$\omega/c_{0}n_{0}$** (-65, 125) **$n_{jj}/n_{0}$** (165, 87)[**$k\Lambda_{T}$**]{} (45, 87)[**$t(s)$**]{} (63, 34) **$n_{ij}/n_{0}$** (-62, 34) **$n_{jj}/n_{0}$** (165, 0)[**$t(s)$**]{} (45, 0)[**$t(s)$**]{} In Sec.II(B) we showed that the polar state of a gas with ferromagnetic interactions has a dynamically unstable mode at long wavelengths. Here we explicitly demonstrate this instability for a trapped gas with ferromagnetic interactions. However we also find that this instability is not observable in current experiments. Given the scattering lengths of $^{87}$Rb, our simulations indicate that the collision times are so short that the coherences necessary to drive this instability will never have time to develop. Additionally we explore details of the instability. We verify that the polar state is always unstable for small $q$, but the final state is dependent on the magnitude of $|c_{2}|/c_{0}$. For small values of this ratio ($\sim 10^{-3}$), finite populations of $m_{f} = \pm 1$ develop, but not all the atoms are transferred out of the $m_{F} = 0$ state. Increasing $c_{2}$ decreases the timescale for the onset of this instability, as well as increasing the fraction of the population transferred. The $m_{F} = 0$ state is stabilized at large Zeeman fields. Note that $S_{z}$ is always a conserved quantity. We start with a polar gas with small coherences, which seed the instability. We consider seeds with two different symmetries: $\textbf{F} = \psi^{*}\psi$ where $\psi = \{\psi_{1}, \psi_{0}, \psi_{-1}\} = \{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$ (nematic) or $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin), where $\epsilon \ll 1$, and we only keep terms ${\cal{O}}(\epsilon)$ in the density matrix. The Wigner functions are assumed to have a Maxwellian form in phase space. First consider $q = 0$ dynamics. As the polar state is unstable to spin fluctuations, an initial spin fluctuation grows exponentially, while a nematic fluctuation merely changes the size of the nematic director, but does not make it unstable (Fig. \[fig:-6\](a)). To enhance the size of the effect, we pick $c_{2} = -c_{0}$, but we observe an exponentially growing spin mode for typical values of $c_{2}$ as well. Although the linear stability analysis predicts an instability, it does not determine the final state, which depends in a complicated way on the long time dynamics. In Fig. \[fig:-6\](a) we show the collisionless evolution of the $m_{F} = 0 (+1)$ populations (solid/dashed) starting in the state $\{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ with $\epsilon = 0.01$. The dashed curve shows that for an initial state with nematic order $\{\frac{\epsilon}{\sqrt{2}}, 1, -\frac{\epsilon}{\sqrt{2}}\}$, no dynamics is seen. In Fig. \[fig:-6\](b) we plot the modes of the polar gas after solving (\[eq:8\]). As before there are two modes at $c_{2} = 0$. One mode has a gap of $(c_{0} - c_{2})n_{0}$ (green curves), which increases as $c_{2} \rightarrow -c_{0}$. The other ungapped mode (blue curves) develops an imaginary part (orange) which increases as $c_{2}$ is made more negative. The quadratic Zeeman effect stabilizes the polar state. Given an initial spin fluctuation about the unmagnetized state, this spin precesses in the Zeeman field at a rate proportional to $q$. If the initial state only has a nematic perturbation, then a finite $q$ is needed to produce the spin fluctuations required to drive an instability. Upon increasing $q$, there are two effects. First there appear wave-vectors where the frequency is real (\[eq:8\]), and oscillations are observed along with an exponential growth of spin populations. Second, the spin vector precesses more rapidly, and averages to zero on a timescale of $\sim |c_{2}|n$, ultimately stabilizing the polar state. In Fig. \[fig:-6\](c) we plot the evolution of the central density of the $m_{F} = 0, +1$ sub-levels in time for $q \sim 10$mHz, in a collisionless gas. We take $c_{2}n \sim 2$Hz, such that $|c_{2}|/c_{0} \sim 0.05$, instead of the $0.005$ in $^{87}$Rb. In Fig. \[fig:-6\](d) the coherences are shown. The timescale for the onset of the instability, as well as the final state depends on the size of the off-diagonal seed, and the nature of the perturbation i.e. whether it is a spin or a nematic fluctuation or both, and the strength of the interactions. Here we have shown one final state, $\psi_{f} = \{\frac{1}{\sqrt{2}}, 0,-\frac{1}{\sqrt{2}} \}$. Note that the final state is not simply an incoherent mixture, unlike in a collisional gas. The distribution functions for the $m_{F} = \pm1$ components are identical, and no local magnetization is observed. Local structures may be observed in an experiment where the trapping potentials are made spin dependent. Performing the simulation with a relaxationn rate approximation, taking into acccount collision times ($\tau \sim 50$ms), and scattering lengths for $^{87}$Rb, we do not observe any dynamics in the polar state. (50, 100)(10, 10) (-50, 15)[![\[fig:-7\] Evolution of central density of $m_{F} = 0$ component (normalized to initial density $n_{0} = n_{00} + n_{11} + n_{-1-1} (t = 0)$) as a function of time for $c_{2} = 0.5 c_{0}$ without (solid) and with the relaxation rate approximation (dashed). The initial state is $\textbf{F} = \psi^{*}\psi$, where $\psi = \{\frac{\epsilon}{\sqrt{2}}, 1, \frac{\epsilon}{\sqrt{2}}\}$ (spin fluctuation), where $\epsilon = 0.01$. Trap parameters: $5 \times 10^{5}$ atoms, $\omega_{r} = 2\pi \times 250$Hz, $\omega_{z} = 2\pi \times 50$Hz. The relaxation time $\tau \sim 50$ms. For definiteness we assumed $^{87}$Rb atoms, even though the scattering lengths in Rb, do not obey this relationship. Owing to the strong interactions, one should be able to observe this instability even for reasonable collision rates.](fig19.eps "fig:")]{} (85, 5)[**$t(s)$**]{} (-60, 60) **$n_{00}/n_{0}$** *Large anti-ferromagnetic interactions*: Next we consider the case where the spin-dependent and spin-independent contact interactions are comparable in magnitude, and both positive. In Fig. \[fig:-7\] we plot the central densities of the $m_{F} = 0$ atoms as a function of time for $c_{2} = 0.5c_{0}$ without (solid curve) and with (dashed) the relaxation time approximation, demonstrating the instability. Due to the large interaction energies, the dynamics is complicated. As Endo and Nikuni [@nikuni] have shown, one must be careful while considering the relaxation time approximation in the $c_{0} \sim c_{2}$ limit where more complicated spin dependent collisions may become important. In order to fully model the physics, more than one “relaxation-rate" may be required. Summary and Outlook =================== We have addressed the role of the spin-dependent interaction in the collisionless dynamics of a thermal spinor gas. By calculating spin and quadrupolar modes, about the magnetized and unmagnetized states, we have shown that the normal state of a spinor has a rich array of spin excitations compared to its well-studied pseudo-spin $\frac{1}{2}$ counterpart. We numerically calculated the dynamics of a non-stationary initial state, finding that the spin-dependent contact interaction drives population dynamics. Finally we provide an explicit demonstration of the instability of the polar state to spin fluctuations. We conclude that many interesting experiments may be done in the *normal* state of a spinor gas. In order to observe some of the physics described here, it will be necessary to attain a limit where the spin dependent and independent interactions are commensurate in magnitude. We hope that this work motivates experiments in that direction. *Acknowledgements*:$-$ S.N. would like to thank Stefan K. Baur and Kaden R.A Hazzard for discussions. We thank Mukund Vengalattore for pointing out [@gerbier] to us, for several helpful discussions and comments on this manuscript. This work was supported by NSF Grant No. PHY-0758104. [99]{} M.Vengalattore, J.M.Higbie, S.R.Leslie, J.Guzman, L.E.Sadler and D.M.Stamper-Kurn, Phys.Rev.Lett., **98** 200801 (2007). B.Sun, W.X.Zhang, S.Yi, M.S.Chapman and L.You, Phys. Rev. Lett., **97** 123201, (2006). J.Kronjäger, C.Becker, P.Navez, K.Bongs and K.Sengstock, Phys.Rev.Lett., **97** 110404 (2006). E.P.Bashkin, Usp. Fiz. Nauk. **148** 433-471 (1986). 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A **73** 041602(R) (2006). M-S Chang, Q. Qin, W. Zhang, L. You and M. S. Chapman, Nature Physics, **1** 111-116, (2005). D.Guéry-Odelin, F.Zambelli, J.Dalibard, and S.Stringari, Phys Rev. A, **60** 4851, (1999). S.S.Natu and E.J. Mueller, Phys. Rev. A, **79** 051601(R), (2009). *Numerical Recipes Third Edition: The Art of Scientific Computing* W.H.Press, S.A. Teukolsky, W.T.Vetterling, and B.P.Flannery, Cambridge University Press, 2007. [^1]: This work was motivated by discussions with Mukund Vengalattore.

--- abstract: 'We present explicit formulas for deformation quantization on the co-adjoint orbits of the real diamond Lie group. From this we obtain quantum half-plans, quantum hyperbolic cylinders, quantum hyperbolic paraboloids via Fedosov deformation quantization and finally, the corresponding unitary representations of this group.' address: 'Haiphong Teacher’s Training College ,Haiphong city,Vietnam' author: - Nguyen Viet Hai date: 'Version of December 25,1999' title: 'Quantum co-adjoint orbits of the real diamond group' --- [^1] Introduction ============ Let us first recall that it was Hermann Weyl(see [@weyl]), who introduced a mapping from classical observables (i.e. functions on the phase space $\mathbb R^{2n}$) to quantum observables (i.e. normal operators in the Hilbert space $L^2(\mathbb R^n)$). The idea was to express functions on $\mathbb R^{2n}$ as Fourier transforms and then, by using the inverse Fourier transforms to correspond this correspondence to functions on characters, i.e. one-dimensional representations of the Heisenberg group, parameterized by the Planck constant $\hbar$ - and finally to present them as elements in the corresponding infinite-dimensional representations of the Heisenberg group. This profound idea was later retrieved by Moyal, who have seen that the symbols of the commutators or of the products of operators are of the the form of $sine$ (or $exponential$) functions of the bidifferential operators (of the Poisson brackets) of the corresponding symbols. In the early 70’s Berezin has treated the general mathematical definition of quantization as a kind of a functor from the category of classical mechanics to a certain category of associative algebras. About the same time as F. A. Berezin, M. Flato, M. G. Fronsdal, F. Bayen, A. Lichnerowicz and D. Sternheimer considered quantization as a deformation of the commutative products of classical observables into a noncommutative $\star $-products which are parameterized by the Planck constant $\hbar$ and satisfy the [*correspondence principle*]{}. They systematically developed the notion of deformation quantization as a theory of $\star$-products and gave an independent formulation of quantum mechanics based on this notion (see[@rt]). It was proved by Gerstenhaber that a formal deformation quantization exists on an arbitrary symplectic manifold, see for example [@fedosov] for a detailed explaination. It is however formal and quite complecate in general. We would like to simplify it in some particular cases. From the orbit method, it is well-known that coadjoint orbits are homogeneous symplectic manifolds with respect to the natural Kirillov structure form on coadjoint orbits. A natural question is to associate in a reasonable way to these orbits some quantum objects, what could be called [*quantum co-adjoint orbits*]{}. In particular, in [@diephai1] and [@diephai2] we obtained “quantum half-planes” and “quantum punctured complex planes", associated with the affine transformation groups of the real or complex straightlines. In this paper we will therefore continue to realize the problem for the real diamond Lie group. This group has a lot of nontrivial 2-dimensional coadjoint orbits, which are the half-planes, the hyperbolic cylinders and the hyperbolic paraboloids. We should find out explicit formulas for each of these orbits. Our main result therefore is the fact that by using $\star$-product we can construct the corresponding quantum half-plans, quantum hyperbolic cylinders, quantum hyperbolic paraboloids and by an [*exact computation*]{} we can find out explicit $\star$-product formulas and then, the complete list of irreducible unitary representations of this group. It is useful to do here a remark that there is a general theory for exponetial and compact groups. But our consideration concerning with non-exponential and noncompact Lie group and associated $G$-homogeneous symplectic manifolds. Let us in few words describe the structure od the paper. We introduce some preliminary results in §2. Then, the adapted chart and in particular, Hamiltonian functions in canonical coordinates of the co-adjoint orbit $\Omega_F$ are exposed in §3. The operators $\hat{\ell}_A$ which define the representations of the real diamond Lie algebra are constructed in §4 and finally, by exponentiating them, we obtain the corresponding unitary representations of the real diamond Lie group $\mathbb R \ltimes \mathbb H_3$. Preliminary results =================== The so called real diamond Lie algebra is the 4-dimensional solvable Lie algebra $\mathfrak g$ with basis X, Y, Z, T satisfying the following commutation relations: $$[X,Y]=Z,[T,X]=-X,[T,Y]=Y,$$ $$[Z,X]=[Z,Y]=[T,Z]=0.$$ These relations show that this real diamond Lie algebra $\mathbb R\ltimes \mathfrak h_3$ is an extension of the one-dimensional Lie algebra $\mathbb R T$ by the Heisenberg algebra $\mathfrak h_3$ with basis $X, Y, Z$, where the action of T on Heisenberg algebra $\mathfrak h_3$ is defined by the matrix $$\ ad_T=\left(\begin{array}{ccc}-1&0&0\\0&1&0\\0&0&0 \end{array}\right).$$ We introduce the following notations. The real diamond Lie algebra is isomorphic to $\mathbb R^4$ as vector spaces. The coordinates in this standard basis is denote by $(a,b,c,d)$. We identify its dual vector space $\mathfrak g^*$ with $\mathbb R^4$ with the help of the dual basis $X^*,Y^*,Z^*,T^* $ and with the local coordinates as $(\alpha,\beta, \gamma,\delta)$. Thus, the general form of an element of $\mathfrak g $ is $U=aX+bY+cZ+dT$ and the general form of an element of $\mathfrak g^* $ is $F=\alpha{X^*} +\beta{Y^*}+\gamma{Z^*}+\delta{T^*}$. The co-adjoint action of $G=\mathbb R \ltimes \mathbb H_3$ on $\mathfrak g^*$ is given (see e.g. [@kirillov1]) by $$\langle K(g)F, U\rangle = \langle F, \Ad(g^{-1})U\rangle,\quad \forall F \in \mathfrak g^*, g \in G \mbox{ and } U \in \mathfrak g=Lie(\mathbb R \ltimes \mathbb H_3).$$ Denote the co-adjoint orbit of $G$ in $\mathfrak g$, passing through $F$ by $$\Omega_F = K(G)F := \{K(g)F \vert g \in G \}.$$ By a direct computation one obtains (see [@dndiep]): - Each point of the line $\alpha=\beta=\gamma =0$ is a 0-dimensional co-adjoint orbit $$\Omega^1=\Omega_{(0,0,0,\delta)}.$$ - The set $\alpha\ne 0,\beta=\gamma=0$ is union of 2-dimensional co-adjoint orbits ,which are just the $$\Omega^2=\{(x,0,0,t)\quad\vert\quad x, t\in \mathbb R , \alpha x>0\}.$$ - The set $\alpha=\gamma=0,\beta\ne 0$ is a union of 2-dimensional co-adjoint orbits, which are $$\Omega^3=\{(0,y,0,t)\quad\vert\quad y, t\in\mathbb R, \beta y >0 \}.$$ - The set $\alpha\beta\ne 0,\gamma=0$ is decomposed into a family of 2-dimensional co-adjoint orbits, which are [**hyperboliccylinders** ]{} $$\Omega^4=\{(x,y,0,t)\quad\vert x,y,t\in\mathbb R\quad\&\quad \alpha x>0,\beta y>0,xy=\alpha\beta\}.$$ - The open set$ \gamma\ne 0$ is decomposed into a family of 2-dimensional co-adjoint orbits ,which are just the [**hyperbolicparaboloids**]{} $$\Omega^5=\{(x,y,\gamma,t)\quad\vert x,y,t \in\mathbb R \quad\&\quad xy- \alpha\beta=\gamma(t-\delta)\}.$$ Thus, the real diamond Lie algebra belongs to the class of $MD_4$-algebras , i.e. every K-orbit of the corresponding Lie group has dimension 0 or maximal (see [@dndiep]). Let us consider now the problem of deformation quantization on half-planes, hyperboliccylinders, hyperbolicparaboloids. In order to do this, we shall construct on each of these orbits a canonical Darboux coordinate system $(p,q)$ and a class of Hamiltonian functions in these coordinates. Hamiltonian functions in canonical coordinates of the orbits $\Omega_F$ ======================================================================= Each element $A\in\mathfrak g$ can be considered as the restriction of the corresponding linear functional $\tilde A$ onto co-adjoint orbits, considered as a subset of $g^*$ ,$\tilde A(F)=\langle F,A\rangle$. It is well-known that this function is just the Hamiltonnian function, associated with the Hamiltonian vector field $\xi_A$, defined by the formula $$(\xi_Af)(x):=\frac {d}{dt}f(x\exp(tA))\vert_{t=0},\forall f\in C^\infty(\Omega _F).$$ It is well-known the relation $\xi_A(f)=\{\tilde A,f\},\forall f\in C^\infty(\Omega_F)$. Denote by $\psi$ the symplectomorphism from $\mathbb R^2$ onto $\Omega_F$ $$(p,q) \in \mathbb R^2 \mapsto \psi(p,q) \in \Omega_F,$$ we have: 1\. Hamiltonian function $ \tilde{A}$ in canonical coordinates $(p,q)$ of the orbit $\Omega_F$ is of the form $$\tilde{A}\circ\psi(p,q) =\cases$$dp+ae\^[-q]{} ,$$& \text{if \quad $ \Omega_F=\Omega^2$}\\$$dp+ be\^q, $$& \text{if \quad $ \Omega_F=\Omega^3$}\\$$ dp+ae\^[-q]{}+be\^q, $$&\text{if \quad $ \Omega_F=\Omega^4$}\\$$(dbe\^q)pae\^[-q]{}b(-)e\^q+c, $$&\text{if \quad $ \Omega_F=\Omega^5$}\\ \endcases$$ 2. In the canonical coordinates $(p,q)$ of the orbit $\Omega_F $, the Kirillov form $\omega$ is coincided with the standard form $ dp \wedge dq$. [*Proof*]{}. 1. We adapt the diffeomorphism $\psi$ to each of the following cases (for 2-dimensional co-adjoint orbits, only) - With $\alpha\ne 0,\beta=\gamma=0$ $$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(\alpha e^{-q},0,0,p) \in \Omega^2$$ Element $F\in \mathfrak g^*$ is of the form $F=\alpha{X^*} +\beta{Y^*}+\gamma{Z^*}+\delta{T^*}$, hence the value of the function $f_A=\tilde A$ on the element $A=aX+bY+cZ+dT$ is $\tilde A(F)=\langle F,A\rangle= $ $$\langle\alpha{X^*} +\beta{Y^*}+\gamma{Z^*}+\delta{T^*},aX+bY+cZ+dT \rangle=\alpha a+\beta b+\gamma c+\delta d .$$ It follows that $$\tilde{A}\circ\psi(p,q) =a\alpha e^{-q} + dp,$$ - With $\alpha=\gamma=0,\beta\ne 0$,$$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(0,\beta e^q,0,p) \in \Omega^3.$$$\tilde A(F)=\langle F,A\rangle =\alpha a+\beta b+\gamma c+\delta d . $ From this, $$\tilde{A}\circ\psi(p,q) =b\beta e^q + dp$$ - With $\alpha\beta\ne 0,\gamma=0$, $$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(\alpha e^{-q},\beta e^{q},0,p) \in \Omega^4.$$$$\tilde{A}\circ\psi(p,q) =a\alpha e^{-q}+b\beta e^q + dp$$ - At last, if $ \gamma\ne 0$, we consider the orbit with the first coordinate $ x>0$ $$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(e^{-q},(\alpha\beta+\gamma p-\gamma\delta)e^q,\gamma,p) \in \Omega^5.$$ We have $$\tilde{A}\circ\psi(p,q) =ae^{-q}+b(\alpha\beta +\gamma p-\gamma\delta)e^q+c\gamma + dp=$$ $$=(d+b\gamma e^q)p+ ae^{-q}+b(\alpha\beta-\gamma\delta)e^q+c\gamma.$$ The case $x<0$ is similarly treated:$$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(-e^{-q},-(\alpha\beta+\gamma p-\gamma\delta)e^q,\gamma,p) \in \Omega^5.$$ $$\tilde{A}\circ\psi(p,q) =-ae^{-q}-b(\alpha\beta +\gamma p-\gamma\delta)e^q+c\gamma + dp=$$ $$=(d-b\gamma e^q)p- ae^{-q}-b(\alpha\beta-\gamma\delta)e^q+c\gamma.$$ 2\. We consider only the following case (the rest are similar): $$(p,q) \in \mathbb R^2 \mapsto \psi(p,q)=(e^{-q},(\alpha\beta+\gamma p-\gamma\delta)e^q,\gamma,p) \in \Omega^5.$$ $$\tilde{A}\circ\psi(p,q) =(d+b\gamma e^q)p+ ae^{-q}+b(\alpha\beta-\gamma\delta)e^q+c\gamma.$$ In canonical Darboux coordinates $(p,q)$ ,$$F'=e^{-q}X^*+(\alpha\beta+\gamma p-\gamma\delta)e^q Y^*+\gamma Z^*+pT^*\quad\in\Omega^5,$$ and for $A=aX+bY+cZ+dT,\quad B=a'X+b'Y+c'Z+d'T$, we have $\langle F',[A,B]\rangle=$ $$=\langle e^{-q}X^*+(\alpha\beta+\gamma p-\gamma\delta)e^q Y^*+\gamma Z^*+pT^*,(ad'-da')X+(db'-bd')Y+(ab'-ba')Z\rangle.$$It follows therefore that $$\langle F',[A,B]\rangle=(ad'-da')e^{-q}+(db'-bd')(\alpha\beta+\gamma p-\gamma\delta)e^q+\gamma(ab'-ba').$$ On the other hand, $$\xi_A(f)=\{\tilde{A},f\}=(d+b\gamma e^q)\frac{\partial f}{\partial q}-[-ae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial f}{\partial p}$$ $$\xi_B(f)=\{\tilde{B},f\} =(d'+b'\gamma e^q)\frac{\partial f}{\partial q}-[-a'e^{-q}+b'(\alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial f}{\partial p}.$$From this, consider two vector fields $$\xi_A=(d+b\gamma e^q)\frac{\partial}{\partial q}-[-ae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial}{\partial p},$$ $$\xi_B=(d'+b'\gamma e^q)\frac{\partial}{\partial q}-[-a'e^{-q}+b'(\alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial}{\partial p}.$$ We have $$\xi_A\otimes \xi_B =(d+b\gamma e^q)(d'+b'\gamma e^q) \frac{\partial}{\partial q}\otimes \frac {\partial}{\partial q}+$$ $$+[(ad'-da')e^{-q}+(db'-d'b)(\alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial}{\partial p}\otimes \frac {\partial}{\partial q}+$$ $$+[-ae^{-q}+b( \alpha\beta+\gamma p-\gamma\delta)e^q][-a'e^{-q}+b'( \alpha\beta+\gamma p-\gamma\delta)e^q]\frac{\partial}{\partial p}\otimes \frac {\partial}{\partial p}$$ From (11) and (12) we conclude that in the canonical coordinates the Kirillov form is just the standard symplectic form $\omega = dp \wedge dq$. The proposition is therefore proved.$\square$ Each chart $\psi^{-1}$ on $\Omega_F$ which satisfy 1. and 2. of proposition 3.1 is called an adapted chart on $\Omega_F $. In the next section we shall see that each adapted chart carries the Moyal $\star$-product from $\mathbb R^2$ onto $\Omega_F$. Moyal $\star$-product and representations of $G=\mathbb R \ltimes \mathbb H_3.$ =============================================================================== Let us denote by $\Lambda$ the 2-tensor associated with the Kirillov standard form $\omega =dp \wedge dq$ in canonical Darboux coordinates. Let us consider the well-known Moyal $\star$-product of two smooth functions $u,v \in C^\infty(\mathbb R^{2n})$ (see e.g [@arnalcortet1],[@diephai1]), defined by $$u \star v = u.v + \sum_{r \geq 1} \frac{1}{r!}(\frac{1}{2i})^r P^r(u,v),$$ where $$P^1(u,v)=\{u,v\}$$ $$P^r(u,v) := \Lambda^{i_1j_1}\Lambda^{i_2j_2}\dots \Lambda^{i_rj_r}\partial^r_{i_1i_2\dots i_r} u \partial^r_{j_1j_2\dots j_r}v,$$ with $$\partial^r_{i_1i_2\dots i_r} := \frac{\partial^r}{\partial x^{i_1}\dots \partial x^{i_r}};\quad x:= (p,q) = (p_1,\dots,p_n,q^1,\dots,q^n)$$ using multi-index notation. It is well-known that this series converges in the Schwartz distribution spaces $\mathcal S (\mathbb R^{2n})$. Furthermore, it was obtained the results (see e.g [@arnalcortet1]): If $u,v \in \mathcal S (\mathbb R^{2n})$, then - $\overline u\star\overline v=\overline{v\star u}$ - $\int(u\star v)(\xi)d\xi=\int uv d\xi $ - $\ell_u : \quad \mathcal S (\mathbb R^{2n}) \longrightarrow \mathcal S (\mathbb R^{2n}) $, defined by $\ell_u(v)= u\star v$ is continuous in $L^2(\mathbb R^{2n},d\xi)$ and then can be extended to a bounded linear operator (still denoted by $\ell_u$ ) on $L^2(\mathbb R^{2n},d\xi)$. We apply this to the special case $n=1$, $x=(x^1,x^2)=(p,q)$ In the above mentioned canonical Darboux coordinates $(p,q)$ on the orbit $\Omega_F$, the Moyal $\star$-product satisfies the relation $$i\tilde{A} \star i\tilde{B} - i\tilde{B} \star i\tilde{A} = i\widetilde{[A,B]}, \forall A, B \in \mathfrak g=Lie(\mathbb R \ltimes \mathbb H_3).$$ [*Proof*]{}. We prove the proposition for the orbit $\Omega^5$, $\tilde{A} =(d+b\gamma e^q)p +ae^{-q}+b(\alpha\beta-\gamma\delta)e^q+c\gamma $ (the other cases are proved similar). Consider the elements $A=aX+bY+cZ+dT,\quad B=a'X+b'Y+c'Z+d'T$, . Then as said above, the corresponding Hamiltonian functions are $$\tilde{A} =(d+b\gamma e^q)p +ae^{-q}+b(\alpha\beta-\gamma\delta)e^q+c\gamma$$ $$\tilde{B} =(d'+b'\gamma e^q)p+a'e^{-q}+b'(\alpha\beta-\gamma\delta)e^q+c'\gamma$$ It is easy then to see that $$\begin{aligned} &&P^0(\tilde{A},\tilde{B}) = \tilde{A}.\tilde{B}\\ && P^1(\tilde{A},\tilde{B}) = \{ \tilde{A},\tilde{B}\} = \partial_p\tilde{A}\partial_q \tilde{B} - \partial_q\tilde{A}\partial_p \tilde{B}=\\ && = (d+b\gamma e^q)[-a'e^{-q}+b'(\alpha\beta+\gamma p-\gamma\delta)e^q]-\\ &&-(d'+b'\gamma e^q)[-ae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^q]= \\ &&=[(ad'-da')e^{-q}+(db'-d'b)(\alpha\beta+\gamma p-\gamma\delta)e^q+(ab'-ba')\gamma]\\ &&P^2(\tilde{A},\tilde{B}) = \Lambda^{12}\Lambda^{12}\partial^2_{pp}\tilde{A} \partial^2_{qq}\tilde{B} + \Lambda^{12}\Lambda^{21}\partial^2_{pq}\tilde{A} \partial^2_{qp}\tilde{B} + \Lambda^{21}\Lambda^{12}\partial^2_{qp}\tilde{A} \partial^2_{pq}\tilde{B} + \\&&+\Lambda^{21}\Lambda^{21}\partial^2_{qq}\tilde{A} \partial^2_{pp}\tilde{B} = -2bb'\gamma^2 e^{2q}\\ && P^3(\tilde{A},\tilde{B}) = \Lambda^{12}\Lambda^{12}\Lambda^{12}\partial^3_{ppp}\tilde{A} \partial^3_{qqq}\tilde{B} + \Lambda^{12}\Lambda^{12}\Lambda^{21}\partial^3_{ppq}\tilde{A} \partial^3_{qqp}\tilde{B} + \\&& + \Lambda^{12} \Lambda^{21}\Lambda^{12}\partial^3_{pqp}\tilde{A} \partial^3_{qpq}\tilde{B} + \Lambda^{21}\Lambda^{12}\Lambda^{12}\partial^3_{qpp}\tilde{A} \partial^3_{pqq}\tilde{B}+ \\&& + \Lambda^{21}\Lambda^{21}\Lambda^{12}\partial^3_{qqp}\tilde{A} \partial^3_{ppq}\tilde{B}+ \Lambda^{21}\Lambda^{12}\Lambda^{21}\partial^3_{qpq}\tilde{A} \partial^3_{pqp}\tilde{B}+\\&& + \Lambda^{12}\Lambda^{21}\Lambda^{21}\partial^3_{pqq}\tilde{A} \partial^3_{qpp}\tilde{B}+\Lambda^{21}\Lambda^{21}\Lambda^{21}\partial^3_{qqq}\tilde{A} \partial^3_{ppp}\tilde{B}=0 . \end{aligned}$$ By analogy, we have $$P^k(\tilde{A},\tilde{B}) = 0, \forall k \geq 4.$$ Thus, $$i\tilde{A} \star i\tilde{B} - i\tilde{B} \star i\tilde{A} = \frac{1}{2i}[P^1(i\tilde{A} , i\tilde{B}) - P^1(i\tilde{B}, i\tilde{A})]$$ $$= i[(ad'-da')e^{-q}+(db'-d'b)(\alpha\beta+\gamma p-\gamma\delta)e^q+(ab'-a'b)\gamma].$$ On the other hand, as $$[A,B] = [aX+bY+cZ+dT,a'X+b'Y+c'Z+d'T]$$ $$=(ad'-da')X+(db'-d'b)Y+(ab'-a'b)Z$$ we obtain $$i[(ad'-da')e^{-q}+(db'-d'b)(\alpha\beta+\gamma p-\gamma\delta)e^q+(ab'-a'b)\gamma]$$ $$=i\widetilde{[A,B]} = i\tilde{A} \star i\tilde{B} - i\tilde{B} \star i\tilde{A} .$$ The proposition is hence proved.$\square$ Consequently, to each adapted chart, we associate a $G$-covariant $\star$-product.Then there exists a representation $\tau$ of $G$ in $\Aut N[[\nu]]$ ,(see [@gutt]) such that (here $\nu=\frac i 2$): $$\tau(g)(u \star v) = \tau(g)u \star \tau(g)v.$$ For each $A \in Lie(\mathbb R\ltimes\mathbb H_3)$, the corresponding Hamiltonian function is $\tilde{A}$ and we can put $\ell_A(u) = i\tilde{A} \star u$,$u\in L^2(\mathbb R^2, \frac{dpdq}{2\pi})^\infty$. It is then continuated to the whole space $L^2(\mathbb R^2, \frac{dpdq}{2\pi})$. Because of the relation in Proposition (4.1), we have $$\ell_{[A,B]}=\ell_A\star\ell_B -\ell_B\star\ell_A:=[\ell_A,\ell_B]^\star$$ This implies that the correspondence $A \in Lie(\mathbb R\ltimes\mathbb H_3) \mapsto \ell_A = i\tilde{A} \star .$ is a representation of the Lie algebra $ Lie(\mathbb R\ltimes\mathbb H_3)$ on the space $N[[\frac{i}{2}]]$ of formal power series in the parameter $\nu = \frac{i}{2}$(i.e $\hbar =1$) with coefficients in $N = C^\infty(M,\mathbb R)$ [@gutt]. Let us denote by $\mathcal F_p(f)$ the partial Fourier transform of the function $f$ from the variable $p$ to the variable $x$(see e.g[@meisevogt]), i.e. $$\mathcal F_p(f)(x,q) := \frac{1}{\sqrt{2\pi}}\int_{\mathbb R} e^{-ipx} f(p,q)dp.$$ Let us denote by $ \mathcal F^{-1}_p(f) (p,q)$ the inverse Fourier transform. 1.$\partial_p \mathcal F^{-1}_p(f) = i \mathcal F^{-1}_p(x.f)$ 2.$\mathcal F_p(p.v) = i \partial_x\mathcal F_p(v) $ 3.$\forall k\geq 2$ ,then $P^k(\tilde{A},\mathcal F^{-1}_p(f))=$ $$=\cases$$ae\^[-q]{}\^k\_[p…p]{}F\^[-1]{}\_p(f) $$&\text{if $\tilde {A}$ is defined by (6) }\\$$(-1)\^k be\^q\^k\_[p…p]{}F\^[-1]{}\_p(f)$$&\text{if $\tilde {A}$ is defined by (7)}\\$$\[ae\^[-q]{}+(-1)\^k be\^q\] \^k\_[p…p]{}F\^[-1]{}\_p(f)$$&\text{if $\tilde {A}$ is defined by (8)} \\$$(-1)\^[k-1]{}k.be\^q \^k\_[qp…p]{}F\_p\^[-1]{}(f)+$$\\$$+ \[a e\^[-q]{}+(-1)\^k b(+p-)e\^q\]\^k\_[p…p]{}F\_p\^[-1]{}(f) $$&\text{if $\tilde {A}$ is defined by (9) }\\ \endcases$$ [*Proof*]{}. The first two formulas are well-known from theory of Fourier transforms. Let us prove 3. Remark that $\Lambda = \left(\begin{array}{cc} 0 & -1\\ 1 & 0 \end{array}\right)$ in the standard symplectic Darboux coordinates $(p,q)$ on the orbit $\Omega_F$, then - If $ \tilde A=a\alpha e^{-q}+dp $ $$\begin{aligned} && P^2(\tilde{A},\mathcal F^{-1}_p(f)) = \Lambda^{12}\Lambda^{12}\partial^2_{pp}\tilde{A} \partial^2_{qq}\mathcal F^{-1}_p(f)) + \Lambda^{12}\Lambda^{21}\partial^2_{pq}\tilde{A} \partial^2_{qp}\mathcal F^{-1}_p(f)) + \\ &&\Lambda^{21}\Lambda^{12}\partial^2_{qp}\tilde{A} \partial^2_{pq}\mathcal F^{-1}_p(f)) +\Lambda^{21}\Lambda^{21}\partial^2_{qq}\tilde{A} \partial^2_{pp}\mathcal F^{-1}_p(f))=a\alpha e^{-q}\partial^2_{pp}\mathcal F^{-1}_p(f)=\\ &&P^3(\tilde A,\mathcal F^{-1}_p(f))=(-1)^6a\alpha e^{-q}\partial^3_{ppp}\mathcal F^{-1}_p(f)=a\alpha e^{-q}\partial^3_{ppp}\mathcal F^{-1}_p(f)\end{aligned}$$ and $P^k(\tilde A,\mathcal F^{-1}_p(f))=a\alpha e^{-q}\partial^k_{p\dots p}\mathcal F^{-1}_p(f)\quad\forall k \geq 4 ,$ - If $\tilde A=b\beta e^q+dp.$ $$P^k(\tilde A,\mathcal F^{-1}_p(f))=(-1)^k b\beta e^q\partial^k_{p\dots p}\mathcal F^{-1}_p(f)$$with $\quad\forall k \geq 2 $ - If $ \tilde{A}=a\alpha e^{-q}+b\beta e^q+dp,$ $$\begin{aligned} &&P^2(\tilde{A},\mathcal F^{-1}_p(f)) = \Lambda^{12}\Lambda^{12}\partial^2_{pp}\tilde{A} \partial^2_{qq}\mathcal F^{-1}_p(f)) + \Lambda^{12}\Lambda^{21}\partial^2_{pq}\tilde{A} \partial^2_{qp}\mathcal F^{-1}_p(f)) + \\ &&\Lambda^{21}\Lambda^{12}\partial^2_{qp}\tilde{A} \partial^2_{pq}\mathcal F^{-1}_p(f)) +\Lambda^{21}\Lambda^{21}\partial^2_{qq}\tilde{A} \partial^2_{pp}\mathcal F^{-1}_p(f))=\\ && =[a\alpha e^{-q}+(-1)^2b\beta e^q]\partial^2_{pp}\mathcal F^{-1}_p(f)\\ &&P^3(\tilde{A},\mathcal F^{-1}_p(f)) = [a\alpha e^{-q}+(-1)^3b\beta e^q]\partial^3_{ppp}\mathcal F^{-1}_p(f)\end{aligned}$$ By analogy we have $$P^k(\tilde{A}, \mathcal F^{-1}_p(f) ) = [a\alpha e^{-q}+(-1)^k b\beta e^q] \partial^k_{p\dots p}\mathcal F^{-1}_p(f)),\quad \forall\quad k \geq 3.$$ - If$ \tilde{A}=(d+b\gamma e^q)p +ae^{-q}+b(\alpha\beta -\gamma\delta)e^q +c\gamma, $ $$\begin{aligned} &&P^2(\tilde{A},\mathcal F^{-1}_p(f)) = \Lambda^{12}\Lambda^{12}\partial^2_{pp}\tilde{A} \partial^2_{qq}\mathcal F^{-1}_p(f)) + \Lambda^{12}\Lambda^{21}\partial^2_{pq}\tilde{A} \partial^2_{qp}\mathcal F^{-1}_p(f)) + \\ &&\Lambda^{21}\Lambda^{12}\partial^2_{qp}\tilde{A} \partial^2_{pq}\mathcal F^{-1}_p(f)) +\Lambda^{21}\Lambda^{21}\partial^2_{qq}\tilde{A} \partial^2_{pp}\mathcal F^{-1}_p(f)) =\\ &&=(-1) 2. b\gamma e^q \partial_{qp}\mathcal F_p^{-1}(f)+[ae^{-q}+(-1)^2b(\alpha\beta+\gamma p-\gamma\delta)e^q]\partial^2_{pp}\mathcal F_p^{-1}(f).\\ &&P^3(\tilde{A},\mathcal F^{-1}_p(f))=(-1)^2.3b\gamma e^q\partial_{qpp}\mathcal F_p^{-1}(f)+\\&&+[ae^{-q}+(-1)^3b(\alpha\beta+\gamma p-\gamma\delta)e^q]\partial^3_{ppp}\mathcal F_p^{-1}(f).\end{aligned}$$ From this we also obtain : $$\begin{aligned} &&P^k(\tilde{A}, \mathcal F^{-1}_p(f) )=\\ &&(-1)^{k-1}.k.b\gamma e^q\partial^k_{qp\dots p}\mathcal F_p^{-1}(f) +\\&&+[ae^{-q}+(-1)^k b(\alpha\beta+\gamma p-\gamma\delta)e^q ]\partial^k_{p\dots p}\mathcal F^{-1}_p(f).\quad\forall\quad k\geq 3 \end{aligned}$$ The lemma is therefore proved.$\square$ We study now the convergence of the formal power series. In order to do this, we look at the $\star$-product of $i\tilde{A}$ as the $\star$-product of symbols and define the differential operators corresponding to $i\tilde{A}$. For each $A \in Lie(\mathbb R \ltimes \mathbb H_3)$ and for each compactly supported $C^\infty$ function $f \in C^\infty_0(\mathbb R^2)$, putting $\hat{\ell}_A(f) := \mathcal F_p \circ \ell_A \circ \mathcal F^{-1}_p(f)$,we have $$\hat{\ell}_A(f) = \cases$$\[d(\_q-\_x)+iae\^[-(q-)]{}\]f $$&\text{if $\tilde {A}$ is defined by (6) }\\$$\[d(\_q-\_x)+ibe\^[(q-)]{}\]f $$&\text{if $\tilde {A}$ is defined by (7)}\\$$\[d(\_q-\_x)+i(ae\^[-(q-)]{}+ be\^[(q-)]{})\]f $$&\text{if $\tilde {A}$ is defined by (8)}$$\ $$[(d+ b\gamma e^{q-\frac{x}{2}})(\frac{1}{2}\partial_q-\partial_x)]f+$$\ $$+i[ae^{-(q-\frac{x}{2})}+b(\alpha\beta-\gamma\delta)e^{q-\frac{x}{2}}+c\gamma]f$$&\ $$[(d- b\gamma e^{q-\frac{x}{2}})(\frac{1}{2}\partial_q-\partial_x)]f+$$\ $$+i[-ae^{-(q-\frac{x}{2})}-b(\alpha\beta-\gamma\delta)e^{q-\frac{x}{2}}+c\gamma]f$$&\ $$ [*Proof*]{}. Applying Lemma (4.3),we have : 1. If $\tilde A=a\alpha e^{-q}+dp$ then $$\hat{\ell}_A(f) := \mathcal F_p \circ \ell_A \circ \mathcal F^{-1}_p(f) = \mathcal F_p(i\tilde{A}\star \mathcal F^{-1}_p(f)) = i\mathcal F_p\left(\sum_{r \geq 0} \left(\frac{1}{2i}\right)^r\frac{1}{r!} P^r(\tilde{A},\mathcal F^{-1}_p(f))\right)=$$ $$\begin{array}{cl}&=i\mathcal F_p\Big\{(a\alpha e^{-q}+dp)\mathcal F_p^{-1}(f)+\frac{1}{1!}\frac{1}{2i}[d\partial_q\mathcal F_p^{-1}(f)+a\alpha e^{-q}\partial_p\mathcal F_p^{-1}(f)]+\\&+\frac{1}{2!}(\frac{1}{2i})^2.a\alpha e^{-q}\partial_pp^2\mathcal F_p^{-1}(f)+ \dots +\frac{1}{r!}(\frac{1}{2i})^r a\alpha e^{-q}\partial^r_{p\dots p}\mathcal F_p^{-1}(f)+\dots \Big\}=\\& =i\Big\{a\alpha e^{-q}f+d\mathcal F_p(p.\mathcal F_p^{-1}(f))+\frac{1}{1!}\frac{1}{2i}[d\partial_qf +a\alpha e^{-q}\mathcal F_p(\partial_p\mathcal F_p^{-1}(f))]+\\& +\frac{1}{2!}(\frac{1}{2i})^2.a\alpha e^{-q}\mathcal F_p(\partial^2_{pp}\mathcal F_p^{-1}(f))+ +\frac{1}{3!}(\frac{1}{2i})^3.a\alpha e^{-q}\mathcal F_p(\partial^3_{ppp}\mathcal F_p^{-1}(f))+\dots\\&+\frac{1}{r!}(\frac{1}{2i})^r.a\alpha e^{-q}\mathcal F_p(\partial^r_{p\dots p}\mathcal F_p^{-1}(f))+\dots \Big\}=\\& =d(\frac 12\partial_q-\partial_x)f+i a\alpha e^{-q}[1+\frac{x}{2}+\frac{1}{2!}(\frac{x}{2})^2+\dots +\frac{1}{r!}(\frac{1}{x})^r+\dots ]f =\\&= d(\frac 12\partial_q-\partial_x)f+i a\alpha e^{-q}e^{\frac{x}{2}}f=d(\frac 12\partial_q-\partial_x)f+ia\alpha e^{-(q-\frac{x}{2})}f\end{array}$$ 2. If $\tilde A=b\beta e^q +dp$ then $$\hat\ell_A(f)=d(\frac 12\partial_q-\partial_x)f+i b\beta e^{q-\frac{x}{2}}f$$ 3. For each $\tilde A=a\alpha e^{-q}+b\beta e^q +dp ,$ we have: $$\begin{array}{cl}&\hat\ell_A=i\mathcal F_p\Big\{(a\alpha e^{-q}+b\beta e^q+dp)\mathcal F_p^{-1}(f)+\frac{1}{2i}[d\partial_q\mathcal F_p^{-1}(f)-(-a\alpha e^{-q}+\\&+ b\beta e^q)\partial_p\mathcal F_p^{-1}(f)]+\frac{1}{2!}(\frac{1}{2i})^2[a\alpha e^{-q}+(-1)^2b\beta e^q]\partial^2_{pp}\mathcal F_p^{-1}(f)+\dots +\\&+\frac{1}{r!}(\frac{1}{2i})^r[a\alpha e^{-q}+(-1)^r b\beta e^q] \partial^r_{p\dots p}\mathcal F_p^{-1}(f)+\dots\Big\}\\ &=ia\alpha e^{-q}.f+id\mathcal F_p\big(p.\mathcal F_p^{-1}(f)\big)+ib\beta e^q f+\frac{1}{2}d\partial_q f+\frac{1}{2}a\alpha e^{-q}\mathcal F_p\big(\partial_p\mathcal F_p^{-1}(f)\big)-\\ &-\frac{1}{2}b\beta e^q\mathcal F_p\big(\partial_p\mathcal F_p^{-1}(f)\big)+\dots i\frac{1}{r!}(\frac{1}{2i})^r a\alpha e^{-q}\mathcal F_p\big(\partial^r_{p\dots p}\mathcal F_p^{-1}(f)\big)+\\ &+i\frac{1}{r!}(\frac{-1}{2i})^r b\beta e^q\mathcal F_p\big(\partial^r_{p\dots p}\mathcal F_p^{-1}(f)\big)+\dots =d(\frac{1}{2}\partial_q-\partial_x)+ia\alpha e^{-q}[1+\frac{x}{2}+\dots\\& +\frac{1}{r!}(\frac{x}{2})^r+\dots]+ib\beta e^q[1+(\frac{-x}{2})+\dots +\frac{1}{r!}(\frac{-x}{2})^r+\dots ]\\& =d(\frac{1}{2}\partial_q-\partial_x)+i[a\alpha e^{-(q-\frac{x}{2})}+b\beta e^{q-\frac{x}{2}}]f.\end{array}$$ 4. For each $\tilde {A}$ is as in (9) , remark that $$P^0(\tilde{A},\mathcal F^{-1}_p(f))=\tilde{A}.\mathcal F^{-1}_p(f);$$ $$P^1(\tilde{A},\mathcal F^{-1}_p(f)) = \{\tilde{A},\mathcal F^{-1}_p(f) \} =$$ $$(d+b\gamma e^q)\partial_q\mathcal F_p^{-1}(f)-[-ae^{-q}+b(\alpha\beta+\gamma p -\gamma\delta)e^q]\partial_p\mathcal F_p^{-1}(f)$$ and applying Lemma (4.3), we obtain: $$\begin{array}{cl}& \hat{\ell}_A(f) = i\Big\{\mathcal F_p\Big([dp+ae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^{q}+c\gamma]\mathcal F_p^{-1}(f)\Big)+\\&+ \frac{1}{2i}\frac{1}{1!}\mathcal F_p\Big([d+b\gamma e^q]\partial_q\mathcal F^{-1}_p(f)-[-ae^{-q}+b(\alpha\beta+\gamma p -\gamma\delta)e^q]\partial_p\mathcal F_p^{-1}(f)\Big)+\\& +(\frac{1}{2i})^2\frac{1}{2!}\mathcal F_p\Big(-2b\gamma e^ {q}\partial^2_{pq}\mathcal F^{-1}_p(f)+ [ae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^q]\partial^2_{pp}(\mathcal F^{-1}_p(f)\Big)+\dots\\& +(\frac{1}{2i})^r\frac{1}{r!}\mathcal F_p\Big((-1)^{r-1}rb\gamma e^q\partial^r _{p \dots pq}\mathcal F^{-1}_p(f)+(-1)^r[(-1)^rae^{-q}+b(\alpha\beta+\gamma p-\gamma\delta)e^q]\times \\& \times \partial^r_{p\dots p}\mathcal F^{-1}_p(f)\Big)+\dots \Big\}=\end{array}$$ $$\begin{array}{cl}&=i\Big\{ae^{-q}f+b(\alpha\beta-\gamma\delta)e^qf+d\mathcal F_p\big(p\mathcal F^{-1}_p(f)\big) +b\gamma e^q\mathcal F_p\big(p\mathcal F^{-1}_p(f)\big)\\& +\frac{1}{2i}\frac{1}{1!} (d+b\gamma e^q)\partial_qf -\frac{1}{2i}[-ae^{-q}ixf +b(\alpha\beta-\gamma\delta)e^qixf+b\gamma e^q\mathcal F_p\big(p\mathcal F^{-1}_p(xf)\big)] +\\&+(\frac{1}{2i})^2\frac{1}{2!}(-2b\gamma e^q)\mathcal F_p\left(\partial_{pq}^2\mathcal F^{-1}_p(f)\right)+(\frac {1}{2i})^2\frac{1}{2!}[ae^{-q}(ix)^2f+ b(\alpha\beta-\gamma\delta)e^q(ix)^2f+\\&+b\gamma e^q \mathcal F_p\left(pi^2\mathcal F^{-1}_p(x^2f)\right)]+\dots +(\frac {1}{2i})^r\frac{1}{r!}(-1)^{r-1}rb\gamma e^q\partial^r_{p\dots pq}\mathcal F^{-1}_p(f) \\&+(\frac{1}{2i})^r\frac{1}{r!}[ae^{-q}(ix)^rf+(-1)^rb(\alpha\beta-\gamma\delta)e^q(ix)^rf+b\gamma e^q\mathcal F_p\left(p(ix)^r\mathcal F^{-1}_p(f)\right)]+\dots \Big\}\\&=i[ae^{-q}(1+\frac{1}{2!}\frac{x}{2}+\dots+\frac{1}{r!}(\frac{x}{2})^r\dots)f ]+i[b(\alpha\beta-\gamma\delta)e^q (1-\frac{1}{2!}\frac{x}{2}+\dots+\\&+(-1)^r\frac{1}{r!}(\frac{x}{2})^r\dots)f]+ic\gamma f+i^2d\partial_xf+\frac{1}{2}d\partial_qf+ib\gamma e^q[i\partial_xf -\frac{1}{2i}\mathcal F_p\left(pi\mathcal F^{-1}_p(xf)\right)+\\& +\dots+(\frac{1}{2i})^r\frac{1}{r!}(-1)^r\mathcal F_p\left(pi^r\mathcal F^{-1}_p(x^rf)\right)+\dots]=d(\frac{1}{2}\partial_q-\partial_x)f+\\&+[iae^{-(q-\frac{x}{2})}+ib(\alpha\beta-\gamma\delta)e^{q-\frac{x}{2}}]f+ic\gamma f+\frac{1}{2}e^{-\frac{x}{2}}b\gamma e^q\partial_qf-b\gamma e^qe^{-\frac{x}{2}}\partial_xf\\&=(d+ b\gamma e^{q-\frac{x}{2}})(\frac{1}{2}\partial_q-\partial_x)f+[iae^{-(q-\frac{x}{2})}+ib(\alpha\beta-\gamma\delta)e^{q-\frac{x}{2}}+ic\gamma]f \end{array}$$ 5. At last, if $\tilde A$ is defined by (10) then : $$\hat\ell_A(f)=(d- b\gamma e^{q-\frac{x}{2}})(\frac{1}{2}\partial_q-\partial_x)f+[-iae^{-(q-\frac{x}{2})}-ib(\alpha\beta-\gamma\delta)e^{q-\frac{x}{2}}+ic\gamma]f$$ The theorem is therefore proved.$\square$ [Setting new variables $s = q - \frac{x}{2}$, $t = q + \frac{x}{2}$, we have $$\hat{\ell} _A(f)=\cases$$(d\_s+iae\^[-s]{})f\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (6) }\\$$(d\_s+ibe\^s)f\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (7)}\\$$(d\_s +i\[ae\^[-s]{}+ be\^s\])f\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (8)}$$\ $$\Big((d+b\gamma e^s)\partial_s+$$\ $$i[ae^{-s}+b(\alpha\beta-\gamma\delta)e^s +c\gamma ]\Big)f\vert_{(s,t)} .&\text{if $\tilde {A}$ is defined by (9)$$}\\$$((d-be\^s)\_s+$$\\$$i\[-ae\^[-s]{}-b(-)e\^s +c\])f\_[(s,t)]{} .& $$ ]{} With above notations we obtain the operators : $$\hat{\ell} _A=\cases$$\_A\^[(2)]{}=(d\_s+iae\^[-s]{})\_[(s,t)]{} $$&\\$$\_A\^[(3)]{}=(d\_s+ibe\^s)\_[(s,t)]{} $$&\\$$\_A\^[(4)]{}=(d\_s+i\[ae\^[-s]{}+ be\^s\])\_[(s,t)]{} $$& \\$$\_A\^[(5)]{}=((d+be\^s)\_s+i\[ae\^[-s]{}+b(-)e\^s+c\])\_[(s,t)]{} .&\ $$\hat\ell_A^{(5')}=\Big((d-b\gamma e^s)\partial_s+i[-ae^{-s}-b(\alpha\beta-\gamma\delta)e^s+c\gamma]\Big)\vert_{(s,t)} \endcases$$ which provides the representations of the Lie algebra $\mathfrak g$=$Lie(\mathbb R\ltimes \mathbb H_3)$. Furthermore, $\forall A,B \in\mathfrak g$, $$\hat\ell_A\circ\hat\ell_ B- \hat\ell_B\circ\hat\ell_ A =\hat\ell_{[A, B]}$$ [*Proof*]{} For each compactly supported $C^\infty$ function $f \in C^\infty_0(\mathbb R^2)$ and for $A,B \in Lie(\mathbb R \ltimes \mathbb H_3)$,we have $$\hat\ell_{(\mu_1A+\mu_2 B)}(f)=\mathcal F_p \circ \ell_{(\mu_1A+\mu_2 B)}\circ \mathcal F^{-1}_p(f)=\mathcal F_p\Big(i(\widetilde{\mu_1A+\mu_2B})\star \mathcal F^{-1}_p\Big)=$$ $$=\mu_1\mathcal F_p\circ \ell_A\circ\mathcal F^{-1}_p(f)+\mu_2\mathcal F_p\circ \ell_B\circ\mathcal F^{-1}_p(f)=\mu_1\hat\ell_A(f)+\mu_2\hat\ell_B(f)\quad\forall \mu_1,\mu_2\in\mathbb R.$$ Moreover, $$\hat\ell_A\circ\hat\ell_B(f)-\hat\ell_B\circ\hat\ell_A(f)=\hat\ell_A \Big(\mathcal F_p \circ \ell_B \circ \mathcal F^{-1}_p(f)\Big)- \hat\ell_B \Big(\mathcal F_p \circ \ell_A \circ \mathcal F^{-1}_p(f)\Big)=$$ $$=\mathcal F_p\Big(i\tilde A\star(i\tilde B\star\mathcal F^{-1}_p(f)\Big)-\mathcal F_p \Big(i\tilde B\star (i\tilde A\star\mathcal F^{-1}_p(f)\Big)=\mathcal F_p\Big(i\widetilde{[A,B]}\star\mathcal F^{-1}_p(f)\Big)=\hat\ell_{[A,B]}(f)$$$\square$ Let $\Omega_F^{\lambda}$ be K-orbits of the real diamond Lie group $G$. With $A$ runs over the Lie algebra $\mathfrak g=Lie(G)$, - $(\Omega^2 ,\hat\ell_A^{(2)}) ;(\Omega^3 ,\hat\ell_A^{(3)})$ are called the quantum half-planes, - $(\Omega^4 ,\hat\ell_A^{(4)})$ - quantum hyperbolic cylinder, - $(\Omega^5 ,\hat\ell_A^{(5)} ,\hat\ell_A^{(5')})$ - quantum hyperbolic paraboloid, with respect to the co-adjoint action of Lie group $G$. In the other words, $(\Omega_F,\hat\ell_A)$, with $A$ running over the Lie algebra $\mathfrak g$ is called a [*quantum co-adjoint orbit*]{} of Lie group $G$. As G=$\mathbb R\ltimes \mathbb H_3$ is connected and simply connected, we obtain a unitary representations $ T$ of G defined by the following formula $$T(\exp A) :=\exp(\hat{\ell}_A);\quad A\in\mathfrak g$$ More detail, $$\exp(\hat{\ell}_A) =\cases$$(d\_s+iae\^[-s]{})\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (6) }\\$$(d\_s+ibe\^s)\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (7)}\\$$(d\_s+i\[ae\^[-s]{}+ be\^s\])\_[(s,t)]{} $$&\text{if $\tilde {A}$ is defined by (8)}$$\ $$\exp ((d+b\gamma e^s)\partial_s+$$\ $$i[ae^{-s}+b(\alpha\beta-\gamma\delta)e^s+c\gamma] )\vert_{(s,t)}.&\text{if $\tilde {A}$ is defined by (9)$$}\\$$((d-be\^s)\_s+$$\\$$i\[-ae\^[-s]{}-b(-)e\^s+c\] )\_[(s,t)]{}.& $$ This means that we refind all the representations $ T(\exp A)$ of the real diamond Lie group $\mathbb R\ltimes \mathbb H_3$, those could implicitly obtained from (induction) orbit method induction. What we did here gives us more precise analytic formulas in this case for orbit method induction. $$\text{ ACKNOWLEDGMENT}$$ The author would like to express his gratitude to Professor Do Ngoc Diep for all his helpfulness and for suggesting many of the topics considered in this paper. The author also thanks Dr. Nguyen Viet Dung for his encouragement. [lblllll]{} D. Arnal and J. C. Cortet, [*$\star$-product and representations of nilpotent Lie groups*]{}, J. Geom. Phys., [**2**]{}(1985), No 2, 86-116. D. Arnal and J. C. Cortet, [*Représentations \* des groupes exponentiels*]{}, J. Funct. Anal. [**92**]{}(1990), 103-135. V. I. Arnold, [*Mathematical Methods of Classical Mechanics*]{}, Springer Verlag, Berlin - New York - Heidelberg, 1984. F.Bayen, M.G.Fronsdai, A.Lichnerovics and D.Sternbeimer, [*Deformation theory and quantization*]{}, Ann.Physics [**111**]{}(1978),61-151. N. Reshetikhin and L. A. Takhtajan, [*Deformation quantization of Kähler manifolds*]{}, math.QA/9907171. Do Ngoc Diep, [*Noncommutative Geometry Methods for Group C\*-Algebras*]{}, Chapman & Hall/CRC Research Notes in Mathematics Series, Vol. 416, 1999. Do Ngoc Diep and Nguyen Viet Hai, [*Quantum half-planes via deformation quantization*]{}, math.QA/9905002, 2 May 1999. Do Ngoc Diep and Nguyen Viet Hai, [*Quantum co-adjoint orbits of the group of affine transformations of the complex straight line*]{}, math.QA/9908046, 11 Aug 1999. B. Fedosov, [*Deformation Quantization and Index Theory*]{}, Akademie der Wissenschaften Verlag, 1993. S. Gutt, [*Deformation quantization*]{}, ICTP Workshop on Representation Theory of Lie groups, SMR 686/14, 1993. I. M. Gelfand and M. A. Naimark, [*Unitary representations of the group of affine transformations of the straight Line*]{}, Dokl. AN SSSR, [**55**]{}(1947), No 7, 571-574. A. A. Kirillov, [*Elements of the Theory of Representation*]{}, Springer Verlag, Berlin - New York - Heidelberg, 1976. A. A. Kirillov, [*Unitary representations of nilpotent Lie groups*]{}, Russian Math. Survey, 1962, 17-52. B. Kostant, [*On Certain unitary representations which arise from a quantization theory*]{}, Lecture Notes in Math., [**170**]{}(1970), 237- ..... R. Meise and D. Vogt, [*Introduction to Functional Analysis*]{}, Clarendon Press, Oxford, 1997. H. Weyl, [*Gruppentheorie und Quantenmechanik*]{}. Leipzig,1928. E. Winger, Phys.Rev,[**40**]{}(1932),749; Math.Ann,[**104**]{}(1931,570-578.) [^1]: This work was supported in part by the Vietnam National Foundation for Fundamental Science Research .

--- abstract: 'We study the Laplacian in deformed thin (bounded or unbounded) tubes in $\R^3$, i.e., tubular regions along a curve $r(s)$ whose cross sections are multiplied by an appropriate deformation function $h(s)> 0$. One the main requirements on $h(s)$ is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.' author: - | César R. de Oliveira[^1] [ and]{} Alessandra A. Verri\ *Departamento de Matemática – UFSCar, *São Carlos, SP, 13560-970 Brazil\ \ ** title: On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes --- [**Keywords:**]{} spectrum; thin tubes; Laplacian; dimensional reduction. [**MSC codes:**]{} 81Q15, 49R50, 35P20, 47B99 Introduction ============ The Laplacian in tubular domains has been studied in various situations [@Bou; @Cr1; @DE; @Kre]. A common tubular region $\Omega$ is as follows: let $I \subseteq \mathbb R$ be an interval of $\mathbb R$, $r:I \to \mathbb R^3$ a curve in $\mathbb R^3$, parametrized by its arc length $s$, and $k(s)$ and $\tau(s)$ denote its curvature and torsion at the point $s \in I$, respectively. Let $S$ be an open, bounded, simply connected and nonempty subset of $\mathbb R^2$. Move the region $S$ along $r(s)$ and at each point $s$ allow the region to rotate by an angle $\alpha(s)$ (see details in Section \[gddh\]). A problem of interest is the description of the spectral properties of the Laplacian in such tubes and weakly effective operators (see the definition just after Theorem \[mainTeor\]) when the region $\Omega$ is “squeezed” to the curve $r(s)$, that is, one considers the sequence of tubes $\Omega_\varepsilon$ generated by the cross section $\varepsilon S$ and analyze the limit $\varepsilon \to 0$. Let $-\Delta_\varepsilon$ be the Dirichlet Laplacian in $\Omega_\varepsilon$. For bounded tubes, i.e., when $I$ is a bounded interval of $\mathbb R$, the spectrum of $-\Delta_\varepsilon$ is purely discrete because in this case its resolvent is compact. In [@Bou] it was analyzed the convergence of the eigenvalues $\{\lambda_i^\varepsilon: i \in \mathbb N\}$ as $\varepsilon \to 0$ and shown that $$\lambda_i^\varepsilon = \frac{\lambda_0}{\varepsilon^2} + \mu_i^\varepsilon, \qquad \mu_i^\varepsilon \to \mu_i,$$ where $\lambda_0$ is the first, i.e., the lowest, eigenvalue of the Laplacian in the Sobolev space ${\mathcal H}_0^1(S)$, and $\mu_i$ are the eigenvalues of the one-dimensional operator $$\label{bbb8} w(s)\;\mapsto \; - w''(s) + \left[ C(S) (\tau(s) + \alpha'(s)) - \frac{k(s)^2}{4} \right] w(s),$$ acting in $\LL^2(I)$. Here $C(S)$ is a nonnegative number depending only on the transverse region $S$ [@Bou]. Note that this effective operator explicitly depends on the geometric shape of the reference curve $r(s)$ (and so of the tube). An interesting problem is to know if there exists a similar result about convergence of eigenvalues for unbounded tubes. For such tubes, in [@Kre] it is shown that if $(\tau + \alpha')(s) = 0$ and $k(s) \neq 0$, then the discrete spectrum is nonempty, whereas if $(\tau + \alpha')(s) \neq 0$ and $k(s) = 0$, then the discrete spectrum is empty. In [@Cr1], by using $\Gamma$-convergence in case of unbounded tubes, a strong resolvent convergence was proven and the same action for the respective effective operator (now acting in $\LL^2(\mathbb R)$) was found as $\varepsilon \to 0$. The Dirichlet Laplacian in strips of $\mathbb R^2$ has been studied in many works [@BF; @Frie; @Sol; @Krej2]. For the case of the constraints of planar motion to curves there are results about the limit operator in [@DellAntTen; @ACF], and the effective potential is written in terms of the curvature. The main novelty, when we pass from planar domain to tubes in $\R^3$, as considered in [@Bou; @Cr1; @Kre; @GJ], is the additional presence of torsion and twisting (i.e., a nonzero $\tau(s) + \alpha'(s)$) in the effective potential, since the case of untwisted tubes has also been previously studied (see, for instance, [@DE; @CDF; @ClaBra; @Ekk; @FreitasKrej; @KS]). In [@Frie; @Sol] the authors consider a family of deformed strips $$\left\{(s, y) \in \mathbb R^2: s \in J, \,0 < y < \varepsilon h(s) \right\},$$ where $J = [-a, b]$, $0< a, b \leq \infty$, and $h(s) > 0$ is a continuous function satisfying: - $h(s)$ is a $\CC^1$ function in $J\setminus\{0\}$ and $\|h'/h\|_\infty<\infty$; - near the origin $h$ behaves as $$\label{vvv1} h(s) = M - s^2 + O( |s|^3), \qquad M > 0,$$ and $s=0$ is a single point of global maximum for $h$; - in case $I=\R$ it is assumed that $ \limsup_{|s| \to \infty} h(s) < M$. In what follows we assume that $h$ satisfies the above conditions. It was shown [@Frie; @Sol] that, for $\varepsilon$ small enough, the discrete spectrum of the Laplacian is always nonempty and the eigenvalues $\lambda_j(\varepsilon)$ have the following behavior $$\mu_j = \lim_{\varepsilon \to 0}\, \varepsilon \left( \lambda_j(\varepsilon) - \frac{\pi^2}{\varepsilon^2 M^2} \right),$$ where $\mu_j$ are the eigenvalues of the operator in $\LL^2(\mathbb R)$ (it acts on a subspace of $\LL^2(\mathbb R)$, independently if the interval $I$ is bounded or not) given by $$(T w)(s) = - w''(s) + 2 \frac{\pi^2 }{M^{3}} \; s^2 \; w(s),$$ so that we say that $T$ is a [*weakly effective operator*]{} (WEO) in such situation. In this work we show that these results hold in a more general setting. We consider a sequence of tubes $\Omega_\varepsilon$ in the space $\mathbb R^3$, as presented at the beginning of this introduction, but we deform them by multiplying their cross sections by the above function $h(s)$. Here the tubes may be bounded or not. Then we analyze the asymptotic behavior of eigenvalues and the weakly effective operators in the limit $\varepsilon\to0$. The situation here differs from [@Frie; @Sol], since besides the different dimensions (we consider regions in $3-$dimensional space), the reference curves defining our tubes are allowed to have nontrivial curvatures and torsion. These tubes, which we shall call [*deformed tubes*]{}, will generically be denoted by $\Lambda_\varepsilon$ (see details in Section \[gddh\]). Our main goal is to study how curvature and torsion of the reference curve, together with the deforming function $h$, influence the WEO and eigenvalues as $\varepsilon \to 0$. To this end, we introduce some notation right now. Recall that $\lambda_0$ is the lowest eigenvalue of the negative Laplacian with Dirichlet conditions in the region $S$, and let $u_0$ be the corresponding (positive) normalized eigenfunction, that is, $$\label{eqU0lambda0} -\Delta u_0 = \lambda_0 u_0, \quad u_0 \in {\mathcal H}_0^1(S), \quad \int_S u_0(y)^2 dy = 1.$$ Furthermore, denote by ${\mathcal L}$ the subspace of $\LL^2(I\times S)$ generated by functions $w(s) u_0(y)$ with $w \in \LL^2(I)$. We study three distinct cases. First, the tubes are bounded since the interval $I$ is of the form $I=[-a,b]$ with $0< a, b < \infty$, and we consider the Dirichlet condition at the boundary $\partial \Lambda_\varepsilon$. In the second case, the tubes are bounded but the Dirichlet condition at the vertical part of the $\partial \Lambda_\varepsilon$, that is, $\{(-a) \times S \cup b \times S\}$, is replaced by Neumann. In the third case we consider $I=\mathbb R$ with Dirichlet condition at $\partial \Lambda_\varepsilon$. If the tubes are not deformed, according to the results of [@Bou; @Cr1], the effective operator  presents an additional potential $$C(S)\left(\tau+\alpha'\right)(s) - k^2(s)/4$$ derived from geometric features of the tube. Hence, here there is a kind of competition between geometric properties of the tube and the behavior at its single maximum of the deformation function $h$. Roughly speaking, it is expected that the behavior of $h$ at the single maximum will control the limit $\varepsilon\to0$, since the geometric effects gives a contribution of order zero, whereas the single maximum of the deformation function $h$ gives a contribution of order $1/\varepsilon$. However, this requires a proof which turns out to be far from trivial, and so for the three cases mentioned in the previous paragraph, we prove the following result: \[mainTeor\] Let $I$ denote either $\R$ or a bounded interval $[-a,b]$ as above; in case $I=\R$ assume that $\lim_{|s| \to \infty} k(s) =0$. If $l_j(\varepsilon)$ denote the eigenvalues of the Dirichlet $-\Delta_\varepsilon$ in the deformed tube $\Lambda_\varepsilon$, then, the limits $$\label{eee3} \mu_j = \lim_{\varepsilon \to 0}\, \varepsilon \left( l_j(\varepsilon) - \frac{\lambda_0}{\varepsilon^2 M^2} \right)$$ exist, where $\mu_j= (2j+1)(2\lambda_0/M^3)^{1/2}$ are the eigenvalues of the self-adjoint operator $T$, acting in $\LL^2(\mathbb R)$, given by $$\label{eqweo} (T u)(s) = - u''(s) + 2\frac{ \lambda_0 }{ M^{3}} \; s^2 \; u(s).$$ Due to the conclusions of Theorem \[mainTeor\], $T$ is a WEO for $-\Delta_\varepsilon$ as $\varepsilon\to0$. Note that $T$ has purely discrete spectrum since the potential $$V(s)=2\frac{\lambda_0}{M^3}\, s^2\to\infty,\qquad |s|\to\infty;$$ in this case it is the harmonic oscillator potential (but see  below). Therefore, for deformed tubes as above, the weakly effective operators $T$ do not depend on some geometric features of the tube, although the curvature of the reference curve must vanish at infinity. The additional potential $V(s)$ is related to the behavior of $h(s)$ near it maximum (at the origin). Hence, the eigenvalues of the Laplacian in quite different deformed tubes are described by the same WEO as $\varepsilon\to0$ ! In Section \[gddh\] we present a detailed construction of the deformed tubes $\Lambda_\varepsilon$. Our study and technique are focused on analyzing the sequence of quadratic forms $$\label{seqfq} F_\varepsilon (\psi) = \int_{\Lambda_\varepsilon} \left(|\nabla \psi|^2 - \frac{\lambda_0}{\varepsilon^2 M^2} |\psi|^2\right) dx, \qquad \dom F_\varepsilon = {\mathcal H}_0^1(\Lambda_\varepsilon).$$ In Section \[ffqq\] it will become clear why we subtract terms of the form $\lambda_0 / (\varepsilon^2 M^2)|\psi|^2$ from the quadratic forms; we think this is in fact a natural choice. In Section \[ffqq\] we also perform a change of variables so that the integration region and the corresponding domains in  remain fixed. In Section \[redim\], we show that our analysis can be restricted to a specific subspace; we will see that this subspace can be identified with the Sobolev space ${\mathcal H}_0^1(I)$, and we call this fact a [*reduction of dimension.*]{} Finally, in Sections \[qqq3\], \[nnnn\] and \[iiii\], we discuss details of the three cases previously mentioned. We remark that although we rely on [@Frie; @Sol], the generalization to our setting is not immediate and different techniques are added to those of the original works. Furthermore, as an alternative to , all results can be easily adapted to cover more general deformation functions $h(s)$, as considered in [@Frie; @Sol], so that near the unique global maximum at the origin they behave as $$\label{aaa6} h(s) = \left\{ \begin{array}{ccc} M - c_+ s^m + O(s^{m+1}), & \hbox{if} & s>0 \\ M - c_- |s|^m + O(|s|^{m+1}), & \hbox{if} & s<0 \end{array}\right.,$$ for some positive numbers $M,m,c_\pm$. For the sake of simplicity, in Equation  we have particularized to $m=2$ and $c_+=c_-=1$. An interesting problem would be if the maximum of $h$ would be reached at an interval of values of the parameter $s$ instead of a single point (see [@BF2] for results in this direction in case of bounded domains, as well as [@BKRS; @Grushin]); we are currently working on a related problem. Geometry of the tubes {#gddh} ===================== Let $I=[-a, b]$, with either $0 < a,b< \infty$ or $a=b=\infty$, be an interval of $\mathbb R$, $r: I \subseteq \mathbb R \rightarrow \mathbb R^3$ a simple $\CC^2$ curve in $\mathbb R^3$ parametrized by its arc length parameter $s$ and, as in the previous section, $k(s)$ is its curvature. The vectors $$T(s) = r'(s), \qquad N(s) = \frac{1}{k(s)} T'(s), \qquad B(s) = T(s) \times N(s),$$ denote, respectively, the tangent, normal and binormal vectors of the curve. We assume that Frenet equations are satisfied, that is, $$\left(\begin{array}{c} T'\\ N'\\ B' \end{array} \right) = \left(\begin{array}{ccc} 0 & k & 0 \\ -k & 0 & \tau \\ 0 & -\tau & 0 \end{array}\right) \left(\begin{array}{c} T\\ N\\ B \end{array} \right),$$ where $\tau(s)$ is the torsion of the curve $r(s)$. Let $S$ be an open, bounded, simply connected and nonempty subset of $\mathbb R^2$. The set $$\Omega = \left\{ x \in \mathbb R^3: x = r(s) + y_1 N(s) + y_2 B(s),\, s \in I, y=(y_1, y_2) \in S \right\}$$ is obtained by translating the region $S$ along the curve $r$. At each point $r(s)$ we allow a rotation of the region $S$ by an angle $\alpha(s)$ with respect to $\alpha(0)=0$, so that the new region is given by $$\Omega^\alpha = \left\{ x \in \mathbb R^3: x = r(s) + y_1 N_\alpha(s) + y_2 B_\alpha(s), \,s \in I, (y_1, y_2) \in S \right\},$$ where $$\begin{aligned} N_\alpha(s) & : = & \cos \alpha(s) N(s) + \sin \alpha(s) B(s), \\ B_\alpha(s) & : = & - \sin \alpha(s) N(s) + \cos \alpha(s) B(s).\end{aligned}$$ Next, for each $0<\varepsilon <1$, we “squeeze” the cross sections of the above region, that is, we consider $$\Omega_\varepsilon^\alpha = \left\{ x \in \mathbb R^3: x = r(s) + \varepsilon y_1 N_\alpha(s) + \varepsilon y_2 B_\alpha(s),\, s \in I, (y_1, y_2) \in S\right \}.$$ Note that $\Omega_\varepsilon^\alpha$ approaches the curve $r(s)$ as $\varepsilon \to 0$. Finally, we consider the function $h(s)$ defined in the Introduction, so that each region $\Omega_\varepsilon^\alpha$ is properly deformed, and the result is $$\Lambda_\varepsilon^\alpha := \left\{ x \in \mathbb R^3: x = r(s) + \varepsilon h(s) y_1 N_\alpha(s) + \varepsilon y_2 h(s) B_\alpha(s),\, s \in I, (y_1, y_2) \in S \right\}.$$ From now on we will omit the symbol $\alpha$ in most notations and write $dx=dsdy_1dy_2$ and $dy=dy_1dy_2$. In this work we study the behavior of a free quantum particle that moves in $\Lambda_\varepsilon$, and initially with Dirichlet boundary condition at the boundary $\partial\Lambda_\varepsilon$. Thus, we initially consider the family of quadratic forms $$\label{ddd6} b_\varepsilon (\psi) := \int_{\Lambda_\varepsilon} |\nabla \psi|^2 dx, \qquad \dom b_\varepsilon = {\mathcal H}_0^1(\Lambda_\varepsilon),$$ which is associated with the Dirichlet Laplacian operator $-\Delta_\varepsilon$ in $\Lambda_\varepsilon$. The symbol $\nabla = (\partial_s, \nabla_y)$, $\nabla_y=(\partial_{y_1},\partial_{y_2})$, denotes the gradient in the coordinates $(s,y_1,y_2)$ in $\mathbb R^3$. Quadratic forms {#ffqq} =============== As usual in this kind of problems, in this section we perform a change of variables so that the integration region in , and consequently the domains, become independent of $\varepsilon > 0$. Then, for the singular limit $\varepsilon\to0$, customary “regularizations” will be employed. Consider the mapping $$\begin{array}{cccl} f_\varepsilon: & I \times S & \to & \Lambda_\varepsilon \\ & (s, y_1, y_2) & \mapsto & r(s) + \varepsilon\, h(s) \left( y_1 N_\alpha(s) + y_2 B_\alpha(s)\right), \end{array}$$ and suppose the boundedness $\| k\|_\infty, \|\tau\|_\infty, \|\alpha'\|_\infty < \infty$. These conditions are to guarantee that $f_\varepsilon$ will be a diffeomorphism. With this change of variables we work with a fixed region for all $\varepsilon > 0$; more precisely, the domain of the quadratic form  turns out to be ${\mathcal H}_0^1(I \times S)$. On the other hand, the price to be paid is a nontrivial Riemannian metric $G= G_\varepsilon^\alpha$ which is induced by $f_\varepsilon$, i.e., $$G=(G_{ij}),\qquad G_{ij} = \langle e_i, e_j \rangle = G_{ji}, \qquad 1 \leq i, j \leq 3,$$ where $$e_1 = \frac{\partial f_\varepsilon}{\partial s}, \qquad e_2 = \frac{\partial f_\varepsilon}{\partial y_1}, \qquad e_3 = \frac{\partial f_\varepsilon}{\partial y_2}.$$ Some calculations show that in the Frenet frame $$\begin{aligned} J & = & \left(\begin{array}{c} e_1 \\ e_2 \\ e_3 \end{array}\right) \\ & = & \left(\begin{array}{ccc} \beta_\varepsilon & -\varepsilon h (\tau+\alpha') \langle z_\alpha^\perp, y \rangle + \varepsilon h' \langle z_\alpha, y \rangle & \varepsilon h (\tau+\alpha') \langle z_\alpha, y \rangle + \varepsilon h' \langle z_\alpha^\perp, y \rangle \\ 0 & \varepsilon h \cos \alpha & \varepsilon h \sin \alpha \\ 0 & -\varepsilon h \sin \alpha & \varepsilon h \cos \alpha \end{array}\right),\end{aligned}$$ where $$\beta_\varepsilon(s, y) = 1 - \varepsilon h(s) k(s) \langle z_\alpha, y \rangle, \quad z_\alpha : = (\cos \alpha, - \sin \alpha), \quad z_\alpha^\perp : = (\sin \alpha, \cos \alpha).$$ The inverse matrix of $J$ is given by $$J^{-1} = \left(\begin{array}{ccc} \frac{1}{\beta_\varepsilon} & \frac{1}{\beta_\varepsilon} \left[(\tau + \alpha') y_2 - \frac{h'}{h} y_1 \right] & \frac{1}{\beta_\varepsilon} \left[ - (\tau + \alpha') y_1 - \frac{h'}{h} y_2 \right] \\ 0 & \frac{\cos \alpha}{\varepsilon h} & \frac{-\sin \alpha}{\varepsilon h} \\ 0 & \frac{\sin \alpha}{\varepsilon h} & \frac{\cos \alpha}{\varepsilon h} \end{array}\right).$$ Note that $J J^t = G$ and $\det J = | \det G|^{1/2} = \varepsilon^2 h^2(s) \beta_\varepsilon(s, y)$. Since $k$ and $h$ are bounded functions, for $\varepsilon$ small enough $\beta_\varepsilon$ does not vanish in $I \times S$. Thus, $\beta_\varepsilon > 0$ and $f_\varepsilon$ is a local diffeomorphism. By requiring that $f_\varepsilon$ is injective (that is, the tube is not self-intersecting), a global diffeomorphism is obtained. Introducing the notation $$\left| \left| \psi \right| \right|_G^2 :=\int_{I \times S} | \psi(s,y) |^2 \varepsilon^2 h^2(s) \beta_\varepsilon(s, y) \,ds dy,$$ we obtain a sequence of quadratic forms $$\tilde{b}_\varepsilon (\psi) : = \left| \left| J^{-1}\nabla \psi \right| \right|_G, \qquad \dom \tilde{b}_\varepsilon = {\mathcal H}_0^1(I \times S, G).$$ More precisely, the above change of coordinates was obtained by a unitary transformation $$\begin{array}{cccl} U_\varepsilon: & \LL^2(\Lambda_\varepsilon) & \to & \LL^2(I \times S, G) \\ & \phi & \mapsto & \phi \circ f_\varepsilon \end{array}.$$ However, we still denote $U_\varepsilon \psi$ by $\psi$. Recall that $\lambda_0$ is the lowest eigenvalue of the negative Laplacian with Dirichlet boundary conditions in the cross section region $S$, and $u_0\ge0$ (see Equation ) the corresponding eigenfunction of this restricted problem. This eigenfunction $u_0$ is directly related to transverse oscillations in $\Lambda_\varepsilon$. Due to this fact, in [@Bou; @Cr1] the authors have remove the diverging energy $\lambda_0 / \varepsilon^2$ from their quadratic forms. In our case, as the boundary of the tubes were multiplied by $h(s)$, we subtract the terms of the form $\lambda_0 /( \varepsilon M)^2$, i.e., since $0 < h(s) \le M$, for all $s \in I$, we eliminate the possible “least transverse energy.” Therefore, we turn to the study of the sequence of quadratic forms $$\tilde{g}_\varepsilon(\psi) : = \left( \left| \left| J^{-1} \nabla \psi \right| \right|_G^2 - \frac{\lambda_0}{\varepsilon^2 M^2} \| \psi\|_G^2 + c \| \psi\|_G^2 \right),$$ where $c$ is a positive constant to be chosen later on. After the norms are written out, we obtain $$\begin{aligned} \tilde{g}_\varepsilon (\psi) & = & \varepsilon^2 \int_{I \times S} \Big( \frac{1}{\beta_\varepsilon^2(s,y)} \left|\psi' + \nabla_y \psi \cdot R y (\tau+\alpha')(s) - \nabla_y \psi \cdot y \,\frac{h'(s)}{h(s)} \right|^2 \\ &+& \frac{ |\nabla_y \psi|^2 }{\varepsilon^2 h(s)^2} - \frac{\lambda_0}{\varepsilon^2 M^2} |\psi|^2 + c |\psi|^2 \Big) h(s)^2 \beta_\varepsilon(s, y)\,ds dy\; .\end{aligned}$$ Note that $\dom \tilde{g}_\varepsilon = {\mathcal H}_0^1(I \times S)$ is a subspace of $\LL^2(I \times S, h(s)^2 \beta_\varepsilon(s, y))$. We observe that the factor $|\nabla_y \psi|^2 / (\varepsilon h(s))^2$ is directly related to transverse oscillations of the particle. This term diverges as $\varepsilon \to 0$, but we control this fact by subtracting $\lambda_0 /( \varepsilon M)^2 |\psi|^2$ from the quadratic form (a renormalization). It will be convenient to work in the space $\LL^2(I \times S, \beta_\varepsilon(s, y))$; so we consider the isometry $$\begin{array}{cccl} & \LL^2(I \times S, \beta_\varepsilon) & \to & \LL^2(I \times S, h(s)^2 \beta_\varepsilon) \\ & v & \mapsto & v h^{-1} \end{array}.$$ This change of variables and the division by the global factor $\varepsilon^2$ (a common singular factor due to the “change of dimension” as $\varepsilon\to0$) leads to $$\begin{aligned} \hat{g}_\varepsilon (v) &: =& \int_{I \times S} \Big( \frac{1}{\beta_\varepsilon(s,y)} \left|v' - v \,\frac{h'(s)}{h(s)} + \nabla_y v \cdot R y (\tau+\alpha')(s) - \nabla_y v \cdot y \,\frac{h'(s)}{h(s)} \right|^2 \\ &+& \frac{\beta_\varepsilon(s,y)}{\varepsilon^2 h(s)^2} |\nabla_y v|^2 - \frac{\beta_\varepsilon(s, y)}{\varepsilon^2 M^2} |v|^2 + c \beta_\varepsilon(s, y) |v|^2 \Big)\,ds dy,\end{aligned}$$ with $\dom \hat{g}_\varepsilon = {\mathcal H}_0^1(I \times S)$, again as a subspace of $\LL^2(I \times S, \beta_\varepsilon(s, y))$. However, this latter space can be identified with $\LL^2(I \times S)$, for all $\varepsilon>0$, since $\beta_\varepsilon(s, y)$ converges uniformly to $1$ as $\varepsilon \to 0$. Hence we introduce the form $$\begin{aligned} g_\varepsilon (v) &: =& \int_{I \times S} \Big( \left|v' - v \,\frac{h'(s)}{h(s)} + \nabla_y v \cdot R y (\tau+\alpha')(s) - \nabla_y v \cdot y \,\frac{h'(s)}{h(s)} \right|^2 \\ &+& \frac{\beta_\varepsilon(s,y)}{\varepsilon^2 h(s)^2} |\nabla_y v|^2 - \frac{\beta_\varepsilon(s, y)}{\varepsilon^2 M^2} |v|^2 + c |v|^2 \Big)\,ds dy,\end{aligned}$$ with $\dom g_\varepsilon = {\mathcal H}_0^1(I \times S)$. Let $\hat{G}_\varepsilon$ and $G_\varepsilon$ be the self-adjoint operators associated with the quadratic forms $\hat{g}_\varepsilon$ and $g_\varepsilon$, respectively. \[teoap\] For $\varepsilon$ small enough, there exists $C > 0$ so that $$\left\| \hat{G}_\varepsilon^{-1} - G_\varepsilon^{-1} \right\| \leq C \varepsilon.$$ This theorem follows basically from the fact that $\beta_\varepsilon (s, y) \to 1$ uniformly as $\varepsilon \to 0$. Its proof is presented in the Appendix. Due to the above changes of variables and Theorem \[teoap\], we may consider the sequence of quadratic forms $g_\varepsilon$ in what follows. Reduction of dimension {#redim} ====================== Recall that $u_0(y)$ is the positive and normalized eigenfunction corresponding to the first eigenvalue $\lambda_0$ of the Laplacian in ${\mathcal H}_0^1(S)$. After the orthogonal decomposition $\LL^2(\mathbb R \times S) = {\mathcal L} \oplus {\mathcal L}^\perp,$ for $\psi \in \LL^2(\mathbb R \times S)$, we can write $$\psi(s, y) = w(s) u_0(y) + \eta(s, y),$$ with $w \in \LL^2(I)$ and $\eta \in {\mathcal L}^\perp$. We observe that $\eta\in {\mathcal L}^\perp$ implies $$\int_S u_0(y) \eta(s, y) dy = 0, \qquad \hbox{a.e.}[s].$$ Note that $wu_0 \in {\mathcal H}_0^1(I \times S)$ if $w \in {\mathcal H}_0^1(I)$. For $\psi \in {\mathcal H}_0^1(\mathbb R \times S)$, write $\psi = w u_0 + \eta$ with $w \in {\mathcal H}_0^1(I)$ and $\eta \in {\mathcal H}_0^1(\mathbb R \times S) \cap {\mathcal L}^\perp$. First we study the quadratic form $g_\varepsilon$ restricted to the subspace For $w \in {\mathcal H}_0^1(I)$, some calculations show that $$\begin{aligned} g_\varepsilon( w u_0) = \int_{I} \left[ |w'|^2 + \vartheta(s) |w|^2 + \zeta_\varepsilon(s, y) \left(\frac{\lambda_0}{\varepsilon^2h^2(s)}-\frac{\lambda_0}{\varepsilon^2M^2}\right) |w|^2 + c |w|^2 \right] ds,\end{aligned}$$ where $$\vartheta(s) = C_1(S) (\tau(s) + \alpha'(s))^2 + (C_2(S) - 1) \left(\frac{h'(s)}{h(s)}\right)^2 - 2 C_3(S) (\tau(s)+\alpha'(s)) \,\frac{h'(s)}{h(s)}$$ and $$\zeta_\varepsilon(s, y) = 1 - \varepsilon\, k(s) h(s) \left\langle z_\alpha(s), F(S) \right\rangle .$$ The constants $C_1(S)$, $C_2(S)$ and $C_3(S)$ that appear in the definition of $\vartheta$ depend only on the region $S$ and are explicitly given by $$C_1(S) = \int_S |\langle \nabla_y u_0, R y \rangle|^2 dy, \qquad C_2(S) = \int_S |\langle \nabla_y u_0, y \rangle|^2 dy,$$ and $$C_3(S) = \int_S \langle \nabla_y u_0, R y \rangle \langle \nabla_y u_0, y \rangle dy.$$ The vector $F(S) = (F_1(S), F_2(S))$ in the definition of $\zeta_\varepsilon$ also depends only on the region $S$, and its components are given by $$F_1(S) = \int_S y_1 |u_0|^2 dy\qquad \hbox{and}\qquad F_2(S) = \int_S y_2 |u_0|^2 dy.$$ Under such restrictions, the quadratic form $b_\varepsilon$ in ${\mathcal H}_0^1(I)$ can be written in terms of the form $t_\varepsilon=t_{\varepsilon,c}$ given by $$\label{eqdefteps} t_\varepsilon(w) : = g_\varepsilon(wu_0)=\int_I \left( |w'|^2 + W_\varepsilon(s) |w|^2 \right) ds,$$ with $$\label{aaa1} W_\varepsilon(s) : = \vartheta(s)+ c + \zeta_\varepsilon(s, y) \left(\frac{\lambda_0}{\varepsilon^2 h^2(s)}-\frac{\lambda_0}{ \varepsilon^2 M^2}\right).$$ We choose the constant $c$ so that $c > \|v\|_\infty + (1 / M^2)\| k(s)^2 / 4\|_\infty$. Since $k(s)$ and $h(s)$ are bounded functions, there exist $\varepsilon_1 > 0$ and $\delta > 0$ so that, for all $s\in I$, $$1 - \varepsilon\, k(s) h(s) \langle z_\alpha(s), F(S) \rangle > \delta \qquad \hbox{and} \qquad 1 - \varepsilon\, k(s) h(s) \langle z_\alpha(s), y \rangle > \delta,$$ for all $\varepsilon < \varepsilon_1$. In what follows, we tacitly assume that $\varepsilon < \varepsilon_1$. The self-adjoint operator associated with $t_\varepsilon$ in $\LL^2(I)$ is $$(T_{\varepsilon,c} w)(s) := - w''(s) + W_\varepsilon(s) w(s), \qquad \dom{T_{\varepsilon,c}} = {\mathcal H}^2(I)\cap {\mathcal H}_0^1(I).$$ From now on we denote by $-\Delta_{\varepsilon,c}$ the operator $-\Delta_\varepsilon + c\Id$ and write $T_\varepsilon=T_{\varepsilon,c}-c\Id$. Next we discuss how the resolvent operator $\left(- \Delta_{\varepsilon,c} - \lambda_0 / \varepsilon^2 M^2 \Id \right)^{-1}$ can be approximated by $T_{\varepsilon,c}^{-1}\oplus 0$, where $0$ is the null operator on the subspace ${\mathcal L}^\perp$. Such result gives a quantitative indication of how $-\Delta_\varepsilon$ is approximated by $T_\varepsilon$. \[ddd10\] [Suppose that $I$ is a bounded interval. Then, there exists $C_6 > 0$ so that $$t_\varepsilon(w) \geq C_6^{-1}\, \varepsilon^{-1} \int_I |w|^2 ds, \qquad \forall w\in {\mathcal H}_0^1(I),\;0 < \varepsilon < \varepsilon_1.$$ ]{} By noting that $$\frac{\varepsilon^2 W_\varepsilon (s)}{s^2} \geq \frac{\lambda_0}{s^2} \delta \left( \frac{1}{h(s)^2} - \frac{1}{M^2} \right),$$ the proof of Lemma \[ddd10\] is similar to the proof of Lemma $2.1$ in [@Frie], and it will not be reproduced here. By following [@Bou], for each $\xi \in \mathbb R^2$, we consider the following perturbed problem $$-{\rm div}[(1 - (\xi \cdot y )) \nabla_y u] = \lambda (1 - (\xi \cdot y)) u, \qquad u \in {\mathcal H}_0^1(S).$$ By taking $\xi= \varepsilon h(s) k(s) z_\alpha$, for $\varepsilon$ small enough, the perturbed operator is positive and with compact resolvent. Denote by $\lambda(\xi) > 0$ its first eigenvalue, i.e., $$\lambda(\xi) = \inf_{\{u \in {\mathcal H}_0^1(S):\: u \neq 0\}} \frac{\int_S (1 - (\xi \cdot y )) |\nabla_y u|^2 dy}{\int_S (1 - (\xi \cdot y )) |u|^2 dy}.$$ Thus, for $v \in {\mathcal H}_0^1(\mathbb R \times S)$, $$\label{ddd8} \frac{1}{\varepsilon^2} \int_S \beta_\varepsilon(s,y) ( | \nabla_y v|^2 - \lambda_0 |v|^2) dy \geq \gamma_\varepsilon(s) \int_S \beta_\varepsilon(s,y) |v|^2 dy \qquad {\rm a.e.}[s],$$ where $$\gamma_\varepsilon(s) := \frac{\lambda(\varepsilon h(s) k(s) z_{\alpha}(s)) - \lambda_0}{\varepsilon^2}.$$ Using the fact that $h(s)$ and $k(s)$ are bounded functions, it is possible to prove that $\gamma_\varepsilon(s)$ converges uniformly as $\varepsilon \to 0$ to a bounded function (see Proposition $4.1$ in [@Bou]). This will be used in the proof of Lemma \[eee4\]. \[eee4\] [Let $I$ denote either $\mathbb R$ or a bounded interval. Then, for $\eta \in {\mathcal H}_0^1(I \times S) \cap {\mathcal L}^\perp$, there exists $C_7\in\R$ so that, for $\varepsilon$ small enough, $$g_\varepsilon(\eta) \geq \frac{C_7}{\varepsilon^2 M^2} \| \eta \|^2.$$]{} Let $\lambda_1$ be the second eigenvalue of the Laplacian in ${\mathcal H}_0^1(S)$, and pick $\eta \in {\mathcal H}_0^1(\mathbb R \times S) \cap {\mathcal L}^\perp$. Since $h(s) \leq M$, for all $s \in I$, we have $$\int_S \beta_\varepsilon(s, y) \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 h(s)^2} - \lambda_1 \frac{|\eta|^2}{\varepsilon^2 M^2} \right)dy \geq \int_S \beta_\varepsilon(s, y) \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 M^2} - \lambda_1 \frac{|\eta|^2}{\varepsilon^2 M^2} \right)dy.$$ By , it follows that $$\int_S \beta_\varepsilon(s, y) \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 M^2} - \lambda_1 \frac{|\eta|^2}{\varepsilon^2 M^2} \right)dy \geq \frac{\gamma_\varepsilon(s)}{M^2} \int_S \beta_\varepsilon(s, y) |\eta|^2 dy.$$ Since $\gamma_\varepsilon(s)$ converges uniformly as $\varepsilon \to 0$, there exists $C_8 \in \mathbb R$ so that, for $\varepsilon$ small enough, $$\frac{\gamma_\varepsilon(s)}{M^2} \geq C_8, \qquad \forall s \in I.$$ Thus, $$\frac{\gamma_\varepsilon(s)}{M^2} \int_S \beta_\varepsilon(s, y) |\eta|^2 dy \geq C_8 \int_S \beta_\varepsilon(s, y) |\eta|^2 dy,$$ and so $$\int_{I \times S} \beta_\varepsilon(s, y) \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 h(s)^2} - \lambda_1 \frac{|\eta|^2}{\varepsilon^2 M^2} \right)dy ds \geq C_8 \int_{I \times S} \beta_\varepsilon(s, y) |\eta|^2 dy ds.$$ Adding and subtracting the term $ \frac{\lambda_0}{\varepsilon^2 M^2} \int_{I \times S} \beta_\varepsilon (s, y) |\eta|^2 dyds$ on the left hand side of the above inequality, we obtain $$\begin{aligned} \int_{I \times S} &\beta_\varepsilon(s, y)& \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 h(s)^2} - \lambda_0 \frac{|\eta|^2}{\varepsilon^2 M^2} \right) dy ds\\ & \geq & C_8 \int_{I \times S} \beta_\varepsilon(s, y) |\eta|^2 dy + \frac{(\lambda_1 - \lambda_0)}{\varepsilon^2 M^2} \int_{I\times S} \beta_\varepsilon(s, y) |\eta|^2 dy ds.\end{aligned}$$ Now, for $\varepsilon$ small enough, there exists $C_9$ so that $$\begin{aligned} g_\varepsilon(\eta) &\geq & \int_{I \times S} \beta_\varepsilon(s, y) \left( \frac{|\nabla_y \eta|^2}{\varepsilon^2 h(s)^2} - \lambda_0 \frac{|\eta|^2 }{\varepsilon^2 M^2} \right) dy ds + c \int_{I \times S} |\eta|^2 dy ds\\ & \geq & C_8 \delta \int_{I \times S} |\eta|^2 dy ds + \frac{(\lambda_1 - \lambda_0)}{\varepsilon^2 M^2} \int_{I \times S} \beta_\varepsilon(s, y) |\eta|^2 dy ds + c \int_{I \times S} |\eta|^2 dy ds\\ & \geq & \frac{C_9}{\varepsilon^2 M^2} \int_{I \times S} \beta_\varepsilon(s, y) |\eta|^2 dy ds\\ & \geq & \frac{C_9}{\varepsilon^2 M^2} \delta \int_{I \times S} |\eta|^2 dy ds.\end{aligned}$$ Finally, it is enough to take $C_7= C_9\, \delta$ to complete the proof of the lemma. ${\hfill \rule {2.5mm} {3mm}\vspace{0.5cm}}$ Now we are ready to state and prove the main result of this section; it will rest on results presented in Section $3$ of [@Sol], combined with the previous lemmas. \[aaa4\] [Let $I$ denote either $\R$ or a bounded interval. Then there exists $C_{10} > 0$ so that, for $\varepsilon$ small enough, $$\left|\left| \left(- \Delta_{\varepsilon,c} - \frac{\lambda_0}{\varepsilon^2 M^2} \Id \right)^{-1} - \left(T_{\varepsilon,c}^{-1} \oplus 0\right) \right| \right| \leq C_{10}\, \varepsilon^{3/2},$$ where $0$ denotes the null operator on the subspace ${\mathcal L}^\perp$. ]{} For $\psi \in {\mathcal H}_0^1(\mathbb R \times S)$ write $$\psi(s,y) = w(s) u_0(y) + \eta(s, y),$$ with $w \in {\mathcal H}_0^1(I)$ and $\eta \in {\mathcal H}_0^1(\mathbb R \times S) \cap {\mathcal L}^\perp$. Thus, the quadratic form $g_\varepsilon(\psi)$ can be rewritten as $$g_\varepsilon(\psi) = t_\varepsilon(w) + g_\varepsilon(\eta) + 2 m_\varepsilon(wu_0, \eta),$$ where $t_\varepsilon(w) = g_\varepsilon(wu_0)$ (see ) and $$\begin{aligned} m_\varepsilon (wu_0, \eta) & = & \int_{I \times S} dy ds \Big[ \left( w' u_0 - w u_0 \,\frac{h'}{h} + w \nabla_y u_0 \cdot Ry (\tau+\alpha') - w \nabla_y u_0 \cdot y \,\frac{h'}{h} \right) \\ & \times & \left(\eta' - \eta \,\frac{h'}{h} + \nabla_y \eta \cdot Ry (\tau+\alpha') - \nabla_y \eta \cdot y \,\frac{h'}{h} \right) \Big]\\ & - & \int_{I \times S} dy ds\; k(s) h(s) \langle z_\alpha, y \rangle w \left(\frac{\nabla_y u_0 \nabla_y \eta}{\varepsilon h^2} - \lambda_0 \frac{u_0 \eta}{\varepsilon M^2} \right).\end{aligned}$$ We are going to show that $t_\varepsilon(w)$, $g_\varepsilon(\eta)$ and $m_\varepsilon(wu_0, \eta)$ satisfy the conditions (3.2), (3.3), (3.4) and (3.5) in Section $3$ of [@Sol], and so the theorem will follow. Conditions (3.2), (3.3) and (3.4) are obtained by applying Lemmas \[ddd10\] and \[eee4\] above. We need only to verify condition (3.5), i.e., that there exists a function $q(\varepsilon)$ so that for each $\psi \in {\mathcal H}_0^1(I \times S)$ $$\label{eee2} |m_\varepsilon(wu_0, \eta)|^2 \,\leq\, q(\varepsilon)^2\, t_\varepsilon(w) \,g_\varepsilon(\eta), \qquad q(\varepsilon) \to 0 \qquad (\varepsilon \to 0).$$ We write $$m_\varepsilon(wu_0, \eta) = m_\varepsilon^1(wu_0, \eta) - m_\varepsilon^2(wu_0, \eta) + m_\varepsilon^3(wu_0, \eta) - m_\varepsilon^4(wu_0, \eta) - m_\varepsilon^5(wu_0, \eta),$$ where $$\begin{aligned} m_\varepsilon^1(w u_0, \eta) & : = & \int_{\mathbb R \times S} w' u_0 \left( \eta' - \eta \,\frac{h'}{h} + \nabla_y \eta \cdot R y (\tau + \alpha') - \nabla_y \eta \cdot y \,\frac{h'}{h} \right) \,ds dy\, , \\ m_\varepsilon^2(wu_0, \eta) & : = & \int_{\mathbb R \times S} w u_0 \,\frac{h'}{h} \left( \eta' - \eta \,\frac{h'}{h} + \nabla_y \eta \cdot R y (\tau + \alpha') - \nabla_y \eta \cdot y \,\frac{h'}{h} \right) \,ds dy\, , \\ m_\varepsilon^3(wu_0, \eta) & : = & \int_{\mathbb R \times S} w \nabla_y u_0 \cdot Ry (\tau+\alpha') \left( \eta' - \eta \,\frac{h'}{h} + \nabla_y \eta \cdot R y (\tau + \alpha') - \nabla_y \eta \cdot y \,\frac{h'}{h} \right) dy ds\, , \\ m_\varepsilon^4(wu_0, \eta) & := & \int_{\mathbb R \times S} w \nabla_y u_0 \cdot y \,\frac{h'}{h} \left( \eta' - \eta \,\frac{h'}{h} + \nabla_y \eta \cdot R y (\tau + \alpha') - \nabla_y \eta \cdot y \,\frac{h'}{h} \right) \,ds dy\, , \\ m_\varepsilon^5(wu_0, \eta) & : = & \int_{I \times S} \frac{k(s) h(s) \langle z_{\alpha} , y \rangle}{\varepsilon} w \left(\frac{\nabla_y u_0 \nabla_y \eta} {h^2} - \lambda_0 \frac{u_0 \eta}{M^2} \right) dy ds.\end{aligned}$$ Now we are going to estimate each of the above terms. Let $$H_1: = \left| \left| \frac{h'}{h} \right| \right|_\infty, \qquad H_2 : = \left| \left| \tau + \alpha ' \right| \right|_\infty,$$ and recall that $$C_1(S) = \int_S |\langle \nabla_y u_0, R y \rangle|^2 dy \qquad \hbox{and} \qquad C_2(S) = \int_S |\langle \nabla_y u_0, y \rangle|^2 dy.$$ By Green identities and some calculations, we get $$\int_S u_0 \langle \nabla_y \eta, Ry \rangle dy = - \int_S \langle \nabla_y u_0, Ry \rangle \eta dy,$$ $$\int_S u_0 \langle \nabla_y \eta, y \rangle dy = - \int_S \langle \nabla_y u_0, y \rangle \eta dy.$$ Hence: $$\begin{aligned} \bullet \quad |m_\varepsilon^1(wu_0, \eta)| & \leq & H_2^{1/2} C_1(S) \left( \int_I |w'|^2 ds \right)^{1/2} \left( \int_{I \times S} |\eta|^2 dy ds\right)^{1/2} \\ & + & H_1^{1/2} C_2(S) \left( \int_I |w'|^2 ds \right)^{1/2} \left( \int_{I \times S} |\eta|^2 dy ds \right)^{1/2} \\ & \leq & C_{11}\, \varepsilon\, t_\varepsilon(w)^{1/2} g_\varepsilon(\eta)^{1/2}; \\\end{aligned}$$ $$\begin{aligned} \bullet \quad | m_\varepsilon^2(wu_0, \eta)| & \leq & \left(\int_{I \times S} |w|^2 |u_0|^2 \left(\frac{h'}{h}\right)^2 dy ds\right)^{1/2} g_\varepsilon(\eta)^{1/2}\\ & \leq & H_1 \left(\int_I |w|^2 ds\right)^{1/2} g_\varepsilon(\eta)^{1/2}\\ & \leq & \frac{H_1}{C_6^{1/2}}\, \varepsilon^{1/2}\, t_\varepsilon(w)^{1/2} g_\varepsilon(\eta)^{1/2};\\\end{aligned}$$ $$\begin{aligned} \bullet \quad |m_\varepsilon^3(wu_0, \eta)| & \leq & \left(\int_{I \times S} |w|^2 |\langle \nabla_y u_0, Ry \rangle|^2 (\tau+\alpha')^2 dy ds \right)^{1/2} g_\varepsilon(\eta)^{1/2} \\ & = & \left(\int_I |w|^2 C_1(S) (\tau+\alpha')^2 ds \right)^{1/2} g_\varepsilon(\eta)^{1/2}\\ & \leq & C_1(S)^{1/2} H_2 \left(\int_I |w|^2 ds\right)^{1/2} g_\varepsilon(\eta)^{1/2} \\ & \leq & C_1(S)^{1/2} \frac{H_2}{C_6^{1/2}}\, \varepsilon^{1/2}\, t_\varepsilon(w)^{1/2} g_\varepsilon(\eta)^{1/2};\\\end{aligned}$$ $$\begin{aligned} \bullet \quad |m_\varepsilon^4(wu_0, \eta)| & \leq & \left(\int_{I \times S} |w|^2 |\langle \nabla_y u_0, y \rangle |^2 \left(\frac{h'}{h}\right)^2 dy ds \right)^{1/2} g_\varepsilon(\eta)^{1/2} \\ & = & \left(\int_I |w|^2 C_2(S) \left(\frac{h'}{h}\right)^2 ds \right)^{1/2} g_\varepsilon(\eta)^{1/2}\\ & \leq & C_2(S)^{1/2} H_1 \left(\int_I |w|^2 ds\right)^{1/2} g_\varepsilon(\eta)^{1/2} \\ & \leq & C_2(S)^{1/2} \frac{H_1}{C_6^{1/2}}\, \varepsilon^{1/2}\, t_\varepsilon(w)^{1/2} g_\varepsilon(\eta)^{1/2}.\end{aligned}$$ Additional calculations show that $$\int_{S} y_1 \langle \nabla_y u_0, \nabla_y \eta \rangle = - \int_{S} f_1(y) \eta dy ds,$$ $$\int_{S} y_2 \langle \nabla_y u_0 \nabla_y \eta \rangle = - \int_{S} f_2(y) \eta dy ds,$$ where $$f_1(y) : = \left(\frac{\partial u_0}{\partial y_1} + y_1 \frac{\partial^2 u_0}{\partial y_1^2} + y_1 \frac{\partial^2 u_0}{\partial y_2^2} \right),$$ $$f_2(y) : = \left(\frac{\partial u_0}{\partial y_2} + y_2 \frac{\partial^2 u_0}{\partial y_2^2} + y_2 \frac{\partial^2 u_0}{\partial y_1^2} \right).$$ Thus, there exist $C_{12}$ and $C_{13}$ so that $$\begin{aligned} |m_\varepsilon^5(wu_0, \eta)| & \leq & \left| \int_{I \times S} k(s) \cos \alpha(s) y_1 w \frac{\nabla_y u_0 \nabla_y \eta}{\varepsilon h} dy ds \right| \\ & + & \left| \int_{I \times S} k(s) h(s) \cos \alpha(s) y_1 \lambda_0 w \frac{u_0 \eta}{\varepsilon M^2} dy ds \right| \\ & + & \left| \int_{I \times S} k(s) \sin \alpha(s) y_2 w \frac{\nabla_y u_0 \nabla_y \eta}{\varepsilon h} dy ds \right| \\ & + & \left| \int_{I \times S} k(s) h(s) \sin \alpha(s) y_2 \lambda_0 w \frac{u_0 \eta}{\varepsilon M^2} dy ds \right| \\ & \leq & \frac{C_{12}}{\varepsilon} \left(\int_\mathbb R |w|^2 ds\right)^{1/2} \left(\int_{\mathbb R \times S} |\eta|^2 dy ds\right)^{1/2}\\ & \leq & C_{13}\, \varepsilon^{1/2}\, t_\varepsilon(w)^{1/2} g_\varepsilon(\eta)^{1/2}.\end{aligned}$$ By the above estimates it follows that there exists $C_{14} > 0$ so that $$|m_\varepsilon(wu_0, \eta)|^2 \leq C_{14}\, \varepsilon\, t_\varepsilon(w)\, g_\varepsilon(\eta),$$ and so is proven. By applying Proposition $3.1$ of [@Sol], it is found that there exists $C_{10}$ so that, for $\varepsilon$ small enough, $$\left| \left| \left(- \Delta_{\varepsilon,c} - \frac{\lambda_0}{\varepsilon^2 M^2} \Id \right)^{-1} - \left(T_{\varepsilon,c}^{-1} \oplus 0\right) \right| \right| \leq C_{10}\, \varepsilon^{3/2}.$$ The proof of the theorem is complete. ${\hfill \rule {2.5mm} {3mm}\vspace{0.5cm}}$ Bounded interval and Dirichlet condition {#qqq3} ======================================== In this section we suppose that $I=[-a,b]$ is a bounded interval and the condition at the boundary $\partial \Lambda_\varepsilon$ is Dirichlet. Since $I$ is bounded, the spectrum of $-\Delta_{\varepsilon,c}$ in $\Lambda_\varepsilon$ is purely discrete and we denote its eigenvalues by $l_j^c(\varepsilon)$. The main result in this section, that is, Theorem \[aaa3\], is a version of Theorem \[mainTeor\] in this context. \[aaa3\] The limits $$\label{eee3new} \mu_j = \lim_{\varepsilon \to 0}\, \varepsilon\left( l_j^c(\varepsilon) - \frac{\lambda_0}{\varepsilon^2 M^2} \right)$$ exist, where $\mu_j$ are the eigenvalues of a self-adjoint operator $T$ (see Equation ) acting in $\LL^2(\mathbb R)$. To prove this theorem we need some previous results; we will follow [@Frie]. Introduce the family of segments $$I_\varepsilon = (-a \varepsilon^{-1/2}, b \varepsilon^{-1/2}), \qquad\, \varepsilon > 0,$$ and the family of unitary operators $J_\varepsilon: \LL^2(I) \to \LL^2 (I_\varepsilon)$ generated by the dilation $s \mapsto s \varepsilon^{1/2}$, that is, $$(J_\varepsilon\psi)(s) = \varepsilon^{1/4}\, \psi(\varepsilon^{1/2}s),$$and identify $\LL^2(I_\varepsilon)$ with the subspace $$\left\{u \in \LL^2(\mathbb R): u(s) = 0\, \hbox{ a.e. in }\, \mathbb R \backslash I_\varepsilon\right \}.$$ Set $$\label{aaa7} \hat{T}_{\varepsilon,c} : = \varepsilon J_\varepsilon T_{\varepsilon,c} J_\varepsilon^{-1},$$ which is a self-adjoint operator acting in $\LL^2(I_\varepsilon)$. \[aaa2\] [In case $I=[-a,b]$ is a bounded interval, one has $$\left| \left| \hat{T}_{\varepsilon,c}^{-1} \oplus 0 - T^{-1} \right| \right| \to 0, \qquad \hbox{as} \qquad \varepsilon \to 0,$$ where $0$ is the null operator on the subspace $\LL^2(\mathbb R \backslash I_\varepsilon)$.]{} We have $\varepsilon J_\varepsilon W_\varepsilon(s) J_\varepsilon^{-1} = \varepsilon W_\varepsilon(\varepsilon^{1/2} s)$, and a direct calculation shows that $$\varepsilon J_\varepsilon W_\varepsilon(s) J_\varepsilon^{-1} = \zeta_\varepsilon(\varepsilon^{1/2} s, y) \lambda_0 \left[ M^{-3} s^2 + \rho (\varepsilon^{1/2} s) s^3 \varepsilon^{1/2} \right] + \varepsilon\, \vartheta(\varepsilon^{1/2} s) + \varepsilon c,$$ with $\rho \in \LL^\infty(I)$. Since $\zeta_\varepsilon(\varepsilon^{1/2} s, y) \to 1$ uniformly as $\varepsilon \to 0$, the proof of Theorem \[aaa2\] is similar to the proof of Theorem $1.3$ in [@Frie], and so it will not be repeated here. [**Proof of Theorem \[aaa3\]:**]{} Let $l_j(T_{\varepsilon,c})$, $l_j(\hat{T}_{\varepsilon,c})$ denote the eigenvalues of $T_{\varepsilon,c}$ and $\hat{T}_{\varepsilon,c}$ respectively. Let $\psi_{j, \varepsilon}^c$ denote the eigenfunction associated with eigenvalue $l_j^c(\varepsilon)$ of $-\Delta_{\varepsilon, c}$. Thus, there exist functions $w_{\varepsilon, c} \in \LL^2(I)$ and $U \in \cal L $ so that $\psi_{j, \varepsilon}^c = w_{j, \varepsilon}^c u_0 + U$. Since $\mathcal L$ is invariant under $(-\Delta_{\varepsilon, c} - \lambda_0 / \varepsilon^2 M^2 \Id)$, it follows that $w_{j, \varepsilon}^c u_0$ is the eigenfunction associated with the eigenvalue $l_j(T_{\varepsilon, c})$. Observe also that the nonzero eigenvalues of $T_{\varepsilon,c}^{-1} \oplus 0$ are exactly the eigenvalues of $T_{\varepsilon,c}^{-1}$. Hence, by Theorem \[aaa4\], we have $$\begin{aligned} \left| \left( l_j^c(\varepsilon) - \frac{\lambda_0}{\varepsilon^2 M^2} \right)^{-1} - l_j^{-1}(T_{\varepsilon,c}) \right| & \leq & \left| \left| \left( -\Delta_{\varepsilon,c} - \frac{\lambda_0}{\varepsilon^2 M^2}\Id \right)^{-1} - T_{\varepsilon,c}^{-1} \oplus 0 \right| \right| \\ & \leq & C_{10}\, \varepsilon^{3/2}.\end{aligned}$$ Thus, $$\left| \frac{1}{\varepsilon} \left( l_j^c(\varepsilon) - \frac{\lambda_0}{\varepsilon^2 M^2} \right)^{-1}- \frac{1}{\varepsilon l_j^{-1}(T_{\varepsilon,c})} \right| \leq C_{10}\, \varepsilon^{1/2}.$$ Since $l_j(\hat{T}_{\varepsilon,c}) = \varepsilon l_j(T_{\varepsilon,c})$, by Theorem \[aaa2\], we find $$\varepsilon l_j(T_{\varepsilon,c}) \to \mu_j, \quad\varepsilon \to 0,$$ and  follows. ${\hfill \rule {2.5mm} {3mm}\vspace{0.5cm}}$ The Neumann case {#nnnn} ================ Here we again consider that $I=[-a,b]$ is a bounded interval, but the Dirichlet condition at the vertical part of the boundary $\partial(I \times S)$, that is, $\{(-a)\times S\cup b\times S\}$, is replaced by Neumann condition. Our point is that the conclusions of Theorem \[aaa3\] also hold true in this case. Although in our case the curvature and torsion can be nontrivial, the proof in this case are similar to the proof of Theorem \[aaa3\] above (and taking into account [@Sol]); for this reason, details will not be presented. The case $I = \mathbb R$ and Dirichlet condition {#iiii} ================================================ In this section we study the case $I = \mathbb R$. First we give sufficient conditions for a nonempty discrete spectrum of the Dirichlet Laplacian, and then discuss the WEO and eigenvalue approximations. The discrete spectrum {#udah} --------------------- Now the spectrum of the Laplacian $- \Delta_{\varepsilon,c}$ in $\Lambda_\varepsilon$ is not necessarily discrete, but in this section we will see that the essential spectrum $\sigma_{\mathrm{ess}}(- \Delta_{\varepsilon,c})$ depends on the behavior of the curvature at infinity; it will then follow that if $k(s)\to 0$ as $|s|\to\infty$, then the discrete spectrum of $- \Delta_{\varepsilon,c}$ is nonempty for $\varepsilon$ small enough. Denote $\nu(\varepsilon) := \inf \sigma_{{\rm ess}} (-\Delta_{\varepsilon,c})$ and let $l_j^c(\varepsilon)$ be the eigenvalues of $-\Delta_{\varepsilon,c}$ (recall the Dirichlet boundary condition). If $I=\mathbb R$ and the curvature satisfies $$\label{eee5} \lim_{|s| \to \infty} k(s) =0,$$ then $\nu(\varepsilon) \to \infty$ as $\varepsilon \to 0.$ Let $N : = \limsup_{|s| \to \infty} h(s) < M$ and $\hat{I} = [-a, a]$ and define $$\Omega_{a, \varepsilon} =\left \{(s, y): s \in \hat{I} \right\} \qquad \hbox{and} \qquad \Omega_{a, \varepsilon}' = \left\{(s, y): s \notin \hat{I}\right \}.$$ Let $-\Delta_{a, \varepsilon, D}^c$, $-\Delta_{a, \varepsilon, D}^{'c}$ be the Dirichlet Laplacian in $\Omega_{a, \varepsilon}$ and $\Omega_{a, \varepsilon}'$ respectively. Similarly, let $-\Delta_{a, \varepsilon, DN}^c$, $-\Delta_{a, \varepsilon, DN}^{'c}$ be the above Laplacian operators but with Neumann condition at the vertical part of the boundaries of $\Omega_{\alpha,\varepsilon}$ and $\Omega_{\alpha,\varepsilon}^{'}$, respectively. Note that $$\label{eee6} - \Delta_{a, \varepsilon, DN}^c + \left(- \Delta_{a, \varepsilon, DN}^{'c}\right) < - \Delta_{\varepsilon, c} < - \Delta_{a, \varepsilon, D}^c + \left(- \Delta_{a, \varepsilon, D}^{'c}\right).$$ Therefore $\inf \sigma_{{\rm ess}} (-\Delta_{\varepsilon, c}) \geq \inf \sigma_{{\rm ess}} (- \Delta_{a, \varepsilon, DN}^{'c})$. Let $q_{a, \varepsilon, DN}'$ the quadratic form associated with the operator $-\Delta_{a, \varepsilon, DN}^{'c}$. Write $K_\varepsilon= \sup_{(s,y) \in \mathbb R \times S} \beta_\varepsilon(s, y)$; we have $$\begin{aligned} q_{a, \varepsilon, DN}' (\psi) & \geq & \left( \inf_{(s,y) \in \mathbb R \times S}{\frac{\beta_\varepsilon(s, y)}{\varepsilon^2 h(s)}} \right) \int_{(\mathbb R \backslash \hat{I}) \times S} |\nabla_y \psi|^2 dy ds \\ & \geq & \lambda_0 \left( \inf_{(s,y) \in \mathbb R \times S}{\frac{\beta_\varepsilon(s, y)}{\varepsilon^2 h(s)}} \right) \int_{(\mathbb R \backslash \hat{I})\times S} |\psi|^2 dy ds \\ & \geq & \lambda_0 \left(\inf_{(s,y) \in \mathbb R \times S}{\frac{\beta_\varepsilon(s, y)}{\varepsilon^2 h(s)}} \right) \frac{1}{K_\varepsilon} \int_{(\mathbb R \backslash \hat{I})\times S} \beta_\varepsilon(s, y) |\psi|^2 dy ds,\end{aligned}$$ for all $\psi \in \dom q_{a, \varepsilon, DN}'$. Since $k$ satisfies , it follows that the essential spectrum of $-\Delta_{a, \varepsilon, DN}^{'c}$ is estimated from below by $\lambda_0$ times a function that converges to $ \frac{1}{\varepsilon^2 N}$ as $a \to \infty$. Since the essential spectrum is a closed subset, it follows that $\nu(\varepsilon) \geq \frac{\lambda_0}{\varepsilon^2 N^2}$ and consequently $\nu(\varepsilon) \to \infty$ as $\varepsilon \to 0$. ${\hfill \rule {2.5mm} {3mm}\vspace{0.5cm}}$ We conclude that, under condition , for $\varepsilon$ small enough the discrete spectrum of $-\Delta_{\varepsilon, c}$ is nonempty. We again stress that we have got another property that does not depend on important geometric features of the tube. Also the WEO $T$ (see also Subsection \[subsecEo\]), which weakly describes the asymptotic behaviors of the eigenvalues of $-\Delta_{\varepsilon, c}$ in the sense of , is not influenced by such geometric features. Weakly effective operator {#subsecEo} ------------------------- The goal of this section is to show that Theorems \[aaa4\], \[aaa3\] and \[aaa2\] have a similar counterpart in case $I = \mathbb R$. In [@Sol] these theorems are proven for two dimensional strips, and here we argue that those proofs can be adapted to our three dimensional setting. The proof of Lemma \[bbb1\] will be postponed to the end of this subsection. \[bbb1\] [There exists $C_6 > 0$ so that, for $\varepsilon$ small enough, $$t_\varepsilon(w) \geq C_6^{-1}\, \varepsilon^{-1} \int_{\mathbb R} |w|^2 ds, \qquad \forall w\in {\mathcal H}_0^1(\mathbb R).$$ ]{} The proof of the next theorem is similar to the proof of Theorem \[aaa4\]; it is enough to take into account Lemma \[eee4\], and then Lemma \[bbb1\] instead of Lemma \[ddd10\]. Recall that ${\mathcal L}$ is the subspace generated by functions $w(s) u_0(y)$ with $w \in \LL^2(\R)$ [Let $I=\mathbb R$. Then, there exists $C_{10} > 0$ so that, for $\varepsilon$ small enough, $$\left|\left| \left(- \Delta_{\varepsilon,c} - \frac{\lambda_0}{\varepsilon^2 M^2} \Id \right)^{-1} - \left(T_{\varepsilon,c}^{-1} \oplus 0\right) \right| \right| \leq C_{10}\, \varepsilon^{3/2},$$ where $0$ denotes the null operator on the subspace ${\mathcal L}^\perp$. ]{} As in the previous section, consider the self-adjoint operators $$\hat{T}_{\varepsilon,c}:= \varepsilon J_\varepsilon T_{\varepsilon,c} J_\varepsilon^{-1},$$ where $J_\varepsilon: \LL^2(\mathbb R) \to \LL^2(\mathbb R)$ is the previously discussed unitary operator generated by the dilation $s\mapsto s \varepsilon^{1/2}$. \[ddd9\] [For $\varepsilon \to 0$ one has $$\left| \left| \hat{T}_{\varepsilon,c}^{-1} - T^{-1} \right| \right| \to 0,$$ where $T$ is the operator . ]{} As in the Section \[qqq3\], we have that $\varepsilon J_\varepsilon W_\varepsilon(s) J_\varepsilon^{-1}$ equals $$\lambda_0 \,\zeta_\varepsilon(\varepsilon^{1/2} s, y) \left[ M^{-3} s^2+ \rho (\varepsilon^{1/2} s) s^3 \varepsilon^{1/2} \right] + \varepsilon\, \vartheta(\varepsilon^{1/2} s) + \varepsilon c.$$ Again, since $\zeta_\varepsilon(\varepsilon^{1/2} s, y) \to 1$ uniformly as $\varepsilon \to 0$, the proof of Theorem \[ddd9\] is similar to the proof of Theorem $1.3$ in [@Frie], and details will be skipped. [**Proof of Lemma \[bbb1\]:**]{} Theorem \[ddd9\] guarantees that $$\varepsilon^{-1} \left| \left| T_{\varepsilon,c}^{-1} \right| \right| \to \left| \left| T^{-1} \right| \right| \qquad (\varepsilon \to 0),$$ and so there exists $C_6> 0$ so that $$\left| \left| T_{\varepsilon,c}^{-1} \right| \right| \leq C_6 \,\varepsilon.$$ The proof is complete. ${\hfill \rule {2.5mm} {3mm}\vspace{0.5cm}}$ Proof of Theorem \[teoap\] ========================== It will be shown that, for $\varepsilon$ small enough, there exists $C_5>0$ so that $$\left\|\hat{G}_\varepsilon^{-1} - G_\varepsilon^{-1}\right \| \leq C_5\, \varepsilon.$$ Remember that $c > \|v\|_\infty + (1 / M^2)\| k(s)^2 / 4\|_\infty$, thus, there exists a number $d > 0$ so that $c = \|v\|_\infty + (1 / M^2)\| k(s)^2 / 4\|_\infty + d$. Since $\zeta_\varepsilon \to 1$ uniformly as $\varepsilon \to 0$, there exist $\varepsilon_1 > 0$ and numbers $\sigma_1, \sigma_2 > 0$ so that $\sigma_1 \leq \beta_\varepsilon \leq \sigma_2$, for all $ \varepsilon < \varepsilon_1$. Thus, $$\hat{g}_\varepsilon(\psi) \geq \sigma_1 d \| \psi\|^2 \qquad \hbox{and} \qquad g_\varepsilon(\psi) \geq d \| \psi\|^2,$$ for all $\varepsilon < \varepsilon_1$. Consequently, $$\|\hat{G}_\varepsilon^{-1}\| \leq \frac{1}{\sigma_1 d} \qquad \hbox{and} \qquad \|G_\varepsilon^{-1}\| \leq \frac{1}{d},$$ for all $\varepsilon < \varepsilon_1$. Since $k, h \in \LL^\infty(\mathbb R)$, $y \in S$ and $S$ is a bounded region, there exist $\varepsilon_0 > 0$ ($\varepsilon_0 < \varepsilon_1$) and $C_1, C_2 >0$ so that $$\left|\left( \frac{1}{\zeta_\varepsilon} - 1 \right)\right| = \left|\frac{\varepsilon\, k(s)h(s) (y \cdot z_\alpha(s))}{\zeta_\varepsilon}\right| \leq C_1\, \varepsilon,$$ and $$c\,|(\zeta_\varepsilon - 1)| \leq C_2\, \varepsilon,$$ for all $\varepsilon < \varepsilon_0$. Under such conditions we have $$\begin{aligned} & & |\hat{g}_\varepsilon(\psi) - g_\varepsilon(\psi)| \\ & \leq & \int_{I \times S} \left|\left( \frac{1}{\zeta_\varepsilon} - 1 \right)\right| \left|\psi' - \psi \,\frac{h'}{h}+ (\nabla_y \psi \cdot R y ) (\tau+\alpha') - (\nabla_y \psi \cdot y ) \frac{h'}{h} \right|^2 dy ds \\ & + & \int_{I \times S} c |(\zeta_\varepsilon-1)| |\psi|^2\,ds dy \\ &\leq & C_1 \varepsilon \int_{I \times S} \left|\psi' - \psi \,\frac{h'}{h}+ (\nabla_y \psi \cdot R y ) (\tau+\alpha') - (\nabla_y \psi \cdot y ) \frac{h'}{h} \right|^2 + C_2 \varepsilon \int_{I \times S} |\psi|^2 dy ds \\ & \leq & C_3 \varepsilon g_\varepsilon(\psi)\end{aligned}$$ for some $C_3> 0$. Hence, $$(1 - C_3 \varepsilon) g_\varepsilon(\psi) \leq \hat{g}_\varepsilon(\psi) \leq (1 + C_3 \varepsilon) g_\varepsilon(\psi),$$ for all $\varepsilon < \varepsilon_0$. The first inequality implies that it is possible to find $\varepsilon_0' > 0$ ($\varepsilon_0' < \varepsilon_0$) and a constant $C_4>0$ so that $$g_\varepsilon (\psi) \leq C_4\, \hat{g}_\varepsilon(\psi),$$ for all $\varepsilon < \varepsilon_0'$. By Schwarz’s Inequality for bilinear forms, we have $$|\hat{g}_\varepsilon(\psi_1, \psi_2)| \leq [\hat{g}_\varepsilon(\psi_1)]^{1/2}\,[\hat{g}_\varepsilon(\psi_2)]^{1/2},$$ $$|g_\varepsilon(\psi_1, \psi_2)| \leq [g_\varepsilon(\psi_1)]^{1/2}\,[g_\varepsilon(\psi_2)]^{1/2},$$ for all $\psi_1, \psi_2 \in {\mathcal H}_0^1(\mathbb R \times S)$. Thus, by using the above estimates, for each pair $\psi_1, \psi_2 \in {\mathcal H}_0^1(\mathbb R \times S)$ we have $$\begin{aligned} \big|\big\langle \hat{G_\varepsilon}^{1/2} \psi_1, \hat{G_\varepsilon}^{1/2} \psi_2 \big\rangle &-& \big\langle G_\varepsilon^{1/2} \psi_1, G_\varepsilon^{1/2} \psi_2 \big \rangle\big| \\ & = & \big|\hat{g}_\varepsilon (\psi_1, \psi_2) - g_\varepsilon (\psi_1, \psi_2)\big| \\ & \leq & C_3 \,\varepsilon\, [g_\varepsilon(\psi_1)]^{1/2}\, [g_\varepsilon(\psi_2)]^{1/2}\\ & \leq & C_3 \sqrt{C_4}\, \varepsilon\, [g_\varepsilon(\psi_1)]^{1/2}\, [\hat{g}_\varepsilon(\psi_2)]^{1/2}.\end{aligned}$$ By picking $\psi_1 = G_\varepsilon^{-1} f$, $\psi_2= \hat{G}_\varepsilon^{-1} g$, where $f,g \in \LL^2(\mathbb R \times S)$ are arbitrary, we obtain $$\begin{aligned} \Big|\big\langle \hat{G_\varepsilon}^{-1}f, g \big\rangle &-& \big\langle G_\varepsilon^{-1}f, g \big\rangle \Big| \\ & \leq & C_3 \sqrt{C_4}\, \varepsilon \left[\big \langle \hat{G}_\varepsilon^{-1} g, g\big \rangle \,\big\langle G_\varepsilon^{-1} g, g \big\rangle \right]^{1/2} \\ & \leq & \frac{C_3 \sqrt{C_4}}{d \sqrt{\sigma_1}} \, \varepsilon\, \|f\| \: \|g\|,\end{aligned}$$ for all $\varepsilon < \varepsilon_0'$. 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--- abstract: 'Recent advances in the development of commercial quantum annealers such as the D-Wave 2X allow solving NP-hard optimization problems that can be expressed as quadratic unconstrained binary programs. However, the relatively small number of available qubits (around 1000 for the D-Wave 2X quantum annealer) poses a severe limitation to the range of problems that can be solved. This paper explores the suitability of preprocessing methods for reducing the sizes of the input programs and thereby the number of qubits required for their solution on quantum computers. Such methods allow us to determine the value of certain variables that hold in either any optimal solution (called strong persistencies) or in at least one optimal solution (weak persistencies). We investigate preprocessing methods for two important NP-hard graph problems, the computation of a maximum clique and a maximum cut in a graph. We show that the identification of strong and weak persistencies for those two optimization problems is very instance-specific, but can lead to substantial reductions in the number of variables.' author: - | Georg Hahn\ Imperial College, London, UK - | Hristo Djidjev (PI)\ Los Alamos National Laboratory title: Reducing Binary Quadratic Forms for More Scalable Quantum Annealing --- Introduction {#sec:intro} ============ Quantum annealing and D-Wave ---------------------------- The recent availability of the first commercial quantum computers of D-Wave Systems Inc. [@dwave2016] provides a novel tool to help finding, by a process known as *quantum annealing*, global solutions to NP-hard problems that seem to be difficult to obtain classically [@Djidjev2017]. As of 2017, the most recent version of the company’s quantum annealer, D-Wave 2000Q, is capable of handling around $2000$ *qubits* of quantum information. Each qubit consists of a superconducting loop on the D-Wave chip which simultaneously encodes a quantum superposition of $-1$ and $+1$ via two superimposed currents in both clockwise and counter-clockwise directions [@Johnson2011; @Bunyk2014]. Upon completion of the annealing process, the system turns classical and each qubit takes a classical (binary) value that can be read and assigned to a variable. D-Wave is designed to minimize an objective function consisting of a sum of linear and quadratic binary contributions, given as $$H=H(x_1,\dots,x_N)=\sum_{i \in V} a_i x_i + \sum_{(i,j) \in E} a_{ij} x_i x_j, \label{eq:hamilt}$$ where $a_i$, $a_{ij} \in \mathbb{R}$ and $V = \{1,\ldots,N\}$, $E=V \times V$ for $N \in {{\mathbb N}}$ qubits [@King2015]. The functional type is called the *Hamiltonian* encoding the energy of a quantum system. In qubits are encoded by variables $x_i$, $E$ denotes the dependency structure between the qubits (the set of interactions), and $a_i$ and $a_{ij}$ encode physical characteristics of the qubits and the links (called *couplers*) between them, respectively. During annealing, the quantum system tries to reach a configuration of minimal energy of $H$. A quadratic form of type with $x_i \in \{-1,1\}$ is called an *Ising model*. By replacing $x_i$ in with $(x_i+1)/2$, one gets a quadratic form of the same type as , but with $x_i \in \{0,1\}$. The resulting problem is called a *quadratic unconstrained binary optimization (QUBO)* problem. Any QUBO or Ising problem can be solved on D-Wave as long as there are enough qubits and couplers between those qubits available on the D-Wave architecture to represent the system . Aim of this work ---------------- Although [@Djidjev2017] show that certain instances designed to fit the D-Wave qubit interconnection network can be solved magnitudes faster with D-Wave than with its classical analogues, the limitation of only around $1000$ qubits and about 3000 couplers available on D-Wave 2X (DW) at Los Alamos National Laboratory poses a severe restriction. For most NP-hard optimization problems, due to the density of the Hamiltonian $H$, the largest problems able to fit DW are of size about only 45. To mitigate the current size limitations of the D-Wave architecture, this work tries to explore the usability of preprocessing methods allowing to determine the value of certain variables in a QUBO or Ising problem. Fixing all such variables then results in a new Hamiltonian of reduced size, consisting only of the unresolved variables. We investigate the usage of partitioning and roof duality methods for two important NP-hard graph problems, the Maximum Clique and the Maximum Cut problem, which have multiple applications including network analysis, bioinformatics, and computational chemistry. Given an undirected graph $G = (V, E)$, a *clique* is a subset $S$ of the vertices forming a complete subgraph, meaning that any two vertices of $S$ are connected by an edge in $G$. The clique size is the number of vertices in $S$, and the *Maximum Clique* problem is to find a clique in $G$ with a maximum number of vertices [@Balas1986]. A cut $S$ of a graph $G$ is the set of (cut) edges between $S$ and $V \setminus S$ in $G$. The *Maximum Cut* problem is to find a maximum cut, which is a cut of maximum size. Previous work ------------- In [@BorosHammer2001], optimization methods for quadratic *pseudo-boolean* functions, which are multi-linear polynomial functions of boolean variables, are considered. The authors analyze a variety of techniques based on roof duality, introduced in [@Hammer1984], in order to determine partial assignments of values or relations between variables in such pseudo-boolean functions. A comprehensive overview of efficient preprocessing techniques for QUBO optimization can be found in [@Boros2006]: Those techniques consists of methods based on first and second order derivatives, roof duality and probing techniques resulting in lower bounds on the minimum energy of a QUBO, the values in either any (strong persistencies) or at least one optimum (weak persistencies), binary relations between variables in some or every optimum, as well as decomposition techniques of the original problem into several smaller pairwise independent QUBO subproblems. A more efficient implementation of the probing techniques of [@Boros2006], combined with the aforementioned roof duality, is presented in [@Rother2007], with a special focus on computer vision applications. The code for the methods investigated in [@Rother2007] are made available for *C++* in the *Qpbo* package of [@Kolmogorov2014]. Bindings of this package for *Python* are available (package *PyQpbo*, which is part of the *PyStruct* package of [@JMLR:v15:mueller14a]) and will be used in the experiments of this article. The article is organized as follows. Section \[sec:background\] gives a detailed overview of the optimization methods used in this report. Moreover, details on the optimization problems for the two NP-hard graph problems we consider, Maximum Clique and Maximum Cut, are given. Section \[sec:maxclique\] focuses on the Maximum Clique problem. We state its formulation as a QUBO, highlight the use of a decomposition method based on graph partitioning methods into smaller subproblems, and present experimental results for Qpbo. For this we employ several test graphs, both stand-alone and in connection with the decomposition approach. Section \[sec:maxcut\] repeats the QUBO formulation and a Qpbo experimental study for the Maximum Cut problem. We conclude with a discussion in Section \[sec:conclusions\]. In the rest of the paper, we denote a graph as $G=(V,E)$, where $V = \{ 1,\ldots,n \}$ is a set of $n$ vertices and $E$ is a set of undirected edges. Background {#sec:background} ========== We briefly overview the techniques used in [@Boros2006] and [@Rother2007] in order to identify strong and weak persistencies. Given a QUBO instance, it is first represented as a (quadratic) *posiform* $$\begin{aligned} \phi(x) = \sum_{T \subseteq {\bf L}} a_T \prod_{u \in T} u, \label{eq:posiform} \end{aligned}$$ where $a_T \geq 0$ for all $T \neq \emptyset$ and $\bf L$ is a set of literals over the $n$ variables $\{1,\ldots,n\}$ and their complements. It is shown in [@Boros2006] that any posiform can be equivalently expressed as a quadratic form $$\Phi(x) = a_0 + \sum_{u \in {\bf L}} a_u u + \sum_{u,v \in {\bf L}} a_{uv} u v.$$ Moreover, [@Boros2006] shows that there is a one-to-one correspondence between posiforms and *implication networks*, which are capacitated networks $G_\Phi=(V,E)$ with $n=|V|$ nodes and non-negative capacities $\gamma_{uv}$ on the edges connecting any two nodes $u,v \in V$, $u \neq v$, $V=\{1,\ldots,n\}$. The implication network $G_\Phi$ has an important application in the computation of the *roof dual*, which is a lower bound on the value of a quadratic posiform [@Hammer1984]. The roof dual is computed with the help of a maximum flow through the implication network $G_\Phi$. More importantly for our work, the flow analysis of the implication network also identifies strong and weak persistencies for problem , which are then transformed for persistencies for problem . As shown in [@Boros2006], the effectiveness of the roof duality method for finding persistencies can be improved by using *probing*, which is a method that assigns values of $0$ and $1$ to some variables and then compares any persistencies found for these alternative assignments. In some cases, such probing analysis can (a) determine additional persistencies for the original problem or (b) identify relationships between variables which must hold in a (local) optimum, meaning that some pairs of variables will have equal or inverted values. This then allows to substitute one of the variables in each pair with the other (or its inverse). The maximum clique problem {#sec:maxclique} ========================== QUBO formulations {#sec:maxclique_qubo} ----------------- We are interested in finding a maximum clique of a graph $G=(V,E)$, where $V=\{1,\ldots,n\}$. We assign a decision variable $x_i$ to each vertex $i$, $i \in \{1,\ldots,n\}$, indicating whether or not vertex $i$ is part of the maximum clique. One way to formulate the Maximum Clique problem as a QUBO is to use the equivalence between Maximum Clique and the maximum independent set problem. The latter asks for a maximum *independent set* of vertices of $G$, that is a set such that no two vertices from it are connected by an edge in $G$. Specifically, an independent set of $H = (V, \overline{E})$ defines a clique in graph $G=(V,E)$, where $\overline{E}$ is the complement of set $E$. This leads to a constrained formulation of Maximum Clique given by $$\begin{aligned} \underset{x_v \in \{0,1\}}{\text{maximize}}~\sum_{v \in V} x_v \qquad \text{subject to}~\sum_{(u,v) \in \overline{E}} x_u x_v=0. \label{eq:mis}\end{aligned}$$ The constraint formulation can equivalently be written in QUBO form as $$\begin{aligned} H = -A\sum_{v \in V} x_v + B\sum_{(u,v) \in \overline{E}} x_u x_v, \label{eq:maxclique_acm}\end{aligned}$$ where the penalty weights can be chosen as $A=1$, $B=2$ [@Lucas2014]. One disadvantage of lies in the fact that $H$ contains $O(n^2)$ quadratic terms for sparse graphs (of $O(n)$ edges), limiting the size of problems that can fit DW. An alternative QUBO formulation of [@Lucas2014] assumes the clique size $K \geq 1$ is known. Its Hamiltonian $$\begin{aligned} H_K = A \bigg( K - \sum_{v \in V} x_v \bigg)^2 + B\bigg(\binom{K}{2} - \sum_{(u,v) \in E} x_u x_v \bigg) \label{eq:maxclique_lucas}\end{aligned}$$ is designed to attain the value zero only for an assignment defining a clique of size $K$, which is achieved if one chooses weights $A=K+1$ and $B=1$ [@Lucas2014]. Problem decomposition {#sec:decomp} --------------------- ![Illustration of the vertex splitting algorithm. $G_1$ contains all neighbors of $v$ (without $v$ itself) and the edges between them; $G_2$ contains all vertices and edges of $G$ except $v$ and the edges incident to $v$.[]{data-label="fig:vertex_splitting"}](vertex_splitting2.pdf){width="50.00000%"} In [@Djidjev2017] we developed several algorithms for decomposing an input graph too large for DW into subgraphs small enough that the (Maximum Clique) QUBO corresponding to each generated subproblem fits DW. After solving each subproblem, the solutions are combined again into a solution to Maximum Clique for the original graph. We describe here an improved version of the most universal of those algorithms, in the sense that it can be applied to all input graphs regardless of their structure. It works as follows. Given a graph $G=(V,E)$, pick any vertex $v$, e.g, with the smallest number of neighbors as in [@Djidjev2017], and define two new graphs: a graph $G_1$ induced by all neighbors of $v$ (but without $v$ itself) and the graph $G_2$ induced by $V\setminus\{v\}$, that is $G$ without $v$ and its incident edges (Figure \[fig:vertex\_splitting\]). Then find recursively the maximum cliques $C_1$ of $G_1$ and $C_2$ of $G_2$. If $|C_1|>|C_2|$, return as a maximum clique for $G$ the subgraph induced by $C_1\cup \{v\}$, otherwise return $C_2$. The recursion stops when $G$ is small enough to fit DW. It is easy to see the recursion is finite, as both $G_1$ and $G_2$ contain no more than $|V|-1$ vertices each, since neither of them contains $v$, and that it correctly finds a clique of maximum size for $G$. Unfortunately, in the worst case it may generate an exponential number of subgraphs (exponential in the size of $G$). We will study in the next subsection how persistency analysis can help to reduce the number of subgraphs generated. Results with Qpbo ----------------- In our experiments, we used the *C++* package Qpbo of [@Rother2007; @Kolmogorov2014] for finding persistencies, developed for computer vision applications. We access Qpbo through a Python interface that we adapted by modifying the PyQpbo package [@JMLR:v15:mueller14a]. This section presents results on the effectiveness of the method for Maximum Clique. We investigate the application of Qpbo to QUBOs and on random as well as special graphs (c-fat and Hamming graphs), both without and in connection with the graph splitting algorithm of Section \[sec:decomp\]. Moreover, we compare the two QUBO formulations and presented in Section \[sec:maxclique\_qubo\]. ### Qpbo on c-fat and Hamming graphs {#sec:qpbo_hamming} We apply Qpbo to the QUBO formulation . The test graphs we use are from the *1993 DIMACS Challenge on maximum cliques, coloring and satisfiabilty* [@johnson-trick-96] and have also been used in [@Boros2006]. Both graph families depend on two parameters: the number of vertices $n$ and an additional internal parameter, precisely the partition parameter $c$ for c-fat graphs and the Hamming distance $d$ for Hamming graphs. We use the generation algorithms of [@Hasselberg1993] for both graph families. Table \[tab:maxclique\] shows the results (where both parameters $c$ and $d$ are summarized as a generic parameter $q$). We see that, in six out of eight cases, Qpbo can find persistencies for 100% of the variables, while in the remaining two cases it cannot find any persistency. Also we observe that probing is the most effective algorithm, while finding strong persistencies is the least effective. {width="45.00000%"}  {width="45.00000%"} In order to observe and compare a wider range of percentages of solved variables with Qpbo, we insert (or delete) edges from c-fat or Hamming graphs with a certain probability $p$ which we will vary. Figure \[fig:plot\_fat500-2\] (left) shows that when adding random edges (with edge probability $p$) to the $(500,2)$ c-fat graph, no strong and weak persistencies are found for both the original graph as well as for almost all dense modified graphs (since the x-axis starts at $p=0.97$). Persistencies (as a percentage of solved variables out of the $n=500$ variables in the QUBO) are only observed for $p \geq 0.97$ and increase to $100\%$ over a very short range. As expected, weak persistencies are found (slightly) earlier than strong ones. The behavior for the Hamming graph with $n=8$ and Hamming distance $d=4$ is similar. {width="45.00000%"}  {width="45.00000%"} Similarly to Figure \[fig:plot\_fat500-2\], Figure \[fig:plot\_ham8-2\] shows strong and weak persistencies as a function of $p$ for the Hamming graph with $n=8$ and $d=4$. However, the behavior of strong and weak persistencies is different in this case. Figure \[fig:plot\_ham8-2\] shows that Qpbo is able to assign a value to $100\%$ of all variables in the QUBO of the original Hamming graph via weak persistencies (see Table \[tab:maxclique\]). Nevertheless, strong persistencies are not found. We therefore add edges with probability $p$ to observe a progression of strong persistencies (Fig. \[fig:plot\_ham8-2\], left) and remove edges with probability $p$ for weak persistencies (Fig. \[fig:plot\_ham8-2\], right). Figure \[fig:plot\_ham8-2\] shows that strong persistencies increase gradually over the entire interval $p \in [0,1]$, whereas the found weak persistencies disappear almost instantly as soon as edges are removed from the graph. ### Comparison of alternative QUBO formulations for Maximum Clique The QUBO formulation was used for all previous investigations. This is due to the fact that, in contrast to , the former does not need the (usually unknown) clique size $K$ as an input parameter. Nevertheless, this section investigates whether the choice of formulation has an effect on the persistencies found and whether changing the QUBO formulation can result in more or fewer persistencies. If this is the case, then it may be worth investing time to search for formulations that are more suitable for the persistency algorithm. In order to find the clique size that is needed for , we first run an alternative algorithm to find the maximum clique size $K$ for each tested graph. We then compute the QUBO coefficients for with parameter $K$ and identify strong and weak persistencies using Qpbo. Table \[tab:qubo\_comparison\] shows results for a Hamming graph with parameters $n=8$ and $d=2$ and a c-fat graph with parameters $n=200$ and $c=1$. As can be seen from the table, formulation results in much larger QUBO problems (which are likely to also have a more complex structure, thus being more difficult to solve for Qpbo) and thus in fewer persistencies compared to , so clearly the latter formulation is preferable, at least for the type of graphs tested. ### Combining Qpbo with the decomposition algorithm {#sec:splitting_random} We apply the graph splitting algorithm described in Section \[sec:decomp\] to solve the Maximum Clique problem on graphs of various sizes. However, before splitting a graph, we look for persistencies (in particular, for strong and weak persistencies since applying probing in each time step took prohibitively long) and, if found, reduce the size of the graph by removing the corresponding variables. We stop splitting when all generated (sub-)graphs are of size small enough to fit DW, which in our case is $45$ vertices. We test the resulting algorithm on random graphs with $n=500$ vertices and an expected number of edges in the interval $[10000,40000]$. ![Proportional saving as a function of the expected number of edges of the random graph when using Qpbo in the graph splitting routine, computed as $(n_\text{qpbo}-n_\text{no-qpbo})/n_\text{no-qpbo}$, where $n_\text{qpbo}$ and $n_\text{no-qpbo}$ denote the number of solver calls when employing graph splitting with and without Qpbo. \[fig:plot\_splitting\_ratio\]](all_splitting_ncalls_ratio){width="50.00000%"} Figure \[fig:plot\_splitting\_ratio\] visualizes the proportional reduction in the number of generated subproblems. For this, we plot the ratio $(n_\text{qpbo}-n_\text{no-qpbo})/n_\text{no-qpbo}$ as a function of $p$, where $n_\text{qpbo}$ and $n_\text{no-qpbo}$ denote the number of solver calls when employing graph splitting with and without Qpbo, respectively. The figure shows that, especially for sparse graphs, substantial reductions with respect to the number of generated subproblems can be achieved. The maximum cut problem {#sec:maxcut} ======================= This section takes a closer look at the Maximum Cut problem introduced in Section \[sec:intro\]. We again state a QUBO formulation for Maximum Cut and present simulation results on $g$ and $U$ graphs, which have already been defined and previously been used for benchmarking Maximum Cut problems in [@Kim2001] and [@Boros2006]. QUBO and Ising formulation -------------------------- We use the QUBO formulation for Maximum Cut of [@BorosHammer1991] who show that the value $f$ of any cut in a graph $G=(V,E)$ is given as $$\begin{aligned} \label{eq:maxcut_QUBO} f(x) = \sum_{(u,v)\in E} \left( x_u (1-x_v) + (1-x_u) x_v \right),\end{aligned}$$ where $x_u \in \{0,1\}$ for all $u \in V$ is the indicator signalling which side of the cut vertex $u$ belongs to. Maximizing over $x \in \{0,1\}^{|V|}$ leads to a maximum cut. The formulation can be simplified when switching the input range of $x_u$ to an Ising model. In , each term $x_u (1-x_v) + (1-x_u) x_v$ is zero if and only if $u$ and $v$ belong to the same side of the cut, and one otherwise. In an Ising formulation, that is when $x_u \in \{-1,+1\}$ for all $u \in V$, the same term can be expressed as $(1-x_u x_v)/2$, leading to the Hamiltonian $$H_0(x) = \sum_{(u,v)\in E} \frac{1}{2} (1-x_u x_v),$$ which has to be maximized (just like ). Equivalently, we can minimize $$\begin{aligned} H(x) &= - \sum_{(u,v)\in E} (1-x_u x_v)= -|E| + \sum_{(u,v)\in E} x_u x_v \nonumber \\ &\sim \sum_{(u,v)\in E} x_u x_v,\label{eq:maxcut_ising}\end{aligned}$$ since $|E|$ is constant. In the following we will solely use the simpler Ising formulation . Ising on Lg and Lg graphs ------------------------- We test Qpbo on the Ising formulation for the Maximum Cut problem applied to $g$ and $U$ graphs. Table \[tab:maxcut\] shows the results. As seen in the table, some weak persistencies are found, but no strong ones. Unlike the results for Maximum Clique, where the found weak persistencies are either $0\%$ or $100\%$, here we observe more varied results. Again, probing is the most effective algorithm. Our results for Maximum Clique are roughly similar to the ones presented in [@Boros2006] (who, however, use a different and possibly more elaborate algorithm than Qpbo, which they did not make available). Additionally, we again studied how the performance of Qpbo varies as the graph becomes denser by adding or removing random edges. These results are similar to the ones already described for Maximum Clique (Figures \[fig:plot\_fat500-2\] and \[fig:plot\_ham8-2\]) and are thus not presented here. Conclusions {#sec:conclusions} =========== This paper studies the use of preprocessing methods to reduce the size of QUBO and Ising integer programs. Such models have recently attracted increased attention since they allow constant time solutions on newly available quantum annealers. However, the small number of available qubits on such devices severely limits the range of solvable problems. This motivates the use of preprocessing methods to reduce the number of required qubits, thus extending the application range of such devices. We use the package Qpbo [@Kolmogorov2014; @JMLR:v15:mueller14a] to compute strong and weak persistencies in integer programs and test the suitability of preprocessing methods on two important NP-hard graph problems, the Maximum Clique and the Maximum Cut problems. Our main findings can be summarized as follows: 1. The observed reduction in variables seems to be very problem-specific, both in terms of the QUBO/ Ising formulation considered as well as the employed test graphs. Test graphs on which Qpbo achieved $100\%$ reductions for Maximum Clique did not lead to any reduction for Maximum Cut or vice versa. We observe $0\%$ and $100\%$ reductions considerably more often than reductions between those extremes. 2. When applied to a decomposition algorithm which divides up a maximum clique computation on graphs of arbitrary size into many smaller subproblems, Qpbo can be used to reduce the number of generated subproblems. This reduction seems especially significant for sparser graphs. 3. Using two different QUBO formulations for Maximum Clique, we show that the performance of Qpbo is not only dependent on the type of problem solved and the input graph, but also on the specific QUBO formulation used. This, therefore, justifies research into finding formulations for different optimization problems that are more suitable for identifying persistencies. This is consistent with the much more general observation in combinatorial optimization practice that the choice of formulation for a given optimization problem can have a huge effect on the time it will take to solve the resulting problem using popular solvers. The effectiveness of Qpbo varied widely between seemingly similar problem instances. An especially interesting and practically important open problem is to be able to characterize input graphs and problem formulations that are more suitable for identification of persistencies. Another interesting open problem is how to best speed up the persistency finding algorithms. For instance, in the case where persistency finding (via Qpbo) is combined with the decomposition algorithm (for Maximum Clique), we currently run Qpbo on every subgraph generated during the graph splitting process. 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--- abstract: 'Within Dyson-Maleev (DM) transformation and self-consistent mean-field treatment, the Néel order/disorder transition is studied for an antiferromagnetic Heisenberg model which is defined on a square lattice with a nearest neighbour exchange $J_1$ and a next-nearest neighbour exchange $J_2$ along only one of the diagonals. It is found that the Néel order may exist up to $J_2/J_1=0.572$, beyond its classically stable regime. This result qualitatively improves that from linear spin-wave theory based on Holstein-Primakoff transformation.' address: | $^1$Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str.38, 01187 Dresden, Germany\ $^2$Pohl Institute of Solid State Physics, Tongji University, Shanghai 200092, P.R.China author: - 'Qingshan Yuan$^{1,2}$' title: 'The Néel order for a frustrated antiferromagnetic Heisenberg model: beyond linear spin-wave theory' --- The two-dimensional (2D) antiferromagnetic Heisenberg models have attracted great interest in recent years, partly because of the fact that the parent compounds of the high temperature superconducting materials are excellent realizations of quasi-2D quantum antiferromagnets [@Chakravarty]. While the unfrustrated Heisenberg model has been well understood, much attention has been paid to the frustrated models such as square-lattice nearest and next-nearest neighbour interaction (so called $J_1$-$J_2$) model, triangular lattice model and [*Kagomé*]{} lattice model etc [@Oitmma; @Hu; @Ivanov; @Mila; @Singh; @Lecheminant]. In these frustrated Heisenberg models the property of the ground state, whether magnetically ordered or disordered, is a subject of considerable interest. For example, the $J_1$-$J_2$ model takes on Néel order at small $J_2/J_1$ and collinear order at large $J_2/J_1$, which are seperated by a region of disordered state [@Oitmma]. Very recently a $S=1/2$ Heisenberg model, which defined on a square lattice with a nearest neighbour antiferromagnetic exchange $J_1$ and a next-nearest neighbour exchange $J_2$ along only [*one*]{} of the diagonals of the lattice as shown in Fig. 1, has been proposed [@Zheng; @Merino]. Its Hamiltonian is written as $$H = J_1\sum_{\langle ij \rangle} {\bf S}_i\cdot {\bf S}_j+J_2 \sum_{\langle lm \rangle} {\bf S}_l\cdot {\bf S}_m \ \ ,\label{H}$$ where the notation $\langle ij \rangle$ denote nearest neighbour bonds and $\langle lm \rangle$ denote next-nearest neighbour bonds along only one diagonal. Topologically this model is equivalent to the Heisenberg antiferromagnet on an anisotropic triangular lattice [@Bhaumik; @Trumper]. In special cases $J_2=0,\ J_1=J_2$ and $J_1=0$, it will recover to unfrustrated square lattice model, isotropic triangular lattice model and decoupled spin chains, respectively. Therefore this model provides a way of interpolating between several well-known one and two-dimensional models. One can study the role of frustration in going from one to two dimensions. On the other hand, this model is of direct relevance to the magnetic phases of some quasi-2D organic superconductors. It has been argued that this model may describe the spin degree of freedom of the insulating phase of the layered molecular crystals $\kappa$-(BEDT-TTF)$_2$X [@McKenzie]. The parameter $J_2/J_1$ for these materials is suggested to be $\sim 0.3-1$ and the magnetic frustration will play an important role. Classically, the ground state of the model can be derived straightforwardly as a function of the ratio $J_2/J_1$ if we assume that the spins lie in the $xz$ plane and are described by a spiral form ${\bf S}_i=S(\sin \theta_i,0,\cos \theta_i)$ as shown in Fig. \[fig\_spin\]. Here the angle $\theta_i={\bf q}\cdot {\bf r}_i$ and the wavevector ${\bf q}=(q,q)$ defines a relative orientation of the spins. Minimization of the classical energy with respect to $q$ gives the result that the ground state take on Néel order (i.e., $q=\pi$) for $J_2/J_1\le 1/2$, and spiral order with $q=arccos(-J_1/2J_2)$ for $J_2/J_1> 1/2$. The quantum model has also been studied numerically and analytically. The series expansions were adopted by Zheng et al. [@Zheng] for numerical calculation. It was found that the Néel order persists up to $J_2/J_1=0.7$. In the region $0.7\le J_2/J_1 \le 0.9$ there is no magnetic order and for larger values of $J_2/J_1$ there is incommensurate or spiral order. It is interesting to note that the Néel order exists beyond its classical result $(J_2/J_1=0.5)$. Analytically the standard and simple linear spin-wave theory (LSW) based on Holstein-Primakoff (HP) transformation was used by Merino et al. [@Merino] to discuss the possible ordered and disordered states. It is helpful to repeat some details here for later discussion. First, for convenience the spin at each site is rotated along its reference direction characterized by the angle $\theta_i$. Such a rotation may be accomplished by the following transformation for spin operators $$\begin{aligned} S_i^x & = & \sin \theta_i \hat{S}_i^z+\cos \theta_i \hat{S}_i^x\\ S_i^y & = & \hat{S}_i^y\\ S_i^z & = & \cos \theta_i \hat{S}_i^z-\sin \theta_i \hat{S}_i^x \ \end{aligned}$$ For new spins $\hat{\bf S}_i$ the reference ground state becomes ferromagnetic. The rotated Hamiltonian becomes $$\begin{aligned} H &=&J_1\sum_{\langle ij \rangle} [\cos \theta_{ij}(\hat{S}_i^x\hat{S}_j^x+ \hat{S}_i^z\hat{S}_j^z)+\sin \theta_{ij}(\hat{S}_i^z\hat{S}_j^x-\hat{S}_i^x\hat{S}_j^z)+\hat{S}_i^y \hat{S}_j^y] \nonumber \\ & & +J_2\sum_{\langle lm \rangle} [\cos \theta_{lm}(\hat{S}_l^x\hat{S}_m^x +\hat{S}_l^z\hat{S}_m^z)+\sin \theta_{lm}(\hat{S}_l^z\hat{S}_m^x-\hat{S}_l^x \hat{S}_m^z)+\hat{S}_l^y\hat{S}_m^y] \ . \label{H:FM}\end{aligned}$$ with $\theta_{ij}=\theta_i-\theta_j=q,\ \theta_{lm}=\theta_l-\theta_m=2q$. Then with HP transformation the above Hamiltonian with only quadratic terms kept may be diagonalized in the momentum space. The final dispersion relation for the spin excitation is $$\omega_{\bf k}=\sqrt{\{[J({\bf k}+{\bf q})+J({\bf k}-{\bf q})]/2-J({\bf q})\}[J({\bf k})-J({\bf q})]}$$ with $J({\bf k})=J_1(\cos k_x+\cos k_y)+J_2\cos (k_x+k_y)$. By calculation of the magnetization $\langle \hat{S}_i^z\rangle$ (see also the dashed line in Fig. \[fig\_Sz\] later) it was suggested by the authors in Ref. [@Merino] that a possible Néel order/disorder transition happens at the point $J_2/J_1\simeq 0.5$. However, we want to point out the inherent limitation for LSW theory here. From the spectrum $\omega_{\bf k}$, it is easy to find that, in order to ensure the argument in the square root always positive in the whole Brillouin zone, the parameter $q$ has to be set as $\pi$ (characterizing Néel order) in the region $J_2/J_1<0.5$, but must [*not*]{} be set as $\pi$ in the region $J_2/J_1>0.5$. This means that within LSW theory the Néel ordered state can never appear beyond its classically stable region, which is in contrast with the numerical result [@Zheng]. As also pointed out by the authors in Ref. [@Merino] themselves, the interaction between spin waves becomes very large at the transition point and it may lead to a completely different picture for the states. So it is very necessary to go beyond the LSW theory to see how the above result will be modified, which is the purpose of this paper. Instead of HP transformation, the Dyson-Maleev (DM) transformation will be adopted to avoid $1/S$ expansion. It has been also recognized that the DM transformation must be prefered if one needs really to go beyond LSW theory within a perturbation scheme [@Canali]. In the treatment of the so called $J_1$-$J_2$ model, it has been shown that the spin-wave theory based on DM transformation gives perfectly consistent result as that from numerical calculation; much better than that from LSW theory [@Hu]. In this work we will not discuss the possible spiral state for large $J_2/J_1$, but focus on the Néel order/disorder transition in the intermediate $J_2/J_1$ region, especially on the problem whether the Néel order may appear beyond its classical region of stability which is one of the most interesting topics for this model. Technically, the Hamiltonian under DM transformation have terms as high as sixth order when spiral state is considered (i.e., for general $q$), which are relatively complicated to treat. When only Néel state is considered the transformed Hamiltonian has no term higher than fourth order, which can be easily treated by mean-field (MF) theory or perturbation theory. Explicitly, one may apply the DM transformation onto the original Hamiltonian (\[H\]); or begin with the rotated Hamiltonian (\[H:FM\]) by setting $\theta_{ij}=\pi,\ \theta_{lm}=2\pi$ and then use the DM transformation for A and B two sublattices in the following form: $$\begin{aligned} \hat{S}_i^{+} = (1-a_i^{\dagger}a_i)a_i,\ \ & \hat{S}_i^{-} = a_i^{\dagger},\ \ & \hat{S}_i^{z} = 1/2-a_i^{\dagger}a_i \\ \hat{S}_j^{-} = b_j^{\dagger}(1-b_j^{\dagger}b_j),\ \ & \hat{S}_j^{+} = b_j,\ \ & \hat{S}_j^{z} = 1/2-b_j^{\dagger}b_j \ , \end{aligned}$$ where $a(a^{\dagger}),\ b(b^{\dagger})$ are bosonic operators for sublattices A and B, respectively. Then the Hamiltonian (\[H:FM\]) is transformed into $$\begin{aligned} H & = & -J_1\sum_{\langle ij \rangle} (1/2-a_i^{\dagger}a_i) (1/2-b_j^{\dagger}b_j)+[(1-a_i^{\dagger}a_i)a_i b_j+ a_i^{\dagger}b_j^{\dagger}(1-b_j^{\dagger}b_j)]/2 \nonumber \\ &+&J_2\{\sum_{\langle lm \rangle \in A} (1/2-a_l^{\dagger}a_l) (1/2-a_m^{\dagger}a_m)+[(1-a_l^{\dagger}a_l)a_l a_m^{\dagger}+ (1-a_m^{\dagger}a_m)a_m a_l^{\dagger}]/2 + \sum_{\langle lm \rangle \in B} a\rightarrow b\}\ . \label{H:DM}\end{aligned}$$ To diagonalize the above Hamiltonian, we treat the quartic terms with a self-consistent MF theory. For those terms which could be decoupled in two ways, we will combine the two kinds of decoupling form together through a weight factor $\lambda$ as introduced by Chu and Shen [@Chu]. For example, the term like $a_i^{\dagger}a_ib_j^{\dagger}b_j$ will be decoupled in the way $$a_i^{\dagger}a_ib_j^{\dagger}b_j\simeq \lambda [\langle a_i^{\dagger}a_i \rangle b_j^{\dagger}b_j+ a_i^{\dagger}a_i \langle b_j^{\dagger}b_j\rangle -\langle a_i^{\dagger}a_i\rangle \langle b_j^{\dagger}b_j\rangle]+ (1-\lambda)[\langle a_i^{\dagger}b_j^{\dagger} \rangle a_i b_j+ a_i^{\dagger}b_j^{\dagger} \langle a_i b_j\rangle -\langle a_i^{\dagger}b_j^{\dagger}\rangle \langle a_i b_j\rangle]\ .$$ The parameter $0\le \lambda\le 1$ reflects the competition between two decoupling ways. Its value may be decided by minimization of the energy, which was found to be 1/2 in Ref. [@Chu] for unfrustrated lattice; or it is required to be equal to 1/2 in order to keep the symmetry of the Hamiltonian before and after decoupling [@Li]. We will take the value $\lambda=1/2$ throughout our calculations. With definition of several parameters: $u=\langle a_i^{\dagger}a_i\rangle=\langle b_i^{\dagger}b_i\rangle,\ v=\langle a_i^{\dagger}b_j^{\dagger}\rangle =\langle a_i b_j\rangle,\ w=\langle a_l^{\dagger}a_m\rangle=\langle b_l^{\dagger}b_m\rangle$, we may obtain a quadratic Hamiltonian in the momentum space: $$\begin{aligned} H & = & \sum_{\bf k} [C_{\bf k} (a_{\bf k}^{\dagger}a_{\bf k}+b_{\bf k}^{\dagger}b_{\bf k})+ E_{\bf k} (a_{\bf k} b_{-\bf k}+a_{\bf k}^{\dagger} b_{-\bf k}^{\dagger})] + {\rm const}\ ,\label{Hk}\\ C_{\bf k} & = & 2J_1(1-u+v)-J_2(1-u+w)[1-\cos (k_x+k_y)] \ ,\nonumber\\ E_{\bf k} & = & -J_1(1-u+v)(\cos k_x+\cos k_y) \ ,\nonumber\\ {\rm const} & = & {N \over 2}[J_1(-1+2u^2+2v^2-4uv)+J_2(1-2u^2-2w^2+4uw)/2]\ , \nonumber\end{aligned}$$ where the summation is over half of the original Brillouin zone and $N$ is total number of lattice sites. Under a Bogoliubov transformation $$\begin{aligned} a_{\bf k} & = & \cosh \lambda_{\bf k}\ \bar{a}_{\bf k} + \sinh \lambda_{\bf k}\ \bar{b}_{-\bf k}^{\dagger}\\ b_{-\bf k}^{\dagger} & = & \sinh \lambda_{\bf k}\ \bar{a}_{\bf k} + \cosh \lambda_{\bf k}\ \bar{b}_{-\bf k}^{\dagger}\end{aligned}$$ with $\tanh 2\lambda _{\bf k}=-E_{\bf k}/C_{\bf k}$, the Hamiltonian (\[Hk\]) may be diagonalized into $$\begin{aligned} H & = & \sum_{\bf k} [\bar{\omega}_{\bf k} (\bar{a}_{\bf k}^{\dagger}\bar{a}_{\bf k}+\bar{b}_{\bf k}^{\dagger}\bar{b}_{\bf k}+1)-C_{\bf k}]+ {\rm const}\end{aligned}$$ with the excitation spectrum $\bar{\omega}_{\bf k}=\sqrt{C_{\bf k}^2-E_{\bf k}^2}$. Correspondingly the self-consistent equations for $u,\ v,\ w$ are expressed as $$\begin{aligned} u & = & {1\over N}\sum_{\bf k}{C_{\bf k}\over\sqrt{C_{\bf k}^2-E_{\bf k}^2}} -1/2 \ ,\\ v & = & -{1\over N}\sum_{\bf k} {E_{\bf k}\cos k_x \over \sqrt{C_{\bf k}^2-E_{\bf k}^2}} \ ,\\ w & = & {1\over N} \sum_{\bf k} {C_{\bf k}\cos (k_x+k_y) \over \sqrt{C_{\bf k}^2-E_{\bf k}^2}}\ .\end{aligned}$$ The magnetization is simply given by $m=1/2-u$ and the ground state energy is $E_0=\sum_{\bf k} (\bar{\omega} _{\bf k}-C_{\bf k}) + {\rm const}$. =4.5cm =4.5cm We show the numerical results for the above self-consistent equations in Figs. \[fig\_Sz\] and \[fig\_vw\]. The magnetization $m$ (derived from the parameter $u$) as a function of $J_2/J_1$ is given by the solid line in Fig. 2, which is the main result in this paper. It is found that the magnetization does not vanish until $J_2/J_1\simeq 0.572$, which gives the Néel order/disorder transition point. As comparison, the result from LSW theory is plotted by the dashed line; the magnetization goes to zero immediately before the classical value $J_2/J_1=0.5$, see also Ref. [@Trumper]. As expected, the current result is closer to the numerical one from series expansions and most importantly, it qualitatively improves the result from LSW theory. As we discussed before, the LSW theory is impossible to deduce a Néel ordered state beyond $J_2/J_1=0.5$. In our treatment the interaction between spin waves is actually partly considered, then the fact that Néel order may exist beyond its classically stable region is shown. It is quite possible that the transition point will shift to larger value if the residual interaction between spin waves is included. This may be also hinted from Fig. \[fig\_vw\], where the parameter $v$, which represents the antiferromagnetic correlation between the two original nearest-neighbour spins, is large near the transition point. For completeness, the ground state energy in the region $0<J_2/J_1<0.57$ is also shown in Fig. \[fig\_GE\] by the solid line, which is close to the result from LSW theory (the dashed line). In the small $J_2/J_1$ region the energy calculated here is a little lower than that of LSW theory, and becomes a little higher with increase of $J_2/J_1$. For $J_2/J_1>0.5$ the energy within LSW theory is derived from spiral state; there is a cusp at point $J_2/J_1=0.5$ [@Merino]. In the current case the state is still Néel ordered and the energy changes smoothly. =4.5cm In summary, within Dyson-Maleev (DM) transformation and self-consistent mean-field treatment, we have studied an antiferromagnetic Heisenberg model on a square lattice which includes a nearest neighbour exchange $J_1$ and a next-nearest neighbour exchange $J_2$ along only one of the diagonals. This model should be of direct relevance to some layered organic superconductors. In this work we focus on the discussion of Néel order/disorder transition for not large $J_2/J_1$. It is found that the Néel order may exist up to $J_2/J_1=0.572$, beyond its classically stable regime. This property is consistent with numerical finding from series expansions, which is one of the most interesting features for this model. Especially, because the interaction between spin waves is partly considered in our treatment, the result derived here qualitatively improves that from LSW theory based on Holstein-Primakoff transformation. It is certainly necessary to continue this work to study the large $J_2/J_1$ region where a spiral order will appear, so that a whole phase diagram may be constructed. The author would like to thank J. Merino for helpful discussion. This work was supported in part by Chinese NSF. S. Chakravarty, B. I. Halperin and D. R. Nelson, Phys. Rev. B [**39**]{}, 2344 (1989). J. Oitmma and Z. Weihong, Phys. Rev. B [**54**]{}, 3022 (1996) and references therein. 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--- author: - 'P.A. Curran' - 'S. Chaty' - 'J.A. Zurita Heras' - 'A. Coleiro' date: 'Received ; accepted' title: 'A search for near infrared counterparts of 3 pulsar wind nebulae[^1]' --- [While pulsar wind nebulae (PWNe) and their associated isolated pulsars are commonly detected at X-ray energies, they are much rarer at near infrared (nIR) and optical wavelengths.]{} [Here we examine three PWN systems in the Galactic plane – [IGRJ14003$-$6326]{}, [HESSJ1632$-$478]{}  and [IGRJ18490$-$0000]{} – in a bid to identify optical/nIR emission associated with either the extended PWNe or their previously detected X-ray point sources.]{} [We obtain optical/nIR images of the three fields with the ESO – New Technology Telescope and apply standard photometric and astrometric calibrations.]{} [We find no evidence of any extended emission associated with the PWNe in any of the fields; neither do we find any new counterparts to the X-ray point sources, except to confirm the magnitude of the previously identified counterpart candidate of [IGRJ18490$-$0000]{}.]{} [Further observations are required to confirm the association of the nIR source to [IGRJ18490$-$0000]{} and to detect counterparts to [IGRJ14003$-$6326]{} and [HESSJ1632$-$478]{}, while a more accurate X-ray position is required to reduce the probability of a chance superposition in the field of the latter.]{} Introduction {#section:intro} ============ Along with the typical supernova remnant (SNR), a rapidly-rotating, highly-magnetised neutron star, or pulsar, is frequently an end product of a supernova explosion. The pulsar has particularly high levels of rotational energy which is dissipated via a highly relativistic particle wind. The interaction of this wind with the surrounding medium, i.e., the ejecta of the supernova explosion itself, causes a continuously refreshed shock wave known as a pulsar wind nebula (PWN). These PWNe emit via synchrotron and inverse Compton processes and are observed across the spectrum from radio to optical to X-ray and higher energies (for a detailed review of PWNe see, e.g. ). The pulsars themselves, as well as being detected as point sources in radio and X-ray where the pulsations are observed, may be detected at optical or near infrared (nIR) wavelengths. However at these wavelengths the emission will be that of an isolated, non-accreting neutron star which is intrinsically dim and hence very few (12 out of $\sim 1800$) have been detected in this regime [@Mignani2011:AdSpR.47]. Here we examine three such PWN systems in the Galactic plane – [IGRJ14003$-$6326]{}, [HESSJ1632$-$478]{}  and [IGRJ18490$-$0000]{} – in a bid to identify extended optical/nIR (OIR) emission associated with the PWNe or optical/nIR counterparts to the X-ray point sources. The intrinsic OIR dimness of both PWNe and isolated neutron stars is further compounded in these cases by the high level of optical extinction in the direction of the Galactic plane [@schlegel1998:ApJ500]; hence observing at nIR wavelengths, where this is less pronounced, may increase the chance of a detection. However, at these positions in the Galactic plane nIR source density is relatively high so even with accurate X-ray positions care must be taken to understand the probability of chance superpositions. In section \[section:observations\] we introduce our observations and reduction method while in section \[section:results\], after briefly introducing each source, we detail the results of those observations. In section \[section:discussion\] we discuss these results and summarise our findings in section \[section:conclusions\]. Throughout, positions (J2000) are given with 90% confidence while all others values, including magnitudes, are given with $1\sigma$ confidence. ESO-NTT observations and data analysis {#section:observations} ====================================== Optical ($B, V, R, i$) and nIR ($J, H, K_s$) data were obtained with the ESO Faint Object Spectrograph and Camera (v.2; EFOSC2) and the Son of ISAAC (SofI) infrared spectrograph and imaging camera on the 3.58m ESO – New Technology Telescope (NTT). Data were obtained on the nights of March 8, 2008 ([IGRJ18490$-$0000]{}), September 16 & 18, 2008 ([IGRJ14003$-$6326]{}; optical & nIR) and March 27, 2010 ([HESSJ1632$-$478]{}). All data used a dithered pattern of $3\times60$seconds or $9\times 10$seconds per final image in the optical and nIR respectively (see Table\[table:observations\]). The NTT data were reduced using the [IRAF]{} package wherein crosstalk correction, flatfielding, sky subtraction, bias-subtraction and frame addition were carried out as necessary. The images were astrometrically calibrated against 2MASS [@Skrutskie2006:AJ.131] or USNO-B1.0 [@Monet2003:AJ125] within the GAIA package and quoted 90% positional errors include a 0.16 2MASS systematic uncertainty. Relative point spread function (PSF) photometry was carried out on the final images using the [DAOPHOT]{} package [@stetson1987:PASP99] within [IRAF]{}. Magnitudes were calibrated against [@Persson1998:AJ116] or [@Landolt1992:AJ.104] photometric standards, observed on the same night (Table\[table:positions\]) and errors include both statistical and calibration errors. The equation, $i-I = (0.247 \pm 0.003) (R-I)$ , was used to transform the cataloged $I$ magnitudes of the standard stars into $i$ magnitudes with which to calibrate the images. Images were inspected visually for any background emission that might be associated with the extended emission of the PWNe; this was done both to the reduced images and to those images after the PSF subtraction of point sources in the field. We find no evidence of any extended emission associated with the PWN in any of the images, but due to uncertainties in the extent of the PWN and the quality of point source subtraction we are unable to put a flux limit on possible emission. Point source upper limits are approximated from the dimmest observable object in the field. ---------------------- --------- ----------- --------- Field Date Filter Exp (s) Seeing () Limit [IGRJ14003$-$6326]{} $Ks$ 90 1.7 $>17.2$ $H$ 90 1.7 $>18.0$ $J$ 90 1.7 $>19.2$ $i$ 180 1.6 $>21.2$ $R$ 180 1.8 $>21.0$ $V$ 180 1.7 $>21.8$ $B$ 180 1.7 $>22.3$ [HESSJ1632$-$478]{} $Ks$ 90 0.9 $>17.7$ [IGRJ18490$-$0000]{} $Ks$ 90 1.6 $>17.7$ ---------------------- --------- ----------- --------- : Observation log.[]{data-label="table:observations"} Results {#section:results} ======= ------------------ ------------- ---------------- ---------- ------------------------------------------------------------------------------------------ (optical source) RA Declination Error () Magnitudes 14:00:45.69 $-$63:25:42.6 0.64 (1) 14:00:45.45 $-$63:25:41.8 0.3 $J = 18.6 \pm 0.2$, $i = 20.90 \pm 0.15$ 16:32:08.8 $-$47:49:01 1.6 (1) 16:32:09.06 $-$47:49:01.6 0.3 $K_S = 16.74 \pm 0.05$ (2) 16:32:08.47 $-$47:49:00.6 0.16 $K_S = 11.35 \pm 0.01$, $K^{\dagger} = 11.695 \pm 0.040, H^{\dagger} = 13.284 \pm 0.065$ 18:49:01.59 $-$00:01:17.73 0.6 (1) 18:49:01.57 $-$00:01:17.6 0.3 $K_S = 17.2 \pm 0.4$ (2) 18:49:01.52 $-$00:01:16.9 0.2 $K_S = 15.58 \pm 0.05$ ------------------ ------------- ---------------- ---------- ------------------------------------------------------------------------------------------ References to X-ray positions: $^a$[@Tomsick2009:ApJ701], $^b$, $^c$[@Ratti2010:MNRAS.408] $^{\dagger}$ 2MASS magnitudes We find no evidence of any extended emission associated with the PWN in any of the images, even after the PSF subtraction of point sources in the field. Neither do we find any nIR counterparts to the X-ray point sources, except to confirm the $K_s$ magnitude of the previously identified counterpart of [IGRJ18490$-$0000]{} [@Ratti2010:MNRAS.408].\ [IGRJ14003$-$6326]{} -------------------- Based on [*Chandra*]{} spectra, [@Tomsick2009:ApJ701] confirm that [IGRJ14003$-$6326]{} [@Keek2006:ATel.810] is an SNR with a PWN, while [@Renaud2010:ApJ.716] discovered a 31.18ms X-ray/radio pulsar at its centre. @Tomsick2009:ApJ701 do not find any higher energy (TeV, GeV) counterparts to the source but @Renaud2010:ApJ.716 obtain radio observations which reveal counterparts to both the point source and the PWN. We find no source within $3\sigma$ of the [*Chandra*]{} poition of [IGRJ14003$-$6326]{}, down to the magnitudes given in Table\[table:observations\]. However there is a dim source detected at $J = 18.6 \pm 0.2$ and marginally at $i = 20.90 \pm 0.15$, at RA, Dec = 14:00:45.45, $-$63:25:41.8 ($\pm 0.3$) or $\sim 5\sigma$ from the [*Chandra*]{} position. We are unable to discern if this is a point or extended source due to its faintness but it is likely unrelated to the X-ray source given the distance discrepancy. [HESSJ1632$-$478]{} ------------------- [HESSJ1632$-$478]{} was initially suggested to be associated with IGRJ16320-4751 [@Aharonian2006:ApJ.636] though this was subsequently rejected after deep XMM-*Newton* observations of the source . It was instead associated with an independent X-ray point source with diffuse emission. These authors also obtained data from radio and high energy archives or catalogs to describe the nature of the source, which they suggest is an energetic PWN with a, yet to be confirmed, central pulsar. We find no source within the XMM error circle of [HESSJ1632$-$478]{}, down to a limiting magnitude of $K_s > 17.7$. However, there is a dim source 2.6$\sim 2.6 \sigma$ to the East (source 1) with a magnitude of $K_S = 16.74 \pm 0.05$, though we cannot propose this as the counterpart with any certainty given the lack of colour information and weak positional agreement. Neither can we compare the colour of this source to other sources in the field to demonstrate that it has similar properties and is thus likely to be a field source, unrelated to the suggested pulsar. There is also a bright 2MASS source (16320846$-$4749005) to the West (source 2), though it is at a distance of $\sim 3.5\sigma$ so is even less likely related to the high energy source. [IGRJ18490$-$0000]{} -------------------- First reported by [@Molkov2004:AstL.30], [IGRJ18490$-$0000]{}  is a PWN (e.g., @Ratti2010:MNRAS.408) with a confirmed 38.5ms pulsar [@Gotthelf2011:ApJ729]. @Ratti2010:MNRAS.408 also suggest a nIR counterpart of magnitude $K_S = 16.4 \pm 0.1$ at RA, Dec = 18:49:01.563, $-$00:01:17.35 ($\pm 0.1$) but find no evidence of any nIR extension as one might expect for a PWN. Within the [*Chandra*]{} error circle of [IGRJ18490$-$0000]{}, we confirm the proposed nIR counterpart of [@Ratti2010:MNRAS.408] at a consistent magnitude of $K_s = 17.2 \pm 0.4$ (source 1). We also note, as those authors did, that the object is heavily blended with a nearby source of magnitude $K_s = 15.58 \pm 0.05$ (source 2). Due to the lack of colour information for the proposed counterpart we are unable to gain any information regarding its equivalent spectral classification, or to compare it to other field sources to demonstrate that it has different properties that may indicate it is associated with the pulsar. Discussion {#section:discussion} ========== The lack of detected compact counterparts for [IGRJ14003$-$6326]{} and [HESSJ1632$-$478]{}, particularly in the optical, is not surprising given the high levels of Galactic extinction [@schlegel1998:ApJ500] towards the sources: $E_{B-V} = 3.83$ ($A_K \sim 1.3$, $A_V \sim 12$; @cardelli1989:ApJ345) and $E_{B-V} = 11.18$ ($A_K \sim 3.9$) respectively. Likewise [IGRJ18490$-$0000]{} suffers significant extinction of $E_{B-V} = 6.62$ ($A_K \sim 2.3$). Note that all these extinctions should be treated with caution as estimates so close to the Galactic plane ($<5\deg$) are unreliable. The magnitude limits of the non-detections are consistent with the $V$ band detections of other isolated neutron stars [@Mignani2011:AdSpR.47]; excluding the Crab at $V=16.6$, these range from 22 to 28 magnitudes (corresponding to $K_s \gtrsim 18.6$ for a spectral index of $-1$), at low optical extinctions ($E_{B-V} \lesssim 0.2$) in the nearby Galaxy ($\lesssim 1$kpc for most). On the other hand, the detection of a proposed counterpart to [IGRJ18490$-$0000]{} at $K_s \sim 17$ is significantly brighter than any other optical identification of an isolated neutron star, except for the Crab; it should also be noted that the proposed counterpart is significantly brighter than a simple power-law extrapolation of the X-ray spectra [@Gotthelf2011:ApJ729] which implies a magnitude of $K_s \sim 23$ (uncorrected for Galactic extinction). While the power-law extrapolation of the X-ray spectra of [HESSJ1632$-$478]{}  is not constraining, that of [IGRJ14003$-$6326]{} [@Renaud2010:ApJ.716] implies that a counterpart should be much dimmer than the observed optical limit, at approximately $K_s \sim 22$. The extrapolations should however be treated with caution as the extracted X-ray spectra themselves may suffer from contamination from the surrounding PWN and thus inaccurate spectral slopes and fluxes. By using a simple power-law extrapolation, we have assumed that the spectra do not evolve between the X-ray and nIR regimes, while it quite possibly breaks to a shallower spectral index or, alternatively, may be described by thermal emission which naturally turns over at lower frequencies. We have also implicitly assumed that the nIR emission originates from the same emission region as the X-ray spectra, which is not necessarily true, but assuming that it is, the above approximations can be treated as lower magnitude limits on how bright we might expect a nIR counterpart to be, before correction for Galactic extinction. To approximate the probability of a chance superposition of the 90% X-ray positions of the compact sources with random sources in the respective fields, we calculate $P \approx 1-\exp^{-(\rho_N \times A_{Err})}$; where $\rho_N$ is the surface area number density of observed sources down to the limiting magnitude and $A_{Err}$ is the area of the X-ray positional error. For [IGRJ14003$-$6326]{} and [IGRJ18490$-$0000]{}, which have positions accurate at the sub-arcsecond level, we find probabilities of chance superpositions of 3% and 5% respectively, making the positional coincidence of [IGRJ18490$-$0000]{} with a nIR object reasonably compelling evidence for its association. However, for [HESSJ1632$-$478]{}, with only an arcsecond accurate position, the probability is much greater at 50%, though the source is in a relatively less densely populated region of the field, where the local probability is $\sim 40\%$. In this case, even if a source had been found within the 90% error circle, its association with the X-ray source would be weak, meaning source 1 at a distance of $\sim 2.6\sigma$ is likely unrelated. To associate a nIR counterpart with this source based on positional coincidence will require a significantly better constrained position from e.g. [*Chandra*]{}. Along with the optical extinction in their directions, the detection of extended emission from the PWNe is complicated by the high level of background in nIR observations as well as the generally high density of field sources in the Galactic plane which contaminate the background. Deep nIR images from larger telescopes better able to resolve field sources, may be able to detect PWN emission, as well as increasing the probability of detecting the compact sources, but in these directions source confusion will always be a major impediment to detection of point or extended sources. Conclusions {#section:conclusions} =========== We find no evidence of any extended nIR emission associated with the PWNe in any of the fields or any new counterparts to the X-ray point sources, but we do confirm the magnitude of the previously suggested counterpart of [IGRJ18490$-$0000]{}. While there is a low probability of chance coincidence of the X-ray position with a nIR object in this field, the candidate source is significantly brighter than most other isolated pulsars and brighter than a simple extrapolation of the X-ray spectra to the nIR, making its association with the X-ray source less certain. If future observations can confirm the association it seems that an additional emission component will be necessary to explain the excess nIR flux. The non-detection of the other two sources, [IGRJ14003$-$6326]{} and [HESSJ1632$-$478]{}, may be understood by the high level of Galactic extinction in their direction and by the intrinsically faint nature of isolated neutron stars, which the compact objects are assumed to be; it may be reasonable to expect the counterparts of these sources to be at magnitudes $K_s \gtrsim 18.6$. We thank the referee for their constructive comments. This work was supported by the Centre National d’Etudes Spatiales (CNES) and is based on observations obtained with MINE: the Multi-wavelength INTEGRAL NEtwork. This work made use of NASA’s Astrophysics Data System. IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under cooperative agreement with the National Science Foundation. GAIA was created by the now closed Starlink UK project funded by the Particle Physics and Astronomy Research Council (PPARC), and has been more recently supported by the Joint Astronomy Centre Hawaii funded by PPARC and now its successor organisation the Science and Technology Facilities Council (STFC). 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Springer-Verlag Berlin Heidelberg, p. 373 P. B., 1987, , 99, 191 J. A., [Chaty]{} S., [Rodriguez]{} J., [Walter]{} R., [Kaaret]{} P., 2009, , 701, 811 [^1]: Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile under ESO programs 080.D-0864, 081.D-0401, 084.D-0535 (P.I. Chaty)

--- abstract: 'The relation of crystal bases with $q$-identities is discussed, and some new results on crystals and $q$-identities associated with the affine Lie algebra $C_n^{(1)}$ are presented.' address: - 'Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan' - 'Department of Mathematics 2-279, M.I.T., Cambridge, MA 02139, U.S.A. andDepartment of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.' - 'Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, VA 24061-0123, U.S.A' author: - Masato Okado - Anne Schilling - Mark Shimozono title: 'Crystal bases and $q$-identities' --- Introduction ============ The purpose of this paper is two-fold. First, we would like to advocate the importance of crystal theory to the theory of $q$-series. In particular crystal base theory provides a unifying and general setting for a large class of $q$-identities. Second, as evidence, some new identities associated to the affine Lie algebra $C_n^{(1)}$ are presented. The emphasis here will not be on the completeness of the results since the field is evolving quite rapidly, but rather on the presentation of the main ideas and techniques used. The Rogers–Ramanujan identities are undoubtably the most famous $q$-series identities. They are given by $$\begin{aligned} \label{RR1} \sum_{n=0}^\infty \frac{q^{n^2}}{(q)_n} &= \prod_{j=1}^\infty \frac{1}{(1-q^{5j-4})(1-q^{5j-1})}\\ \label{RR2} \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q)_n} &= \prod_{j=1}^\infty \frac{1}{(1-q^{5j-3})(1-q^{5j-2})}\end{aligned}$$ where $(q)_n=(1-q)(1-q^2)\cdots(1-q^n)$. We will view them here as identities of formal power series, meaning that in the expansion as series in the formal variable $q$ the coefficients of $q^N$ match on both sides for all $N\ge 0$. What contributes to their beauty is that these coefficients can be interpreted combinatorially. The coefficient of $q^N$ on the left-hand side of is the number of partitions of $N$ for which the difference between any two parts is at least two. The coefficient of $q^N$ of the right-hand side of on the other hand is the number of partitions of $N$ with parts congruent to 1 or 4 modulo 5. Similarly, the coefficients of $q^N$ on the left and right side of can be interpreted as the number of partitions of $N$ for which the difference between any two parts is at least two and the smallest part is greater than one, and the number of partitions of $N$ with parts congruent to 2 and 3 modulo 5, respectively. Many of the ideas regarding crystals and $q$-identities can already be demonstrated in terms of the Rogers–Ramanujan identities. The point of focus here shall be the debut of the Rogers–Ramanujan identities on the mathematical physics stage, in particular their appearance in the hard hexagon model in a paper by Baxter [@B81] in 1981. In this setting the Rogers–Ramanujan identities can be viewed as two different evaluations of the generating function of certain paths which are coined bosonic and fermionic evaluations. Details are discussed in section \[sec:HHM\]. As it turns out the relation between the Rogers–Ramanujan identities and the hard hexagon model is only part of a much bigger picture. In terms of representation theory, the paths that occur in the hard hexagon model are elements of tensor products of crystals associated with the affine Lie algebra $\hat{\mathfrak{sl}}_2$. Crystal bases were introduced by Kashiwara [@K1; @K2] and roughly speaking are bases of representations of quantum universal enveloping algebras $U_q({\mathfrak{g}})$ as the parameter $q$ (not to be confused with the $q$ in the $q$-series!) tends to zero. Here ${\mathfrak{g}}$ is any symmetrizable Kac–Moody algebra. As in the Rogers–Ramanujan case, there are two different ways to evaluate generating functions of tensor products of crystals, thereby giving rise to $q$-identities. Hence crystal base theory provides a natural framework for $q$-identities. Crystal bases, path spaces and their generating functions are discussed in section \[sec:crystal\]. The two different ways to evaluate the paths generating functions are subject of sections \[sec:bose\] and \[sec:fermi\], respectively. In particular in section \[sec:lev C\] we present new fermionic formulas for level-restricted paths. We close in section \[sec:open\] with some outstanding open problems. Before indulging in the fascinating theory of crystal bases there is one important point that needs to be addressed. Applying Jacobi’s triple product identity $$\sum_{n=-\infty}^\infty z^n q^{n^2} = \prod_{n=0}^\infty (1-q^{2n+2})(1+z q^{2n+1})(1+z^{-1} q^{2n+1})$$ the right-hand sides of and can be rewritten as alternating sums yielding the identities $$\begin{aligned} \label{bf1} \sum_{n=0}^\infty \frac{q^{n^2}}{(q)_n} &= \frac{1}{(q)_\infty} \sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+1)}\\ \label{bf2} \sum_{n=0}^\infty \frac{q^{n(n+1)}}{(q)_n} &= \frac{1}{(q)_\infty} \sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+3)}.\end{aligned}$$ The two different evaluations of generating functions of crystal paths that were mentioned above really yield (polynomial) analogues of and rather than and . Identities relating sums to alternating sums are more general than identities relating sums to products. Only in special cases can the alternating sums be evaluated as products, namely when Jacobi’s triple product identity or more generally the Macdonald identities [@Mac] can be applied. Acknowledgements {#acknowledgements .unnumbered} ---------------- We are grateful to Tim Baker for discussions. A.S. would like to thank the organizers Bruce Berndt and Ken Ono for the invitation to present these results at the conference on $q$-Series with Applications to Combinatorics, Number Theory, and Physics held in Urbana-Champaign in October 2000. Thanks for this exciting conference! The Rogers–Ramanujan identities and the Hard Hexagon model {#sec:HHM} ========================================================== The hard hexagon model is a two-dimensional lattice model of a gas of hard or non-overlapping particles. The particles are placed on a triangular lattice such that no two particles can occupy two adjacent sites. If one views each particle as the center of a hexagon, then the condition that no two particles can be adjacent translates into the condition that the hexagons cannot overlap. This explains the name of the model. An example of an allowed particle configuration is shown in Figure \[fig:hhconfig\]. (260,140)(0,0) (0,0)(260,0) (0,34)(260,34) (0,67)(260,67) (0,102)(260,102) (0,134)(260,134) (0,67)(40,134) (0,0)(80,134) (40,0)(120,134) (80,0)(160,134) (120,0)(200,134) (160,0)(240,134) (200,0)(260,102) (240,0)(260,34) (40,0)(0,67) (80,0)(0,134) (120,0)(40,134) (160,0)(80,134) (200,0)(120,134) (240,0)(160,134) (260,34)(200,134) (260,102)(240,134) (60,102)[4]{}[Black]{}[Black]{} (60,34)[4]{}[Black]{}[Black]{} (120,67)[4]{}[Black]{}[Black]{} (200,67)[4]{}[Black]{}[Black]{} (40,67)(80,67) (80,67)(100,102) (100,102)(80,134) (80,134)(40,134) (40,134)(20,102) (20,102)(40,67) (40,0)(80,0) (80,0)(100,34) (100,34)(80,65) (80,65)(40,65) (40,65)(20,34) (20,34)(40,0) (102,34)(140,34) (140,34)(160,67) (160,67)(140,102) (140,102)(102,102) (102,102)(82,67) (82,67)(102,34) (180,34)(220,34) (220,34)(240,67) (240,67)(220,102) (220,102)(180,102) (180,102)(160,67) (160,67)(180,34) To pack the lattice densely all particles must lie on one of the three sublattices corresponding to the three corners of the triangles of the lattice. Hence one of the sublattices is distinguished from the others. At low particle density the probability for a particle to be on a particular site is equal for sites on all three sublattices (assuming either an infinite or sufficiently large lattice so that boundary effects can be neglected). Let $\rho_a$ for $a=1,2,3$ be the probability that there is a particle at a fixed site on sublattice $1,2,3$, respectively. If the boundary conditions of the model are such that at close packing all particles are on sublattice 1 then it is intuitively clear that the order parameter defined as $R=\rho_1-\rho_2$ must undergo a phase transition. At low densities $R$ is zero, but at some critical density $R$ will become positive until at high densities it is one. Baxter [@B81] managed to determine the precise point at which the phase transition occurs exactly by using corner transfer matrices. In essence, the corner transfer matrix method reduces the two-dimensional problem to a one-dimensional problem. The precise details are beyond the scope of this paper and can be found in section 14 of Baxter’s book [@B82]. The one-dimensional problem that Baxter encountered and which turns out to be of importance to the Rogers–Ramanujan identities is the following. Consider $L+1$ points on a line labeled by $i=0,1,2,\ldots,L$. Assign to each point a height variable $\sigma_i$ which takes on the values $0$ or $1$. In addition the height variables satisfy the restrictions $\sigma_0=\sigma_L=0$ and $\sigma_i\sigma_{i+1}=0$. An allowed configuration of height variables for a given length $L$ is called a *path* of length $L$, and the set of all paths of length $L$ is denoted by ${\mathcal{D}}_L$. One can illustrate a path graphically by drawing all points $(i,\sigma_i)$ and connecting adjacent points by straight lines. An example for a path with $L=9$ is given in Figure \[fig path\]. (200,50)(0,0) (10,20)[(1,1)[20]{}]{} (50,20)[(-1,1)[20]{}]{} (90,20)[(1,1)[20]{}]{} (130,20)[(-1,1)[20]{}]{} (150,20)[(1,1)[20]{}]{} (190,20)[(-1,1)[20]{}]{} (8,9)[0]{} (28,9)[1]{} (48,9)[2]{} (68,9)[3]{} (88,9)[4]{} (108,9)[5]{} (128,9)[6]{} (148,9)[7]{} (168,9)[8]{} (188,9)[9]{} (10,20)[(1,0)[190]{}]{} (10,20) (30,40) (50,20) (70,20) (90,20) (110,40) (130,20) (150,20) (170,40) (190,20) The condition $\sigma_i\sigma_{i+1}=0$ requires that the paths consist of a certain number of non-overlapping triangles (this condition comes directly from the condition in the two-dimensional hard hexagon model requiring that no two particles can be on adjacent sites). To each path $\sigma=(\sigma_0,\ldots,\sigma_L)$ one may assign an energy $E(\sigma)$ by summing up the positions of the peaks, that is $$E(\sigma)=\sum_{j=1}^L j \sigma_j.$$ The energy of the path in Figure \[fig path\] is $E(\sigma)=1+5+8=14$. The generating function of paths of length $L$ which is also called a one dimensional configuration sum is defined as $$\label{path} X(L)=\sum_{\sigma\in {\mathcal{D}}_L} q^{E(\sigma)}.$$ The path picture immediately implies that $X(L)$ satisfies the following initial conditions and recurrence which completely specify it $$\label{rec} \begin{split} X(0)&=X(1)=1\\ X(L)&=X(L-1)+q^{L-1}X(L-2). \end{split}$$ The aim is to find explicit expressions for $X(L)$. We will describe two ways to obtain such an expression which will be related to the two sides of . Fermionic formula {#subsec:fermi} ----------------- Interpret each peak in a path as a particle, that is, there is a particle at site $i$ if $\sigma_i=1$. Fix the number of particles to be $n$. The ground state path ${\sigma_{\mathrm{G}}}$ with minimal energy is the path with particles at positions $1,3,\ldots,2n-1$. The energy of ${\sigma_{\mathrm{G}}}$ is $E({\sigma_{\mathrm{G}}})=1+3+\cdots+2n-1=n^2$. An example of the ground state path with $3$ particles is shown in Figure \[fig:ground\]. (200,50)(0,0) (10,20)[(1,1)[20]{}]{} (50,20)[(-1,1)[20]{}]{} (50,20)[(1,1)[20]{}]{} (90,20)[(-1,1)[20]{}]{} (90,20)[(1,1)[20]{}]{} (130,20)[(-1,1)[20]{}]{} (8,9)[0]{} (28,9)[1]{} (48,9)[2]{} (68,9)[3]{} (88,9)[4]{} (108,9)[5]{} (128,9)[6]{} (148,9)[7]{} (168,9)[8]{} (188,9)[9]{} (10,20)[(1,0)[190]{}]{} (10,20) (30,40) (50,20) (70,40) (90,20) (110,40) (130,20) (150,20) (170,20) (190,20) All other paths with $n$ particles can be obtained from ${\sigma_{\mathrm{G}}}$ by moving particles to the right in such a way that the particles never overtake each other. If the length of the path is $L$, the rightmost particle can move at most $L-2n$ positions to the right. Hence the paths of length $L$ with $n$ particles are in one-to-one correspondence with partitions with at most $n$ parts not exceeding $L-2n$. If a path $\sigma$ corresponds to partition ${\lambda}$ then its energy is $E(\sigma)=E({\sigma_{\mathrm{G}}})+|{\lambda}|=n^2+|{\lambda}|$. The generating function of partitions with at most $n$ parts not exceeding $m$ is the $q$-binomial coefficient defined as $${\genfrac{[}{]}{0pt}{}{m+n}{n}}=\frac{(q)_{m+n}}{(q)_m(q)_n}$$ for $m,n\in{\mathbb{N}}$ and zero otherwise. This implies the following explicit expression for $X(L)$ $$\label{RR fermi} X(L)=\sum_{n=0}^\infty q^{n^2} {\genfrac{[}{]}{0pt}{}{L-n}{n}}.$$ Because of its interpretation in terms of non-overlapping particles this expression is called a quasiparticle or fermionic formula. Bosonic formula {#subsec:bose} --------------- As opposed to there is another expression for $X(L)$ given by $$\label{RR bose} X(L)=\sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+1)} {\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j}{2}\rfloor}}$$ where $\lfloor x\rfloor$ is the largest integer not exceeding $x$. It can be proven by showing that the right-hand side satisfies the recurrence and initial condition . However, formula can also be interpreted in terms of paths. For this purpose the above path picture needs to be slightly reformulated. The state diagram of the paths in the hard hexagon model is $$\begin{picture}(20,40)(-10,0) \put(0,20){\line(0,1){20}} \put(0,10){\circle{20}} \put(0,20){\circle*{4}} \put(0,40){\circle*{4}} \end{picture}$$ This diagram is to be understood as follows. If the system is in the bottom state at position $i$, that is $\sigma_i=0$, then at position $i+1$ it can either be in the bottom state again so that $\sigma_{i+1}=0$ as indicated by the loop or it can be in the top state which means $\sigma_{i+1}=1$ as indicated by the line. On the other hand, if the system is in the top state at position $i$ then at the next position it has to be in the bottom state, meaning that $\sigma_i=1$ implies $\sigma_{i+1}=0$. This corresponds to condition on paths that $\sigma_i\sigma_{i+1}=0$ for all $i$. Up to a ${\mathbb{Z}}_2$ symmetry this state diagram is isomorphic to $$\label{equiv} \raisebox{-.3cm}{ \begin{picture}(20,40)(-10,0) \put(0,20){\line(0,1){20}} \put(0,10){\circle{20}} \put(0,20){\circle*{4}} \put(0,40){\circle*{4}} \end{picture} } \quad\cong\quad \raisebox{-1cm}{ \begin{picture}(4,60)(-2,0) \put(0,0){\line(0,1){60}} \put(0,0){\circle*{4}} \put(0,20){\circle*{4}} \put(0,40){\circle*{4}} \put(0,60){\circle*{4}} \end{picture}} \;\;\Big{/}\;\; {\mathbb{Z}}_2.$$ Under this correspondence the paths of the hard hexagon model become paths in a strip of height 4 with steps going up or down at each position. For example, with the convention that the paths start at height three, the path in Figure \[fig path\] becomes the path in Figure \[fig:newpath\]. (210,80)(-10,0) (10,60)[(1,1)[20]{}]{} (30,80)[(1,-1)[20]{}]{} (50,60)[(1,-1)[20]{}]{} (70,40)[(1,1)[20]{}]{} (90,60)[(1,1)[20]{}]{} (110,80)[(1,-1)[20]{}]{} (130,60)[(1,-1)[20]{}]{} (150,40)[(1,-1)[20]{}]{} (170,20)[(1,1)[20]{}]{} (8,9)[0]{} (28,9)[1]{} (48,9)[2]{} (68,9)[3]{} (88,9)[4]{} (108,9)[5]{} (128,9)[6]{} (148,9)[7]{} (168,9)[8]{} (188,9)[9]{} (-3,17)[1]{} (-3,37)[2]{} (-3,57)[3]{} (-3,77)[4]{} (10,20)[(1,0)[180]{}]{} (10,80)[(1,0)[180]{}]{} (10,20)[(0,1)[60]{}]{} (190,20)[(0,1)[60]{}]{} (10,60) (30,80) (50,60) (70,40) (90,60) (110,80) (130,60) (150,40) (170,20) (190,40) Let us now consider the following set of paths. Let $p=(p_1,p_2,\ldots,p_L)$ be a sequence of $1$’s and $2$’s and let ${\lambda}=({\lambda}_1,{\lambda}_2)$ be the content of $p$ where ${\lambda}_i$ is the number of $i$’s in $p$. Denote by ${\mathcal{P}}_{L,{\lambda}}$ the set of all $p=(p_1,\ldots,p_L)$ with content ${\lambda}$. Another way to represent $p$ is by height variables $\sigma=(\sigma_0,\sigma_1,\ldots,\sigma_L)$ where by convention $\sigma_0=3$ and $\sigma_i=\sigma_{i-1}+1$ if $p_i=1$ and $\sigma_i=\sigma_{i-1}-1$ if $p_i=2$ for $1\le i\le L$. We will use $\sigma$ and $p$ interchangeably. We are actually mostly concerned here with the set ${\mathcal{P}}_L:={\mathcal{P}}_{L,{\lambda}}$ where ${\lambda}=(\lfloor \frac{L}{2}\rfloor,\lfloor \frac{L+1}{2}\rfloor)$. Consider the following subsets of ${\mathcal{P}}_L$. Let ${\overline{\mathcal{P}}}_L$ be the subset of ${\mathcal{P}}_L$ consisting of all paths in the strip of height four, that is all $\sigma\in{\mathcal{P}}_L$ such that $1\le \sigma_i\le 4$ for all $1\le i\le L$. Furthermore, let ${\mathcal{P}}_L^{\uparrow,j}$ be the set of all paths $\sigma\in{\mathcal{P}}_L$ such that there exist indices $1\le i_1<i_2<\cdots<i_j\le L$ so that $\sigma_{i_1},\sigma_{i_3},\sigma_{i_5},\ldots$ are greater than four and $\sigma_{i_2},\sigma_{i_4},\sigma_{i_6},\ldots$ are less than one. Similarly let ${\mathcal{P}}_L^{\downarrow,j}$ be the set of all paths $\sigma\in{\mathcal{P}}_L$ such that there exist indices $1\le i_1<i_2<\cdots<i_j\le L$ such that $\sigma_{i_1},\sigma_{i_3},\sigma_{i_5},\ldots$ are less than one and $\sigma_{i_2},\sigma_{i_4},\sigma_{i_6},\ldots$ are greater than four. By inclusion-exclusion we have $${\overline{\mathcal{P}}}_L=\Bigl({\mathcal{P}}_L \cup \bigcup_{j\ge 1} ( {\mathcal{P}}_L^{\uparrow,2j} \cup {\mathcal{P}}_L^{\downarrow,2j}) \Bigr) \setminus \bigcup_{j\ge 1} ({\mathcal{P}}_L^{\uparrow,2j-1} \cup {\mathcal{P}}_L^{\downarrow,2j-1}).$$ Now define the energy function $E$ for $\sigma\in{\mathcal{P}}_L$ as $$E(\sigma)=\sum_{i=1}^{L-1} i\cdot h(\sigma_{i-1},\sigma_i,\sigma_{i+1})$$ where $h$ is the local energy function given by $$h(\sigma_{i-1},\sigma_i,\sigma_{i+1})= \begin{cases} 1 & \text{if $\sigma_{i-1}=\sigma_i-1=\sigma_{i+1}$ and $\sigma_i>3$,}\\ & \text{or $\sigma_{i-1}=\sigma_i+1=\sigma_{i+1}$ and $\sigma_i<2$,}\\ 0&\text{else.} \end{cases}$$ The term $q^{\frac{j}{2}(5j+1)}{\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j}{2} \rfloor}}$ in can then be interpreted as the generating function of ${\mathcal{P}}_L^{\downarrow,j}$ given by $\sum_{\sigma\in{\mathcal{P}}_L^{\downarrow,j}} q^{E(\sigma)}$ if $j>0$ and the generating function of ${\mathcal{P}}_L^{\uparrow,j}$ given by $\sum_{\sigma\in{\mathcal{P}}_L^{\uparrow,j}} q^{E(\sigma)}$ if $j<0$. The term $j=0$ in is the generating function of ${\mathcal{P}}_L$. In section \[sec:bose\] we will encounter more general inclusion-exclusion arguments in terms of operations on crystals. Identities ---------- Equations and yield the following polynomial identity $$\sum_{n=0}^\infty q^{n^2} {\genfrac{[}{]}{0pt}{}{L-n}{n}}= \sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+1)} {\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j}{2}\rfloor}}.$$ Using $\lim_{L\to\infty} {{\textstyle\genfrac{[}{]}{0pt}{}{L}{m}}}=\frac{1}{(q)_m}$ this identity implies in the limit $L\to\infty$. All discussion so far focused on . The second Rogers–Ramanujan identity can in fact be treated in a very similar fashion. Define the set ${\mathcal{D}}'_L$ analogous to ${\mathcal{D}}_L$ with the only difference that now $\sigma_0=1$ instead of 0. The generating function is defined to be $X'(L)=\sum_{\sigma\in {\mathcal{D}}'_L} q^{E(\sigma)}$. Analogous arguments to those in sections \[subsec:fermi\] and \[subsec:bose\] yield the following two expressions for $X'(L)$ $$X'(L)=\sum_{n=0}^\infty q^{n(n+1)} {\genfrac{[}{]}{0pt}{}{L-n-1}{n}} =\sum_{j=-\infty}^\infty (-1)^j q^{\frac{j}{2}(5j+3)} {\genfrac{[}{]}{0pt}{}{L}{\lfloor \frac{L-5j-1}{2}\rfloor}}.$$ In the limit $L\to\infty$ this polynomial identity implies . Crystal bases {#sec:crystal} ============= In this section we review the main results about crystal bases and explain how they provide a general setting for the definition of one-dimensional configuration sums. The quantized universal enveloping algebra $U_q({\mathfrak{g}})$ associated with a symmetrizable Kac–Moody Lie algebra ${\mathfrak{g}}$ was introduced independently by Drinfeld [@D] and Jimbo [@J] in their study of two dimensional solvable lattice models in statistical mechanics. The parameter $q$ corresponds to the temperature of the underlying model. Kashiwara [@K] showed that at zero temperature or $q=0$ the representations of $U_q({\mathfrak{g}})$ have bases, which he coined crystal bases, with a beautiful combinatorial structure and favorable properties such as uniqueness and stability under tensor products. The existence and uniqueness of crystal bases for integrable highest weight modules for an arbitrary symmetrizable Kac–Moody algebra was given in [@K1]. Axiomatic definition of crystals -------------------------------- Let ${\mathfrak{g}}$ be a symmetrizable Kac-Moody algebra, $P$ the weight lattice, $I$ the index set for the vertices of the Dynkin diagram of ${\mathfrak{g}}$, $\{\alpha_i\in P \mid i\in I \}$ the simple roots, and $\{h_i\in P^*=\mathrm{Hom}_{{\mathbb{Z}}}(P,{\mathbb{Z}})\mid i\in I \}$ the simple coroots. Let $U_q({\mathfrak{g}})$ be the quantized universal enveloping algebra of ${\mathfrak{g}}$. A $U_q({\mathfrak{g}})$-crystal is a nonempty set $B$ equipped with maps ${\mathrm{wt}}:B\rightarrow P$ and $e_i,f_i:B\rightarrow B\cup\{\emptyset\}$ for all $i\in I$, satisfying $$\begin{aligned} &f_i(b)=b' \Leftrightarrow e_i(b')=b \text{ if $b,b'\in B$} \\ &{\mathrm{wt}}(f_i(b))={\mathrm{wt}}(b)-\alpha_i \text{ if $f_i(b)\in B$} \\ &{\langle h_i\,,\,{\mathrm{wt}}(b)\rangle}=\varphi_i(b)-\epsilon_i(b)\end{aligned}$$ where $\langle \cdot,\cdot\rangle$ is the natural pairing. Here for $b\in B$, $$\label{eq:eps phi def} \begin{split} \epsilon_i(b)&= \max\{n\ge0\mid e_i^n(b)\not=\emptyset \} \\ \varphi_i(b) &= \max\{n\ge0\mid f_i^n(b)\not=\emptyset \}. \end{split}$$ (It is assumed that $\varphi_i(b),\epsilon_i(b)<\infty$ for all $i\in I$ and $b\in B$.) A $U_q({\mathfrak{g}})$-crystal $B$ can be viewed as a directed edge-colored graph (the crystal graph) whose vertices are the elements of $B$, with a directed edge from $b$ to $b'$ labeled $i\in I$, if and only if $f_i(b)=b'$. The element $b$ is said to be highest weight if $e_i(b)=\emptyset$ for all $i\in I$. Let $K$ be a subset of $I$. Then the $K$-component of the crystal $B$ is the graph obtained by only considering edges colored by $i\in K$. We also define the crystal reflection operator $s_i:B\rightarrow B$ by $$s_i(b) = \begin{cases} f_i^{\varphi_i(b)-\epsilon_i(b)}(b) & \text{if $\varphi_i(b)>\epsilon_i(b)$} \\ b & \text{if $\varphi_i(b)=\epsilon_i(b)$} \\ e_i^{\epsilon_i(b)-\varphi_i(b)}(b) & \text{if $\varphi_i(b)<\epsilon_i(b)$}. \end{cases}$$ It is obvious that $s_i$ is an involution. Tensor products of crystals --------------------------- Given two crystals $B$ and $B'$, there is also a crystal obtained by taking the tensor product $B\otimes B'$. As a set $B\otimes B'$ is just given by the Cartesian product of the sets $B$ and $B'$. The weight function ${\mathrm{wt}}$ for $b\otimes b'\in B\otimes B'$ is ${\mathrm{wt}}(b\otimes b')={\mathrm{wt}}(b)+{\mathrm{wt}}(b')$ and the raising and lowering operators $e_i$ and $f_i$ act as follows $$\label{tensor} \begin{split} e_i(b\otimes b')&=\begin{cases} e_i b \otimes b' & \text{if $\epsilon_i(b)>\phi_i(b')$,}\\ b \otimes e_i b' & \text{otherwise,} \end{cases}\\ f_i(b\otimes b')&=\begin{cases} f_i b \otimes b' & \text{if $\epsilon_i(b)\ge\phi_i(b')$,}\\ b \otimes f_i b' & \text{otherwise.} \end{cases} \end{split}$$ The reader is warned that this convention is different from Kashiwara’s convention. The order of the tensor factors is interchanged. Finite and infinite crystals ---------------------------- Let us fix some notation. From now on let ${\mathfrak{g}}$ be a simple complex Lie algebra and ${\hat{{\mathfrak{g}}}}$ be the associated untwisted affine algebra. That is, let ${\hat{{\mathfrak{g}}}}'={\mathfrak{g}}\otimes{\mathbb{C}}[t,t^{-1}] \oplus {\mathbb{C}}c$ be the central extension of the loop algebra of ${\mathfrak{g}}$ and ${\hat{{\mathfrak{g}}}}={\hat{{\mathfrak{g}}}}'\oplus{\mathbb{C}}d$ where $d$ is the degree derivation. Let $J$ (resp. $I=J\cup\{0\}$) be a set indexing the vertices of the Dynkin diagram of ${\mathfrak{g}}$ (resp. ${\hat{{\mathfrak{g}}}}$ and ${\hat{{\mathfrak{g}}}}'$). A classical weight is a weight with respect to the algebra ${\mathfrak{g}}$. Denote by $U_q({\mathfrak{g}})$, $U'_q({\hat{{\mathfrak{g}}}})$, and $U_q({\hat{{\mathfrak{g}}}})$ the quantized universal enveloping algebras of ${\mathfrak{g}}$, ${\hat{{\mathfrak{g}}}}'$, and ${\hat{{\mathfrak{g}}}}$ respectively. There are two main categories of crystals. The first one contains the crystal bases of irreducible integrable $U_q({\hat{{\mathfrak{g}}}})$-modules $V({\Lambda})$ of highest weight ${\Lambda}\in P^+$ where $P^+$ denotes the set of dominant weights of ${\hat{{\mathfrak{g}}}}$. These crystals are infinite-dimensional. The second one contains finite-dimensional crystals corresponding to $U_q'({\hat{{\mathfrak{g}}}})$. Since the set $B$ is finite in this case these crystals are called finite crystals. The level of a finite crystal $B$ is defined as $${\mathrm{lev}}B = \min\{ \langle c,\epsilon(b) \rangle\mid b\in B\}$$ where $\epsilon(b)=\sum_{i\in I} \epsilon_i(b) {\Lambda}_i$ and $\{{\Lambda}_i\mid i\in I\}$ is the ${\mathbb{Z}}$-basis of the weight lattice of ${\hat{{\mathfrak{g}}}}'$. A $U'_q({\hat{{\mathfrak{g}}}})$-crystal can be viewed as a $U_q({\mathfrak{g}})$-crystal by restricting to the $J$-component. The crystal of each integrable highest weight $U_q({\hat{{\mathfrak{g}}}})$-module can be realized in terms of a semi-infinite tensor product of perfect crystals [@KKMMNN; @KKMMNN2; @KKM]. Perfect crystals are finite crystals with some additional properties (for details see [@KKMMNN Definition 4.6.1]). At least one perfect crystal for each integrable $U_q({\hat{{\mathfrak{g}}}})$-module for ${\hat{{\mathfrak{g}}}}=A_n^{(1)}, B_n^{(1)}, C_n^{(1)}, D_n^{(1)}, A_n^{(2)}$ and $D_{n+1}^{(2)}$ was given in [@KKMMNN2]. Kashiwara and Nakashima [@KN] constructed the finite $U_q({\mathfrak{g}})$-crystals for ${\mathfrak{g}}=A_n$, $B_n$, $C_n$ and $D_n$ explicitly in terms of tableaux. The cases ${\mathfrak{g}}=A_n$ and $C_n$ are discussed in more detail in the next subsection. Finite crystals of type $A_n$ and $C_n$ {#sec:finite AC} --------------------------------------- The finite crystals associated with ${\mathfrak{g}}=A_n$ and $C_n$ are presented in more detail since they will be our main examples throughout the paper. Let $\Lambda_i$ for $i\in J$ be the fundamental weights of ${\mathfrak{g}}$. For later purposes it will be convenient to give the root and weight structure of $A_n$ and $C_n$ explicitly. Let $(\cdot|\cdot)$ be the standard bilinear form normalized such that $(\alpha_i\mid\alpha_i)=2$ for the long roots $\alpha_i$. \[ex:roots\] For either ${\mathfrak{g}}=A_n$ or ${\mathfrak{g}}=C_n$ the index set for the simple roots is $J=\{1,2,\dotsc,n\}$. For ${\mathfrak{g}}=A_n$ the weight lattice is embedded in the subspace of ${\mathbb{R}}^{n+1}$ orthogonal to the vector $e=\sum_{j=1}^{n+1} \epsilon_j$, where $\epsilon_i$ is the $i$-th standard basis vector of ${\mathbb{R}}^{n+1}$. The simple roots are $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for all $i\in J$. The fundamental weights are ${\Lambda}_i=\sum_{j=1}^i \epsilon_j -\frac{i}{n+1}e$ for $i\in J$. The half-sum of positive roots is $\rho=(n,n-1,\dotsc,1,0)-\frac{n}{2}e$. The scalar product is the standard one: $(\alpha|\beta)=\alpha\cdot\beta$. For ${\mathfrak{g}}=C_n$ the simple roots are given by the short roots $\alpha_i=\epsilon_i-\epsilon_{i+1}$ for $1\le i<n$ and the long root $\alpha_n=2\epsilon_n$ where $\epsilon_i$ is the $i$-th unit vector in ${\mathbb{R}}^n$. The fundamental weights are ${\Lambda}_i=\epsilon_1+\cdots+\epsilon_i$ for $i\in J$. Half the sum of all positive roots is $\rho=(n,n-1,\ldots,1)$. The bilinear form is given by $(\alpha|\beta)=\alpha\cdot \beta/2$. For both type $A_n$ and $C_n$ we will identify dominant weights, that is weights of the form ${\Lambda}={\Lambda}_{k_1}+\cdots+{\Lambda}_{k_m}$, with partitions. A partition ${\lambda}=({\lambda}_1,\ldots,{\lambda}_{n+1})$ (with ${\lambda}_{n+1}=0$ for type $C_n$) corresponds to the weight ${\Lambda}=\sum_{i\in J}({\lambda}_i-{\lambda}_{i+1}){\Lambda}_i$. For each dominant weight $\Lambda$ there is a finite classical crystal $B(\Lambda)$ [@KN]. The crystals $B(\Lambda_1)$ for type $A_n$ and $C_n$ are given in Table \[table: classical crystals\] where the arrow $\stackrel{i}{\longrightarrow}$ stands for $f_i$. $A_n$ ------- -- $C_n$ : \[table: classical crystals\]Classical crystals $B({\Lambda}_1)$ The finite crystals $B({\Lambda}_k)$ for $k\in J$ can be obtained in the following way. Let $u_{{\Lambda}_k}= \raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7) \BBoxc(7,0)(14,14)\Text(7,0)[]{$k$}\end{picture}}} \otimes \cdots \otimes \raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7) \BBoxc(7,0)(14,14)\Text(7,0)[]{$2$}\end{picture}}} \otimes \raisebox{-0.14cm}{\scalebox{0.8}{\begin{picture}(14,14)(0,-7) \BBoxc(7,0)(14,14)\Text(7,0)[]{$1$}\end{picture}}}$ be the unique highest weight vector of weight ${\Lambda}_k$ in $B({\Lambda}_1)^{\otimes k}$. Then $B({\Lambda}_k)$ is the connected component of $B({\Lambda}_1)^{\otimes k}$ containing $u_{{\Lambda}_k}$. In [@KN] the elements in this connected component were identified with certain one-column tableaux. The finite crystal $B({\Lambda})$ for a dominant weight ${\Lambda}={\Lambda}_{k_1}+\cdots+ {\Lambda}_{k_m}$ with $k_1\ge k_2\ge \cdots \ge k_m$ is isomorphic to the connected component in $B({\Lambda}_{k_1})\otimes \cdots \otimes B({\Lambda}_{k_m})$ which contains the highest weight element $u_{{\Lambda}_{k_1}}\otimes \cdots \otimes u_{{\Lambda}_{k_m}}$. Combining the two embeddings it follows that $B({\Lambda})$ is isomorphic to a certain connected component in $B({\Lambda}_1)^{\otimes M}$ where $M=k_1+\cdots+k_m$. For type $A_n$ the elements in $B({\Lambda})$ can be identified with semi-standard Young tableaux of shape ${\lambda}$ where ${\lambda}$ is the partition corresponding to the weight ${\Lambda}$. The paths encountered in section \[sec:HHM\] are sequences of 1’s and 2’s. Viewed as single box Young tableaux over the alphabet $\{1,2\}$, these are exactly the elements of the crystal $B({\Lambda}_1)$ of type $A_1$. Simple crystals --------------- Simple crystals were introduced by Akasaka and Kashiwara [@AK]. As will be explained in the next section they have an isomorphism and energy function which are required for the definition of the one-dimensional configuration sums. Let ${\hat{W}}$ be the Weyl group of the affine Kac–Moody algebra ${\hat{{\mathfrak{g}}}}$ generated by the simple reflections $r_i$ for $i\in I$ defined as $$\label{reflections} r_i(\beta)=\beta - \langle h_i,\beta\rangle \alpha_i$$ where $\beta$ is a root. Let $B$ be the crystal graph of an integrable $U_q({\hat{{\mathfrak{g}}}})$-module. Say that $b\in B$ is an extremal vector of weight $\Lambda\in P$ provided that ${\mathrm{wt}}(b)=\Lambda$ and there exists a family of elements $\{b_w\mid w\in {\hat{W}}\}\subset B$ such that 1. $b_w=b$ for $w=e$. 2. If ${\langle h_i\,,\,w{\Lambda}\rangle}\ge 0$ then $e_i(b)=0$ and $f_i^{{\langle h_i\,,\,w{\Lambda}\rangle}}(b_w)=b_{r_i w}$. 3. If ${\langle h_i\,,\,w{\Lambda}\rangle}\le 0$ then $f_i(b)=0$ and $e_i^{{\langle h_i\,,\,w{\Lambda}\rangle}}(b_w)=b_{r_i w}$. Say that a $U'_q({\hat{{\mathfrak{g}}}})$-crystal $B$ is *simple* if 1. $B$ is the crystal base of a finite-dimensional integrable $U'_q({\hat{{\mathfrak{g}}}})$-module. 2. There is a dominant weight $\Lambda$ with respect to the weight lattice of the classical algebra ${\mathfrak{g}}$ such that $B$ has a unique vector (denoted $u(B)$) of weight $\Lambda$, and the weight of any extremal vector of $B$ is contained in $W\Lambda$. Here $W$ is the Weyl group corresponding to the classical algebra ${\mathfrak{g}}$. \[thm:simple\] [@AK] 1. Simple crystals are connected as graphs. 2. The tensor product of simple crystals is simple. Let $B$ be a simple $U'_q({\hat{{\mathfrak{g}}}})$-crystal, equipped with a function $D=D_B:B\rightarrow{\mathbb{Z}}$, called its intrinsic energy, which is required to be constant on $J$-components and defined up to a global additive constant. Call the pair $(B,D)$ a graded simple $U'_q({\hat{{\mathfrak{g}}}})$-crystal. We normalize the intrinsic energy function by the requirement that $$D_B(u(B))=0.$$ Finite dimensional affine crystals ---------------------------------- Recently, new families of crystals of the finite dimensional representations of $U_q({\hat{{\mathfrak{g}}}})$ were conjectured [@HKOTY; @K3] where ${\hat{{\mathfrak{g}}}}$ is a untwisted affine Lie algebra. [@HKOTY; @K3]\[conj:Brs\] For each $r\in J$ and $s\ge1$, there exists an irreducible finite-dimensional integrable $U'_q({\hat{{\mathfrak{g}}}})$-module $W^{(r)}_s$ with simple crystal base $B^{r,s}$ generated a unique extremal vector $u(B^{r,s})$ of weight $s{\Lambda}_r$, and a prescribed $U_q({\mathfrak{g}})$-crystal decomposition of the form $B^{r,s} \cong B(s{\Lambda}_r)\oplus B$, where $B$ is a direct sum of $U_q({\mathfrak{g}})$-crystals of the form $B({\Lambda})$ where ${\Lambda}$ is a classical dominant weight and $s {\Lambda}_r \vartriangleright {\lambda}$. Here ${\Lambda}'\trianglerighteq{\Lambda}$ if and only if ${\Lambda}'-{\Lambda}\in\bigoplus_{i\in J} {\mathbb{N}}\alpha_i$. Moreover there is a prescribed intrinsic energy function $D=D_{B^{r,s}}:B^{r,s}\rightarrow {\mathbb{Z}}$, that is constant on $J$-components, such that $0=D(u)>D(b)$ where $u$ is the $J$-highest weight vector of weight $s{\Lambda}_r$ in $B^{r,s}$, and $b$ is any element not in the $J$-component of $u$. Combinatorial $R$-matrix and energy function -------------------------------------------- Suppose $B_1$ and $B_2$ are simple $U'_q({\mathfrak{g}})$-crystals. Then there is a unique isomorphism of $U'_q({\mathfrak{g}})$-crystals $\sigma:B_2\otimes B_1\rightarrow B_1\otimes B_2$. There is also a function $H=H_{B_2,B_1}:B_2\otimes B_1\rightarrow{\mathbb{Z}}$, called local energy function which is unique up to global additive constant, such that for all $b_2\otimes b_1\in B_2\otimes B_1$ with $b_1'\otimes b_2'= \sigma(b_2 \otimes b_1)$ $$\label{eq:local energy} H(e_i(b_2\otimes b_1))= H(b_2\otimes b_1)+ \begin{cases} -1 & \text{if $i=0$, $\epsilon_0(b_2)>\varphi_0(b_1)$, $\epsilon_0(b_1')>\varphi_0(b_2')$} \\ 1 & \text{if $i=0$, $\epsilon_0(b_2)\le\varphi_0(b_1)$, $\epsilon_0(b_1')\le \varphi_0(b_2')$} \\ 0 & \text{otherwise.} \end{cases}$$ The pair $(\sigma,H)$ is called the combinatorial $R$-matrix. It is convenient to normalize the local energy function $H$ by requiring that $$\label{eq:extremal H} H(u(B_2)\otimes u(B_1)) = 0.$$ With this convention it follows by definition that $$\label{eq:HR=H} H_{B_1,B_2} \circ \sigma_{B_2,B_1} = H_{B_2,B_1}$$ as operators on $B_2\otimes B_1$. Let $(B_j,D_j)$ be graded simple $U'_q({\hat{{\mathfrak{g}}}})$-crystals for $1\le j\le L$ and set $B=B_L\otimes\dotsm\otimes B_1$. Following [@NY] define the energy function $E_B:B\rightarrow{\mathbb{Z}}$ by $$\label{eq:NY} E_B = \sum_{1\le i<j\le L} H_i\sigma_{i+1}\sigma_{i+2}\dotsm\sigma_{j-1}$$ where $H_i$ (resp. $\sigma_i$) is the local energy function (resp. isomorphism) acting on the $i$-th and $(i+1)$-st tensor factor. By the normalization assumption it follows that $$\label{eq:extremal NY} E_B(u(B))=0.$$ As shown in [@OSS], the intrinsic energy $D_B$ for the $L$-fold tensor product $B=B_L\otimes\dotsm\otimes B_1$ is given by $$\label{eq:energy} D_B = E_B + \sum_{j=1}^L D_j \sigma_1\sigma_2\dotsm\sigma_{j-1}$$ where $D_j$ acts on the rightmost tensor factor which is $B_j$. The coenergy and intrinsic coenergy are defined as $${\overline{E}}_B=-E_B \qquad\text{and}\qquad {\overline{D}}_B=-D_B.$$ One-dimensional configuration sums ---------------------------------- There are three different sets of paths that we consider. Let $B=B_L\otimes\cdots\otimes B_1$ where all $B_i$ are simple crystals. For a classical weight $\Lambda$ the set of unrestricted paths is defined as $$\label{p} {\mathcal{P}}(B,{\Lambda})=\{ b\in B \mid {\mathrm{wt}}(b)={\Lambda}\}.$$ For a dominant classical weight ${\Lambda}$ the set of classically restricted paths is $$\label{p'} {\mathcal{P}}'(B,{\Lambda})=\{ b\in B \mid \text{${\mathrm{wt}}(b)={\Lambda}$ and $e_ib=0$ for all $i\in J$} \}$$ and the set of level-restricted paths for $\ell\in{\mathbb{N}}$ is $$\label{pl} {\mathcal{P}}^\ell(B,{\Lambda})=\{ b\in B \mid \text{${\mathrm{wt}}(b)={\Lambda}$, $e_ib=0$ for all $i\in J$ and $e_0^{\ell+1}b=0$}\}.$$ The corresponding one-dimensional configuration sums are the generating functions of these sets of paths with energy/coenergy statistics. The one-dimensional sums $$\label{super} \begin{split} S(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}(B,{\Lambda})} q^{D_B(b)}\\ {\overline{S}}(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}(B,{\Lambda})} q^{{\overline{D}}_B(b)} \end{split}$$ are called supernomials, whereas $$\label{kostka} \begin{split} X(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}'(B,{\Lambda})} q^{D_B(b)}\\ {\overline{X}}(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}'(B,{\Lambda})} q^{{\overline{D}}_B(b)} \end{split}$$ are the classically restricted configuration sums or generalized Kostka polynomials, and $$\label{lkostka} \begin{split} X^\ell(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}^\ell(B,{\Lambda})} q^{D_B(b)}\\ {\overline{X}}^\ell(B,{\Lambda})&=\sum_{b\in {\mathcal{P}}^\ell(B,{\Lambda})} q^{{\overline{D}}_B(b)} \end{split}$$ are the level-$\ell$ restricted configuration sums or $\ell$-generalized Kostka polynomials. The classically restricted configurations sums are graded tensor product multiplicities. The level-restricted configuration sums are graded level $\ell$ fusion coefficients. Let $B_{{\Lambda}'}$ denote the crystal corresponding to the affine irreducible highest weight representation $V({\Lambda}')$. By the Verlinde formula [@V], the fusion coefficient is the coefficient of $B_{{\Lambda}'}$ of weight ${\Lambda}' ={\Lambda}+\ell{\Lambda}_0$ in the decomposition of $B \otimes B_{\ell{\Lambda}_0}$. The affine highest weight vectors of weight ${\Lambda}'$, whose number is the above multiplicity, are the summands of ${\overline{X}}^\ell(B,{\Lambda})$. It will be shown in sections \[sec:bose\] and \[sec:fermi\] that there are two different evaluations of ${\overline{X}}(B,{\Lambda})$ and ${\overline{X}}^\ell(B,{\Lambda})$ which give rise to $q$-identities. Bosonic evaluation {#sec:bose} ================== Here we present the bosonic evaluation of ${\overline{X}}(B,\Lambda)$ and ${\overline{X}}^\ell(B,\Lambda)$ as defined in and . Similarly to the bosonic evaluation of $X(L)$ of section \[subsec:bose\], the bosonic evaluation of ${\overline{X}}(B,\Lambda)$ and ${\overline{X}}^\ell(B,\Lambda)$ can be obtained by inclusion-exclusion arguments as shown in [@SS]. We discuss the main ideas and techniques of sign-reversing involutions. Classically-restricted case --------------------------- Let $B=B_L\otimes \cdots \otimes B_1$ be a $U_q({\mathfrak{g}})$-crystal and let ${\Lambda}$ be a classical weight. The Weyl group $W$ of ${\mathfrak{g}}$ which is generated by the simple reflections $r_i$ as in with $i$ restricted to $i\in J$. The bosonic expression for the generating function of classically-restricted paths ${\overline{X}}(B,{\Lambda})$ as defined in is given by $$\label{bose cr} {\overline{X}}(B,{\Lambda})=\sum_{\omega \in W} (-1)^\omega \; {\overline{S}}(B,\omega({\Lambda}+\rho)-\rho)$$ where $(-1)^\omega$ is the sign of $\omega$ and $\rho$ is half the sum of all positive roots. This formula follows directly from Weyl’s character formula. As a warm-up for the level-restricted case, we would like to briefly sketch for types $A_n$ and $C_n$ how can be derived via a sign-reversing involution. \[ex:Weyl\] For ${\mathfrak{g}}=A_n$ the Weyl group $W$ is generated by the reflections $r_1,\ldots,r_n$ which act on ${\lambda}\in{\mathbb{Z}}^{n+1}$ as follows $$r_i({\lambda})=({\lambda}_1,\ldots,{\lambda}_{i+1},{\lambda}_i,\ldots,{\lambda}_{n+1}).$$ Hence on ${\mathbb{Z}}^{n+1}$ the Weyl group acts as the symmetric group $S_{n+1}$. For ${\mathfrak{g}}=C_n$ the Weyl group is generated by $$\begin{aligned} r_i({\lambda})&=({\lambda}_1,\ldots,{\lambda}_{i+1},{\lambda}_i,\ldots,{\lambda}_n) \qquad \text{for $1\le i<n$}\\ r_n({\lambda})&=({\lambda}_1,\ldots,{\lambda}_{n-1},-{\lambda}_n) \end{aligned}$$ so that $W$ on ${\mathbb{Z}}^n$ acts by all permutations and sign changes. Set $$\mathcal{S}=\{(\omega,b)\in W\times B \mid \omega({\mathrm{wt}}(b)+\rho)={\Lambda}+\rho\}$$ On the set $\mathcal{S}$ we define an involution $\Phi:\mathcal{S}\to\mathcal{S}$ with the properties that the fixed points are the pairs $(1,b)$ with $b\in {\mathcal{P}}'(B,{\Lambda})$ which recall are the paths underlying ${\overline{X}}(B,{\Lambda})$ (see ). Furthermore, if $\Phi(\omega,b)=(\omega',b')$ with $(\omega,b)\neq (\omega',b')$ then $\omega$ and $\omega'$ have opposite signs. Let $\mathcal{S}_i$ for $i\in J$ be the set of pairs $(\omega,b)\in \mathcal{S}$ such that $\epsilon_i(b)>0$. Define $\Phi_i:\mathcal{S}_i\to\mathcal{S}_i$ by $\Phi_i(\omega,b)=(\omega r_i,s_i e_i(b))$. Define the set $\mathcal{S}'=\mathcal{S}-\{(1,b)\mid b\in{\mathcal{P}}'(B,{\Lambda})\}$ so that $\mathcal{S}'=\bigcup_{i\in J} \mathcal{S}_i$. Then $\Phi$ is given by $$\Phi(\omega,b)=\begin{cases} (\omega,b) & \text{if $(\omega,b)\in \mathcal{S}\setminus \mathcal{S}'$}\\ \Phi_i(\omega,b) & \text{if $(\omega,b)\in \mathcal{S}'$ and $i=v(\omega,b)$} \end{cases}$$ where $v$ is some functions $v:\mathcal{S}'\to J$. To show that $\Phi$ indeed exists and is an involution it needs to be shown that if $v(\omega,b)=i$ then $$\label{v prop} (\omega,b)\in\mathcal{S}_i \qquad \text{and} \qquad v(\Phi_i(\omega,b))=i.$$ As mentioned in section \[sec:finite AC\], the crystal $B=B_L\otimes\cdots\otimes B_1$ of type $A_n$ or $C_n$ can be embedded into $B({\Lambda}_1)^{\otimes M}$ for some $M$. Let $p=p_M\otimes\cdots\otimes p_1$ be the image of $b\in B$ under this embedding. Then $v$ can be defined as follows. Let $k$ be minimal such that $\epsilon_i(p_k\otimes\cdots\otimes p_1)>0$ for some $i\in J$. Then it is clear from Table \[table: classical crystals\] that there is a unique $i$ satisfying $\epsilon_i(p_k\otimes\cdots\otimes p_1)>0$. Set $v(\omega,b)=i$. It follows from equation and Table \[table: classical crystals\] that the first $k$ tensor factors of $p$ stay invariant under $\Phi_i$ since there are no strings of length greater than one. This ensures . Hence inclusion-exclusion implies . Level-restricted case --------------------- The bosonic expression for the level-restricted generating function ${\overline{X}}^\ell(B,{\Lambda})$ defined in can also be found by a sign-reversing involution. The difference is that one needs to consider elements $\omega$ in the affine Weyl group ${\hat{W}}$ which is generated by $r_i$ with $i\in I$ (rather than $i\in J$). For ${\hat{{\mathfrak{g}}}}=A_n^{(1)}$ the affine Weyl group ${\hat{W}}$ is generated by the reflections $r_1,\ldots,r_n$ as in example \[ex:Weyl\] and $$r_0({\lambda})=({\lambda}_{n+1}+\ell+n+1,{\lambda}_2,\ldots,{\lambda}_n,{\lambda}_1-(\ell+n+1)).$$ For ${\hat{{\mathfrak{g}}}}=C_n^{(1)}$ the affine Weyl group ${\hat{W}}$ is generated by the reflections $r_1,\ldots,r_n$ as in example \[ex:Weyl\] and $$r_0({\lambda})=(-{\lambda}_1+2(\ell+n+1),{\lambda}_2,\ldots,{\lambda}_n).$$ The affine Weyl group is isomorphic to ${\hat{W}}\cong T \rtimes W$ where $T$ is the set of certain translations $t_\alpha$ indexed by $\alpha\in M$ for a particular set $M$ (for more details see [@Kac Section 6]). Then it was shown in [@SS] that [^1] $$\begin{gathered} \label{bose lr} {\overline{X}}^\ell(B,{\Lambda})=\sum_{\omega\in W} \sum_{\alpha\in M} (-1)^\omega q^{\frac{1}{2}a_0 (\alpha|\alpha)(\ell+h^\vee)-a_0(\rho+{\Lambda}|\alpha)}\\ \qquad \times {\overline{S}}(B,\omega({\Lambda}+\rho-(\ell+h^\vee)\alpha)-\rho)\end{gathered}$$ where $h^\vee$ is the dual Coxeter number of ${\hat{{\mathfrak{g}}}}$ and $a_0$ is the label of the zeroth node in the Dynkin diagram of ${\hat{{\mathfrak{g}}}}$. Let us give more explicitly for ${\hat{{\mathfrak{g}}}}=A_n^{(1)}$ and $C_n^{(1)}$. In both cases the dual Coxeter number is $h^\vee=n+1$ and $a_0=1$. For type $A_n^{(1)}$ the set $M$ is given by all $\beta\in {\mathbb{Z}}^{n+1}$ such that $|\beta|:=\beta_1+\cdots+\beta_{n+1}=0$ so that $$\begin{gathered} {\overline{X}}^\ell(B,{\lambda})=\sum_{\omega\in W} \sum_{\substack{\beta\in{\mathbb{Z}}^{n+1}\\ |\beta|=0}} (-1)^\omega q^{\frac{1}{2} \beta\cdot \beta(\ell+n+1)-(\rho+{\lambda})\cdot \beta}\\ \times {\overline{S}}(B,\omega({\lambda}+\rho-(\ell+n+1)\beta)-\rho)\end{gathered}$$ where $W=S_{n+1}$ is the set of permutations. For type $C_n^{(1)}$ we have $M=2{\mathbb{Z}}^n$. Hence $$\begin{gathered} {\overline{X}}^\ell(B,{\lambda})=\sum_{\omega\in W} \sum_{\beta\in 2{\mathbb{Z}}^n} (-1)^\omega q^{\frac{1}{4}\beta\cdot \beta (\ell+n+1)-\frac{1}{2}(\rho+{\lambda})\cdot \beta}\\ \times {\overline{S}}(B,\omega({\lambda}+\rho-(\ell+n+1)\beta)-\rho)\end{gathered}$$ where the Weyl group $W$ in this case is the set of all permutations and sign changes. The extra factor of $1/2$ in the exponent of $q$ comes from the normalization of $(\cdot|\cdot)$ as alluded to in example \[ex:roots\]. The arguments in [@SS] involve a sign-reversing involution. Similarly to the classically restricted case set $$\mathcal{S}=\{(\omega,b)\in {\hat{W}}\times B \mid \omega({\mathrm{wt}}(b)+\rho)={\Lambda}+\rho\}.$$ For $i\in J$ define $\mathcal{S}_i$ and $\Phi_i$ as before. In addition, let $\mathcal{S}_0$ be the subset of all pairs $(\omega,b)$ in $\mathcal{S}$ such that $\epsilon_0(b)>\ell$ and define $\Phi_0(\omega,b)=(\omega r_0, s_0 e_0^{\ell+1} b)$. One can find a sign-reversing involution $\Phi$ with fixed point set being the set of level-$\ell$ restricted paths ${\mathcal{P}}^\ell(B,{\Lambda})$ by again specifying a function $v:\mathcal{S}-\{(1,b)\mid b\in {\mathcal{P}}^\ell(B,{\Lambda})\}\to I$ satisfying . The existence of such a function $v$ was proven in [@SS] for a large class of crystals. Supernomial coefficients ------------------------ The bosonic formulas and involve the supernomial coefficients defined in . To obtain truely explicit expressions it is still necessary to give formulas for the supernomial coefficients. These are not yet known in general. Here we give a few examples for which formulas exist. The supernomial coefficients for type $A_n^{(1)}$ for single columns were given in [@HKKOTY; @Ki]. Let $B=B^{\mu_L,1}\otimes \cdots \otimes B^{\mu_1,1}$ be the tensor product of crystals of type $A_n^{(1)}$ corresponding to the partition $\mu=(\mu_1,\ldots,\mu_L)$. Furthermore, let ${\lambda}\in {\mathbb{N}}^{n+1}$ be a composition. Then $$\label{super sc} {\overline{S}}(B,{\lambda})=\sum_\nu \prod_{\substack{1\le a\le n\\ 1\le i\le \mu_1}} {\genfrac{[}{]}{0pt}{}{\nu_i^{(a+1)}-\nu_{i+1}^{(a+1)}}{\nu_i^{(a)}-\nu_{i+1}^{(a+1)}}}$$ where the sum is over all sequences of partitions $\nu=(\nu^{(1)},\ldots,\nu^{(n)})$ such that $$\begin{split} &\emptyset=\nu^{(0)}\subset \nu^{(1)}\subset \cdots \subset \nu^{(n+1)} =\mu^t\\ &\text{$\nu^{(a)}/\nu^{(a-1)}$ is a horizontal ${\lambda}_a$-strip}. \end{split}$$ A horizontal $p$-strip is a skew shape with $p$ boxes such that each column contains at most one box. For $\mu=(1^{|{\lambda}|})$ equation reduces to the $q$-multinomial coefficient $$\label{multi} {\genfrac{[}{]}{0pt}{}{L}{{\lambda}_1,\ldots,{\lambda}_{n+1}}}=\frac{(q)_L}{(q)_{{\lambda}_1}\cdots (q)_{{\lambda}_{n+1}}} \qquad \text{if $|{\lambda}|=L$}$$ and zero otherwise. The supernomials for type $A_n^{(1)}$ for single rows were also given in [@HKKOTY; @Ki]. Let $\mu=(\mu_1,\ldots,\mu_L)$ be a partition, $B=B^{1,\mu_L}\otimes \cdots \otimes B^{1,\mu_1}$ and ${\lambda}\in{\mathbb{N}}^{n+1}$ a composition. Then $$\label{super sr} {\overline{S}}(B,{\lambda})=\sum_\nu q^{\phi(\nu)} \prod_{\substack{1\le a\le n\\ 1\le i\le \mu_1}} {\genfrac{[}{]}{0pt}{}{\nu_i^{(a+1)}-\nu_{i+1}^{(a)}}{\nu_i^{(a)}-\nu_{i+1}^{(a)}}}$$ where the sum is over all sequences of partitions $\nu=(\nu^{(1)},\ldots,\nu^{(n)})$ such that $$\begin{split} &\emptyset=\nu^{(0)}\subset \nu^{(1)}\subset \cdots \subset \nu^{(n+1)} =\mu^t\\ &\text{$|\nu^{(a)}|={\lambda}_1+\cdots+{\lambda}_a$ for all $1\le a \le n$.} \end{split}$$ Here $\phi(\nu)$ is defined as $$\phi(\nu)=\sum_{\substack{1\le a\le n\\ 1\le i\le \mu_1}} \nu_{i+1}^{(a)}(\nu_i^{(a+1)}-\nu_i^{(a)}).$$ As in the previous example this reduces to the $q$-multinomial coefficient if $\mu=(1^{|{\lambda}|})$. For type $A_1$ this formula coincides with that in [@SW]. The supernomials of type $C_n$ for single boxes can be obtained by the following arguments. Let $B=(B^{1,1})^{\otimes L}$ and ${\lambda}\in{\mathbb{Z}}^n$ with $\|{\lambda}\|:=|{\lambda}_1|+\cdots+|{\lambda}_n|\le L$. This means that there are at least ${\lambda}_i$ letters $i$ if ${\lambda}_i\ge 0$ and at least ${\lambda}_i$ letter $\overline{\imath}$ if ${\lambda}_i<0$. If $\|{\lambda}\|<L$ then there have to be $(L-\|{\lambda}\|)/2$ pairs of barred and unbarred letters in order to have weight ${\lambda}$. Hence we have $${\overline{S}}(B,{\lambda})=\sum_{\substack{\mu\in {\mathbb{N}}^n\\ 2|\mu|=L-\|{\lambda}\|}} {\genfrac{[}{]}{0pt}{}{L}{|{\lambda}_1|+\mu_1,\ldots,|{\lambda}_n|+\mu_n,\mu_1,\ldots,\mu_n}}$$ where ${\genfrac{[}{]}{0pt}{}{L}{p_1,\ldots,p_k}}$ is the $q$-multinomial as defined in . Explicit formulas also exist for level one cases for $B_n^{(1)}$, $D_n^{(1)}$ [@DJKMO] and $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $D_{n+1}^{(2)}$ [@KMOTU]. Fermionic evaluation {#sec:fermi} ==================== The derivation of fermionic evaluations of the classically- and level-restricted configuration sums and is in general much more intricate than for the hard-hexagon model. There exists a vast literature on conjectures and proofs of fermionic formulas, most of which deal with the case ${\hat{{\mathfrak{g}}}}=A_1^{(1)}$ or $A_n^{(1)}$. A relatively complete list of references can be found in [@HKOTY]. Fermionic formulas for all untwisted quantum affine algebras were recently conjectured in [@HKOTY]. For type $A_n^{(1)}$ these are proven for the classically-restricted case in [@KSS] and the level-restricted case in [@SSa]. For type $C_n^{(1)}$ the classically-restricted formulas are proven in [@OSS]. We will present these results here and also derive the fermionic level-restricted formulas for type $C_n^{(1)}$ in section \[sec:lev C\]. Interestingly, Kirillov and Reshetikhin [@KR] conjectured that the coefficients of the decomposition of the representations of $U'_q({\hat{{\mathfrak{g}}}})$ naturally associated with multiples of the fundamental weights into direct sums of irreducible representations of $U_q({\mathfrak{g}})$ are given by the fermionic formulas at $q=1$. Chari [@Chari] proved this conjecture for a single tensor factor for ${\mathfrak{g}}$ a simple Lie algebra of classical type and also for some exceptional cases. Classically-restricted case {#sec:classic} --------------------------- We will state here the fermionic formulas conjectured in [@HKOTY]. Let ${\hat{{\mathfrak{g}}}}=X_n^{(1)}$ with $X=A,B,C,D$ or $E$ for $n=6,7,8$ or $F$ for $n=4$ or $G$ for $n=2$. Let $B=\bigotimes_{a=1}^n \bigotimes_{i\ge 1} (B^{a,i})^{\otimes L_i^{(a)}}$ where $L_i^{(a)}\in{\mathbb{N}}$ for all $i\ge 1$ and $1\le a\le n$ and only finitely many $L_i^{(a)}$ are nonzero. Define the following polynomial in $q$ depending on $B$ and a dominant weight ${\Lambda}$ $$\label{fermi kostka} {\overline{F}}(B,{\Lambda})=\sum_{\{m\}} q^{{cc}(\{m\})} \prod_{a=1}^n \prod_{i\ge 1} {\genfrac{[}{]}{0pt}{}{m_i^{(a)}+p_i^{(a)}}{m_i^{(a)}}}$$ where the sum is over all $\{m_i^{(a)}\in{\mathbb{N}}\mid 1\le a\le n,\; i\ge 1\}$ subject to the constraints $$\label{m restr} \sum_{a=1}^n \sum_{i\ge 1} i m_i^{(a)} \alpha_a = \sum_{a=1}^n \sum_{i\ge 1} iL_i^{(a)} {\Lambda}_a - {\Lambda}.$$ The variables $p_i^{(a)}$ and the exponent ${cc}(\{m\})$ are defined as $$\begin{aligned} \label{vac} p_i^{(a)} &= \sum_{j\ge 1} L_j^{(a)} \min(i,j) - \sum_{b=1}^n (\alpha_a|\alpha_b) \sum_{k\ge 1} \min(t_b i,t_a k) m_k^{(b)}\\ \label{cc} {cc}(\{m\}) &= \frac{1}{2} \sum_{a,b=1}^n (\alpha_a|\alpha_b) \sum_{j,k\ge 1} \min(t_b j,t_a k) m_j^{(a)} m_k^{(b)}\end{aligned}$$ where $t_a=\frac{2}{(\alpha_a\mid\alpha_a)}$. Recall that $(\cdot\mid\cdot)$ is normalized such that $(\alpha_a\mid\alpha_a)=2$ if $\alpha_a$ is a long root. Then it was conjectured [@HKOTY] that $${\overline{X}}(B,{\Lambda})={\overline{F}}(B,{\Lambda}).$$ For type $A_n^{(1)}$ and general $B$ this is proven in [@KSS] and for type $C_n^{(1)}$ and $B=\bigotimes_{a=1}^n (B^{a,1})^{\otimes L_1^{(a)}}$ a proof is given in [@OSS]. Parts of the proofs given in [@KSS; @OSS] are quite technical, but we would like to highlight the general ideas of the proof which also give more insight into the fermionic formulas. Rigged configurations --------------------- Rigged configurations provide a combinatorial interpretation of the fermionic formula . We will focus first on type $A_n^{(1)}$. The sum over the variables $\{m_i^{(a)}\}$ in subject to the restriction can be interpreted as follows. Let $\nu=(\nu^{(1)},\ldots,\nu^{(n)})$ be a sequence of partitions with constraints on their sizes given by $$\label{nu restr} |\nu^{(a)}|=-\sum_{j=1}^a {\lambda}_j+\sum_{i\ge 1}\sum_{b=1}^n i\; L_i^{(b)} \min(a,b)$$ where ${\lambda}$ is the partition corresponding to the weight ${\Lambda}$. Then and are equivalent provided that $m_i^{(a)}$ is interpreted as the number of parts of $\nu^{(a)}$ of size $i$. In terms of $\nu$ the definitions and read $$\begin{aligned} P_i^{(a)}(\nu)&=Q_i(\nu^{(a-1)})-2Q_i(\nu^{(a)})+Q_i(\nu^{(a+1)}) +\sum_{j\ge 1} L_j^{(a)} Q_i(j)\\ cc(\nu)&=\sum_{a=1}^n \sum_{i\ge 1} \alpha_i^{(a)}(\alpha_i^{(a)} -\alpha_i^{(a+1)})\end{aligned}$$ where $Q_i(\mu)$ is the number of boxes in the first $i$ columns of the partition $\mu$, $\alpha_i^{(a)}$ is the size of the $i$-th column of $\nu^{(a)}$ and $p_i^{(a)}=P_i^{(a)}(\nu)$. To interpret combinatorially one uses the fact the $q$-binomial coefficient ${{\textstyle\genfrac{[}{]}{0pt}{}{m+p}{m}}}$ is the generating function of partitions in a box of size $m\times p$. More precisely, these are the partitions with at most $m$ parts each not exceeding $p$. Hence can be restated as $${\overline{F}}(B,{\Lambda})=\sum_{(\nu,J)\in{\mathrm{RC}}(B,{\Lambda})} q^{{cc}(\nu,J)}$$ where ${\mathrm{RC}}(B,{\Lambda})$ is the set of all $(\nu,J)$ where $\nu$ is a sequence of partitions satisfying and $J$ is a double sequence of partitions $$J=\{J^{(a,i)}\}_{\substack{1\le a\le n\\ i\ge 1}}.$$ The partition $J^{(a,i)}$ has to fit in a box of size $m_i^{(a)}(\nu) \times P_i^{(a)}(\nu)$ where $m_i^{(a)}(\nu)=m_i^{(a)}$ is the number of parts of size $i$ in $\nu^{(a)}$. In particular this requires that $P_i^{(a)}(\nu)\ge 0$ for all $i\ge 1$ and $1\le a\le n$. The exponent is defined as $${cc}(\nu,J)={cc}(\nu)+\sum_{a=1}^n\sum_{i\ge 1} |J^{(a,i)}|.$$ The elements in ${\mathrm{RC}}(B,{\Lambda})$ are called rigged configurations. Originally, rigged configurations were introduced in papers by Kerov, Kirillov and Reshetikhin [@KKR; @KR] in their study of the XXX model using Bethe Ansatz techniques. Rigged configurations index the solutions to the Bethe Ansatz equations. In [@KSS] the fermionic formula was proven by showing that there is a statistic preserving bijection between classically restricted paths and rigged configurations. It should be noted that for type $A_n^{(1)}$ the intrinsic energy is equal to the energy, ${\overline{D}}_B={\overline{E}}_B$. [@KSS]\[thm:kss\] For $B=\bigotimes_{a=1}^n \bigotimes_{i\ge 1} (B^{a,i})^{\otimes L_i^{(a)}}$ a crystal of type $A_n^{(1)}$, there is a bijection $\phi:{\mathcal{P}}'(B,{\Lambda})\to {\mathrm{RC}}(B,{\Lambda})$ such that for $b\in {\mathcal{P}}'(B,{\Lambda})$ we have ${\overline{E}}_B(b)={cc}(\theta\circ\phi(b))$. Here $\theta:{\mathrm{RC}}(B,{\Lambda})\to{\mathrm{RC}}(B,{\Lambda})$ maps $(\nu,J)$ to $(\nu,\tilde{J})$ where $\tilde{J}$ is obtained from $J$ by complementing each $J^{(a,i)}$ in the box of dimensions $m_i^{(a)}(\nu)\times P_i^{(a)}(\nu)$. The bijection $\phi$ is given explicitly in [@KSS], [@OSS Section 5.4]. The fermionic formula for type $C_n^{(1)}$ crystals of the form $B_C=\bigotimes_{a=1}^n (B_C^{a,1})^{\otimes L_1^{(a)}}$ was proven in [@OSS] using an embedding of type $C_n^{(1)}$ crystals into type $A_{2n-1}^{(1)}$ crystals. Let $$\Psi(B_C^{r,1}) = \begin{cases} B_A^{2n-r,1} \otimes B_A^{r,1} & \text{if $1\le r<n$}\\ B_A^{n,2} & \text{if $r=n$.} \end{cases}$$ Baker [@B] showed that there is an embedding $\Psi^{r,1}:B_C^{r,1}\to \Psi(B_C^{r,1})$. It can be defined by requiring that the $U_q(C_n)$-highest weight vector $u_C^{r,1}$ in $B_C^{r,1}$ is mapped to $u_A^{2n-r,1}\otimes u_A^{r,1}$ where $u_A^{r,s}$ is the $U_q(A_{2n-1})$-highest weight vector in $B_A^{r,s}$, and $$\begin{aligned} \Psi^{r,1} \circ f_i^C &= f_{2n-i}^A \circ f_i^A \circ \Psi^{r,1}\\ \Psi^{r,1} \circ e_i^C &= e_{2n-i}^A \circ e_i^A \circ \Psi^{r,1}.\end{aligned}$$ For a tensor product, define $\Psi_L:B_C^{r_L,1}\otimes\cdots\otimes B_C^{r_1,1}\to \Psi(B_C^{r_L,1})\otimes\cdots\otimes\Psi(B_C^{r_1,1})$ by $\Psi_L=\Psi^{r_L,1}\otimes\cdots\otimes\Psi^{r_1,1}$. [@OSS]\[thm:emb\] Let $B_C=B_C^{r_L,1}\otimes\cdots\otimes B_C^{r_1,1}$. The image ${\mathrm{Im}}(\phi\circ\Psi_L)$ of $\phi\circ\Psi_L: {\mathcal{P}}'(B_C,\cdot) \to {\mathrm{RC}}(\Psi(B_C),\cdot)$ is characterized by the set of rigged configurations $(\nu,J)$ satisfying: 1. $(\nu,J)^{(k)}=(\nu,J)^{(2n-k)}$. 2. All parts of $\nu^{(n)}$ are even. 3. All riggings in $(\nu,J)^{(n)}$ are even. This characterization of the image of $\phi\circ\Psi_L$ suggests the following definition of type $C$ rigged configurations. Let ${\lambda}$ be a partition and $B_C=\bigotimes_{a=1}^n (B_C^{a,1})^{\otimes L_1^{(a)}}$, and let $\nu=(\nu^{(1)},\ldots,\nu^{(n)})$ be a sequence of partitions with the properties $$\label{nu C} \begin{split} &|\nu^{(a)}| = -\sum_{j=1}^a {\lambda}_j + \sum_{b=1}^n L_1^{(b)} \min(a,b) \qquad \text{for $1\le a\le n$}\\ &\text{$\nu^{(n)}$ has only even parts.} \end{split}$$ Define the vacancy numbers as $$\label{P C} \begin{split} P_i^{(a)}(\nu) &= Q_i(\nu^{(a-1)}) - 2Q_i(\nu^{(a)}) + Q_i(\nu^{(a+1)}) + L_1^{(a)} \quad \text{for $1\le a<n$,} \\ P_i^{(n)}(\nu) &= Q_i(\nu^{(n-1)}) - Q_i(\nu^{(n)}) + \frac{1}{2} L_1^{(n)} Q_i(2). \end{split}$$ The set of rigged configurations of type $C$ corresponding to a weight $\Lambda$ with associated partition ${\lambda}$ and crystal $B_C$, denoted by ${\mathrm{RC}}_C(B_C,{\Lambda})$, is given by $(\nu,J)$ where $\nu$ is a sequence of partitions satisfying and $J$ is a double sequence of partitions $$J=\{J^{(a,i)}\}_{\substack{1\le a\le n\\ i\ge 1}}$$ where $J^{(a,i)}$ is a partition in a box of size $m_i^{(a)}(\nu) \times P_i^{(a)}(\nu)$ with $P_i^{(a)}(\nu)$ as in and $m_i^{(a)}(\nu)$ the number of parts of $\nu^{(a)}$. It is shown in [@OSS] that ${\overline{D}}_A\circ \Psi_L=2{\overline{D}}_C$. Hence using Theorem \[thm:emb\] the statistics of type $C$ rigged configurations becomes $$\begin{split} {cc}_C(\nu,J) &= {cc}_C(\nu)+\sum_{a=1}^n\sum_{i\ge 1} |J^{(a,i)}| \\ \text{where}\quad {cc}_C(\nu) &= \sum_{i\ge 1} \Bigl( \sum_{a=1}^{n-1} \alpha_i^{(a)}(\alpha_i^{(a)}-\alpha_i^{(a+1)}) +\frac{1}{2} \alpha_i^{(n) \;2} \Bigr) \end{split}$$ which implies that $${\overline{X}}(B_C,\Lambda) = \sum_{(\nu,J)\in{\mathrm{RC}}_C(B_C,\Lambda)} q^{{cc}_C(\nu,J)}.$$ It is also not so hard to show that $${\overline{F}}(B_C,\Lambda) = \sum_{(\nu,J)\in{\mathrm{RC}}_C(B_C,\Lambda)} q^{{cc}_C(\nu,J)}$$ by identifying $$\begin{split} p_i^{(a)}&=\begin{cases} P_i^{(a)}(\nu) & \text{for $1\le a<n$}\\ P_{2i}^{(n)}(\nu) & \text{for $a=n$} \end{cases}\\ m_i^{(a)}&=\begin{cases} m_i(\nu^{(a)}) & \text{for $1\le a<n$}\\ m_{2i}(\nu^{(n)}) & \text{for $a=n$.} \end{cases} \end{split}$$ This proves that ${\overline{X}}(B_C,\Lambda)={\overline{F}}(B_C,\Lambda)$. Level-restricted case --------------------- Fermionic formulas for the level-restricted one-dimensional configuration sums ${\overline{X}}^\ell(B,\Lambda)$ were conjectured in [@HKOTY] for all ${\hat{{\mathfrak{g}}}}$ as in section \[sec:classic\] and special weight $\Lambda=0$. Let $\ell\in{\mathbb{N}}$ and define the following polynomial in $q$ depending on $B$ $$\label{fermi lev kostka} {\overline{F}}^\ell(B)=\sum_{\{m\}} q^{{cc}^\ell(\{m\})} \prod_{(a,i)\in H^\ell} {\genfrac{[}{]}{0pt}{}{m_i^{(a)}+p_i^{(a)}}{m_i^{(a)}}}$$ where $H^\ell=\{(a,i)\mid 1\le a\le n, 1\le j\le t_a\ell\}$ and the sum is over all $\{m_i^{(a)}\in{\mathbb{N}}\mid (a,i)\in H^\ell\}$ subject to the constraints $$\label{m lev restr} \sum_{(a,i)\in H^\ell} i\; m_i^{(a)} \alpha_a = \sum_{(a,i)\in H^\ell} i\; L_i^{(a)} {\Lambda}_a.$$ The variables $p_i^{(a)}$ and the exponent ${cc}^\ell(\{m\})$ are defined as $$\begin{aligned} \label{p ell} p_i^{(a)} &= \sum_{j=1}^{t_a\ell} L_j^{(a)} \min(i,j) - \sum_{(b,k)\in H^\ell} (\alpha_a|\alpha_b) \min(t_b i,t_a k) m_k^{(b)}\\ \label{cc ell} {cc}^\ell(\{m\}) &= \frac{1}{2} \sum_{(a,j),(b,k)\in H^\ell} (\alpha_a|\alpha_b) \min(t_b j,t_a k) m_j^{(a)} m_k^{(b)}.\end{aligned}$$ Then it was conjectured [@HKOTY] that $$\label{conj level} {\overline{X}}^\ell(B,0)={\overline{F}}^\ell(B).$$ Level-restricted case: type $A_n^{(1)}$ --------------------------------------- For type $A_n^{(1)}$ the conjecture and its generalization to arbitrary weights $\Lambda$ was proven in [@SSa]. These formulas can again be understood in terms of rigged configurations. We will explain this here since it will enable us to derive the level-restricted fermionic formulas for type $C_n^{(1)}$ in the next section. Let ${\lambda}=({\lambda}_1,\ldots,{\lambda}_{n+1})$ be the partition corresponding to the dominant integral weight $\Lambda$. A partition ${\lambda}$ is restricted of level $\ell$ if ${\lambda}_1-{\lambda}_{n+1}\le \ell$. Define ${\tilde{\ell}}= \ell-({\lambda}_1-{\lambda}_{n+1})$, which is nonnegative by assumption. Set ${\lambda}'=({\lambda}_1-{\lambda}_{n+1},\ldots,{\lambda}_n-{\lambda}_{n+1})^t$ (where ${}^t$ stands for transpose) and denote the set of all column-strict tableaux of shape ${\lambda}'$ over the alphabet $\{1,2,\ldots,{\lambda}_1-{\lambda}_{n+1}\}$ by ${\mathrm{CST}}({\lambda}')$. Define a table of modified vacancy numbers depending on a sequence of partitions $\nu$ and $t\in{\mathrm{CST}}({\lambda}')$ by $$\label{t vacancy} P_i^{(a)}(\nu,t) = P_i^{(a)}(\nu) - \sum_{j=1}^{{\lambda}_a-{\lambda}_{n+1}} \chi(i\ge{\tilde{\ell}}+t_{j,a}) + \sum_{j=1}^{{\lambda}_{a+1}-{\lambda}_{n+1}} \chi(i\ge{\tilde{\ell}}+t_{j,a+1})$$ where $t_{j,a}$ denotes the entry in $t$ in row $j$ and column $a$ and $\chi(S)=1$ if the statement $S$ is true and $\chi(S)=0$ otherwise. \[def:levrc\] Let $B=\bigotimes_{a=1}^n \bigotimes_{i=1}^\ell (B^{a,i})^{\otimes L_i^{(a)}}$ be a crystal of type $A_n^{(1)}$ and ${\Lambda}$ a dominant integral weight with corresponding partition ${\lambda}$. Say that $(\nu,J)\in{\mathrm{RC}}(B,{\Lambda})$ is restricted of level $\ell$ provided that 1. $\nu_1^{(a)} \le \ell$ for all $a$. 2. There exists a tableau $t\in{\mathrm{CST}}({\lambda}')$, such that for every $i,a\ge 1$, the largest part of $J^{(a,i)}$ does not exceed $P_i^{(a)}(\nu,t)$. Denote by ${\mathrm{RC}}^\ell(B,{\Lambda})$ the set of $(\nu,J)\in{\mathrm{RC}}(B,{\Lambda})$ that are restricted of level $\ell$. Note in particular that the second condition requires that $P_i^{(a)}(\nu,t)\ge 0$ for all $i,a\ge 1$. \[ex:vacuum\] Let ${\lambda}=(m^{n+1})$ be rectangular with $n+1$ rows. Then ${\lambda}'=\emptyset$ and $P_i^{(a)}(\nu,\emptyset)=P_i^{(a)}(\nu)$ for all $i,a\ge 1$ so that the modified vacancy numbers are equal to the vacancy numbers. [@SSa Theorem 8.2] \[thm:lev image A\] The bijection $\phi:{\mathcal{P}}'(B,{\Lambda})\to {\mathrm{RC}}(B,{\Lambda})$ restricts to a well-defined bijection $\phi:{\mathcal{P}}^\ell(B,{\Lambda})\to {\mathrm{RC}}^\ell(B,{\Lambda})$. Since ${\overline{E}}_B={cc}\circ\theta\circ\phi$ by Theorem \[thm:kss\] it follows from Theorem \[thm:lev image A\] that for type $A_n^{(1)}$ $$\label{lev comb A} {\overline{X}}^\ell(B,{\Lambda})=\sum_{(\nu,J)\in {\mathrm{RC}}^\ell(B,{\Lambda})} q^{{cc}(\theta(\nu,J))}.$$ Note that ${cc}(\theta(\nu,J))={cc}(\nu)+\sum_{a=1}^n \sum_{i=1}^\ell P_i^{(a)}(\nu)m_i^{(a)}(\nu)-\sum_{a=1}^n \sum_{i=1}^\ell |J^{(a,i)}|$. Let ${\mathrm{SCST}}({\lambda}')$ be the set of all nonempty subsets of ${\mathrm{CST}}({\lambda}')$. Since the $q$-binomial ${{\textstyle\genfrac{[}{]}{0pt}{}{m+p}{m}}}$ is the generating function of partitions with at most $m$ parts each not exceeding $p$ it follows by inclusion-exclusion that $$\label{lev type A} {\overline{X}}^\ell(B,{\Lambda})=\sum_{S\in{\mathrm{SCST}}({\lambda}')} (-1)^{|S|+1} \sum_{\{m\}} q^{{c}^\ell(\{m\})} \prod_{(a,i)\in H^\ell} {\genfrac{[}{]}{0pt}{}{m_i^{(a)}+p_i^{(a)}(S)}{m_i^{(a)}}}_{1/q}$$ where the sum is over all $\{m_i^{(a)}\in{\mathbb{N}}\mid (a,i)\in H^\ell\}$ subject to the constraints $$\sum_{(a,i)\in H^\ell} i m_i^{(a)} \alpha_a = \sum_{(a,i)\in H^\ell} i L_i^{(a)} {\Lambda}_a-{\Lambda}.$$ Also $p_i^{(a)}(S)$ and ${c}^\ell(\{m\})$ are defined as $$\begin{aligned} p_i^{(a)}(S)&=p_i^{(a)}+\min_{t\in S}\Bigl\{ - \sum_{j=1}^{{\lambda}_a-{\lambda}_{n+1}} \chi(i\ge{\tilde{\ell}}+t_{j,a}) + \sum_{j=1}^{{\lambda}_{a+1}-{\lambda}_{n+1}} \chi(i\ge{\tilde{\ell}}+t_{j,a+1})\Bigr\}\\ {c}^\ell(\{m\})&={cc}^\ell(\{m\})+\sum_{(a,i)\in H^\ell} p_i^{(a)} m_i^{(a)}\end{aligned}$$ with $p_i^{(a)}$ as in and ${cc}^\ell(\{m\})$ as in . ${\genfrac{[}{]}{0pt}{}{m+p}{m}}_{1/q}$ is the $q$-binomial with $q$ replaced by $1/q$. In particular if ${\lambda}=(m^{n+1})$ as in Example \[ex:vacuum\] so that $\Lambda=0$ and $p_i^{(a)}(\nu,\{\emptyset\}) =p_i^{(a)}$ the fermionic form reduces to since ${\genfrac{[}{]}{0pt}{}{m+p}{m}}_{1/q}=q^{-mp}{\genfrac{[}{]}{0pt}{}{m+p}{m}}$. Further details can be found in [@SSa]. It should be remarked that even though contains explicit signs, it is clear from the equivalent combinatorial formula that it is a nonnegative polynomial in $q$. Level-restricted case: type $C_n^{(1)}$ {#sec:lev C} --------------------------------------- In this section we will show how the level-restricted fermionic formulas for type $C_n^{(1)}$ can be obtained from Theorems \[thm:lev image A\] and \[thm:emb\]. Under the embedding $\Psi$ a dominant weight $\Lambda_C=\sum_{k=1}^m \Lambda_{i_k}^C$ of type $C_n^{(1)}$ becomes the weight $\Lambda_A=\sum_{k=1}^m (\Lambda_{i_k}^A+\Lambda_{2n-i_k}^A)$ of type $A_{2n-1}^{(1)}$ where all $1\le i_k\le n$. In terms of the corresponding partitions ${\lambda}^A$ and ${\lambda}^C$ this implies that $$\begin{aligned} {\lambda}_a^A-{\lambda}_{2n}^A&={\lambda}_1^C+{\lambda}_a^C && \text{for $1\le a\le n$}\\ {\lambda}_a^A-{\lambda}_{2n}^A&={\lambda}_1^C-{\lambda}_{2n+1-a}^C && \text{for $n< a\le 2n$.}\end{aligned}$$ Let $B_C=\bigotimes_{a=1}^n (B_C^{a,1})^{\otimes L_1^{(a)}}$. Hence, under the embedding $\Psi_L:B_C\to \Psi(B_C)$, the conditions for level-restriction for rigged configurations of type $A$ as given in Definition \[def:levrc\] become the following. For a partition ${\lambda}^C$ define $({\lambda}^C)'=(2{\lambda}_1^C,{\lambda}_1^C+{\lambda}_2^C, \ldots,{\lambda}_1^C+{\lambda}_n^C,{\lambda}_1^C-{\lambda}_n^C,{\lambda}_1^C-{\lambda}_{n-1}^C,\ldots, {\lambda}_1^C-{\lambda}_1^C)^t$. Let ${\mathrm{CST}}(({\lambda}^C)')$ be the set of all semi-standard tableaux of shape $({\lambda}^C)'$ in the alphabet $\{1,2,\ldots,2{\lambda}_1^C\}$. For $t\in {\mathrm{CST}}(({\lambda}^C)')$ set $$f_i^{(a)}(t)= \begin{cases} -\sum_{j=1}^{{\lambda}_1^C+{\lambda}_a^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,a})&\\ \quad+\sum_{j=1}^{{\lambda}_1^C+{\lambda}_{a+1}^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,a+1}) &\text{for $1\le a<n$}\\ -\sum_{j=1}^{{\lambda}_1^C+{\lambda}_n^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,n})&\\ \quad+\sum_{j=1}^{{\lambda}_1^C-{\lambda}_n^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,n+1}) &\text{for $a=n$}\\ -\sum_{j=1}^{{\lambda}_1^C-{\lambda}_{2n+1-a}^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,a})&\\ \quad+\sum_{j=1}^{{\lambda}_1^C-{\lambda}_{2n-a}^C} \chi(i\ge 2\ell-2{\lambda}_1^C+t_{j,a+1}) &\text{for $n<a<2n$.} \end{cases}$$ Define modified vacancy numbers as $$\label{P C lev} P_i^{(a)}(\nu,t)=P_i^{(a)}(\nu)+\begin{cases} \min\{f_i^{(a)}(t),f_i^{(2n-a)}(t)\} & \text{for $1\le a<n$}\\ \frac{1}{2}f_i^{(n)}(t) & \text{for $a=n$} \end{cases}$$ with $P_i^{(a)}(\nu)$ as defined in . Let ${\Lambda}_C$ be a dominant weight and ${\lambda}^C$ the corresponding partition. Let $B_C=\bigotimes (B_C^{a,1})^{\otimes L_1^{(a)}}$ be a crystal of type $C_n^{(1)}$. Say that $(\nu,J)\in{\mathrm{RC}}_C(B_C,{\Lambda}_C)$ is restricted of level $\ell$ provided that 1. $\nu_1^{(a)} \le 2\ell$ for all $a$. 2. There exists a tableau $t\in{\mathrm{CST}}(({\lambda}^C)')$, such that for every $i,a\ge 1$, the largest part of $J^{(a,i)}$ does not exceed $P_i^{(a)}(\nu,t)$ defined in . Denote by ${\mathrm{RC}}_C^\ell(B_C,{\Lambda}_C)$ the set of $(\nu,J)\in{\mathrm{RC}}_C(B_C,{\Lambda}_C)$ that are restricted of level $\ell$. It follows that $${\overline{X}}^\ell(B_C,{\Lambda}_C)=\sum_{(\nu,J)\in {\mathrm{RC}}_C^\ell(B_C,{\Lambda}_C)} q^{{cc}_C(\theta(\nu,J))}.$$ Let ${\mathrm{SCST}}(({\lambda}^C)')$ be the set of all nonempty subsets of ${\mathrm{CST}}(({\lambda}^C)')$. By the same arguments as in the type $A$ we find $$\label{lev type C} {\overline{X}}^\ell(B_C,{\Lambda}_C)=\sum_{S\in{\mathrm{SCST}}(({\lambda}^C)')} (-1)^{|S|+1} \sum_{\{m\}} q^{{c}^\ell(\{m\})} \prod_{(a,i)\in H^\ell} {\genfrac{[}{]}{0pt}{}{m_i^{(a)}+p_i^{(a)}(S)}{m_i^{(a)}}}_{1/q}$$ where the sum is over all $\{m_i^{(a)}\in{\mathbb{N}}\mid (a,i)\in H^\ell\}$ subject to the constraints $$\sum_{(a,i)\in H^\ell} i m_i^{(a)} \alpha_a = \sum_{(a,1)\in H^\ell} L_1^{(a)} {\Lambda}_a^C-{\Lambda}_C.$$ The variable $p_i^{(a)}(S)$ is defined as $$\begin{aligned} p_i^{(a)}(S)&=p_i^{(a)}+\begin{cases} \min_{t\in S}\{f_i^{(a)}(t),f_i^{(2n-a)}(t)\} & \text{for $1\le a<n$}\\ \frac{1}{2}\min_{t\in S}\{f_i^{(n)}(t)\} & \text{for $a=n$} \end{cases}\\ {c}^\ell(\{m\})&={cc}^\ell(\{m\})+\sum_{(a,i)\in H^\ell} p_i^{(a)} m_i^{(a)}\end{aligned}$$ with $p_i^{(a)}$ as in and ${cc}^\ell(\{m\})$ as in . In particular if ${\lambda}^C=\emptyset$ so that $\Lambda_C=0$ the fermionic form reduces to . Summary and open problems {#sec:open} ========================= Equating the bosonic and fermionic evaluations for ${\overline{X}}(B,{\Lambda})$ and ${\overline{X}}^\ell(B,{\Lambda})$ as given in sections \[sec:bose\] and \[sec:fermi\] yields polynomial identities in $q$. One may take limits of these identities in various ways to obtain $q$-series identities. We refer the interested reader to [@HKKOTY; @HKOTY; @SSa; @SW] for details. It is still an outstanding problem to prove the conjectured fermionic formulas and for general ${\hat{{\mathfrak{g}}}}$. There is strong evidence that the rigged configuration approach will still be applicable in this case. Part of this program also requires the proof of the existence and structure of the crystals $B^{r,s}$ for general ${\hat{{\mathfrak{g}}}}$ as mentioned in Conjecture \[conj:Brs\]. The astute reader will have noticed that the energy function introduced in section \[sec:crystal\] does not actually specialize to the statistics on path of section \[sec:HHM\] which yield the Rogers–Ramanujan identities. This is due to the fact that the Rogers–Ramanujan identities correspond to a non-unitary physical model (more precisely the minimal model $M(2,5)$) whereas the crystal base theory introduced in section \[sec:crystal\] is related to unitary physical models. To really embed the Rogers–Ramanujan identities into the framework of crystal base theory it is hence necessary to generalize the definition of the (intrinsic) energy function of section \[sec:crystal\] to include the non-unitary setting. For ${\hat{{\mathfrak{g}}}}=\widehat{\mathfrak{sl}}_2$ results are available [@FB; @BMS; @FW; @FLPW]. However for general ${\hat{{\mathfrak{g}}}}$ details are not yet known in general. Some results in this direction can be found in [@N]. [99]{} T. Akasaka and M. 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For the examples of type $A_n^{(1)}$ with $B_i=B^{r_i,s_i}$ and $C_n^{(1)}$ with $B_i=B^{r_i,1}$, for which we will consider fermionic formulas in the next section, these conditions are all satisfied.

--- abstract: 'Post-quantum cryptography studies the security of classical, i.e. non-quantum cryptographic protocols against quantum attacks. Until recently, the considered adversaries were assumed to use quantum computers and behave like classical adversaries otherwise. A more conservative approach is to assume that also the communication between the honest parties and the adversary is (partly) quantum. We discuss several options to define secure encryption and authentication against these stronger adversaries who can carry out *superposition attacks*. We re-prove a recent result of Boneh and Zhandry, stating that a uniformly random function (and hence also a quantum-secure pseudorandom function) can serve as a message-authentication code which is secure, even if the adversary can evaluate this function in superposition.' author: - Maria Velema bibliography: - 'thesisbMVelema.bib' date: 'July 12, 2013' title: '[Classical Encryption and Authentication under Quantum Attacks]{}' ---

--- abstract: 'Manufacturing requires consistent production rate and task success for sustainable operation. Some manufacturing tasks require a semi-autonomous approach, exploiting the combination of human adaptability and machine precision and speed, to be cost effective. The main contribution of this paper is a new approach to determine the level of autonomy for human-machine shared control based on the automation uncertainty. Moreover, the haptic feedback is scaled by the level of autonomy to indicate machine confidence to the operator. Experimentation results, with a human-robot peg-in-a-hole testbed, show more than 5 times improvement in the error tolerance for task completion with the shared control approach when compared to a purely autonomous method.' author: - 'Parker Owan, Joseph Garbini, and Santosh Devasia' bibliography: - 'IEEEabrv.bib' - 'hmi.bib' title: '**Uncertainty-based Arbitration of Human-Machine Shared Control**' --- Introduction ============ Fully autonomous systems are becoming increasingly prevalent in manufacturing facilities throughout the world, some of which operate with minimal human supervision [@Null2003]. Some manufacturing operations such as aircraft production have been unable to adopt a fully autonomous policy because, according to Felder [@Felder2011]: (i) new aircraft design tends to be a modification of previous designs where autonomous production was not a primary design consideration; (ii) the deliverable product is larger than most of the machines used in its assembly; and (iii) the delivery rate is much lower than required rates of facilities that have transitioned to fully-autonomous production. In order to transition from the current state to increased autonomy in aircraft manufacturing, there is interest in semi-autonomous solutions wherein the human shares some aspect of control with the machine. Such semi-autonomous approach can allow for more cost-effective solutions to support aircraft manufacturing when compared to the fully autonomous approach. For such semi-autonomous manufacturing, there is a need to develop shared control strategies wherein both the machine and the human are simultaneously managing a task. Shared human-machine control requires an arbitration approach to select the relative amount of human and machine control. One approach is for the human (or humans) to fully guide the machine as in teleoperation [@Nudehi2005; @Khademian2012]. Alternatively, the machine might facilitate human operation, e.g., through artificial potential fields [@Khatib1986] for lane keeping or hazard avoidance [@Gerdes2001] and through virtual fixtures [@Rosenberg1993] to guide task completion [@OMalley2006]. In contrast to these human-centered or machine-centered approaches, blending of both the human and the machine input [@Yu2003; @Hansson2010] can leverage both the adaptability of the human as well as the computing power and bandwidth of the machine controller. One approach to blend the human input $\bm{q_h}$ and the machine input $\bm{q_m} $ is through an adjustable level of autonomy (LOA) $\alpha \in [0,1]$ [@Yu2003], e.g., to determine the reference input $\bm{q_{ref}}$ to the system (such as the desired position of a robot) $$\label{eqn:reference} \bm{q_{ref}} = \alpha \bm{q_m} + (1 - \alpha) \bm{q_h},$$ as represented in Fig. \[fig:sharedControl\]. As opposed to a fixed level [@You2012; @Parasuraman1997] or discrete levels of autonomy $\alpha$ [@Kortenkamp2000], this work considers the more general shared-control case where the level of autonomy $\alpha$ is allowed to *slide* continuously as in [@Desai2005; @Dias2008], which has been used in applications such as active mobility devices [@Yu2003; @Wang2014], control of multi-agent UAV systems [@Franchi2012], semi-autonomous operation of large manipulator arms [@Hansson2010], and active driver assistance systems for lane keeping [@Saleh2013]. {width="\textwidth"} The main contribution of this paper is a new approach to determine the level of autonomy for human-machine shared control based on the automation uncertainty, e.g., to reflect the level of machine confidence in the goal prediction, which can be used to determine when the automation can take over from a human [@Dragan2013]. In aerospace manufacturing operations, there can be uncertainty in the location of the obstacles and the goal location for a manufacturing operation such as drilling. If these uncertainties are substantial, then a human might need to take over the manufacturing operation. The current work proposes a method for using apriori knowledge of automation uncertainty and the probability of failure to arbitrate the level of autonomy in shared control. For example as the probability $P(\mathcal{E})$ of the failure event $\mathcal{E}$ increases, the level of autonomy (LOA) tends to zero, $\alpha \rightarrow 0$, and more control authority is relinquished to the human. Conversely, the level of autonomy becomes larger, $\alpha \rightarrow 1$, when the probability of a failure event is low, giving more control authority to the machine. Additionally, the proposed level of autonomy $\alpha$ is also used to scale the haptic feedback [@Hannaford1989; @Hogan1985] assisting the user in task completion [@Griffiths2004] to convey the level of automation confidence to the operator [@Abbink2012]. Experimentation results, with a human-robot peg-in-a-hole testbed, show more than 5 times improvement in the error tolerance for task completion with the proposed shared control approach when compared to a purely autonomous method. In Section \[sec:application\] an application is described. The control approach for the application is identified in Section \[sec:control\]. In Section \[sec:autonomySelection\] machine confidence is generalized to principal modes of uncertainty, and a policy is derived for shifting authority based on environmental uncertainty. Haptic virtual interaction forces are discussed in Section \[sec:haptics\]. In Section \[sec:results\], experimental results are shown comparing the novel shared control scheme with a purely autonomous process. ![A smooth time-based automation trajectory $\bm{q_m}$ based on manual waypoints, Bezier curves, and acceleration and velocity limits.[]{data-label="fig:machine"}](machine.pdf){width="48.00000%"} Parameter Value Units ------------------------ ------- --------- Velocity at end-points 0 m/s Max acceleration 2.0 m/s$^2$ Max velocity 0.2 m/s : Machine Trajectory \[tbl:machine\] Application and Assumptions {#sec:application} =========================== Peg in a Hole Task ------------------ A peg-in-hole task is studied in this work to illustrate the proposed uncertainty-based arbitration. A peg (representative of a drill bit) fixed to the end of a serial manipulator as shown in Fig. \[fig:task\] is to be inserted into a hole at location $\bm{q_d}$ on a planar surface. A minimum of five degrees of freedom (DOF) endpoint control are needed for this task (3 DOF in translation and 2 DOF in normal orientation to a plane). The robot used is a Kinova MICO with 6-joints, sufficient for completing the peg-in-hole task. Machine Trajectory Input ------------------------ A time-dependent machine trajectory $\bm{q_m}(t)$ is constructed offline using waypoints and Bezier curves [@Choi2014], where the final point of the trajectory (at $t = t_f$) is the nominal goal location, i.e., $\bm{q_m}(t_f) = \bm{\hat{q}_d}$. The automation trajectory remains identical for the entire experiment. Fig. \[fig:machine\] shows waypoints and the smoothed trajectory in workspace coordinates. Parameters of the acceleration profile include starting velocity, velocity limits, and acceleration limits, summarized in Table \[tbl:machine\]. Uncertainty and Human Input --------------------------- Because the machine trajectory is planned for the nominal hole location $\bm{\hat{q}_d}$, a large error $\Delta_e$ in the actual hole location $\bm{q_d}$ could result in an automation failure. While the actual location of the hole is unknown to the robot, the uncertainty in the hole location is assumed to be known as a Gaussian distribution $\bm{q_d} ~\sim \mathcal{N}(\bm{\hat{q}_d},\sigma_e^2)$, with the nominal hole location $\bm{\hat{q}_d}$ as in Fig. \[fig:task\]a, and variance of the distribution is $\sigma_e^2$. The error of the actual hole location versus the nominal hole location is $\Delta_e = \bm{q_d} - \bm{\hat{q}_d}$. To manage this error in the hole location, human input is included in the control loop. The human input $\bm{q_h}$ is received via a Force Dimension haptic interface, shown in Fig. \[fig:task\] b). The operator input $\bm{q_h}$ is combined with the machine input $\bm{q_m}(t)$ according to . Human Interface --------------- The human operator is presented with virtual interaction forces via the Force Dimension haptic display. These forces include a virtual stiffness centered around the machine trajectory as well as the obstacles in the environment. The haptic display does not include measured forces from the robot. The platform used for providing haptic rendering is the CHAI3D library, operating at 2 kHz. Real camera feedback of the actual environment at the end of the arm, shown in Fig. \[fig:task\] c), is provided to the user real-time. The hole is in the field of view (FOV) when the robot is near the goal $\bm{q_d}$. This view allows for the human to close the loop around the true goal. Virtual views of the robot in the environment are also presented to the user, similar to Fig. \[fig:task\] a). The operator interface and high-level control is handled by Virtual Robotic Experimentation Platform (V-REP) software at a workstation isolated from the operating environment of the robot. The workstation and software are shown in Fig. \[fig:task\] b). This software runs at $\sim$20 Hz, and provides a virtual simulation of the robot in the environment. V-REP also provides a positive estimated scalar distance $\hat{d}_e$ of the robot end-effector to the nearest obstacle or environmental surface, available via nominal CAD (computer aided design) data $\hat{\mathcal{U}}$. This CAD data contains the feature location of the nominal hole $\bm{\hat{q}_d}$. When the robot is near the goal, the distance $\hat{d}_e$ approximates the distance of the peg to the nominal hole $\bm{\hat{q}_d}$. Control Approach {#sec:control} ================ In human machine interaction, shared control is formulated as automation sharing control with a human. Fig. \[fig:comahi\] shows a block diagram for the control approach for this paper. In this experiment, it is assumed that the machine and human share a common goal $\bm{q_d}$. Machine error $\Delta_e$ is introduced into the system as an automation error due to environment uncertainty. To keep hardware and sensor costs practical, knowledge of the behavior of this error is used to determine when to incorporate human intervention, e.g. when an error $\Delta_e$ in the environment is high, and as such the automation cannot complete a task, human adaptability can be incorporated in the control. The blended input trajectory $\bm{q_{ref}}$ is passed through the Inverse Kinematics (IK) of the robot model, using a damped least squares method [@Wampler1986], to generate a set of desired joint angles $\bm{\theta}$. These desired joint angles $\bm{\theta}$ are then passed through a nonlinear dynamics simulation of the torque-controlled robot, as shown in Fig. \[fig:dynamics\], to generate the joint commands $\bm{\theta_{ref}}$. This simulation tends to filter high frequency commands, and provides a means to generate a set of feasible joint commands $\bm{\theta_{ref}}$. Closed-loop motor control to achieve the reference trajectory $\bm{\theta_{ref}}$ is handled by the software provided with the robot. When automation uncertainties are present in the system, the human input $\bm{q_h}$ provides closed-loop control based on visual feedback $\mathcal{V}$ to correct for errors $\Delta_e$ in the estimated goal location $\hat{\bm{q_d}}$. Shared Control -------------- The human input $\bm{q_h}$ is applied through a haptic input device. The machine input $\bm{q_m}$ is queried from the path planning module in Fig. \[fig:machine\]. An autonomy selection policy defines the level of autonomy $\alpha$. Fig. \[fig:trajectory\] shows how the reference trajectory changes as the level of autonomy (LOA) slides from a fully autonomous mode to a tele-operation mode. Autonomy Selection Policy {#sec:autonomySelection} ------------------------- Previous works [@Dragan2013] have used the confidence interval in a machine prediction task as the level of autonomy $\alpha$, e.g. $$\label{eqn:dragan} \alpha = \max \left(0, 1 - \frac{d}{D}\right),$$ where $d$ is the distance to a goal and $D$ is a threshold defining the point of $\alpha = 0$, i.e. no confidence in the machine. This concept is extended in this work to define a general autonomy selection based on uncertainty in a task. In particular, if $P(\mathcal{E})$ is the probability of a machine failure mode $\mathcal{E}$ occurring, then the level of autonomy $\alpha$ is chosen as $$\label{eqn:probAlpha} \alpha ~= 1-P(\mathcal{E}),$$ where $P(\mathcal{E})$ is the probability of an environmental machine failure occurring. As the probability of the failure event increases, $\alpha \rightarrow 0$, i.e. more control authority is relinquished to the human. Conversely, the level of autonomy $\alpha \rightarrow 1$ when the probability of failure event is low, giving more control authority to the machine. ### Failure Estimation for Arbitrating Level of Autonomy {#sec:arbitrationDerivation} Consider the 1-dimensional case of the robot tip $q_t$ approaching a goal $q_d$, with initial $q_t > q_d$. The nominal goal is located at $\hat{q}_d$. From section \[sec:application\], the probability density function (pdf) of the goal location $q_d$ is normal, i.e., $q_d \sim \mathcal{N}(\hat{q}_d,\sigma_e^2)$, and given by $$\label{eqn:envrPDF} f(q_d) = \frac{1}{\sigma_e \sqrt{2\pi}} \exp \left\{ -\frac{(q_d - \hat{q}_d)^2}{2\sigma_e^2} \right\}.$$ The complementary cumulative distribution function $\bar{F}(q)$ is $$\bar{F}(q) = \int_q^{\infty} f(s) ds = \left. -\frac{1}{2} \operatorname{erf}\left\{ \frac{\hat{q}_d - s}{\sigma_e \sqrt{2}} \right\} \right|_q^{\infty},$$ which reduces to $$\bar{F}(q) = \frac{1}{2} \left[1 + \operatorname{erf}\left\{ \frac{\hat{q}_d - q}{\sigma_e \sqrt{2}} \right\} \right].$$ Since the robot tip approaches with $\hat{q}_d > q$, the probability $P(\mathcal{E})$ of encountering the hole at $q_t$ is the complementary cumulative distribution $\bar{F}(q_t)$ $$\label{eqn:cdfTip} \begin{array}{rcl} P(\mathcal{E}) ~= \bar{F}(q_t) & = & \frac{1}{2} \left[1 + \operatorname{erf}\left\{ \frac{\hat{q}_d - q_t}{\sigma_e \sqrt{2}} \right\}\right] \\[0.5em] & = & \frac{1}{2} \left[ 1 - \operatorname{erf}\left\{ \frac{\hat{d}_e}{\sigma_e \sqrt{2}} \right\}\right] \end{array}$$ where $\hat{d}_e = q_t - \hat{q}_d$ is the measured distance of the robot tip $q_t$ from the nominal goal location $\hat{q}_d$. Note that if the goal is located directly on a geometric surface (e.g., a wall or obstacle), then the probability of encountering the goal is the probability of encountering an obstacle. From and , the level of autonomy $\alpha$ can be rewritten as $$\label{eqn:policy} \alpha = 1 - P(\mathcal{E}) = \frac{1}{2}\left[1 + \operatorname{erf}\left\{ \frac{\hat{d}_e}{\sigma_e \sqrt{2}} \right\}\right].$$ In this work, the robot approaches the goal from above (orthogonal to the surface in which the hole is located), so the probability of encountering the surface is $P(\mathcal{E})$, where the error in hole location is in the $z$ direction, i.e., $\Delta_{e_z}$. The robot will collide with the surface if the hole centerline is off by some error $\Delta_{e_x}$, so experimentally $\Delta_{e_z}$ and $\Delta_{e_x}$ must be varied to evaluate . This representation allows the level of autonomy $\alpha$ to slide toward human intervention when the likelihood of encountering the goal (and potential failure due to uncertainty in the goal location) is high. This is different from typical cases where the level of autonomy $\alpha$ shifts toward machine control when the distance to the goal $\hat{d}_e$ is small because the robot becomes confident in its prediction of the goal [@Dragan2013]. An advantage of the approach in  is that in cases with multiple independent failure modes $\mathcal{E}_i$ due to uncertainty, the selection of the level of autonomy $\alpha$ can be generalized as $$\label{eqn:prob} \alpha = 1 - \prod_i P({\mathcal{E}}_i).$$ Thus, the proposed uncertainty-based approach can be used to manage different uncertainty modes in shared human-machine control such as: (i) *environmental uncertainty* that arises when there is a change in an environment from the apriori model such as tolerance variation in an assembly from its geometric design; (ii) *method uncertainty* that occurs when automation fails to find a solution to a task such as a path planning module failing to find a trajectory; and (iii) *measurement uncertainty* arises when a sensor has substantial errors due to external noise. ![Without the filter in , a chattering effect on the level of autonomy $\alpha$ is noticeable in experimentation when the difference between human input and machine input $|\bm{q_h} - \bm{q_m}|$ is relatively large. Enabling a filter in  on the level of autonomy $\alpha$ at 9.3 seconds removes the chattering phenomenon.[]{data-label="fig:chattering"}](chattering.pdf){width="48.00000%"} ### Chattering effect Rapid switching between the human control mode and the machine control mode is possible close to the location of a potential failure. This is similar to chattering found in sliding mode control (SMC) architectures [@Young1999]. Fig. \[fig:chattering\] shows experimental data with chattering for level of autonomy $\alpha$ defined by . The chattering mode is excited in experimentation when: (i) there is a large difference between human input and machine input $|\bm{q_h} - \bm{q_m}|$ and (ii) the distance to the nominal goal location $\hat{d}_e$ is close to variance $\sigma_e$ of the uncertainty in the goal location. The chattering effect is reduced by passing changes in the level of autonomy $\alpha$ through a lowpass (first order) filter, i.e., $$\label{eqn:filtPolicy} \xi \dot{\alpha} + \alpha = \frac{1}{2}\left[1 + \operatorname{erf}\left\{ \frac{\hat{d}_e}{\sigma_e \sqrt{2}} \right\}\right],$$ where $\xi$ is the filter time constant. The filter time constant $\xi$ is chosen to be $0.08$s, which is just faster than a the human neuromuscular reflex time constant of 0.1 s [@Saleh2013]. Fig. \[fig:chattering\] presents experimental results that show that enabling the filter (at $t = 9.3$s) mitigates the chattering phenomenon. Haptics Generator {#sec:haptics} ----------------- The haptic interaction force $\bm{F_{\mathcal{H}}}$ applied to the user is the linear combination of virtual fixtures $\bm{F_f}$ for guiding the operator along the machine trajectory [@Rosenberg1993] and potential fields $\bm{F_v}$ for keeping the operator input away from collisions [@Khatib1986] $$\bm{F_{\mathcal{H}}} = \bm{F_f}(\alpha,\bm{q_h},\bm{\dot{q}_h}) + \bm{F_v}(\alpha,\bm{q_h}). \label{eqn:haptics}$$ ### Virtual Fixtures A virtual fixture comprised of variable linear stiffness $k$ and viscous damping $b$ is incorporated into the system. The virtual fixture force $\bm{F_f}$ applied to the user is $$\bm{F_f} = -k(\bm{q_h} - \bm{q_m}) - b \bm{\dot{q}_h},$$ where $\bm{q_h}$ is the input from the human, and $\bm{q_m}$ is the input from the machine. Griffiths and Gillespie’s premise of haptically conveying machine confidence [@Griffiths2005] is accomplished by varying stiffness $k$ as a function of level of autonomy $\alpha$ $$\label{eqn:stiffness} k = \alpha(k_{max}-k_{min})+k_{min},$$ where $k_{min}$ and $k_{max}$ are the minimum and maximum stiffness values imposed on the system, respectively. Using the arbitration factor $\alpha$ to scale stiffness resolves trade-off issues associated with parameter selection for virtual fixtures. In cases of high certainty ($\alpha \rightarrow 1$), the virtual fixture is strongly imposed on the user. Conversely, in cases of low certainty ($\alpha \rightarrow 0$), the virtual fixture is weakly imposed. ### Selection of Haptics Parameters The haptic feedback gains $k_{min}$, $k_{max}$, and $b$ are selected experimentally as in [@Marayong2004]. A user familiar with operating the haptic interface moves a cursor to a visual goal constrained by different haptic gains. These gains are held constant for the duration of each test, and parameters are evaluated for a no goal error case and a large goal error case, similar to Marayong and Okamura’s method. The settling time for different selections of stiffness parameters $k_{max}$, $k_{min}$, and damping $b$ is shown in Fig. \[fig:parameters\]. ![Settling time response characteristics are shown for parameter variations in haptic time constant $\tau = b/k$ and stiffness $k$ for a) cases with no goal error, and b) cases with a large goal error $\Delta$ of 30 mm. Error bars show standard deviation for a minimum of 10 runs.[]{data-label="fig:parameters"}](parameters.pdf){width="48.00000%"} The parameter $k_{min}=10$N/m was selected to minimize the settling time $T_s$ for a large goal error $\Delta = 30$mm, and the parameters $k_{max}=75$ N/m and $\tau = b/k_{max} =0.1$s were selected to minimize the settling time $T_s$ with no goal error $\Delta = 0$ mm as seen from Fig. \[fig:parameters\]. This results in a constant damping $b= \tau k_{max} = 7.5$ N-m/s. ### Potential Fields Potential fields $\bm{F_v}$ are also implemented as obstacles in the haptic workspace with a variable stiffness $k_v$ normal to the obstacle mesh plane $\bm{q_{obs}}$, such that $$\bm{F_v} = -k_v(\bm{q_h} - \bm{q_{obs}}),$$ and $k_v$ is generated by $$\label{eqn:stiffness2} k = \alpha(k_{v_{max}}-k_{v_{min}})+k_{v_{min}},$$ where $\alpha$ is the level of autonomy, and $k_{v_{max}}$ and $k_{v_{min}}$ are maximum and minimum virtual fixture stiffness, respectively. Unless contact is made with these potential field regions, this force is not felt by the user. The goal of the potential field is to show the user where obstacles might be, and although the user may need to overcome the field, it should less compliant than the virtual fixture. The maximum stiffness for the haptic device before it exhibits instabilities was experimentally found to be $k_{v_{max}} = 1000$ N/m. A minimum stiffness $k_{v_{min}}$ of $200$ N/m was selected to provide a guiding field that the user can easily perceive but also overcome. Table \[tbl:haptics\] shows the virtual fixture parameters used in the experiments. Parameter Value Units ------------------------------- --------------- ------- ------- Virtual Fixture Max Stiffness $k_{max}$ 75.0 N/m Virtual Fixture Min Stiffness $k_{min}$ 10.0 N/m Virtual Fixture Damping $b$ 7.5 N-s/m Potential Field Max Stiffness $k_{v_{max}}$ 1000 N/m Potential Field Min Stiffness $k_{v_{min}}$ 200 N/m : Selected Haptics Parameters \[tbl:haptics\] Experimental Results {#sec:results} ==================== The proposed shared approach is compared to the case without shared control by experimentally evaluating the task completion time and the task success rate. In the first case, the machine autonomously attempts to complete the task with no operator intervention, i.e., without the shared control. In the second case, the proposed shared control approach is used, and the operator can intervene to complete the task. In both instances, an error $\Delta_{e_z}$ between the actual $\bm{q_d}$ and nominal hole $\hat{\bm{q_d}}$ location height is generated randomly, according to the normal distribution assumed in Section \[sec:application\], with a standard deviation of $\sigma_e = 10$ mm. To exercise the surface collision event considered in Section \[sec:arbitrationDerivation\], 7 discrete error values in the hole centerline $\Delta_{e_x}$ are spread from $0$ to $3\sigma_e$ mm. These values are randomized to minimize human anticipation, with $N$ = 10 runs to capture completion time variance and success rate. A user familiar with operating the system performed 70 runs of the semi-autonomous case, and data was collected for 70 runs of the purely autonomous case. ![Experimental Results comparing task time and success rate to tolerance error for both pure automation (red) and the uncertainty-based shared control scheme (blue).[]{data-label="fig:ph_results"}](results.pdf){width="48.00000%"} Completion times, with and without the shared control, are shown in Fig. \[fig:ph\_results\] with 10 runs for each data point. Error bars show $\pm$ 1 standard deviation for indicated data points. A typical time history of the peg for a large goal error ($\Delta_{e_x} = 20$ mm) is shown in Fig. \[fig:timehistdata\] for the case with shared control. Initially, the human input (blue) closely follows the machine input (red) due to haptic interaction forces asserted on the operator. Notice the slide in level of autonomy toward human authority ($\alpha \rightarrow 0$) as the tool tip reference $q_{ref}$ (cyan) begins to near the goal around $t > 3.5$ s. During this phase, haptic virtual fixture forces are strongly imposed on the operator to keep the haptic tooltip following the machine trajectory. The hole is not in the feedback camera field of view. At about $t = 4.2$ s, the operator realizes that the machine trajectory is incorrect via the camera feed, and uses this visual feedback to adjust the trajectory accordingly. Because the robot is near an obstacle, the virtual fixture forces are only weakly imposed on the operator. The potential field around the environment is also felt by the user, and the operator corrects for the incorrect machine trajectory by overcoming this field. The peg is plugged into the hole successfully at $t = 5.4$ s. ![Experimental results showing time trajectories of the peg with the shared control approach. Notice the actual goal location is offset from the nominal goal location.[]{data-label="fig:timehistdata"}](timehistdata.pdf){width="48.00000%"} Fig. \[fig:3dtraj\] compares shared control trajectories for no goal error and a large goal error. With no goal error, the human is quickly able to complete the task since the true goal aligns with the estimated goal of the machine. In the case of a large goal error however ($\Delta_{e_x} = 30$ mm), the operator realizes the machine trajectory is incorrect and compensates via visual feedback. ![Experimental Results showing human $q_h$ and machine $q_m$ trajectories for a) no goal error $\Delta_e = 0$ mm and b) a large goal error $\Delta_e = 30$ mm. Haptic interaction forces allow the operator input to closely track the machine input when a probability of collision is low, but sliding the level of autonomy gives the operator control authority when adaptation is needed.[]{data-label="fig:3dtraj"}](trajectories3d.pdf){width="48.00000%"} Mean completion time for the the purely autonomous system over all goal errors was $4.52$ s, with a standard deviation of $0.18$ s. Mean completion time for the shared control method over all goal errors was $5.79$ s with a standard deviation of $0.93$ s. A successful task is registered when the robot inserts the peg into the hole. The autonomous system began to register more failed tasks than successes for goal errors $\Delta_{e_x} > 6$ m, while the shared control method had 100 % successes over the entire experiment space as summarized in Table \[tbl:results\]. Autonomous Shared Control Units -------------------------------- ------------ ---------------- ------- Error $\Delta_{e_x}$ tolerance 6 30 mm Mean Completion Time 4.52 5.79 s Completion Time StDev 0.18 0.93 s : Experimental Results Summary \[tbl:results\] Conclusion {#sec:conclusion} ========== The probabilistic approach presented in this paper offers a method for incorporating human intervention in the case of automation uncertainty to increase task success with minimal impact to completion time. This is accomplished by using apriori knowledge of uncertainties to arbitrate human-machine shared control. Furthermore, haptic interaction virtual forces are scaled by the arbitration factor to continuously convey automation confidence to the operator. Results for a peg-in-hole application show that the sliding autonomy shared control approach increases error tolerance by at least $5\times$ over pure feed-forward automation. The cost paid in completion time is $28$% ($\sim$ $1.2$ s) of the mean completion time exhibited by the autonomous method. Future work should consider estimating uncertainties online such that the system can achieve rapid completion time when uncertainty is low, but incorporate human intervention and maintain desired success rates when uncertainty is high. Acknowledgments =============== This work was supported by Boeing Research and Technology (BR&T) at the Boeing Advanced Research Center (BARC), University of Washington in Seattle.

--- abstract: 'Two new bright X-ray sources in globular clusters, and many less luminous ones in globular clusters and in old open clusters, have been discovered with ROSAT. Accurate positions obtained with ROSAT help identification with optical objects, which however is still very incomplete in globular clusters. One dim globular cluster source has been identified with a recycled radio pulsar; several others may be cataclysmic variables. The four brightest X-ray sources identified in the old open cluster M$\,$67 are puzzling, as we do not understand why they emit X-rays. In comparison with the two old open clusters studied so far, globular clusters are remarkably underluminous in X-rays.' author: - Frank Verbunt nocite: - '[@jvh95]' - '[@hmg+92]' - '[@zvh+98]' - '[@zhb+98]' - '[@hcn+96]' - '[@vbhj95]' - '[@hg83]' - '[@cgb+95]' - '[@jv96]' - '[@cgkb93]' - '[@hjv94]' - '[@vh98]' - '[@jvh94]' - '[@bai95]' - '[@hgb93]' - '[@jkp92]' - '[@dkt94]' - '[@dksk97]' - '[@vbrp97]' - '[@hcjv97]' - '[@gak97]' - '[@ako89]' - '[@egg+96]' - '[@mmp+97]' - '[@cgkb93]' - '[@gcc+95]' - '[@cgc+95]' - '[@gbd+91]' - '[@bvs93]' - '[@bvm98]' - '[@pba94]' - '[@flbb97]' - '[@mlg90]' - '[@san77]' - '[@vp95]' - '[@lab+97]' - '[@bvm98a]' - '[@bvm98b]' - '[@lbn+98]' - '[@gkm+92]' - '[@ver96d]' - '[@am96]' - '[@phv97]' - '[@phmv97]' - '[@sr94]' - '[@sbg+96]' title: 'X-ray sources in old stellar clusters: the contribution of ROSAT' --- 0. First {#first .unnumbered} ======== I am grateful to the organizers of this meeting for the opportunity to contribute to this celebration of the 65$^{\rm th}$ birthday of Joachim Trümper. During my years (1985-1989) at the Max Planck Institut für extraterrestrische Physik, I have always felt very much at home, as if an ‘honorary German’, and I thank Joachim and my other colleagues for this. I was impressed with the width and depth of Joachim Trümper’s interests, and thoroughly enjoyed my many discussions with him. The deadline for the first Announcement of Opportunity for observing with ROSAT was on a Sunday evening, at midnight (in checking the date, I find it was on my birthday!). Joachim Trümper was in his office that night, to sign proposals until just before the deadline, or almost, as 11 copies needed to be made for submission. Some of the proposals he signed that night contributed to the results discussed in this review. Introduction ============ Two types of stellar clusters are commonly discriminated. [*Globular clusters*]{} are distributed spherically around the Galactic Center, have metal abundances ${\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}0.1$ of the solar abundances, contain each up to $10^6-10^7$ stars, and are very old (in danger in fact of being older than the Universe...), $11.5\pm1.5$Gyr (Chaboyer et al. 1998). [*Open*]{} or [*galactic clusters*]{} are located in the disk of the Milky Way, have metallicities comparable to solar, contain each some $10^3-10^4$ stars, and in age range from very young ($\sim$Myr) to fairly old ${\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}6$Gyr). Open clusters have been found only relatively nearby, and it is quite possible that open clusters exist with star numbers and ages closer to those of globular clusters. The study of clusters with different ages has taught us much about the evolution of single stars; with large numbers of binaries now being detected in clusters we may hope to learn much also about the evolution of binaries. There is an interesting interplay between the evolution of a cluster as a whole and the evolution of the individual stars in it (as reviewed by Hut et al. 1992). For example, sudden mass loss by many of the stars (e.g. due to supernovae in a very young cluster) can dissolve the cluster. Close encounters between single stars and binaries can increase the velocity dispersion of the stars in the cluster. And finally, collisions between stars or exchange encounters (in which a single star kicks a binary star out of its orbit and takes its place) can lead to peculiar stars and binaries that could not arise from isolated (binary) evolution. The high stellar density in globular clusters implies that encounters between stars in them are common; the lower density in open clusters imply that such encounters are less frequent there, although they may have occurred more often in the past. In this article I will illustrate how X-ray observations with ROSAT contribute to the study of and comparison between binaries in globular clusters and in old open clusters. X-ray emission identifies stars which are, or have been, interacting with other stars. Followup studies at optical and ultraviolet wavelengths may explain the nature of the interaction. In Sect. 2 I discuss the observations of globular clusters and the detection of bright and of less luminous X-ray sources in them. 47$\,$Tuc and NGC$\,$6397 are discussed as examples of efforts to optically identify dim X-ray sources in the cores of globular clusters. In Sect. 3 I discuss the discovery of X-ray sources in the old open clusters M$\,$67 and NGC$\,$188, and of followup observational and theoretical work. Sect. 4 summarizes the conclusions. Globular clusters ================= Bright X-ray sources -------------------- Since the discovery of the first bright ($L_x{\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$>$}}}}10^{36}\,$erg/s) X-ray sources in globular clusters with UHURU, twelve such sources have been found, five of which are transients (see the review by Hut et al. 1992). An X-ray source at these luminosities must be a neutron star or a black hole accreting from a binary companion. X-ray bursts due to thermonuclear fusion on the surface of a neutron star have been detected in ten cluster sources, the last one with SAX (in’t Zand et al. 1998b).[^1] Binary periods have been found for four sources, the last one in NGC$\,$6712 (Homer et al. 1996). Remarkably, two of these are extremely short, 11.4 min (source in NGC6624) and 13.2 or 20.6 minutes (in NGC6712), indicating that the mass donor is a white dwarf with mass $M_{wd}<0.1M_{\odot}$. No such short-period X-ray binaries have been found in the Galactic Disk. Repeated ROSAT observations show that the lightcurve of the 11.4 minute binary is variable, complicating the effort to derive a period derivative for this system (Van der Klis et al. 1993). The globular cluster system contains only $\sim0.1$% of the stars of our Galaxy, but 10% of the bright X-ray sources. This indicates that bright X-ray sources are formed in globular clusters by stellar collisions or by exchange encounters between neutron stars and binaries. In agreement with this suggestion is the observation that the bright X-ray sources are located in the cores of the clusters. ROSAT has increased the number of accurate source positions on which this statement is based (Figure 1). Dim X-ray sources ----------------- ### Census of dim sources The Einstein satellite discovered low-luminosity or dim ($L_x{\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}10^{35}\,$erg/s) X-ray sources in the cores of nine clusters, one of which (in NGC$\,$6440) may be the quiescent counterpart of the bright transient in this cluster (Hertz & Grindlay 1983).[^2] Sources were also discovered outside the cores of $\omega\,$Cen and NGC$\,$6656 but these have turned out to be fore- or background sources (Cool et al. 1995a). ROSAT observations added substantially to this sample, bringing the total to more than thirty (for a list, see Johnston & Verbunt 1996), and has found multiple sources in the cores of NGC$\,$6397 (Cool et al. 1993), 47$\,$Tuc (Hasinger et al. 1994, Verbunt & Hasinger 1998), $\omega\,$Cen (Johnston et al. 1994), and NGC$\,$6652 (Bailyn 1995). Combination of the detections with the upper limits (many of which are from the Rosat All Sky Survey) gives the most likely X-ray luminosity distribution of these sources, $dN(L_x)\propto {L_x}^{-1.5}dL_x$, down to $\sim 10^{31.5}\,$erg/s (Figure 2). This distribution implies that the total X-ray luminosity of each cluster is dominated by a few bright sources rather than by large numbers of undetected less luminous ones (Johnston & Verbunt 1996). This is confirmed by the observation of the unresolved X-ray flux in spatially resolved cores, which is low or even undetectable. Several sources have been shown to be variable, including one with an extremely soft spectrum ($kT\sim40\,$eV) in M$\,$3 $=$ NGC$\,$5272 (Hertz et al. 1993). The number $N$ of sources in a cluster scales with core mass $M_c$ and density of its core $\rho_c$ as $N\propto M_c{\rho_c}^{0.5}$ (Johnston & Verbunt 1996). This dependence of the number of sources is between the proportionalities with mass $M_c$ and that expected for pure tidal capture $M_c\rho_c$, a result that was earlier found to hold for radio pulsars in globular clusters (Johnston et al. 1992). This may be the consequence of the destruction of binaries formed by close encounters in subsequent encounters in the densest clusters; or because the sources are a mixture of binaries evolved from primordial binaries and binaries formed by close encounters. ### Nature of the dim sources The cores of globular clusters are known to harbour soft X-ray transients in the low state (Sect. 2.1), cataclysmic variables, and millisecond radio pulsars; they probably also harbour chromospherically active close binaries, RS CVn systems. All of these have been suggested as possible counterparts for the dim X-ray sources. One way to investigate these suggestions is by comparing the X-ray luminosities of the dim sources in globular clusters with those of individually identified quiescent transients, cataclysmic variables, millisecond radio pulsars and RS CVn systems in the Galactic Disk (Figure 3). It is seen that the relatively bright X-ray sources, at luminosities $L_x{\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$>$}}}}10^{32}\,$erg/s) are matched in luminosity only by the quiescent X-ray transients. Millisecond radio pulsars, cataclysmic variables and RS CVn systems are found in the Galactic Disk at $L_x{\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}10^{32}\,$erg/s. Indeed, at least part of the X-ray emission of the globular cluster M$\,$28 is due to the millisecond pulsar in it (Danner et al. 1994, 1997). Dim X-ray sources have also been detected at positions compatible with the known dwarf nova in M$\,$5 ($=$ NGC$\,$5904) and the known old nova T$\,$Sco 1860 in M$\,$80 ($=$ NGC$\,$6093); if these are indeed the counterparts the X-ray to optical flux ratio of the dwarf nova in M$\,$5 falls in the range of cataclysmic variables in the Galactic Disk, but that of T$\,$Sco 1860 is rather higher (Figure 4). Efforts to identify the multiple dim sources in 47$\,$Tuc and NGC$\,$6397 are described in the following subsections. 47$\,$Tuc --------- The globular cluster 47$\,$Tuc ($=$ NGC$\,$104) has been observed six times with the ROSAT HRI between 1992 and 1996. Using four X-ray sources – not related to the cluster – detected in most of these observations to align them precisely, Verbunt & Hasinger (1998) obtain an added 58,000$\,$s image of the core of 47$\,$Tuc. Contour plots of this image are shown in Figure 5. A multiple-source algorithm detects five significant sources in the core. Two of the four sources outside the cluster which were used in the alignment can be identified optically (with a G5 V star and with a galaxy, Geffert et al. 1997), which provides a tie of the X-ray coordinate frame to the optical J$\,$2000 coordinate system with an estimated accuracy of ${\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}2''$. This reduces the number of possible optical counterparts to the dim X-ray sources in the core, as compared to the number previously allowed by the $5''$ positional accuracy. In particular, the remarkable ultraviolet variable and binary AKO$\,$9 (Minniti et al. 1997) is not a counterpart to any of the individually detected X-ray sources. Looking for counterparts among the blue stragglers, blue variables, and eclipsing binaries, we find that the suggested cataclysmic variables V$\,$1 and V$\,$2 are possible counterparts for X$\,$9 and X$\,$19 respectively, whereas an eclipsing binary, star 12 from Edmonds et al. (1996), is a possible counterpart to X$\,$10. It should be noted, however, that there is a sizable probability that all coincidences are due to chance: for the three X-ray sources in the central $20''\times20''$ region of the cluster, 22 counterparts are investigated, each acceptable as counterpart in an area $4''\times4''$. NGC$\,$6397 ----------- The globular cluster NGC$\,$6397 also contains multiple sources in the center, as shown by the X-ray contours in Figure 6. Cool et al.(1993) analyze the 1991-1992 data, and conclude that there are (at least) three sources close to the core; another source is about 25$''$ to the North of the cluster center. Three H$\,\alpha$ emission stars, and various blue stars, are suggested as possible counterparts by Cool et al. (1995b). Spectroscopy with the Hubble Space Telescope shows that the H$\,\alpha$ emission stars have spectra like (magnetic) cataclysmic variables (Grindlay et al. 1995). In Figure 6 I show the optical positions of the proposed counterparts on top of the X-ray contours, in a hitherto unpublished 1995 observation. Within the likely error of $\sim 2''$ two H$\,\alpha$ candidates are acceptable; the third (rightmost) one is less obvious. Old open clusters ================= M$\,$67 ------- Following the discovery of a cataclysmic variable of the AM Her type in the old open cluster M$\,$67 (Gilliland et al. 1991), ROSAT observations of this cluster were obtained, which did indeed detect the variable, and in addition detected some twenty other cluster members (Belloni et al. 1993, 1998). The brightest X-ray sources in the ROSAT band are listed in Table 1. They are all proper-motion and radial-velocity members of M$\,$67. The location of identified X-ray sources in the colour magnitude diagram of M$\,$67 is shown in Figure 7. The X-ray emission of several of these sources can be explained in terms of known source categories. One very soft X-ray source, for example, is a hot white dwarf (Pasquini et al. 1994, Fleming et al. 1997). Two sources are contact binaries whose X-ray emission is due to magnetic activity. Magnetic activity is also likely to explain the X-ray emission of three binaries with circular orbits and binary periods $P_b{\mathrel{\hbox{\rlap{\lower.55ex \hbox {$\sim$}} \kern-.3em \raise.4ex \hbox{$<$}}}}10\,$d (Belloni et al. 1998). However, there are several sources whose X-ray emission is unexplained. Remarkably, none of the four brightest sources in M$\,$67 is a predicted type of X-ray source. All four are in locations in the colour magnitude diagram away from the isochrone of single M$\,$67 members, i.e. they are special stars photometrically. $M_{\rm V}$ B-V $L_{\rm x}$(erg/s) $P_{\rm b}$(d) $e$ type ------------- ------ --------------------- ---------------- ------ --------------------- 11.52 0.88 $7.8\times 10^{30}$ 42.83 0.00 giant$+$white dwarf 13.52 1.07 $7.3\times 10^{30}$ 18.39 0.22 sub-subgiant 13.59 1.08 $7.3\times 10^{30}$ 2.82 0.03 sub-subgiant 11.25 0.41 $7.2\times 10^{30}$ blue straggler 12.47 0.66 $6.8\times 10^{30}$ SB RS CVn? 13.27 0.60 $6.5\times 10^{30}$ 14.38 0.73 $6.2\times 10^{30}$ SB RS CVn? 12.60 0.78 $5.3\times 10^{30}$ 10.06 0.00 RSCVn 13.34 0.80 $2.8\times 10^{30}$ 0.36 0.00 W Uma 15.03 0.81 $2.5\times 10^{30}$ RS CVn?? 13.98 0.61 $2.5\times 10^{30}$ 2.66 0.00 RS CVn : The eleven brightest X-ray sources in M$\,$67 (all with membership probability $\geq86$%), in order of X-ray luminosity. For each X-ray source counterpart the magnitude and colour are listed, the X-ray luminosity in $0.1-2.4\,$keV band, orbital period, orbital eccentricity, and the binary type. All X-ray luminosities assume an X-ray spectrum typical for emission of a magnetically active star. SB indicates a spectroscopic binary from the survey of Latham and Mathieu (see Mathieu et al. 1990) without a determined orbit solution. Adapted from Belloni et al. (1998). ### S$\,$1040: giant and white dwarf The brightest source is a binary that consists of a giant star and a white dwarf, in a circular orbit around one another. It is star 1040 in Sanders’ (1977) catalogue. Verbunt & Phinney (1995) showed that the currently visible giant cannot have circularized the orbit, if the standard circularization theory is correct, and predicted that the unseen companion of the giant would be a white dwarf, whose progenitor circularized the binary in its giant stage. This white dwarf was subsequently found in the ultraviolet; it is undermassive, at $\sim 0.2M_{\odot}$, and has an effective temperature $T_{\rm eff}=16160\,$K (Landsman et al. 1997). The white dwarf is too cool to be responsible for the X-ray emission. The ultraviolet spectrum shows the Mg$\,$II$\,\lambda$2800 line in emission (Landsman et al.), and a blue optical spectrum shows emission cores in the Ca$\,$II$\,$H and K lines (Van den Berg et al. 1998b), both a sign of chromospheric activity on the giant star. This indicates that the X-ray emission is due to magnetic activity; however, S1040 doesn’t rotate fast, and it is not clear why it would be magnetically active. ### S$\,$1063 and S$\,$1113: stars below the subgiant branch The next two brightest X-ray sources in M$\,$67 are two stars which are located below the subgiant branch in the colour magnitude diagram of M$\,$67, number 1063 and 1113 in Sanders’ list. Both are binaries, one with a circular orbit, and one with an eccentric orbit (Table 1). The binary periods are too long, and the binaries are too far above the main sequence, for these stars to be contact binaries. Optical spectra show emission cores in the Ca$\,$II$\,$H and K lines of both binaries (Van den Berg et al. 1998b), suggesting that the X-ray emission could be due to magnetic activity. In addition there is evidence for H$\,\alpha$ emission (Figure 8). We do not understand the location of these stars below the subgiant branch. In principle, mass transfer can cause a star to become subluminous (because readjusting hydrodynamical equilibrium for a mass-losing star drains its stellar luminosity), but one expects very rapid circularization for a Roche-lobe-filling star (Verbunt & Phinney 1995). Interestingly, the periastron distance in S$\,$1063 is such that tidal forces can have brought a slightly evolved star into corotation at periastron without (as yet) having circularized the orbit (Van den Berg et al. 1998a); this could explain the chromospheric emission of the spun-up star, but not why it is underluminous. The X-ray flux of S$\,$1063 is variable, being $8.1\pm0.9$ counts/ksec in November 1991, and $4.7\pm0.6$ counts/ksec in April 1993 (Belloni et al. 1998). ### S$\,$1072 and S$\,$1237: wide eccentric binaries Two X-ray sources are wide binaries, with orbital periods of about 1500 and 700$\,$d respectively, and with significant eccentricities. Both binaries are in the yellow straggler region of the colour magnitude diagram. Their optical spectra do not show significant emission in the Ca$\,$II$\,$H and K lines (Van den Berg et al. 1998b). The eccentricity of the orbits indicate that no significant tidal interaction is occurring or has occurred in these binaries. We do not understand why these two binaries would emit X-rays. ### S$\,$1082: a blue straggler binary Star 1082 in Sanders’ list is a blue straggler, the only one in this cluster to be detected in X-rays. From an ultraviolet excess, Landsman et al. (1998) conclude that it is a binary with a subdwarf O star. Optical spectra taken by Van den Berg et al. (1998b) show variability in the H$\,\alpha$ line, which may be related to the binarity. The radial velocity of the star shows some variation (Mathieu et al. 1986), but no orbital period has been found so far; the orbital period of about 1 day suggested from photometric data by Goranskii et al. (1992) in particular is not present in the radial velocity data, contrary to expectation in a short-period eclipsing binary. Again, we do not know why this binary emits X-rays. NGC$\,$188 ---------- Another old open cluster, NGC$\,$188, has also been observed with ROSAT, and two sources have been detected in it (Belloni et al. 1998). Their location in the colour magnitude diagram of NGC$\,$188 is shown in Figure 9. The brightest source, at $L_x({\rm 0.1-2.4\,keV})\simeq1.7\times10^{31}\,$erg/s, is a rapidly rotating single giant, a star of the FK Com type. Such stars possibly are the result of a merger of two stars into a single, rapidly rotating star. The other X-ray source, three times less luminous, is identified with a star which is a short-period velocity variable, as well as a rapid rotator. The X-rays of both detected members of NGC$\,$188 can therefore be explained in terms of magnetic activity of rapidly rotating cool stars. Summary, questions and outlook ============================== Whereas the nature of the bright X-ray sources in globular clusters is clear – at least 11 of the 12 known are accreting neutron stars – the nature of many of the less luminous sources is still unknown. At least one dim source is a radio pulsar; one (in M5) is likely a dwarf nova, and several others (in 47Tuc and in NGC6397) quite possibly are cataclysmic variables. The comparison of the old open clusters with the globular clusters shows that the open clusters are surprisingly bright in X-rays. The added X-ray luminosity of the M$\,$67 sources in the $0.5-2.5\,$keV band is about $3\times 10^{31}\,$erg/s. Note that this doesn’t include the cataclysmic variable, which emits only at energies $<0.5\,$keV. If a typical globular cluster contains more than a thousand times as many stars, with most binaries in the core, simple scaling from M$\,$67 would lead to a predicted X-ray luminosity for a globular cluster well in excess of $10^{34}\,$erg/s, in clear contradiction to many of the upper limits determined in the ROSAT All Sky Survey (see Figure 2). Apparently, the X-ray sources of the types seen in M$\,$67 are relatively under-represented in the cores of globular clusters (Verbunt 1996). The same reasoning also shows that globular cluster cores do not contain thousand FK Com type stars with X-ray luminosities like that of the one in NGC$\,$188. Another remarkable fact uncovered by the analysis of M$\,$67 is that none of the four brightest X-ray sources in this old open cluster is of a known type of X-ray emitter. This raises two questions for future research. First, we would like to understand what causes the X-ray emission in each of the four brightest X-ray sources in M$\,$67. In particular it will be interesting to know whether the X-ray emission of these stars is the consequence of more-or-less ordinary binary evolution or whether the peculiar properties of these binaries are the consequence of close encounters between binaries and/or single stars in the past history of M$\,$67. N-body computations combined with stellar evolution of open clusters may provide ideas (Aarseth & Mardling 1996; Portegies Zwart et al. 1997a,b). Second, we would like to understand why X-ray sources like the brightest ones in M$\,$67 and NGC$\,$188 are not present in their thousands in the cores of globular clusters. In fact, even the less bright sources in the old open clusters, like the RS CVn sources and the cataclysmic variables, cannot be present in very large numbers in globular clusters. The absence in globular clusters of wide binaries, and of binaries (like cataclysmic variables) that have evolved from wide binaries, can be understood from the efficiency with which wide binaries are ionized in encounters with third stars (e.g. Davies 1997). Theoretical formation and evolution scenarios of cataclysmic variables in globular clusters (Di Stefano & Rappaport 1994) nonetheless predict much larger numbers than appear to be present according to optical or ultraviolet observations (Shara et al. 1996). This indicates that our understanding of the formation (via tidal capture) and of the evolution of cataclysmic variables, as well as their phenomenology is far from complete. The absence of close binaries, like RS CVn systems, also is not readily understood. An alternative line of reasoning is that open clusters like M67 have already lost many of the stars originally in them, but retained a relatively larger fraction of the binaries. Investigation into this line of reasoning again can be made with N-body computations that include stellar evolution. The near future will bring X-ray satellites that can study more X-ray sources in old open and globular clusters, like AXAF, or that can study the X-ray spectra of these sources, like XMM. Together with theoretical and optical followup, these observations will hopefully contribute answers to the questions about X-ray sources in old clusters, that have been raised thanks to ROSAT. 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--- abstract: 'Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time $\tilde{O}(2^k {\mathrm{poly}}(k) M n^\omega) = O^*(2^k M)$. If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega(\log\log M + 1/\varepsilon))$. For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time $\tilde{O}(2^k {\mathrm{poly}}(k) M n^3) $ and a $(1+\varepsilon)$-approximate solution of running time $\tilde{O}(2^k {\mathrm{poly}}(k) n^3(\log\log M + 1/\varepsilon))$. All of the above algorithms are randomized with a polynomially-small error probability.' author: - 'Avinatan Hassidim[^1]' - Orgad Keller - 'Moshe Lewenstein[^2]' - Liam Roditty bibliography: - 'range.bib' title: 'Finding the Minimum-Weight $k$-Path' --- Introduction ============ Given an $n$-vertex graph $G=(V,E)$ and a parameter $k$, in the *$k$-path* problem we wish to find a path in $G$ consisting of $k$ vertices, if such exists. The $k$-path problem can be easily shown to be NP-complete: when $k=n$, it is exactly the Hamiltonian path problem. While a trivial $O^*(n^k)$ solution[^3] is to try all $\binom{n}{k}$ combinations of $k$ vertices, better can be obtained; Monien was the first to show an improvement [@Monien85], with an $O^*(k!)$ algorithm. In their seminal result, Alon, Yuster, and Zwick [@AYZ95] introduced the *color-coding* technique. They used it to present a randomized $O^*((2e)^k)$ algorithm for this problem, which can be derandomized, replacing the $2e$ term with a large constant. Their result thus shows that the <span style="font-variant:small-caps;">logpath</span> problem of determining whether a graph contains a path of length $\log n$ can be solved in polynomial time. Later, two independent results [@KMRR06; @CLSZ07] presented randomized $O^*(4^k)$ algorithms, again with larger constants when derandomized, having running times of $O^*(16^k)$ [@KMRR06] and $O^*(12.5^k)$ [@CLSZ07]. While these results were combinatorial in nature, the next improvements used algebraic techniques: Koutis [@Koutis08] presented an algorithm solving the problem in $O^*(2.83^k)$ time. His method was perfected by Williams [@Williams09], reducing the running time to $O^*(2^k)$. This had somewhat closed the gap between the $k$-path problem and the best method known for the specific case of finding a Hamiltonian path in a directed graph, which is $O^*(2^n)$ (though the latter is combinatorial in nature). For undirected graphs, recent results presented $O^*(1.657^n)$ [@Bjorklund10] and later $O^*(1.657^k)$ [@BHKK10; @AB13] running times for Hamiltonian path and $k$-path, respectively. It is worthwhile to focus on Koutis’ and Williams’ techniques, as they are the basis to this paper. They reduce $k$-path and other problems to the problem of determining whether a given $n$-variable polynomial contains a $k$-multilinear-monomial (that is, a term which is the multiplication of $k$ distinct variables) in its sum-product expansion. The problem is then solved by (roughly) evaluating this polynomial over random values taken from an adequate choice of an algebraic structure. In a later result [@KW09] they both show that, in the evaluation framework they use, their technique for finding a $k$-multilinear-monomial is essentially optimal, as any choice of an algebraic structure for the polynomial evaluation would require that the elements in this structure have an $\Omega(2^k/k)$-sized representation. One of the most natural generalizations coming to mind, is the *minimum-weight $k$-path* problem: in this scenario, the graph edges are weighted and we wish to find a $k$-path having minimum weight in the graph. In [@Williams09] this was referred to as the *short cheap tour* problem and mentioned that while the $O^*(4^k)$ methods can be easily extended to accommodate for this version, the algebraic methods do not seem to support such extension, and left this as an open problem. We solve this problem for the specific case in which the edge weights are integers in the range $[-M,M]$, incurring a running time which also has a superlinear dependency on $M$. If the weights are reals in $[1,M]$ (or can be normalized to this range, as is the case if they are in the range $[\ell,h]$ for $0 < \ell < h$), we provide a $(1+\varepsilon)$-approximation which reduces this dependency to $\log \log M$. Notice that by this we conform to the important line of research in recent years, of discussing variants of distance problems in which edge-weights are integers taken from a bounded range, see e.g., [@Zwick02; @CGS12]. Another problem that generalizes $k$-path is presented in [@KW09]: in the $k$-tree problem, given an $n$-vertex graph $G$ and a $k$-node tree $T$, find a copy of $T$ in $G$. For a similar generalization of this problem to *minimum-weight $k$-tree*, and under similar restrictions on the edge weights, we show similar exact and approximate results. #### Paper Organization. In Section \[sec:method\], we first present an $\tilde{O}(2^k {\mathrm{poly}}(k) M n^\omega )$ algorithm for computing the weight of the minimum-weight $k$-path when edge weights are integers in $[-M,M]$, where $\omega < 2.3727$ stands for the matrix multiplication exponent [@Williams12]. In Section \[sec:actual\], we show how to find the path itself, incurring an $O(k \cdot {\mathrm{poly}}\log n)$ multiplicative overhead for the above algorithm. Finally, in Section \[sec:approx\], for the case of real edge-weights in $[1,M]$, we provide a $(1+\varepsilon)$-approximation algorithm that reduces the dependency on $M$ to $\log\log M$, by using a technique of careful adaptive scaling of the edge weights. The overall running time of this algorithm is $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega(\log\log M + 1/\varepsilon))$. In Section \[sec:tree\] we turn to the $k$-tree problem, and show similar results: we present an $\tilde{O}(2^k {\mathrm{poly}}(k) M n^3)$ algorithm for finding the minimum-weight $k$-tree when edge weights are integers in $[-M,M]$, and for the case the edge-weights are reals in $[1,M]$, provide a $(1+\varepsilon)$-approximation algorithm having running time $\tilde{O}(2^k {\mathrm{poly}}(k) n^3(\log\log M + 1/\varepsilon))$. Preliminaries {#sec:prelim} ============= We follow Williams’ notation [@Williams09]. Let ${\mathbb{F}}$ be a field and $G$ be a multiplicative group. The group algebra ${\mathbb{F}}[G]$ is defined over the set of elements of the form $$\sum_{g \in G}a_g g$$ where $a_g \in {\mathbb{F}}$ for all $g \in G$, i.e., on the set of sums of elements from $G$ with coefficients from ${\mathbb{F}}$. Addition is computed component-wise as $$\sum_{g \in G}a_g g + \sum_{g \in G}b_g g = \sum_{g \in G}(a_g + b_g) g \enspace,$$ multiplication is defined in the form of a convolution: $$\left(\sum_{g \in G}a_g g\right) \left(\sum_{g \in G}b_g g\right) = \sum_{g,h \in G}a_g b_h gh = \sum_{g \in G} \left(\sum_{h \in G} a_h b_{h^{-1} g} \right) g \enspace,$$ (since $G$ is a multiplicative group, the expression $h^{-1} g$ here replaces the expression of the type $g-h$ which is usually found in a convolution definition) and multiplication by a scalar $c \in {\mathbb{F}}$ as $$c\left(\sum_{g \in G}a_g g\right) = \sum_{g \in G} c a_g g \enspace.$$ Let $0_{\mathbb{F}}, 1_{\mathbb{F}}$ be the addition and multiplication identities of ${\mathbb{F}}$, respectively. Let $1_G$ be the identity of $G$. It is easy to verify that ${\mathbb{F}}[G]$ is a ring where the addition identity element $0_{{\mathbb{F}}[G]}=\sum_{g \in G}0_{\mathbb{F}}\cdot g$ is the element having all coefficients taken as $0_{\mathbb{F}}$, and the multiplication identity element $1_{{\mathbb{F}}[G]} = 1_{\mathbb{F}}\cdot 1_G = 1_G$. For ease of notation, hereafter $0$ and $1$ will denote $0_{{\mathbb{F}}[G]}$ and $1_{{\mathbb{F}}[G]}$, respectively. Let $z$ be a symbolic variable. Our computations are done on the set $({{{\mathbb{F}}[G]}})[z]$ of univariate polynomials on $z$ with coefficients in ${\mathbb{F}}[G]$. Notice that the set of polynomials with coefficients in a ring is a ring by itself. For our algorithm, we follow Williams and choose $G$ to be ${\mathbb{Z}}_2^k$ (i.e., the set of binary vectors of dimension $k$) with multiplication between elements of ${\mathbb{Z}}_2^k$ defined as entry-wise addition modulo $2$. It follows that $1_G$ is the $k$-dimensional all-zeros vector. Notice that for all $u,v \in {\mathbb{Z}}_2^k$, $u \cdot v = 1_G$ iff $u = v$. We also choose ${\mathbb{F}}={\mathrm{GF}}(2^\ell)$ for $\ell = \log k + 3$. Notice that since ${\mathbb{F}}= {\mathrm{GF}}(2^\ell)$ has characteristic $2$, it holds that for all $c \in {\mathbb{F}}$, $c + c = 0_{\mathbb{F}}$, and therefore that for all $v \in {{{\mathbb{F}}[G]}}$, $v + v = 0$. Method {#sec:method} ====== Given a weighted, directed or undirected graph $H=(V,E,w)$ on the vertex-set $V = \{1,\ldots,n\}$, with integer edge-weights in $[-M,M]$, we first show how to compute the weight of the minimum-weight $k$-path with high probability. We can assume the edge weights are actually in $[0,M]$, otherwise we re-define $w(i,j) \gets w(i,j) + M$ for each $(i,j) \in E$ and then $M \gets 2M$: as this process incurs a penalty of $(k-1)M$ for each $k$-path, it maintains the order relation on $k$-path weights. Define a *$k$-walk* to be a walk in the graph comprised of $k$ (not necessarily distinct) vertices, and let $I = \langle i_1,\ldots,i_k \rangle$ be some arbitrary $k$-walk in $H$. With a slight abuse of notation, we will also use $I$ to denote the set of edges participating in the walk. We define a collection $\{B_c\}_{c=1}^{k-1}$ of polynomial matrices $B_c$ as follows: $$B_c[i,j]=\begin{cases} y_{i,j,c} \cdot x_i \cdot z^{w(i,j)} & \text{if $(i,j) \in E$,}\\ 0 & \text{if $(i,j) \notin E$;}\\ \end{cases}$$ where each variable $y_{i,j,c}$ shall be assigned with a randomly selected value from ${\mathbb{F}}$ and each $x_i$ will be assigned with a value chosen from ${{{\mathbb{F}}[G]}}$ by a method to be described shortly. Notice that each $x_i$ corresponds to vertex $i$. Assume the values $\{y_{i,j,c}\}_{{i,j,c}}$ have already been chosen. Recall that $z$ is a symbolic variable. We define the polynomial $P$ as follows: $P(x_1, \ldots, x_n, z) = \vec{1} \cdot B_1 \cdots B_{k-1} \cdot\vec{x}$, where $\vec{1}$ is the $n$-dimensional all-ones vector and $\vec{x}$ is the vector $(x_1, \ldots, x_n)$. Re-writing $P$ as its sum-product expansion we get: $$P(x_1, \ldots, x_n, z) = \sum_{\substack{I\\I=\langle i_1,\ldots,i_k \rangle \text{ is a walk in }H}}\left(\prod_{c=1}^{k-1} B_c[i_c,i_{c+1}]\right)x_{i_k}\enspace,$$ that is, $P$ is an aggregate sum over all $k$-walks in $H$, where each walk $I=\langle i_1,\ldots,i_k \rangle $ is represented by the product of its corresponding components in $B_1,\ldots, B_{k-1}$, finally multiplied by $x_{i_k}$ which corresponds to the final vertex of the walk. By substituting the $B_c[i_c,i_{c+1}]$’s for their values, and re-arranging the walk’s product such that the $y_{i,j,c}$ terms appear first, then the $x_i$ terms, and finally the $z$ term, it follows that $$P(x_1, \ldots, x_n, z) = \sum_{\substack{I\\I=\langle i_1,\ldots,i_k \rangle \text{ is a walk in }H}} y^I \cdot x^I \cdot z^{w(I)}\enspace,$$ where $y^I = \prod_{c=1}^{k-1} y_{i_c, i_{c+1}, c}$, $x^I = x_{i_1} \cdots x_{i_k}$, and $w(I) = \sum_{e \in I}w(e)$ is the weight of walk $I$. Algorithm --------- Given $H$, randomly choose all values $y_{i,j,c} \in {\mathbb{F}}$, and randomly pick $n$ vectors $v_1,\ldots,v_n$ from $G={\mathbb{Z}}_2^k$. Now compute the polynomial $P'(z)=P(1_G+v_1,\ldots,1_G+v_n,z)$. Let ${\mathrm{coeff}}_z^d P'(z)$ be the $d$-th degree term coefficient of $P'(z)$, and let $d'=\min\{d \mid {\mathrm{coeff}}_z^d P'(z) \text{ is not $0$} \}$ (if such exists). If $d'$ exists, return it. Otherwise output “no $k$-path exists in $H$”. Proof of Correctness -------------------- Let $I$ be the minimum-weight $k$-simple-path in $H$, and notice that $w(I)$ is represented in $P$ by the term $z^{w(I)}$ in the product corresponding to $I$. Notice that while no degrees $d < w(I)$ occur in $P$, it might be that the $w(I)$-th degree term of $P$ was eliminated when (partially) evaluating $P$. Our goal is to show that this happens with low probability. For a walk $I$, notice that if $I$ is simple, i.e., it visits every node at most once, then $x^I$ is multilinear, or equivalently, square-free, since each variable $x_i$ appears in it at most once. On the other hand, if $I$ is non-simple, then $x^I$ must contain some square $x_j^2$. Therefore, in order to prove the algorithm correct, we need to show that w.h.p., (a) products corresponding to non-simple paths vanish, (b) products corresponding to simple-$k$-paths do not vanish by their evaluation, and that (c) products corresponding to simple-$k$-paths are not eliminated when they are summed with other (same-degree) products. These notions are captured by the following propositions, which are similar to the ones in [@Williams09]. Due to lack of space and for completeness, proofs are detailed in the appendix. \[pro:vanish\] If $x^I$ is non-multilinear, it vanishes. Let $J=\sum_{v \in G}v$ be the sum of all vectors from $G={\mathbb{Z}}_2^k$ (addition here is the addition of ${{{\mathbb{F}}[G]}}$). \[thr:independent\] Let $I = \langle i_1,\ldots,i_k \rangle$ be a $k$-walk. If $x^I$ is multilinear (i.e., $I$ is a $k$-path), then if the vectors $v_{i_1}, \ldots , v_{i_k} \in {\mathbb{Z}}_2^k$ are linearly independent w.r.t. entry-wise addition modulo $2$, then $x^I = J$. \[cor:2\] Let $I = \langle i_1,\ldots,i_k \rangle$ be a $k$-walk. If $x^I$ is multilinear (i.e., $I$ is a $k$-path), then with probability at least $0.28$ it does not vanish. We have shown that with at least constant probability, multilinear terms do not vanish when they are assigned values as described. However, it still might happen that such multilinear terms will get eliminated when they are summed up with other multilinear terms. The next two propositions show that this can happen with at most constant probability. \[thr:dependent\] Let $I = \langle i_1,\ldots,i_k \rangle$ be a $k$-walk. If the variables $v_{i_1}, \ldots , v_{i_k} \in {\mathbb{Z}}_2^k$ are linearly dependent w.r.t. entry-wise addition modulo $2$, then $x^I$ vanishes. Recall that $P(x_1, \ldots, x_n, z)$ is a polynomial in $z$ and therefore can be viewed as $$P(x_1, \ldots, x_n, z) = \sum_{d=0}^{kM}\sum_{\substack{I\\I=\langle i_1,\ldots,i_k \rangle \text{ is a walk in }H\\w(I)=d}} y^I \cdot x^I \cdot z^{d}\enspace.$$ It is therefore easy to see that the minimum-degree term in $P$ corresponds to minimum-weight $k$-paths in $H$. Let $d'$ be the minimum degree of $P$ and let $${\mathrm{coeff}}_z^{d'} P(x_1, \ldots, x_n, z) = \sum_{\substack{I\\I \text{ is a walk in }H\\w(I)=d'}} y^I \cdot x^I$$ be its corresponding coefficient. Our goal is to show that with at least constant probability, ${\mathrm{coeff}}_z^{d'} P$ does not vanish when it is evaluated. \[pro:high-pro\] ${\mathrm{coeff}}_z^{d'} P'(z)$ does not vanish with probability at least $1/5$. Running Time Analysis --------------------- The running time of the algorithm is dominated by $k$ matrix multiplications, where the basic arithmetic operations are done over the polynomial ring $({{{\mathbb{F}}[G]}})[z]$. Therefore, we need to account for the the cost of each such operation. Notice that for any arithmetic operation in $({{{\mathbb{F}}[G]}})[z]$ performed by our algorithm, the maximum degree of the operand polynomials and resulting polynomial, is at most $kM$. We can therefore focus on the set $R$ of polynomials in $({{{\mathbb{F}}[G]}})[z]$ with degree at most $kM$. By treating the polynomials in $R$ as periodic with period $kM$ (since there will be no carry or overflow to greater degrees), $R$ continues to be a ring. Let $T$ be the upper-bound on the time required for an arithmetic operation in $R$; trivially, $T = \Omega(2^k\cdot kM \log{\lvert {\mathbb{F}}\rvert})$. It follows that the algorithm requires $O(k n^\omega T)$ time, and it remains to compute $T$. #### Addition. Addition of two polynomials can be easily done component-wise in time $O(kM \cdot 2^k \cdot \log {\lvert {\mathbb{F}}\rvert}) = O(2^k {\mathrm{poly}}(k) M)$. #### Multiplication. Multiplication is trickier and is done by employing a multidimensional fast Fourier transform-type approach.[^4] We now describe the multiplication process in more detail. The multiplication process will be easier to describe on the ring ${\mathbb{F}}[{\mathbb{Z}}_2^k\times [kM]]$ which is isomorphic to $R$, as will be shown immediately. Given a vector $v =(v_1,\ldots,v_k) \in {\mathbb{Z}}_2^k$ and an integer $d \in [kM]$, let $(v;d)$ denote the vector $(v_1,\ldots,v_k,d)\in {\mathbb{Z}}_2^k\times [kM]$. A polynomial $p \in R$ can be uniquely described as a sum $\sum_{v,d}a_{(v;d)}\cdot(v;d)$ of at most $N = 2^k kM$ summands, where each $a_{(v;d)} \in {\mathbb{F}}$ is the coefficient of $v$ appearing in ${\mathrm{coeff}}_z^d p$ (i.e., if ${\mathrm{coeff}}_z^d p = \sum_{v \in G} b_v v$, then $a_{(v;d)} = b_v$). Our definition of multiplication over $G={\mathbb{Z}}_2^k$ can be naturally extended to ${\mathbb{Z}}_2^k\times [kM]$: multiplication still corresponds to entry-wise addition, only that now addition is done modulo $2$ for dimensions $1,\ldots,k$ and modulo $kM$ for dimension $k+1$. With that in mind, our definitions of addition, multiplication, and identity elements for $R$ are extended appropriately, thus forming the ring ${\mathbb{F}}[{\mathbb{Z}}_2^k\times [kM]]$. The bottom line is that now any $p \in R$ can be viewed as a sum of elements with coefficients taken from a multidimensional array indexed by values from ${\mathbb{Z}}_2^k\times [kM]$ and that multiplication is still a convolution, an important fact to be used later. Moving to ${\mathbb{F}}={\mathrm{GF}}(2^\ell)$, being a finite field, all elements in ${\mathbb{F}}$ can be represented in the usual manner as a degree-$\ell$ polynomials with coefficients in ${\mathbb{Z}}_2={\mathrm{GF}}(2)$ and operations that are done modulo some predefined irreducible polynomial of degree $\ell$ (this irreducible polynomial can even be found naïvely as $\ell=\log k + 3$). For the purpose of using FFT, we treat polynomials in ${\mathbb{Z}}_2[x]$ as if they were actually in ${\mathbb{C}}[x]$, i.e., the set of univariate polynomials over the complex numbers. At the end of the multiplication process, we will appropriately convert polynomials in ${\mathbb{C}}[x]$ back to ${\mathrm{GF}}(2^\ell)$ as will be described shortly. By the above arguments, given two polynomials $p,q \in R$ to be multiplied, they can be taken as the sums $\sum_{v,d}p_{(v;d)}\cdot(v;d)$ and $\sum_{v,d}q_{(v;d)}\cdot(v;d)$, respectively, where $p_{(v;d)}, q_{(v;d)} \in {\mathbb{C}}[x]$ for each $v\in {\mathbb{Z}}_2^k$ and $d\in [kM]$. As the multiplication corresponds to a convolution, by the convolution theorem, it holds that $p * q = {\mathrm{DFT}}^{-1}({\mathrm{DFT}}(p) \cdot {\mathrm{DFT}}(q))$, where $*$ denotes a convolution, $\cdot$ denotes point-wise multiplication, and ${\mathrm{DFT}}$ denotes the $(k+1)$-dimensional discrete Fourier transform for values indexed by vectors of type $(v_1,\ldots,v_k,d)\in {\mathbb{Z}}_2^k\times [kM]$. Let $D(\ell)$ denote the time required for an arithmetic operation on degree-$\ell$ polynomials in ${\mathbb{C}}[x]$—including converting them back to ${\mathrm{GF}}(2^\ell)$ by division by an irreducible polynomial—and notice that $D(\ell) = O(\ell^2) = O({\mathrm{poly}}\log k)$ as multiplication and division here are quadratic by nature. Then the above DFT operations can be computed efficiently in time $O(N \log N \cdot D(\ell)) = \tilde{O}(2^k k^2 M)$ by using the multidimensional FFT algorithm. Once we have computed ${\mathrm{DFT}}(p)$ and ${\mathrm{DFT}}(q)$, thus obtaining for each of them $N$ values in ${\mathbb{C}}[x]$ (indexed as well by vectors in ${\mathbb{Z}}_2^k\times [kM]$), we point-wise multiply them, obtaining a sum $w = {\mathrm{DFT}}(p) \cdot {\mathrm{DFT}}(q)$, and compute ${\mathrm{DFT}}^{-1}(w)$, again by using FFT on multidimensional coefficients in ${\mathbb{C}}[x]$. Finally, we reduce ${\mathbb{C}}[x]$ terms (which are actually in ${\mathbb{Z}}[x]$, as convolution over integer values returns integer values) by dividing them by the irreducible polynomial used before and the appropriate modulo operations. We conclude that multiplication of polynomials in $R$ can be performed in time $\tilde{O}(2^k {\mathrm{poly}}(k) M )$, and therefore $T = \tilde{O}(2^k {\mathrm{poly}}(k) M)$. Finding the Actual Path {#sec:actual} ======================= Let $G=(V,E,w)$ be a weighted graph with integer edge-weights in $[-M,M]$. Given the algorithm from the previous section, we show that it is possible to find the minimum-weight $k$-path itself with only $O(k {\mathrm{poly}}\log n)$ multiplicative overhead w.r.t. the previous algorithm and with a polynomially small error probability. We denote by $\mathcal{A}$ the algorithm from the previous section, amplified by running $O(\log n)$ iterations of it and choosing the minimal result, such that its error probability is bounded by $1/n^{c'}$ for some constant $c'$. The algorithm for finding the actual path uses $\mathcal{A}$ as a sub-routine. Its pseudo-code is provided as Algorithm \[alg:finding\]. The rest of this section is deferred to Section \[sec:actual:app\] of the appendix due to lack of space. \[alg:finding\] $d \gets \mathcal{A}(G,k)$ Approximation {#sec:approx} ============= The main drawback of the previous algorithm is that its running time has a superlinear dependency in $M$, the bound on an edge weight. If the weights are in $[1,M]$ (or can be normalized to this range), we show that if we settle for a $(1+\varepsilon)$-approximation algorithm to the problem, this dependency can be brought down to $\log\log M$, by using a technique of careful adaptive scaling of the edge weights, thus bringing the overall running time to $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega(\log\log M + 1/\varepsilon))$. Our techniques are in the spirit of the FPTAS of Erg[ü]{}n et al. [@ESZ02] for the restricted shortest path problem. We start with the following proposition: \[lemma:bound\] Given a graph $G$ with integer edge-weights in $[0,M]$, a parameter $k$, and a value $B$, it is possible to find an exact solution to the minimum-weight $k$-path problem of weight at most $B$, if such exists, or to return that no such solution exists, in time $\tilde{O}(2^k {\mathrm{poly}}(k) B n^\omega) = O^*(2^k B)$ and polynomially-small error probability.[^5] The algorithm is identical to the previous one, except that as a first step, edges of weight greater than $B$ are deleted from the graph, and that when multiplying two polynomials in $({{{\mathbb{F}}[G]}})[z]$ of degree at most $B$, we truncate from the resulting polynomial any term of degree greater than $B$, thus keeping all polynomials throughout the algorithm at degree of at most $B$. As every polynomial multiplication now takes $\tilde{O}(2^k {\mathrm{poly}}(k) B)$ time, the running time analysis follows. We denote with $\mathcal{B}$ the algorithm that finds an exact solution to the $k$-path problem of weight at most $B$, if such exists, or to returns that no such solution exists. We will use it as a sub-routine in our approximation algorithm. Define $k'=k-1$ (the number of edges in a $k$-path), and let $OPT$ be the minimum-weight $k$-path. Our approximation algorithm starts by defining an upper and a lower bound, $U$ and $L$, respectively, to the weight of $OPT$. At first, $U=k'M$ and $L=k'$. It then iteratively fine-tunes $U$ and $L$ to the point where the ratio $U/L$ is less than or equal to $2$, while maintaining the invariant that $L \leq w(OPT) \leq U$. This fine tuning is done as follows. At each iteration we let the value $X=\sqrt{LU}$ be the geometric mean of $L$ and $U$, and define the value $\delta = (L/U)^{1/3}-\sqrt{L/U}$ which will serve as a scaling coefficient. Notice that $\delta > 0$ as $U>L$. We then scale-down the edge weights by a factor of $\delta U / k'$, thus defining a new weight $w'(i,j) = \left\lfloor{\frac{w(i,j)}{\delta U /k'}}\right\rfloor$ for each edge $(i,j)$, and let $G'=(V,E,w')$ be the graph with the new weights. Ideally, we would like to test whether the weight of the optimal solution is less than or greater than $X$ by calling $\mathcal{B}(G', k, \frac{X}{\delta U/k'})$; here notice that the value $\frac{X}{\delta U/k'}$ is the scaled-down equivalent of $X$ in $G'$. However, while the scaling guarantees that this test can be done without incurring a high running time cost, it also introduces a loss of precision due to the floor function in the scaling: define ${w_{\mathrm{eff}}}(i,j)=(\delta U /k')w'(i,j)$ as the effective weight $w'(i,j)$ simulates, then we have that ${w_{\mathrm{eff}}}(i,j) \leq w(i,j) \leq {w_{\mathrm{eff}}}(i,j) + \delta U /k'$, and therefore for a $k$-path $P$, we have that ${w_{\mathrm{eff}}}(P) \leq w(P) \leq {w_{\mathrm{eff}}}(P) + \delta U$. Therefore, in the case $w'(OPT) > \frac{X}{\delta U/k'}$ we have that $w(OPT) \geq {w_{\mathrm{eff}}}(OPT) > X$, but if $w'(OPT) \leq \frac{X}{\delta U/k'}$ (and therefore ${w_{\mathrm{eff}}}(OPT) \leq X$) then all we can assert is that $w(OPT) \leq X + \delta U$. Therefore, a $k$-path returned by a call to $\mathcal{B}(G', k, \frac{X}{\delta U/k'})$ has weight at most $X + \delta U$ (and not $X$) w.r.t. the original graph. According to the outcome of the call to $\mathcal{B}(G', k, \frac{X}{\delta U/k'})$, we redefine $U$ and $L$: if $\mathcal{B}(G', k, \frac{X}{\delta U/k'})$ returned a result, we set $U \gets X+\delta U$; otherwise we set $L \gets X$. When the main loop is done (convergence is shown to exist below), we again redefine a new weight function: $w'(i,j) = \left\lfloor{\frac{w(i,j)}{\varepsilon L /k'}}\right\rfloor$ for each edge $(i,j)$, the graph $G'=(V,E,w')$, and return the result of a call to $\mathcal{B}(G',k, \frac{U}{\varepsilon L/k'})$. The full algorithm pseudo-code is given as Algorithm \[alg:approx\]. \[alg:approx\] $k' \gets k-1$ $L \gets k'$ $U \gets k'M$ Define $w' \colon E \to \mathbb{N}$ such that $w'(i,j) = \left\lfloor{\frac{w(i,j)}{\varepsilon L /k'}}\right\rfloor$\[line:weights-final\] $G' \gets (V,E,w')$ \[line:return\] #### Running-Time. We first show that the main loop performs $O(\log\log M)$ iterations. Let $L_i,U_i$ be the respective values of $L,U$ at the start of iteration $i$; we will show that $U_{i+1}/L_{i+1} \leq (U_{i}/L_{i})^{2/3}$. At the end of each iteration $i$, we have that either $L_{i+1}\gets L_i$ and $U_{i+1}\gets X+\delta U_i$, or that $L_{i+1}\gets X$ and $U_{i+1}\gets U_i$, where $X=\sqrt{L_iU_i}$ and $\delta = (L_i/U_i)^{1/3}-\sqrt{L_i/U_i}$. In the former case we have that $$\frac{U_{i+1}}{L_{i+1}}=\frac{X+\delta U_i}{L_i}=\frac{\sqrt{L_iU_i} + \left(\left(\frac{L_i}{U_i}\right)^{1/3}-\sqrt{\frac{L_i}{U_i}}\right)U_i}{L_i} \\ =\frac{\left(\frac{L_i}{U_i}\right)^{1/3}U_i}{L_i}=\left(\frac{U_i}{L_i}\right)^{2/3}\enspace,$$ and in the latter $$\frac{U_{i+1}}{L_{i+1}}=\frac{U_i}{X}=\frac{U_i}{\sqrt{L_iU_i}}=\sqrt{\frac{U_i}{L_i}} \leq \left(\frac{U_i}{L_i}\right)^{2/3}\enspace.$$ In both cases we have that $U_{i+1}/L_{i+1} \leq (U_{i}/L_{i})^{2/3}$. Therefore it converges to a constant after $O(\log\log M)$ iterations. Notice that an invocation of $\mathcal{B}(G', k, \frac{X}{\delta U/k'})$ costs $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega)$ by Proposition \[lemma:bound\], with the bound $B= \frac{X}{\delta U/k'}$ which is $O(k)$, as $\delta U = \Omega(X)$. We conclude that the overall cost of the main loop is $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega \log\log M)$. As for the final call to $\mathcal{B}(G',k, \frac{U}{\varepsilon L/k'})$, we have that its running time is $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega /\varepsilon)$ by Proposition \[lemma:bound\], with the bound $B= \frac{U}{\varepsilon L/k'}$ which is $O(k/\varepsilon)$ since at this stage $U \leq 2L$. We conclude that the overall running time of the approximation algorithm is $\tilde{O}(2^k {\mathrm{poly}}(k) n^\omega(\log\log M + 1/\varepsilon))$. #### Correctness. Throughout the execution, the algorithm maintains the invariant that $L < X < X+\delta U< U$. That can be easily seen by substituting $X$ and $\delta$ for their values and observing that $L < \sqrt{LU} < L^{1/3}U^{2/3} < U$. Assume there exist a $k$-path in $G$, and let $OPT$ be the minimum-weight $k$-path. By the scaling arguments, and the fact that we have brought the loss of precision due to scaling into consideration when redefining $U$ and $L$, we have that the invariant $L \leq w(OPT) \leq U$ always holds. Due to the running-time argument, when the main loop is done we have $U/L \leq 2$. Let $P^*$ be the result of the call to $\mathcal{B}(G',k, \frac{U}{\varepsilon L/k'})$ at line \[line:return\] of the pseudo-code, and notice the the weights defined at line \[line:weights-final\] incur an $\varepsilon L /k'$ loss of precision per edge, or equivalently $\varepsilon L $ per $k$-path. By the call to the exact algorithm, we have that $w'(P^*) \leq w'(OPT)$ and therefore also ${w_{\mathrm{eff}}}(P^*) \leq {w_{\mathrm{eff}}}(OPT)$. Accounting for the loss of precision, we have that $w(P^*) \leq {w_{\mathrm{eff}}}(P^*) + \varepsilon L \leq {w_{\mathrm{eff}}}(OPT) + \varepsilon L \leq (1+\varepsilon)w(OPT)$. $k$-tree {#sec:tree} ======== In [@KW09], they provide a solution to the $k$-tree problem: given an $n$-vertex graph $G$ and a $k$-node tree $T$, is there a (not necessarily induced) copy of $T$ in $G$. Again their solution is based on a reduction to the question of is there a $k$-multilinear-monomial in the sum-product expansion of a given polynomial. We show how to handle the *minimum-weight $k$-tree* problem—in which we are given a weighted graph $G$, and wish to find a minimum-weight copy of $T$ in it, across all copies of $T$ in it—again, when the weights are integers in a given range $[-M,M]$. \[thr:tree\] Given a graph $G$, if the edge-weights are integers in $[-M,M]$, the minimum-weight $k$-tree can be found in $\tilde{O}(2^k {\mathrm{poly}}(k) M n^3)$ time. If the edge-weights are reals in $[1,M]$, the problem can be approximated within $(1+\varepsilon)$ in $\tilde{O}(2^k {\mathrm{poly}}(k) n^3(\log\log M + 1/\varepsilon))$ time. Let $N_G(i)$ be the neighbor-set of vertex $i$ in $G$, and let $X=\{x_1,\ldots,x_n\}$ be a variable-set corresponding to $V(G)$. We use the following polynomial on $X$, implemented as an arithmetic circuit: Let $V(G)=[n]$ and $V(T)=[k]$. The polynomial $C_{T,i,j}(x_1,\ldots,x_n)$ is defined as follows. If ${\lvert V(T) \rvert}=1$, then $C_{T,i,j}=x_j$. Otherwise, $C_{T,i,j}$ is defined recursively: let $\{T_{i,\ell} \mid \ell \in N_T(i)\}$ be the subtrees of $T$ created by removing node $i$ from $T$, where $T_{i,\ell}$ is the subtree containing $\ell$. Then $$C_{T,i,j} = \prod_{\ell \in N_T(i)}\left( \sum_{j'\in N_G(j)} y_{(i,\ell), (j,j')} \cdot z^{w(j,j')} C_{T_{i,\ell},\ell,j'} \right)\enspace,$$ where as before, $z$ is a symbolic variable, and the values $\{y_{e, e'} \mid e \in E(T), e' \in E(G)\}$ are random values drawn from ${\mathbb{F}}$.[^6] Finally, define the polynomial $Q = \sum_{j\in V(G)} C_{T,1,j}$. Each $C_{T,1,j}$ is a circuit containing at most ${\lvert E(T) \rvert}\cdot {\lvert E(G) \rvert}$ addition and multiplication gates and therefore $Q$ contains $n \cdot {\lvert E(T) \rvert}\cdot {\lvert E(G) \rvert} = O(n^3 k)$ such gates. $Q$ is a sum over all homomorphisms from $T$ to subgraphs of $G$ of size at most $k$: specifically $C_{T,i,j}$ aggregates over all homomorphisms that map $i\in V(T)$ to $j \in V(G)$ (proof can be found in [@KW09][^7]). Therefore, a monomial $x_{j_1}\cdots x_{j_k}$ appears in the sum-product expansion of $Q$ if an only if there is a homomorphism mapping $V(T)$ to $\{j_1,\ldots, j_k\}$ such that if $(i,\ell)\in E(T)$, then $(j_i, j_\ell) \in E(G)$. If such a monomial is multilinear, it corresponds to such a homomorphism in which $j_1,\ldots, j_k$ are distinct vertices, i.e., a vertex in $G$ was not used more than once for the sake of a single mapping. From this point, the same algorithms given before follow (only this time, evaluating $Q$ over $({{{\mathbb{F}}[G]}})[z]$), and propositions similar to Propositions \[pro:vanish\]–\[pro:high-pro\] apply. Full proofs are deferred to the full version of the paper. We obtain that the minimum-weight $k$-tree problem with integer edge-weights in $[-M,M]$ can be solved in $\tilde{O}(2^k {\mathrm{poly}}(k) M n^3)$ time and that if the edge-weights are reals in $[1,M]$, it can be approximated within $(1+\varepsilon)$ in $\tilde{O}(2^k {\mathrm{poly}}(k) n^3(\log\log M + 1/\varepsilon))$ time. Acknowledgments =============== We would like to thank Ryan Williams and Danny Raz for helpful comments. Appendix {#appendix .unnumbered} ======== #### Proof of Proposition \[pro:vanish\]. Assume $x^I$ contains some square $x_j^2$. Since $x_j$ was assigned with $1_G+v_j$, it holds that $x_j^2 = (1_G+v_j)^2 = 1_G^2 + 2 \cdot 1_G\cdot v_j + v_j^2 = 1_G + 2 \cdot 1_G \cdot v_j + 1_G = 2 \cdot 1_G+ 2 \cdot 1_G\cdot v_j = 0+0=0$ where the third equality holds since for all $v \in G$, $v\cdot v = 1_G$, and the fifth equality holds since ${\mathbb{F}}$ has characteristic $2$ and therefore for all $c \in {\mathbb{F}}$, $2c = 0_{\mathbb{F}}$. #### Proof of Proposition \[thr:independent\]. If the $k$ vectors $v_{i_1}, \ldots , v_{i_k} \in {\mathbb{Z}}_2^k$ are linearly-independent, then they form a basis $B = \{v_{i_1}, \ldots , v_{i_k}\}$ for ${\mathbb{Z}}_2^k$. Notice that $x^I = \prod_{c=1}^k (1_G + v_{i_c}) = \sum_{S \subseteq B}\prod_{v \in S} v$, i.e., $x^I$ is the sum of every possible combination of vectors from $B$, multiplied together. Hence, the sum covers all vectors in the span of $B$, that is, $\sum_{S \subseteq B}\prod_{v \in S} v = \sum_{v \in {\mathrm{span}}(B) }v = \sum_{v \in {\mathbb{Z}}_2^k}v = J$. #### Proof of Corollary \[cor:2\]. The values $v_{i_1}, \ldots , v_{i_k} \in {\mathbb{Z}}_2^k$ were chosen randomly and independently. It is known that a random $k \times k$ matrix of values from ${\mathbb{Z}}_2$ has full rank with probability at least $0.28$ [@BK95]. #### Proof of Proposition \[thr:dependent\]. Recall that $x^I = \sum_{S \subseteq \{v_{i_1},\ldots,v_{i_k}\}}\prod_{v \in S} v$. If the $k$ vectors $v_{i_1}, \ldots , v_{i_k} \in {\mathbb{Z}}_2^k$ are linearly-dependent, then there exists a set $T \subseteq \{v_{i_1}, \ldots , v_{i_k}\}$ such that $\prod_{v \in T} = 1_G$. Since, as mentioned, for $u,v \in G$ it holds that $uv = 1_G$ iff $u = v$, we get that for all $S' \subseteq T$, $\prod_{v \in S'}v = \prod_{v \in T\setminus S'}v$. It follows that every value $r = \prod_{v \in S}v$ occurs twice in the sum, one time as $r = \prod_{v \in S}v$, and one time as $r = \prod_{v \in (S \setminus T) \cup (T \setminus S)}v$. Since $2r=0_{\mathbb{F}}\cdot r$ as ${\mathbb{F}}$ has characteristic $2$, all terms are eliminated in the sum. #### Proof of Proposition \[pro:high-pro\]. By Propositions \[thr:independent\] and \[thr:dependent\], it holds that $${\mathrm{coeff}}_z^{d'}P'(z) = J \cdot \sum_{\substack{I\\I \text{ is a walk in }H\\w(I)=d'\text{ and $I$ survived}}} y^I \ .$$ Let $$Q = \sum_{\substack{I\\I \text{ is a walk in }H\\w(I)=d'\text{ and $I$ survived}}} y^I \ .$$ $Q$ is a degree-$k$ polynomial in the variables $\{y_{i,j,c}\}_{i,j,c}$. With probability at least $0.28$ at least one minimum-weight $k$-path $I$ had survived and therefore $Q$ is not identically zero. In this case, by the Schwartz-Zippel lemma [@Schwartz80; @Zippel79; @DL78], when assigning random values from ${\mathrm{GF}}(2^\ell)$ to the variable set $\{y_{i,j,c}\}_{i,j,c}$, $Q$ evaluates to zero with probability at most $k/2^\ell=1/8$. Therefore $Q$ (and hence, ${\mathrm{coeff}}_z^{d'} P'(z)$) does not vanish with probability at least $0.28 \cdot 7/8 > 1/5$. Finding the Actual Path {#sec:actual:app} ----------------------- Let $G=(V,E,w)$ be a weighted graph. We first run $\mathcal{A}(G,k)$ on the graph. Let $d$ be the value returned by it, i.e., the weight of the minimum-weight $k$-path. If ${\lvert V \rvert} > 10k$, repeat the following procedure $\Theta(\log n)$ times:[^8] remove each of the graph vertices with probability $1/k$. If $\Omega({\lvert V \rvert}/k)$ vertices were removed, run $\mathcal{A}$ on the resulting graph and $k$. If the algorithm had returned a result $d' = d$, then keep the vertices discarded indefinitely and stop, otherwise return them back to the graph. If after the $\Theta(\log n)$ iterations no vertices were discarded indefinitely, output “Fail”. The above procedure is repeated as long as ${\lvert V \rvert} > 10k$. Once ${\lvert V \rvert} \leq 10k$, we perform an ordinary self reduction: each time we remove a different vertex and query $\mathcal{A}$ with the resulting graph and $k$; if the result stays the same, we keep this vertex discarded, otherwise, we return it to the graph. Once ${\lvert V \rvert}=k$, we return the edge-set $E$ as the resulting path. This algorithm’s pseudo-code is given as Algorithm \[alg:finding\]. #### Error probability. Let $P$ be the minimum-weight $k$-path in $G$, and assume $k \geq 3$, otherwise the problem is trivial. Let $T$ be the set of vertices removed from $G$ in an iteration of the for loop. The probability $T$ does not include any of the vertices of $P$ is $(1-1/k)^k \geq 1/4$. Now assume it does not, in that case it holds that $E[{\lvert T \rvert}] = \frac{{\lvert V(G) \rvert}-k}{k} \geq \frac{9{\lvert V(G) \rvert}}{10k}$, and that $Var[{\lvert T \rvert}]=({\lvert V(G) \rvert}-k)(1/k)(1-1/k) < {\lvert V(G) \rvert}/k$. According to Chebyshev’s inequality, ${\lvert T \rvert} = \Omega ({\lvert V(G) \rvert}/k)$ with probability of at least a constant. It follows that the probability to pick $T$ that does not hit any of the vertices in $P$ and at the same time is $\Omega ({\lvert V(G) \rvert}/k)$ is at least a constant $\alpha > 0$. We define this event as a “success”. Since we perform at most $\Theta(\log n)$ trials at each iteration of the while loop, the probability of failing in all of them is $(1-\alpha)^{\Theta(\log n)}$ which can be made at most $ 1/n^c$ for some constant $c$. By using the union-bound over the $k \ln n$ iterations of the while loop, we get a polynomially-small error probability of at most $k \ln n / n^c$. Since the probability to fail any invocation of $\mathcal{A}$ is less than $1 / n^{c'}$, by a similar union-bound argument the probability to fail in any of the calls to $\mathcal{A}$ is $O(k \log^2 n / n^{c'})$. We obtain an overall polynomially-small error probability. #### Running time. Each non-failed iteration of the while loop in Algorithm \[alg:finding\] discards $\Omega({\lvert V(G) \rvert}/k)$ vertices and therefore reduces the number of vertices in the graph by a multiplicative factor of $(1-\Omega(1/k))$. As this happens until ${\lvert V(G) \rvert} \leq 10k$, $O(k\ln n)$ iterations are enough for getting the number of vertices to $10k$. As each iteration invoked $\mathcal{A}$ at most $O(\log n)$ times, the $O(k {\mathrm{poly}}\log n)$ multiplicative factor follows for this stage of the algorithm. As the for-each loop incurs only $O(k) < O(k {\mathrm{poly}}\log n)$ calls to $\mathcal{A}$, the running-time analysis follows. [^1]: Research supported by ISF grant 1241/12 and by GIF Young. [^2]: Research supported by BSF grant 2010437, a Google Research Award and GIF grant 1147/2011. [^3]: Here and throughout, the $O^*$ notation discards all factors that are polynomial in $n$, $k$, and $\log M$ from the running time. Similarly, the $\tilde{O}$ expressions discard poly-logarithmic factors. [^4]: Here, as opposed to Williams [@Williams09], the Walsh-Hadamard transform is not an adequate choice anymore due to the existence of the variable $z$ which can have a degree up to $kM$. [^5]: $B$ does not have to be an integer, but the effect in this case is as if $\lfloor B \rfloor$ is used. [^6]: In [@KW09], the $y$-values are implicit and come from the multiplication of the output of each multiplication gate with a random value taken from ${\mathbb{F}}$. [^7]: Their arithmetic circuit is defined as $Q = \sum_{i \in V(T),j\in V(G)} C_{T,i,j}$, however, it seems to contain redundancy. [^8]: For the sake of brevity, in this section we do not give full details of the underlying constants that are required.

--- abstract: 'Motivated by stringy considerations, Randall & Sundrum have proposed a model where all the fields and particles of physics, save gravity, are confined on a 4-dimensional brane embedded in 5-dimensional anti-deSitter space. Their scenario features a stable bound state of bulk gravity waves and the brane that reproduces standard general relativity. We demonstrate that in addition to this zero-mode, there is also a discrete set of metastable bound states that behave like massive 4-dimensional gravitons which decay by tunneling into the bulk. These are resonances of the bulk-brane system akin to black hole quasinormal modes—as such, they give rise to the dominant corrections to 4-dimensional gravity. The phenomenology of braneworld perturbations is greatly simplified when these resonant modes are taken into account, which is illustrated by considering gravitational radiation emitted from nearby sources and early universe physics.' author: - | Sanjeev S. Seahra[^1]\ Institute of Cosmology & Gravitation\ University of Portsmouth\ Portsmouth, PO1 2EG, UK title: | **Metastable massive gravitons from\ an infinite extra dimension[^2]** --- String theory, one of the leading candidates for the ‘theory of everything,’ tells us that the universe has more than four dimensions, but everyday experience seems to suggest otherwise. Over the years, there have been several attempts to explain why extra dimensions are hidden, including the conventional assumption that they have a compact topology with radii on the order of the Planck length. But recently, certain developments in non-perturbative string theory have provided alternatives to this ‘Kaluza-Klein’ compactifaction. The key ingredient is so-called $d$-branes, which are $(d+1)$-dimensional hypersurfaces on which the standard model particles and fields can be consistently confined. This raises the possibility that the observable universe is a 3-brane, i.e., we are living on a ‘braneworld’ [@Maartens:2003tw review]. But in these scenarios, gravity is free to propagate in the full higher-dimensional ‘bulk’ manifold. This is potentially worrisome, since the force of gravity we are familiar with behaves in an entirely 4-dimensional manner. For example, in a higher-dimensional universe we would expect deviations from Newton’s inverse square law at large distances, and such deviations have never been measured. We could again invoke compactification to rescue the model, but there is a different intriguing strategy: What if we were able to find scenarios where the bulk graviton was, in some sense, *dynamically* bound to the brane? Could we then allow the extra dimensions to be infinite? In 1999, Randall & Sundrum [@Randall:1999vf] proposed a phenomenological realization of this idea in 5-dimensional anti-deSitter space. Their model has the line element $$ds^2 = e^{-2|y|/\ell} \eta_{\alpha\beta} dx^\alpha dx^\beta + dy^2, \quad G^{(5)}_{ab} = (6/\ell^2) g^{(5)}_{ab},$$ where $\ell$ is the anti-deSitter length scale. The brane is identified with the intrinsically flat geometric defect at $y = 0$, and the Israel junction conditions imply that it is actually a thin sheet of vacuum energy. Fluctuations of this model can be written as $\eta_{\alpha\beta} \rightarrow \eta_{\alpha\beta} + h_{\alpha\beta}$ with $$h_{\alpha\beta} = e^{-|y|/2\ell} \psi_k(t,y) e^{-i\mathbf{k} \cdot \mathbf{x}} \epsilon_{\alpha\beta}, \quad \epsilon_{\alpha\beta} = \text{constant},$$ where $\psi_k$ satisfies a one-dimensional wave equation $$\label{wave equation} -\ell^2 {\partial}_t^2 \psi_k = -{\partial}_z^2 \psi_k + V_k(z) \psi_k, \quad z = e^{y/\ell} \ge 1.$$ Here, we have restricted attention to one half of the bulk. The potential $V_k$ is shown in Figure \[fig:potential\]. The delta function enforces the boundary condition ${\partial}_z (z^{3/2} \psi_k) = 0$ at $z = 1$, which ensures that the matter content of the brane is unaltered by the perturbation. The attractive nature of the delta function allows for a normalizable, stable bound state of the brane and the 5-dimensional graviton $$\psi^{(0)}_k = e^{i\omega t} z^{-3/2}, \quad \omega = k.$$ From the point of view of 4-dimensional brane observers, the fluctuation behaves exactly like a massless spin-2 field propagating on a flat background. Hence, this so-called ‘zero-mode’ reproduces 4-dimensional weak field gravity on the brane, and shows how one can recover standard general relativity with an infinite extra dimension. ![The potential governing gravity waves in the Randall-Sundrum braneworld[]{data-label="fig:potential"}](potential.eps) However, $\psi^{(0)}_k$ is not the only solution to (\[wave equation\]). There is a whole spectrum of modes parameterized by a separation constant $m$: $$\label{massive mode solution} \psi^{(m)}_k = e^{i \omega t} \sqrt{z} \left[ {\text{H}^{(1)}_{1}}(m\ell) {\text{H}^{(2)}_{2}}(m\ell z) - {\text{H}^{(2)}_{1}}(m\ell) {\text{H}^{(1)}_{2}}(m\ell z) \right], \quad \omega^2 = k^2 + m^2,$$ where ${\text{H}^{(n)}_{\nu}}$ are Hankel functions. The dispersion relation on the right implies that each mode looks like a massive spin-2 field to a brane observer. So, in addition to the bound state solution that reproduces general relativity, there is a continuum of massive graviton ‘Kaluza-Klein’ modes that predict deviations from it. A complete description of brane gravity must include both. What is the essential behaviour of the massive modes on the brane, and under what circumstances are they important relative to the zero-mode? To be sure, a direct attack on this problem with Green’s functions can be mounted, but we pursue a more physical approach by asking a different, more subtle question: Does the potential shown in Figure \[fig:potential\] treat each value of $m$ democratically, or do certain masses tend to dominate the gravity wave spectrum? That is, are any *resonant* massive modes within the Kaluza-Klein continuum? Resonant phenomena play a prominent role in many branches of physics, and can often be used to simply characterize the most important features of a given system’s dynamics. There are at least two examples of this that are useful to highlight: In black hole perturbation theory, the master equations governing gravity wave propagation support resonant solutions known as *quasinormal modes* [@Nollert], which tend to dominate the late time behaviour of scattered gravity waves, irrespective of the initial configuration of the perturbation. A second example comes from accelerator physics, where resonances in scattering cross sections allows one to identify bound states of elementary particles *without* having a complete theoretical grasp on the interactions between them. In both cases, we do not need to actually solve the equations, subject to a given source, to predict the behaviour of the system—the resonant effects are predominant. These model problems suggest that the key to gaining physical intuition about the massive modes in the braneworld is to find the resonant solutions of (\[wave equation\]). Asymptotically far from the brane, the exact solution (\[massive mode solution\]) reduces to a superposition of travelling waves, and the ratio of the coefficients of the outgoing and incoming contributions defines the scattering matrix. As in the two example problems cited above, the resonant modes of our system are defined by the poles of this scattering matrix.[^3] A simple calculation [@Seahra:2005wk] yields that these correspond to a discrete set of *complex* masses $$\label{resonant masses} \{m_j\ell\} = \{\mu_j\} = \{0.419 + 0.577\,i,\, 3.832 + 0.355\,i,\, 7.016 + 0.350 \,i\, \cdots\}.$$ If $k$ is real, the dispersion relation $\omega^2 = k^2 + m^2$ implies that these modes have $\text{Im}\,\omega > 0$, i.e., they are exponentially damped in time. In this sense, they are exactly analogous to the quasinormal modes of perturbed black holes. Because these solutions decay in time, it is sensible to call them the metastable bound states of the brane and the bulk graviton. A useful interpretation comes from recalling Gamov’s classic 1928 treatment of the radioactive alpha decay [@Gasiorowicz], where the $\alpha$-particle is thought to be trapped in a potential well surrounding a much heavier partner. In that problem, the metastable bound states represent solutions where the $\alpha$-particle is mostly localized near the daughter nucleus, but there is a finite probability of it tunnelling through the potential barrier and out to infinity. In this spirit, we see that the resonant masses (\[resonant masses\]) represent a spectrum of massive 4-dimensional gravitons mostly localized on the brane, but subject to decay by tunnelling into the bulk. This opens up a whole new way of looking at perturbative gravity in the Randall-Sundrum scenario. Before, one thought of a zero-mass graviton accompanied by continuum of massive spin-2 fields. Now, we realize that the essential behaviour is that of a stable massless graviton augmented by a discrete family of quasi-bound massive cousins. To gain a better understanding of the phenomenology of these ‘particles,’ we consider a wavepacket of gravitational radiation on the brane. We assume motion in the $x$-direction, and a momentum space profile $\alpha(k)$. The pulse’s evolution on the brane will dominated by contributions from the zero-mode and resonant masses: $$\label{pulses} \delta h_{\alpha\beta} \sim \epsilon_{\alpha\beta} \int dk \, \alpha(k) \left[ \exp{ik(t-x)} + \sum_j c_j \exp{ik\left(\frac{t}{n_j}-x\right)} \right].$$ Here, $j$ runs over the resonances, the $c_j$ expansion coefficients are determined from the initial extra-dimensional pulse profile, and $n_j$ is the complex reflective index [@Jackson] $$n_j = n_j(k) = \frac{k}{\omega_j(k)} = \frac{k\ell}{\sqrt{(k\ell)^2+\mu_j^2}}.$$ Hence, $h_{\alpha\beta}$ is a superposition of a discrete pulses corresponding to the zero-mode and massive gravitons. Since $n_j$ has a nonzero real and imaginary parts, each of the massive pulses behaves like it is travelling in an absorptive medium, slower than the speed of light. On the other hand, the zero mode acts like it is propagating in a vacuum. If $\alpha(k)$ is peaked about some $k = k_0$, we can sensibly ask: How fast do the massive pulses travel? And how far do they get before decaying? We define the group velocity $v_j$ and lifetime $\tau_j$ of each of the modes in the usual way: $$v_j = \text{Re} \, \omega_j'(k_0) = \text{Re}\,n_j(k_0), \quad \tau_j = \frac{1} {\text{Im} \, \omega_j(k_0)} = \frac{\text{Im}\,n_j(k_0)}{k_0}.$$ Together, these give an attenuation length $d_j$, which is the distance a given massive mode travels before its amplitude decreases by a factor of $e$: $$\frac{d_j}{\ell} = \frac{v_j \tau_j}{\ell} = \frac{k_0\ell}{\text{Re}\,\mu_j\,\text{Im}\,\mu_j}.$$ The denominator on the right is of order unity or larger, so we see that modes with $k_0 \ell \gg 1$ can travel for large distances, while modes with $k_0 \ell \ll 1$ have very short streaming lengths on the brane. Now, tabletop experiments of Newton’s law limit $\ell \lesssim 0.1 \text{ mm}$, which immediately dashes any hopes of seeing any bulk effects in the gravity waves emitted from nearby sources. The reasoning is as follows: Astrophysical systems have sizes much larger than 0.1 mm, which means that any gravitational radiation will primarily be composed of partial waves with $k\ell \ll 1$. Thus, the signal from the massive modes will only propagate for a minuscule distance $d \ll \ell$ along the brane before decaying away, making their direct detection impossible.[^4] The situation is quite different when we consider cosmological braneworld scenarios, where the motion of the brane in the extra dimension accounts for the expansion of the universe. In the early universe epoch $H\ell \sim 1$ of such a model, all sub-horizon modes will have $k\ell \gtrsim 1$. Hence, the attenuation length of the massive gravitons will be of order the horizon size or larger, implying that they should have an important effect of the gravity wave background. However, our calculations to this point have been for a static brane, and it is unclear how general brane motion affects the quasi-particle excitations. But some (heuristic) progress can be made by considering the small-scale fluctuations, which have $$\label{dispersion} k\ell \gg 1 \quad \Rightarrow \quad \omega_j(k) \approx k \left[ 1 + \frac{\mu_j^2}{2(k\ell)^2} \right] \quad \Rightarrow \quad |\omega_j(k)| \gg H.$$ The last inequality means that the typical oscillation timescale is much less than the speed of cosmological expansion, so it is safe to say that these modes ‘see’ the brane to be stationary, and are still valid resonant solutions. Hence, in the early universe the high frequency component of the gravity wave background will effectively be governed by a discrete spectrum of spin-2 fields obeying the above dispersion relation. As the universe expands the ‘fingerprint’ of these fields expands with it, eventually decoupling from the massive modes when its size grows larger than $\ell$. Therefore, this ‘back of the envelope’ calculation predicts that the relic gravity wave background carries within it primordial signatures of the extra dimension, encoded in $\omega_j(k)$ and thereby the discrete set of complex numbers $\{\mu_j\}$. So, what has our knowledge of the resonances between bulk gravity waves and the brane achieved for us? We have seen that the late time behaviour of metric fluctuations is dominated by a discrete set of spin-2 fields with complex masses plus the zero-mode. Kinematically, the resonant massive modes behave like gravitons travelling in an absorptive medium—the dissipation is due to the tunnelling of gravitational radiation into the extra dimension. For a given AdS length scale $\ell$ in the bulk, we have calculated the lifetime and streaming-length of these particles on the brane, which are of order $\ell$ and $\ell/c$ respectively. With $\ell \lesssim 0.1$ mm, we see that the massive part of the spectrum cannot play a large role astrophysical processes in the nearby universe, but may be significant in earlier epochs where $H\ell \sim 1$. This highlights the ephemeral nature of bulk effects on the brane in the Randall-Sundrum scenario, and why such a model is credible description of the *real* world. By and large, we see that Einstein’s theory of gravitation can—in principle—live peacefully with infinite extra dimensions, and the regions of conflict are neatly parameterized by discrete spectrum of massive decaying gravitons. I would like to thank Chris Clarkson, Roy Maartens, and David Wands for helpful discussions and encouragement, and <span style="font-variant:small-caps;">NSERC</span> for financial support. [1]{} Roy Maartens. “Brane-world gravity.” , 7:7, 2004. gr-qc/0312059. Lisa Randall and Raman Sundrum. “An alternative to compactification.” , 83:4690–4693, 1999. hep-th/9906064. Hans-Peter Nollert. “Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars.” , 16:R159, 1999. John R. Taylor. . Wiley, New York, 1972. L. D. Landau and E. M. Lifshitz. , volume 3 of [ *Course of Theoretical Physics*]{}. Elsevier, London, 3rd edition, 1977. Sanjeev S. Seahra. “Ringing the [R]{}andall-[S]{}undrum braneworld.” 2005. hep-th/0501175. Stephen Gasiorowicz. . Wiley, New York, 2nd edition, 1996. John David Jackson. . Wiley, New York, 3rd edition, 1999. [^1]: Email: [email protected] [^2]: This essay received an honourable mention in the 2005 Gravity Research Foundation essay competition [^3]: For a comprehensive introduction to the theory of one-dimensional scattering and the associated resonant phenomena, see the textbooks by Taylor [@Taylor] or Landau and Lifshitz [@LL]. [^4]: A complimentary result can be derived by integrating over real values of $\omega$ in (\[pulses\]) and assuming that $k$ is complex. Then, one finds that the attenuation length becomes long only when the source involves frequencies with $\omega\ell \gg 1$, i.e., with $\omega \gg 10^{12} \text{ sec}^{-1}$.

--- abstract: 'This work incorporates the multi-modality of the data distribution into a Gaussian Process regression model. We approach the problem from a discriminative perspective by learning, jointly over the training data, the target space variance in the neighborhood of a certain sample through metric learning. We start by using data centers rather than all training samples. Subsequently, each center selects an individualized kernel metric. This enables each center to adjust the kernel space in its vicinity in correspondence with the topology of the targets — a multi-modal approach. We additionally add descriptiveness by allowing each center to learn a precision matrix. We demonstrate empirically the reliability of the model.' address: - ' Computer Vision Lab, Delft University of Technology, Delft, Netherlands\' - ' Intelligent Sensory Information Systems, University of Amsterdam, Amsterdam, Netherlands\' author: - 'S. L.' - 'J. C.' - 'A. W. M.' bibliography: - 'agp.bib' title: Asymmetric kernel in Gaussian Processes for learning target variance --- Gaussian Process ,kernel metric learning ,asymmetric kernel distances ,regression. Introduction {#sec:intro} ============ Departing from the standard Gaussian Process, we introduce a regression approach that incorporates the multi-modality of the data distribution. While in the Gaussian Process model we have a global kernel metric that is shared by all the samples [@rasmussen2006gaussian], here we propose to define a set of training data centers considerably smaller than the number of training samples. Subsequently, we learn from the numerous training samples an individualized kernel metric per training data center. By doing so, we are able to use a smaller training kernel matrix computed only on the training data centers while retaining the descriptive power of the model. This is highly efficient at test-time as it limits the size of the kernel matrix. In our method we introduce two main changes to the standard Gaussian Process regressor: (i) we define a number of centers over the training data, by clustering or sampling; (ii) we learn individual kernel metric parameters per data center, discriminatively through metric learning, giving rise to a multi-modal approach with an asymmetric kernel matrix. Figure \[fig:abstract\] illustrates these differences when comparing with the standard Gaussian Process trained on either data centers or on all samples: we use fewer samples in the kernel computation, while enhancing the descriptiveness of the model by learning individualized metrics per center. We additionally expand the kernel parameters to a precision matrix, also learned through metric learning per center, giving rise to a multivariate multi-modal approach. The individual steps of our model are not specifically novel, yet their combination is what gives strength. Clustering of the data in Gaussian Processes has been previously proposed [@mycviu; @snelson2007local]. Asymmetric kernels have been studied in works such as [@mackenzie2004asymmetric; @wua2010asymmetric]. While a multivariate lengthscale parameter has been used in [@kersting2007most; @lazaro2011variational; @mycviu] for improved descriptiveness. Here, we combine all these ideas into a new approach which is suitable for learning target data variance. If we consider each training sample to be a data center, and enforce that all samples share the same kernel metric, and assume a univariate lengthscale in the kernel metric, we recover the standard Gaussian Process definition. We evaluate the proposed approach by gradually enabling these changes: center-based Gaussian Process, univariate multi-modal asymmetric Gaussian Process, and multivariate multi-modal asymmetric Gaussian Process. The experiments validate our models on the regression datasets *So2* and *Temp* used in [@titsias2013variational], the large scale *Airlines* dataset of [@hoang2015unifying], and two realistic image datasets: UCSD [@chan2012counting] and VOC-2007 [@pascal-voc-2007] for pedestrian and generic-object counting, respectively. Related Work ============ Mixtures of Gaussian Processes ------------------------------ Noteworthy work has been focusing on mixtures of Gaussian Processes [@lazaro2012overlapping; @meeds2006alternative; @nguyen2014fast; @rasmussen2002infinite; @tresp2000mixtures; @yuan2009variational]. In [@tresp2000mixtures] a mixture of Gaussian Processes is proposed to effectively deal with large data. [@meeds2006alternative; @rasmussen2002infinite] extend this idea to an infinite mixture of Gaussian Processes. Somewhat similarly, [@li2014gaussian] splits the problem into subproblems in a divide-and-conquer fashion and solves each such problem in a Gaussian Process model. [@yuan2009variational] proposes an elegant variational Bayesian algorithm for training the mixture of Gaussian Process experts. An effective model is proposed in [@lazaro2012overlapping], where the authors simplify the mixture of experts so no gating function is used to assign samples to components and rather, trajectory clustering is employed. [@nguyen2014fast] gracefully combines the mixture of Gaussian Process experts with the idea of inducing points, providing fast approximate Gaussian Process models. Unlike these works, where the final prediction entails a combination of predictions, each obtained within the metric space of individual components, we learn the hyper-parameters associated with each training data centers in a single Gaussian Process. We do so by employing an asymmetric kernel. Therefore, at test-time for an input test sample, rather than computing $N \times M$ kernel distances, where $N$ is the number of components and $M$ is the number of training samples, we only compute $N$ distances. In our case $N$ is the number of data centers, considerably smaller than $M$. Efficiency in Gaussian Processes -------------------------------- The work of [@quinonero2005unifying] reviews the sparse approximations of Gaussian Process from a unified perspective by analyzing the implied prior of different methods. [@csato2002sparse] proposes learning iteratively, online, the sparse set of inducing points in a Bayesian formalism by minimizing the KL divergence. Inspired from metric learning techniques, [@lawrence2003fast] uses forward selection to obtain a sparse and time-efficient model. [@snelson2005sparse] proposes a graceful solution of learning a set of sparse pseudo-inputs through gradient based optimization. A combination between sparse methods based on inducing points and local regression based on a multitude of experts describing locally the target space, is proposed in [@snelson2007local]. In [@titsias2009variational; @titsias2010bayesian] variational approaches are used to learn sparse representations. [@titsias2009variational] jointly learns the inducing points and the kernel hyper-parameters by minimizing a lower bound through KL divergence. The robust method of [@hensman2013gaussian] decomposes the Gaussian Process model, variationally, such that it is factorized based on a set of global inducing variables. [@bo2012greedy; @ranganathan2011online] focus on iterative updates of the Gaussian Process. [@rodner2012large] proposes the use of parameterized histogram intersection kernels to bypass the hyper-parameter estimation. [@cao2013efficient] proposes a method to speed up the hyper-parameter estimation by inducing sparsity in the model. Somewhat similar to these methods, we only retain a set of data centers as informative training samples. Yet, unlike the above approaches, we subsequently add extra information into the Gaussian Process model by treating the data centers differently. Descriptiveness in Gaussian Processes ------------------------------------- Full matrices in the kernel definition have been proposed in [@mycviu; @vivarelli1999discovering], to make the model more descriptive. Here, we also learn precision matrices, in the kernel metric definition. However in our work, each center has an individualized precision matrix. [@paciorek2004nonstationary] proposes nonstationary covariance matrices in the Gaussian Process model, tying the kernel metrics to the input samples. However, the final kernel matrix is symmetric as it is defined using symmetric combinations of per-sample covariances, similar to RVM (Relevance Vector Machine). [@kersting2007most; @lazaro2011variational] propose well-founded approaches to adding descriptiveness by extending the Gaussian Process definition to a heteroscedastic approach, by modeling the noise distribution to be dependent on the training data. [@kuss2005approximate] proposes EP (Expectation Propagation) as an effective manner to train these models. Similarly, we also start with the assumption that the kernel metric should be data dependent and learn an individualized kernel metric. [@titsias2013variational] proposes a compelling method for adjusting the kernel distances by assuming the data is mapped in a feature space based on the Mahalanobis kernel distance, estimated through variational inference. Unlike this work, we learn both the shape and the scale of the kernels per data center by minimizing the predictive loss. Asymmetric Kernels ------------------ The use of asymmetric kernel distances is not a recent idea but, rather, a well-matured topic [@kulis2011you; @mackenzie2004asymmetric; @tsuda1999support; @wua2010asymmetric]. In [@tsuda1999support] asymmetric kernels are proposed in the context of SVM classification. [@wua2010asymmetric] shows how similarity functions commonly used in real-life applications, can be related to asymmetric kernels, and gives a formal definition for the mathematical space described by asymmetric kernels. The work in [@mackenzie2004asymmetric] proposes asymmetric kernel regression in the context of neural networks and shows that such models are better behaved around the data boundaries. The recent work of [@kulis2011you] learns asymmetric distances for visual domain adaptation in the context of object recognition. Similar to these methods, we also use asymmetric kernel distances, as these prove to have more descriptive power when limiting the number of samples in the training kernel computation. Metric Learning --------------- [@rahimi2007random] learns a lower dimensional mapping of data while maintaining the distances between the data samples — the kernel distances remain approximately equal to the ones of the original features. In our work, we learn the kernel metric given the targets, rather than the feature representation. In [@weinberger2007metric] the kernel metric minimizes the leave-one-out regression error. [@jain2012metric] combines kernel learning with metric learning by employing a linear transformation. In this work, we use a fixed kernel — the squared exponential kernel, we additionally expand the parameters of the kernel to full precision matrices. [@globerson2005metric; @weinberger2009distance; @xing2002distance] represent pioneering work in the field of metric learning. [@xing2002distance] is the first paper to pose metric learning as a convex optimization problem learned from similar/dissimilar pairs of points. [@globerson2005metric] is one of the first works to propose Mahalanobis distances for metric learning. In [@weinberger2009distance] the Mahalanobis distance is learned in a nearest neighbor classifier, which induces a large-margin separation of classes. [@kedem2012non; @kostinger2012large; @weinberger2009distance] are recent works focusing on metric learning for classification with kernels, while [@huang2013kernel] focuses on sparse kernel learning for regression. In this work we employ metric learning rather than estimating the optimal model hyper-parameters through marginal likelihood [@rasmussen2006gaussian]. We do so, as each data center has an associated lengthscale in the proposed model and, thus, the marginal likelihood optimization is not straightforward in our case. Asymmetric Kernel for Gaussian Processes ======================================== We redefine the Gaussian Process model by allowing each training data center to learn an individualized kernel metric. This entails that the kernel matrix ceases to be symmetric in our case. However, this comes at extra gain in descriptive power, as despite using a small set of samples in the training kernel matrix, we optimize the individualized kernel metrics over the numerous available training samples. Standard Gaussian Process Revisited ----------------------------------- We shortly revisit the standard Gaussian Process formulation, to unify the notations. The mean of the predictive distribution is [@rasmussen2006gaussian]: $$\begin{aligned} {1} f(\mathbf{x}^*) &= k(\mathbf{X},\mathbf{x}^*)^T \left( k(\mathbf{X},\mathbf{X}) + \sigma^2\mathbf{I} \right)^{-1} \mathbf{y}, \label{eq:inverse}\end{aligned}$$ Where $\mathbf{x}^*$ represents an input test sample, $\mathbf{X}$ represents the training samples used for the training kernel matrix computation, $\mathbf{y}$ represents the training targets, $f(\mathbf{x}^*)$ is the prediction over the input $\mathbf{x}^*$ and $k(\cdot, \cdot)$ is the kernel metric used for estimating sample distances, and $\sigma$ is the noise hyper-parameter. Center-based Gaussian Process ----------------------------- As equation \[eq:inverse\] indicates, the training procedure requires the computation of the inverse of the training kernel-matrix, $k(\mathbf{X},\mathbf{X})$, which is prohibitive on larger datasets. The first alteration of the Gaussian Process model that we investigate, is considering a set of data centers rather than individual training samples. We do so by either sampling the data or clustering it into a set of centers, $\overline{\mathbf{X}}$. Despite its simplicity, this is very effective in getting a fair overview over the variation in the training data while not having to use all samples during training. More principled manners of defining data centers such as effective sampling techniques are possible. However, the focus here is not on the center definition, which is just meant as a first step towards reducing the size of the training kernel matrix. The strength of our model comes from allowing these centers to learn individualized metrics. Multi-modal Asymmetric Kernel ----------------------------- Given that we have sparsified the training data by keeping only the training data centers, we lost information regarding the smoothness or variability of the target function in different regions of the data space. Therefore, we allow each training center, $\overline{\mathbf{x}}_i \in \overline{\mathbf{X}}$, to define individualized kernel metrics in its data neighborhood. The lengthscale hyper-parameter is the one defining the size of the kernel space, thus, we propose individualized lengthscale hyper-parameters, $l_i$ for each center $\overline{\mathbf{x}}_i$. This entails the second alteration of the standard Gaussian Process model. In this case the prediction function uses training and test kernel terms with per-center metrics: $$\begin{aligned} {1} f(\mathbf{x}^*) &= \hat{k}(\overline{\mathbf{X}},\mathbf{x}^*)^T \left( \hat{k}(\overline{\mathbf{X}},\overline{\mathbf{X}})+ \sigma^2I \right)^{-1} \mathbf{y}, \label{eq:pred_center}\end{aligned}$$ where $\hat{k}(\cdot,\cdot)$ is a non-symmetric kernel whose size depends on its corresponding training center — $\hat{k}(\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j) = k_i (\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j)$, and $\overline{\mathbf{x}}_i, \overline{\mathbf{x}}_j \in \overline{\mathbf{X}}$ are data centers. Thus, the distance from a training center to the others is computed within the associated kernel space of that center. At test-time $\hat{k}(\overline{\mathbf{X}},\mathbf{x}^*) = \left(k_1(\overline{\mathbf{x}}_1,\mathbf{x}^*), k_2(\overline{\mathbf{x}}_2,\mathbf{x}^*),.. k_N(\overline{\mathbf{x}}_N,\mathbf{x}^*)\right)$, where $N$ is the number of training centers, and $\mathbf{x}^*$ is a test sample. In this work we restrain our focus to the squared exponential kernel distance: $$\begin{aligned} {1} \hat{k}(\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j) = k_i(\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j) &= \text{exp}\left( -\frac{1}{2 l^2_i} (\overline{\mathbf{x}}_i-\overline{\mathbf{x}}_j)(\overline{\mathbf{x}}_i-\overline{\mathbf{x}}_j)^T \right). \label{eq:univariate}\end{aligned}$$ where $l_i$ is the lengthscale associated with data center $\overline{\mathbf{x}}_i$. Given the use of individualized metrics per data center, the kernel ceases to be symmetric. Therefore, we can no longer employ the standard Cholesky decomposition for estimating the kernel-matrix inverse. We compute the kernel matrix inversion through SVD (Singular Value Decomposition). Despite this drawback, the individualized kernel metrics allow us to optimize the scale of the kernel locally, in the neighborhood of each data center. Multivariate Multi-modal Asymmetric Kernel ------------------------------------------ By allowing each center to define its own kernel metric, we change the model such that we can locally resize the kernel space to better map the target space. As highlighted in [@chen2013stress], not only the size of the kernel space is important but also the shape. Therefore, we also consider a multivariate extension of the model that allows for optimizing also the kernel shape per training center. $$\begin{aligned} {2} \hat{k}(\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j) = k_i(\overline{\mathbf{x}}_i,\overline{\mathbf{x}}_j) &= \text{exp}\left( -\frac{1}{2} (\overline{\mathbf{x}}_i-\overline{\mathbf{x}}_j) \bm{P}_i (\overline{\mathbf{x}}_i-\overline{\mathbf{x}}_j)^T \right). \label{eq:multivariate}\end{aligned}$$ where $\bm{P}_i$ is the precision matrix associated with the data center $\overline{\mathbf{x}}_i$, to be learned from training data samples other than the ones defining data centers. Kernel Metric Optimization {#ssec:kernel} -------------------------- Standardly in the literature, the Gaussian Process hyper-parameters are learned through gradient methods by maximizing the marginal likelihood over the weights. Given that we aim to optimize a kernel metric, following the metric learning literature [@chen2013stress; @globerson2005metric; @weinberger2009distance; @xing2002distance] we approach this problem discriminatively, and optimize the hyper-parameters with respect to the squared loss. Thus, we learn the appropriate kernel metric for each data center, $\overline{\mathbf{x}}_i \in \overline{\mathbf{X}}$, from training samples other than the data centers, $\mathbf{x}_n \in \mathbf{X}\setminus \overline{\mathbf{X}}$. We add a regularization term to the squared loss weighted by $\mu$. We use the regularized squared loss over the targets as the function to be minimized and we employ SGD (Stochastic Gradient Descent) by estimating the gradients with respect to each per-center lengthscale, $l_i$ in the univariate case and $\mathbf{P}_i$ in the multivariate case. $$\begin{aligned} {1} \mathcal{L}(f, \mathbf{y}^*) &= \left\{ \begin{array}{ll} \sum_{i=1}^N \left( \sum_{\mathbf{x}_n \in \mathbf{X} \setminus \overline{\mathbf{X}}} ( f(\mathbf{x}_n) - y_n^*)^2 + \mu l_i^2 \right), & \text{\scriptsize if univariate;}\\ \sum_{i=1}^N \left( \sum_{\mathbf{x}_n \in \mathbf{X} \setminus \overline{\mathbf{X}}} ( f(\mathbf{x}_n) - y_n^*)^2 + \mu \mid\mid \bm{P}_i \mid\mid \right), & \text{\scriptsize multivariate.} \end{array} \right. \label{eq:loss}\end{aligned}$$ We denote by $\mathbf{y}^*$ the training target vector composed of values $y_n^*$ for input training samples $\mathbf{x}_n$, where $\mathbf{x}_n \in \mathbf{X}\setminus \overline{\mathbf{X}}$ are not data centers, and $N$ is the number of data centers, $\mid\mid \cdot \mid\mid$ denotes the Frobenius norm in the multivariate case, and $f(\cdot)$ is the predictive function following eq \[eq:pred\_center\]. At each iteration we perform one gradient update step for all hyper-parameters, therefore, allowing them to be jointly learned. ---------------------------------------------- ------------------------------------------------- --------------------------------------------------- ----------------------------------------------------- --------------------------------------------- [NRMSE:]{} [82.99%]{} [65.32%]{} [**56.26%**]{} {width="16.00000%"} {width="16.00000%"} {width="16.00000%"} {width="16.00000%"} {width="23.00000%"} ---------------------------------------------- ------------------------------------------------- --------------------------------------------------- ----------------------------------------------------- --------------------------------------------- \ ### Univariate Multi-modal Kernel Optimization {#sssec:unikernel} The derivative of the loss with respect to the lengthscale per center, $l_i$, for the univariate case is given by the following formulation: $$\begin{aligned} {1} \frac{\partial\mathcal{L}(f, \mathbf{y}^*)}{\partial{l_i}} &= \sum_{i=1}^N \left\{ \sum_{\mathbf{x}_n \in \mathbf{X} \setminus \overline{\mathbf{X}} } 2 (f(\mathbf{x}_n) - y_n^*) \right. \notag \\ & \left. \left[ \frac{\partial{\hat{k}}(\cdot,\mathbf{x}_n)}{\partial {l_i}} \alpha_i + \hat{k}(\cdot,\mathbf{x}_n) \left( -\hat{\mathbf{K}}^{-1} \frac{\partial{\hat{\mathbf{K}}}}{\partial{l_i}} \hat{\mathbf{K}}^{-1} \mathbf{y} \right) \right] + 2\mu l_i \right\}, \label{eq:lossUni0}\end{aligned}$$ $$\begin{aligned} {3} \frac{\partial{\hat{k}}(\cdot,\mathbf{x}_n)}{\partial {l_i}} &= \left\{ \begin{array}{ll} \frac{1}{l_i^3}(\overline{\mathbf{x}}_j - \mathbf{x}_n)(\overline{\mathbf{x}}_j - \mathbf{x}_n)^T \hat{k}(\overline{\mathbf{x}}_j,\mathbf{x}_n), & \overline{\mathbf{x}}_j \in \overline{\mathbf{X}}, j = i;\\ 0, & \overline{\mathbf{x}}_j \in \overline{\mathbf{X}}, j \neq i. \end{array} \right.\\ \frac{\partial{\hat{\mathbf{K}}}}{\partial {l_i}} &= \left( \frac{\partial{\hat{k}}(\cdot,\overline{\mathbf{x}}_m)}{\partial {l_i}} \right)_{\overline{\mathbf{x}}_m\in\overline{\mathbf{X}}},\\ \hat{\mathbf{K}}^{-1} &= \left(\hat{k}(\overline{\mathbf{X}},\overline{\mathbf{X}}) + \sigma^2\mathbf{I}\right)^{-1},\\ \bm{\alpha} &= \left(\hat{k}(\overline{\mathbf{X}},\overline{\mathbf{X}}) + \sigma^2 \mathbf{I}\right)^{-1} \mathbf{y}, \label{eq:lossUni1}\end{aligned}$$ where we denote by $\overline{\mathbf{X}}$ the data centers, $\mathbf{\hat{K}}$ represents the asymmetric training kernel matrix, and $\mathbf{y}^*$ is a vector of training targets $y^*_n$ for training samples $\mathbf{x}_n \in \mathbf{X}\setminus \overline{\mathbf{X}}$ that are not data centers. The lengthscale hyper-parameters, $l_i$, are estimated per training data center rather than globally. [^1] ### Multivariate Multi-modal Kernel Optimization In the multivariate case we learn a precision matrix, $\bm{P}_i$, rather than a scalar lengthscale per data center, $\overline{\mathbf{x}}_i \in \overline{\mathbf{X}}$. Therefore, we have to ensure that the precision matrix learned is symmetric. For this, in the gradient computation, we apply the derivations for symmetric matrices. $$\begin{aligned} {1} \frac{\partial\mathcal{L}(f, \mathbf{y}^*)}{\partial\bm{P}_i} &= \left[\frac{\partial\mathcal{L}(f, \mathbf{y}^*)}{\partial\bm{P}_i}\right]+ \left[\frac{\partial\mathcal{L}(f,\mathbf{y}^*)}{\partial\bm{P}_i}\right]^T-\text{diag} \left[\frac{\partial\mathcal{L}(f,\mathbf{y}^*)}{\partial\bm{P}_i}\right] \label{eq:lossMulti0}\end{aligned}$$ For gradient computation, the only change in the multivariate case is in the derivative of the kernels with respect to the per-center precision matrices, $\mathbf{P}_i$: $$\begin{aligned} {3} \frac{\partial{\hat{k}}(\cdot,\mathbf{x}_n)}{\partial\bm{P}_i} &= \left\{ \begin{array}{ll} - \frac{1}{2}(\overline{\mathbf{x}}_j - \mathbf{x}_n)^T(\overline{\mathbf{x}}_j - \mathbf{x}_n) \hat{k}(\overline{\mathbf{x}}_j,\mathbf{x}_n), & \overline{\mathbf{x}}_j \in \overline{\mathbf{X}}, j = i;\\ 0, & \overline{\mathbf{x}}_j \in \overline{\mathbf{X}}, j \neq i. \end{array} \right.\\ \frac{\partial{\hat{\mathbf{K}}}}{\partial\bm{P}_i} &= \left( \frac{\partial{\hat{k}}(\cdot,\overline{\mathbf{x}}_m)}{\partial\bm{P}_i} \right)_{\overline{\mathbf{x}}_m\in\overline{\mathbf{X}}}. \label{eq:lossMulti1}\end{aligned}$$ In this case, an additional cone projection step [@weinberger2009distance] is applied after each update to ensure that the precision matrix remains positive definite. Model Properties ---------------- We analyze the properties considered in [@bellet2015metric], from the metric learning perspective:\ – *Learning paradigm.* Fully supervised, as we learn from training data the best lengthscale hyper-parameters with respect to the $L_2$ prediction loss.\ – *Form of metric.* Non-linear and local with respect to the predictive function. Different metrics are learned for different regions in the target space.\ – *Scalability.* Scales with the number of data centers in the univariate case, because the kernel matrix is computed over the data centers. Thus, is more efficient than using all training samples. In the multivariate case, we learn a precision matrix, and the method scales also with the number of data dimensions, making it more difficult to optimize.\ – *Optimality of the solution.* Our learning formulation, given in equations \[eq:lossUni0\] and \[eq:lossMulti0\], is a non-convex function with respect to the hyper-parameters and a global optimum is not guaranteed. For this reason we use an additional validation set during training on which we select among the local optima. Illustrative Results {#sec:illu} ==================== ![Numeric consistency: In red, the Euclidian distance between the estimated hyper-parameters by our method, using two data centers, and the optimal hyper-parameter of the Gaussian Process we have sampled from. We plot standard deviation over 3 repetitions. In blue, the distance between the optimal hyper-parameter and the one of the standard GP on the two centers. With more data, our estimated lengthscales approach the optimal value. []{data-label="fig:consistency"}](img/blobs/consistency){width="37.00000%"} Figure \[fig:blobs\] illustrates the need for the asymmetric model. If all samples share the same kernel metric, figures \[fig:blobs\].(ii) and \[fig:blobs\].(iii) would not be possible. When restricting our attention to few training samples, we lose the information of how the target function varies between the samples, yet the per-center kernel metrics help recover this information. In this illustration, we fix the training centers to the centers of the two ellipses. The $[0,1]$-normalized pixel coordinates are the input features, and the pixel intensities are the targets. We visualize three models: (i) *center-GP* — standard GP on the 2 data centers, and the optimal lengthscale hyper-parameter is found through cross validations over 100 randomly sampled training pixels; (ii) *univariate-AGP* — using equation \[eq:univariate\] with the optimal lengthscale per center estimated as in subsection \[sssec:unikernel\], and (iii) *multivariate-AGP* — using equation \[eq:multivariate\]. In figure \[fig:blobs\] the univariate asymmetric model estimates better the sizes of the two blobs, when compared to its center-based counterpart, while the multivariate asymmetric model has the lowest error, as it learns both the sizes and the shapes of the blobs. We also show the training and validation losses in figure \[fig:blobs\].(iv). We additionally analyze the consistency of our approach, numerically. For this, we sample on a uniform grid a standard 1$D$ Gaussian Process with a fixed lengthscale set to $13.5$. The aim is to test if the lengthscales estimated by our method tend to the true lengthscale of the Gaussian Process from which we have sampled, as we add more training data. We use two data centers for the kernel computation. Figure \[fig:consistency\] shows in red the Euclidian distance between the two estimated hyper-parameters by our method and the optimal one. We report standard deviation over 3 repetitions. In blue, we show the distance between the optimal hyper-parameter and the one estimated by the standard GP through grid search, using two data centers. With more training samples, our estimated lengthscale parameters approach the optimal value. The distance does not converge to 0 as we use only two data centers in the training kernel matrix. Experiments =========== Experimental Setup ------------------ Baseline \#Trainval \#Test \#Features --------------------------------------- ------------ -------- ------------ Temp [@titsias2013variational] 7,117 3,558 106 So2 [@titsias2013variational] 15,304 7,652 27 Airlines [@hoang2015unifying] 2,055 K 102 K 8 UCSD [@chan2012counting] 1,200 2,800 1,000 VOC-2007 [@chattopadhyay2017counting] 5,011 4952 1,000 : Datasets statistics. \[tab:datasets\] **Data splitting and center selection.** Table \[tab:datasets\] depicts the specifics of each dataset used. Each dataset is split into trainval and test, following the standard way. Given the non-convexity of the problem, we evaluate the hyper-parameters on a small validation set after each training epoch, and keep the best. For all datasets the validation set is obtained as 100 random samples taken from the trainval dataset. These samples are not used for defining the data centers or for training data statistics. Different center selection approaches are considered in section \[ssec:center\]. We standardize the data to zero mean and unit variance per dimension, by extracting statistics over the training data excluding the validation set.\ **Parameter setting.** In the SGD, the initial learning rate is set by looking at the plots of validation and training losses during training. We use a starting learning rate of $1.0e-5$ for *So2*, *Temp* and *Airlines* datasets, and an initial learning rate of $1.0e-11$ for *UCSD* and *VOC-2007*, where the data dimensionality is considerably higher. We use batch-SGD with mini-batches of 64 randomly selected training samples. Given the non-convexity of the solved problem, we make use of momentum and set it to $0.9$ as advised in [@sutskever2013importance]. The regularization term in the loss function, $\mu$, is set to $1.0e-5$. For the standard GP models as well as for *center-GP* — Gaussian Process trained on data centers only — we estimate the model hyper-parameters by performing grid-search and evaluating on the validation samples. We use the same procedure for initializing our per-center lengthscales, before optimizing them in the SGD.\ **Evaluation metric.** For comparison with existing work we report MSE (Mean Squared Error), RMSE (Root Mean Squared Error), or NRMSE (Normalized Root Mean Squared Error) defined as: $$\begin{aligned} {2} \text{NRMSE}(\mathbf{y},\mathbf{y}^*) &= \sqrt{\frac{1}{N} \sum_n^N \frac{( y_n - y^*_n)^2}{ \text{var}(\mathbf{y}_\text{train}) } }, \label{eq:nrms}\end{aligned}$$ where $\text{var}(\mathbf{y}_\text{train})$ is the label variance on the training data, $\mathbf{y}^*$ are the test targets, and $\mathbf{y}$ the predictions. ------------ --------- ---------- ---------- ------- -- \# Centers (r)[2-5]{} K-Means Sampling Spectral GMM 10 0.602 0.557 0.579 0.606 50 0.482 0.467 0.482 0.523 ------------ --------- ---------- ---------- ------- -- : The effect of the center selection method when considering: K-means, random sampling, spectral clustering and GMM center definition on the *Temp* dataset. \[tab:center\] -------------------------- -------------------------- -------- -------- -------- PIC FITC DTC (r)[1-2]{} Univariate Multivariate **30.093** ($\pm$ 2.285) **30.805** ($\pm$ 2.950) 33.351 39.530 39.531 -------------------------- -------------------------- -------- -------- -------- : Evaluation of the proposed models — *univariate-AGP* and *multivariate-AGP* on the large scale dataset used in [@hoang2015unifying]. Our proposed models are trained on 50 data centers. We compare our results with the methods analyzed in [@hoang2015unifying]: PIC, FITC, DTC. \[tab:airlines\] [c@[1.5in]{}c]{} \# Samples NRMSE ------------------ ------------ ----------- Titsias 2013 100 1.004 GP 100 0.985 center-GP 50 **0.984** univariate-AGP 50 **0.846** multivariate-AGP 50 **0.863** center-GP 10 0.985 univariate-AGP 10 **0.818** multivariate-AGP 10 **** & \# Samples NRMSE ------------------ ------------ ----------- Titsias 2013 100 0.489 GP 100 0.533 center-GP 50 0.559 univariate-AGP 50 **0.482** multivariate-AGP 50 **** center-GP 10 0.642 univariate-AGP 10 0.602 multivariate-AGP 10 0.493 \ (a) *So2* data evaluation. & (b) *Temp* data evaluation.\ \[tab:res2\] Center Selection {#ssec:center} ---------------- Here we analyze the effect of the center selection method on the overall performance of our method. For this we use the *Temp* dataset which has $106$ dimensions per sample. We consider four center selection methods: K-Means, random sampling, spectral clustering and GMM (Guassian Mixture Model). We test on our *univariate-AGP* — using univariate individualized lengthscales — with 10 and 50 centers, respectively. Table \[tab:center\] indicates that the choice of the centers is not essential as all methods perform similar. Given that the strength of the model is in learning individualized hyper-parameters and less in the method used for defining centers, in our subsequent experiments we rely on k-means clustering. Multi-modal Approach Evaluation ------------------------------- Table \[tab:res2\] depicts the results of our approaches on the *So2* and *Temp* datasets when compared to [@titsias2013variational] and with the standard Gaussian Process model. The gain brought by the multi-modal asymmetric methods over the center-based Gaussian Process and the standard Gaussian Process is more obvious for the *Temp* dataset. This can be explained by the larger number of dimensions to learn from, in the multivariate case. On both datasets, the proposed models outperform [@titsias2013variational], while using only 50 data centers. The performance decreases slightly with the increase in the number of data centers for the multivariate asymmetric models. This is due to the model being trained for the same number of iterations as the univariate case, while having to learn a larger number of parameters — a precision matrix of size $D \times D$. With the increase in data dimensionality, the multivariate model becomes considerably slower and harder to optimize. This represents a drawback of the proposed multivariate approach. The variability in the target space also affects the ease with which a good solution is found. -------------------------------------------------------- -------------------- [@chattopadhyay2017counting] glance-noft-2L 0.50 ($\pm$ 0.02) [@chattopadhyay2017counting] glance-sos-2L 0.51 ($\pm$ 0.02) [@chattopadhyay2017counting] aso-sub-ft-1L-3$\times$3L 0.43 ($\pm$ 0.01) [@chattopadhyay2017counting] seq-sub-ft-3$\times$3 0.42 ($\pm$ 0.01) AGP-25 Univariate 0.43 ($\pm$ 0.002) -------------------------------------------------------- -------------------- : RMSE results on the VOC-2007 general object counting dataset. The first two “glance" models use global image features learned in a deep learning framework, which is similar to us. The last two models use local information by dividing the image into a 3$\times$3 grid and extracting deep learning features from each cell. Our method outperforms the models relying on global image features. \[tab:voc\] State-of-the-art Comparison --------------------------- ### Large Scale Regression Problem Here, we consider a more challenging task where the number of training samples is markedly high. We compare against effective state-of-the-art methods that focus on the same problem as we do — representing the data using an informative subset while retaining the descriptiveness of the model [@hoang2015unifying]. We use the *Airlines* dataset of [@hoang2015unifying] containing over $2,000,000$ training samples. The models considered are: DTC (Deterministic Training Conditional) [@seeger2003fast], FITC (Fully Independent Training Conditional) [@snelson2005sparse], and PIC (Partially Independent Conditional) [@snelson2007local]. For evaluating our models we repeat each experiment three times and report the mean RMSE (Root Mean Squared Error) together with the standard deviation. Table \[tab:airlines\] shows that our proposed *AGP* models perform well when dealing with a prohibitive number of training samples. Our approach outperforms existing methods [@seeger2003fast; @snelson2005sparse; @snelson2007local] while using only a limited number of data centers. ### Realistic Data: Counting from Images [c]{} (a) Squared losses on UCSD.\ ![(a) MSE results on the UCSD pedestrians counting dataset when compared with three prior works [@dalal2005histograms; @felzenszwalb2008discriminatively; @chan2012counting]. We obtain comparable performance with prior work, though we use only global deep learning features, while [@chan2012counting] relies on motion segmentation masks. (b) The training and validation squared losses on the UCSD pedestrian counting dataset.[]{data-label="fig:ucsd"}](img/results/losses_ucsd0 "fig:"){width="35.00000%"}\ (b) MSE on UCSD.\ ------------------------------------- -------------------- [@dalal2005histograms] 39.75 [@felzenszwalb2008discriminatively] 24.72 [@chan2012counting] 9.95 AGP-10 Univariate 16.37 ($\pm$ 0.41) AGP-25 Univariate 13.90 ($\pm$ 0.05) ------------------------------------- -------------------- \ We test our regression approach on two realistic image datasets — UCSD [@chan2012counting] and VOC-2007 [@pascal-voc-2007] — for people and generic object counting, respectively. Given the image data, we rely on deep learning features. We extract 1,000 dimensional features as the output of the fully-connected layer of the ResNet-50 [@he2016deep] pretrained on ImageNet [@russakovsky2015imagenet]. For computational efficiency here we use only the univariate version of our approach with 25 centers, since the multivariate version requires optimizing a precision matrix of size $1,000 \times 1,000$. **Pedestrian counting.** Figure \[fig:ucsd\] shows the results of our approach on the realistic problem of pedestrian counting from images on the UCSD dataset, together with the training and validation losses. We compare with [@dalal2005histograms] that uses low level image features, [@felzenszwalb2008discriminatively] relying on a person detection method specifically trained for the task, and [@chan2012counting] which employs motion segmentation masks. Unlike these methods, we do not use either motion segmentation masks or class specific detectors. We use only global image features extracted from a pretrained deep network, and we manage to obtain comparable performance with [@chan2012counting], while greatly outperforming [@dalal2005histograms; @felzenszwalb2008discriminatively]. **Generic object counting.** In table \[tab:voc\] the goal is generic object counting on the VOC-2007 generic object dataset. We compare with the set of models proposed in the very recent deep learning method of [@chattopadhyay2017counting]. Our features are extracted from a pretrained deep learning model, while theirs are specifically fine-tuned for this counting task. We outperform their *“glance"* models, which similar to us, rely on global image features. We additionally obtain comparable performance to *aso-sub-ft-1L-3$\times$3* and *seq-sub-ft-3$\times$3* which rely on local image information, as they divide each image into a 3$\times$3 grid and extract features from each cell. It is worthwhile noting that we use only 25 data centers for computing the kernel matrix, and achieve comparable performance with methods relying on stronger features. These results support our approach. Conclusions =========== This work brings forth an asymmetric kernel for the Gaussian Process model. This encompasses three components: (i) training on training centers only, (ii) learning individualized kernel metrics per center and, (iii) extending the lengthscale hyper-parameter to a precision matrix, thus learning not only the appropriate size but also the shape in the kernel metric. Due to the limitations imposed by the dependency between per-center hyper-parameters, we discriminatively solve the problem through metric learning. Individualized kernel metrics entail the loss of the symmetry in the kernel matrix. However, this has the gain of better describing the target function in the neighborhood of the center points, used for kernel computation. [ **Acknowledgements** Research supported by the Dutch national program COMMIT. Thanks to dr. Mijung Park and dr. Marco Loog for the insightful suggestions and advice. ]{} [^1]: A simple univariate torch implementation can be found at:\ *https://silvialaurapintea.github.io/code/gp.lua*.

--- author: - 'Elton J. G. Santos, Andrés Ayuela and Daniel Sánchez-Portal' title: ' First-Principles Study of the Electronic and Magnetic Properties of Defects in Carbon Nanostructures' --- Introduction {#intro} ============ The experimental discovery of graphene, a truly two-dimensional crystal, has led to the rapid development of a very active line of research. Graphene is not only a fundamental model to study other types of carbon materials, but exhibits many uncommon electronic properties governed by a Dirac-like wave equation (Novoselov et al. 2004 and 2005) Graphene, which exhibits ballistic electron transport on the submicrometre scale, is considered a key material for the next generation of carbon-based electronic devices (Geim et al 2007, Castro Neto et al. 2009). In particular, carbon-based materials are quite promising for spintronics and related applications due to their long spin relaxation and decoherence times owing to the spin-orbit interaction and the hyperfine interaction of the electron spins with the carbon nuclei, both negligible  (Hueso et al. 2008, Trauzettel et al. 2007, Tombros et al. 2007, Yazyev 2008). In addition, the possibility to control the magnetism of edge states in nanoribbons and nanotubes by applying external electric fields introduces an additional degree of freedom to control the spin transport (Son et al. 2006, Ma[ñ]{}anes et al. 2008). Nevertheless, for the design of realistic devices, the effect of defects and impurities has to be taken into account. Indeed, a substantial amount of work has been devoted to the study of defects and different types of impurities in these materials. The magnetic properties of point defects, like vacancies, adatoms or substitutionals, have been recognized by many authors (Lehtinen et al. 2003, Palacios et al. 2008, Kumazaki et al. 2008, Santos et al. 2010a, Santos et al. 2010b, Fuhrer et al. 2010, Ugeda et al. 2010, Yazyev et al. 2007). It has now become clear that the presence of defects can affect the operation of graphene based devices and can be used to tune their response. In the present Chapter, we provide a review of some recent computational studies on the role of some particular type of defects in determining the electronic and, in particular, the magnetic properties of graphene and carbon nanotubes (Santos et al. 2008, 2010a, 2010b, 2010c, 2011, 2012a, 2012b). We will consider two types of defects: Substitutional transition metals and covalently bonded adsorbates. For the substitutional transition metals, we first present some of the existing experimental evidence about the presence of such impurities in graphene and carbon nanotubes. Then, we summarize our results for the structural, electronic and magnetic properties of substitutional transition metal impurities in graphene. We show that all these properties can be explained using a simple model based on hybridization between the $d$ shell of the metal atoms and the defects states of an unrelaxed carbon vacancy. We also show that it is possible to change the local spin moment of the substitutional impurities by applying mechanical deformations to the carbon layer. This effect is studied in detail in the case of Ni substitutionals. Although these impurities are non-magnetic at a zero strain, we demonstrate that it is possible to switch on the magnetism of Ni-doped graphene either by applying uniaxial strain or curvature to the carbon layer. Subsequently, we explore the magnetic properties induced by covalent functionalization of graphene and carbon nanotubes. We find that the magnetic properties in this case are universal, in the sense that they are largely independent of the particular adsorbate: As far the adsorbate is attached to the carbon layer through a single C-C covalent bond (or other weakly-polar covalent bond), there is always a spin moment of 1 $\mu_B$ associated with each adsorbate. We show that this result can be understood in terms of a simple model based on the so-called $\pi-$vacancy, i.e. one $p_z$ orbital removed from a $\pi$-tight-binding description of graphene. This model captures the main features induced by the covalent functionalization and the physics behind. In particular, using this model we can easily predict the total spin moment of the system when there are several molecules attached to the carbon layer simultaneously. Finally, we have also studied in detail the magnetic couplings between Co substitutional impurities in graphene. Surprisingly, the Co substitutional impurity is also well described in terms of a simple $\pi$-vacancy model. Substitutional Transition Metal Impurities in Graphene {#substitutionals} ======================================================= Experimental Evidences {#substitutionals-exp} ---------------------- Direct experimental evidence of the existence of substitutional impurities in graphene, in which a single metal atom substitutes one or several carbon atoms in the layer, has been recently provided by Gan and coworkers (Gan et al. 2008). Using high-resolution transmission electron microscopy (HRTEM), these authors could visualize individual Au and Pt atoms incorporated into a very thin graphitic layer probably consisting of one or two graphene layers. From the real-time evolution and temperature dependence of the dynamics, they obtained information about the diffusion of these atoms. Substantial diffusion barriers ($\sim$2.5 eV) were observed for in-plane migration, which indicates the large stability of these defects and the presence of strong carbon-metal bonds. These observations indicate that the atoms occupy substitutional positions. In another experiment using double-walled carbon nanotubes (DWCNT) (Rodriguez-Manzo et al. 2010), Fe atoms were trapped at vacancies likewise the previous observations for graphene layers. In these experiments, the electron beam was directed onto a predefined position and kept stationary for few seconds in order to create a lattice defect. Fe atoms had been previously deposited on the nanotube surface before the defect formation. After irradiation, a bright spot in the dark-field image was observed. A quantitative analysis of the intensity profile showed an increase of the scattered intensity at the irradiated position relative to the center of the pristine DWCNT. This demonstrates that at the defect position, on the top or bottom side of the DWCNT, a Fe atom was trapped. Recent evidence was also reported for substitutional Ni impurities in single-walled carbon nanotubes (SWCNT) (Ushiro et al. 2006) and graphitic particles (Banhart et al. 2000). Ushiro and co-authors (Ushiro et al. 2006) showed that Ni substitutional defects were present in SWCNT samples synthesized using Ni-containing catalysts even after careful purification. According to their analysis of X-ray absorption data (XANES), the most likely configuration for these defects has a Ni atom replacing a carbon atom. The presence of substitutional defects in the samples can have important implications for the interpretation of some experiments. For example, substitutional atoms of transition metals are expected to strongly influence the magnetic properties of graphenic nanostructures. Interestingly, transition metals like Fe, Ni or Co are among the most common catalysts used for the production of SWCNTs (Dresselhaus et al. 2001). Furthermore, the experiments by Banhart and coworkers (Rodriguez-Manzo et al. 2009) have demonstrated that it is possible to create individual vacancies at desired locations in carbon nanotubes using electron beams. This experiment, combined with the observed stability of substitutional impurities, opens a route to fabricate new devices incorporating substitutional impurities at predefined locations. These devices would allow for the experimental verification of the unusual magnetic interactions mediated by the graphenic carbon network that have been predicted recently (Brey et al. 2007, Kirwan et al. 2008, Santos et al. 2010a). This becomes particularly interesting in the light of the recent finding that the spin moment of that impurities could be easily tuned by applying uniaxial strain and/or mechanical deformations to the carbon layer (Santos et al. 2008, Huang et al. 2011, Santos et al. 2012a). In spite of this, the magnetic properties of substitutional transition-metal impurities in graphenic systems were not studied in detail until very recently. Few calculations have considered the effect of this kind of doping on the magnetic properties of the graphenic materials, and this will be the main topic of the following sections. Structure and Binding {#substitutionals-theory1} ---------------------- ![\[fig:fig1-njp\] Typical geometry of transition and noble substitutional metal atoms in graphene. The metal atom moves upwards from the layer and occupies, in most cases, an almost perfectly symmetric three-fold position with C$_{3v}$ symmetry. ](figures_arXiv/fig1_black_v010812ink.eps){width="3.250in"} In Fig. \[fig:fig1-njp\] we show the typical geometry found in our calculations for a graphene layer where one carbon atom has been substituted by a metal impurity. The metal atom appears always displaced from the carbon layer. The height over the plane defined by its three nearest carbon neighbors is in the range 1.7-0.9 Å. These three carbon atoms are also displaced over the average position of the graphene layer by 0.3-0.5 Å. The total height (h$_z$) of the metal atom over the graphene plane is the sum of these two contributions and ranges between 1.2-1.8 Å, as shown in panel (c) of Fig. \[fig:fig2-njp\]. In most cases the metal atom occupies an almost perfectly symmetric configuration with C$_{3v}$ symmetry. Exceptions are the studied noble metals, that are slightly displaced from the central position, and Zn that suffers a Jahn-Teller distortion in its most stable configuration. However, we have found that it is also possible to stabilize a symmetric configuration for Zn with a binding energy only $\sim$150 meV smaller. This configuration will be referred to as Zn$_{{\rm C}_{3v}}$ throughout this chapter. Figures \[fig:fig2-njp\](a)-(c) present a summary of the structural parameters of substitutional 3$d$ transition metals, noble metals and Zn in graphene. Our calculations are in very good agreement with the results of a similar study performed by Krasheninnikov et al. (Krasheninnikov et al. 2009), although they overlooked the existence of the high-spin Zn$_{{\rm C}_{3v}}$ configuration. Solid circles correspond to calculations using the [Siesta]{} code (Soler et al. 2002) with pseudopotentials (Troullier and Martins 1993) and a basis set of atomic orbitals (LCAO), while open squares stand for [Vasp]{} (Kresse and Hafner 1993, Kresse and Furthm[ü]{}ller 1996) calculations using plane-waves and PAW potentials (Bl[ö]{}chl 1994). As we can see, the agreement between both sets of calculations is excellent. Data in these figures correspond to calculations using a 4$\times$4 supercell of graphene. For several metals we have also performed calculations using a larger 8$\times$8 supercell and find almost identical results. This is particularly true for the total spin moments, which are less dependent on the size of the supercell, but require a sufficiently dense k-point sampling to converge. The behavior of the metal-carbon bond length and h$_z$ reflect approximately the size of the metal atom. For transition metals these distances decrease as we increase the atomic number, with a small discontinuity when going from Mn to Fe. The carbon-metal bond length reaches its minimum for Fe (d$_{\rm{C-Fe}}$=1.76 Å), keeping a very similar value for Co and Ni. For Cu and Zn the distances increase reflecting the fully occupied 3$d$ shell and the large size of the 4$s$ orbitals. Among the noble metals we find that, as expected, the bond length largely increases for Ag with respect to Cu, but slightly decreases when going from Ag to Au. The latter behavior is understood from the compression of the 6$s$ shell due to scalar relativistic effects. [@lccc]{}\ & d$_{\rm{C-M}}$(Å) & h$_z$(Å) & $\theta$ ($^{\circ}$)\ \ Cu & 1.93, 1.90, 1.90 & 1.40 & 88.9, 88.9, 95.2\ Ag & 2.23, 2.19, 2.19 & 1.84 & 71.7, 71.7, 76.7\ Au & 2.09, 2.12, 2.12 & 1.71 & 78.0, 78.0, 81.6\ Zn & 2.06, 1.89, 1,89 & 1.54 & 88.3 88.3 103.9\ Zn$_{{\rm C}_{3v}}$ & 1.99 & 1.67 & 87.9\ \ As already mentioned, noble metals and Zn present a distorted configuration. In Table \[tab:geometals\] we find the corresponding structural parameters. For Cu and Ag one of the metal-carbon bond lengths is slightly larger than the other two, whereas for Au one is shorter than the others. However, the distortions are rather small with variations of the bond lengths smaller than 2 %. The larger scalar relativistic effects in Au give rise to slightly smaller metal-carbon bond lengths for this metal as compared to Ag. In the case of Zn atoms, the distorted configuration presents one larger Zn-C bond (by $\sim$3.5%) and two shorter bonds ($\sim$5%) compared with the bond length (1.99 Å) of the undistorted geometry. The distorted configuration is more stable by 160 meV (with SIESTA, this energy difference is reduced to 120 meV using [VASP]{}). This rather small energy difference between the two configurations might point to the appearance of non-adiabatic electronic effects at room temperature. ![\[fig:fig2-njp\] Structural parameters and binding energies of substitutional transition and noble metals in graphene. Bond lengths and angles have been averaged for the noble metals. The data presented for Zn correspond to the high-spin solution with C$_{3v}$ symmetry, and are very close to the averaged results for the most stable distorted solution. Adapted from (Santos et al. 2010b). ](figures_arXiv/all_metals_parameters1_2_3_v040113.eps){width="3.250in"} The binding energies of the studied substitutional metal atoms in graphene are shown in Fig. \[fig:fig2-njp\](d). In general, the binding energy correlates with the carbon-metal bond length, although the former exhibits a somewhat more complicated behavior. The binding energies for transition metals are in the range of 8-6 eV. Subtitutional Ti presents the maximum binding energy, which can be easily understood since for this element all the metal-carbon bonding states (Santos et al. 2010b) are fully occupied. One could expect a continuous decrease of the binding energy as we move away from Ti along the transition metal series, and the non-bonding 3$d$ and the metal-carbon antibonding levels become progressively populated. However, the behavior is non-monotonic and the smaller binding energies among the 3$d$ transition metals are found for Cr and Mn, and a local maximum is observed for Co. This complex behavior is explained by the simultaneous energy down-shift and compression of the 3$d$ shell of the metal impurity as we increase the atomic number (Santos et al. 2010b). In summary, the binding energies of the substitutional 3$d$ transition metals are determined by two competing effects: ([*i*]{}) as the 3$d$ shell becomes occupied and moves to lower energies the hybridization with the carbon vacancy states near the Fermi energy (E$_F$) is reduced, which decreases the binding energy; ([*ii*]{}) the transition from Mn to late transition metals is accompanied by a reduction of the metal-carbon bond length by $\sim$0.1 Å, which increases the carbon-metal interaction and, correspondingly, the binding energy. The binding energies for noble metals are considerably smaller than for transition metals and mirror the reverse behavior of the bond lengths: 3.69, 1.76 and 2.07 eV, respectively, for Cu, Ag and Au. The smallest binding energy ($\sim$1 eV) among the metals studied here is found for Zn, with both $s$ and $d$ electronic shells filled. Spin Moment Formation: Hybridization between Carbon Vacancy and 3$d$ Transition Metal Levels {#substitutionals-spins} -------------------------------------------------------------------------------------------- ![\[fig:fig3-njp\] Spin moment of substitutional transition and noble metals in graphene as a function of the number of valence electrons (Slater-Pauling-type plot). Black symbols correspond to the most stable configurations using GGA. Results are almost identical using [Siesta]{} and [Vasp]{} codes. Three main regimes are found as explained in detail in the text: ([*i*]{}) filling of the metal-carbon bonding states gives rise to the non-magnetic behavior of Ti and Sc; ([*ii*]{}) non-bonding $d$ states are filled for V, Cr and Mn giving rise to high spin moments; ([*iii*]{}) for Fe all non-bonding levels are occupied and metal-carbon antibonding states start to be filled giving rise to the observed oscillatory behavior for Co, Ni, Cu and Zn. Open and red symbols correspond, respectively, to calculations of Fe using GGA+U and artificially increasing the height of the metal atom over the graphene layer (see the text). Symbol marked as Zn$_{{\rm C}_{3v}}$ corresponds to a Zn impurity in a high-spin symmetric C$_{3v}$ configuration. Adapted from (Santos et al. 2010b). ](figures_arXiv/Figure-diagram_spin_vs_Z_v062011.eps){width="3.950in"} Our results for the spin moments of substitutional transition and noble metals in graphene are shown in Fig. \[fig:fig3-njp\] (Santos et al. 2010b). Similar results have been found by several authors (Krasheninnikov et al. 2009, Huang et al. 2011). We have developed a simple model that allows to understand the behavior of the spin moment, as well as the main features of the electronic structure, of these impurities (Santos et al. 2010b). Our model is based on the hybridization of the 3$d$-states of the metal atom with the defect levels of a carbon vacancy in graphene. In brief, we can distinguish three different regimes according to the filling of electronic levels: - [**bonding regime**]{}: all the carbon-metal bonding levels are filled for Sc and Ti and, correspondingly, their spin-moments are zero; - [**non-bonding regime**]{}: non-bonding 3$d$ levels become populated for V and Cr giving rise to a spin moment of, respectively, 1 and 2 $\mu_B$ with a strong localized $d$ character. For Mn one additional electron is added to the antibonding $d_{z^2}$ level and the spin moment increases to 3 $\mu_B$; - [**antibonding regime**]{}: finally, for Fe and heavier atoms all the non-bonding 3$d$ levels are occupied and the spin moment oscillates between 0 and 1 $\mu_B$ as the antibonding metal-carbon levels become occupied. The sudden decrease of the spin moment from 3 $\mu_B$ for Mn to 0 $\mu_B$ for Fe is characterized by a transition from a complete spin-polarization of the non-bonding 3$d$ levels to a full occupation of those bands. However, this effect depends on the ratio between the effective electron-electron interaction within the 3$d$ shell and the metal-carbon interaction (Santos et al. 2010b). If the hybridization with the neighboring atoms is artificially reduced, for example by increasing the Fe-C distance, Fe impurities develop a spin moment of 2 $\mu_B$ (see the red symbol in Fig. \[fig:fig3-njp\]). Our results also show that it is possible to switch on the spin moment of Fe by changing the effective electron-electron interaction within the 3$d$ shell. These calculations were performed using the so-called GGA+U method. For a large enough value of U (in the range 2-3 eV), Fe impurities develop a spin moment of 1 $\mu_B$. It is noteworthy that this behavior is unique to Fe: using similar values of U for other impurities does not modify their spin moments. S$_M$($\mu_B$) S$_C$ ($\mu_B$) S$_{tot}$ ($\mu_B$) --------------------- ---------------- ------------------- --------------------- V 1.21 -0.09 1.0 Cr 2.53 -0.20 2.0 Mn 2.91 -0.10 3.0 Co 0.44 0.06 1.0 Cu 0.24 -0.03, 0.31, 0.31 1.0 Ag 0.06 -0.31, 0.54, 0.54 1.0 Au 0.16 -0.28, 0.50, 0.50 1.0 Zn$_{{\rm C}_{3v}}$ 0.23 0.37 2.0 : \[tab:spinmoment\] Mulliken population analysis of the spin moment in the central metal impurity (S$_M$) and the carbon nearest neighbors (S$_C$) for different substitutional impurities in graphene. S$_{tot}$ is the total spin moment in the supercell. At the level of the GGA calculations, Fe constitutes the border between two different trends of the spin moment associated with the substitutional metal impurities in graphene. For V, Cr and Mn the spin moment is mainly due to the polarization of the 3$d$ shell of the transition metal atoms. The strongly localized character of the spin moment for those impurities, particularly for V and Cr, is corroborated by the Mulliken population analysis shown in Table \[tab:spinmoment\]. For Co, Ni, the noble metals and Zn the electronic levels close to the E$_F$ have a much stronger contribution from the carbon neighbors. Thus, for those impurities we can talk about a “defective graphene"-like magnetism. Indeed, it is possible to draw an analogy between the electronic structure of the late transition, noble metals and Zn substitutional impurities and that of the isolated unreconstructed (D$_{3h}$) carbon vacancy (Santos et al. 2008, Santos et al. 2010a, Santos et al. 2010b). The stronger carbon contribution and delocalization in the distribution of the spin moment for Co, the noble metals and Zn impurities is evident in Table \[tab:spinmoment\]. ![\[fig:fig5-njp\] (a) Scheme of the hybridization between the 3$d$ levels of Ti and the localized impurity levels of the D$_{3h}$ carbon vacancy. Only $d$ levels of Ti are represented since our calculations show that, at least for transition metals, the main contribution from $s$ levels appears well above E$_F$. Levels with A symmetry are represented by gray (green) lines, while those with E symmetry are marked with black lines. The region close to E$_F$ is highlighted by a square. (b) Schematic representation of the evolution of the electronic structure near E$_F$ for several substitutional transition metals in graphene. The spin moment (S) is also indicated. Substitutional Sc impurities act as electron acceptors, causing the p-doping of the graphene layer. Adapted from (Santos et al. 2010b)](figures_arXiv/Mn_levels-scheme_diagram_v020512.eps){width="3.75250in"} In the following we present the “hybridization” model that allows to distinguish the three regimes of the spin-moment evolution described before, corresponding to the filling of levels of different character. We have found that the electronic structure of the substitutional impurities can be easily understood as a result of the interaction of two entities: (i) the localized defect levels associated with a symmetric $D_{3h}$ carbon vacancy and, (ii) the 3$d$ states of the metal atom, taking also into account the down shift of the 3$d$ shell as the atomic number increases. We considered explicitly the 3$d$ states of the metal atom since our calculations show that, at least for transition metals, the main contribution from 4$s$ orbitals appears well above E$_F$. To illustrate the main features of our model in Figure \[fig:fig5-njp\] (a) we present a schematic representation of the hybridization of the 3$d$ levels of Ti with those of an unreconstructed $D_{3h}$ carbon vacancy in graphene. The interested reader can see (Santos et al. 2010b) for an extension of the model for the other metals and technical details. The defect levels of the unreconstructed D$_{3h}$ vacancy can be easily classified according to their $sp$ or $p_z$ character and whether they transform according to A or E-type representations. A scheme of the different level can be found in Fig. \[fig:fig5-njp\], while the results of a DFT calculation are depicted in Fig. \[fig1-co\] (b) (see also Santos et al 2010b and Amara et al 2007). Close to the E$_F$ we can find a fully symmetric A $p_z$ level (thus belonging to the A$^{\prime\prime}_2$ irreducible representation of the D$_{3h}$ point group) and two degenerated defect levels with E symmetry and $sp$ character (E$^\prime$ representation). Approximately 4 eV below E$_F$ we find another defect level with A $sp$ character (A$^\prime_1$ representation). Due to the symmetric position of the metal atom over the vacancy the system has a $C_{3v}$ symmetry and the electronic levels of the substitutional defect can still be classified according to the $A$ or $E$ irreducible representations of this point group. Of course, metal and carbon vacancy states couple only when they belong to the same irreducible representation. Thus, occupied $A$ $p_z$ and $A$ $sp$ vacancy levels can only hybridize with the 3$d_{z^2}$ orbitals ($A_1$ representation), while all the other 3$d$ metal orbitals can only couple to the unoccupied $E$ $sp$ vacancy levels. With these simple rules in mind and taking into account the relative energy position of carbon and metal levels, that changes as we move along the transition metal series, we can understand the electronic structure of substitutional transition metals in graphene as represented in Fig. \[fig:fig5-njp\] (a) and (b). Some parameters in the model can be obtained from simple calculations. For example, a rough estimate of the position of the 3$d$ shell of the metal atom respect to the graphene E$_F$ is obtained from the positions of the atomic levels. The relative strengths of the different carbon-metal hoppings can be estimated from those of the corresponding overlaps. With this information it is already possible to obtain most of the features of the model in Fig. \[fig:fig5-njp\]. However, some uncertainties remain, particularly concerning the relative position of levels with different symmetry. To solve these uncertainties the simplest approach is to compare with first-principles calculations. The details of the model presented in Fig. \[fig:fig5-njp\] have been obtained from a thorough analysis of our calculated band structures (Santos at al. 2010b). In particular, we have used the projection of the electronic states into orbitals of different symmetry as an instrumental tool to classify the levels and to obtain the rational that finally guided us to the proposed model. In contrast, it is interesting to note that some features that derive from our way to understand the electronic structure of these defects are very robust and could actually be guessed without direct comparison with the calculated band structures. For example, the fact that for V we start to fill the non-bonding 3$d$ states, and this impurity, as well as Cr and Mn, develops a spin moment, can be argued from simple symmetry and electron-counting arguments. According to our model for the substitutional metals, there are three localized defect levels with $A_1$ character and three twofold-degenerate levels with $E$ character. Two of these $E$ levels correspond to bonding-antibonding $sp$-$d$ pairs, while the third one corresponds to 3$d$ non-bonding states. For Sc-Mn the three $A_1$ levels can be pictured as a low lying bonding level with $A$ $sp$-$d_{z^2}$ character and a bonding-antibonding pair with $A$ $p_z$-$d_{z^2}$ character. As shown in Fig. \[fig:fig5-njp\] we have four metal-vacancy bonding levels (two $A$ and one doubly-degenerate $E$ levels) that can host up to eight electrons. For instance, Ti contributes with four valence electrons, and there are four electrons associated with the localized carbon-vacancy levels. Ti has the bonding states completely occupied. Consequently, Ti presents the highest binding energy among all 3$d$ transition metals and has a zero spin moment. Figure \[fig:fig5-njp\] (b) shows the situation for other impurities in the series Sc-Mn. Substitutional Sc impurities have zero spin-moment because they act as electron acceptors. Note that all the bonding levels are also filled for Sc, causing a p-doping of the graphene layer. As already mentioned V, Cr and Mn present an increasing spin moment due to the filling of the non-bonding levels, while for Fe the non-bonding shell is completely filled. Late transition, noble metals and Zn substitutional impurities have the filled levels coming from an antibonding interaction between the carbon vacancy and the metal states. The character and spatial localization of those levels are very similar to those of the levels of the D$_{3h}$ vacancy close to E$_F$. Co substitutionals present a singly-degenerate half-occupied defect level at E$_F$. As we will discuss in more detail in the next section, this level is reminiscent of the state that appears at E$_F$ associated with a single carbon vacancy in a $\pi$-tight-binding description of graphene (Palacios et al. 2008). A second electron occupies this level for Ni impurities, and the spin polarization is lost (Santos et al. 2008). An additional electron is added for noble metal impurities. This electron populates a doubly-degenerate level coming from the antibonding interaction of the 2$sp^{2}$ lobes in the nearest carbon neighbors, with the orbitals of d$_{xz}$ and d$_{yz}$ symmetries in the metal impurity. This state is reminiscent of E $sp$ level of the D$_{3h}$ carbon vacancy. The occupation of this two-fold degenerate state with only one electron explains both, the observed 1 $\mu_B$ spin moment and the structural distortion of the noble metal impurities (Santos et al. 2010b). As we will see in Section \[strain-tensile\], the E $sp$ impurity levels also play a crucial role to explain the switching on of the magnetism of Ni impurities under mechanical deformations and uniaxial strain. For Zn two electrons occupy the two-fold degenerate E $sp$ level. As a consequence, the system suffers a Jahn-Teller distortion and has a zero spin moment. However, it is possible to stabilize a symmetric configuration (Zn$_{{\rm C}_{3v}}$) with a moment of 2 $\mu_B$ and only slightly higher in energy (Santos et al. 2010b). Co Substitutionals in Graphene as a Realization of Single $\pi$-vacancies {#Co-analogy} --------------------------------------------------------------------------- ![(a) Isosurface of the spin density induced by a Co$_{sub}$ defect. Positive and negative spin densities correspond to light and dark surfaces with isovalues of $\pm$0.008 e$^-$/Bohr$^3$, respectively. Panel (b) presents the spin-unpolarized band structure of an unreconstructed $D_{3h}$ carbon vacancy. Panel (c) and (d) show, respectively, the band structure of majority and minority spins for a Co$_{sub}$ defect in a similar cell. The size of filled symbols in panel (b) indicate the contribution of the $p_{z}$ orbitals of the C atoms surrounding the vacancy, whereas empty symbols correspond to the $sp^{2}$ character. In panels (c) and (d), the filled and empty circles denote the contribution of hybridized Co $3d_{z^2}$-C $2p_{z}$ and Co $3d$-C $2sp^{2}$ characters, respectively. E$_F$ is set to zero. Adapted from (Santos et al. 2010a).[]{data-label="fig1-co"}](figures_arXiv/Cosub_spin-density_book_springer_v040113.eps "fig:"){width="3.1502400in"}\ ![(a) Isosurface of the spin density induced by a Co$_{sub}$ defect. Positive and negative spin densities correspond to light and dark surfaces with isovalues of $\pm$0.008 e$^-$/Bohr$^3$, respectively. Panel (b) presents the spin-unpolarized band structure of an unreconstructed $D_{3h}$ carbon vacancy. Panel (c) and (d) show, respectively, the band structure of majority and minority spins for a Co$_{sub}$ defect in a similar cell. The size of filled symbols in panel (b) indicate the contribution of the $p_{z}$ orbitals of the C atoms surrounding the vacancy, whereas empty symbols correspond to the $sp^{2}$ character. In panels (c) and (d), the filled and empty circles denote the contribution of hybridized Co $3d_{z^2}$-C $2p_{z}$ and Co $3d$-C $2sp^{2}$ characters, respectively. E$_F$ is set to zero. Adapted from (Santos et al. 2010a).[]{data-label="fig1-co"}](figures_arXiv/bands_Co_defect_v040113.eps "fig:"){width="3.627300in"} In this section we examine in detail the analogy that can be established between substitutional Co atoms in graphene (Co$_{sub}$) and the simplest theoretical model trying to account for the properties of a carbon vacancy in graphene. The electronic structure and magnetic properties of a Co$_{sub}$ impurity are analogous to those of a vacancy in a simple $\pi$-tight-binding description of graphene. This toy model system, the $\pi-$vacancy, has been extensively studied in the graphene literature due to its very interesting magnetic properties directly related to the bipartite character of the graphene network (Castro Neto et al. 2009, Palacios et al. 2008). We begin by looking at the spin density of the Co$_{sub}$ impurity as shown in Figure \[fig1-co\] (a). The spin polarization induced in the carbon atoms has a $p_z$-like shape and decays slowly as we move away from the impurity. The sign of the spin polarization follows the bipartite character of graphene: the polarization aligns parallel (antiparallel) to the spin moment located in the Co impurity for carbon atoms in the opposite (same) sublattice. The value of the total spin moment is 1.0 $\mu_{B}$ per defect. Using Mulliken population analysis, the moment on the Co atom has a value of 0.44 $\mu_{B}$; the three first carbon neighbors have 0.18 $\mu_{B}$; and there are 0.38 $\mu_{B}$ delocalized in the rest of the layer. Therefore, the total spin moment has contribution from both Co and carbon orbitals. To understand the origin of this spin polarization, we now analyze in detail the band structure. Figures \[fig1-co\] (c) and (d) present the results for a Co$_{sub}$ defect in a 4$\times$4 graphene supercell. Similar results are obtained using a 8$\times$8 cell. For comparison, panel (b) shows the spin-compensated band structure of a single unreconstructed $D_{3h}$ carbon vacancy. For the $D_{3h}$ vacancy, there are three defect states in a range of $\sim$0.7 eV around E$_F$. Two states appear above E$_F$ at 0.3 eV at $\Gamma$ and have a large contribution from the $sp^{2}$ lobes of the C atoms surrounding the vacancy. These levels correspond to the two degenerate E $sp$ states appearing in Figure \[fig:fig5-njp\]. Another state at 0.35 eV below E$_{F}$ shows a predominant $p_z$ contribution and corresponds to the A $p_z$ level in Figure \[fig:fig5-njp\]. This last level represents the defect state that appears at E$_{F}$ for a vacancy using a $\pi$-tight-binding description. For a Co$_{sub}$, the defect states of the vacancy described above hybridize with the Co $3d$ states. The two $2sp^{2}$ defect bands, now an antibonding combination of Co $3d$ and the original C $2sp^{2}$ vacancy levels, are pushed at higher energies, $\sim$1.0 eV above E$_{F}$ (see Fig. \[fig1-co\] (c) and (d)). The singly occupied $p_{z}$ state, now hybridized mainly with the Co $3d_{z^2}$ orbital, remains at E$_{F}$ and becomes almost fully spin-polarized. The Co$_{sub}$ impurity becomes thus analogous to a vacancy in the $\pi$-tight binding model of graphene. This analogy is a powerful one, since it brings our results for the magnetism of Co$_{sub}$ impurities into contact with Lieb’s theorem for a half-filled Hubbard model (Lieb 1989), where the spin polarization is an intrinsic property of the defective bipartite lattice. Applying this theorem and our analogy, we can expect that the total spin of an array of Co$_{sub}$ impurities can be described according to the simple rule S=$|N_{A}-N_{B}|$, where $N_{A}$ and $N_{B}$ indicate the number of Co substitutions in A and B sublattices, respectively. In Section \[substitutionals-coupling\] we will show results from first-principles calculations that confirm this behavior. However, Lieb’s theorem is global, in the sense that it refers to the total spin moment of the system, and does not enter into the local description of the magnetic interactions. This will be described in more detail in Section \[substitutionals-coupling\], where we will compute the exchange couplings between Co$_{sub}$ defects. Other realistic defects, besides Co$_{sub}$ impurities, can also be mapped onto the simple $\pi$-vacancy model. In Section \[covalent-spin\_formationNT\] we will see that complex adsorbates chemisorbed on carbon nanotubes and graphene generate a spin polarization. The magnetism due to such a covalent functionalization displays a behavior similar to that of the $\pi$-vacancies. Some concepts already used here will be again invoked to explain the main features of the magnetism associated with these defects, leading to a universal magnetic behavior independent of the particular adsorbate. Tuning the Magnetism of Substitutional Metals in Graphene with Strain {#strain} ====================================================================== In the previous section, we have considered in detail the formation of local spin-moments induced by a particular class of defects in graphene, substitutional transition metals. Although this is an important subject, other aspects are also crucial to understand and control the magnetism associated with this kind of doping. For example, one needs to explore the characteristics of the couplings between local moments, as well as the possibility to control such couplings, and the size of the local moments, using external parameters. This kind of knowledge is instrumental in possible applications in spintronics and quantum information devices. The subject of the magnetic couplings will be postponed until Section \[couplings\]. In this Section we analyze how the structural, electronic and magnetic properties of substitutional defects in carbon nanostructures can be controlled using strain. We focus on Ni substitutionals and conclude that externally applied strain can provide a unique tool to tune the magnetism of Ni-doped graphene. Strain provides a frequently used strategy to modify the properties of materials. For example, strain is intentionally applied to improve mobility in modern microelectronic devices. This so-called strain engineering has taken a key position over years. Recently, strain effects have also been proposed as a route to control the electronic properties of pristine graphene, which had a deep impact on the physics of this material (Guinea et al. 2009, Pereira et al. 2009). Here, we show that the application of uniaxial strain can be used to switch on the magnetism of graphene doped with Ni substitutional impurities (Ni$_{sub}$) (Santos et al. 2012a). Whereas Ni$_{sub}$ defects are non-magnetic in flat graphene, we find that their spin moment changes from zero when no strain in applied, up to 1.9 $\mu_{B}$ at $\sim$7.0% strain. This strong variation stems from the modifications of the local structure of the defect, which cause changes in the electronic structure of the defect that can be related to those of the unreconstructed carbon vacancy in graphene under strain. The similarities between the electronic structure of the D$_{3h}$ vacancy and that of Ni$_{sub}$ were already stressed in the previous section. We also show that substitutional metallic impurities in carbon nanotubes display a different magnetic behavior from that observed in flat graphene. Using Ni$_{sub}$ dopants as an example we demonstrate that the intrinsic curvature of the carbon layer in the SWCNTs can be used to switch on the magnetism of Ni substitutionals (Santos et al. 2008). The defect electronic structure is modified by curvature in a similar way as by uniaxial strain. In addition, we find a strong dependence of the spin moment on the impurity distribution, tube metallicity and diameter of the nanotube. Switching On the Magnetism of Ni-Doped Graphene with Uniaxial Strain {#strain-tensile} -------------------------------------------------------------------- In this subsection, we study the electronic structure of Ni$_{sub}$ defects in graphene under uniaxial strain. According to the analysis presented in Section \[substitutionals-spins\], at a zero strain Ni$_{sub}$ defects are non-magnetic in flat graphene (Santos et al. 2008, Santos et al. 2010b). However, we find that under moderate uniaxial strain these impurities develop a non-zero spin moment, whose size increases with that of the applied strain. This magnetoelastic effect might be utilized to design a strain-tunable spin device based on defective graphene. ![Spin moment as a function of the applied strain along (a) the $(n,n)$ and (b) the $(n,0)$ directions. In panels (a) and (b), filled squares indicate results obtained using geometries from a non-spin polarized calculation using a DZ basis. The spin moment and electronic structure is calculated using a DZP basis using such geometry. Open squares indicate a similar calculation, but the geometries have been obtained from a spin-polarized calculation. The triangles represent full calculations (geometry and spin moment) with DZP basis set. Panels (c) and (d) show the results for the structural parameters as a function of the applied strain along $(n,n)$ and $(n,0)$, respectively. Bond lengths between the different C atoms are denoted d$_{12}$ and d$_{22}$, while the bond lengths between Ni and C atoms are d$_{Ni-C1}$ and d$_{Ni-C2}$. The structural information was calculated using a DZ basis. Adapted from (Santos et al. 2012a). []{data-label="fig2-ni-strain"}](figures_arXiv/Nisub_strain_nn_n0_book_springer_v020512.eps){width="5.00in"} Figures \[fig2-ni-strain\] (a)-(b) show the spin moment of a Ni$_{sub}$ defect as a function of the applied strain along the $(n,n)$ and $(n,0)$ directions, respectively. The curves with filled squares show simulations using geometries from a non-spin polarized calculation with a DZ basis set (see references Soler et al. 2002 and Artacho et al. 1999 for a description of the different basis sets). The spin moment and electronic structure are always calculated using a more complete DZP basis. The open squares indicate systems that were calculated using the previous procedure, i.e. a DZ basis, but the geometries have been obtained from spin-polarized calculations. The triangles display calculations with DZP basis set for both geometry and spin moment. At zero strain the Ni$_{sub}$ defect is non-magnetic. As the uniaxial tension is applied, the system starts to deform. At $\sim$3.5% strain, the system becomes magnetic with a spin moment that evolves nearly linearly with the uniaxial strain up to values of $\sim 0.30 - 0.40$ $\mu_{B}$ at $\sim$6.0%. The magnetism of the system using different basis set is very similar. At $\sim$6.8% the spin moment increases sharply from $\sim$0.40 $\mu_{B}$ to $\sim$1.9 $\mu_{B}$. This transition takes place for both directions, although it is somewhat more abrupt along the $(n,n)$ direction (Figure \[fig2-ni-strain\](a)) where no intermediate steps are observed. Thus, the magnetic properties depend on the local defect geometry and, to a much lesser extent, on the defect orientation relative to the applied strain. Panels (c) and (d) in Figure \[fig2-ni-strain\] present the local defect geometry. When the strain is applied, the triangle formed by the three C neighboring atoms to the Ni impurity deforms. The C-C distances along the strain direction increase, whereas distances along the perpendicular direction decrease in response to such elongation. The distance of the Ni atom to the first carbon neighbors also increases, but this bond length changes for the studied strains are less than $\sim$5.0% ( averaged over both strain directions) in comparison with $\sim$20.0% for the C-C distances. The analysis of distances suggests that the carbon neighbors and the central Ni impurity interact strongly, which is also reflected in the high stability of the defect with a binding energy $\sim$7.9 V to the carbon vacancy. ![ Density of states (DOS) of the Ni$_{sub}$ defect under 0.0%, 2.25%, 5.30% and 7.26% strain applied along the $(n,n)$ direction. Symbols $A$ and $E$ indicate the character and symmetries of the defect states, with large weight of Ni hybridized with C states. $A$ corresponds to Ni $3d_{z^2}-$C $2p_{z}$, and $E$ represents Ni $3d_{xz}, 3d_{yz}-$C $2sp$. At 5.30% and 7.26 % strains, the open squares (green curve) represent the spin up channel and filled squares (red curve) the spin down. For clarity, the curves have been shifted. The Fermi energy is marked by the dashed (gray) line and is set to zero. Adapted from (Santos et al. 2012a).[]{data-label="fig3-dos-ni-strain"}](figures_arXiv/Nisub_dos_book_springer_v011612.eps){width="3.200in"} In order to understand the magnetic moment in Ni$_{sub}$ defects, the density of states (DOS) around E$_F$ under strains of 0.0%, 2.2%, 5.3% and 7.2% are shown in Figure \[fig3-dos-ni-strain\]. The strain is along the $(n,n)$ direction although the qualitative behavior is similar to other directions. Several defect levels around E$_F$ have Ni and C mixed character. As pointed out before (Section \[substitutionals\]), the metal atom over the vacancy has a $C_{3v}$ symmetry at a zero strain, and the electronic levels are classified according to the $A$ or $E$ irreducible representations of this point group. Essentially, these three defect states and their evolution as a function of the applied strain determine all the observed physics. One of them with $A$ character is occupied and appears around $\sim$0.50 eV below E$_F$ at a zero strain. This level comes from a fully symmetric linear combination of the $2p_z$ orbitals (z-axis normal to the layer) of the nearest C neighbors interacting with the $3d_{z^2}$ orbital of Ni. The other twofold-degenerate levels with $E$ character, coming from the hybridization of the in-plane $sp^2$ lobes of the carbon neighbors with the Ni $3d_{xz}$ and $3d_{yz}$ orbitals, appear at 0.50 eV above E$_F$ at a zero strain. Because this electronic structure has the Ni $3d$ states far from E$_F$ and no flat bands crossing E$_F$, the spin moment of the Ni$_{sub}$ impurity in graphene is zero. Interestingly, these three levels that appear close to E$_F$ in Figure \[fig3-dos-ni-strain\] are reminiscent of those found for the unreconstructed carbon vacancy in graphene as we have already seen in Section \[substitutionals\]. The energy position of the three levels shifts as a function of the applied strain. When the strain is applied, the degeneracy between Ni $3d_{xz}-$ C $2sp$ and Ni $3d_{yz}-$ C $2sp$ states is removed and a gradual shift towards E$_F$ of one of them is observed. This level becomes partially populated, and the system starts to develop a spin moment. The Ni $3d_{z^2} -$C $2p_{z}$ state also changes its position approaching $E_F$, although for small values of the strain this level does not contribute to the observed magnetization. However, around a 7% strain both the Ni $3d_{z^2}-$ C $2p_{z}$ and the Ni $3d_{xz,yz}-$ C $2sp$ levels become fully polarized and the system develops a moment close to 2.00 $\mu_{B}$. Figure \[fig3-dos-ni-strain\] also shows the resulting spin-polarized DOS at 5.3% and 7.2% strain (upper part of the panel). The exchange splittings of the $3d_{xz}$ and $3d_{yz}$ levels are, respectively, $\sim$0.29 eV and $\sim$0.13 eV at 5.3%, increasing with the applied strain and the associated spin moment. The energy gain with respect to the spin-compensated solutions develops from 13.9 meV at 5.3% to 184.1 meV at 7.26%. Thus, a moderate variation of the strain applied to the graphene layer changes the spin state and enhances the stability of the defect-induced moment. According to these results, if it is possible to control the strain applied to the graphene layer, as shown in recent experiments (Mohiuddin et al. 2009, Kim et al. 2009), the magnetism of Ni-doped graphene could be turned [*on*]{} and [*off*]{} at will, like switches used in [*magnetoelastic devices*]{}, however, with no applied magnetic field. This suggests a sensitive and effective way to control the magnetic properties of graphene, which is interesting for its possible applications in nanoscale devices (Santos et al. 2012a). ![(a)-(b) Spin densities for Ni$_{sub}$ defects at strains of 5.30% and 7.26% along the $(n,n)$ direction. The strain direction is marked by the arrows in both panels. The isovalue cutoff at (a) and (b) panels is $\pm$0.035 and $\pm$0.060 e$^-$/Bohr$^3$. Adapted from (Santos et al. 2012a). []{data-label="fig4-ni-strain"}](figures_arXiv/spin-density_5.30_7.26_inkscape_v020512.eps){width="5.00in"} Figure \[fig4-ni-strain\](a)-(b) shows the spin magnetization patterns induced by the presence of a Ni$_{sub}$ defect under two different magnitudes of uniaxial strain applied along the $(n,n)$ direction. The spin polarization induced in the neighboring carbon atoms has shape and orbital contributions that depend sensitively on strain. At 5.30% the spin density is mainly localized at the Ni impurity and at the C atom bonded to Ni along the strain direction. The anti-bonding character of the $E$ defect state that originates the magnetization is clear (see the node in the bond direction). The spin density at this C atom shows a $2sp$-like shape to be contrasted with that at 7.26% strain, in which apart from the $2sp$-like shape, a $2p_z$ component is clearly observed. At this larger strain farther neighboring-carbon atoms also contribute to the spin density with mainly $2p_z$ character. This additional contribution to the spin polarization pattern corresponds to the Ni $3d_{z^2}-$C $2p_{z}$-defect state at E$_F$ for strains above $\sim$7%, as explained in the previous section using the DOS. ![Experimental setup that could be utilized to measure the effect of strain on the magnetic properties of Ni-doped graphene. The layer is deposited on a stretchable substrate which keeps a large length-to-width ratio in order to obtain a uniform tensile strain in the graphene film. Spectroscopy measurements using scanning tunnelling microscope (STM) would allow to identify the shift of the different defect levels. If the magnetic anisotropy is large enough or there is an external magnetic field, it could be also possible to measure the presence and orientation of a magnetic moment at the defect site using a spin-polarized tip. []{data-label="fig-exp-ni-strain"}](figures_arXiv/fig1a_experiment_v060811.eps){width="3.500in"} Figure \[fig-exp-ni-strain\] shows a possible experimental setup that could be used to test our predictions. This is similar to a mechanically controlled break junction setup with an elastic substrate (Mohiuddin et al. 2009, Kim et al. 2009). Graphene is deposited in the center of such a substrate in order to obtain a uniform strain. Bending or stretching the substrate causes an expansion of the surface, and the deposited graphene will follow this deformation. In principle, the modifications on the electronic structure can be detected using a scanning tunneling microscope (STM) since the defect levels that are involved are localized around the Fermi energy. For example, Ugeda [*et al.*]{} were able to measure using STM the energy position and spatial localization of the defect levels associated with a carbon vacancy in the surface of graphite (Ugeda et al. 2010). If the magnetic anisotropy of the defect is high enough, at sufficiently low temperatures, a preferential orientation of the moment would be stabilized and, in principle, a STM with a spin polarized tip (Spin-STM) could allow to monitor the evolution of the magnetic properties of the Ni-doped graphene with strain. Instead, an external magnetic field may be used to align the magnetic moments of the defects and define the hard/easy axis of the system. It is noteworthy that the break junction-like setup has already been successfully used (Standley et al. 2008). Ni substitutionals in Carbon Nanotubes: Curvature Induced Magnetism {#strain-NiNT} ------------------------------------------------------------------- ![ (a) Relaxed geometry of a substitutional Ni (Ni$_{sub}$) impurity in a (5,5) SWCNT, and (b) isosurface ($\pm$0.002 e$^-$/Bohr$^3$) of the magnetization density with light (gray) and dark (blue) surfaces corresponding, respectively, to majority and minority spin. Adapted from (Santos et al. 2008). []{data-label="fig:fig1-Ni-tubes"}](figures_arXiv/t55-16cells_spin-density_book_spinger_v040113.eps){width="11.50cm"} Although Ni$_{sub}$ impurities are non-magnetic in flat graphene, their magnetic moment can be switched on by applying curvature to the structure. To understand why, we will begin looking at the equilibrium structure of Ni$_{sub}$ for a (5,5) SWCNT. The Ni atom is displaced $\sim$0.9 Å from the carbon plane. Although both outward and inward displacements were stabilized, the outward configuration is more stable. The calculated Ni-C distances (d$_{Ni-C}$) are in the range 1.77-1.85 Å in good agreement with experiment (Ushiro et al. 2006, Banhart et al. 2000). Armchair tubes exhibit two slightly shorter and one larger values of d$_{Ni-C}$, the opposite happens for zigzag tubes, whereas for graphene we obtain a threefold symmetric structure with d$_{Ni-C}$=1.78 Å. Ni adsorption inhibits the reconstruction (Amara et al. 2007) of the carbon vacancy. Furthermore, we have checked that for a vacancy in graphene, a symmetric structure is obtained after Ni addition even when starting from a relaxed vacancy. Figure \[fig:fig1-Ni-tubes\] (b) shows the magnetization density profile for a Ni$_{sub}$ defect in a (5,5) metallic nanotube at large dilution (0.3 % Ni concentration). The total spin moment is 0.5 $\mu_B$. The magnetization is on the Ni atom and its C neighbors. However, it also extents considerably along the tube, particularly in the direction perpendicular to the tube axis. This profile indicates that the spin polarization follows some of the delocalized electronic states in the metallic nanotube. Indeed, as we clarify below, the magnetism in substitutionally Ni-doped SWCNTs only appears associated with the curvature and the metallicity of the host structure. ![ Band structure of a (5,5) nanotube containing a Ni$_{sub}$ impurity in four unit cells for (left panel) a paramagnetic calculation (PAR), and for (middle panel) majority and (right panel) minority spins. Circles and diamonds correspond respectively to the amount of Ni 3$d_{yz}$ and 3$d_{xz}$ character. X-axis is parallel to the tube axis and y-axis is tangential. Adapted from (Santos et al. 2008). []{data-label="fig:fig3-Ni-tubes"}](figures_arXiv/4cells_par_up_down_book_springer_v011612i.eps){width="11.50cm"} The basic picture described in Section \[substitutionals\] is still valid for the electronic structure of the Ni$_{sub}$ impurity in SWCNTs. However, the modifications that appear due to the curvature of the carbon layer are responsible for the appearance of a magnetic moment. Figure \[fig:fig3-Ni-tubes\] (a) shows the band structure of a paramagnetic calculation of a (5,5) SWCNT with a Ni$_{sub}$ impurity every four unit cells. Comparing the band structure in Figure \[fig:fig3-Ni-tubes\] (a) with the electronic structure of the Ni$_{sub}$ impurity in flat geographer (lower curve in Figure \[fig3-dos-ni-strain\]), we appreciate the effects of curvature. The degeneracy between d$_{xz}$ and d$_{yz}$ states is removed (x-axis taken along the tube axis and y-axis along the tangential direction at the Ni site). The d$_{yz}$ contribution is stabilized by several tenths of eV and a quite flat band with strong d$_{yz}$ character is found [*pinned*]{} at E$_{F}$ close to the Brillouin-zone boundary. Under these conditions, the spin-compensated solution becomes unstable and a magnetic moment of 0.48 $\mu_B$ is developed. Figures \[fig:fig3-Ni-tubes\] (b) and (c) show, respectively, the band structure for majority and minority spins. The exchange splitting of the d$_{yz}$ level is $\sim$0.4 eV and the energy gain with respect to the paramagnetic solution is 32 meV. In general, whenever a flat band with appreciable Ni 3$d$ character becomes partially filled we can expect the appearance of a spin moment. The population of such an impurity level will take place at the expense of the simultaneous depopulation of some of the delocalized carbon $2p_z$ levels within the host structure. For this reason, the development of a spin moment is more likely for Ni$_{sub}$ impurities in metallic structures like the armchair tubes. The crucial role of the host states also explains the delocalized character of the magnetization density depicted in Fig. \[fig:fig1-Ni-tubes\] (b). However, it is important to stress that the spin moment associated with a Ni$_{sub}$ impurity in SWCNTs forms driven by the local curvature of the carbon layer, because the energy position of one of the impurity levels shifts downwards until it crosses E$_{F}$. A schematic representation of this phenomenon is shown in Fig. \[fig:fig4-Ni-tubes\] where we also emphasize the similarities between the levels of the Ni$_{sub}$ defect and those of the unreconstructed carbon vacancy. Notice the similarities with the effects of uniaxial strain described in the previous Section. At large tube diameters, the limit of flat graphene with zero spin moment (see Section \[substitutionals\]) is recovered. ![ Effect of curvature (anisotropic strain) on Ni$_{sub}$ in (n,n) tubes. Upper panel: Illustration of the dominant hoppings at the defect site in graphene. The equivalence between the electronic structure of a Ni$_{sub}$ impurity and a carbon vacancy is stressed here. The carbon sheet is rolled around the $(n,0)$ direction in order to form the armchair tubes. Lower panel: Scheme of the main Ni$_{sub}$ impurity energy levels as a function of curvature. One of the impurity levels with antibonding C 2$sp$-Ni 3$d$ character is shifted downwards and, for large enough curvatures, becomes partially populated and spin-polarized. Adapted from (Santos et al. 2008). []{data-label="fig:fig4-Ni-tubes"}](figures_arXiv/energy_vacancy-strain_book_springer_v021412.eps "fig:"){width="9.0cm"} ![ Effect of curvature (anisotropic strain) on Ni$_{sub}$ in (n,n) tubes. Upper panel: Illustration of the dominant hoppings at the defect site in graphene. The equivalence between the electronic structure of a Ni$_{sub}$ impurity and a carbon vacancy is stressed here. The carbon sheet is rolled around the $(n,0)$ direction in order to form the armchair tubes. Lower panel: Scheme of the main Ni$_{sub}$ impurity energy levels as a function of curvature. One of the impurity levels with antibonding C 2$sp$-Ni 3$d$ character is shifted downwards and, for large enough curvatures, becomes partially populated and spin-polarized. Adapted from (Santos et al. 2008). []{data-label="fig:fig4-Ni-tubes"}](figures_arXiv/spin_splitting_energy_Ni_book_springer_v011612.eps "fig:"){width="12.0cm"} For semiconducting tubes, the situation is somewhat different. The $d_{xz}$ and $d_{yz}$ -like levels remain unoccupied unless their energies are shifted by a larger amount that pushes one of them below the top of the valence band. If the tube has a large enough gap, the spin moment is zero irrespective of the tube diameter. We have explicitly checked that a zero spin moment is obtained for (8,0) and (10,0) semiconducting tubes for Ni concentrations ranging from 1.5% to 0.5%. The different magnetic behavior of Ni$_{sub}$ impurities depending on the metallic and semiconducting character of the host structure provides a route to experimentally identify metallic armchair tubes. Magnetic Coupling Between Impurities {#couplings} ==================================== In previous Sections, we have considered the formation of local moments associated with defects in carbon nanostructures, as well as the use of mechanical deformations to tune the sizes of such local moments. Here, we present calculations of the exchange couplings between the local moments in neighboring defects. This is a necessary step to elucidate whether it is possible to induce magnetic order in these materials, which is crucial in the application of carbon-based nanostructures in spintronics. We focus on defects that can be mapped onto the simple model provided by the fictitious $\pi$-vacancy. According to the results presented in Section \[Co-analogy\], Co substitutional impurities belong to this class of defects. In this Section we present another type of impurities that behave according to the same analogy: molecules attached to graphene and carbon nanotubes through weakly-polar covalent bonds. A $\pi$-vacancy corresponds to a missing $p_z$ orbital in a graphene plane described using a $\pi$-tight-binding model. The magnetic properties of the $\pi$-vacancies have been extensively studied (Castro Neto et al. 2009, Palacios et al. 2008). Among other interesting properties, the magnetism of the $\pi$-vacancy model reflects faithfully the bipartite character of the graphene lattice. For example, the total spin of the system is S=$|N_A-N_B|$, where $N_A$ and $N_B$ are the number of $\pi$-vacancies in each of the graphene sublattices. This behavior can be traced back to Lieb’s theorem for a half-filled Hubbard model in a bipartite lattice (Lieb 1989). In the following, we will see that the calculated data for Co substitutionals and covalently chemisorbed molecules indeed follow the predictions of Lieb’s theorem. In addition, we analyze in detail the spatial decay of the exchange couplings. Magnetic Couplings Between Co Substitutional Impurities in Graphene {#substitutionals-coupling} ------------------------------------------------------------------- ![(a) Schematic representation of the geometry used to calculate the relative stability of ferromagnetic (FM), antiferromagnetic (AFM) and spin compensated (PAR) solutions as a function of the positions of two Co$_{sub}$ impurities. Sublattices A and B are indicated by blue and red circles, respectively. One of the impurities is fixed at a central A-type site, whereas the other is moved along the $(n,n)$ and $(n,0)$ directions. Panels (b) and (c) show the results of the energy differences for $(n,n)$ and $(n,0)$ configurations, respectively. Solid squares at positive values indicate FM spin alignments, while solid triangles at negative values correspond to AFM ones. The empty circles represent spin compensated solutions and the full circles for AA substitutions correspond to a fit of a Heisenberg model (see text for details). Adapted from (Santos et al. 2010a).[]{data-label="fig2-co"}](figures_arXiv/Co_coupling_path_energy_v011512.eps){width="5.000in"} Here, we consider the magnetic couplings between Co$_{sub}$ defects. For this purpose we perform calculations using a large 8$\times$8 supercell with two Co$_{sub}$ impurities. We calculate the energy difference between spin alignments as a function of the relative position of the defects. Figure \[fig2-co\] shows the results along with a schematic representation of our notation. Positive values indicate ferromagnetic (FM) spin alignment while negative values are antiferromagnetic (AFM) ones. Several observations from spin couplings in Fig. \[fig2-co\] can be made: ([*i*]{}) when the impurities are located in the same sublattice (AA systems) the FM configuration is more stable than the AFM one; ([*ii*]{}) if the Co atoms are in opposite sublattices (AB systems) it is very difficult to reach a FM solution, [^1] instead the system finds either a spin-compensated (PAR) or an AFM solution; ([*iii*]{}) at short distances ($<$ 3.0 Å) the systems always converge to spin compensated solutions.  for the nomenclature). Positive and negative spin densities are indicated by light (gray) and dark (blue) isosurfaces corresponding to $\pm$0.001 e$^-$/Bohr$^3$, respectively. Adapted from (Santos et al. 2010a).[]{data-label="fig3-co"}](figures_arXiv/Co_spin_coupling_density_v040113.eps){width="3.800in"} The FM cases of Fig. \[fig2-co\] always have total spin magnetization about 2.00 $\mu_{B}$. The spin population on every Co atom remains almost constant $\sim$0.50 $\mu_{B}$ and it is $\sim$0.30 $\mu_{B}$ on the three C nearest-neighbors. In other cases the total spin is zero. Thus, the total spin moment of the system follows the equation $S=|N_{sub}^{A}-N_{sub}^{B}|$, where N$_{sub}^{A(B)}$ is the number of Co$_{sub}$ defects in the A(B) sublattices. Our total moment is consistent with Lieb’s theorem for bipartite lattices (Lieb 1989). This finding supports the analogy, presented in Section \[Co-analogy\], between the electronic structure of Co$_{sub}$ defects and single vacancies in a simplified $\pi$-tight-binding description of graphene. Some selected configurations have their spin magnetization densities plotted in Fig. \[fig3-co\]. Although the spin is quite localized on the Co atom and the neighboring C atoms, part of the magnetization density is delocalized with alternated signs in both graphene sublattices. The triangular spin patterns reflect the three-fold symmetry of the layer with different orientations for A and B substitutions. This explains the anisotropic AB interaction along the $(n,n)$ direction seen in Fig. \[fig2-co\] (b): the energy difference between AFM and PAR solutions for $(n,n)^{AB}$ configurations strongly depends on the relative position of the impurities, showing such a directionality. Similar patterns have already been observed experimentally (Kelly et al. 1998, Mizes et al. 1989, Rutter et al. 2007, Ruffieux et al. 2000) for point defects in graphene using STM techniques and theoretically discussed for $\pi$-vacancies (Yazyev 2008, Palacios et al. 2008, Pereira et al. 2008). For Co$_{sub}$, similar STM experiments should display the topology of the spin densities given in Fig. \[fig3-co\]. We can also investigate the magnetic interactions within the framework of a classical Heisenberg model: $$H= \sum_{i<j} J_{AA/AB}({\bf r}_{ij}){\bf S}_{i}{\bf S}_{j} \label{heisenberg}$$ \ where ${\bf S}_{i}$ is the local moment for a Co$_{sub}$ impurity at site $i$. The angular dependence of the exchange $J ({\bf r}_{ij})$ is taken from an analytical RKKY coupling as given in (Saremi 2007). We fit the exponent for the distance decay to our [*ab initio*]{} results. The exchange interaction for AA systems can be fitted with a $|r_{ij}|^{-2.43}$ distance dependence (see the full circles in Fig.\[fig2-co\] (b) and (c)). This distance dependence is in reasonable agreement with the $|r_{ij}|^{-3}$ behavior obtained with analytical models for substitutional defects and voids (Saremi 2007, Vozmediano et al. 2005). In the case of AB systems a simple RKKY-like treatment fails to describe accurately the interactions, at least for the short distances between defects considered in our calculations. We next explain how PAR solutions appear in Fig. \[fig2-co\]. The PAR solutions are stable because defect states in neighboring impurities interact strongly for AB systems. This interaction opens an appreciable [*bonding-antibonding*]{} gap in the $p_z$ defect band. [^2] For AA systems, however, the bipartite character of the graphene lattice makes the interaction between the defects much smaller. This explains why AA configurations show a local spin polarization. Even for AA configurations, when the impurities are very close, non-magnetic solutions are stabilized because a larger defect-defect interaction opens a large gap. It is interesting to point out that similar behaviors have been observed for vacancies described within a $\pi$-tight-binding model (see Section 1.6.1) (Yazyev 2008, Kumazaki et al. 2007, Palacios et al. 2008). Covalent Functionalization Induces Magnetism: Universal Properties {#covalent-spin_formationNT} ------------------------------------------------------------------- ![Total spin polarized density of states (t-DOS) for (a) (5,5) and (b) (10,0) SWCNTs with a single adsorbate of different types chemisorbed on top of one carbon atom in the supercell. Positive and negatives t-DOS correspond to spin up and spin down, respectively. The t-DOS for pristine (5,5) and (10,0) SWCNTs is also shown for comparison. For clarity, the curves in panel (a) and (b) have been shifted and smoothed with a Lorentzian broadening of 0.12 eV. Fermi energy is marked by the dashed lines and is set to zero in both panels. []{data-label="fig1-adsorbates2"}](figures_arXiv/t-dos_t55-4cels-x_adsorbates_up_down_DZiv.eps "fig:"){width="3.200in"} ![Total spin polarized density of states (t-DOS) for (a) (5,5) and (b) (10,0) SWCNTs with a single adsorbate of different types chemisorbed on top of one carbon atom in the supercell. Positive and negatives t-DOS correspond to spin up and spin down, respectively. The t-DOS for pristine (5,5) and (10,0) SWCNTs is also shown for comparison. For clarity, the curves in panel (a) and (b) have been shifted and smoothed with a Lorentzian broadening of 0.12 eV. Fermi energy is marked by the dashed lines and is set to zero in both panels. []{data-label="fig1-adsorbates2"}](figures_arXiv/t100_dos_up_down_book_springer_v011512.eps "fig:"){width="3.200in"} In this Section we show that, apart from playing an increasingly important role in technological applications, chemical functionalization can be also used to induce spin moments in carbon nanostructures. Here, we focus on SWCNT and demonstrate that, when a single C-C bond is established on the carbon surface by covalent functionalization, a spin moment is induced into the system. This moment has a universal value of 1.0 $\mu_B$ and is independent of the particular adsorbate. In our recent work (Santos et al. 2011) we showed that this effect occurs for a wide class of organic and inorganic molecules of different biological and chemical activity (e.g. alkanes, polymers, diazonium salts, aryl and alkyl radicals, nucleobases, amido and amino groups, acids). Furthermore, we have recently found that a similar universal behavior is obtained for covalent functionalization of graphene (Santos et al. 2012b). We have also found that, either for metallic and semiconducting SWCNTs or for graphene, only when neighboring adsorbates are located at the same sublattice a spin moment is developed. For metallic tubes and graphene the local moments align ferromagnetically, while for semiconducting tubes we have almost degenerate FM and AFM spin solutions (Santos et al. 2011, Santos et al. 2012b). To understand the origin of the spin moment induced when a covalent bond is formed in the tube wall, we analyze the total spin-polarized density of states (t-DOS) when different adsorbates are attached to metallic and semiconducting tubes (see Figure \[fig2-tube-adsorbate\] for the structure of some of these systems). Figure \[fig1-adsorbates2\] presents results for (a) (5,5) and (b) (10,0) SWCNT’s. We consider first the well-known case of the adsorption of atomic hydrogen. In both cases, when a single H atom chemisorbs on top of a C atom a defect state appears pinned at E$_{F}$ with full spin polarization. This state is mainly composed of the $p_z$ orbitals at the nearest C neighbors of the defect site, with almost no contribution from the adsorbate. A detailed Mulliken analysis of this $p_z$-defect state assigns a contribution of the adsorbate of about $\sim$1$\%$. Thus, the adsorbate has a primary role in creating the bond with the nanotube, and the associated defect level, but it does not appreciably contribute to the spin moment. More complex adsorbates, notwithstanding their biological and chemical activity (e.g. alkanes, polymers, diazonium salts, aryl and alkyl radicals, nucleobases, amido and amino groups, acids), show a similar behavior. This is observed in the t-DOS curves corresponding to other adsorbates in metallic (5,5) and semiconducting (10,0) SWCNTs as shown in Figure \[fig1-adsorbates2\](a) and \[fig1-adsorbates2\](b), respectively. Several common points are worth mentioning: (i) All molecules induce a spin moment of 1.0 $\mu_{B}$ localized at the carbon surface; (ii) The origin of the spin polarization corresponds to a $p_{z}$-defect state with a character and a spatial distribution similar to those of the state appearing at E$_F$ for a $\pi$-vacancy defect; (iii) The t-DOS around E$_{F}$ follows the same pattern in all cases. This match demonstrates that the spin moment induced by covalent functionalization is largely independent of the particular type of adsorbate. These results also demonstrate the complete analogy between a single C-H bond and more complex adsorbates linked to graphene through a single C-C bond (or other weakly-polar covalent bonds) (Santos et al. 2011, Santos et al. 2012b). Such similarity is not obvious and could not be easily anticipated. ![Isosurface of the magnetization density induced by some adsorbates at the SWCNT surface: (a) Pmma and (b) Adenine derivative in a (10,0) tube; and (c) C$_{6}$H$_{4}$F and (d) CH$_{3}$ in a (5,5) tube. Majority and minority spin densities correspond respectively to light and dark surfaces, which alternate on the honeycomb lattice with a long decaying order in all cases. The cutoff is at $\pm$0.013 $e^-/bohr^3$. Adapted from (Santos et al. 2011). []{data-label="fig2-tube-adsorbate"}](figures_arXiv/spin-density_tubes.eps){width="4.8400in"} Next, we study the spin polarization texture induced by the adsorbates on the carbon nanotube wall. The analysis of local spin moments for all the adsorbates assigns general trends to both types, metallic and semiconducting, of SWCNTs. The C atoms that participate directly in the bond formation, at either the molecule or the surface, show a local spin moment smaller than $\sim 0.10 \mu_{B}$. However, the wall carbon atoms contribute with 0.40 $\mu_{B}$ in the three first C nearest-neighbors, -0.10 $\mu_{B}$ in the next nearest-neighbors, 0.20 $\mu_{B}$ in the third-neighbors. The adsorbate removes a $p_{z}$ electron from the adsorption site, and leaves the $p_z$ states of the nearest carbon neighbors uncoordinated. This gives rise to a defect state localized in the carbon layer, reminiscent of that of a vacancy in a $\pi$-tight-binding model of graphenic nanostructures. The carbon spins polarize parallel (antiparallel) respect to the C atom that binds to the surface when sitting in the opposite (same) sublattice. Figure \[fig2-tube-adsorbate\] shows the magnetization density in semiconducting (10,0) and metallic (5,5) SWCNTs for several molecules: (a) Pmma polymer chain (Haggenmueller et al. 2000), (b) Adenine group nucleobase (Singh et al. 2009), (c) C$_6$H$_4$F salt (Bahr et al. 2001) and (d) CH$_3$ molecule (Saini et al. 2002). The spin density in the metallic (5,5) (Figure \[fig2-tube-adsorbate\](c) and \[fig2-tube-adsorbate\](d)) is more spread over the whole surface than in the semiconducting (10,0) (Figure \[fig2-tube-adsorbate\](a) and \[fig2-tube-adsorbate\](b)). Thus, the electronic character of the nanotube wall plays a role in determining the localization of the defects states and, as will be seen below, in mediating the interaction between adsorbates. ![Variation of total energy as a function of the relative adsorption positions of two H atoms on (a) a (5,5) and (c) a (7,0) SWCNT. for different magnetic solutions. One of the adsorbates moves parallel to the axis of the tube, while the other remains at the origin. In (b) and (d), the light (yellow) and dark (blue) squares correspond to PAR spin solutions at AB and AA sublattices, respectively; the circles and triangles indicate the FM and AFM solutions, respectively, at the same sublattice. Adapted from (Santos et al. 2011). []{data-label="fig3-tube-adsorbate"}](figures_arXiv/energy_t100_t55_spin_coupling_book_springer_v040113.eps){width="5.5200in"} Now we deal with the relative stability of the different magnetic solutions when two molecules are simultaneously adsorbed on the walls of CNTs. Due to the universal character of the magnetism associated with covalent functionalization of SWCNTs, and in order to alleviate the computational effort, we have considered here hydrogen atoms. However, we have explicitly checked for some configurations for the case of SWCNTs, and for flat graphene (Santos et al. 2012b), that identical results are obtained when using CH$_3$ instead of H. For the metallic (5,5) and semiconducting (7,0) single wall CNTs, we calculate the variation of the total energy for several spin alignments as a function of the distance between the adsorbates. The relative positions of the adatoms along the tube are schematically shown in Figures \[fig3-tube-adsorbate\](a) and \[fig3-tube-adsorbate\](c). One H is sited at the origin, and the other sites in different positions along of the tube axis. Several observations can be first made on the stability when two adsorbates are located at the same sublattice (AA configurations). In the metallic (5,5), the FM configuration is most stable than the non-magnetic one (PAR). The energy difference between these two spin solutions along the tube axis oscillates and no AFM solution could be stabilized at all. In the semiconducting (7,0), the FM and AFM solutions are almost degenerate, with a small energy difference (exchange coupling). If the two molecules are now located at different sublattices (AB configurations), we were not able to stabilize any magnetic solution for both nanotubes. Instead the systems are more stable without magnetic polarization. This behavior for adsorbates at opposite sublattices is related to the interaction between the defect levels. As already pointed out for Co substitutionals, while for AA configurations the interaction is negligible, for AB ones this interaction opens a bonding-antibonding gap around $E_{F}$ in the $p_z$ defect band and, thus, contributes to the stabilization of PAR solutions. If the gap is larger than the spin splitting of the majority and minority spin defect bands the system will be non-magnetic (see Section \[substitutionals-coupling\]). In fact, our detailed analysis of the band structure fully confirmed such an explanation. However, it is worth noting that AB adsorption seems to be always more stable in our calculations. This indicates that if the adsorption takes place at random sites, the magnetic solutions will only be stable for low density functionalization. Conclusions =========== In this Chapter we have reviewed the structural, electronic and magnetic properties of two types of defects, substitutional metal impurities and $sp^3$-type covalent functionalization, in carbon nanostructures. We have focused on their role to induce and control magnetism in graphene and carbon nanotubes. Density functional theory was the main tool used to compute the properties of the studied systems. We also developed simple models to understand the observed trends. For instance, substitutional dopants in graphene were understood in terms of the hybridization of the $d$ states of the metal atoms with those of an unreconstructed carbon vacancy. The main ingredients of the model are the assumption of a three-fold symmetric bonding configuration and the approximate knowledge of the relative energy positions of the levels of the carbon monovacancy and the $d$ shell of the metal impurity as we move along the transition series. With this model, we understood the variations of the electronic structure, the size and localization of the spin moment, and the binding energy of transition, noble metals and Zn substitutional impurities in graphene (Santos et al 2010b). Our model also allowed us to draw an analogy between substitutionals of the late transition metals and the symmetric D$_{3h}$ carbon vacancy. As a result of our analysis, a particularly powerful analogy was established between the substitutional Co impurity and the fictitious $\pi$-vacancy in graphene (Santos et al 2010a). The $\pi$-vacancy corresponds to a missing $p_z$ orbital in a simple description of graphene using a $\pi$-tight-binding model. The magnetic properties of the $\pi$-vacancies have been extensively studied. This analogy brings our results for the magnetism of Co$_{sub}$ defects into contact with the predictions of Lieb’s theorem for a half-filled Hubbard model in a bipartite lattice. We found that, according to Lieb’s theorem, the total spin of the system is S=$|N_A-N_B|$, where $N_A$ and $N_B$ are the number of substitutions performed in each of the graphene sublattices. Thus, the couplings between local moments for AA substitutions are ferromagnetic and predominantly antiferromagnetic for AB substitutions. We have also used a simple RKKY-model to extract the distance decay of the couplings. Adsorbates attached to graphene or SWCNTs through covalent bonds, particularly if the bonds are weakly polar, constitute another example of defects whose magnetism is analogous to that of the $\pi$-vacancy. We have analyzed the magnetic properties induced by such a covalent functionalization using many types of adsorbates: polymers, diazonium salts, aryl and alkyl radicals, nucleobases, amide and amine groups, sugar, organic acids, for SWCNTs (Santos et al. 2011) and graphene (Santos et al. 2012b). A universal spin moment of 1.00 $\mu_{B}$ is induced on the carbon surface when a single C-C bond is formed between an adsorbate and the graphenic layer. In metallic carbon nanotubes and graphene, molecules chemisorbed at the same sublattice (AA adsorption) have their local moments aligned ferromagnetically. In semiconducting nanotubes, FM and AFM solutions are almost degenerate even for AA adsorption. For two molecules in different sublattices (AB adsorption), we could not stabilize any magnetic solution, and the system is more stable without a local spin moment. We have also explored the possibility to control the magnetism induced by substitutional impurities using mechanical deformations. We have found that the spin moment of substitutionally Ni-doped graphene can be controlled by applying mechanical deformations that break the hexagonal symmetry of the layer, like curvature or uniaxial strain. Although Ni$_{sub}$ impurities are non-magnetic in flat graphene, we have observed that stretching the layer by a few percents along different crystalline directions is enough to turn the non-magnetic ground state of Ni atoms embedded in graphene to a magnetic state (Santos et al. 2012a). The spin moment slowly increases as a function of the applied strain. However, at a critical strain value of 6.8%, a sharp transition to high spin ($\sim$2 $\mu_B$) state is observed. This transition is independent of the orientation of the applied strain. A detailed analysis indicates that this strain-tunable spin moment is the result of changes of the positions of three defect levels around Fermi energy which are antibonding combinations of the Ni $3d$ states and the $2p_{z}$ and $2sp^2$ orbitals of the neighboring C atoms. This tunable magnetism observed in Ni$_{sub}$ defects via strain control may play an interesting role in flexible spintronics devices. Our calculations show that Ni$_{sub}$ magnetism can also be switched on by applying curvature (Santos et al. 2008). For metallic carbon nanotubes the curvature of the carbon layer around the defect can drive the transition of the Ni$_{sub}$ impurities to a magnetic state. For semiconducting tubes, the Ni$_{sub}$ impurities remain non-magnetic irrespective of the tube diameter. We have analyzed in detail the origin and distribution of the magnetic moment. We found that the spin moment associated with Ni$_{sub}$ impurities forms accompanied by the simultaneous polarization of delocalized electronic states in the carbon layer. Furthermore, the spin moment of Ni$_{sub}$ is a signature of the metallicity of the structure: only metallic tubes develop a moment that depends on the tube diameter and Ni concentration. Our work predicts a complex magnetic behavior for transition metals impurities in carbon nanotubes and graphene. This investigation is highly relevant in the interpretation of experimental results since it has been proposed that appreciable amounts of metal atoms could be incorporated into the carbon network, forming this type of substitutional defects in the course of synthesis, and are very difficult to eliminate afterwards. Our results also indicate that covalent functionalization provides a powerful route to tune the magnetism of graphene and carbon nanostructures. This is particularly attractive due to the recent successful synthesis of different graphene derivatives using surface chemical routes ( Cai et al. 2010, Treier et al. 2010). Thus, the synthesis of carbon nanostructures with functional groups at predefined positions, for example starting from previously functionalized monomers, seems plausible nowadays. According to our results, this could be applied to synthesize magnetic derivatives of graphene that behave according to well studied theoretical models like the $\pi$-vacancy. [^1]: When we could stabilize a FM solution, it lies at higher energy, around 0.2 eV above the PAR one. [^2]: For the AB systems, we find bonding-antibonding gaps in the impurity bands ranging from 0.3 eV to 0.9 eV for the $(1,1)^{AB}$ and the $(-1,-1)^{AB}$ configuration, respectively. These values are similar to the $\sim$0.5 eV spin-splitting of the Co$_{sub}$ defect. In fact, all those AB systems with gaps larger than 0.4 eV converge to PAR solutions.

--- address: 'Department of Mathematics, Faculity of Science, Kyoto University, 606-01, Japan ' author: - | Kōta Yoshioka\ Dept. of Math. Kyoto Univ.\ Fax number 75-753-3711\ running title\ Moduli of stable sheaves on a ruled surface title: | Chamber structure of polarizations and\ the moduli of stable sheaves on a ruled surface --- \[section\] \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Corollary]{} \[section\] \[section\] [^1] Introduction ============ Let $X$ be a smooth projective surface defined over $\Bbb C$ and $H$ an ample divisor on $X$. Let $M_H(r;c_1,c_2)$ be the moduli space of stable sheaves of rank $r$ whose Chern classes $(c_1,c_2) \in H^2(X,\Bbb Q)\times H^4(X,\Bbb Q)$ and $\overline{M}_H(r;c_1,c_2)$ the Gieseker-Maruyama compactification of $M_H(r;c_1,c_2)$. When $r=2$, these spaces are extensively studied by many authors. When $r \geq 3$, Drezet and Le-Potier \[D1\],\[D-L\] investigated the structure of moduli spaces on $\Bbb P^2$, and Rudakov \[R\] treated moduli spaces on $\Bbb P^1 \times \Bbb P^1$. In this paper, we shall consider moduli spaces of rank $r \geq 3$ on a ruled surface which is not rational. In particular, we shall compute the Picard group of $\overline{M}_H(r;c_1,c_2)$. Let $\pi:X \to C$ be the fibration, $f$ a fibre of $\pi$ and $C_0$ a minimal section of $\pi$ with $(C_0^2)=-e$. We assume that $e>2g-2$, where $g$ is the genus of $C$. Then $K_X$ is effective, and hence $(K_X,H)<0$ for any ample divisor $H$. In particular, $M_H(r;c_1,c_2)$ is smooth with the expected dimension $2r^2 \Delta-r^2(1-g)+1$. In section 2, we shall generalize the chamber structure of Qin \[Q2\]. As an application, we shall consider the difference of Betti numbers of moduli spaces on a ruled surface. Although we cannot generalize the method in \[Y2, 0\] directly, by using Qin’s method we can generalize it to any rank case. In \[Y2\], we computed the number of $\mu$-semi-stable sheaves of rank 2 on a ruled surface defind over $\Bbb F_q$. So, in principle, we can compute the Betti numbers of $M_H(3;c_1,c_2)$ on $\Bbb P^2$. Combining chamber structure with another method, Göttsche \[Gö\] also considered the difference of Hodge numbers (and hence Betti numbers) of moduli spaces of rank 2. Matsuki and Wentworth \[M-W\] also generalized the chamber structure of polarizations. Combining another chamber structure, they showed that the rational map between two moduli spaces is factorized to a sequence of flips. In sections 4 and 5, we assume that $X$ is a ruled surface which is not rational. Then, in the same way as in \[Q1\], we can give a condition for the existence of stable sheaves. Since we had computed the Picard group $Pic(\overline{M}_H(r;c_1,c_2))$ in case of $(c_1,f)=0$ \[Y3\], we assume that $0<(c_1,f)<r$. In section 5, we shall compute the Picard group of $\overline{M}_H(r;c_1,c_2)$, which is a generalization of \[Q1\] to $r \geq 3$. The proof is the same as that in \[D-N\]. As is well known, it is difficult to treat the moduli spaces on rational ruled surfaces (cf. \[D-L\], \[R\]). However we can also check that $M_H(r;c_1,c_2)$ is emply or not in principle. I would like to thank Professor S. Mori for valuable suggestions. chamber structure ================= Notation Let $X$ be a smooth projective surface defined over $\Bbb C$. Let $NS(X)$ be the Neron-Severi group of $X$ and $Num(X)=NS(X)/\text{torsion}$. Let $C(X) \subset Num(X) \otimes _{\Bbb Z}\Bbb R$ be the ample cone. We denote the moduli space of stable sheaves of rank $r$ with Chern classes $(c_1,c_2) \in H^2(X,\Bbb Q)\times H^4(X,\Bbb Q)$ by $M_H(r;c_1,c_2)$ and the Gieseker-Maruyama compactification of $M_H(r;c_1,c_2)$ by $\overline{M}_H(r;c_1,c_2)$. We denote the open subscheme of $M_H(r;c_1,c_2)$ consisting of $\mu$-stable sheaves by $M_H(r;c_1,c_2)^{\mu}$ and the open subscheme consisting of $\mu$-stable vector bundles by $M_H(r;c_1,c_2)^{\mu}_0$. For a torsion free sheaf $E$ on $X$, we set $\mu(E)=\frac{c_1(E)}{{\operatorname{rk}}(E)} \in H^2(X,\Bbb Q)$ and $\Delta(E)=\frac{1}{{\operatorname{rk}}(E)}(c_2(E)-\frac{{\operatorname{rk}}(E)-1}{2{\operatorname{rk}}(E)}c_1(E)^2) \in H^4(X,\Bbb Q)$. For a $x \in H^2(X,\Bbb Q)$, we set $P(x)=(x,x-K_X)/2+\chi(\cal O_X)$. For a scheme $S$, we denote the projection $X \times S \to S$ by $p_S$. In this section, we shall generalize the chamber structure of polarizations in \[Q2\]. For a torsion free sheaf $E$, we set $\gamma(E):=({\operatorname{rk}}(E),\mu(E),\Delta(E)) \in H^0(X,\Bbb Q) \times H^2(X,\Bbb Q) \times H^4(X,\Bbb Q)$. For $\gamma \in \prod_{i=0}^2 H^{2i}(X, \Bbb Q)$, let $M_H^{\gamma}$ be the set of torsion free sheaves $E$ defined over $\Bbb C$ with $\gamma(E)=\gamma$ which is $\mu$-semi-stable with respect to $H$. \[lem:1\] Let $E$ be a torsion free sheaf which is defined by an extension $ 0 \to F_1 \to E \to F_2 \to 0$. Then $\Delta(E)=\frac{{\operatorname{rk}}(F_1)}{{\operatorname{rk}}(E)}\Delta(F_1)+\frac{{\operatorname{rk}}(F_2)}{{\operatorname{rk}}(E)}\Delta(F_2)- \frac{{\operatorname{rk}}(F_1){\operatorname{rk}}(F_2)}{2{\operatorname{rk}}(E)^2}((\mu(F_1)-\mu(F_2))^2)$. Let $B$ be a subset of $C(X)$. Let $\cal F_B(\gamma)$ be the set of filtrations $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ which satisfies (1) $\gamma(E)=\gamma$, (2) $\Delta_i=\Delta(F_i/F_{i-1}) \geq 0$ and (3) there is an element $H \in B$ with $(\mu(F_{i-1})-\mu(F_i),H)=0$ for $2 \leq i \leq s$. If $B$ is compact, then $S_B(\gamma)=\{(\gamma(F_1),\cdots,\gamma(F_s))| \text{$F_i$ is the $i$-th filter of $F \in \cal F_B(\gamma)$}\}$ is a finite set. We denote $\mathrm{gr}_i(F):=F_i/F_{i-1}$ by $E_i$. By using Lemma \[lem:1\] successively, we see that $$\label {eq:2.1} \Delta(E)=\sum_{i=1}^s \frac{{\operatorname{rk}}(E_i)}{{\operatorname{rk}}(E)}\Delta(E_i)-\sum_{i=2}^s\frac{{\operatorname{rk}}(E_{i-1})}{2 {\operatorname{rk}}(E_i) {\operatorname{rk}}(E)}((\mu(F_{i-1})-\mu(F_i))^2).$$ By the Hodge index theorem, we get $-((\mu(F_{i-1})-\mu(F_i))^2) \geq 0$ and $-((\mu(F_{i-1})-\mu(F_i))^2)=0$ if and only if $\mu(F_{i-1})-\mu(F_i)=0$. By \[F-M, II, Lemma 1.4\], the set of ${c_1(F_i)}$ is finite. Hence ${\Delta_i}$ is finite. Therefore $S_B(\gamma)$ is a finite set. For a filtration $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ which belongs to $\cal F_B(\gamma)$, $F':0 \subset F_i \subset F_s$ belongs to $\cal F_B(\gamma)$ for $1 \leq i \leq s-1$. In fact implies that $\Delta(F_i)\geq 0$ and $\Delta(F_s/F_i)\geq 0$. For an element $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ of $\cal F_{C(X)}(\gamma)$, we define a wall $W^F:=\cup_{i}\{ H \in C(X)|(\mu(F_s)-\mu(F_{i}),H)=0 \}$, where $i$ runs for $1 \leq i \leq s-1$ with $\mu(F_s)-\mu(F_i) \ne 0$. By the above lemma, $\cup_F W^F$ is locally finite. We shall call the connected component of $C(X) \setminus \cup_F W^F$ by chamber. \[lem:2\] Let $H$ and $H'$ be ample divisors which belong to a chamber $\cal C$. Let $E$ be a $\mu$-semi-stable sheaf with respect to $H$. Then $E$ be also $\mu$-semi-stable with respect to $H'$, and hence we may denote $M_H^{\gamma}$ by $M_{\cal C}^{\gamma}$. Assume that $E$ is not $\mu$-semi-stable with respect to $H'$. We shall construct a wall which separates $H'$ from $H$. There is a filtration $F$ of $E$ such that $(\mu(F_{i-1})-\mu(F_i),H')>0$, $2\leq i \leq s$ and $\Delta({\operatorname{gr}}_i(E))\geq 0$, $1 \leq i \leq s$. In fact, let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ be the Harder-Narasimhan filtration of $E$ with respect to $H'$. Then, the Bogomolov-Gieseker inequality implies that $\Delta(F_i/F_{i-1}) \geq 0$. Let $H_t=H'+t(H-H'), 0 \leq t \leq 1$ be a line segment joining $H$ and $H'$. There is a $t_1 \in \Bbb Q$ such that $(\mu(E_i)-\mu(E_{i+1}),H_t)>0$ for $t<t_1$ and $(\mu(E_j)-\mu(E_{j+1}),H_{t_1})=0$ for some $j$. Let $\{F_1',F_2',\dots,F_{s(t_1)}' \}$ be a subset of $\{F_1,F_2,\dots,F_s \}$ such that $({\operatorname{rk}}(F_i'),(F_i',H_{t_1})) \in \Bbb Q \times \Bbb Q$, $1 \leq i \leq s(t_1)$ are vertices of the convex hull of $\{({\operatorname{rk}}(F_i),(F_i,H_{t_1})) \}_{i=1}^{s(t_1)}$. By using Lemma \[lem:1\], we see that $\Delta(F_i'/F_{i-1}') \geq 0$. Assume that $s(t_1) \ne s$. Applying this argument successively, we obtain a filtration $F'':0 \subset F_1'' \subset F_2'' \subset \cdots \subset F_u''=E$ such that $\Delta(F_i''/F_{i-1}'')\geq 0$ and $(\mu(F_i'')-\mu(F_{i+1}''),H_{t'})=0$ for some $t'$ with $0 < t' \leq 1$, moreover $\mu(F_i'')-\mu(F_{i+1}'')\ne 0$. This implies that $H_{t'}$ belongs to a wall, which is a contradiction. Let $W$ be a wall and $\cal C$ a chamber such that $\overline{\cal C}$ intersects $W$. Let $H$ be an ample divisor belonging to $\overline{\cal C} \cap W$ and $H_1$ an ample divisor which belongs to $\cal C$. $V_{H,\cal C}^{\gamma}$ be the set of $\mu$-semi-stable sheaves with respect to $H$ such that $E$ is not $\mu$-semi-stable with respect to $H_1$ and $\gamma(E)=\gamma$. We shall investigate the set $V_{H,\cal C}^{\gamma}$. We set $H_t=H_1+t(H-H_1)$, and $B=\{H_t| 0 \leq t \leq 1 \}$. For an element $E$ of $V_{\cal C,H}^{\gamma}$, $S_B(\gamma)$ is a finite set. Hence $S=S_B(\gamma) \cup {\operatornamewithlimits{\cup}}\limits_{F \in \cal F_B(\gamma)}\cup_i S_B(\gamma({\operatorname{gr}}_i(E)))$ is a finite set. Then there is a number $t'$ such that for all $t$ with $t' \leq t < 1$, $F$ is the Harder-Narasimhan filtration of $E$ with resrect to $H_t$ if and only if $F$ is that with respect to $H_{t'}$. In fact, let $W^G$ be a wall defined by a $(\gamma(G_1),\cdots,\gamma(G_s))\in S$ and $I=\{H_t|t' \leq t \leq 1\}$ an interval which is contained in $B \setminus \cup_{G}W^G$. Let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_{s-1} \subset F_s=E$ be the Harder-Narasimhan filtration of $E$ with respect to $H_{t'}$. In the same way as in the proof of Lemma \[lem:2\], there is a subset $\{F_1',F_2',\dots,F_{s'}' \}$ of $\{F_1,F_2,\dots,F_s \}$ such that $F':0 \subset F_1' \subset F_2' \subset \cdots \subset F_{s'}'=E$ belongs to $\cal F_I(\gamma)$. By the choice of $t'$, we get $(\mu(F_i')-\mu(F_{i+1}'),H)=0$. Moreover since $S_I(\gamma(F'_i/F'_{i-1}))$ is a subset of $S$, $\{F_1',F_2',\dots,F_{s'}' \}$ must be $\{F_1,F_2,\dots,F_s \}$. Thus $F$ belongs to $\cal F_B(\gamma)$. If ${\operatorname{gr}}_i(E)$ is not $\mu$-semi-stable with respect to some $H_t$ with $t'<t \leq 1$, then $t'$ and $t$ are separated by a wall (Lemma 2.3), which is a contradiction. Thus $F$ is the Harder-Narasimhan filtration of $E$ for all $H_t$, $t' \leq t<1$. Therefore we get the following proposition. Let $C$ be a 2-dimensional vector space such that $C \cap \cal C \ne \emptyset$ and $H \in C$. (1)There is an element $H_1 \in C$ and $V_{H,\cal C}^{\gamma}$ is the set of torsion free sheaves $E$ such that $E$ has the Harder-Narasimhan filtration $F$ with respect to $H_1$ which is also Harder-Narasimhan filtration with respect to $H_t, 0 \leq t<1$, and $F$ belongs to $\cal F_{\{H\}}(\gamma)$. \(2) $M_H^{\gamma}=M_{\cal C}^{\gamma} \amalg V_{H,\cal C}^{\gamma}$. Equivariant cohomology of $M_H^{\gamma}$ ========================================= Let $C$ be a smooth projective curve with genus $g$ and $\pi:X \to C$ a ruled surface. Let $C_0$ be a minimal section of $\pi$ with $(C_0^2)=-e$. We assume that $e>\chi=2g-2$. In this section, we shall define a cohomology of $M_H^{\gamma}$ and consider the effect of change of polarizations. For a scheme $S$, we denote the projection $S \times X \to S$ by $p_S$. Let $D=nH$, $n \gg 0$ be an ample divisor such that for an element $E \in M_H^{\gamma}$, $E(D)$ is generated by global sections and $H^j(X,E(D))=0$ $j>0$. Let $Q^{\gamma}$ be an open subscheme of $Quot_{\cal O_X(-D)^{\oplus N}/X/\Bbb C}$ such that for a quotient $\cal O_X(-D)^{\oplus N} \to E$, $E$ belongs to $M_H^{\gamma}$, $H^0(X, \cal O_X^{\oplus N}) \cong H^0(X, E(D))$ and $H^j(X, E(D))=0 ,j>0$. Since $(K_X,H)<0$, $Q^{\gamma}$ is smooth. Let $H^*_{GL(N)}(Q^{\gamma},\Bbb Q):=H^*(Q^{\gamma} \times_{GL(N)}E(GL(N)),\Bbb Q)$ be the equivariant cohomology of $Q^{\gamma}$, where $E(GL(N))$ is the universal $GL(N)$-bundle over the classifying space. $H^*_{GL(N)}(Q^{\gamma},\Bbb Q)$ does not depend on the choice of $Q^{\gamma}$. We denote this cohomology by $\tilde H^*(M_H^{\gamma},\Bbb Q)$ and the Poincaré polynomial $\sum_i \dim \tilde{H}^i(M_H^{\gamma},\Bbb Q) z^i$ by $\tilde P (M_H^{\gamma},z)$. Let $Q_i^{\gamma}$ $(i=1,2)$ be an open subscheme of $Quot_{\cal O_X(-D_i)^{\oplus N_i}/X/\Bbb C}$ which satisfies the above conditions. Let $q_i:\cal O_{Q_i^{\gamma} \times X}(-D_i)^{\oplus N_i} \to \cal U_i$ be the universal quotient on $Q_i^{\gamma} \times X$. From the construction, $p_{Q_1^{\gamma}*}\cal U_1(D_2)$ is a locally free sheaf on $Q_1^{\gamma}$. Let $\varphi: \Bbb V=\Bbb V(\cal Hom(\cal O_{Q_1^{\gamma}}^{\oplus N_2}, p_{Q_1^{\gamma}*}\cal U_1(D_2))^{\vee}) \to Q_1^{\gamma}$ be a vector bundle over $Q_1^{\gamma}$ and $h_1:\cal O_{\Bbb V}^{\oplus N_2} \to \varphi^*p_{Q_1^{\gamma}*}\cal U_1(D_2)$ the universal homomorphism. Let $\cal G$ be the open subscheme of $\Bbb V$ such that $h_1$ is an isomorphism. Then $\cal G$ is a principal $GL(N_2)$-bundle over $Q_1^{\gamma}$ and there is a surjection $\cal O_{\cal G \times X}(-D_2)^{\oplus N_2} \to \cal U_1$. For a $S$-valued point of $\cal G$, there is a flat family of quotients $\cal O_{S \times X}(-D_1)^{\oplus N_1} \to \cal E$ and an isomorphism $\cal O_S^{\oplus N_2} \cong p_{S*}\cal E(D_2)$. It defines a surjection $q:\cal O_{S \times X}(-D_2)^{\oplus N_2} \to \cal E$, and conversely for a surjection $q$, it defines an isomorphism $\cal O_S^{\oplus N_2} \cong p_{S*}\cal E(D_2)$. Thus we obtain the following. $$\cal G(S)=\left.\left\{(\cal E,q_1,q_2)\left| \begin{aligned} & \text{$\cal E$ is a flat family of coherent sheaves which belong to $M_H^{\gamma}$,}\\ & \text{ and $q_i:\cal O_{S \times X}(-D_i)^{\oplus N_i} \to \cal E$ is a surjective homomorphism.} \end{aligned} \right. \right\}\right/\sim$$ where $(\cal E,q_1,q_2) \sim (\cal E',q_1',q_2')$ if and only if there is an isomorphism $\psi:\cal E \to \cal E'$ with $q_i'=\psi \circ q_i$. Hence $\cal G$ can be regarded as a $GL(N_1)$-bundle over $Q_2^{\gamma}$. For simplicity, we denote $GL(N_i)$ by $G_i$. $\cal G$ has a natural $G_1\times G_2$-action and $\cal G \times_{G_1 \times G_2} EG_1 \cong Q_1^{\gamma} \times_{G_1} EG_1$. Therefore $\cal G \times_{G_1 \times G_2}EG_1 \times EG_2 \to Q_1^{\gamma} \times _{G_1}EG_1$ is a $EG_2$-bundle. Thus $H^*_{G_1\times G_2}(\cal G,\Bbb Q) \cong H^*_{G_1}(Q_1^{\gamma},\Bbb Q)$. In the same way $H^*_{G_1\times G_2}(\cal G,\Bbb Q) \cong H^*_{G_2}(Q_2^{\gamma},\Bbb Q)$. Hence $H^*_{G_1}(Q_1^{\gamma},\Bbb Q) \cong H^*_{G_2}(Q_2^{\gamma},\Bbb Q)$, which implies that $\tilde{H}^*(M_H^{\gamma},\Bbb Q)$ is well defined. Let $\cal C$ be a chamber and $W$ a wall with $\overline{\cal C} \cap W \ne \emptyset $. Let $H$ be an ample divisor on $W$ and $H' \in \cal C$ an ample divisor which is sufficiently close to $H$. For a sequence of $\gamma_i=(r_i,\mu_i,\Delta_i)$, $1 \leq i \leq s$ with $(\mu_i-\mu_{i+1},H)=0$ and $(\mu_i-\mu_{i+1},H')>0$, we set $$V_{H,\cal C}^{\gamma_1,\cdots,\gamma_s}=\left\{E \left| \begin{aligned} & \text{$E$ is not $\mu$-semi-stable with respect to $H'$ and for the Harder-}\\ & \text{Narasimhan filtration $F:0 \subset F_1 \subset \cdots \subset F_s=E$, $\gamma(F_i/F_{i-1})= \gamma_i$.} \end{aligned} \right. \right \}.$$ Let $Q^{\gamma_1,\cdots,\gamma_s}$ be the subscheme of $Q^{\gamma}$ such that for a quotient $\cal O_X(-D)^{\oplus N} \to E$, $E$ belongs to $V_{H,\cal C}^{\gamma_1,\cdots,\gamma_s}$, and $\Gamma_{H,\cal C}$ the set of sequence $(\gamma_1,\cdots,\gamma_s)$. By \[D-L, 1\], $Q^{\gamma_1,\cdots,\gamma_s}$ is a smooth locally closed subscheme of $Q^{\gamma}$. For the same $D$, let $Q^{\gamma_i}$ be an open subscheme of $Quot_{\cal O_X(-D)^{\oplus N_i}/X/ \Bbb C}$ a quotient $\cal O_X(-D)^{\oplus N_i} \to E$ is contained in $Q^{\gamma_i}$ if and only if $E$ belongs to $M_H^{\gamma_i}$. In the same way as in \[Y2, Appendix\] (cf. \[A-B\],\[K\]), we obtain the following theorem. \[thm:1\] (1) $$\begin{aligned} H_{GL(N)}^*(Q^{\gamma_1,\cdots,\gamma_s},\Bbb Q) & \cong \otimes_iH_{GL(N_i)}^*(Q^{\gamma_i},\Bbb Q)\\ & \cong \otimes_i\tilde H^*(M_{\cal C}^{\gamma_i},\Bbb Q).\end{aligned}$$ (2) $d_{\gamma_1,\cdots,\gamma_s}:={\operatorname{codim}}Q^{\gamma_1,\cdots,\gamma_s}= -\sum_{i<j}r_ir_j(P(\mu_j-\mu_i)-\Delta_i-\Delta_j)$. (3)$$\tilde P(M_H^{\gamma},z)=\tilde P(M_{\cal C}^{\gamma},z)+\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}z^{2d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s\tilde P(M_{\cal C}^{\gamma_i}, z).$$ The proof is the same as that in \[Y2, Appendix\], so we shall give a sketch of the proof. Let $q_i:\cal O_{Q^{\gamma_i} \times X}(-D)^{\oplus N_i} \to \cal F_i$ be the universal quotient. We set $Z=\prod_{i=1}^sQ^{\gamma_i}$ and denote the $i$-th projection by $\varpi_i$. Then the quotient $\oplus_i q_i:\oplus_{i=1}^s \varpi_i^* \cal O_{Q^{\gamma_i} \times X}(-D)^{\oplus N_i} \to \oplus_{i=1}^s \varpi_i^* \cal F_i$ defines a morphism $Z \to Q^{\gamma}$, which is an immersion. We set $Y_1=Z$. We shall define a sequence of schemes $Y_s \to \cdots \to Y_2 \to Y_1$ and quotients $\oplus _{j=1}^i\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i}$ $1 \leq i \leq s$ as follows. Let $\psi_2:Y_2 \to Y_1$ be the vector bundle defined by a locally free sheaf ${\operatorname{Hom}}_{p_{Y_1}}(\varpi_2^*(\ker q_2), \varpi_1^* \cal F_1)$. There is a family of quotients $q_{1,2}:\oplus_{i=1}^2 \cal O_{Y_2 \times X}(-D)^{\oplus N_i} \to \cal F_{1,2}$, which induces $q_1$ and $q_2$. For $\psi_i:Y_i \to Y_{i-1}$ and $\oplus _{j=1}^i\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i}$, ${\operatorname{Hom}}_{p_{Y_i}}(\ker q_{i+1}, \cal F_{1,2,\cdots,i})$ is a locally free sheaf on $Y_i$. Let $q_{i+1}:Y_{i+1} \to Y_i$ be the associated vector bundle on $Y_i$. Then there is a quotient $\oplus _{j=1}^{i+1}\cal O_{Y_j \times X}(-D)^{\oplus N_j} \to \cal F_{1,2,\cdots,i+1}$. Let $P_{\gamma_1, \cdots ,\gamma_s}$ be the parabolic subgroup of $GL(N)$ which preserves the filtration $0 \subset \cal O_{Z \times X}(-D)^{\oplus N_1} \subset \oplus _{i=1}^2 \cal O_{Z \times X}(-D)^{\oplus N_i} \subset \cdots \subset \oplus_{i=1}^s \cal O_{Z \times X}(-D)^{\oplus N_i}$. Then $Q^{\gamma_1, \cdots ,\gamma_s} \cong GL(N) \times_{ P_{\gamma_1, \cdots ,\gamma_s}}Y_s$. The assertions follow from this (cf. \[A-B, 7\]). Let $\cal C'$ be another chamber with $\overline{\cal C'} \cap W \ne \emptyset$. Then $$\tilde P(M_{\cal C'}^{\gamma},z)=\tilde P(M_{\cal C}^{\gamma},z)+\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}\left\{z^{2d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s\tilde P(M_{\cal C}^{\gamma_i}, z) -z^{2d_{\gamma_s,\gamma_{s-1},\cdots,\gamma_1}}\prod_{i=1}^s\tilde P(M_{\cal C'}^{\gamma_i}, z) \right \}.$$ In the same way, we denote the set of $\mu$-semi-stable sheaves defined over $\Bbb F_q$ by $M_{\cal C}^{\gamma}(\Bbb F_q)$. By using \[D-R\], we see that $$\begin{gathered} \label{eq:7} \sum_{E \in M_{\cal C'}^{\gamma}(\Bbb F_q)}\frac{1}{\# {\operatorname{Aut}}(E)}= \sum_{E \in M_{\cal C}^{\gamma}(\Bbb F_q)}\frac{1}{\# {\operatorname{Aut}}(E)}\\ +\sum_{(\gamma_1,\cdots,\gamma_s) \in \Gamma_{H,\cal C}}\left\{q^{d_{\gamma_s,\gamma_{s-1},\cdots,\gamma_1}} \prod_{i=1}^s \sum_{E \in M_{\cal C}^{\gamma_i}(\Bbb F_q)}\frac{1}{\#{\operatorname{Aut}}(E)} -q^{d_{\gamma_1,\cdots,\gamma_s}}\prod_{i=1}^s \sum_{E \in M_{\cal C'}^{\gamma_i}(\Bbb F_q)}\frac{1}{\#{\operatorname{Aut}}(E)} \right \}.\end{gathered}$$ By using the Weil conjectures \[De\] and results of Kirwan \[K\], we can also obtain this corollary (cf. \[Y2, Proposition 4.3\]). By using , \[Y2, Theorem 0.1\] and analoguous argument to the proof of \[Y1, Proposition 0.3\], in principle, we can compute the Betti numbers of the moduli spaces of stable sheaves of rank 3 on $\Bbb P^2$ (in case of $c_1=0$, see \[Y2, 4\]). For example, we obtain the following. $$\begin{aligned} P(M_H(3;1,2),z) &=1+z^2+z^4,\\ P(M_H(3;1,3),z) &=1+2z^2+5z^4+8z^6+10z^8+8z^{10}+5z^{12}+2z^{14}+z^{16}, \\ P(M_H(3;1,4),z) &=1+2z^2+6z^4+12z^6+24z^8+38z^{10}+54z^{12}+59z^{14}\\ &\qquad \qquad+54z^{16}+38z^{18}+24z^{20}+12z^{22}+6z^{24}+2z^{26}+z^{28}.\end{aligned}$$ Let $X$ be a K3 or an Abelian surface and assume that Bogomolov-Gieseker inequality holds. Then also holds. By using induction on $r$, we see that $\sum\limits_{E \in M^{\gamma}_{\cal C}(\Bbb F_q)}\frac{1}{\#{\operatorname{Aut}}(E)}$ does not depend on $\cal C$ (cf. \[Gö\]). The existence of stable sheaves. ================================ In this section, we assume that $X$ is not rational, that is, $g \geq 1$ and assume that $e>-\chi=2g-2$. We denote $C_0+xf$ by $H_x$. Let $W_x$ be a wall containing $H_x$ and let $\cal C_x^+$ (resp. $\cal C_x^-$) a chamber containing $H_x+\epsilon f$ (resp. $H_x-\epsilon{}f$) with $0< \epsilon \ll 1$. For $(\gamma{}_1,\cdots,\gamma{}_s) \in \Gamma{}_{H_x,\cal C_x^-}$, we shall prove that $d_{\gamma{}_1,\cdots,\gamma{}_s} \geq 2$. Since $(\mu{}_j-\mu{}_i)^2<0$ and $\Delta{}_i \geq 0$, it is enough to prove that $r_ir_j(\mu{}_j-\mu{}_i,K_X/2) \geq 1$ for $i <j$. We denote $r_ir_j(\mu{}_j-\mu{}_i)$ by $aC_0-bf$, and then $a$ and $b$ are positive integer. A simple calculation shows that $r_ir_j(\mu{}_j-\mu{}_i,K_X/2)=b+(g-1+e/2)a \geq 3/2$. Therefore $d_{\gamma{}_1,\cdots,\gamma{}_s} \geq 2$. In particular, if $M_{\cal C_x^+}^{\gamma{}}$ is not empty, then $M_{\cal C_x^-}^{\gamma{}}$ is not empty. From this we obtain the following proposition. \[prop:1\] For a triplet $\gamma=(r,\mu,\Delta)$ with $0<(\mu,f)<1$, there exists a $\mu$-semi-stable sheaf $E$ of $\gamma(E)=\gamma$ with respect to $H_x$ if and only if $x \leq \frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$. Assume that $M_{H_x}^{\gamma}$ is not emty. Since $(K_X+f,H)<0$, the deformation theory implies that there is a $\mu$-semi-stable sheaf $E$ and an exact sequence $$0 \to F_1(C_0) \to E \to F_2 \to 0,$$ where $F_1$ and $F_2$ are torsion free sheaves with $(\mu(F_1),f)=(\mu(F_2),f)=0$. We denote ${\operatorname{rk}}(F_i), \mu(F_i)$ and $\Delta(F_i)$ by $r_i,\mu_i$ and $\Delta_i$ respectively. $(\mu(F_1(C_0))-\mu(F_2),H)=(\mu_1-\mu_2+C_0,C_0+xf)=(\mu_1-\mu_2,C_0)-e+x \leq 0$. Thus $x \leq -(\mu_1-\mu_2,C_0)+e$. On the other hand, $\Delta(E)=\frac{r_1}{r} \Delta(F_1(C_0))+\frac{r_2}{r} \Delta_2 -\frac{r_1r_2}{2r^2}((\mu(F_1(C_0))-\mu(F_2))^2) \geq -\frac{r_1r_2}{2r^2}(-e+2(\mu_1-\mu_2,C_0))$. Thus $\Delta(E) \geq \frac{r_1r_2}{2r^2}(x-e/2)$, and hence $x \leq \frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$. We shall next prove that the above condition is sufficient. Let $E$ be a vector bundle defined by the following exact sequence. $$0 \to F_1(C_0) \to E \to F_2 \to 0,$$ where $F_1$ (resp. $F_2$) is the pull-back of a semi-stable vector bundle of rank $r_1$ (resp. $r_2$) on $C$ with degree $d_1=r_1d+\frac{r_1^2-r_1}{2}e-c_2$ (resp. $d-d_1$). Then $E$ is $\mu$-semi-stable with respect to $H'=C_0+(\frac{e}{2}+\frac{r^2}{r_1r_2} \Delta(E))f$. For general $H_x$, the claim follows immediately. \[cor:1\] If $H$ is not on a wall, then there is a $\mu$-stable sheaf. Moreover, ${\operatorname{codim}}(\overline{M}_H(r;c_1,c_2) \setminus M_H(r;c_1,c_2)^{\mu}) \geq 2$. We may assume that $r \geq 4$. Let $F:0 \subset F_1 \subset F_2 \subset \cdots \subset F_s=E$ be a Jordan-Hölder filtration of a $\mu$-semi-stable sheaf $E$. We set $E_i=F_i/F_{i-1}$. Then $-\chi(E_i,E_j)={\operatorname{rk}}(E_i){\operatorname{rk}}(E_j)(g-1+\Delta(E_i)+\Delta(E_j))$. By Proposition \[prop:1\], we see that ${\operatorname{rk}}(E_i)\Delta(E_i) > e/4$. Hence $-\chi(E_i,E_j)>({\operatorname{rk}}(E_i)+{\operatorname{rk}}(E_j))e/4\geq e \geq 1$. The claim follows from this, (cf. \[D-L\] and \[Y3, Proposition 2.3\]). \[cor:2\] If $H$ is not on a wall, then $\overline{M}_H(r,c_1,c_2)$ is locally factorial. This follows from the above corollary and \[D2\]. \[rem:2\] For $(\gamma_1,\cdots ,\gamma_s) \in \Gamma_{H_x,\cal C_x^-}$, $d_{\gamma_1,\cdots ,\gamma_s} \geq 3$. To prove this assertion, we may assume that $s=2$. In the same way, we denote $r_1r_2(\mu_2-\mu_1)$ by $aC_0-bf$. Since $r_1r_2(\mu_2-\mu_1,K_X/2)=b+((g-1)+e/2)a$, the assertion holds unless $a=b=1$. Assume that $a=b=1$, and then $(\mu_1,f)$ or $(\mu_2,f)$ is not an integer. Hence we may assume that $(\mu_1,f)$ is not an integer. By Proposition \[prop:1\], $r_1 \Delta_1 \geq \frac{1}{2}(x-\frac{e}{2})$. Since $x=b/a+e=e+1$, $r_1 \Delta_1 >1/2$. Therefore, we get $d_{\gamma_1,\gamma_2} \geq 3$. The Picard group of $\overline{M}_H(r;c_1,c_2)$ =============================================== In this section, we shall compute the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$ under the assumption that $H_x$ does not lie on a wall. By Corollary \[cor:1\] and \[cor:2\], $Pic(\overline{M}_{H_x}(r;c_1,c_2))=Pic(M_{H_x}(r;c_1,c_2)^{\mu})$. By using \[Y1, Theorem 0.4\], we see that ${\operatorname{codim}}(M_{H_x}(r;c_1,c_2)^{\mu} \setminus M_{H_x}(r;c_1,c_2)^{\mu}_0) \geq r-1$, and hence we shall compute $Pic(M_{H_x}(r;c_1,c_2)^{\mu}_0)$. For a $\mu$-stable vector bundle $E \in M_{H_x}(r;c_1,c_2)^{\mu}_0$, $E^{\vee}$ is $\mu$-stable. Hence $Pic(M_{H_x}(r;c_1,c_2)^{\mu}_0) \cong Pic(M_{H_x}(r;-c_1,c_2)^{\mu}_0)$. Therefore we may assume that $0<(r,f) \leq r/2$. For $c_1=r_1C_0+df$, we set $r_2=r-r_1,d_1=r_1d+\frac{r_1^2-r_1}{2}e-c_2$ and $d_2=d-d_1$. We shall first define a morphism $\overline{M}(r;c_1,c_2) \to J^{d_1} \times J^{d_2}$. Let $\cal Q$ be an open subscheme of quot-scheme $Quot_{V/X/\Bbb C}$ such that $\overline{M}_{H_x}(r;c_1,c_2) =\cal Q/PGL(N)$ and $V \otimes \cal O_{\cal Q \times X} \to \cal U$ the universal quotients. $\cal L=\det((1_{\cal Q} \times \pi)_! \cal U(-C_0))$ is a line bundle on $\cal Q \times C$. It defines a morphism $\lambda_{\cal Q}:\cal Q \to J^{d_1}$. It is easy to see that $\lambda_{\cal Q}$ is $PGL(N)$-invariant. Thus we get a morphism $\lambda:\overline{M}(r;c_1,c_2) \to J^{d_1}$. The line bundle $\det \cal U \otimes \cal O(-r_1C_0)\otimes \cal L^{\vee}$ defines a morphism $\nu: \overline{M}(r;c_1,c_2) \to J^{d_2}$. Therefore we obtain the required morphism $\lambda \times \nu:\overline{M}(r;c_1,c_2) \to J^{d_1} \times J^{d_2}$. For simplicity, we denote $M_H(r;c_1,c_2)_0^{\mu}$ by $M_0^{\mu}$. We set $$M^0=\{E|E \in M_0^{\mu} \text{ and }E_{|\pi{}^{-1}(P)} \cong \cal O_{\pi{}^{-1}(P)}(1)^{\oplus r_1} \oplus \cal O_{\pi{}^{-1}(P)}^{\oplus r_2} \text{ for all $P \in C$} \}.$$ Assume that $r_1 \ne 1$. Since $e >\chi$, we get $(K_X+f,H_x)<0$. Then, by the deformation theory, we see that ${\operatorname{codim}}(M_0^{\mu} \setminus M^0)\geq 2$. Next we assume that $r_1=1$. For a fibre $l$, we set $$Z_l=\{E|E \in M_0^{\mu},\;E_l \cong \cal O_l(1)^{\oplus 2} \oplus \cal O_l(-1) \oplus \cal O_l^{\oplus r_2-2} \}.$$ Then we see that ${\operatorname{codim}}Z_l=2$ unless $Z_l=\emptyset$. We set $Z=\cup_{P \in C}Z_{\pi{}^{-1}(P)}$. $Z$ is a locally closed subscheme of $M_0^{\mu}$. For a point $E$ of $Z$, there is an exact sequence $0 \to L(C_0) \to E \to F \to 0$ , where $L$ is a line bundle with $c_1(L)=(d_1+1)f$ and $F$ is a torsion free sheaf with $c_1(F)=(d_2-1)f$ and $c_2(F)=1$. We set $x_0=\frac{e}{2}+\frac{r^2}{r_1r_2} \Delta$ and $x_1=\frac{e}{2}+\frac{r^2}{r_1r_2} (\Delta-\frac{1}{r})$. If $x_1<x<x_0$, then $(H_x,\mu{}(L(C_0)))>(H_x,\mu{}(E))$, which is a contradiction. Thus $Z=\emptyset$. Assume that $x<x_1$. Then for some $L$ and $F$ such that $F$ is semi-stable, there is an exact sequence $$\label{eq:2} 0 \to L(C_0) \to E \to F \to 0$$ such that $E$ is semi-stable (see the proof of Proposition \[prop:1\]). Thus $Z$ is not empty. Therefore, to compute the Picard group of $\overline{M}_{H_x}(r;c_1,c_2)$, it is enough to consider $Pic(M^0 \cup (Z^0 \cap M_0^{\mu}) )$, where $Z^0$ is the subscheme of $M_H(r;c_1,c_2)$ consisting of stable sheaves which are defined by the exact sequence . We set $V_i=\cal O_X(-nH_x)^{\oplus N_i}$, $(i=1,2$). Let $Quot_{V_1/X/\Bbb C}^{\gamma_1}$ ( resp. $Quot_{V_2/X/\Bbb C}^{\gamma _2}$) be a quot-scheme parametrizing all quotients $V_1\to F_1$ (resp. $V_2 \to F_2$) such that $\gamma(F_1)=(r_1,C_0+\frac{d_1}{r_1}f,0)$ (resp. $\gamma(F_2)=(r_2,\frac{d_2}{r_2}f,0)$). Let $Q_i$ ($i=1,2$) be the open subscheme of $Quot_{V_i/X/\Bbb C}^{\gamma_i}$ consisting all quotients $V_i \to F_i$ which satisfy 1. $F_i$ is $\mu$-semi-stable with respect to $H_x$, 2. $F_{i|\pi^{-1}(\eta)}$ is a semi-stable vector bundle, where $\eta$ is the generic point of $C$, 3. $H^0(X,V_i(nH_x)) \cong H^0(X,F_i(nH_x))$, $H^j(X,F_i(nH_x))=0,j>0$. Let $V_i \otimes \cal O_{Q_i \times X} \to \cal F_i$ be the universal quotient, and $\cal K_i$ the universal subsheaf. If we choose a sufficiently large integer $n$ and a suitable $N_i$, then all $\mu$-semi-stable sheaves which satisfy (ii) are parametrized by $Q_i$. We set $\cal F_1'=\cal F(-C_0)$ and $\cal F_2'=\cal F_2$. Let $g_i:G_i= Gr(p_{Q_i*}(\cal F_i'),r_i-1) \to Q_i$ be the grassmannian bundle over $Q_i$ parametrizing rank $r_i-1$ subbundle of $p_{Q_i*}(\cal F_i')$ and $\cal U_i$ the universal subbundle of rank $r_i-1$. Since $p_{Q_i*}(\cal F_i')$ is $PGL(N_i)$-linearized, $G_i$ is $PGL(N_i)$-linearized. Let $G_i'=\{x \in G_i|\cal U_x \otimes \cal O_X \text{ is a subbundle of }(\cal F_i')_x \}$. Let $h_i:D_i=\Bbb P(\cal Hom (\cal O_{G'_i}^{(r_i-1)},\cal U)^{\vee})\to G'_i$ be a projective bundle and $\cal O_{D_i}(1)$ the tautological line bundle on $D_i.$ On $D_i$, there is a homomorphism $\delta:\cal O_{D_i}^{\oplus (r_i-1)} \to h_i^*\cal U \otimes \cal O_{D_i}(1)$. Let $D_i'=\{x \in D_i |\delta_x \text{ is an isomorphism} \}$ be an open set of $D_i$. Setting $\widetilde{\cal F_i'}=(g_i \circ h_i \times 1_X)^*\cal F_i'$ and $\widetilde{\cal U}=p^*_{D_i'} h_i^* \cal U$, there is an injective homomorphism on $D_i' \times X$: $\cal O_{D_i' \times X}^{\oplus (r_i-1)} \to \widetilde{\cal U} \otimes p_{D_i'}^*\cal O_{D_i'}(1) \to \widetilde{\cal F_i'} \otimes p_{D_i'}^*\cal O_{D_i'}(1)$. The quotient $\widetilde{\cal F_i'}\otimes p_{D_i'}^*\cal O_{D_i'}(1)/ \cal O^{\oplus (r_i-1)}_{D_i' \times X}$ is a flat family of line bundles of degree $d_i$. Thus we obtain an extension $$0 \to \cal O^{\oplus (r_i-1)}_{D_i' \times X} \to \widetilde{\cal F_i'}\otimes p_{D_i'}^*\cal O_{D_i'}(1) \to \det(\widetilde{\cal F_i'}\otimes p^*_{D_i'}\cal O_{D_i'}(1)) \to 0. \label{eq:1}$$ We set $Q=Q_1 \times Q_2$, $D=D_1' \times D_2'$ and $I=PGL(N_1)\times PGL(N_2)$. Then, in the same way as in \[D-N, 7.3.4\], we obtain the following exact sequence: $$0 \to Pic^{I}(Q_1 \times Q_2) \to Pic^{I}(D_1 \times D_2) \to T \to 0, \label{eq:5}$$ where $T$ is a finite abelian group with $\#T=\frac{(r_1-1)d_1}{n_1}\frac{(r_2-1)d_2}{n_2}$. Let $\cal P_i$ be a poincaré line bundle of degree $d_i$ on $J^{d_i}\times X$. Let $\cal V_i:={\operatorname{Ext}}^1_{p_{J^{d_i}}}(\cal P_i,\cal O_{J^{d_i} \times X}^{\oplus (r_i-1)})$ be the relative extension sheaf on $J^{d_i}$. The base change theorem implies that $\cal V_i$ is locally free. Let $\mu_i:\Bbb P_i=\Bbb P(\cal V_i^{\vee}) \to J^{d_i}$ be the projection and $\cal O_{\Bbb P_i}(1)$ the tautological line bundle on $\Bbb P_i$. On $\Bbb P_i$, there is a universal family of extensions: $$0 \to \cal O_{\Bbb P_i \times X}^{\oplus (r_i-1)} \to \cal E_i \to \mu_i^* \cal P \otimes \cal O_{\Bbb P_i}(-1) \to 0. \label{eq:4}$$ We set $\Bbb P_i^{ss}=\{y \in \Bbb P_i|\text{$(\cal E_i)_y$ is semi-stable}\}$. The extension gives a morphism $k':D_i' \to \Bbb P_i^{ss}$ such that the pull–back of is . Since $k'$ is $PGL(N_i)$-invariant and each fibre of $k'$ is an orbit of $PGL(N_i)$, \[M-F-K, Proposition 0.2\] implies that $\Bbb P_i^{ss}$ is a geometric quotient of $D_i'$ by $PGL(N_i)$. By \[Y3, 4.2\], $k'$ has a local section. Since the action of $PGL(N_i)$ is set-theoretically free, Z.M.T. implies that $D_i'$ is a Zariski locally trivial fibre bundle. In particular, $Pic^{I}(D)=Pic(\Bbb P_1^{ss}\times \Bbb P_2^{ss})$ (\[SGA I, 8\]). The base change theorem implies that $\cal V={\operatorname{Ext}}^1_{p_Q}(\cal F_2,\cal F_1)$ is a locally free sheaf on $Q$. Let $\Bbb P_Q=\Bbb P(\cal V^{\vee}) \to Q$ be the projective bundle associated to $\cal V^{\vee}$ and $\cal O_{\Bbb P_Q}(1)$ the tautological line bundle. On $\Bbb P(\cal V^{\vee}) $, there is a universal extension $$0 \to \cal F_1 \to \cal E \to \cal F_2 \otimes \cal O_{\Bbb P_Q}(-1) \to 0. \label{eq:6}$$ $I$ acts on $\Bbb P(\cal V^{\vee})$ and $\cal E$ is a $GL(N_1) \times PGL(N_2)$-linearized. Let $\Bbb P_Q^s$ be the open subscheme of $\Bbb P(\cal V^{\vee})$ parametrizing stable sheaves. Then there is a surjective morphism $\lambda:\Bbb P_Q^s \to M^0$. It is easy to see that $\lambda$ is $I$-invariant and each fibre is an orbit of this action. Thus $M^0$ is a geometric quotient of $\Bbb P_Q^s$ by $I$. Let $S \to M^0$ be a smooth and surjective morphism such that there is a universal family $\cal E$. Then there is an exact sequence $0 \to \cal G_1(C_0) \to \cal E \to \cal G_2 \to 0$, where $\cal G_1=\pi^*\pi_* \cal E(-C_0)$ and $\cal G_2=\cal E/\cal G_1(C_0)$. There is an open covering $\{U_i \}$ of $S$ such that $p_{U_i*} \cal G_1$ and $p_{U_i*} \cal G_2$ are free $\cal O_{U_i}$-module. Then it defines a morphism $U_i \to Q$ and hence we get $U_i \to \Bbb P_Q^s$. Therefore we get a local section of $\Bbb P_Q^s \times _{M^0}S \to S$. Since $S \to M^0$ is a smooth morphism, $\Bbb P_Q^s \times _{M^0}S$ is smooth. By using Z.M.T., we see that $\Bbb P_Q^s \times _{M^0}S \to Q$ is a Zariski locally trivial $I$-bundle. By the descent theory (\[SGA I, 8\]), $\Bbb P_Q^s \to M^0$ is a $I$-bundle. We shall prove that ${\operatorname{codim}}(\Bbb P_Q\setminus \Bbb P^s_Q) \geq 2$. In the same way as in the proof of Theorem\[thm:1\], we define $Y_2=\Bbb V({\operatorname{Hom}}_{p_Q}(\cal K_2,\cal F_1)^{\vee})$. Then, there is a quotient $\cal O_{Y_2\times X}(-D)^{\oplus N} \to \cal F_{1,2}$. It defines a closed immersion $Y_2 \hookrightarrow \cal Q$. Let $Y_2^s$ be the open subscheme of $Y_2$ parametrizing all quotients which are stable with respect to $H_x$. Corollary \[cor:1\] implies that ${\operatorname{codim}}(Y_2\setminus Y_2^s)\geq 2$. Note that each fibre of $Y_2 \to \Bbb V({\operatorname{Ext}}_{p_Q}(\cal F_2,\cal F_1)^{\vee})$ is contained in an orbit of $P_{\gamma_1,\gamma_2}$, where $P_{\gamma_1,\gamma_2}$ is the parabolic subgroup of $GL(N)$ defined in the proof of Theorem \[thm:1\]. From this, we obtain that ${\operatorname{codim}}(\Bbb P_Q \setminus \Bbb P^s_Q) \geq 2$. On $D_i$, there is a $GL(N_i)$-linearized line bundle such that the action of the center $\Bbb C^{\times}$ is multiplication by constants. In the same way, we obtain the following exact and commutative diagram: $$\begin{CD} @[email protected]@.0@. \\ @.@VVV @VVV @VVV @. \\ 0 @>>> Pic^{I}(Q) @>>> Pic^{I}(\Bbb P_Q^s) @>>> \frac{n_1 n_2}{n} \Bbb Z @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> Pic^{I}(D) @>>> Pic^{I}(\Bbb P_D^s) @>>> \Bbb Z @>>> 0 \\ @. @VVV @VVV @VVV @. \\ 0 @>>> T @>>> T' @>>> \Bbb Z/ \frac{n_1 n_2}{n}\Bbb Z @>>> 0 \\ @.@VVV @VVV @VVV @. \\ @.0 @[email protected]@. \end{CD}$$ where $T'$ is a finite abelian group with $\#T'=\frac{(r_1-1)d_1(r_2-1)d_2}{n}$. Let $K(X)$ be the Grothendieck group of $X$. Let $K^0(X)$ be the subgroup of $K(X)$ which is generated by $\cal O_X-\cal O_X(-D)$ and $\cal O_C-\cal O_C(-D)$, $D,D' \in Pic^0(X)$. Then $K^0(X) \cong Pic^0(X)\oplus Alb(X)$. We shall represent the class in $K(X)$ of $\cal O_X,\cal O_X(-f), \cal O_X(-C_0)$ and $\cal O_X(-C_0-f)$ by $e_1,e_2,e_3$ and $e_4$ respectively. Then $K(X) \cong K^0(X) \oplus L$, where $L$ is the free $\Bbb Z$-module of rank 4 generated by $e_i$, $1 \leq i \leq 4$. Let $\varepsilon$ be the class in $K(X)$ of a torsion free sheaf of rank $r$ with Chern classes $c_1,c_2$ and let $K(r;c_1,c_2)$ be the kernel of a homomorphism $K(X) \to \Bbb Z:x \mapsto \chi(\varepsilon \otimes x)$. Let $\cal E$ be a family of stable sheaves of rank $r$ with Chern classes $c_1,c_2$ parametrized by a smooth scheme $S$. Then $\det(p_{S!}(\cal E \otimes x))$, $x \in K(r;c_1,c_2)$ defines a line bundle on $S$. Thus we obtain a homomorphism $\kappa{}_S:K(r;c_1,c_2) \to Pic(S)$. We can also define $\kappa{}:K(r;c_1,c_2) \to Pic(M_H(r;c_1,c_2))$, (see \[Y3, 4.3\]). $K(r;c_1,c_2)=K^0(X) \oplus K$ where $K=K(r;c_1,c_2) \cap L$. \[lem:6\] If $r_1 \ne 1$, then $K(r;c_1,c_2) \to Pic(M_H(r;c_1,c_2))/Pic(J^{d_1} \times J^{d_2})$ is surjective. We denote $Pic^I(\Bbb P_D)/Pic(J^{d_1} \times J^{d_2})$ by $N$. Since $Pic^I(D)/Pic(J^{d_1} \times J^{d_2}) \cong Pic(\Bbb P_1 \times \Bbb P_2)/Pic(J^{d_1} \times J^{d_2}) \cong \Bbb Z^{\oplus 2}$, we get $N \cong \Bbb Z^{\oplus 3}$. We shall prove that $\#(N/{\operatorname{im}}(\kappa_{\Bbb P_D}))=\#T'$. We denote the image of $\cal O_{\Bbb P_1}(1), \cal O_{\Bbb P_2}(1)$ and $\cal O_{\Bbb P_D}(1)$ to $N$ by $\nu_1,\nu_2$ and $\nu$ respectively. Let $\theta:Pic^I(\Bbb P_D) \to N$ be the quotient homomorphism. We shall define $A_i$ $(1 \leq i \leq 4)$ as follows: $$\left\{ \begin{split} A_1 &:=\theta (\det p_{\Bbb P_D!}\cal E)=-((2\chi+2d_1-e)\nu_1+(\chi+d_2)\nu_2+(r_2\chi+d_2)\nu),\\ A_2 &:=\theta (\det p_{\Bbb P_D!}\cal E(-f))=-((2\chi+2d_1-2-e)\nu_1+(\chi+d_2-1)\nu_2+(r_2\chi+d_2-r_2)\nu),\\ A_3 &:=\theta (\det p_{\Bbb P_D!}\cal E(-C_0))=-(\chi+d_1)\nu_1,\\ A_4 &:=\theta (\det p_{\Bbb P_D!}\cal E(-C_0-f))=-(\chi+d_1-1)\nu_1. \end{split} \right.$$ Let $\phi:L \to N $ be a homomorphism such that $\phi(e_i)=A_i$. Then a simple calculation shows that $\phi(K)={\operatorname{im}}(\kappa_{\Bbb P_D})$. There is the following exact and commutative diagram $$\begin{CD} @[email protected]@.0@. \\ @.@.@VVV @VVV @. \\ @.@. \ker \phi @= \ker \phi @. \\ @.@.@VVV @VVV @. \\ 0 @>>> K @>>> L @> \psi>> n\Bbb Z @>>> 0 \\ @. @| @VV{\phi}V @VVV @. \\ 0 @>>> K @>>> \phi(L) @>>> \phi(L)/K @>>> 0 \\ @.@.@VVV @VVV @. \\ @.@. 0 @.0@. \end{CD}$$ It is easy to see that $\ker \phi$ is generated by $(\chi+d_1-1)e_3-(\chi+d_1)e_4$ and $N/\phi(L) \cong \Bbb Z/(r_2-1)d_2 \Bbb Z$. Hence $\psi(\ker \phi)=(r_1-1)d_1 \Bbb Z$. Therefore $\#N/K=\#(N/\phi(L))\# (\phi(L)/K)=\frac{(r_1-1)d_1(r_2-1)d_2}{n}$, and hence $\# N/K=\#T'$. Thus, we obtain our lemma. \[lem:14\] The restriction of $\kappa:K(r;c_1,c_2) \to Pic(M(r;c_1,c_2))$ to $K^0(X)$ is injective and its image is $(\lambda \times \det)^*(Pic^0(J^{d_1} \times J^{d_2}))$. If $D=\sum_ia_i \pi^*(R_i), a_i \in \Bbb Z$, then we see that $\kappa_{\Bbb P_D}(\cal O_X(D)-\cal O_X)= \otimes_i (\cal P_{1R_i}^{\otimes 2}\otimes \cal P_{2R_i})^{\otimes a_i}$ and $\kappa_{\Bbb P_D}(\cal O_{C_0}(D)-\cal O_{C_0}) =\otimes_i \cal P_{1R_i}^{\otimes a_i}$. The assertion follows immediately from this. We shall first consider the case of $g \geq 2$. Assume that $g \geq 2$ and $H_x$ does not lie on a wall. \(1) If $r_1 \ne 1$, then $Pic(\overline{M}_{H_x}(r;c_1,c_2)) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}$. \(2) If $r_1=1$, then $$Pic(\overline{M}_{H_x}(r;c_1,c_2)) \cong \left\{ \begin{aligned} & Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 2}, \;x_1<x<x_0\\ & Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}, \;x<x_1. \end{aligned} \right.$$ \(3) $Pic(\overline{M}_{H_x}(r;c_1,c_2))/Pic(J^{d_1}\times J^{d_2})$ is generated by the image of $\kappa$. In the same way as in the proof of \[Y3, Lemma 4.2\], we see that ${\operatorname{codim}}(\Bbb P_i^{ss}) \geq 2$, ($i=1,2$) for a sufficiently large $d$. Then (1) follows immediately from Lemma \[lem:6\] and Lemma \[lem:14\]. We shall treat the case $r_1=1$. In the same way, we can define $Q_1,Q_2,D_1$ and $D_2$, where $D_1=Q_1$. Moreover we can define a projective bundle $\Bbb P_D$, where $D=D_1 \times D_2$. Then it is easy to see that $Pic^I(\Bbb P_D)/Pic(J^{d_1} \times J^{d_2}) \cong \Bbb Z^{\oplus 2}$ and $\# (Pic^I(\Bbb P_D)/Pic^I(\Bbb P_{Q_1 \times Q_2}))=\frac{(r_2-1)d_2}{n_2}$. We denote $Pic^I(\Bbb P_D)/Pic^I(\Bbb P_{Q_1 \times Q_2})$ by $N$. In the same way as in Lemma \[lem:6\], we denote the image of $\cal O_{\Bbb P_2}(1)$ and $\cal O_{\Bbb P_D}(1)$ to $N$ by $\nu_2$ and $\nu$ respectively. Let $\phi:L \to N$ be the homomorphism such that $\phi(e_1)=-((\chi+d_2)\nu_2+(r_2 \chi+d_2)\nu), \phi(e_2)=-((\chi+d_2-1)\nu_2+(r_2 \chi+d_2-r_2)\nu),\phi(e_3)=\phi(e_4)=0$. Then $\phi(K)={\operatorname{im}}(\kappa_{\Bbb P_D})$. There is the following exact and commutative diagram: $$\begin{CD} @[email protected]@.0@. \\ @.@VVV @VVV @VVV @. \\ 0 @>>> K \cap \ker \phi @>>> \ker \phi @>>> \psi(\ker \phi) @>>> 0\\ @.@VVV @VVV @VVV @. \\ 0 @>>> K @>>> L @> \psi>> \Bbb Z @>>> 0 \\ @. @VVV @VV{\phi}V @VVV @. \\ 0 @>>> \phi(K) @>>> \phi(L) @>>> \phi(L)/K @>>> 0 \\ @.@VVV @VVV @VVV @. \\ @. 0@. 0 @.0@. \end{CD}$$ Since $\ker \phi=\Bbb Z e_3 \oplus \Bbb Z e_4$, we get $\Bbb Z/\psi(\ker \phi)=0$, and hence $\phi(K)=\phi(L)$. A simple calculation shows that $\# N/\phi(L)=\frac{(r_2-1)d_2}{n_2}$, therefore $K \to Pic(M^0)/Pic(J^{d_1} \times J^{d_2})$ is surjective. From this we get $Pic(M_{\cal C_{x_0}^-}(r;c_1,c_2))=Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 2}$, and hence we obtain the assertion for $x_1<x<x_0$. We shall next prove the claim for $x<x_1$. It is sufficient to compute $Pic(M_{\cal C_{x_1}^-}(r;c_1,c_2))$. We set $V_i=\cal O_X(-nH_{x_1})^{\oplus N_i}$, $(i=3,4$). Let $Quot_{V_3/X/\Bbb C}^{\gamma_3}$ ( resp. $Quot_{V_4/X/\Bbb C}^{\gamma _4}$) be a quot-scheme parametrizing all quotients $V_3\to F_3$ (resp. $V_4 \to F_4$) such that $\gamma(F_3)=(r_1,C_0+\frac{d_1+1}{r_1}f,0)$ (resp. $\gamma(F_4)=(r_2,\frac{d_2-1}{r_2}f,\frac{1}{r_2})$). Let $Q_i$ ($i=3,4$) be the open subscheme of $Quot_{V_i/X/\Bbb C}^{\gamma_i}$ consisting quotients $V_i \to F_i$ which satisfy 1. $F_i$ is $\mu$-semi-stable with respect to $H_x$, 2. $F_{i|\pi^{-1}(\eta)}$ is a semi-stable vector bundle, 3. $H^0(X,V_i(nH_{x_1})) \cong H^0(X,F_i(nH_{x_1}))$, $H^j(X,F_i(nH_{x_1}))=0,j>0$. Let $V_i \otimes \cal O_{Q_i \times X} \to \cal F_i$ be the universal quotient, and $\cal K_i$ the universal subsheaf. If we choose a sufficiently large integer $n$ and a suitable $N_i$, then all $\mu$-semi-stable sheaves which satisfy (ii) are parametrized by $Q_i$. We set $R=Q_3 \times Q_4$. The base change theorem implies that $\cal W={\operatorname{Ext}}^1_{p_R}(\cal F_4,\cal F_3)$ is a locally free sheaf on $R$. Let $\Bbb P_R=\Bbb P(\cal W^{\vee}) \to R$ be the projective bundle associated to $\cal W^{\vee}$ and $\cal O_{\Bbb P_R}(1)$ the tautological line bundle. On $\Bbb P(\cal W^{\vee}) $, there is a universal extension $$0 \to \cal F_3 \to \cal E \to \cal F_4 \otimes \cal O_{\Bbb P_R}(-1) \to 0. \label{eq:3}$$ $PGL(N_3)\times PGL(N_4)$ acts on $\Bbb P(\cal W^{\vee})$ and $\cal E$ is a $GL(N_3) \times PGL(N_4)$-linearized. Let $\Bbb P_R^s$ be the open subscheme of $\Bbb P(\cal W^{\vee})$ parametrizing stable sheaves. Then there is a surjective morphism $\lambda:\Bbb P_R^s \to Z^0 \subset M(r;c_1,c_2)$. It is easy to see that $\lambda$ is $PGL(N_3)\times PGL(N_4)$-invariant and each fibre is an orbit of this action. In the notation of [**5.1**]{}, we denote the pull–back of $Z^0$ to $\cal Q$ by $\cal Z^0$. \[D-L, 1\] implies that $\cal Z^0$ is smooth. Hence $Z^0$ is a geometric quotient of $\Bbb P_R^s$ by $PGL(N_3)\times PGL(N_4)$. On $\cal Z^0$, there is an exact sequence $ 0 \to \cal G_1(C_0) \to \cal U_{|\cal Z^0 \times X} \to \cal G_2 \to 0$, where $\cal G_1$ is a flat family of line bundles with $c_1=(d_1+1)f$ and $\cal G_2$ a flat family of torsion free sheaves of rank $r-1$ with Chern classes $((d_2-1)f,1)$. Then the normal bundle $\cal O_{\cal Z^0}(\cal Z^0)$ is isomorphic to ${\operatorname{Ext}}^1_{p_{\cal Z^0}}(\cal G_1(C_0),\cal G_2)$. It is easy to see that the pull–back of ${\operatorname{Ext}}^1_{p_{\cal Z^0}}(\cal G_1(C_0),\cal G_2)$ to $\cal Z^0 \times _{Z^0} \Bbb P_R^s$ is isomorphic to the pull–back of ${\operatorname{Ext}}^1_{p_{\Bbb P^s_R}}(\cal F_3,\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$ to $\cal Z^0 \times _{Z^0} \Bbb P_R^s$. Since $Pic(\Bbb P_R^s) \to Pic(\cal Z^0 \times _{Z^0} \Bbb P_R^s)$ is injective, the pull–back of $\cal O_{Z^0}(Z^0)$ to $\Bbb P_R^s$ is isomorphic to ${\operatorname{Ext}}^1_{p_{\Bbb P^s_R}}(\cal F_3,\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$. By virtue of Remark \[rem:2\], we get ${\operatorname{codim}}(\Bbb P_R \setminus \Bbb P_R^s) \geq 2$. Let $\lambda' \times \det:\overline{M}(r_2;(d_2-1)f,1) \to C \times J^{d_2-1}$ be the morphism defined in \[Y3\]. Let $J^{d_1+1}\times C \times J^{d_2-1} \to J^{d_1} \times J^{d_2}$ be the morphism sending $(L,P,L')$ to $(L \otimes \cal O_C(-P),L' \otimes \cal O_C(P))$. Then the composition $Z^0 \to \overline{M}(r_1;(d_1+1)f,0) \times \overline{M}(r_2;(d_2-1)f,1) \to J^{d_1+1}\times C \times J^{d_2-1} \to J^{d_1} \times J^{d_2}$ is the same as the restriction of $\lambda \times \nu$ to $Z^0$. It is easy to see that $K \cap \ker \phi=\Bbb Z((\chi+d_1-1)e_3-(\chi+d_1)e_4)$, and hence $L:=\kappa((\chi+d_1-1)e_3-(\chi+d_1)e_4))$ can be written as $\cal O(nZ^0)\otimes \cal L$, where $\cal L$ is the pull–back of a line bundle on $J^{d_1} \times J^{d_2}$. We shall prove that $n=-1$. Since $\det(\cal E(-C_0))=\det(\cal F_3(C_0))\otimes \det(\cal F_4 \otimes \cal O_{\Bbb P_R}(-1))$, the restriction of $\det(\cal E(-C_0))$ to a fibre of $\Bbb P_R \to R$ is $\cal O(1)$. From this, we see that $n=-1$. (3) follows from the proof of (1) and (2). We shall treat the case of $g=1$. We assume that $(r_1,d_1) \ne 1$ and $(r_2,d_2)=1$, and then there is an integers $r_2'$ and $d_2'$ such that $r_2d_2'-r_2'd_2=1$ and $0<r_2'<r_2$. We set $r_2''=r_2-r_2'$ and $d_2''=d_2-d_2'$. Let $W$ be the subset of $M^0$ whose element $E$ has the following filtration $F:0 \subset F_1 \subset F_2 \subset F_3=E$ such that 1. ${\operatorname{rk}}(F_1)=r_1$, $c_1(F_1(-C_0))=d_1f$ and $c_2(F_1(-C_0))=0$, 2. ${\operatorname{rk}}(F_2/F_1)=r_2'$, $c_1(F_2/F_1)=d_2'f$ and $c_2(F_2/F_1)=0$, 3. ${\operatorname{rk}}(F_3/F_2)=r_2''$, $c_1(F_3/F_2)=d_2''f$ and $c_2(F_3/F_2)=0$. If $\frac{r}{r_1}\left(\frac{d_2''}{r_2''}-\frac{d}{r} \right) < x < \frac{d_2}{r_2}-\frac{d}{r}$, then for a general element $E$ of $W$, the Harder-Narasimhan filtration is $F':0 \subset F_2 \subset F_3=E$. Let $Q_2'$ be an open subscheme of $Quot_{V_2/X/\Bbb C}$ whose point $y$ satisfies that $\cal F_y$ is semi-stable or the Harder-Narasimhan filtration of $\cal F_y$ is $0 \subset G_1 \subset \cal F_y$, where $G_1$ is a semi-stable vector bundle of rank $r_2'$ and degree $d_2'$. We shall replace $Q_2$ by $Q_2'$, and construct $\Bbb P_D$ and $\Bbb P_D^s$. Let $\widetilde{W}$ be the open subscheme of $\Bbb P_D \setminus \Bbb P_D^s$ whose point defines an element of $W$. Then, there is an exact sequence $ \Bbb Z \widetilde{W} \to Pic^I(\Bbb P_D) \to Pic^I(\Bbb P_D^s) \to 0.$ In the same way, we see that $\cal O_{\widetilde{W}}(\widetilde{W})$ is a primitive element of $Pic(\widetilde{W})$. Note that $Pic^I(\Bbb P_D)$ is isomorphic to $Pic(\overline{M}_{H_{x'}}(r,c_1,c_2))$, $x'<\frac{r}{r_1}(\frac{d_2''}{r_2''}-\frac{d}{r})$. Therefore $Pic^I(\Bbb P_D^s) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus 3}/\Bbb Z \widetilde{W} \cong Pic(J^{d_1} \times J^{d_2})\oplus \Bbb Z^{\oplus 2}$. In the same way, we obtain the following theorem If $g=1$, then $Pic(\overline{M}_H(r;c_1,c_2)) \cong Pic(J^{d_1} \times J^{d_2}) \oplus \Bbb Z^{\oplus a}$, $a=1,2$ or $3$. [\[M-F-K\]]{} Atiyah, M. F., Bott, R. [*The Yang-Mills equations over Riemann surfaces,*]{} Philos. Trans. Roy. Soc. London Ser. A [**308**]{} (1982), pp. 523–615 Deligne, P., [*La conjecture de Weil. I,*]{} Inst. Hautes Etudes Sci. Publ.Math. [**43**]{} (1974) pp. 273–307 Drezet, J.-M., [*Groupe de Picard des variétés de modules de faisceaux semi-stables sur $\Bbb P^2(\Bbb C)$,*]{} Ann. Inst. Fourier [**38**]{} (1988), pp. 105–168 Drezet, J.-M., [*Points non factoriels des variétés de modules de faisceaux semi-stable sur une surface rationelle,*]{} J. reine angew. Math. [**413**]{} (1991), pp. 99–126 Drezet, J.-M., Le-Potier, J., [*Fibrés stables et fibrés exceptionnels sur $\Bbb P^2$,*]{} Ann. scient. Éc. Norm. Sup., $4^e$ série, t. [**18**]{} (1985), pp. 193–244 Drezet, J.-M., Narasimhan, M. S., [*Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algégriques,*]{} Invent. math. [**97**]{} (1989), pp. 53–94 Desale, U. V., Ramanan, S. [*Poincaré polynomials of the variety of stable bundles,*]{} Math. Ann. [**216**]{} (1975) pp. 233–244 Friedman, R., Morgan, J., [*On the diffeomorphism types of certain algebraic surface I,*]{} J. Differential Geometry [**27**]{} (1988), pp. 371–398 Gieseker, D., [*On a theorem of Bogomolov on Chern classes of stable bundles,*]{} Amer. J. Math. [**101**]{} (1979), pp. 77–85 Göttsche, L., [*Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces,*]{} MPI 94-9 (1994) Grothendieck, G., [*Séminaire de géométrie algébrique I,*]{} Springer Lecture Notes [**224**]{} (1971) Kirwan, F., [*Cohomology of quotients in symplectic and algebraic geometry,*]{} Princeton Math. Notes 31 Princeton N.J. (1984) Maruyama, M., [*Moduli of stable sheaves II,*]{} J. Math. Kyoto Univ. [**18**]{} (1978), pp. 557–614 Mumford, D., Fogarty, J., Kirwan, F., [*Geometric Invariant Theory,*]{} Springer-Verlag Matsuki, K., Wentworth, R., [*Mumford-Thaddeus principle on the moduli space of vector bundles on an algebraic surface,*]{} preprint Qin, Z. B., [*Moduli of stable sheaves on ruled surfaces and their Picard groups,*]{} J. reine angew. Math. [**433**]{} (1992), pp. 201–219 Qin, Z. B., [*Equivalence classes of polarizations and moduli spaces of sheaves,*]{} J. Differential Geometry [**37**]{} (1993) pp. 397–413 Rudakov, A. N., [*A description of Chern classes of semistable sheaves on a quadric surface,*]{} J. reine angew. Math. to appear Strømme, S. A., [*Ample Divisors on Fine Moduli spaces on the Projective Plane,*]{} Math. Z. [**187**]{} (1984), pp. 405–423 Yoshioka, K., [*The Betti numbers of the moduli space of stable sheaves of rank 2 on $\Bbb P^2$,*]{} J. reine angew. Math. to appear Yoshioka, K., [*The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface,*]{} Math. Ann. to appear Yoshioka, K., [*The Picard group of the moduli space of stable sheaves on a ruled surface,*]{} preprint [^1]: Supported by JSPS Research Fellowships for Young Scientists

--- abstract: 'We present a statistical analysis of 132 dayside (LT 0700-1700) bow shock crossings of the AMPTE/IRM spacecraft. We perform a superposed epoch analysis of plasma and magnetic field parameters as well as of low frequency magnetic power spectra some minutes upstream and downstream of the bow shock by dividing the events into categories depending on the angle $\theta_{Bn}$ between bow shock normal and interplanetary magnetic field and on the plasma-$\beta$, i.e., the ratio of plasma to magnetic pressure. On average, the proton temperature is nearly isotropic downstream of the quasi-parallel bow shock ($\theta_{Bn} < 45^\circ$) and it is clearly anisotropic with $T_{p\perp}/T_{p\|} \approx 1.5$ downstream of the quasi-perpendicular bow shock ($\theta_{Bn} > 45^\circ$). In the foreshock upstream of the quasi-parallel bow shock, the power of magnetic fluctuations is roughly 1 order of magnitude larger ($\delta B \sim 4$nT for frequencies 0.01–0.04Hz) than upstream of the quasi-perpendicular bow shock. There is no significant difference in the magnetic power spectra upstream and downstream of the quasi-parallel bow shock, only at the bow shock itself magnetic power is enhanced by a factor of 4. This enhancement may be due to an amplification of convecting upstream waves or due to wave generation at the shock interface. On the contrary, downstream of the quasi-perpendicular bow shock the magnetic wave activity is considerably higher than upstream. Downstream of the quasi-perpendicular low-$\beta$ bow shock we find a dominance of the left-hand polarized component at frequencies just below the ion cyclotron frequency with amplitudes of about 3nT. These waves are identified as ion cyclotron waves which grow in a low-$\beta$ regime due to the proton temperature anisotropy. We find a strong correlation of this anisotropy with the intensity of the left-hand polarized component. Downstream of some nearly perpendicular ($\theta_{Bn} \approx 90^\circ$) high-$\beta$ crossings mirror waves are identified. However, there are also cases where the conditions for mirror modes are met downstream of the nearly perpendicular shock, but no mirror waves are observed.' author: - 'A. Czaykowska, T. M. Bauer, R. A. Treumann, and W. Baumjohann' title: 'Average observed properties of the Earth’s quasi-perpendicular and quasi-parallel bow shock' --- Introduction ============ Since the beginning of the space age, the Earth’s bow shock is of particular interest because it serves as a unique laboratory for the study of shock waves in collisionless plasmas. Most of our understanding of structure, dynamics, and dissipation processes of such shocks has come from in situ spacecraft measurements crossing the bow shock. Early observations of waves and particles upstream of the bow shock can be found in the special issue of the [*Journal of Geophysical Research*]{}, [*86*]{}, pp. 4317–4536 \[1981\]. A collection of observational and theoretical work on the bow shock is contained in [*Stone & Tsurutani*]{} \[1985\] and [*Tsurutani & Stone*]{} \[1985\]. Reviews focusing on the dissipation processes taking place at the Earth’s bow shock have been given by, e.g., [ *Kennel et al*]{}. \[1985\] and, more recently, by [*Omidi*]{} \[1995\]. Plasma wave observations across the bow shock in the high frequency range have been reviewed by [*Gurnett*]{} \[1985\]. [*Schwartz et al*]{}. \[1996\] have reviewed results concerning low frequency waves in the magnetosheath region behind the bow shock. Many of the previous measurements have demonstrated that a large variety of nonthermal particles is generated at the bow shock. While nonthermal electrons can act as a source for high frequency waves, nonthermal ions can be responsible for low frequency waves. Electrons and ions reflected at the shock stream sunward along the interplanetary magnetic field, thus forming the electron and ion foreshock, respectively. Structure, dynamics, and dissipation processes of the bow shock vary considerably depending on the angle $\theta_{Bn}$ between the upstream magnetic field and the shock normal, on the plasma $\beta$, i.e., the ratio of plasma to magnetic pressure in the upstream region, and on the Mach numbers $M_A$ or $M_{ms}$, i.e., the ratios of the solar wind velocity along the shock normal to the upstream Alfvén or magnetosonic speed. For quasi-perpendicular shocks with $\theta_{Bn} > 45^\circ$, the main transition from the solar wind to the magnetosheath is accomplished at a sharp ramp. In contrast, quasi-parallel shocks with $\theta_{Bn} < 45^\circ$ consist of large-amplitude pulsations extending into the foreshock region. For larger Mach numbers this pulsating structure continuously re-forms by virtue of collisions between convecting upstream waves and the shock \[[*Burgess*]{}, 1989\] or due to an instability at the interface between solar wind and heated downstream plasma \[[*Winske et al*]{}., 1990\]. Two-fluid theories of shocks have indicated the presence of a critical magnetosonic Mach number, $M^*$, above which ion reflection is required to provide the necessary dissipation. However, it has been demonstrated by observations \[[*Greenstadt & Mellott*]{}, 1987; [ *Sckopke et al*]{}., 1990\] that ion reflection occurs also below $M^*$ and that the distinction between subcritical ($M_{ms}<M^*$) and supercritical ($M_{ms}>M^*$) shocks is not sharp. Whereas the ramp of quasi-perpendicular shocks at high Mach numbers is preceded by a foot and followed by an overshoot, these features are less prominent at low Mach numbers \[[*Mellott & Livesey*]{}, 1987\]. While quasi-parallel shocks are steady at low Alfvén Mach numbers, $M_A\le2.3$, they become unsteady for higher Mach numbers, where they continuously re-form \[[*Krauss-Varban & Omidi*]{}, 1991\]. An important role in the dissipation process is played by ions reflected at the bow shock. At quasi-parallel shocks they can escape from the shock into the foreshock region and drive ion beam instabilities. These instabilities may excite large-amplitude waves observed in the foreshock region, e.g., by [*Le & Russell*]{} \[1992\] and [*Blanco-Cano & Schwartz*]{} \[1995\]. At quasi-perpendicular shocks the reflected ions gyrate back to the shock and enter the downstream region, where their presence leads to a strong perpendicular temperature anisotropy \[[*Sckopke et al*]{}., 1983\]. This anisotropy leads to the generation of ion cyclotron and mirror waves \[e.g., [*Price et al.*]{}, 1986; [*Gary et al.*]{}, 1993\]. These waves have been observed in the Earth’s magnetosheath, e.g., by [*Sckopke et al.*]{} \[1990\] and [*Anderson et al.*]{} \[1993,1994\]. Closer to the magnetopause the mechanism of field line draping leads to the formation of anisotropic ion distributions and the formation of a plasma depletion layer. Waves in this environment have also been described by [*Anderson et al.*]{} \[1993,1994\]. Large-amplitude mirror waves have been observed in planetary magnetospheres, e.g., by [ *Bavassano-Cattaneo et al.*]{} \[1998\] in Saturn’s magnetosphere, where the ion temperature anisotropies are due to both shock heating and field line draping. In the present study we investigate the average behavior of plasma and magnetic field parameters including the low frequency magnetic wave power as measured by AMPTE/IRM during a fairly large number of bow shock crossings. We show that, as expected, quasi-perpendicular and quasi-parallel bow shocks behave differently even in their average properties. This difference has been quantified in our investigation. Section 2 provides a short description of the available data. It is followed by Section 3 which compares the properties of quasi-perpendicular and quasi-parallel shocks. In Section 4 low and high-$\beta$ bow shock crossings are compared for the quasi-perpendicular shock, and it is outlined why a classification by $\beta$ is preferred to a classification by Mach number. Finally, Section 5 presents concluding remarks. Data description ================ The present analysis uses data from the AMPTE/IRM satellite. From the periods when the apogee of AMPTE/IRM was on the Earth’s dayside (August – December 1984 and August 1985 – January 1986), we have selected all crossings of the satellite through the Earth’s bow shock in the local time interval 0700 – 1700, whenever there was a reasonable amount of data measured on both sides of the bow shock, i.e., at least 2min upstream and 4min downstream. Altogether this gives 132 events, with some events belonging to multiple crossings due to the fast movement of the bow shock relative to the slowly moving satellite. Due to the satellite’s orbital parameters, all crossings occurred at low latitudes, i.e., in the interval $\pm 30^\circ$ from the ecliptic plane. We analyze the data from the triaxial fluxgate magnetometer described by [*Lühr et al.*]{} \[1985\] which gives the magnetic field vector at a rate of 32 samples per second. In addition, we use the plasma moments calculated from the three-dimensional particle distribution functions measured once every spacecraft revolution ($\sim 4.3$ s) by the plasma instrument \[[*Paschmann et al.*]{}, 1985\]. In Fig. \[positions\] we show the locations of the individual bow shock crossings rotated into the ecliptic along meridians. Cases where the angle $\theta_{Bn}$ is less than $45^\circ$, i.e., quasi-parallel events, and cases where the angle $\theta_{Bn}$ is greater than $45^\circ$, i.e., quasi-perpendicular events, are shown in addition to the best fit hyperbola of [*Fairfield*]{} \[1971\] using data from the Imp 1 to 4 and Explorer 33 and 35 spacecraft. It is found that most of the AMPTE/IRM bow shock crossings occurred closer to the Earth than Fairfield’s average bow shock. Since we analyze only bow shock crossings on the dayside, the best fit hyperbola derived from our data is not reliable at the flanks. However, the distance of the subsolar point is well defined. We find a value of 12.3 $R_E$, which is more than 2 $R_E$ closer to the Earth than the value of 14.6 $R_E$ found by Fairfield. In his study, [*Formisano*]{} \[1979\] analyzed 1500 bow shock crossings. He normalized the observed distance $R_{\rm obs}$ of these crossings to an average value of the solar wind dynamical pressure using $$\label{form} R_{\rm norm}=R_{\rm obs}\left( \frac{n_{\rm obs}v_{\rm obs}^2}{n_0v_0^2}\right)^{1/6}$$ with a typical value of the solar wind speed $v_0=450$ km/s and particle density $n_0=9.4$ cm$^{-3}$. He derived a distance of the subsolar point of 11.9 $R_E$. Applying the same normalization to the AMPTE/IRM data, we find a value of 11.7 $R_E$, which is in good agreement with the result of [*Formisano*]{} \[1979\]. This indicates that the difference of the distance of the subsolar point between Fairfield’s and our study is due to different average solar wind dynamical pressure. We interpret this finding as a solar cycle effect since the AMPTE/IRM data are obtained close to solar activity minimum, whereas Fairfield’s data set is from the years 1964-1968, when solar activity increased from minimum to maximum. In solar minimum the Earth is hit more frequently by high speed solar wind streams than in solar maximum. The high speed solar wind has, although less dense, a higher dynamical pressure than the slow solar wind. Hence, the solar wind dynamical pressure is usually higher on average during solar minimum than during solar maximum \[[*Fairfield*]{}, 1979\]. In addition, during solar minimum, the heliospheric plasma sheet described by [*Winterhalter et al.*]{} \[1994\] is fairly flat, i.e., near the ecliptic plane. With its very high densities it can enhance the solar wind pressure although the solar wind velocity is only around 350 km/s. In a more recent study, [*Peredo et al.*]{} \[1995\] investigated 1392 bow shock crossings from 17 spacecraft during the years 1963–1979, i.e., one and a half solar cycles. They found a dependence on the Alfvén Mach number $M_A$. With the average Alfvén Mach number $M_A = 5.6 \pm 2.9$ of the AMPTE/IRM data set the distance of the subsolar point should be in the range of 14.0-14.9 $R_E$. Performing a normalization with the average values of $n_0=7.8$ cm$^{-3}$ and $v_0=454$ km/s used by [*Peredo et al.*]{} \[1995\], the distance of the subsolar point of the AMPTE/IRM data set is 12.1 $R_E$. The results of [*Peredo et al.*]{} \[1995\] are thus not in agreement with our results and those of [*Formisano*]{} \[1979\]. [*Peredo et al.*]{} \[1995\] explain this disagreement with the fact that the study of [ *Formisano*]{} \[1979\] is biased by the dominance of the high latitude HEOS 2 bow shock crossing. However, our data are low-latitude and agree well with the results of [*Formisano*]{} \[1979\]. In principle, the bow shock position depends on the magnetopause position and on the standoff distance between the magnetopause and the bow shock. Whereas the magnetopause position depends only on the solar wind dynamical pressure, the standoff distance at a given Mach number depends on the polytropic index $\gamma$ ([*Spreiter et al.,*]{} 1966). Since our data were sampled at typical solar minimum conditions, the polytropic index might be different than in other phases of the solar cycle. This might contribute to the discrepancy. Comparison of quasi-perpendicular and quasi-parallel bow shock crossings ======================================================================== We divided the 132 events into 92 quasi-perpendicular ($\theta_{Bn} > 45^\circ$) and 40 quasi-parallel ($\theta_{Bn} <45^\circ$) cases and compared the average behavior of plasma and magnetic field parameters and low frequency magnetic fluctuations of the two groups. Actually, for the quasi-parallel bow shock crossings $\theta_{Bn}$ varies substantially with time in the dynamic foreshock region. For these events the angle $\theta_{Bn}$ had to be averaged over a time interval of about 20 s further upstream to identify them with quasi-parallel shock crossings. The high level of fluctuations in the region upstream of the quasi-parallel bow shock is well known \[e.g., [*Hoppe et al.*]{}, 1981, [*Greenstadt et al.,*]{} 1995\]. In order to obtain the average behavior of plasma and magnetic field parameter at the bow shock, one would ideally need average spatial profiles of these parameters. However, with just one satellite and in a region with strong plasma flows and strong motions of the region itself, it is not unambiguously possible to translate the time profiles into spatial profiles. Therefore we perform a superposed epoch analysis by averaging time profiles centered on the bow shock crossing time and consider the result as an approximation for the average spatial behavior. The time series are aligned on the keytime with the upstream always preceding, i.e., for outbound crossings the time sequence had to be reversed. The keytime, i.e., the bow shock crossing time, is identified with the steepest drop in the proton velocity. This drop is well defined for the quasi-perpendicular cases and corresponds, of course, to the shock ramp. Due to the large-amplitude pulsations in the foreshock, the keytime cannot as easily be found in the quasi-parallel cases. We therefore applied, as an additional criterion for the quasi-parallel events, that no solar wind-like plasma is allowed to be visible in the downstream region. As noted in Section 1, quasi-parallel shocks consist of large-amplitude pulsations associated with a sequence of partial transitions from solar wind-like to magnetosheath-like plasma and vice versa. Thus our definition of the keytime implies that the keytime of quasi-parallel shocks corresponds to the downstream end of this pulsating transition region. We use data from 2min upstream to 4min downstream for the analysis of the plasma and magnetic field parameters and data from 3 min upstream to 9min downstream for the low frequency fluctuations, although not for all events such long time profiles are available. One has to be aware that superposed epoch analysis can mask small scale structures, in particular if they are not visible in all events, like magnetic foot and overshoot structures. These and other features can be smeared out due to different bow shock velocities with respect to the satellite for different crossings. This effect becomes worse the farther away from the keytime the data are averaged. Nevertheless, superposed epoch analysis is useful to reveal the average plasma and magnetic field parameters and the typical features in the vicinity of the key-structure as has been shown by, e.g., [*Paschmann et al.*]{} \[1993\], [*Phan et al.*]{} \[1994\], and [*Bauer et al.*]{} \[1997\] in there studies at the magnetopause. Plasma and magnetic field parameters ------------------------------------ The time series adjusted in the way described above are superimposed by averaging 10-s bins. The averages are performed geometrically in order to reduce the dominance of cases with large dynamical ranges. The result of the superposition is shown in Fig. \[ppsp1\] and Fig. \[ppsp2\]. The averages of the downstream and upstream values, excluding the 4 bins closest to the bow shock on both sides, are given in Table \[mittelwerte\], together with the ratios of the downstream to the upstream values. ------------------------------------------------------------------------------------------------------------------ Upstream Downstream Ratio ------------------------------------------------ ----------- ----------------- ----------------- ----------------- q-$\perp$ $9.4\pm 0.1$ $28.5\pm 0.8$ $3.0\pm 0.1$ \[-2.0ex\][$B$ \[nT\]]{} q-$\|$ $10.9\pm 0.8$ $19.6\pm 1.1$ $1.8\pm 0.2$ q-$\perp$ $436\pm 6$ $198\pm 7$ $0.45\pm 0.03$ \[-2.0ex\][$v_p$ \[km/s\]]{} q-$\|$ $332\pm 16$ $171\pm 17$ $0.52\pm 0.08$ q-$\perp$ $-401\pm 9$ $-122\pm 7$ $0.30\pm 0.02$ \[-2.0ex\][$v_n$ \[km/s\]]{} q-$\|$ $-283\pm 18$ $-96\pm 16$ $0.34\pm 0.08$ q-$\perp$ $0.6\pm 0.1$ $2.3\pm 0.5$ $4.0\pm 1.4$ \[-2.0ex\][$\delta B_2$ \[nT\]]{} q-$\|$ $1.9\pm 0.3$ $3.2\pm 0.5$ $1.7\pm0.5$ q-$\perp$ $7.4\pm 0.1$ $20.1\pm $2.7\pm 0.1$ 0.4$ \[-2.0ex\][$N_e^*$ \[cm$^{-3}$\]]{} q-$\|$ $7.8\pm 0.3$ $14.4\pm 1.1$ $1.8\pm 0.2$ q-$\perp$ $0.15\pm 0.01$ $0.38 \pm 0.03$ $ 2.6\pm 0.3$ \[-2.0ex\][$N_{2p}$ \[$10^{-2}$ cm$^{-3}$\]]{} q-$\|$ $4.0\pm 2.6$\pm 0.5$ $0.7\pm 0.2$ 0.5$ q-$\perp$ $27.5\pm 0.4$ $71.3\pm $2.6\pm 0.1$ 0.7$ \[-2.0ex\][$T_e$ \[$10^{4}$ K\]]{} q-$\|$ $43\pm 1$ $73\pm 3$ $1.7\pm0.1$ q-$\perp$ $1.51\pm 0.06$ \[-2.0ex\][$T_{p\perp}/T_{p\|}$]{} q-$\|$ \[-2.0ex\][—]{} $1.05\pm 0.03$ \[-2.0ex\][—]{} ------------------------------------------------------------------------------------------------------------------ The top panel of Fig. \[ppsp1\] shows the magnitude $B$ of the magnetic field. It increases steeply by a factor of 3 at the quasi-perpendicular bow shock and gradually by a factor of 1.8 at the quasi-parallel bow shock. The proton bulk velocity $v_p$, shown in the next panel, decreases to somewhat less than half of its solar wind value for the quasi-perpendicular bow shock and to slightly more than half for the quasi-parallel bow shock. The third panel shows the proton velocity parallel to the shock normal vector. The latter is calculated from the Fairfield bow shock model \[[*Fairfield*]{}, 1971\]. For both categories $v_n$ decreases by more than the magnitude of the proton velocity $v_p$, indicating that the plasma is deflected away from the bow shock normal direction in order to flow around the magnetopause. In the last panel we show the root mean square amplitude $\delta B_2$ of the high resolution magnetic field measurements during one spin period: $$\label{edeltab} \delta B_2=\left[\sum_{k=1}^3 \frac{1}{n}\sum_{i=1}^n(B_{k,i}-\bar{B}_k)^2 \right]^{1/2}$$ where $i=1,..., n$ counts the measurements during one spin period, and $B_k$, $k=1,2,3$ denote the magnetic field vector components. $\bar{B}_k$ is the average of $B_k$ taken during one spin period. At the keytime, $\delta B_2$ is approximately half the jump of the magnitude of the magnetic field. Further upstream and downstream it is a measure of wave activity with Doppler-shifted periods shorter than the spin period of about 4.3 s. Although the magnetic field increases only gradually at the quasi-parallel bow shock, $\delta B_2$ is strongly enhanced at the bow shock crossing time and has a maximum immediately downstream of the shock. This jump in $\delta B_2$, as a parameter measured independently of the velocity, is a confirmation that the selection of the crossing times with the help of the velocity jump (Section 2) is reasonable. In the foreshock region upstream of the quasi-parallel bow shock, $\delta B_2$ is considerably higher than in the solar wind regime upstream of the quasi-perpendicular bow shock. The first panel of Fig. \[ppsp2\] shows the electron density $N_e^*$, approximately corrected for the low-energy cut-off of the plasma instrument \[[*Sckopke et al.*]{}, 1990\]. Like the magnetic field, the density rises sharply at the quasi-perpendicular bow shock, whereas it increases gradually at the quasi-parallel bow shock. The second panel shows $N_{2p}$, the proton density in the energy interval 8-40 keV. $N_{2p}$ increases by a factor of 2.6 at the quasi-perpendicular bow shock. In the foreshock region upstream of the quasi-parallel bow shock $N_{2p}$ is an order of magnitude higher than upstream of the quasi-perpendicular bow shock and decreases by a factor of about 0.7 in the downstream region. In the next panel the electron temperature $T_e$ is shown. Downstream of the quasi-parallel bow shock crossing it has about the same value as downstream of the quasi-perpendicular bow shock. However, upstream of the quasi-parallel bow shock $T_e$ is about a factor of 1.6 higher than upstream of the quasi-perpendicular bow shock. The reason for this is again the foreshock region where the solar wind kinetic energy is already partly transformed into thermal energy. The last panel shows the proton temperature anisotropy $T_{p\perp}/T_{p\|}$. The plasma instrument did not resolve the cold, supersonic distributions of the solar wind ions. The calculated proton densities and temperatures are therefore not reliable in the solar wind regime. Hence, the proton temperature anisotropy cannot be determined in the upstream region and is therefore set to 1. Whereas the proton temperature anisotropy downstream of the quasi-parallel bow shock is insignificant, there is a strong proton temperature anisotropy, $T_{p\perp}/T_{p\|} > 1.4$, downstream of the quasi-perpendicular bow shock, with a maximum value of more than 2 immediately behind the shock. Comparing the downstream values of the electron and proton temperatures (not shown), we find $T_p \approx 4.3 \times 10^6\,{\rm K} \approx 6\,T_e$ for the quasi-parallel bow shock and $T_p \approx 2.9 \times 10^6\,{\rm K} \approx 4\,T_e$ for the quasi-perpendicular bow shock. Whereas the electron temperature is slightly anisotropic, $T_{e\perp}/T_{e\|}\approx 0.9$, upstream and downstream of quasi-perpendicular shocks, no significant anisotropy is observed at quasi-parallel shocks. Low frequency magnetic fluctuations ----------------------------------- In order to analyze the low frequency magnetic fluctuations we perform a spectral analysis of the magnetic field using a cosine-bell filter \[see, e.g., [*Bauer et al.*]{}, 1995\]. The Fourier transform is taken over a time interval of 4 min. Fig. \[flucsp1\] and Fig. \[flucsp2\] show the resulting power spectra of the compressive and the right- and left-hand polarized modes, respectively. In each graph the center time (-3,-1,1,3,5,7) of the transformed time interval is given in minutes relative to crossing time. A cross marks the proton cyclotron frequency $f_{cp}=eB/2\pi m_p$ with $m_p$ the proton mass. The solar wind (SW) spectrum of Fig. \[flucsp1\], 3min upstream of the quasi-perpendicular bow shock, shows a structureless decrease to higher frequencies following the power law $S \sim f^{-1.3}$. The compressive mode lies below the transverse modes that represent Alfvén waves frequently observed in the interplanetary medium. They are thought to have their origin in the vicinity of the Sun \[[*Belcher & Davis*]{}, 1971\]. The spectrum upstream of the quasi-parallel bow shock has much more power than that upstream of the quasi-perpendicular bow shock. It also has a different structure: For lower frequencies it shows a flatter decrease ($S \sim f^{-0.5}$), while for higher frequencies it decreases more steeply ($S \sim f^{-2.0}$). The kink in the spectrum lies below the proton cyclotron frequency. The next power spectra (-1, SW/BS) contain magnetic field data from the upstream region and the bow shock itself. At the quasi-perpendicular bow shock the compressive mode at low frequencies is one order of magnitude higher compared to the transverse modes and follows a power law of $S \sim f^{-2.1}$. This represents simply the spectrum of the jump of the magnetic field across the shock filtered with a cosine bell function. The spectra of the transverse modes are a little higher than 2min earlier and the decrease with frequency is not constant any more. At the quasi-parallel bow shock the increase of the magnetic field is not visible, which is not surprising since the magnetic field increases only gradually. Level and structure of the spectrum are similar to those calculated 2 min earlier. The spectra (1, SW/BS) contain magnetic field data from again the bow shock itself and from the magnetosheath just downstream of the bow shock. At the quasi-perpendicular bow shock the compressive mode behaves similar to the spectrum 2min earlier, whereas the spectral power of the transverse modes is higher than 2min earlier. Just below the proton cyclotron frequency first indications of a plateau are visible. At the quasi-parallel bow shock wave activity is significantly enhanced by a factor of 4 compared to the upstream spectra. Again all three modes behave similar. The proton cyclotron frequencies increase according to the magnetic field increase by a factor of 3 at the quasi-perpendicular bow shock and by a factor of 2 at the quasi-parallel bow shock. Figure \[flucsp2\] shows that for both categories of shock crossings the spectra do not change much in the interval 2 to 8min downstream of the bow shock. Below $f_{cp}$ the compressive and the right-hand polarized modes downstream of the quasi-perpendicular bow shock follow a power law $S \sim f^{-1.1}$, whereas the spectral energy of the left-hand polarized mode is clearly enhanced in the frequency interval from about 0.1Hz to about the proton cyclotron frequency. This fact is investigated more carefully in Section 4. Downstream of the quasi-parallel bow shock the spectral energy is again higher than downstream of the quasi-perpendicular bow shock. For $f<f_{cp}$ it follows the power law $S\sim f^{-0.8}$. 3min after the crossing the spectral energy is higher than in the 2 later spectra but already lower than directly at the bow shock. For both categories the spectral energy decreases steeply ($S\sim f^{-2.6}$) for $f>f_{cp}$. Discussion ---------- The processes in the bow shock transition region itself cannot be described in terms of magnetohydrodynamics (MHD). However, if one considers the bow shock as an infinitesimally thin discontinuity one can derive from the MHD equations the Rankine-Hugoniot jump conditions (see, e.g., [*Siscoe*]{}, 1983), which are relations for the conditions in the plasma upstream and downstream of the discontinuity under the assumption of time independence. For mass continuity the jump condition is $$\label{con} n_2 v_{n2} - n_1 v_{n1} = 0$$ where the subscripts 1 and 2 denote the upstream and downstream values of the corresponding parameters, respectively. According to Eq. (\[con\]) the product of the downstream-to-upstream ratio of the electron density $N_e$ and the normal proton velocity $v_n$, which is equivalent to the plasma bulk speed normal to the discontinuity, must be unity. For the quasi-parallel events this product is observed to be about 0.6. This fact is not surprising, since the considered upstream time profiles are not taken from the quiet solar wind regime but from the dynamic foreshock region, which is highly time dependent. For the quasi-perpendicular events this product is observed to be about 0.8, which means that the jump condition Eq. (\[con\]) is not fully satisfied. This could be explained by the fact that we have not measured the exact electron density, but only the density of electrons with energies between 15 eV and 30 keV. Electrons with higher energies are negligible since already in the energy range of 1.8 - 30 keV the upstream density is of the order of $10^{-5}$ cm$^{-3}$. However, particularly in the cold solar wind, electrons with energies below the instrument cut-off contribute an essential part to the total electron density. Therefore we use the corrected electron density $N_e^*$. The correction is calculated with the assumption of a Maxwellian distribution of the measured temperature. However, as has been measured, e.g., recently by the Wind spacecraft \[Fig. 4 of [*Lin*]{}, 1997\], the quiet solar wind flow cannot be described by a single Maxwellian distribution over its whole energy range. Therefore the electron density could easily be overestimated by the correction, especially in the cold solar wind regime where the low energy electrons are more important than in the warmer magnetosheath regime. In order to satisfy the jump condition Eq. (\[con\]) we can estimate, that the value of the corrected electron density upstream of the quasi-perpendicular bow shock is about 20% too high if we assume that the error in the downstream value is not of importance. Other factors for the apparent deviation from mass continuity could be the unknown bow shock motion, which changes the downstream plasma velocity by a higher percentage than the upstream plasma velocity, and the uncertainty of the shock normal vector. For an exactly parallel shock wave one derives from the Rankine-Hugoniot jump condition that the magnetic field remains constant in magnitude and direction and one is left with the equations for a purely hydrodynamic shock wave. In the case of an exactly perpendicular shock wave the jump conditions give the relation $$\label{esenk} \frac{n_2}{n_1}=\frac{B_2}{B_1}=\frac{v_1}{v_2}$$ For the limit of high Mach numbers, i.e., the Alfvén Mach number $M_A \rightarrow \infty$ and the sonic Mach number $M_{s} \rightarrow \infty$, Eq. (\[esenk\]) has a value of 4 for $\gamma=5/3$. Since we do not observe the extreme cases of exactly perpendicular and exactly parallel geometry but quasi-perpendicular and quasi-parallel shock wave crossings with finite Mach numbers the numbers given above are not reached. For the quasi-perpendicular bow shock the ratio Eq. (\[esenk\]) is about 3 for the average magnetic field $B$ and proton normal velocity $v_n$, and somewhat less for the corrected electron density $N_e^*$ according to the above mentioned probable overestimation of this parameter in the solar wind regime. For the quasi-parallel bow shock the change in the magnitude of the magnetic field is significantly lower than for the quasi-perpendicular bow shock, which is consistent with theory. In their study of subcritical quasi-perpendicular shocks, [*Thomsen et al.*]{}, \[1985\] found that the downstream electron temperature is nearly isotropic, $T_{e\perp 2}/T_{e\| 2}\approx 0.9$ , and that the downstream-to-upstream ratio $T_{e\perp 2}/T_{e\perp 1}$ of the perpendicular temperature is approximately equal to the ratio $B_2/B_1$ of the magnetic field strength. From the latter result they concluded that the net heating is adiabatic, although $T_{e\perp}/B$ is not constant. Obtaining averages, $T_{e\perp 2}/T_{e\| 2}\approx 0.9$, $T_{e\perp 2}/T_{e\perp 1}\approx 2.6$, and $B_2/B_1\approx 3.0$, we can confirm these results for our data set of (subcritical and supercritical) quasi-perpendicular shocks. Moreover, isotropy of the downstream electron temperature and adiabatic net heating is also found for our data set of quasi-parallel shocks, for which we obtain averages $T_{e\perp 2}/T_{e\| 2}\approx 1.0$, $T_{e\perp 2}/T_{e\perp 1}\approx 1.7$, and $B_2/B_1\approx 1.8$. An interesting question of shock physics is how the dissipated bulk flow energy of the solar wind is partitioned amongst ion and electron heating. For the average proton-to-electron temperature ratio we found $T_p/T_e\approx 6$ downstream of the quasi-parallel bow shock and $T_p/T_e\approx 4$ downstream of the quasi-perpendicular bow shock. Hence, for quasi-parallel shocks proton heating is more favored with respect to electron heating than for quasi-perpendicular shocks. Let us turn to the low frequency waves of Fig. \[flucsp1\] and Fig. \[flucsp2\]. At quasi-parallel bow shocks the wave power observed 3min upstream of the keytime is much higher than the power of the interplanetary Alfvén waves observed upstream of quasi-perpendicular shocks. This enhanced power reflects upstream waves generated in the foreshock. Observations of upstream waves have recently been reviewed by [*Greenstadt et al.*]{} \[1995\] and [*Russell & Farris*]{} \[1995\]. The nonlinear steepening of the shock leads to whistler precursors phase standing in the shock frame. The interaction between ions reflected at the shock and the incoming solar wind can drive ion beam instabilities. These are probably the source of large-amplitude waves observed at periods around 30s. Finally, there are upstream propagating whistlers with frequencies around 1Hz, which seem to be generated directly at the shock. The most striking feature in the average spectrum of upstream waves observed 3min before the keytime is the kink at $0.04\,{\rm Hz}\approx f_{cp}/3$. The average power measured in the flat portion 0.01–0.04Hz of the spectrum corresponds to a mean square amplitude $\delta B_1\approx 4\,nT$ or $\delta B_1/B\approx 0.4$. Large-amplitude waves observed in this frequency range \[[*Le & Russell*]{}, 1992; [*Blanco-Cano & Schwartz*]{}, 1995\] have been interpreted as upstream propagating magnetosonic waves excited by the right-hand resonant ion beam instability, upstream propagating Alfvén/ion cyclotron waves excited by the left-hand resonant ion beam instability, or downstream propagating magnetosonic waves excited by the non-resonant instability. Whereas the upstream propagating magnetosonic waves should be left-hand polarized in the shock frame (and also in the spacecraft frame), the other two wave types should be right-hand polarized. This might explain why none of the two circular polarizations dominates in our average spectra. Moreover, the compressional component is comparable to the two transverse components. This shows that the waves propagate at oblique angles to the magnetic field. For oblique propagation, low frequency waves have only a small helicity \[[*Gary*]{}, 1986\]. Thus they are rather linearly than circularly polarized. The power spectra presented by [*Le & Russell*]{} \[1992\] exhibit clear peaks at $f\approx f_{cp}/3$. Looking into the spectra of individual time intervals, we find that sometimes the IRM data exhibit similar spectral peaks. However, most of the individual spectra do not have clear peaks, but are rather flat in the range 0.01–0.04Hz like the average spectrum of Fig. \[flucsp1\]. The steep decrease of the power above about $f\approx f_{cp}/3$ is common to our spectra and those reported previously. In fact, the maximum growth rate of the ion beam generated waves is expected for frequencies below the proton cyclotron frequency \[e.g., [*Scholer et al.*]{}, 1997\]. In Fig. \[ppsp1\] we saw that magnetic fluctuations above 0.23Hz in the foreshock have root mean square amplitudes $\delta B_2\approx 1.5$nT. This is comparable to typical amplitudes of the upstream propagating whistlers. In individual time intervals these narrow-band waves lead to clear spectral peaks at frequencies around 1Hz. However, since the frequency varies from event to event, no such peak appears in the average spectra. At the keytime of quasi-parallel bow shocks we observed a clear enhancement of the wave power. This enhancement can either be due to an amplification of the upstream waves or due to wave generation at the shock interface. Wave generation at the shock due to the interface instability has been found in hybrid simulations of [*Winske et al.*]{} \[1990\] and [*Scholer et al.*]{} \[1997\]. This instability is driven by the interaction between the incoming solar wind ions and the heated downstream plasma at the shock interface. Amplification of upstream waves has been predicted by [*McKenzie & Westphal*]{} \[1969\] who analyzed the transmission of MHD waves across a fast shock. They found that the amplitude of Alfvén waves increases by a factor of 3. For compressional waves the amplification can even be stronger. However, the hybrid simulations of [*Krauss-Varban*]{} \[1995\] show that the transmission of waves across the shock is complicated by mode conversion. The proton temperature anisotropy downstream of the quasi-perpendicular bow shock serves as a source of free energy. According to both observations and simulations this kind of free energy drives two modes of low frequency waves under the plasma conditions in the magnetosheath: the ion cyclotron wave and the mirror mode (see e.g., [*Sckopke et al.*]{}, 1990, [*Hubert et al.*]{}, 1989 and [*Anderson et al.*]{}, 1994 for observations, [*Price et al.*]{}, 1986, and [*Gary et al.*]{}, 1993 for simulations, and [*Schwartz et al.*]{} 1996 for a review). Which of these waves grow under which conditions is investigated in Section 4, where we divide the crossings of quasi-perpendicular shocks into cases with low and high upstream $\beta$, respectively. It turned out that for this classification by $\beta$ the differences become somewhat clearer than for a classification by upstream Mach number. The critical Mach number $M^*$ above which ion reflection is required to provide the necessary dissipation is strongly dependent on the plasma-$\beta$ \[[*Edmiston & Kennel*]{}, 1984\]. We have calculated the ratio $M_{ms} / M^*$ for our shock crossings and have found that all subcritical shocks are low-$\beta$, i.e., that the classification low-$\beta$ versus high-$\beta$ is more or less identical to the classification subcritical versus supercritical. The reason for this is that $M^* \rightarrow 1$ for $\beta \gg 1$. As the excitation of mirror and ion cyclotron waves depends on $\beta$, the results of Section 4 should be interpreted as the effect of the plasma-$\beta$ and not as an effect of subcritical or supercritical Mach numbers. For the quasi-parallel bow shocks, we could not investigate the difference between subcritical and supercritical shocks, because no subcritical quasi-parallel shock was identified in the data set. Trying higher thresholds for the division into low and high Mach numbers, we did not find any qualitative differences. In this context it should be noted that only one of the cases in our data set has an Alfvén Mach number in the range $M_A\le 2.3$, for which quasi-parallel shocks are steady according to the hybrid simulations of [*Krauss-Varban & Omidi*]{} \[1991\]. Comparison of quasi-perpendicular low-$\beta$ and high-$\beta$ bow shock crossings ================================================================================== In order to reveal the origin of the left-hand polarized component in the power spectra downstream of the quasi-perpendicular bow shock (Fig. \[flucsp2\]) we divide the crossings into classes with low ($< 0.5$) and high ($> 1.0$) upstream $\beta$ and compare these two classes. There are 20 low-$\beta$ and 47 high-$\beta$ cases. The crossings with $0.5 \le \beta \le 1.0$ are not included in this analysis in order to emphasize the differences between the low-$\beta$ and high-$\beta$ regimes. Plasma and magnetic field parameters ------------------------------------ In Fig. \[pplh\] we show some interesting differences between the low-$\beta$ and high-$\beta$ categories. Of course the plasma parameter $\beta$ differs essentially. In the upstream region $\beta$ is derived by setting the proton density to the corrected electron density and the proton temperature to $10^{-5}$ K, the long term average of the proton temperature, since proton distribution functions are not well measured in the cold solar wind (Section 3.1). The classification for the low-$\beta$ and high-$\beta$ categories is derived from the estimated plasma $\beta$ in the upstream region. The first panel of Fig. \[pplh\] shows that the same classification could be obtained using the plasma-$\beta$ of the downstream region with the limits shifted to larger values. The most striking differences between the low-$\beta$ and high-$\beta$ bow shock are shown in the next two panels, i.e., the proton temperature anisotropy $T_{p\perp}/T_{p\|}$ and the mirror wave instability criterion. Both parameters are again determined only in the downstream region. The instability criterion for almost perpendicular propagation of the mirror mode in its general form \[[*Hasegawa*]{}, 1969\] is given by $$\label{mirin} -1 + \sum_j \beta_{j\perp} \left ( \frac{\beta_{j\perp}} {\beta_{j\|}}-1\right ) > 0.$$ The subscript $j$ denotes the particle species ($j$ = $e$, $p$ for electrons and protons, respectively). Downstream of the quasi-perpendicular low-$\beta$ bow shock the proton temperature anisotropy is very high, $T_{p\perp}/T_{p\|}\approx 2.5$, immediately behind the shock and remains high, $T_{p\perp}/T_{p\|} > 2$, throughout the whole magnetosheath interval investigated. Downstream of the quasi-perpendicular high-$\beta$ bow shock the proton temperature anisotropy is also significant, but lower compared to the low-$\beta$ bow shock, i.e., $T_{p\perp}/T_{p\|}\approx 1.8$ just behind the bow shock and $T_{p\perp}/T_{p\|}\approx 1.3$ further downstream. The mirror instability criterion is only marginally satisfied immediately downstream of the quasi-perpendicular low-$\beta$ bow shock and is not satisfied at later times, since the instability criterion does not only depend on the particle temperature anisotropy but also on the absolute value of $\beta$. Downstream of the quasi-perpendicular high-$\beta$ bow shock the mirror instability criterion is satisfied in the entire interval of 4min behind the shock. Extremely high values of the left-hand side of Eq. (\[mirin\]) are occasionally observed immediately behind the shock. Low frequency magnetic fluctuations ----------------------------------- Fig. \[fluclh\] shows the magnetic power spectra downstream of the quasi-perpendicular low-$\beta$ (left) and high-$\beta$ (right) bow shock, 3, 5, and 7min after the crossing time. Now it becomes obvious that the left-hand polarized mode dominates only behind the quasi-perpendicular bow shock with low-$\beta$ in a frequency interval below the proton cyclotron frequency. 5 and 7 min downstream, the left-hand polarized mode has up to one order of magnitude more power spectral density than the compressive and the right-hand polarized components. Below the frequency range where the left-hand polarized component dominates in the low-$\beta$ cases, the spectrum downstream of the quasi-perpendicular high-$\beta$ bow shock shows a weaker gradient than downstream of the low-$\beta$ bow shock. In addition, the compressive mode lies on the same level as the transverse modes and occasionally at somewhat higher levels, whereas 5 and 7min downstream of the low-$\beta$ bow shock the compressive mode lies clearly below the transverse modes.Due to the higher magnetic field the proton cyclotron frequency is a factor of about 2.3 higher downstream of the low-$\beta$ bow shock ($B \approx$ 45 nT) than downstream of the high-$\beta$ bow shock ($B \approx$ 23 nT). At this point, let us collect some numbers for typical wave amplitudes. For that purpose we use the root mean square amplitude $\delta B_2$ given in Eq. (\[edeltab\]), which has already been shown in Fig. \[ppsp1\] and can be obtained by integrating the power spectra for frequencies above 0.23Hz, and we use the root mean square amplitude $\delta B_1$ of fluctuations in the frequency range 0.01–0.04Hz. The range 0.01–0.04Hz has been chosen, because it corresponds to the flat portion of the spectrum observed 3min upstream of the keytime of quasi-parallel bow shocks. These upstream waves have $\delta B_1\approx 4$nT or $\delta B_1/B\approx 0.4$. In contrast, the Alfvén waves seen in the solar wind upstream of quasi-perpendicular bow shocks have $\delta B_1\approx 0.8$nT or $\delta B_1/B\approx 0.09$. At the keytime of quasi-parallel bow shocks we observed a clear enhancement of the wave power, which leads to $\delta B\approx 8$nT or $\delta B/B\approx 0.5$. In this section we saw that the characteristics of the wave activity downstream of quasi-perpendicular shocks depend on $\beta$. For low $\beta$, the magnetic fluctuations above 0.23Hz are dominated by left-hand polarized fluctuations with $\delta B_2\approx 3$nT or $\delta B_2/B\approx 0.08$, which will be interpreted as ion cyclotron waves. For high $\beta$, typical amplitudes are $\delta B_2\approx 1.5$nT or $\delta B_2/B\approx 0.08$. Discussion ---------- [*Sckopke et al.*]{} \[1990\] have performed a case study of low-$\beta$ subcritical bow shock crossings, using the AMPTE/IRM data of September 5, and November 2, 1984. These events are also included in our quasi-perpendicular low-$\beta$ data set: There are 8 events from September 5 and 3 events from November 2, 1984. [*Sckopke et al.*]{} \[1990\] identified the dominating left-hand polarized component with the ion cyclotron wave which can be generated by a proton temperature anisotropy (e.g., [*Hasegawa*]{}, 1975). The growth rate of the ion cyclotron wave is positive when the instability criterion is satisfied. This holds for the resonance with protons when $$\label{eion} \frac{T_{p\perp}}{T_{p\|}}> \frac{f_{cp}}{f_{cp}-f}.$$ The AMPTE/IRM plasma instrument did not resolve ion masses, therefore all ions are assumed to be protons. Figure \[corrl\] shows the ratio of the left-hand polarized to the right-hand polarized component in the frequency band 0.3–$0.8 f_{cp}$ for 32 2-min intervals from 4 to 8min downstream of the quasi-perpendicular low-$\beta$ bow shock as a function of the proton temperature anisotropy. There is a clear correlation found between these two ratios with a correlation coefficient of 0.8. This shows that the wave intensity more than 4min downstream of the quasi-perpendicular bow shock depends strongly on the local temperature anisotropy. The temporal evolution of the correlation coefficients, calculated for 2-min intervals downstream of the quasi-perpendicular low-$\beta$ bow shock, is shown in Fig. \[corrver\]. The value of the correlation coefficient 11 minutes downstream is not reliable since only a limited data set extends so far downstream. Although the temperature anisotropy is highest immediately downstream of the bow shock, the best correlation is found around 5min downstream. This shows that the ion cyclotron waves need a certain time to develop in the moving plasma. Since downstream of the quasi-perpendicular high-$\beta$ bow shock the mirror instability criterion is satisfied on average, we have looked more carefully for this highly compressive non-propagating mode. The fact that the compressive mode has a higher power spectral density downstream of the high-$\beta$ than downstream of the low-$\beta$ shock might indicate the existence of mirror modes. We therefore perform a superposed epoch analysis for 9 cases for which the mirror instability criterion is particularly well satisfied, 7min downstream of the high-$\beta$ bow shock. The resulting spectrum is shown in Fig. \[mmir\]. In this spectrum we find a compressive mode slightly dominating at some frequencies well below the proton cyclotron frequency. Contrary to the correlation of the intensity of the left-hand polarized ion cyclotron waves to the proton temperature anisotropy for quasi-perpendicular low-$\beta$ cases, our investigation does not reveal a clear correlation of the intensity of the compressive mode to any parameter for the quasi-perpendicular high-$\beta$ cases. One reason for this could be that the highly compressive mirror mode, which is expected to exist under the observed high-$\beta$ conditions is a purely growing mode with frequencies being pure Doppler-shifted frequencies. Consequently, the waves do not appear in a fixed frequency interval and can be smeared out in the superposition. Therefore we have looked into the individual spectra of intervals 5, 7, and 9min downstream of the quasi-perpendicular high-$\beta$ bow shock when the mirror criterion is fulfilled. This is the case in 34 of the quasi-perpendicular high-$\beta$ events (72 %). Only in 4 cases (12 % of the quasi-perpendicular high-$\beta$ events where the mirror criterion is fulfilled) mirror waves can clearly be identified in the magnetosheath and are visible in several consecutive spectra. These events and the means of identification of the mirror modes are described in [*Czaykowska et al.*]{} \[1998\]. The 4 events have in common that the angle $\theta_{Bn}$ is larger than $80^\circ$. But there are also almost perpendicular high-$\beta$ events with large values of the left-hand side of Eq. (\[mirin\]) where no indication for mirror waves is visible. Thus a more complex dependence on different parameters seems to determine the growth of the mirror wave. For 14 of the high-$\beta$ events (41 %) where the mirror instability criterion is fulfilled, the compressive component is at least slightly dominating at several frequencies. Several of these events have an angle $\theta_{Bn} < 80^\circ$. In addition, 2 high-$\beta$ events show ion cyclotron waves in the consecutive spectra taken 5, 7, and 9min downstream. It is well known \[[ *Price et al.*]{}, 1986; [*Gary et al.*]{}, 1993\] that the growths of the ion cyclotron and mirror waves are competing processes. In the magnetosheath plasma the crucial parameters for this competition are the $\alpha$-particle concentration and the plasma-$\beta$. However, in our data set we have found many events in the high-$\beta$ regime where none of the two wave modes can be identified although the proton temperature anisotropy is high. This seems to indicate that the dominance of the growing mode does not persist long enough to be visible in one spectrum. The energy of the growing mode might be transferred to other modes by nonlinear effects. Statistical studies of measurements in the magnetosheath suggest that a relation of the form $$\label{anisobeta} \frac{T_{p\perp}}{T_{p\|}} -1 = \frac{S}{(\beta_{p\|})^{\alpha}}$$ exists between the proton temperature anisotropy and the ratio $\beta_{p\|}$ of field-aligned proton pressure and magnetic pressure. Analyzing AMPTE/CCE data, [*Anderson et al.*]{} \[1994\] determined $S=0.85$ and $\alpha=0.48$ and [*Fuselier et al.*]{} \[1994\] determined $S=0.83$ and $\alpha=0.58$. Using AMPTE/IRM data, [*Phan et al.*]{} \[1994\] obtained $S=0.58$ and $\alpha=0.53$. We performed a similar analysis on our data set of quasi-perpendicular bow shock crossings. For this analysis we computed 2-min averages of all measurements of $T_{p\perp}/T_{p\|}$ and $\beta_{p\|}$ taken between the keytime and 8min downstream of the keytime. We find a reasonable fit to Eq. (\[anisobeta\]) with $S=0.43\pm 0.03$ and $\alpha=0.58\pm 0.05$, which is not too different from the result of [*Phan et al.*]{} \[1994\]. A relation of the form of Eq. (\[anisobeta\]) is regarded as the consequence of the combined action of ion cyclotron and mirror waves, which grow due to the temperature anisotropy and reduce this anisotropy by means of pitch angle scattering. The growth of the waves depends on $T_{p\perp}/T_{p\|}$ and $\beta_{p\|}$ and it is expected that the anisotropy is reduced until the growth rate, $\gamma_m$, of the most unstable wave falls below some threshold. In fact, [ *Anderson et al.*]{} \[1994\] showed that Eq. (\[anisobeta\]) with their values of $S$ and $\alpha$ corresponds approximately to the threshold $\gamma_m/2\pi f_{cp}=0.01$. Hence, the validity of Eq. (\[anisobeta\]) indicates that the magnetosheath plasma reaches a state near marginal stability of the waves driven by the temperature anisotropy. As noted above, we found that data obtained less than 8min downstream of quasi-perpendicular bow shocks satisfy a relation of the form Eq. (\[anisobeta\]) with values of $S$ and $\alpha$ that are not too different from those determined by [*Phan et al.*]{} \[1994\] for the entire magnetosheath. This indicates that the state near marginal stability is already reached close to the shock. Figure \[fluclh\] shows that the largest amplitudes of the ion cyclotron waves at quasi-perpendicular low $\beta$ shocks are observed on average about 5min downstream of the keytime (see also Fig. \[corrver\]). The bow shock moves relative to the spacecraft at speeds of 10–100km/s. Taking a typical speed of 30km/s, we can translate 5min to a downstream distance of 9000km. According to Fig. \[ppsp1\], the plasma velocity, $v_{pn}$, normal to quasi-perpendicular shock is 120km/s on average. Thus the plasma needs about 75s to flow 9000km downstream. Since the ion cyclotron waves are convected with the plasma while they are growing, these 75s can serve as a rough estimate for the time $\tau$ that the waves need to reach their maximum amplitudes and saturate. In terms of gyro-periods, we have $\tau\sim 75\,{\rm s}\approx 50/f_{cp}$. Moreover, we find that on the same time scale $\tau$ the temperature anisotropy is reduced from about 2.5 immediately downstream of the low $\beta$ shock to about 2.1 (Fig. \[pplh\]) and that the ion cyclotron waves typically reach amplitudes of $\delta B/B\sim 0.07$. These results can be compared with two-dimensional hybrid simulations of [*McKean et al.*]{} \[1994\]. These authors examined a plasma with $\beta_{p\|}=1$ and $T_{p\perp}/T_{p\|}=3$. Under these conditions the ion cyclotron mode is found to be the dominant mode and the waves saturate after $\tau\approx5/f_{cp}$ and reach amplitudes of $\delta B/B\sim 0.15$. The proton temperature anisotropy is reduced on the same time scale $\tau$ from 3 to about 1.8. For our data set of low $\beta$ bow shocks $T_{p\perp}/T_{p\|}$ is on average 2.5 and $\beta_{p\|}$ is on average 0.5 immediately downstream of the shock. Since these values are considerably lower than the initial values used by [*McKean et al.*]{} \[1994\], the plasma simulated by [*McKean et al.*]{} \[1994\] is initially much farther away from the state of marginal stability. Thus it is not surprising that the waves grow faster, reach larger amplitudes and therefore lead to a stronger reduction of the anisotropy by means of pitch angle scattering. [*McKean et al.*]{} \[1994\] also examined a plasma with $\beta_{p\|}=4$ and $T_{p\perp}/T_{p\|}=3$. Under these conditions the ion cyclotron mode dominates for low $\alpha$-particle concentration, whereas the mirror mode dominates for high $\alpha$-particle concentration. The waves saturate after $\tau\approx 10/f_{cp}$ and reach amplitudes of $\delta B/B\sim 0.2$. The proton temperature anisotropy is reduced on the same time scale $\tau$ from 3 to about 1.5. This can be compared with data obtained at the quasi-perpendicular high $\beta$ shock. For our data set of high-$\beta$ bow shocks $T_{p\perp}/T_{p\|}$ is on average 1.8 and $\beta_{p\|}$ is on average 5 immediately downstream of the shock. Again, the plasma simulated by [*McKean et al.*]{} \[1994\] is initially much farther away from the state of marginal stability. Fig. \[pplh\] shows that a reduction of the anisotropy to 1.3 is observed 30s downstream of the keytime. This can again be translated to a downstream distance and used to estimate the time span that passes while the plasma travels this distance. This estimate gives $8\,{\rm s}\approx 2.5/f_{cp}$. Finally, it should be noted that the mirror waves analyzed by [*Czaykowska et al.*]{} \[1998\] have amplitudes of $\delta B/B\sim 0.2$, which is comparable to those found in the simulations of [*McKean et al.*]{} \[1994\]. Conclusions =========== We have analyzed the plasma and magnetic field parameters as well as low frequency magnetic fluctuations at 132 dayside AMPTE/IRM bow shock crossings. The average distance of the subsolar point, which results from the coordinates of the investigated bow shock crossings, is considerably smaller than in other studies, even when normalized to the average solar wind dynamical pressure. A reason for this discrepancy might be a variation of the polytropic index with the solar cycle since our observations are performed during typical solar minimum conditions. The position of the Earth’s bow shock still seems to be a matter of discussion. A superposed epoch analysis has been carried out by averaging particle and magnetic field data as well as low frequency magnetic spectra upstream and downstream of the bow shock. We have performed this analysis by dividing the events into different categories, i.e., quasi-perpendicular and quasi-parallel events as well as quasi-perpendicular low-$\beta$ and high-$\beta$ events. The particle and magnetic field data show that upstream of the quasi-parallel bow shock, in the foreshock region, the plasma is already heated compared to the undisturbed solar wind. Moreover, there are more energetic protons in the foreshock region, and the magnetic field is highly variable. Downstream of the quasi-perpendicular bow shock, a proton temperature anisotropy is found, which is higher on average downstream of the quasi-perpendicular low-$\beta$ than downstream of the quasi-perpendicular high-$\beta$ bow shock. Concerning the low frequency magnetic fluctuations we find that upstream of the quasi-perpendicular bow shock the solar wind spectrum is undisturbed with transverse Alfvén waves surpassing the compressive spectral component. Upstream of the quasi-parallel bow shock largely enhanced wave activity is detected in the turbulent foreshock region. These upstream waves are convected downstream, experiencing an enhancement at the bow shock itself. Downstream of the quasi-perpendicular bow shock the observed proton temperature anisotropy leads to the generation of left-hand polarized ion cyclotron waves under low-$\beta$ conditions and in some cases to the generation of mirror waves under high-$\beta$ conditions. A clear correlation has been observed between the intensity of the left-hand polarized component of the magnetic power spectrum relative to the right-hand polarized component and the proton temperature anisotropy. 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--- abstract: 'We present the *Sejong Suite*, an extensive collection of state-of-the-art high-resolution cosmological hydrodynamical simulations spanning a variety of cosmological and astrophysical parameters, primarily developed for modeling the Lyman-$\alpha$ (Ly$\alpha$) forest. The suite is organized into three main categories ([*Grid Suite*]{}, [*Supporting Suite*]{}, and [*Systematics Suite*]{}), addressing different science targets. Adopting a particle-based implementation, we follow the evolution of gas, dark matter (cold and warm), massive neutrinos, and dark radiation, and consider several combinations of box sizes and number of particles. With additional enhancing techniques, we are able to reach an equivalent resolution up to $3 \times 3328^3 = 110$ billion particles in a $(100 h^{-1} {\rm Mpc})^3$ box size, ideal for current and future surveys (e.g., eBOSS, DESI). Noticeably, for the first time, we simulate extended mixed scenarios describing the combined effects of warm dark matter, neutrinos, and dark radiation, modeled consistently by taking into account the neutrino mass splitting. In addition to providing multicomponent snapshots from $z=5.0$ to $z=2.0$ in intervals of $\Delta z=0.2$ for all of the models considered, we produced over 288 million Ly$\alpha$ skewers in the same $z$-range and extended parameter space. The skewers are well suited for Ly$\alpha$ forest science studies, for mapping the high-$z$ cosmic web and the matter-to-flux relation and bias, and for quantifying the critical role of baryons at small scales. We also present a first analysis of the suite focused on the matter and flux statistics, and show that we are able to accurately reproduce the 1D flux power spectrum down to scales $k=0.06~{\rm [km/s]^{-1}}$ as mapped by recent high-resolution quasar data, as well as the thermal history of the intergalactic medium. The simulations and products described here will be progressively made available.' author: - Graziano Rossi title: 'The *Sejong Suite*: Cosmological Hydrodynamical Simulations with Massive Neutrinos, Dark Radiation, and Warm Dark Matter' --- Introduction {#sec_introduction} ============ State-of-the-art experiments such as the Sloan Digital Sky Survey (SDSS; York et al. 2000; Dawson et al. 2016; Blanton et al. 2017; Abolfathi et al. 2018), the WiggleZ Dark Energy Survey (Drinkwater et al. 2010, 2018), the Dark Energy Survey (DES; The Dark Energy Survey Collaboration 2005; Abbott et al. 2018a, 2018b, 2019), and the cosmic microwave background (CMB) Planck mission (Planck Collaboration et al. 2018a, 2018b) have been pivotal in constraining with high accuracy the basic parameters of the standard spatially flat $\Lambda$CDM cosmology, dominated by collisionless cold dark matter (CDM) and a dark energy (DE) component in the form of a cosmological constant ($\Lambda$), with baryons representing only $\bf \sim 5\%$ of the total matter-energy content. The $\Lambda$CDM model is a relatively simple and successful framework centered on the assumption that general relativity (GR) is the correct theory of gravity on cosmological scales, and characterized by six minimal parameters, a power-law spectrum of adiabatic scalar perturbations, and primordial Gaussian density fluctuations. This scenario is consistent with spatial flatness to percent-level precision, in excellent agreement with big bang nucleosynthesis (BBN) and the standard value of the number of effective neutrino species $N_{\rm eff}$, with no evidence for dynamical DE and deviations from Gaussianity. However, a pure CDM framework stemmed from Gaussian initial conditions is incomplete, as some level of primordial non-Gaussianity (PNG) is generically expected in all inflation models, and conclusions regarding the concordance $\Lambda$CDM model are mainly drawn from large-scale structure (LSS) observations – while small scales remain poorly explored to date. In addition, GR may not be holding at cosmological scales, the nature of DE and dark matter (DM) still needs to be explained, and it has now been experimentally confirmed that neutrinos are massive particles and should be accounted for in the cosmological framework. All of these considerations point to necessary extensions of the $\Lambda$CDM scenario, and call for a substantial reexamination in our understanding of the [*dark sector*]{}: precisely the dark side of the universe, and in particular, massive neutrinos, dark radiation, and the nature of DM are the central focus of the present work – the primary target of which is the Lyman-$\alpha$ (Ly$\alpha$) forest. {width="85.00000%"} In this respect, pursuing the physics associated with neutrino mass is currently considered one of the five major science drivers – as featured in the report of the 2014 USA Particle Physics Project Prioritization Panel (P5; see Abazajian et al. 2015a, 2015b), and in the Decadal Survey on Astronomy and Astrophysics (Astro2020; Panel on Cosmology). It is also the main motivation behind our study, which aims at producing new high-resolution cosmological hydrodynamical simulations principally for the [*dark sector*]{} at small scales, as mapped by high-redshift quasars. Determining the absolute neutrino mass scale, hierarchy, and $N_{\rm eff}$ are within reach using cosmological measurements, rather than laboratory experiments: upper bounds on the total neutrino mass $\sum m_{\nu}$ from cosmology are now approaching the minimum value allowed by the inverted hierarchy (IH; $\sum m_{\nu} = 0.097~{\rm eV}$), while in the normal hierarchy (NH) scenario, $\sum m_{\nu} = 0.057~{\rm eV}$ (see, e.g., Capozzi et al. 2017; Planck Collaboration 2018b). Future experiments and next-generation CMB missions (i.e., CMB-S4; Abazajian et al. 2016, 2019; Abitbol et al. 2017) in combination with 21 cm surveys are expected to significantly improve the present bounds on $\sum m_{\nu}$ and $N_{\rm eff}$. For example, the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et al. 2016a, 2016b) will be able to measure $\sum m_{\nu}$ with an uncertainty of $0.020$ eV for $k_{\rm max} < 0.2~h {\rm Mpc^{-1}}$, sufficient to make the first direct detection of $\sum m_{\nu}$ at more than $3 \sigma$ significance, and rule out the IH at the $99\%$ confidence level (CL) if the hierarchy is normal and the masses are minimal. Upper bounds on massive neutrinos from cosmology are usually obtained from the CMB (the most direct route), and via other baryonic tracers of the LSS sensitive to neutrino properties. Here we focus on the Ly$\alpha$ forest, namely the absorption lines in the spectra of high-redshift quasars due to neutral hydrogen in the intervening photoionized intergalactic medium (IGM); for recent studies on the forest, see, for instance, du Mas des Bourboux et al. (2019), Garzilli et al. (2019), and Porqueres et al. (2019). Being a direct tracer of the underlying DM fluctuations and of the baryonic matter distribution over a wide range of scales and redshifts, and having radically different systematics than those of other lower-$z$ probes, the Ly$\alpha$ forest is characterized by an excellent sensitivity to neutrino mass, as thoroughly quantified in Rossi (2017). The current work is primarily centered around an accurate modeling of the forest, and of the high-$z$ cosmic web. Indeed, the most competitive neutrino mass constraints so far have been derived by combining Ly$\alpha$ data with additional tracers, and future forecasts including the forest as mapped by planned large-volume experiments are even more promising. To this end, Figure \[fig\_neutrino\_properties\] summarizes the present status of the most competitive neutrino mass constraints derived from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013) combined with Planck CMB data (yellow and orange zones in the panels), and highlights predicted achievements by the SDSS-IV Extended Baryon Oscillation Spectroscopic Survey (eBOSS; Dawson et al. 2016) as expected in its final data release (DR16; see the forecasts by Zhao et al. 2016) and DESI (lighter and darker green zones, respectively) – always in synergy with CMB data.[^1] Specifically, the left panel shows individual neutrino masses $m_{\rm i}$ (indicated with three different colors) in the two possible configurations of NH (long-dashed lines) and IH (dotted lines), versus the summed neutrino mass. The right panel displays the total neutrino mass as a function of the lightest one, in the two hierarchy configurations: the long-dashed line is again used for the NH, while the dotted line shows the IH. The shaded cyan and purple areas in the figure represent the $3\sigma$ regions allowed by both configurations. The absolute neutrino mass scale (lightest $m_{\nu}$) has a significant effect on the combined constraints; this parameter governs the lower bound on the joint mixing matrix from neutrino oscillation data, and if it is near the current upper cosmological limit, direct search experiments in the foreseeable future may have a chance to reach this lower bound. Clearly, mass splitting effects are quite small and may be too challenging to be measured. However, as is evident from the figure, DESI (and partly eBOSS but at lower CL), in combination with CMB data, should be able to rule out the IH at more than $3\sigma$ under the most standard assumptions previously mentioned, by placing the tightest $\sum m_{\nu}$ upper bound – a measurement that would otherwise be much more challenging from particle physics experiments. Another interesting aspect related to the *dark sector*, which we examine in this work, is the possibility of extra light relic particles in the universe that may shed light on the physics beyond the standard model, since recently particle detector experiments have hinted at the possible existence of additional species of neutrinos (see Gelmini et al. 2019, and references therein). Generically, all of these nonstandard models are termed [*dark radiation*]{}, expressing deviations $\Delta N_{\rm eff}$ from the canonical value of $N_{\rm eff}=3.046$ – which point to ‘hidden’ neutrino interactions mediated by a scalar or a vector. Departures of $N_{\rm eff}$ from $3.046$ would imply nonstandard neutrino features or the contribution of other relativistic relics. Our goal is also to carry out, for the first time, a full hydrodynamical treatment for dark radiation cosmologies – as we explain in detail in Sections \[sec\_sejong\_suite\_general\] and \[sec\_sejong\_suite\_components\]. Moreover, DM (or at least part of it) may be warm rather than cold: in this respect, hypothetical sterile neutrinos could account for the baryon asymmetry in the universe and neutrino flavor mixing, and at the same time they may constitute convenient candidates for warm dark matter (WDM). In the present work, we also explore this possibility by considering early decoupled thermal relics, and assuming that all the DM is warm when massive neutrinos are not present (i.e., all the CDM is turned into WDM, known as ‘pure’ $\Lambda$WDM models). Plausible candidates are keV right-handed neutrinos or sterile neutrinos. In addition, we also reproduce mixed scenarios where massive neutrinos and WDM, or dark radiation and WDM, are both present – another novelty in the literature. Even in this case, the Ly$\alpha$ forest is an excellent probe for detecting dark radiation and WDM imprints, thanks to significant attenuation effects on the matter and flux power spectra at small scales, jointly with a characteristic nonlinear and $z$-dependent suppression of power (Rossi 2017). Competitive constraints on $N_{\rm eff}$ and on the WDM mass from Ly$\alpha$ have already appeared in numerous works (e.g., Rossi et al. 2015; Ir[š]{}i[č]{} et al. 2017b; Archidiacono et al. 2019). Modeling nonlinear evolution and the complex effects of baryons at small scales as required by the Ly$\alpha$ forest, as well as the presence of additional components such as massive neutrinos, dark radiation, and WDM, is only possible via sophisticated high-resolution cosmological hydrodynamical simulations. Traditionally, such simulations are developed following two basic flavors: smoothed particle hydrodynamics (SPH; Gingold & Monaghan 1977; Lucy 1977), and grid-based methods; there are also more sophisticated combinations of the two categories. The SPH technique, which uses particles to represent the various components, is the choice adopted in this study – although both approaches have been successfully used in the literature to model the Ly$\alpha$ forest. Moreover, realistic numerical simulations able to reproduce and control the various observational surveys are indispensable for interpreting high-quality data as those expected from upcoming cosmological experiments, and for controlling systematics that can spoil parameter constraints: a complete hydrodynamical treatment is mandatory to reach the precision that data are now beginning to show. A vast number of $N$-body gravity-only simulations is available in the literature. Among the plethora of realizations, we recall, e.g., the [*Hubble Volume*]{} (Evrard et al. 2002), [*Millennium*]{} (Springel et al. 2005), [*MultiDark*]{} and [*Bolshoi*]{} (Riebe et al. 2013; Klypin et al. 2016), [*Dark Sky*]{} (Skillman et al. 2014), [*UNIT*]{} (Chuang et al. 2019), and [*HACC*]{} (Heitmann et al. 2019a, 2019b) simulations. In comparison, less has been already developed in terms of the hydrodynamical counterpart – but see, e.g., the remarkable [*Illustris*]{} and [*IllustrisTNG*]{} (Vogelsberger et al. 2014; Nelson et al. 2019), [*Horizon-AGN*]{} (Dubois et al. 2014), [*EAGLE*]{} (Schaye et al. 2015), [*Magneticum*]{} (Dolag 2015), [*MassiveBlack-II*]{} (Khandai et al. 2015), [*Sherwood*]{} (Bolton et al. 2017), and [*Borg Cube*]{} (Emberson et al. 2019) simulations; it is even more so in relation to the [*dark sector*]{}. This is mainly due to the high computational demand and considerable costs in producing a large number of hydrodynamical runs, and because of the complexity in resolving baryonic physics at small scales and the challenges associated with the modeling of exotic species – such as neutrinos and dark radiation. Within this context, and motivated by all of these reasons, we have undertaken a new long-term challenging effort, which aims at pushing further and upgrading the significant work carried out in Rossi et al. (2014, 2015) and in Rossi (2017). To this end, we have produced an extensive number of state-of-the-art and more refined cosmological hydrodynamical simulations (over 300 runs) termed the [*Sejong Suite*]{}, developed primarily for Ly$\alpha$ forest studies. The realizations span a variety of cosmological and astrophysical parameters including those that represent the [*dark sector*]{}, and are particularly well suited for the high-$z$ cosmic web as seen in the Ly$\alpha$ forest, in the redshift interval $5.0 \le z \le 2.0$. The entire suite has been organized into three main categories, addressing different scientific and technical aspects: the [*Grid Suite*]{}, useful for cosmological parameter constraints especially regarding massive and sterile neutrinos; the [*Supporting Suite*]{}, aimed at characterizing the impact of exotic particles on the high-$z$ cosmic web; and the [*Systematics Suite*]{}, meant to quantify various systematic effects. The [*Sejong Suite*]{} features a number of improvements and novelties with respect to our previously released simulations, related to [*technical*]{}, [*modeling*]{}, and [*innovative*]{} aspects, which we discuss throughout the paper and summarize in the final part. Noticeably, we have expanded the parameter space for the [*Grid Suite*]{} and tighten their variation range; we have included, for the first time, extended mixed scenarios describing the combined effects of WDM, dark radiation, and neutrinos. We also address a series of nontrivial systematics, and we have produced more than 288 million Ly$\alpha$ forest skewers mapping an extended parameter space. In particular, reaching a very high sensitivity on small scales and resolving baryonic physics are essential aspects for improving neutrino, $N_{\rm eff}$, and [*dark sector*]{} constraints, and for breaking degeneracies – as demanded by future high-quality surveys. This work is primarily intended as a presentation, technical description, and guide to the usage of the various simulations and of the related post-processing products. It also represents the foundation for future extensions in the modeling the Ly$\alpha$ forest along novel paths of improvement. Moreover, we also undertake a first analysis of the [*Sejong Suite*]{}, mainly focused on the matter and flux statistics. The layout of the paper is organized as follows. Section \[sec\_sejong\_suite\_general\] provides a general description and philosophy of the entire suite: in particular, we present the overall suite structure, several technical specifics about the production of hydrodynamical simulations, the post-processing procedure, and some basic aspects regarding the numerical implementation of massive neutrinos, dark radiation, and WDM – within the SPH framework. Section \[sec\_sejong\_suite\_components\] illustrates in more detail the individual categories comprising the [*Sejong Suite*]{}, namely the [*Grid Suite*]{}, the [*Supporting Suite*]{}, and the [*Systematics Suite*]{}; a comparison to previous studies in terms of resolution and box-size requirements is also included. Section \[sec\_sejong\_suite\_first\_results\] describes selected results from the first analysis of the simulations, focused on the matter and flux statistics. In particular, we show that we are able to accurately reproduce the 1D Ly$\alpha$ flux power spectrum down to scales $k=0.06~{\rm [km/s]^{-1}}$ as mapped by recent high-resolution quasar data, as well as the thermal history of the IGM. Finally, we conclude in Section \[sec\_conclusions\], where we summarize the various products available and highlight the improvements and novelties of the entire suite – along with ongoing applications and future avenues. The [*Sejong Suite*]{}: Structure, Philosophy, Technical Details, and Modeling Aspects {#sec_sejong_suite_general} ====================================================================================== In this section we introduce the [*Sejong Suite*]{}. We begin by presenting the general structure and organization of the suite along with technicalities regarding codes and simulation specifics (shared by all of the realizations), as well as the post-processing procedure. We then highlight our implementation of massive neutrinos, dark radiation, and WDM within the hydrodynamical framework. Finally, we provide additional details on the supercomputers where the simulations were performed, and a brief description of the new pipeline developed for producing all of the runs. General Structure of the *Sejong Suite* --------------------------------------- The [*Sejong Suite*]{} is an extensive collection of state-of-the-art high-resolution cosmological hydrodynamical simulations, covering a wide range in cosmological and astrophysical parameters and spanning a variety of cosmologies including scenarios with massive neutrinos, dark radiation, and WDM. These new simulations are akin in philosophy and strategy to those developed in Rossi et al. (2014), but they contain several improvements at all levels and represent an upgrade with respect to our previous runs. In particular, for the first time, massive neutrinos, dark radiation, and WDM are consistently [*jointly*]{} modeled, allowing one to explore their combined effects deep in the nonlinear regime. The suite is organized into three main categories, addressing different scientific and technical aspects: (1) the [*Grid Suite*]{} ($76 \times 3 = 228$ simulations), targeted primarily for cosmological parameter constraints especially regarding massive and sterile neutrinos and the [*dark sector*]{}, exploiting the small-scale flux power spectrum – and they represent our leading effort and major deliverable; (2) the [*Supporting Suite*]{} (114 simulations), aimed at studying the detailed physical effects of exotic particles and dark radiation models, as well as their impact on the high-redshift cosmic web; and (3) the [*Systematics Suite*]{} (35 realizations), meant to address several systematic effects, ranging from numerical challenges until parameter degeneracies. Primarily, the simulations are targeted to explore the high-$z$ cosmic web as seen in the Ly$\alpha$ forest (we cover in fact the redshift interval $5.0 \le z \le 2.0$), although they can be of much broader usage. Building upon this main framework, we plan to extend the number of realizations in future works. 2.0pt Parameter Value ----------------------------------- -------------------------- [*Cosmological*]{} $n_{\rm s}$ 0.968 $\sigma_8 (z=0)$ 0.815 $\Omega_{\rm m}$ 0.308 $H_0$ \[km s$^{-1}$Mpc$^{-1}$\] 67.8 $N_{\rm eff}$ 3.046 $\sum m_{\nu}$ \[eV\] 0.0 $m_{\rm WDM}$ \[keV\] 0.0 [*Astrophysical*]{} $T_0 (z=3)$ \[K\] 15000 $\gamma(z=3)$ 1.3 $\tau_{\rm A}$ 0.0025 $\tau_{\rm S}$ 3.7 [*Fixed*]{} $\Omega_{\rm b}$ 0.048424 $\Omega_{\rm tot}$ 1.0 $A_{\rm S}$ 2.139 $\times$ $10^{-9}$ $z_{\rm re}$ 8.8 [*Derived*]{} $\Omega_{\rm \Lambda}$ 0.692 $\Omega_{\rm c}$ 0.25958 $\Omega_{\rm c} h^2$ 0.119324 $\Omega_{\rm b} h^2$ 0.02226 \[table\_baseline\_params\_sims\] : Baseline Parameters of the ‘Sejong Suite’. The box sizes considered range from $25h^{-1}{\rm Mpc}$ to $100h^{-1}{\rm Mpc}$, and the resolution varies from $208^3$ up to $1024^3$ particles/type. For the [*grid*]{} runs, adopting a splicing technique (McDonald 2003) we are able to achieve an effective resolution up to $3 \times 3328^3 = 110$ billion particles within a $(100h^{-1} {\rm Mpc})^3$ box size, which improves on our previous realizations and is ideal for reproducing the main aspects of the Ly$\alpha$ forest as mapped by eBOSS and DESI. The various simulations were performed with periodic boundary conditions and for the majority of the runs with an equal number of DM, gas, and neutrino particles. Taking into account our largest $100h^{-1}{\rm Mpc}$ box size, the average spacing between sightlines is of the order of $10h^{-1}{\rm kpc}$ – much smaller than the scale probed by the Ly$\alpha$ forest. Output snapshots are created at regular intervals $\Delta z$ in redshift within the range $5.0 \le z \le 2.0$, where $\Delta z = 0.2$. For some visualization runs, we also reach $z = 0$ and/or produce additional snapshots at every redshift interval of $\Delta z = 0.1$. The fundamental parameters considered for the baseline cosmology of the entire suite (indicated as [*‘Best Guess’*]{} or BG) along with their central values are listed in Table \[table\_baseline\_params\_sims\]. They have been organized into two main categories: cosmological (first block from the top) and astrophysical (second block). For the majority of the runs, they are consistent with Planck 2015 results (Planck Collaboration et al. 2016) – i.e., TT+lowP+lensing $68\%$ limits – and with the SDSS DR12 flat-$\Lambda$CDM cosmology; although, we also have performed some realizations using the latest Planck 2018 reported measurements for systematic studies (Planck Collaboration et al. 2018b). As cosmological parameters, we adopt the spectral index of the primordial density fluctuations $n_{\rm s}$, the amplitude of the matter power spectrum $\sigma_8$, the matter density $\Omega_{\rm m}$, the Hubble constant $H_0$, the number of effective neutrino species $N_{\rm eff}$, the total neutrino mass $\sum m_{\nu}$ if nonzero (expressed in eV), and the WDM mass $m_{\rm WDM}$ (expressed in keV). As astrophysical parameters, we consider the IGM normalization temperature $T_0(z)$ and the logarithmic slope $\gamma (z)$ at $z = 3$ – although practically $T_0$ and $\gamma$ are set by two related quantities that alter the amplitude and density dependence of the photoionization heating rates. Moreover, we take into account two additional parameters that do not enter directly in the simulations but that are used in a subsequent post-processing phase to provide the normalization of the flux power spectrum via the effective optical depth, namely, $\tau_{\rm A}$ and $\tau_{\rm S}$. The lower blocks of the same table show some relevant fixed or derived parameters, such as the baryon, CDM, cosmological constant, and total densities ($\Omega_{\rm b}, \Omega_{\rm c}, \Omega_{\rm \Lambda}, \Omega_{\rm tot}$), the initial amplitude of primordial fluctuations $A_{\rm S}$, and the reionization redshift $z_{\rm re}$. Flatness is always assumed, so that the total density parameter $\Omega_{\rm tot} \equiv \Omega_{\rm m} + \Omega_{\Lambda} =1$. For the [*Grid Suite*]{}, these baseline parameters are varied in turn, as we explain in detail in Section \[sec\_sejong\_suite\_components\]. Technical Details: Main Codes and Simulation Specifications ----------------------------------------------------------- All of the simulations are produced with a customized version of GAlaxies with Dark matter and Gas intEracT (Gadget-3; Springel et al. 2001; Springel 2005) for evolving Euler hydrodynamical equations and primordial chemistry, along with cooling and an externally specified ultraviolet (UV) background, supplemented by the code for Anisotropies in the Microwave Background (CAMB; Lewis et al. 2000) and a modified parallel version of second-order Lagrangian perturbation theory (2LPT; Crocce et al. 2006) for setting the initial conditions. Gadget-3 is a widely used massively parallel tree-SPH code for collisionless and gas-dynamical cosmological simulations originally developed by Volker Springel, written in ANSI C, and exploiting the standardized message passing interface (MPI) along with several open-source libraries. The code is efficiently parallelized via a space decomposition achieved with a space-filling curve (i.e. the Peano-Hilbert decomposition), and reaches a high-level optimization along with being efficient in memory consumption and communication bandwidth – with a nearly optimal scalability and work-load balance. Gravitational interactions are computed with a hierarchical multipole expansion using the standard $N$-body method, and gas-dynamics is followed with SPH techniques and fully adaptive smoothing lengths, so that energy and entropy are conserved. Besides gas and DM, we model similarly massive neutrinos, dark radiation, and WDM within the SPH framework (see more details in Section \[subsec\_nu\_dark\_rad\_wdm\_modeling\]). Short-range forces are treated with the tree method, and long-range forces with Fourier techniques. For our realizations, we set the number of mesh cells of the particle-mesh (PM) grid equal to the number of particles. When massive neutrinos are included, we neglect the small-scale neutrino clustering, and their short-range gravitational tree force in the TreePM scheme is not computed: hence, the spatial resolution for the neutrino component is of the order of the grid resolution used for the PM force calculation. Generally, in all of the various realizations, the gravitational softening is set to 1/30 of the mean interparticle spacing for all of the species considered. Using CAMB, we compute transfer functions per each individual component and corresponding power spectra, and for the majority of the runs, initial conditions are fixed at $z = 33$ with 2LPT. We adopt the entropy formulation of SPH proposed by Springel & Hernquist (2002), with the gas assumed of primordial composition photoionized and heated by a spatially uniform ionizing background, having an helium mass fraction of $Y = 0.24$ and no metals or evolution of elementary abundances. Regarding star formation, we utilize the same criterion assumed in Rossi et al. (2014, 2015) and include the most relevant effects of baryonic physics that impact the IGM, so that the thermal history in the simulations is consistent with recent temperature measurements (see Section \[section\_igm\_properties\]). This criterion speeds up the calculations considerably, and it has been shown to have negligible effects on the Ly$\alpha$ flux statistics. Moreover, we disable feedback options and neglect galactic winds, as suggested by the results of Bolton et al. (2008), who found that winds have a negligible effect on the Ly$\alpha$ forest. Post-processing {#subsec_postprocessing} --------------- From simulated Gadget-3 snapshots within the redshift range $5.0 \le z \le 2.0$, we construct Ly$\alpha$ skewers fully spanning all of the various simulated cosmologies. This is achieved through an elaborate pipeline, which eventually allows us to obtain averaged Ly$\alpha$ flux power spectra and temperature-density relations. The primary step in this process requires the extraction of simulated quasar sightlines: for the majority of all of our runs and for all the redshift intervals considered, we extract 100,000 randomly placed sightlines through the simulation box per redshift, using 2048 bins along the line of sight (LOS) – unless specified otherwise. We then estimate the absorption due to each SPH particle near the sightline from the positions, velocities, densities, and temperatures of all of the SPH particles at a given redshift (i.e., Theuns et al. 1998, 2002). With this procedure, we obtain a number of simulated quasar spectra that we subsequently smooth with a 3D cubic spline kernel. The photoionization rate is then fixed, by requiring the effective optical depth at each redshift to follow the empirical power law $\tau_{\rm eff}(z) = \tau_{\rm A}(1 + z)^{\tau_{\rm S}}$, with $\tau_{\rm A} = 0.0025$ and $\tau_{\rm S} = 3.7$, as done in Rossi et al. (2014, 2015) and in Rossi (2017). This procedure is routinely adopted in a series of other studies, as the instantaneous IGM temperature only depends on the spectral shape of the UV background and the gas density. Finally, all of the spectra are rescaled by a constant so that the mean flux across all spectra and absorption bins matches the observed mean flux at redshift $z$. After performing the normalization procedure, the mean over all of the rescaled spectra is used as the extracted flux power spectrum for a given box. In addition to fluxes, we also extract particle samples for studying the IGM temperature-density relation ($T_0-\gamma$) in the presence of massive neutrinos, dark radiation, and WDM. This is performed for every simulation and for each considered redshift interval, and we generally extract 100,000 particles per type per redshift. Implementation of Massive Neutrinos, Dark Radiation, and Warm Dark Matter {#subsec_nu_dark_rad_wdm_modeling} ------------------------------------------------------------------------- {width="76.00000%"} As in our previous releases, we adopt a particle-based implementation of massive neutrinos following the SPH formalism: in essence, neutrinos are treated as an additional particle component, on top of gas and DM. Our methodology is now common to other recent related studies (e.g., Viel et al. 2010; Castorina et al. 2014, 2015; Villaescusa et al. 2014, 2018, 2019; Carbone et al. 2016; Ruggeri et al. 2018), and represents a neat way of incorporating massive neutrinos within the SPH framework. In fact, in the Ly$\alpha$ regime and for the range of neutrino masses considered in our simulations, the neutrino free-streaming condition is always satisfied (Lesgourgues & Pastor 2006; Rossi 2017) so that neutrinos can simply be treated as collisionless particles similarly to DM, with a phase space distribution isotropic following a Fermi-Dirac distribution and a nonzero anisotropic stress. Despite more demanding computational requirements due to an extra particle component, this implementation is ideal for our main goals as we need to resolve small-scale physics deep in the nonlinear regime, and in particular reproduce with high accuracy all of the main aspects of the Ly$\alpha$ forest – at the level of eBOSS and DESI, in a $k$-range where nonlinear evolution of cosmological neutrinos cannot be neglected –as well as the scale dependence of the total matter power spectrum due to nonlinear evolution. We do not rely on any linear approximation nor interchange between grid- and particle-based methods, since nonlinear evolution at small scales is not properly reproduced by the grid method. With this implementation, we are able to precisely quantify the response of the power spectrum to isolated variations in individual parameters and varying neutrino masses, and also disentangle the $\sum m_{\nu}-\sigma_8$ degeneracy, as well as the effects of baryons at small scales. This latter aspect is particularly relevant for the usage of the 1D and 3D flux power spectra for cosmological studies. We have already proven this implementation to be robust, and capable of providing competitive constraints on massive neutrinos and dark radiation (i.e., Rossi et al. 2015). Recent alternative implementations can be found in Ali-Haimoud & Bird (2013), Banerjee & Dalal (2016), Upadhye et al. (2016), Emberson et al. (2017), and Mummery et al. (2017). The variety of techniques adopted in the literature reflects the fact that neutrinos can be treated either as a fluid or as an ensemble of particles, and one may describe their evolution with linear theory or perform a full nonlinear treatment. While all of these complementary methods may be faster in terms of central processing unit (CPU) hours and production runs than ours, we need to accurately reproduce the nonlinear evolution at small scales in order to characterize the Ly$\alpha$ forest with high-fidelity – hence our implementation choice. Regarding dark radiation (i.e., any deviation from the canonical value of $N_{\rm eff} =3.046$ –which remains still poorly studied in the literature), in this work we consider models with four neutrinos, where three are massive and active while the fourth one is massless, sterile, and thermalized with the active ones. Unlike in Rossi et al. (2015) where we adopted an analytic remapping technique to simulate such models, here, for the first time, we consistently carry out a full hydrodynamical treatment – combining three massive neutrinos of degenerate mass with a massless sterile neutrino so that $N_{\rm eff}=4.046$. This represents a novelty in the literature, and such new simulations were already used in Rossi (2017) to study the combined effect of baryons, DM, neutrinos, and dark radiation at high-$z$ and small scales, to quantify the impact of extra dark components on Ly$\alpha$ forest observables as a function of scale and redshift. In particular, in Rossi (2017), we have shown that departures of $N_{\rm eff}$ from the canonical value of $3.046$, due to nonstandard neutrino features or to the contribution of other relativistic relics, have a non-negligible repercussion for the subsequent cosmological structure formation: the CMB power spectrum shows a suppressed first peak and enhancements in the the others, making the situation clearly distinguishable from the case of three massive neutrinos only. As far as WDM, we focus here on two implementations, following different methodologies. In both cases, we only consider early decoupled thermal relics, and assume that all the DM is warm when massive neutrinos are not present (i.e., all the CDM is turned into WDM, also known as ‘pure’ $\Lambda$WDM models): suitable candidates are keV right-handed neutrinos or sterile neutrinos. Thermal relics are the most basic model of WDM particles, analogous to neutrinos but with a larger particle mass (indicated as $m_{\rm WDM}$ in our notation), and characterized by WDM with a simple thermal history – so that it is possible to calculate their free streaming. Noticeably, for early decoupled thermal relics, there is a direct correspondence between their mass $m_{\rm WDM}$ (always expressed in keV), and the mass $m_{\rm s}$ of a non-thermalized nonresonant massive sterile neutrino in the standard Dodelson-Widrow mechanism (DW; Dodelson & Widrow 1994): hence, our WDM runs are useful for studying and eventually constraining both pure WDM candidates and/or nonresonant massive sterile neutrinos. Specifically, it is possible to introduce a re-parameterization via a non-thermalized massive sterile neutrino as follows: $$m_{\rm s} = 4.43 \Big ( {m_{\rm WDM} \over {\rm keV}} \Big )^{4/3} \Big ( {\omega_{\rm WDM} \over 0.1225} \Big )^{-1/3}~{\rm keV}$$ where $\omega_{\rm WDM} = \Omega_{\rm WDM} h^2$, and the relation holds for the nonresonant DW production mechanism. And, in terms of $\Delta N_{\rm eff}$, one can write: $$m^{\rm eff} = \chi m_{\rm s} \equiv \Delta N_{\rm eff} m_{\rm s}, \label{eq_sterile_cea_2}$$ where $m_{\rm s} = m_{\rm WDM} / (\Delta N_{\rm eff})^{1/4}$. In the context of the previous formalism, in our first prescription, WDM is implemented by modifying the linear matter power spectrum to mimic the presence of thermal relics. This is achieved by introducing a small-scale cut-off at the level of 2LPT (when we generate our initial conditions) to the BG CAMB linear power spectrum $P_{\rm \Lambda CDM}$. Namely: $$T^2_{\rm lin} (k) = {P_{\rm WDM} (k) \over P_{\rm \Lambda CDM} (k) } = \Big [1 + (\alpha k)^{2 \nu} \Big ]^{-10/\nu} \label{eq_viel_1}$$ where $\Omega_{\rm WDM} = \Omega_{\rm m} - \Omega_{\rm b} \equiv \Omega_{\rm c}$ and $$\alpha \equiv \alpha (m_{\rm WDM}) = a_1 \Big( {1~ {\rm keV} \over m_{\rm WDM}} \Big )^{a_2} \Big ( {\Omega_{\rm WDM} \over 0.25 } \Big )^{a_3} \Big( {h \over 0.7} \Big )^{a_4} \label{eq_viel_2}$$ with $\alpha$ in units of $h^{-1}{\rm Mpc}$, and ($\nu, a_1, a_2, a_3, a_4$) = ($1.12, 0.049, 1.11, 0.11, 1.22$) following Viel et al. (2005, 2012, 2013) or in a different prescription assuming ($\nu, a_1, a_2, a_3, a_4$) = ($1.2, 0.048, 1.15, 0.15, 1.3$) – the latter according to Bode, Ostriker & Turok (2001). This is a good approximation at $k< 5\div 10 ~h {\rm Mpc}^{-1}$, and below one needs a more complex function plus the addition of acoustic oscillations. For $m_{\rm WDM} \sim 1$ keV, the characteristic cutoff in the power spectrum is at $ k \sim 1.5 ~h {\rm Mpc}^{-1}$. Initial velocities for WDM are drawn from a Fermi-Dirac distribution and added to the proper velocity assigned by linear theory. In our second prescription, WDM is implemented starting directly from a properly designed CAMB power spectrum, by setting $\Delta N_{\rm eff}$ to express departures from the canonical $N_{\rm eff}$ value according to the relic mass – where $N_{\rm eff} = 3.046 + \Delta N_{\rm eff}$. These runs are simply indicated as WDM in Table \[table\_supporting\_sims\_wdm\], and we adopt this implementation for our [*Grid Suite*]{}, since it is more physically motivated and it is also straightforward (although delicate) to include massive neutrinos within this framework. The inclusion is simply achieved by accounting for the neutrino mass degeneracy, where: $$\Delta N_{\rm eff} = \Big ( {93.14 \cdot \Omega_{\rm WDM} h^2 \over m_{\rm WDM} } \Big )^{4/3}.$$ We eventually use the one-to-one mapping between thermal relics and keV sterile neutrinos to simulate massive sterile neutrino cosmologies in the DW mechanism – as previously explained. Within the second prescription, we also simulate for the first time cosmologies with both massive active and sterile neutrinos, in a consistent fashion. This is achieved by introducing two neutrino eigenstates (one for the active ones, and one for the sterile counterpart) in a three-component simulation, and then by considering a neutrino mass splitting – so that $\Omega_{\nu} = \Omega_{\rm m} - \Omega_{\rm b}$ now contains two distinct contributions. In essence, three active massive-with-degenerate-mass neutrinos contribute as $N_{\rm eff} =3.046$, while a non-thermalized massive sterile neutrino contributes with an additional $\Delta N_{\rm eff}$ to the number of effective neutrino species, directly proportional to its mass. Such runs represent a unique novelty in the literature, and allow us to study the combined effects of massive and sterile neutrinos, as well as to put constraints on pure WDM scenarios directly from Ly$\alpha$ forest data. The latter simulations are indicated as $\sum m_{\nu}^{+} m_{\rm WDM}^{+}$ in Table \[table\_grid\_sims\_cross\_terms\] ([*Grid*]{} runs), and as [*NU\_WDM*]{} in Table \[table\_supporting\_sims\_wdm\]. Finally, note that we always include massless neutrinos in WDM-only simulations. Additional Technical Details: Machines and Automated Pipeline ------------------------------------------------------------- All of the simulations presented here have been produced at the Korea Institute of Science and Technology Information (KISTI) via the Tachyon 2 supercomputer, under allocations KSC-2016-G2-0004, KSC-2017-G2-0008, and KSC-2018-G3-0008 (for a total of 11.25 million CPU hours), which allowed us to use up to 2176 dedicated cores for 9 months on an exclusive queue, and also up to 8192 cores on a regular queue – exploiting the MPI parallel architecture. Tachyon 2 is an SUN Blade 6275 machine with Intel Xeon x5570 Nehalem 2.93 GHz CPUs on an Infiniband 4x QDR network, having a total of 25,408 cores grouped into eight-core nodes, with 24 GB of memory per node and 300 TFlops of peak performance. Our standard runs required all 2176 cores, with an average clock time of about 3–4 days per simulation (excluding post-processing). Our highest-resolution runs used 8192 cores. The post-processing of the simulations (i.e., extraction of LOS skewers and particle samples) demanded dedicated fat nodes on different architectures, and it was performed with the KISTI KAT System under allocations KSC-2018-T1-0017, KSC-2018-T1-0033, and KSC-2018-T1-0061. Specifically, we used the KISTI/TESLA ‘Skylake’ architecture, an Intel Xeon Skylake (Gold 6140) at 2.30GHz (18-core and two socket), with 192GB DDR4 memory, and the KISTI/TESLA ‘Bigmem’ architecture, an Intel Xeon Westmere (E7-4870) at 2.40GHz (10-core, four socket) with 512GB DDR3 memory. Finally, some further post-processing was carried out with our new cluster system at Sejong University, composed of a Xeon Silver 4114 master node and a Xeon Gold 6126 computing node architecture. For performing all our novel simulations and post-processing, we devised a new portable and efficient pipeline, able to produce end-to-end simulations with a single integrated script, in order to avoid human error in the process. Figure \[fig\_pipeline\_workflow\_details\] provides a self-explanatory workflow diagram of the pipeline: in particular, blue circles indicate the various products stored at different stages, with the most computationally demanding post-processing step requiring fat node architecture (green arrows in the diagram). All of the details of the simulation are preselected by the user (including directory settings, choices of initial conditions, as well as simulation, astrophysical, cosmological, neutrino, WDM, and post-processing parameters), and all of the specifics of a simulation are organized by categories with a unique identifier associated per simulation. The elegance of this pipeline relies in its portability and generality (not tied to a specific machine), and on the overall simplicity in performing in a fully automated way a rather complex sequence of tasks – which represents a considerable improvement from the pipeline developed in Rossi et al. (2014). In this way, we are able to run a generic simulation (including post-processing) with just a few preparatory steps and minimal effort from the user, who is only required to set initially the specifics of the simulation to run, and subsequently simply launch the desired production run. This improvement was necessary, given the large amount of simulations and heavy post-processing involved here (exceeding 200 TB of data). Our pipeline is also straightforward to modify or extend for future use. The [*Sejong Suite*]{}: Components and Products {#sec_sejong_suite_components} =============================================== In this section we describe the three main categories of simulations that constitute the *Sejong Suite* – denoted as the [*Grid*]{}, [*Supporting*]{}, and [*Systematics Suites*]{}, respectively. Moreover, we present the entire list of simulations performed in Tables \[table\_grid\_sims\_base\]-\[table\_systematics\_sims\_all\], provide some visualizations from selected snapshots, and also include a comparison to previous studies in terms of resolution and box-size requirements. We conclude this part with a schematic list of all of the products available. The *Grid Suite* {#subsec_grid_suite} ---------------- The hydrodynamical [*Grid Suite*]{} has been devised with the intent of providing a high-resolution mapping – including all of the effects of baryonic physics at small scales – of a wide landscape defined by a variety of cosmological and astrophysical parameters, centered on what we indicate as BG or ‘reference’ cosmology. Here the BG model is set by Planck 2015 results, i.e., TT+lowP+lensing $68\%$ limits (see Table \[table\_baseline\_params\_sims\] for the central parameters adopted). The primary motivation in developing such a grid of simulations is the construction of a Taylor expansion model for the flux power spectrum, a key theoretical quantity useful for building the Ly$\alpha$ forest likelihood (see, e.g., Rossi et al. 2015; Rossi 2017). With the Ly$\alpha$ forest likelihood in hands, it is possible to provide competitive constraints on cosmological parameters, neutrino masses, WDM, and more generally on the [*dark sector*]{}, exploiting the synergy between the Ly$\alpha$ forest as a unique high-$z$ probe, lower-$z$ tracers, and the CMB. ![Typical structure of the [*Grid Suite*]{}, a subset of the [*Sejong Suite*]{}, for a generic combination ($p_1, p_2$) of cosmological and/or astrophysical parameters.[]{data-label="fig_grid_structure"}](grid_structure_cartoon.pdf){width="43.00000%"} 2.0pt Parameter Central Value Variation ------------------------------------- --------------- -------------- $n_{\rm s}$ 0.968 $\pm$ 0.018 $\sigma_8 (z=0)$ 0.815 $\pm$ 0.027 $\Omega_{\rm m}$ 0.308 $\pm$ 0.036 $H_0$ \[km s$^{-1}$Mpc$^{-1}$\] 67.8 $\pm$ 2.7 $N_{\rm eff}$ 3.046 $\pm$ 1.000 $\sum m_{\nu}$ \[eV\] 0.0 +0.3 $\sum m_{\rm WDM}$ \[keV\] 0.0 +3.0 $T_0 (z=3)$ \[K\] 15000 $\pm$ 5000 $\gamma(z=3)$ 1.3 $\pm$ 0.3 $\tau_{\rm A}$ 0.0025 $\pm$ 0.0015 $\tau_{\rm S}$ 3.7 $\pm$ 0.4 \[table\_grid\_params\_variations\] : Allowed variations of the key grid parameters. 2.0pt Simulation Name $M_{\rm \nu}$ \[eV\] $m_{\rm WDM}$ \[keV\] $N_{\rm eff}$ $\sigma_8(z=0)$ Boxes \[Mpc/h\] $N_{\rm p}^{1/3}$ Mean Part. Sep. \[Mpc/h\] Softening \[kpc/h\] ----------------------------- ---------------------- ----------------------- --------------- ----------------- ----------------- ------------------- --------------------------- --------------------- BEST GUESS a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^-$ a/b/c 0.0 0.0 3.0460 0.788 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^-$ a/b/c 0.0 0.0 2.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $m_{\rm WDM}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $T_0^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $T_0^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\gamma^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\gamma^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\tau_{\rm A}^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\tau_{\rm S}^-$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 \[table\_grid\_sims\_base\] As reported in Table \[table\_baseline\_params\_sims\], our grid is constructed around seven cosmological parameters ($n_{\rm s}$, $\sigma_8$, $\Omega_{\rm m}$, $H_0$, $N_{\rm eff}$, $\sum m_{\nu}$, and $m_{\rm WDM}$) plus four astrophysical parameters ($T_0[z=3]$, $\gamma[z=3]$, $\tau_{\rm A}$, and $\tau_{\rm S}$). The baseline BG model does not contain any massive neutrinos, WDM, or dark radiation (i.e., $\Delta N_{\rm eff} =0$), but only a massless neutrino component in the form of three massless active neutrinos compatible with the standard model of particle physics, such that $N_{\rm eff} = 3.046$. The overall structure of the grid is exemplified in Figure \[fig\_grid\_structure\]. Moving from the BG fiducial simulation and considering in turn distinct pairs of parameters ($p_1, p_2$) selected from those of the grid, we produce two runs per parameter altering their reference central values via symmetric positive and negative shifts, indicated as $p_1^{+}, p_2^{+}, p_1^{-}, p_2^{-}$ in the figure – and marked with red triangles or blue squares, respectively. These realizations are useful for evaluating second-order derivatives in the Taylor expansion of the flux. We then carry out an additional joint run where both $p_1$ and $p_2$ are (simultaneously) positively altered, namely $p_1^+ p_2^+$ (upper right corner in the figure), in order to evaluate their cross-derivative terms, which also enter in the Taylor expansion of the flux. This procedure is repeated for all the possible sets of parameter pairs describing the grid; hence, using $n$ to indicate the number of parameters, the comprehensive simulations required to compute the Taylor expansion coefficients are \[$1+2n+n(n-1)/2]$. Note also that all of the derivatives are approximated to second order, except for the cross-terms, which are of the first order; else, we would need $[n(n - 1)/2]$ supplementary simulations. Essentially, the structure of the grid is determined by positive and negative variations of the key parameters around their reference values, plus all the possible combinations of pairs having both positive shifts. This means that a mapping defined by 11 parameters requires $76$ simulations to cover the entire space – excluding two unphysical runs characterized by negative mass variations (i.e., $\sum m_{\nu}^{-}$ and $m^{-}_{\rm WDM}$). In practice, the astrophysical parameters $\tau_{\rm A}$ and $\tau_{\rm S}$ are only varied during post-processing, thus reducing the effective total number of required simulations. The chosen variations of the cosmological and astrophysical grid parameters are reported in Table \[table\_grid\_params\_variations\]. These fluctuations are tighter than those allowed in our previous simulation grid, and more stringent in terms of cosmology – in order to avoid wider excursions in their values, which may lead to interpolation errors and eventually impact parameter constraints. Specifically, we consider changes of $\pm 0.018$ in $n_{\rm s}$, variations of $\pm 0.027$ in $\sigma_8$, variations of $\pm 0.036$ in $\Omega_{\rm m}$, while we alter the Hubble constant $H_0$ by $\pm 2.7$. The number of effective neutrino species $N_{\rm eff}$ is varied by $\pm 1$ with respect to its canonical value of $3.046$. For massive neutrinos, we assume a central total upper mass limit of $\sum m_{\nu} =0.3~{\rm eV}$, while for WDM interpreted as massive sterile neutrinos, we take $m_{\rm WDM} =3.0~{\rm keV}$ as the reference value. Regarding astrophysical parameters, we alter both $T_0$ and $\gamma$, the former by $\pm 5000$ K and the latter by $\pm 0.3$. The parameters $\tau_{\rm A}$ and $\tau_{\rm S}$ are instead changed only in the post-processing phase by $\pm 0.0015$ and $\pm 0.4$, respectively. Regarding interpolation between models, in earlier works (see, e.g., Rossi et al. 2014, 2015; Palanque-Delabrouille et al. 2015) we have investigated and assessed the precision of the classical Taylor expansion of the flux (i.e., Viel et al. 2010; Wang et al. 2013) resulting from an analogous simulation grid. In particular, by analyzing simulations for several sets of parameters with values different from those used in the grid and comparing the power spectrum derived to the one predicted by the Taylor expansion, we found good agreement – as long as the tested parameters remained within the range of values used to compute the expansion. For these novel simulations, the variations of the grid parameters are less extreme compared to our previous runs, particularly in terms of $n_{\rm s}$, $\sigma_8$ and neutrino mass (which are expected to affect $P^{\rm 1D}$ more significantly), in order to improve the accuracy of the interpolation between models. For every cosmological scenario defined by an individual grid parameter or a grid parameter pair, our standard simulation set consists of three realizations (rather than just one), characterized by different box sizes and numbers of particles; their combinations determine the lowest and highest $k$-modes that can be resolved in terms of power spectra, our major observables. This setting is motivated by the application of the [*splicing*]{} technique proposed by McDonald (2003), which allows us to considerably reduce computational time by avoiding the production of single high-resolution and computationally prohibitive numerical simulations (see in particular Section \[subsec\_resolution\_requirements\] for more details on this aspect). Hence, in effect we always generate one run having the largest box size and the lowest resolution considered, a second run characterized by the smallest box size and the highest resolution adopted, and a third run with the resolution of the first simulation and the box size of the second one. In this way, we are able to correct the larger-box simulation for the lack of resolution and the small box for the lack of nonlinear coupling between the highest and lowest $k$-modes, virtually mimicking an equivalent single larger-box highest-resolution (and more expensive) simulation. We chose to adopt similar box-size settings as in Rossi et al. (2014), while increasing the number of particles in the various realizations – since extensive convergence and resolution tests in support of our setup have already been carried out (see Rossi et al. 2014, and references therein). Specifically, we assume a box size of $100~h^{-1} {\rm Mpc}$ for large-scale power with a number of particles per component $N_{\rm p} = 832^3$, and a box size of $25~h^{-1}{\rm Mpc}$ for small-scale power, in this case with $N_{\rm p} = 832^3$ or $208^3$, respectively. In a series of works, we have already proven analogous choices to be considerably successful in constraining the summed neutrino mass and the amount of dark radiation, and in accurately reproducing the measured Ly$\alpha$ flux power spectrum. Thanks to these new simulations and via *splicing*, we are thus able to achieve an effective resolution up to $3 \times 3328^3 = 110$ billion particles in a $(100 h^{-1} {\rm Mpc})^3$ box size, while abating computational expenses: this effective resolution is ideal for reproducing all of the Ly$\alpha$ forest key features at the quality of eBOSS, and also for upcoming DESI high-$z$ quasar observations –see also Section \[subsec\_resolution\_requirements\] for more details on resolution requirements. By construction, all of the grid simulations have been tuned to have $\sigma_8$ at $z=0$ matching the corresponding BG central value, given by the Planck 2015 baseline cosmology as indicated in Table \[table\_baseline\_params\_sims\]. This normalization choice, namely $\sigma_8$ at the present epoch consistent with the value derived in the reference model, differs from the convention adopted for most of the [*Supporting Suite*]{} realizations; we will return to this aspect in the next subsection. Moreover, the grid simulations have been performed with the same random seed, starting at $z = 33$ with initial conditions fixed by 2LPT, and carried out until $z=2.0$; output snapshots are created at regular $\Delta z = 0.2$ intervals, within the range $5.0 \le z \le 2.0$. The full list of [*Grid Simulations*]{} produced is shown in Tables \[table\_grid\_sims\_base\] and \[table\_grid\_sims\_cross\_terms\]. Specifically, Table \[table\_grid\_sims\_base\] reports the central reference model (BG) and all the runs involving variations in individual parameters that define the grid – as indicated in Table \[table\_grid\_params\_variations\] and exemplified in Figure \[fig\_grid\_structure\] using the symbols $p^{\pm}$. Table \[table\_grid\_sims\_cross\_terms\] contains the cross-terms comprising all of the allowed combinations, for the cosmological and astrophysical parameters considered in the grid. The tables also display some technical details regarding the various simulations, such as box sizes, resolutions (number of particles $N_{\rm p}$ per type), mean particle separation, and gravitational softening, along with essential information on neutrino and WDM masses ($\sum m_{\nu} \equiv M_{\nu}$; $m_{\rm WDM}$), the number of effective neutrino species $N_{\rm eff}$, and the value of $\sigma_8$. 2.0pt Simulation Name $M_{\rm \nu}$ \[eV\] $m_{\rm WDM}$ \[keV\] $N_{\rm eff}$ $\sigma_8(z=0)$ Boxes \[Mpc/h\] $N_{\rm p}^{1/3}$ Mean Part. Sep. \[Mpc/h\] Softening \[kpc/h\] ---------------------------------------- ---------------------- ----------------------- --------------- ----------------- ----------------- ------------------- --------------------------- --------------------- $n_{\rm s}^+\sigma_8^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+\Omega_{\rm m}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+H_{\rm 0}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+\sum m_{\nu}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+N_{\rm eff}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+m_{\rm WDM}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+T_0^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm s}^+\gamma^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm S}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $n_{\rm S}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+\Omega_{\rm m}^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+H_{\rm 0}^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+\sum m_{\nu}^+$ a/b/c 0.3 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+N_{\rm eff}^+$ a/b/c 0.0 0.0 4.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+m_{\rm WDM}^+$ a/b/c 0.0 3.0 3.0466 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+T_0^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_8^+\gamma^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_{\rm 8}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sigma_{\rm 8}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.842 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+H_{\rm 0}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+\sum m_{\nu}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+N_{\rm eff}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+m_{\rm WDM}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+T_0^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+\gamma^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\Omega_{\rm m}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+\sum m_{\nu}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+N_{\rm eff}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+m_{\rm WDM}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+T_0^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+\gamma^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $H_{\rm 0}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+N_{\rm eff}^+$ a/b/c 0.3 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+m_{\rm WDM}^+$ a/b/c 0.3 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+T_0^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+\gamma^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+\tau_{\rm A}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\sum m_{\nu}^+\tau_{\rm S}^+$ a/b/c 0.3 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+m_{\rm WDM}^+$ a/b/c 0.0 3.0 4.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+T_0^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+\gamma^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $N_{\rm eff}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 4.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $m_{\rm WDM}^+T_0^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $m_{\rm WDM}^+\gamma^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $m_{\rm WDM}^+\tau_{\rm A}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $m_{\rm WDM}^+\tau_{\rm S}^+$ a/b/c 0.0 3.0 3.0466 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $T_0^+\gamma^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $T_{\rm 0}^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $T_{\rm 0}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\gamma^+\tau_{\rm A}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\gamma^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 $\tau_{\rm A}^+\tau_{\rm S}^+$ a/b/c 0.0 0.0 3.0460 0.815 25/25/100 208/832/832 0.12019/0.03005/0.12019 4.01/1.00/4.01 \[table\_grid\_sims\_cross\_terms\] {width="90.00000%"} {width="90.00000%"} Figure \[fig\_visualization\_1\] shows visual examples of full-size projections at $z =2.0$ of the density field along the $x$- and $y$-directions (and across $z$) from one of the BG reference simulations, having a box size of $25h^{-1}{\rm Mpc}$ and the resolution defined by $N_{\rm p} = 832^3$ particles per type – as reported in Table \[table\_grid\_sims\_base\]. Specifically, the left panel displays the gas distribution, the central panel is for the DM distribution, and the right panel shows both components over the entire extension of the simulation box. Figure \[fig\_visualization\_1\_bis\] is a progressive zoom into the BG gas distribution as seen in the left panel of Figure \[fig\_visualization\_1\], when the original box size is $25 h^{-1} {\rm Mpc}$ and $N_{\rm p} = 832^3$ particles per type. The first inset is a $\sim 2.7 \times 5.4$ (Mpc/h)$^2$ area out of the full $25 \times 25$ (Mpc/h)$^2$ projected surface, while the second inset considered in the central panel – and fully visualized in the right one – is an enlargement of a $\sim 1.2 \times 3.9$ (Mpc/h)$^2$ patch containing a rich and complex structure. The main point of this visualization is to show that the resolution quality of the *Grid Suite* allows one to resolve quite accurately small-scale structures, even below megaparsec (Mpc) scales. Noticeably, the grid contains models with massive neutrinos (indicated as $\sum m_{\nu}^{+}$), WDM (termed with $m_{\rm WDM}^{+}$), and dark radiation (labeled as $N_{\rm eff}^{+}$). See Section \[subsec\_nu\_dark\_rad\_wdm\_modeling\] for technical implementation details. We just recall here that when we include massive neutrinos, we always keep $\Omega_{\Lambda}+ \Omega_{\rm m}$ fixed to give a flat geometry (i.e., $\Omega_{\rm tot} = 1$ with $\Omega_{\rm m} = \Omega_{\rm b} + \Omega_{\rm c} + \Omega_{\nu}$), and vary the additional massive neutrino component $\Omega_{\nu}$ to the detriment of the DM component $\Omega_{\rm c}$. Moreover, for the first time, we also modeled consistently combined scenarios with massive neutrinos and WDM (indicated as $\sum m_{\nu}^{+} m_{\rm WDM}^{+}$ in Table \[table\_grid\_params\_variations\]), scenarios with massive neutrinos and dark radiation ($ \sum m_{\nu}^{+} N_{\rm eff}^{+}$), and cosmologies with dark radiation and WDM ($ N_{\rm eff}^{+} m_{\rm WDM}^{+}$): their implementation in the $N$-body setting is nontrivial, and it is also briefly addressed in Section \[subsec\_nu\_dark\_rad\_wdm\_modeling\]. {width="90.00000%"} {width="90.00000%"} Finally, Figure \[fig\_visualization\_3C\] shows a comparison between the BG run (top panels) and a massive neutrino cosmology realization characterized by a total neutrino mass of $\sum m_{\nu} =0.3$ eV (bottom panels – run denoted as $\sum m_{\nu}^{+}$ in Table \[table\_grid\_sims\_base\]). Similarly as in Figures \[fig\_visualization\_1\] and \[fig\_visualization\_1\_bis\], the plot displays full-size projections of the density field at $z =2.0$ along the $x$- and $y$-directions (and across $z$). All of the components are now shown simultaneously, namely, gas, DM, and massive neutrinos – whenever present. The resolution of the simulations is $N_{\rm p} = 832^3$ particles/type, over a $25h^{-1}{\rm Mpc}$ box size. From left to right, there is a progressive enlargement focused on two square patches: the first is an $\sim 8.0 \times 8.0$ (Mpc/h)$^2$ zoom (middle panels), and the second represents a $3.3 \times 3.3$ (Mpc/h)$^2$ amplification (right panels). In the bottom panels, the neutrino component is explicitly rendered using a white/brighter color scale, in order to make the overall details of the neutrino clustering visually appreciable. In actuality, because of significant free streaming, the impact of massive neutrinos on the gas and DM distribution is a relatively small effect that is hard to detect and, hence, not easily distinguishable from the BG scenario – at least visually. However, as shown in Rossi (2017), the growth of structures is less evolved in the simulation with neutrinos (i.e., the voids are less empty) since their suppressed clustering slows down the growth of perturbations in the overall matter density, and this in turn does affect the properties of the gas and DM. Hence, the presence of massive neutrinos does induce detectable changes in the thermal state of the gas and in the LSS clustering of DM, impacting both the properties of the gas as well as the overall DM clustering features. In particular, variations in the gas thermal state are relevant for the well-known $T_0-\gamma$ power-law relation, which arises from the competition between photo-heating and cooling due to the adiabatic expansion of the universe, following reionization. Typical neutrino fluctuations at the largest scales are about $10\%$ around the mean, while for gas and DM, the fluctuations are usually much stronger. The *Supporting Suite*: Massive Neutrinos, Dark Radiation, Warm Dark Matter {#subsec_supporting_suite} --------------------------------------------------------------------------- 2.0pt Simulation Name $M_{\rm \nu}$ \[eV\] $N_{\rm eff}$ $\sigma_8(z=0)$ Boxes \[Mpc/h\] $N_{\rm p}^{1/3}$ Mean Par. Sep. \[Mpc/h\] Softening \[kpc/h\] ---------------------------------------- ---------------------- --------------- ----------------- ----------------- ------------------- -------------------------- --------------------- Best Guess Grid a/b/c 0.0 3.046 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 BG\_NORM a/b/c 0.0 3.046 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 BG\_UN a/b 0.0 3.046 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_NORM 01 a 0.1 3.046 0.8150 25 256 0.0976 3.25 NU\_NORM 02 a 0.2 3.046 0.8150 25 256 0.0976 3.25 NU\_NORM 03 Grid a/b/c 0.3 3.046 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 NU\_NORM 03 a 0.3 3.046 0.8150 25 256 0.0976 3.25 NU\_NORM 04 a 0.4 3.046 0.8150 25 256 0.0976 3.25 NU\_UN 01 Grid-Like a/b/c 0.1 3.046 0.7926 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 NU\_UN 01 a/b/c 0.1 3.046 0.7926 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 NU\_UN 02 a 0.2 3.046 0.7674 25 256 0.0976 3.25 NU\_UN 03 Grid-Like a/b/c 0.3 3.046 0.7423 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 NU\_UN 03 a/b/c 0.3 3.046 0.7423 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 NU\_UN 04 a 0.4 3.046 0.7179 25 256 0.0976 3.25 DR\_NORM BG a 0.0+s 4.046 0.8150 25 256 0.0976 3.25 DR\_NORM 01 a 0.1+s 4.046 0.8150 25 256 0.0976 3.25 DR\_NORM 02 a 0.2+s 4.046 0.8150 25 256 0.0976 3.25 DR\_NORM 03 Grid a/b/c 0.3+s 4.046 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 DR\_NORM 03 a 0.3+s 4.046 0.8150 25 256 0.0976 3.25 DR\_NORM 04 a 0.4+s 4.046 0.8150 25 256 0.0976 3.25 DR\_UN BG a 0.0+s 4.046 0.7583 25 256 0.0976 3.25 DR\_UN 01 Grid-Like a/b/c 0.1+s 4.046 0.7375 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 DR\_UN 01 a/b/c 0.1+s 4.046 0.7375 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 DR\_UN 02 a 0.2+s 4.046 0.7140 25 256 0.0976 3.25 DR\_UN 03 Grid-Like a/b/c 0.3+s 4.046 0.6908 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.00/1.00/4.00 DR\_UN 03 a/b/c 0.3+s 4.046 0.6908 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 DR\_UN 04 a 0.4+s 4.046 0.6682 25 256 0.0976 3.25 BG\_VIS\_NORM a 0.0 3.046 0.8150 25 208 0.1202 4.01 NU\_VIS\_NORM 03 a 0.3 3.046 0.8150 25 208 0.1202 4.01 NU\_VIS\_UN 03 a 0.3 3.046 0.7423 25 208 0.1202 4.01 DR\_VIS\_UN 03 a 0.3+s 4.046 0.6908 25 208 0.1202 4.01 \[table\_supporting\_sims\_dark\_rad\] The hydrodynamical [*Supporting Suite*]{} has been developed to characterize the physics of the small-scale high-$z$ cosmic web in several nonstandard cosmological scenarios, in order to identify unique signatures and preferred scales where such models may differ significantly from the baseline $\Lambda$CDM framework. In particular, we focus here on cosmologies with massive neutrinos, dark radiation, and WDM – along with their combinations, which represent a novelty in the literature. When dark radiation is included, it is implemented in the form of a massless sterile neutrino thermalized with active neutrinos. Regarding WDM, we only consider thermal relics – recalling that there is a direct correspondence between thermal relics and massive sterile neutrinos. For the various implementation details within the SPH formalism, see Section \[subsec\_nu\_dark\_rad\_wdm\_modeling\]. The primary aim in developing these runs is to carefully study and quantify the effects of neutrinos, dark radiation, and WDM on structure formation at small scales, with a closer eye on Ly$\alpha$ forest observables and on the high-$z$ cosmic web. This is an essential step for obtaining robust parameter constraints free from systematic biases. In what follows, we briefly highlight the main aspects of the [*Supporting Suite*]{}, while we will present detailed analyses of these simulations in forthcoming studies. Table \[table\_supporting\_sims\_dark\_rad\] lists all of the [*supporting*]{} simulations related to massive neutrinos and dark radiation, along with corresponding details such as resolution, box size, mean particle separation, gravitational softening length, values of $\sigma_8$ at $z=0$, neutrino mass, and $N_{\rm eff}$. As specified in the table, we consider different box sizes and resolutions, varying from $25h^{-1}$Mpc to $100h^{-1}$Mpc, and a maximum number of particles characterized by $3 \times 832^3$. We run a total of 56 simulations for this subset, including BG runs. Aside from the standard reference cosmology, all these realizations contain different degrees of summed neutrino mass (runs abbreviated with ‘NU’) and additional dark radiation contributions (runs indicated as ‘DR’), the latter in the form of a massless sterile neutrino thermalized with active neutrinos. The presence of a sterile neutrino is denoted by ‘$s$’ in Table \[table\_supporting\_sims\_dark\_rad\]. Specifically, while our central model has only a massless neutrino component (i.e., $\sum m_{\nu}=0.0~{\rm eV}, N_{\rm eff}=3.046$), the other scenarios incorporate three degenerate massive neutrinos with $\sum m_{\nu} =0.1, 0.2, 0.3, 0.4 $ eV, respectively, and whenever indicated, they also contain an additional massless sterile neutrino – so that $N_{\rm eff}=4.046$. The runs with the extra label ‘VIS’ are small-box simulations only made for visualization purposes, and carried out until $z=0$ with increased redshift outputs. In terms of normalization, all the realizations labeled as ‘NORM’ share the convention highlighted in Section \[sec\_sejong\_suite\_general\]; namely, $\sigma_8$ is rescaled so that its value at $z=0$ is consistent with that of the reference cosmology (BG) at the present epoch. Instead, all the runs termed ‘UN’ have the same primordial amplitude value $A_{\rm s}$ of the baseline model, but they differ in terms of $\sigma_8$ at $z=0$. This subset of [*supporting*]{} simulations has already been used in Rossi (2017) for studies related to Ly$\alpha$ forest observables, and in particular, to accurately measure the tomographic evolution of the shape and amplitude of the small-scale matter and flux power spectra in massive neutrino and dark radiation cosmologies, and to characterize the corresponding thermal state of the IGM through the temperature-density relation – seeking for unique signatures at small scales (see their work for extensive details). {width="95.00000%"} Figure \[fig\_supporting\_suite\_visualizations\_neutrinos\_A\] is a visual example of the LSS cosmic web as seen in the neutrino component alone from a [*supporting*]{} simulation with a $25h^{-1}$ Mpc box size and resolution $N_{\rm p} = 832^3$ particles per type. The plot displays the density projection at $z = 2.0$, when the total summed neutrino mass is $\sum m_{\nu} = 0.3$ eV. From left to right, the progressive enlargements show an $\sim 8.0 \times 8.0$ (Mpc/h)$^2$ square patch (middle panel), and a $3.3 \times 3.3$ (Mpc/h)$^2$ square patch (right panel). While neutrinos do not cluster at small scales because of their high free-streaming velocities, their presence does induce noticeable changes in the global thermal state of the gas, and in the LSS clustering of DM – with a measurable impact on the high-$z$ cosmic web. It is also quite interesting to notice – even just visually – how neutrinos trace the overall structure of the cosmic web. 2.0pt Simulation Name $M_{\rm \nu}$ \[eV\] $m_{\rm WDM}$ \[keV\] $N_{\rm eff}$ $\sigma_8(z=0)$ Boxes \[Mpc/h\] $N_{\rm p}^{1/3}$ Mean Par. Sep. \[Mpc/h\] Softening \[kpc/h\] -------------------------------------------- ---------------------- ----------------------- --------------- ----------------- ----------------- ------------------- -------------------------- --------------------- Best Guess Grid a/b/c 0.0 0.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 Best Guess a/b/c 0.0 0.00 3.0460 0.8150 25/25/100 128/256/512 0.1953/0.0976/0.1953 6.51/3.25/6.51 BG\_UN a/b/c 0.0 0.00 3.0460 0.8150 25/25/100 128/256/512 0.1953/0.0976/0.1953 6.51/3.25/6.51 WDM$^{*}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDM$^{*}$\_0.25\_keV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM$^{\dagger}$\_UN\_0.25\_keV a 0.0 0.25 3.0460 0.8305 25 128 0.1953 6.51 WDM$^{\dagger}$\_0.25\_keV a 0.0 0.25 3.0460 0.8150 25 128 0.1953 6.51 WDM\_UN\_0.25\_keV a 0.0 0.25 3.0617 0.8269 25 128 0.1953 6.51 WDM\_0.25\_keV a 0.0 0.25 3.0617 0.8150 25 128 0.1953 6.51 WDM$^{\dagger}$\_1.00\_keV a/b/c 0.0 1.00 3.0460 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM$^{\dagger}$\_2.00\_keV a/b 0.0 2.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV a/b 0.0 3.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM$^{\dagger}$\_3.00\_keV Grid-Like a/b/c 0.0 3.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM$^{\dagger}$\_4.00\_keV a/b 0.0 4.00 3.0460 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM\_1.00\_keV a/b/c 0.0 1.00 3.0485 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 WDM\_2.00\_keV a/b 0.0 2.00 3.0469 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM\_3.00\_keV a/b 0.0 3.00 3.0466 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 WDM\_3.00\_keV Grid a/b/c 0.0 3.00 3.0466 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 WDM\_4.00\_keV a/b 0.0 4.00 3.0464 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_01\_WDM\_3.00\_keV a/b/c 0.1 3.00 3.0466 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 NU\_02\_WDM\_3.00\_keV a/b 0.2 3.00 3.0466 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_03\_WDM\_3.00\_keV a/b 0.3 3.00 3.0466 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_03\_WDM\_3.00\_keV Grid a/b/c 0.3 3.00 3.0466 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 NU\_04\_WDM\_3.00\_keV a/b 0.4 3.00 3.0466 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_03\_WDM\_1.00\_keV a/b/c 0.3 1.00 3.0484 0.8150 25/100/100 256/512/832 0.0976/0.1953/0.1202 3.25/6.51/4.01 NU\_03\_WDM\_2.00\_keV a/b 0.3 2.00 3.0469 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 NU\_03\_WDM\_4.00\_keV a/b 0.3 4.00 3.0464 0.8150 25/100 256/512 0.0976/0.1953 3.25/6.51 \[table\_supporting\_sims\_wdm\] Analogously to Table \[table\_supporting\_sims\_dark\_rad\], Table \[table\_supporting\_sims\_wdm\] lists all of the [*supporting*]{} simulations related to WDM and massive neutrinos, along with corresponding details such as resolution, box size, mean particle separation, and gravitational softening length. As explained in Section \[subsec\_nu\_dark\_rad\_wdm\_modeling\], WDM is implemented using two different procedures, as indicated by different names/symbols in the table. Specifically, in one implementation, WDM is included by modifying the linear matter power spectrum to mimic the presence of early decoupled thermal relics; the BG power spectrum is altered at the 2LPT level, by introducing a small-scale cutoff. The cutoff can have different functional forms: we indicate with [*“WDM$^{*}$”*]{} the form originally proposed by Bode, Ostriker & Turok (2001), and with [*“WDM$^{\dagger}$”*]{} the updated version by Viel et al. (2005, 2012, 2013). The considered masses of thermal relics are specified explicitly using corresponding names, and expressed in keV. In this implementation, all the CDM is turned into WDM, which is valid only for a nonresonant DW production mechanism. Note again the direct correspondence between the mass of WDM early decoupled relics and the mass of a non-thermalized non-resonant DW sterile neutrino –while the resonant production is more complex. An alternative implementation of early decoupled thermal relics is indicated simply with [*“WDM”*]{} in Table \[table\_supporting\_sims\_wdm\]; also in this case all the DM is warm, since we consider pure WDM particles with suitable candidates such as keV right-handed neutrinos or sterile neutrinos. For this latter modeling, we introduce a shift $\Delta N_{\rm eff}$ from the canonical value $N_{\rm eff}=3.046$ at the level of CAMB, and enforce a neutrino mass splitting if massive neutrinos are included. There is a direct correspondence with a thermal relic $m_{\rm X} \equiv m_{\rm WDM}$ and the DW sterile neutrino of mass $m_{\rm s}$; hence, we only need simulations with thermal relics $m_{\rm X}$, and then $m_{\rm s}$ of a massive sterile neutrino will be readily determined via a direct mapping procedure using $\Delta N_{\rm eff}$, $h$, and $\Omega_{\rm c}$. This latter WDM implementation is also used in our [*Grid Suite*]{} to simulate WDM scenarios, since it is more physically motivated. To this end, note in particular that with this implementation, the value of $N_{\rm eff}$ actually changes according to the addition of $\Delta N_{\rm eff}$, depending on the thermal relic mass. In all of these WDM runs, we consider five values of the relic mass, namely, $m_{\rm WDM} =0.25, 1.00, 2.00, 3.00, 4.00~{\rm keV}$, as well as a range of box sizes and resolutions – as specified in Table \[table\_supporting\_sims\_wdm\]. The runs where massive neutrinos are also present in addition to WDM, indicated as [*NU\_0\#\_WDM*]{} with the label $0\#$ used to specify the total neutrino mass in eV (for example, ‘NU\_03’ corresponds to $\sum m_{\nu} =0.3~{\rm eV}$), are quite interesting and represent a novelty in the literature. Their implementation is nontrivial, as it requires two separate eigenstates and a mass splitting: namely, we model three degenerate massive neutrinos of total mass $\sum m_{\nu}$ in one eigenstate, and a non-thermalized massive sterile neutrino (interpreted as nonresonant WDM relic) in a different eigenstate. For these joint runs, we consider two scenarios: in the first one, we keep the WDM relic mass fixed to be $m_{\rm WDM} =3.00~{\rm keV}$ and vary the total neutrino mass as $\sum m_{\nu} =0.1, 0.2, 0.3, 0.4~{\rm eV}$, respectively. In the second one, we fix the total neutrino mass to be $\sum m_{\nu} =0.3~{\rm eV}$ and allow $m_{\rm WDM}$ to vary as $1.00$, $2.00$, and $4.00~{\rm keV}$, respectively. Note also that we adopt similar normalization conventions as in the massive neutrino/dark radiation *Supporting Suite*. In total, we run 58 [*supporting*]{} simulations related to WDM and massive neutrinos (including BG runs), and their detailed analysis is the subject of a companion publication. {width="49.00000%"} {width="49.00000%"} As a visual example drawn from Table \[table\_supporting\_sims\_wdm\], we show in Figure \[fig\_wdm\_visualization\] a projected $(5.0 \times 6.0)$ $[h^{-1} {\rm Mpc}]^2$ patch from simulations having a box size of $25h^{-1}$ Mpc and a relatively low resolution of $256^3$ particles/type, describing the gas internal energy at $z=2.0$. The left panel displays a complex structure as seen in the BG reference model, while the right panel highlights the same structure as seen in the WDM cosmology when $m_{\rm WDM} = 2.00~{\rm keV}$. Once again, differences are tiny and therefore hard to be detected visually, although the impact of an $m_{\rm WDM} = 2.00~{\rm keV}$ relic on the high-redshift LSS is significant. With 56 simulations describing massive neutrino/dark radiation cosmologies and 58 WDM/massive neutrino realizations, our [*Supporting Suite*]{} consists in total of 114 simulations (including BG reference runs) and represents a valuable resource to study and accurately characterize the high-$z$ cosmic web including the [*dark sector*]{}, as well as baryonic effects at small scales. The *Systematics Suite* {#subsec_systematics_suite} ----------------------- 2.0pt Simulation Name $\sum m_{\nu}$ \[eV\] $N_{\rm eff}$ $\sigma_8(z=0)$ Boxes \[Mpc/h\] $N_{\rm p}^{1/3}$ Mean Par. Sep. \[Mpc/h\] Softening \[kpc/h\] ----------------------------------- ----------------------- --------------- ----------------- ----------------- ------------------- -------------------------- --------------------- Best Guess Grid a/b/c 0.00 3.0460 0.8150 25/25/100 192/768/768 0.1302/0.0325/0.1302 4.34/1.08/4.34 Best Guess Grid a/b/c 0.00 3.0460 0.8150 25/25/100 224/896/896 0.1116/0.0279/0.1116 3.72/0.93/3.72 BG\_CONV a 0.00 3.0460 0.8150 10 128 0.0781 2.60 BG\_CONV b 0.00 3.0460 0.8150 10 256 0.0391 1.30 BG\_CONV c 0.00 3.0460 0.8150 10 336 0.0298 0.99 BG\_CONV d 0.00 3.0460 0.8150 10 512 0.0195 0.65 BG\_A$_{\rm s}^{+}$ a/b/c 0.00 3.0460 0.8684 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 BG\_A$_{\rm s}^{-}$ a/b/c 0.00 3.0460 0.7907 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 BG\_FINER 0.00 3.0460 0.8150 100/100 512/832 0.1953/0.1202 6.51/4.01 BGM 0.06 3.0460 0.8150 25 256 0.0976 3.25 BGT\_UN 0.01 3.0460 0.8150 25 256 0.0976 3.25 NU\_SN\_01 a 0.10 3.0460 0.8150 25 256/512 0.0976/0.0488 3.25/1.63 NU\_SN\_01 b 0.10 3.0460 0.8150 100 512/1024 0.1953/0.0976 6.51/3.25 DR\_SN\_01 a 0.10 4.0460 0.8150 25 256/512 0.0976/0.0488 3.25/1.63 BG\_SEED\_1 0.00 3.0460 0.8150 100 832 0.1202 4.01 BG\_SEED\_2 0.00 3.0460 0.8150 100 832 0.1202 4.01 BG\_SEED\_3 0.00 3.0460 0.8150 100 832 0.1202 4.01 BG\_SEED\_4 0.00 3.0460 0.8150 100 832 0.1202 4.01 BG\_SEED\_5 0.00 3.0460 0.8150 100 832 0.1202 4.01 BG\_PLANCK18 a/b/c 0.00 3.0460 0.8150 25/25/100 208/832/832 0.1202/0.0300/0.1202 4.01/1.00/4.01 NU03\_z33 0.30 3.0460 0.8150 100 832 0.1202 4.01 NU03\_z50 0.30 3.0460 0.8150 100 832 0.1202 4.01 BG High Res 0.00 3.0460 0.8150 100 1024 0.0976 3.25 NU\_01 High Res 0.10 3.0460 0.8150 100 1024 0.0976 3.25 DR\_01 High Res 0.10 4.0460 0.8150 100 1024 0.0976 3.25 \[table\_systematics\_sims\_all\] The third component of the *Sejong Suite* is termed [*Systematics Suite*]{}. This collection of simulations has been performed with the main purpose of studying and characterizing several systematic effects that can impact parameter constraints, which are related to the modeling of additional species (i.e., massive neutrinos, dark radiation, WDM), or simply that are intrinsic to the numerical nature of our simulations. What we denote as ‘systematics’ thus varies from convergence and resolution tests, to numerical artifacts, theoretical degeneracies, and much more. We report here a first set of systematic runs performed, and we will expand along these lines in dedicated forthcoming publications – including full analyses. This first list of realizations serves in fact as a general reference and guideline, since the various systematic runs share similar aspects that are common to all of the other simulations presented here. It will be then convenient for future work to expand around this framework and add more targeted simulations, while using the current setup as the standard ‘backbone’ of the [*Sejong Suite*]{}. Table \[table\_systematics\_sims\_all\] lists our first set of systematic runs, addressing different modeling aspects as we briefly explain next. Specifically, the ‘[*Best Guess Grid*]{}’ simulations are constructed using the same parameters of the BG (which is part of the [*Grid Suite*]{}), but their resolution is different. In the first case, it is lower than that of the grid (with a maximum of $2 \times 768^3$ particles), while in the second case, it is higher (for a maximum of $2 \times 896^3$ particles). These runs allow one to explore the effect of resolution on the various Ly$\alpha$ observables, while maintaining analogous box-size selections. The simulations termed ‘BG\_CONV’ are small-box realizations used for convergence and resolution studies, as explained in the next Section \[subsec\_resolution\_requirements\]. The simulations denoted as “BG\_A$_{\rm s}^{+}$" and “BG\_A$_{\rm s}^{-}$" explore the effects of altering the overall normalization given by the initial amplitude of the primordial fluctuations $A_{\rm s}$, which is well known to be degenerate with a variation in $\sigma_8$, which in turn can mimic a small neutrino mass. The realizations termed “BG\_FINER" are runs where the time-integration steps are forced to be much smaller than what is assumed for the grid simulations, and their overall performance requires almost twice the execution time requested for an equivalent simulation but with grid-like integration steps. These simulations allow one to test the impact of finer time-integration steps on quantities such as the nonlinear matter and flux power spectra, and ultimately the impact on parameter constraints. BGM is a run having a BG-like cosmology setup, but also containing a minimal neutrino mass of $0.06$ eV, which is essentially the baseline model adopted in the Planck 2015 and 2018 analyses. BGT\_UN is a control simulation containing a tiny neutrino mass of $\sum m_{\nu} =0.01$ eV, used to test the consistency and convergence of our neutrino implementation – as done in Rossi et al. (2014). The simulations “NU\_SN\_01" and “DR\_SN\_01" have been developed to study shot-noise (“SN" in simulation names) effects in the presence of massive neutrinos and dark radiation (with an enhanced number of particles for neutrino modeling). In detail, “NU\_SN\_01" contains a total neutrino mass of $\sum m_{\nu}=0.10$ eV and has twice as many particles for modeling the neutrino component as the corresponding gas and DM components ($512^3$ or $1024^3$ when the related DM and gas particles are $256^3$ and $512^3$, respectively). “DR\_SN\_01" is similar to the previous run, but with the addition of a thermalized massless sterile neutrino, so that $N_{\rm eff}=4.046$. The series of “BG\_SEED\_\#" explores the effect of cosmic variance, as the initial random seeds of the simulations are varied with respect to the one used for grid simulations. “BG\_PLANCK18" is a BG-like run, but with Planck 2018 best-fit parameters. “NU03\_z33" and “NU03\_z50" are two realizations sharing the same massive neutrino cosmology with $\sum m_{\nu}=0.3{~\rm eV}$, but differing in the initial redshift. The first realization starts at $z=33$, while the second one begins at $z=50$ –both with 2LPT. Finally, simulations with [*“BG High Res"*]{}, [*“NU\_01 High Res"*]{}, and [*“DR\_01 High Res"*]{} involve high-resolution runs, where the number of particles per species is increased to $1024^3$. Specifically, BG has a BG-like setting, [*NU\_01*]{} has a $\sum m_{\nu} =0.10~{\rm eV}$ mass, and [*DR\_01*]{} has in addition a massless sterile neutrino, so that $N_{\rm eff} =4.046$. These realizations are carried out to evaluate the global computational cost, and to estimate the advantages/disadvantages of such settings. Overall, we performed 35 runs for the first set of [*systematics simulations*]{}. A detailed analysis of this interesting suite – as well as additional systematic runs within this framework, such as effects of active galactic nucleus (AGN) feedback on massive neutrinos, and much more –will be presented in forthcoming dedicated studies. Characterizing systematic effects is in fact an essential component for obtaining unbiased parameter estimates, particularly when constraints are derived from small-scale measurements. Resolution and Box-Size Requirements: Comparisons to Previous Studies {#subsec_resolution_requirements} --------------------------------------------------------------------- {width="85.00000%"} Achieving numerical convergence in the modeling of the Ly$\alpha$ forest is challenging, especially because most of the signal comes from poorly resolved under-dense regions –not necessarily in local hydrostatic equilibrium. Two primary factors are driving the resolution level that simulations need to reach, for an accurate reproduction of the forest: the experimentally attainable Ly$\alpha$ forest range, and the measurement errors on the targeted Ly$\alpha$ forest observables. Clearly, convergence requirements will always depend on the physical process under consideration, as well as on the precision of the observational data with which the simulations are confronted. Moreover, one needs to find an optimal compromise between the simulation box size, the total number of particles used in the runs, and the overall CPU consumption. In particular, assuming the power spectrum as the main targeted observable, the largest $k$-mode achievable ($k_{\rm max}$) is bounded by the Nyquist-Shannon limit –i.e., $k_{\rm Nyquist} = \pi/ \Delta v$, where $\Delta v$ is the pixel width of the measured spectra by a given survey. The smallest $k$-mode ($k_{\rm min}$) is instead driven, at least in principle, by the overall extension of the Ly$\alpha$ forest. In reality, instrumental constraints make it difficult to reach $k_{\rm Nyquist}$, and in addition, one needs a sufficiently large box size $L$ to prevent missing modes. Numerically, $L$ determines the smallest $k$-mode (i.e. $k_{\rm min} = 2 \pi / L$), while the largest $k$-mode depends on the specific computational algorithm adopted. In fact, within the SPH formalism, over-densities (in local hydrostatic equilibrium, thus bounded by the Jeans scale) are sampled with higher spatial resolution than average – with a particle spacing significantly smaller than $L/N$, where $N^3$ is the total number of particles per species. This is not always the case for under-dense regions, often out of local hydrostatic equilibrium, for which convergence tests ensure that simulations are able to resolve the smallest structures in the transverse direction – useful for an accurate estimate of the 1D flux power spectrum. To this end, we have extensively addressed convergence and resolution requirements within our SPH formalism, in order to accurately reproduce the main Ly$\alpha$ forest observables as demanded by state-of-the-art redshift surveys. Based on convergence studies, our conclusions in terms of resolution requirements are challenging, pointing toward computationally expensive runs. For instance, in previous releases (see, e.g., Rossi et al. 2014), we have developed realizations with an equivalent resolution of $N^3=3072^3$ particles per species over a $100 h^{-1}{\rm Mpc}$ box (corresponding to a mean grid resolution of $\sim 33 h^{-1}{\rm kpc}$), a choice that guarantees a power spectrum convergence within $\sim 2.0\%$ for every redshift in the range $2.2 \le z \le 4.4$ and in the $k$-range $1 \times 10^{-3} \le k [{\rm km/s}]^{-1} \le 2 \times 10^{-2} $ (or approximately $0.1 \le k {\rm [{\it h}Mpc^{-1}]} \le 2.0$ at $z \sim 3$). The previous $k$-range represents the optimal coverage provided by the SDSS-III/BOSS and SDSS-IV/eBOSS surveys. This is obtained by assuming a constant pixel width of $\Delta v = 69 {\rm [km/s]^{-1}}$ common for SDSS quasar coadded spectra, and by considering instrumental constraints and additional observational limitations – so that, within a given Ly$\alpha$ forest, the redshift span is $\Delta z =0.2$. Regarding box size, $100h^{-1}{\rm Mpc}$ represents an optimal configuration choice to safely reach $k_{\rm min} = 1.0 \times 10^{-3} {\rm [km/s]^{-1}}$. Having eBOSS and DESI Ly$\alpha$ forest data as our primary target, as well as high-resolution spectra such as those provided by the VLT/XSHOOTER legacy survey (XQ-100; L[ó]{}pez et al. 2016), with the [*Sejong Grid Suite*]{} we have improved the effective resolution of the simulations. In fact, although the Ly$\alpha$ forest range covered by eBOSS is similar to that of BOSS (with an identical pixel width of $\Delta v = 69 {\rm [km/s]^{-1}}$) but spans over 35 modes from $k = 10^{-3}{\rm [km/s]^{-1}}$ to $k = 0.02{\rm [km/s]^{-1}}$ beyond which the SDSS spectrograph resolution cuts the power by a factor of 10, better estimations of the SDSS pipeline noise and a more statistically significant sample of high-$z$ quasars have reduced the measurement errors on Ly$\alpha$ forest observables. Therefore, an increased mass resolution in simulations is necessary to match the quality of these new data. Moreover, DESI is pushing further the observable Ly$\alpha$ forest extension, and it is expected to provide data with almost twice the spectral resolution and a higher signal-to-noise ratio for quasars at $z > 2.1$ compared to eBOSS. In this respect, a minimal requirement to meet current and upcoming high-quality Ly$\alpha$ forest data is being able to cover at least $100h^{-1}{\rm Mpc}$ with an equivalent mean grid resolution of $30 h^{-1}{\rm kpc}$. This conclusion is supported by the convergence study for Ly$\alpha$ flux statistics shown in Figure \[fig\_convergence\], based on the simulations indicated as ‘BG\_CONV’ in Table \[table\_systematics\_sims\_all\] from the [*Systematics Suite*]{}. These realizations are characterized by the same cosmology of the reference BG, but they cover only a small periodic box of $10h^{-1}{\rm Mpc}$. Their mass resolution varies as $N^3=128^3$, $256^3$, $336^3$, and $512^3$ – comparable to an equivalent grid resolution of $78, 39, 30, 20$ $h^{-1}{\rm kpc}$, respectively. The corresponding DM particles at those resolutions are $3.435 \times 10^7$, $4.294 \times 10^6$, $1.899 \times 10^6$, and $5.367 \times 10^5$ $h^{-1}M_{\odot}$; and, in terms of gas mass particles, we have $6.409 \times 10^6$, $8.011 \times 10^5$, $3.543 \times 10^5$, and $1.001 \times 10^5$ $h^{-1}M_{\odot}$. The left panel of Figure \[fig\_convergence\] shows flux power spectra at $z=2, 3, 4$ computed from $10,000$ LOS skewers extracted from all of the 10$h^{-1}{\rm Mpc}$ runs at each targeted redshift. Different line styles and colors refer to the varying mass resolutions, as specified in the plot. The right panels display 1D flux power spectrum ratios for realizations having $N^3=128^3, 256^3, 336^3$, respectively, estimated with respect to the highest-resolution run (i.e. “L10\_N512", having an equivalent grid resolution of $20h^{-1}{\rm kpc}$), expressed in percentage, and with identical colors and line styles as in the left panel. Error bars are 1-$\sigma$ deviations computed from $10,000$ simulated skewers randomly extracted at each redshift from the simulation boxes, and the gray areas highlight the $1\%$ level. As can be clearly seen, 1D flux power spectra obtained from the $N^3=336^3$ realization (i.e. $30h^{-1}{\rm kpc}$ equivalent grid resolution) are within $\sim 1\%$ in the $k$-range covered by eBOSS (up to $k=2 \times 10^{-2} [{\rm km/s}]^{-1}$), and closer to the same degree of accuracy for the expected extension of DESI Ly$\alpha$ forest data. Clearly, achieving sub-percentage convergence at higher-$z$ is progressively challenging, but a grid of $30h^{-1}{\rm kpc}$ is sufficient for a satisfying convergence even at those redshifts. The previous rationale is precisely what has driven the overall architecture of the [*Sejong Grid Suite*]{}, designed to achieve an equivalent resolution up to $3 \times 3328^3$ = 110 billion particles in a $(100h^{-1}{\rm Mpc})$ box, corresponding to a $30 h^{-1}{\rm kpc}$ mean grid resolution that ensures a convergence on Ly$\alpha$ flux statistics closer to the desired $\sim 1.0\%$ level that the final DESI data will provide. Clearly, running a very large number of simulations with $2 \times 3328^3$ or $3 \times 3328^3$ elements over a $(100h^{-1}{\rm Mpc})$ box is still computationally challenging, particularly when neutrinos are included as particles. The global computational cost in performing an entire suite at such resolution would easily require $\sim 100$ million CPU hours. Therefore, as in our previous release, we adopt a splicing technique. In this respect, the three simulations per fixed parameter set of the [*Sejong Grid Suite*]{} should be regarded as one single realization having the equivalent mean grid resolution of $30 h^{-1}{\rm kpc}$ over a $100h^{-1}{\rm Mpc}$ box size. To this end, we note that using simulations from Rossi et al. (2014) with lower resolution than the [*Sejong Grid Suite*]{}, along with splicing (and despite the limitations intrinsic to the splicing method), we were able to provide among the most competitive neutrino mass and dark radiation bounds to date, by combining Ly$\alpha$ forest data with additional tracers and adopting a Taylor expansion approach for the Ly$\alpha$ forest flux. With this new release, other than several technical improvements, we have increased the resolution to achieve a better modeling of the 1D flux power spectrum at high-$z$ (see Figure \[fig\_1d\_flux\_ps\_data\_16D\_paper\]), and therefore, we expect tighter upper bounds on the summed neutrino mass along with more stringent limits on dark radiation. In this context, a very interesting and meticulous study on resolution requirements for the modeling of the Ly$\alpha$ forest has been carried out by Lukic et al. (2015). Their conclusions are also quite stringent, and in particular their findings point to a grid resolution of $20h^{-1}{\rm kpc}$ to reproduce $1\%$ convergence on the Ly$\alpha$ flux statistics up to $k=10 {\rm {\it h}Mpc^{-1}} $, and a box size greater than $40h^{-1}{\rm Mpc}$ to suppress numerical errors to a sub-percent level and overcome missing modes. A direct comparison with Lukic et al. (2015) is not straightforward, since their results are obtained with an Eulerian hydrodynamical code, while our convergence findings are based on SPH-Lagrangian techniques. In fact, within the SPH formalism, the gravitational resolution is much higher for the same grid ($L/N$), providing $\sim 10$ times higher gravitational resolution than the grid codes for an identical grid configuration, while hydrodynamical quantities are smoothed on scales of $\sim 2$ times the mean interparticle spacing for gas around the mean density. Nevertheless, our requirements are consistent with their prescriptions. The simulations of the [*Grid Suite*]{} – useful for parameter constraints via a Taylor expansion model for the flux – should in fact be seen as realizations with $N^3=3328^3$ particles/type over a 100 $h^{-1}{\rm Mpc}$ box, thus spanning a sufficiently large size to guarantee an accurate reproduction of the Ly$\alpha$ flux statistics – in line with their suggested safe choice of $80 h^{-1}{\rm Mpc}$. In terms of mass resolution, the [*Grid Suite*]{} is designed to reach an equivalent mean grid resolution of $30h^{-1}{\rm kpc}$. While this effective grid resolution is larger than the one proposed by Lukic et al. (2015), eBOSS can only map $k$-modes up to $2 {\rm {\it h}Mpc^{-1}} $ and DESI will expand the range (up to $ k \sim 3 {\rm {\it h}Mpc^{-1}} $) but certainly will not reach $k=10{\rm {\it h}Mpc^{-1}} $. Therefore, our effective resolution is able to guarantee convergence on Ly$\alpha$ flux statistics approaching the desired $\sim 1\%$ level within the targeted $k$-range for those surveys. Note also that our $10 h^{-1}{\rm Mpc}$ run with $N^3=512^3$ used for convergence studies (see Table \[table\_systematics\_sims\_all\] and Figure \[fig\_convergence\]) is comparable to their realization termed “L10\_N512", which, according to their results, looks virtually identical to the “L10\_N1024" simulation in terms of flux values, and agrees to better than $1\%$ even beyond $k=0.1{\rm [km/s]^{-1}}$ in terms of flux statistics. Clearly, the constraining power of the Taylor-based approach for modeling the Ly$\alpha$ flux is ultimately limited by the accuracy of splicing. To this end, Lukic et al. (2015) have reported a $5-10\%$ precision level related to this technique, in terms of 1D flux power spectrum. Even in this case, a comparison with their results is not direct. Nevertheless, in our studies we have found the method to be more accurate, and in addition at the level of parameter constraints, the residual biases introduced by splicing are corrected via nuisance parameters, and eventually marginalized over. Finally, we note that the stringent resolution requirements discussed in this section are primarily relevant for the targeted science associated with the [*Grid Suite*]{}, while the [*Supporting*]{} and [*Systematics Suites*]{} are mostly developed for relative comparisons and for the study of systematics, where having high resolution is not essential. With less memory-intensive and more optimized codes becoming available, running larger-volume high-resolution hydrodynamical simulations will be progressively less computationally prohibitive. Hence, either performing a large number of runs at the challenging resolution previously discussed, or abandoning splicing and interpolation techniques and adopting emulator-based strategies will soon be feasible. We aim at extending the [*Sejong Suite*]{} in this direction, and leave the task to future releases. Summary of Available Products {#subsec_list_available_products} ----------------------------- Schematically, all of the products available from the entire [*Sejong Suite*]{} and illustrated in this section are listed below, for ease of accessibility. - Full simulation snapshots (boxes) in Gadget format (type 2), containing all of the species considered (gas, DM, neutrinos whenever present), from $z=5.0$ to $z=2.0$ in intervals of $\Delta z = 0.2$, for all of the models reported in Tables \[table\_grid\_params\_variations\]-\[table\_systematics\_sims\_all\]. - Initial conditions (2LPT) for all the runs. - Initial CAMB total matter power spectra and transfer functions per components for all the runs. - Nonlinear matter power spectra (total and individual components) from $z=5.0$ to $z=2.0$, in intervals of $\Delta z = 0.2$, for all of the models described in Tables \[table\_grid\_params\_variations\]-\[table\_systematics\_sims\_all\]. - Ly$\alpha$ forest skewers/mocks from $z=5.0$ to to $z=2.0$ in intervals of $\Delta z = 0.2$ for all of the models considered, containing information about flux density, column density, gas temperature, velocity, optical depth, and pixel position. At each $z$-interval, we provide 100,000 skewers randomly selected from the simulated boxes for any given cosmology (see Section \[subsec\_postprocessing\] for details). - Particle samples from $z=5.0$ to $z=2.0$ in intervals of $\Delta z = 0.2$ for all of the models considered. For each $z$-interval and selected model, we provide subsets of 100,000 particles. The [*Sejong Suite*]{}: First Results {#sec_sejong_suite_first_results} ===================================== {width="95.00000%"} ![Comparison between the [*‘Best Guess’*]{} Sejong Suite and the Horizon simulations (Dubois et al. 2014, 2016), in terms of total matter power spectrum at $z=3$. \[Top panel\] Total matter power spectrum from the Sejong BG 100/832 run (red dashed line), the Sejong BG 25/832 run (green dashed line), the Horizon run containing full baryonic physics (“HR AGN"; blue long-dashed-dotted line), and the Horizon run lacking AGN physics (“HR NO AGN"; pink short-dashed-dotted line). The corresponding linear and Halofit-based predictions are also shown, with solid and dotted lines, respectively. \[Bottom panels\] Zoom into the $k$-range $1 \le k {\rm {[} {\it h} \rm Mpc^{-1}]} \le 10$, showing that the “HR NO AGN" realization appears to be in reasonably good agreement (within error bars) with the Sejong 25/832 simulation, while the “HR AGN" total matter power spectrum already shows deviations and signs of suppression at $z=3$ due to AGN feedback.[]{data-label="fig_3d_ps_A_bis"}](3d_total_matter_ps_bg_SEJONG_SUITE_z_3_RESOLUTION_EFFECT_HR_PAPER_ZOOM_BIS.pdf){width="46.00000%"} In this section we present a first analysis of the *Sejong Suite*, mainly focused on selected topics related to the matter and flux statistics. In particular, we show that we are able to accurately reproduce the 1D flux power spectrum down to scales $k=0.06~{\rm [km/s]^{-1}}$ as mapped by recent high-resolution quasar data, as well as the thermal history of the IGM. The small-scale 1D flux power spectrum is a key quantity for constructing the Ly$\alpha$ forest likelihood, and an excellent probe for detecting neutrino and dark radiation imprints. Moreover, as shown in Rossi (2017), the IGM at $z \sim 3$ provides the best sensitivity to active and sterile neutrinos. While we discuss here only a few scientific applications, we anticipate detailed studies based on the *Sejong Suite* in forthcoming publications – addressing a variety of aspects, particularly in relation to the [*dark sector*]{}. Matter Power Spectrum: Shape and Tomography {#subsec_3d_matter_ps} ------------------------------------------- {width="85.00000%"} {width="70.00000%"} The 3D total matter power spectrum ($P^{\rm 3D}_{\rm t}$) represents a key cosmological observable, among the main targets of large-volume galaxy surveys. Its overall shape and amplitude across a variety of $k$-ranges and redshift intervals (i.e., tomography) have been widely used in the literature, especially for obtaining cosmological parameter constraints. To this end, Figure \[fig\_3d\_ps\_A\] shows examples of $P^{\rm 3D}_{\rm t}$ as predicted or derived from the BG model – the baseline cosmology of the *Sejong Suite*. Specifically, in the top panels of the figure, solid lines display linear theory expectations, dashed lines with error bars are measurements from simulated *grid* snapshots having $832^3$ particles/type over a $100h^{-1}{\rm Mpc}$ box size (denoted as 100/832), and dotted lines are nonlinear halo model predictions obtained with *Halofit* (Takahashi et al. 2012). The tomographic evolution spanning $z=2.0$, $3.0$, $4.0$ is shown from left to right, respectively, as indicated in the plot. The $1\sigma$ error bars displayed are related to the hydrodynamical measurements; they are estimated by considering the expected statistical (Gaussian) error consisting of sample variance and a Poisson shot-noise contribution (see, e.g., Schneider et al. 2016). In addition, the highlighted green areas refer to the $k$-regime relevant for the Ly$\alpha$ forest as mapped, for example, by eBOSS. Departures from linear theory in that regime are clearly significant, and they cannot be neglected. Indeed, the DM nonlinear clustering as well as its tomographic evolution plays a key role in shaping $P^{\rm 3D}_{\rm t}$. The bottom panels of the same figure show ratios between measurements derived from the high-resolution hydrodynamical simulations and corresponding power spectra predicted by *Halofit* – indicated with solid lines, and expressed in percentages. The extended horizontal colored areas in the plot highlight the $5\%$ scatter level. Not surprisingly, standard halo model predictions neglecting baryonic effects are generally inaccurate, showing deviations up to [**$\sim 10-15\%$**]{}, especially in the Ly$\alpha$ region of interest, for all the redshift intervals considered. Hence, our multicomponent high-resolution realizations are essential – particularly in the Ly$\alpha$ regime, where nonlinear effects and baryonic physics play a crucial role. Modeling the impact of nonlinear late-time baryonic physics is in fact essential for a percent-level knowledge of the total matter power spectrum at scales $k \le 1 {\rm {\it h} Mpc^{-1}}$, which are particularly relevant for Ly$\alpha$ forest and weak lensing surveys. To this end, a number of works in the literature have addressed the topic theoretically or via hydrodynamical simulations, and several groups have quantified the impact of baryons on the matter power spectrum from their runs reporting different results (see, in particular, Chisari et al. 2018; Springel et al. 2018; Schneider et al. 2019). In general, the inclusion of baryons can suppress the matter power spectrum by $10\%$ or more at small scales. However, the precise values and scales relevant for baryonic effects rely on the simulations adopted. And comparing among different simulations is nontrivial, because all of the results depend on the specific cosmological model, resolution, volume, sub-grid baryonic physics, implementation of the hydrodynamics, specifics of the feedback model, and the actual physics treated. In particular, recently Springel et al. (2018) presented a detailed analysis of the impact of baryons on the clustering of galaxies and matter in the IllustrisTNG simulations, a set of cosmological hydrodynamical runs spanning different volumes and physics implementations –with an updated AGN feedback recipe compared to the previous Illustris release (Vogelsberger et al. 2014). Also, Chisari et al. (2018) reported accurate measurements of the total matter power spectrum from the Horizon cosmological hydrodynamical simulations (Dubois et al. 2014, 2016) including the effects of AGN feedback. Their new AGN sub-grid model is similar to the one implemented in the IllustrisTNG, and differs from that of the Illustris simulation. In addition, Chisari et al. (2018) compared their results with the OWLS (van Daalen et al. 2011), EAGLE, Illustris, and IllustrisTNG simulations, and found that the impact of baryonic processes on the total matter power spectrum are smaller at $z=0$ in the Horizon runs. According to their results, while the qualitative behavior of all simulations is essentially similar, with a power suppression at $k \sim 10 {\rm {\it h} Mpc^{-1}}$ due to the effect of AGN feedback on the gas, the exact scale and strength of the suppression varies between them, even up to 30% at $\sim 5 {\rm {\it h} Mpc^{-1}}$. For example, the new IllustrisTNG runs show a much lower impact of baryons on the distribution of matter, with a reduction of the overall amplitude of the effect and a restriction to smaller scales compared to Illustris. Moreover, Springel et al. (2018) found that baryonic effects increase the clustering of DM on small scales and damp the total matter power spectrum on scales up to $k \sim 10 {\rm {\it h} Mpc^{-1}}$ by $\sim 20\%$. Despite all of the differences related to cosmology, resolution, implementation of hydrodynamics, and sub-grid physics, in Figure \[fig\_3d\_ps\_A\_bis\], we attempt a comparison with the Horizon simulations at $z=3$. Specifically, the top panel shows the total matter power spectrum as measured in the Sejong BG 100/832 run (red dashed line), in the Sejong BG realization characterized by $N^3 = 832^3$ particles/type over a $25h^{-1}{\rm Mpc}$ box (denoted as “25/832"; green dashed line), in the Horizon run containing full baryonic physics (termed “HR AGN", blue long dot-dashed line), and in the Horizon run lacking AGN physics (indicated as “HR NO AGN", pink short-dashed-dotted line). The linear and [*Halofit*]{} corresponding predictions are also displayed, with solid and dotted lines, respectively. The bottom panels of the same figure are enlargements in the $k$-range $1 \le k {\rm {[} {\it h} \rm Mpc^{-1}]} \le 10$, showing that the “HR NO AGN" realization appears to be in reasonably good agreement within error bars with the Sejong 25/832 run (as expected, since AGN feedback is switched off in the Sejong simulations), while the ‘HR AGN’ total matter power spectrum already shows deviations and signs of suppression at $z=3$ due to AGN feedback, particularly relevant at smaller scales. Note also that the 25/832 run is in better agreement with the ‘HR NO AGN’ realization than the 100/832 simulation, since its resolution is four times higher, allowing one to resolve small scales more accurately. In general, the effects of baryons on the total matter power spectrum are non-monotonic throughout $z=0-5$ due to an interplay between AGN feedback (if present), gas pressure, and the growth of structure (see again Chisari et al. 2018). As noted in Rossi (2017), while at the linear level and in absence of massive neutrinos, extra dark radiation, and WDM, the global shape of $P^{\rm 3D}_{\rm t}$ is redshift-independent for modes well inside the Hubble radius, this is no longer the case when massive neutrinos or any [*dark sector*]{} particles are introduced. In fact, even at the linear level, their addition induces a scale-dependent distortion of the matter power spectrum shape, along with a combined evolution of the amplitude; the effect is then amplified in a nontrivial way in the nonlinear regime – see their extensive discussion about the ‘spoon-like’ mechanism. In particular, the amplitude and position of the maximal suppression of the ‘spoon-like’ feature caused by neutrinos and dark radiation on $P^{\rm 3D}_{\rm t}$ defines a characteristic nonlinear scale, which can potentially be used to constrain the properties of neutrinos and $N_{\rm eff}$ from LSS observables. Moreover, baryonic effects can mimic neutrino- or dark-radiation-induced suppressions, especially when $\sum m_{\nu} = 0.1$ (a limit that is approaching the normal mass hierarchy regime), and linear theory is unable to capture several key aspects of the small-scale evolution. Therefore, neglecting baryons and using linear theory extrapolations in this regime are incorrect approximations that could potentially mislead cosmological results. Finally, we recall that at $k \sim 5h {\rm Mpc}^{-1}$ the suppression on the matter power spectrum induced by $\sum m_{\nu} = 0.1~{\rm eV}$ neutrinos can reach $\sim 4\%$ at $z\sim 3$ when compared to a massless neutrino cosmology, and $\sim 10\%$ if a massless sterile neutrino is included (Rossi 2017). Next, we turn to the flux statistics and focus on the main Ly$\alpha$ forest observables. Ly$\alpha$ Forest Spectra and Skewers {#subsec_lya_spectra_skewers} ------------------------------------- Mapping the properties of the gas at various redshift slices while considering different combinations of cosmological parameters is an essential step, in order to characterize the complex matter-to-flux relation and eventually compute the bias of Ly$\alpha$ forest quasars. In this view, our high-resolution skewers described in Sections \[subsec\_postprocessing\] and \[sec\_sejong\_suite\_components\] represent a valuable asset, as they allow one to accurately quantify the variations of the gas key properties within a wide range in parameter space – particularly when massive or sterile neutrinos, dark radiation, and WDM are included. Figure \[fig\_LyA\_observables\_A\] shows a number of key physical properties of the gas, as inferred from such Ly$\alpha$ mocks. These skewers, randomly drawn from snapshots at different redshifts, are constructed from $25~h^{-1}{\rm Mpc}$ box simulations that are part of the *Supporting Grid* (see Table \[table\_supporting\_sims\_dark\_rad\]). In the various panels, the normalization is adjusted such that $A_{\rm S}$ is kept fixed as in the fiducial (BG) cosmology. Hence, the values of $\sigma_8$ at the present epoch differ from the Planck 2015 central value, depending on the degree of neutrino mass and dark radiation components; we refer to this convention as ‘UN’ in the previous tables. For display purposes, we consider identical skewers (i.e., LOS drawn with the same random seed) across three different cosmologies, namely, the BG (solid orange lines), a scenario with three degenerate massive neutrinos having $\sum m_{\nu} = 0.3~{\rm eV}$ (dotted green lines), and a model where a massless sterile neutrino is added to the three active massive ones, so that $N_{\rm eff} = 4.046$ (dashed-dotted blue lines). The physical quantities displayed, ordered from top to bottom, are the neutral atomic Hydrogen (HI) density fraction, the HI temperature (in kelvins), the HI peculiar velocity (in km/s), the HI optical depth $\tau_{\rm HI}$, and the HI transmitted flux ${{\cal{F}}_{\rm HI}}$ – where ${\cal{F}}_{\rm HI} = \exp [- \tau_{\rm HI}]$. The redshift evolution is also shown: specifically, in the left panels $z=2.0$, in the central panels $z=3.0$, and the right panels have $z=4.0$. Even visually, variations in all of the key physical properties among the three distinctive cosmologies are clearly noticeable. These differences will eventually manifest in the major Ly$\alpha$ forest observables, and in particular in the 1D flux power spectrum – as well as in the overall structure of the cosmic web traced by the gas component. In addition, note that departures from the BG are more significant when a massless sterile neutrino is included (blue dashed-dotted lines in the panels), and thus the Ly$\alpha$ forest flux is also an excellent high-$z$ probe for detecting hypothetical sterile neutrinos. Figure \[fig\_LyA\_observables\_B\] highlights the effects of our normalization conventions on the HI transmitted flux. As an illustrative example, we consider a massive neutrino cosmology with $\sum m_{\nu} =0.3~{\rm eV}$ (left panel), and a dark radiation scenario having $N_{\rm eff} =4.046$ (right panel). Two different random skewers per model are displayed, at $z=2.0$. Solid black lines are used for the ‘UN’ normalization convention; dotted or dashed colored lines are used for the ‘NORM’ convention, where $\sigma_8(z=0)$ is enforced to be consistent with Planck 2015 expectations for all the realizations. The main point of the plot is to highlight the impact of distinct normalization choices on the HI transmitted flux: differences are visible, and they will eventually manifest in the various Ly$\alpha$ forest observables. Flux Power Spectrum {#subsec_flux_ps} ------------------- ![Examples of 1D flux power spectra as inferred from the [*Sejong Suite*]{}, as well as from recent high-$z$ quasar catalogs. Green triangles are measurements reported by Irsic et al. (2017b) using the VLT/XSHOOTER legacy survey, and orange circles indicate those of Yeche et al. (2017), which also include data from the SDSS-III BOSS survey. Three redshift intervals are displayed: $z=3.2$ (top panel), $z=3.6$ (middle panel), and $z=4.0$ (bottom panel). In the figure, black solid lines show the [*“Best Guess”*]{} [*spliced*]{} flux power spectrum obtained by combining a large-scale power run having a box size of $100~h^{-1} {\rm Mpc}$ (gold dashed lines; $N_{\rm p} = 832^3$/type) with two small-scale power realizations characterized by a $25~h^{-1} {\rm Mpc}$ box size (blue dashed-dotted and red dotted lines; $N_{\rm p} = 832^3$ and $N_{\rm p} = 208^3$ per type, respectively), so that the equivalent resolution is $2 \times 3328^3 \sim 74$ billion particles in a $(100 h^{-1} {\rm Mpc})^3$ box size. All of the synthetic power spectra are averaged over 100,000 mock quasar absorption spectra extracted from individual simulations at the corresponding redshifts. Overall, there is a remarkable consistency between data measurements and predictions from the [*Sejong Suite*]{} high-resolution hydrodynamical simulations.[]{data-label="fig_1d_flux_ps_data_16D_paper"}](1d_flux_ps_bg_data_16Ca_paper.pdf){width="49.00000%"} Next, we consider the 1D flux power spectrum ($P^{\rm 1D}_{{ \cal{F}}}$), a key Ly$\alpha$ forest observable highly sensitive to cosmological parameters, neutrino masses, WDM, and additional dark radiation components such as sterile neutrinos – via significant attenuation effects especially at small scales. The neutrino free streaming induces in fact a characteristic redshift- and mass-dependent suppression of power that affects the properties of the transmitted flux fraction; hence, accurately measuring the full shape, amplitude, and tomographic evolution of the small-scale $P^{\rm 1D}_{{ \cal{F}}}$ is fundamental for inferring key properties on the formation and growth of structures at high-$z$, and for constraining cosmological parameters and the neutrino mass. Because of nonlinearities and baryonic physics, a careful small-scale modeling of $P^{\rm 1D}_{{ \cal{F}}}$ is only possible via sophisticated high-resolution hydrodynamical simulations such as those presented in the *Sejong Suite*. In fact, at small scales the Ly$\alpha$ flux distribution depends on the complex IGM spatial distribution, peculiar velocity field, and thermal properties of the gas, making the connection with the total matter power spectrum $P^{\rm 3D}_{\rm t}$ rather complex. In this view, a detailed knowledge of the gas-to-matter and peculiar velocity biases, of the nature of the ionizing background radiation, and on the fluctuations in the temperature-density relation is necessary. In previous works, using a refined technique that involves a model based on a Taylor expansion of the flux (evaluated numerically), we have already successfully used the information contained in $P^{\rm 1D}_{{ \cal{F}}}$ to obtain among the strongest individual and joint bounds on neutrino masses and $N_{\rm eff}$, exploiting the Ly$\alpha$ forest in synergy with the CMB and low-$z$: perfecting this technique using new simulations from the *Sejong Suite* is the subject of ongoing work. To this end, Figure \[fig\_1d\_flux\_ps\_data\_16D\_paper\] displays a few BG 1D flux power spectra as derived from our [*Grid Suite*]{} (i.e., Table \[table\_grid\_sims\_base\]), confronted with recent $P^{\rm 1D}_{{ \cal{F}}}$ measurements obtained from the SDSS-III BOSS survey and from the VLT/XSHOOTER legacy survey (XQ-100; L[ó]{}pez et al. 2016) – the latter is observed with the X-Shooter spectrograph on the Very Large Telescope (Vernet et al. 2011). In the plot, green triangles refer to the observational measurements reported by Irsic et al. (2017b), while orange circles indicate those of Yeche et al. (2017). Three redshift intervals are considered: namely, $z=3.2$ (top panel), $z=3.6$ (middle panel), and $z=4.0$ (bottom panel), respectively. In all of the panels, black solid lines display the BG [*spliced*]{} flux power spectrum – corresponding to the effective resolution of a single simulation characterized by $2 \times 3328^3 \sim 74$ billion particles in a $(100 h^{-1} {\rm Mpc})^3$ box size for the reference cosmology. The three individual flux power spectra used to derive the spliced $P^{\rm 1D}_{{ \cal{F}}}$ are also shown, with different line styles. Specifically, gold dashed lines refer to the large-scale power run with a number of particles per component $N_{\rm p} = 832^3$ over a box size of $100~h^{-1} {\rm Mpc}$, while blue dashed-dotted and red dotted lines are for the small-scale power realizations (i.e., $25~h^{-1} {\rm Mpc}$ box size) having $N_{\rm p} = 832^3$ and $N_{\rm p} = 208^3$ per type, respectively. Note that all of the different simulated power spectra displayed in the figure are obtained by averaging 100,000 mock quasar absorption spectra extracted from each individual simulation at the corresponding redshift, as explained in Section \[subsec\_postprocessing\], and that the wavevector $k$ is now expressed in ${\rm [km/s]}^{-1}$. Overall, there is a remarkable consistency between data measurements and predictions from our high-resolution hydrodynamical simulations. {width="75.00000%"} {width="34.00000%"} {width="30.90000%"} {width="31.30000%"} As a further insight into the McDonald (2003) [*splicing*]{} technique to overcome resolution demands in simulations, Figure \[fig\_1d\_flux\_ps\_theory\_4\] quantifies the scatter (in percentage) between the BG [*spliced*]{} flux power spectrum and its individual components – assuming the reference cosmology of the [*Sejong Suite*]{}. Line styles and colors are similar to those adopted in Figure \[fig\_1d\_flux\_ps\_data\_16D\_paper\]. Namely, gold dashed lines are used for the $100~h^{-1} {\rm Mpc}$ run with $N_{\rm p} = 832^3$/type, blue dashed-dotted lines show the $25~h^{-1} {\rm Mpc}$ box-size realization having $N_{\rm p} = 832^3$/type, and red dotted lines display the $25~h^{-1} {\rm Mpc}$ box-size simulation with resolution $N_{\rm p} = 208^3$/type. We examine the same redshift intervals as in Figure \[fig\_1d\_flux\_ps\_data\_16D\_paper\] (i.e., $z=3.2, 3.6, 4.0$). The $y$-axis reports the ratios of individual flux power spectrum components, normalized by the [*spliced*]{} $P^{\rm 1D}_{{ \cal{F}}}$ – expressed in percentage. The extended horizontal cyan band highlights the $\pm 5\%$ scatter level. As evident from the figure, the highest-resolution run (blue dashed-dotted lines, $25~h^{-1} {\rm Mpc}$ side, $N_{\rm p} = 832^3$/type) lies within the $5\%$ range, and it is quite accurate at small scales (large $k$), while the larger-box realization ($100~h^{-1} {\rm Mpc}$) generally lacks in resolution. To this end, the ‘transition’ run (i.e., red dotted lines, $25~h^{-1} {\rm Mpc}$ side, $N_{\rm p} = 208^3$/type) is used to correct the larger-box simulation for the lack of resolution, and the small box for the lack of nonlinear coupling between the highest and lowest $k$-modes. Regarding [*dark sector*]{} physics, Figure \[fig\_1d\_flux\_ps\_theory\_10\] shows an example of the tomographic evolution of the small-scale $P^{\rm 1D}_{{ \cal{F}}}$ in the presence of massive neutrinos and dark radiation, as inferred from simulations belonging to the [*Supporting Suite*]{} (Table \[table\_supporting\_sims\_dark\_rad\]). For illustrative purposes, we consider runs with $25~h^{-1} {\rm Mpc}$ box sizes and a resolution $N_{\rm p} = 256^3$/type. From left to right, the redshift intervals examined are $z=2.0, 3.0, 4.0$, respectively. Here, the normalization convention is the one previously indicated as “UN", where $A_{\rm s}$ is fixed as in the reference cosmology – in order to clearly isolate the effects of neutrinos and dark radiation from other possible degeneracies. In dimensionless units, the top panels display $P^{\rm 1D}_{{ \cal{F}}}$ for the BG (black solid line), for a model with three degenerate massive neutrinos having a total summed mass $\sum m_{\nu} = 0.1~{\rm eV}$ (blue dotted line) or $\sum m_{\nu} = 0.3~{\rm eV}$ (cyan dashed line), and for a corresponding dark radiation cosmology with $N_{\rm eff} =4.046$ (red dotted-dashed or orange long dashed-dotted lines, respectively). The evolution of the flux power spectrum shape across different cosmic epochs is clearly seen, along with the characteristic global suppression of power induced by massive and/or sterile neutrinos – more prominent for larger neutrino masses, and more enhanced when sterile neutrinos are also present. The bottom panels display flux power spectrum ratios for the same non-canonical models, normalized by the reference BG simulation – using similar line styles as in the top panels. In this case, all of the flux power spectra measurements are averaged over 10,000 mock quasar absorption spectra, and in the bottom plots, error bars are $1 \sigma$ deviations computed from those simulated skewers at each redshift interval. As thoroughly examined in Rossi (2017), the characteristic signature of massive and/or additional sterile neutrinos on the transmitted Ly$\alpha$ forest flux is a global suppression of power at small scales, manifesting as a typical ‘spoon-like’ feature. This effect is evident in the bottom panels of Figure \[fig\_1d\_flux\_ps\_theory\_10\], where one can clearly appreciate the small-scale $P^{\rm 1D}_{{ \cal{F}}}$ suppression along with its tomographic evolution. Similarly as in Rossi (2017), we confirm that the most significant deviations from the BG reference model occur around $z \sim 3$, with a maximal departure at scales $k \sim 0.005~[{\rm km/s}]^{-1}$ corresponding to $k\sim0.575 ~h {\rm Mpc}^{-1}$ in a Planck 2015 cosmology. These preferred nonlinear scales are characterized by a maximal sensitivity to massive neutrinos and dark radiation in terms of the 1D flux statistics. Luckily, this regime falls in the middle of the standard Ly$\alpha$ forest extension; hence, the Ly$\alpha$ forest has a remarkable potential in constraining neutrino masses and possible deviations from the canonical $N_{\rm eff} = 3.046$ value – especially around $z \sim 3$ – in synergy with complementary lower-$z$ probes. Finally, Figure \[fig\_grid\_coverage\] provides a few selected illustrations of the parameter space coverage, as described by the flux power spectrum inferred from simulations belonging to the [*Grid Suite*]{} (Tables \[table\_grid\_sims\_base\] and \[table\_grid\_sims\_cross\_terms\]). Specifically, the plot shows the effect of changing two individual parameters ($n_{\rm s}$ and $N_{\rm eff}$) and two joint parameter combinations ($n_{\rm s}$ and $H_0$, or $\sum m_{\nu}$ and $m_{\rm WDM} $, respectively) on $P^{\rm 1D}_{{ \cal{F}}}$ (expressed in percentage), with respect to the BG cosmology – as indicated in Table \[table\_grid\_params\_variations\]. The simulations considered are those with the highest resolution, having $832^3$ particles/type over a $25h^{-1}{\rm Mpc}$ box. The various flux power spectra are obtained by averaging 100,000 LOS skewers extracted from the corresponding runs at $z=3.0$, and error bars are 1$\sigma$ deviations, with the colored horizontal band representing the $1\%$ level. Generally, flux variations with respect to the baseline cosmology fluctuate from 1% to 10% or more, depending on the combinations of cosmological and astrophysical parameters, thus covering a sensible range. In particular, the flux statistics is quite sensitive to changes in $n_{\rm s}$, as well as in modifications of the astrophysical parameters. Also, the individual or combined parameter variations are scale-dependent and impact the flux power spectrum differently, allowing us to build a Taylor expansion model of the flux power spectrum. Moreover, note that for all of these realizations, the power spectrum normalization is such that $\sigma_8=0.815$ at $z=0$; this is not the case for the [*Supporting Suite*]{}, where we use instead a different (theoretically motivated) convention. The sensitivity of $P^{\rm 1D}_{{ \cal{F}}}$ under variations of different cosmological and astrophysical parameters has also been addressed in our previous studies. In particular, in Rossi (2017), we have proven that suppressions on the matter – rather than on the flux – power spectrum induced by a $0.1$ eV neutrino are even more significant, and can reach  4% at $z \sim 3$ when compared to a massless neutrino cosmology, and $\sim 10\%$ if a massless sterile neutrino is included. Furthermore, the highest response to free-streaming effects is achieved in the Ly$\alpha$ forest regime, thus making it an ideal place for constraining the neutrino mass. ![Sensitivity of $P^{\rm 1D}_{{ \cal{F}}}$ under variations of individual or combined cosmological parameters (see Table \[table\_grid\_params\_variations\]), as inferred from the [*Grid Suite*]{} for simulations with $832^3$ particles/type over a $25h^{-1}{\rm Mpc}$ box. The various flux power spectra are obtained by averaging 100,000 skewers extracted from the corresponding simulations at $z=3.0$, with error bars showing 1$\sigma$ deviations. The colored horizontal band in the panel represents the $1\%$ level.[]{data-label="fig_grid_coverage"}](PAPER_summary_1dps_parameter_coverage_grid_10000_LOS_A4_BIG.pdf){width="50.00000%"} IGM Properties {#section_igm_properties} -------------- ![Gas mean temperature $T_0$ as inferred from particle subsamples extracted from the [*‘Best Guess’*]{} at different $z$ (black filled dots), contrasted with the well-known measurement by Becker et al. (2011) of the IGM temperature at the mean density when $\gamma = 1.3$ (empty blue dots with error bars). Our central $T_0$ values are consistent with their reported measurements, as well as the robust detection of an increase in $T_0$ between $z=5.0$ and $z=2.0$.[]{data-label="fig_igm_properties_1"}](T0_gamma_relation_paper_1.pdf){width="48.00000%"} {width="31.00000%"} {width="32.50000%"} {width="31.00000%"} Uncertainties in the IGM thermal status are among the main causes of systematics in the flux power spectrum determinations, and therefore accurately reproducing the thermal and ionization history of the IGM is essential for improving the robustness of all Ly$\alpha$-based studies. The IGM probed by the Ly$\alpha$ forest, highly ionized at high-$z$ and progressively neutral with decreasing redshift, consists of mildly nonlinear gas density fluctuations; the gas traces the DM distribution, and it is generally assumed to be in photoionization equilibrium with the UV background produced by galaxies and quasars. In low-density regions, the IGM gas density ($\rho$) and temperature ($T$) are closely connected, via a redshift-dependent polytropic power-law relation. In simple form, the relation can be expressed as a function of redshift as: $$\log T(z) = \log T_0(z) +[\gamma(z) -1] \log \delta(z), \label{eq_IGM_temperature_density_relation}$$ where $T_0$ is the gas mean temperature, $\delta = \rho/ \rho_{\rm c}$, $\rho_{\rm c}$ is the critical density, and $\gamma$ is a redshift-dependent parameter, which is also related to the reionization history model and spectral shape of the UV background. To this end, Figure \[fig\_igm\_properties\_1\] compares the gas mean temperature $T_0$ as inferred from particle subsamples extracted from the BG at different redshifts ([*Grid Suite*]{}, Table \[table\_grid\_sims\_base\]) via computation of the relevant IGM quantities that enter in Equation (\[eq\_IGM\_temperature\_density\_relation\]) using standard SPH techniques, with the well-known measurement by Becker et al. (2011) of the IGM temperature at the mean density when $\gamma = 1.3$. This corresponds to a mild flattening of the $T-\rho$ relation as expected during an extended He II reionization process. In the figure, black filled dots are results from the BG, while empty blue dots with error bars refer to the measurements performed by Becker et al. (2011). Note that, for our baseline cosmology, we set $\gamma(z=3)=1.3$ and $T_0 (z=3)=15000~{\rm K}$ – see Table \[table\_baseline\_params\_sims\]. Moreover, $T_0$ is largely insensitive to $\gamma$ at $z > 4$, as highlighted by the same authors. The main point of the figure is to show the consistency of our central $T_0$ values with their reported measurements, and to confirm the finding of a similar trend – namely, a robust detection of an increase in $T_0$ between $z=5.0$ and $z=2.0$. Measuring $T_0$ as a function of redshift may be informative on the amount of heating or cooling that is occurring for a relatively consistent population of baryons in the IGM. The additional presence of massive neutrinos and exotic particles such as sterile neutrinos (or more generally dark radiation) does affect the status and basic properties of the IGM at high $z$. Potentially, this may also be seen as a source of systematics, or perhaps as a way to detect such particles through their effects on the $T-\rho$ relation, and eventually on the formation of structures at Galactic scales. Hence, quantifying the impact of massive neutrinos and dark radiation on the IGM is important, particularly in relation to galaxy formation studies. Figure \[fig\_igm\_properties\_2\] illustrates the extent of such impact. Specifically, the left panel shows an example of the $T-\rho$ relation at $z=2.0$ as measured from selected high-resolution hydrodynamical simulations from the [*Supporting Suite*]{} (Table \[table\_supporting\_sims\_dark\_rad\]). Aside from the baseline BG model (black solid lines), we consider a neutrino cosmology characterized by three degenerate massive neutrinos with $\sum m_{\nu} =0.3~{\rm eV}$ (orange dashed lines), and a dark radiation scenario with $N_{\rm eff} =4.046$ (cyan dashed-dotted lines). Contour levels in the figure are isodensity surfaces that are $1/2^{\rm n}$ times the height of the maximum value of the density of sources, with $n$ ranging from $1$ to $7$. The straight lines having analogous styles and colors are simple linear fits to simulated data, in order to infer the best-fit values of $T_0$ and $\gamma$ for the three different models. As is evident from the panel, the impact of massive neutrinos and dark radiation on $T-\rho$ is rather small. To this end, the other two panels of Figure \[fig\_igm\_properties\_2\] quantify (in percentage, and normalized by the BG reference cosmology) the tomographic effect in the redshift range $2.0 \le z \le 5.0$ on $T_0$ (middle panel) and $\gamma$ (right panel), due to the presence of massive and sterile neutrinos. Besides the two nonstandard models also shown in the left panel, we display here two more scenarios: a neutrino cosmology with three degenerate massive neutrinos but now with a total mass of $\sum m_{\nu} =0.1~{\rm eV}$ (red dotted lines), and a corresponding dark radiation model with $N_{\rm eff}=4.046$ (blue dashed-dotted lines). In all of the cases, the tomographic variations of $T_0$ and $\gamma$ across the models are always at the sub-percent level, and on average within $0.5\%$ with respect to the reference baseline cosmology; hence, they are hard to detect. Therefore, the temperature-density relation is only marginally affected, showing minor modifications at small scales. {width="90.00000%"} ![Systematics related to the starting redshift choice. Two simulations, one initiated at $z_{\rm in}=50$ and the other at $z_{\rm in}=33$ (both with 2LPT), and characterized by a $100 h^{-1}{\rm Mpc}$ box and $832^3$ particles/type in a massive neutrino cosmology with $\sum m_{\nu}=0.3~{\rm eV}$, are compared in terms of 1D flux statistics – with quantities expressed in power spectrum ratios. The colored horizontal band represents the 0.5% level. Points in the figure are slightly displaced along the $x$-axis, for ease of visibility. Error bars are $1\sigma$ deviations computed from 100,000 skewers extracted at different final $z$-intervals, as indicated in the panel. The starting redshift effect is a sub-percent systematics, impacting the flux statistics at the $\simeq 0.5\%$ level.[]{data-label="fig_starting_z"}](TEST_SUMMARY_1d_nu03_starting_redshift_impact_ALL_RATIOS_F.pdf){width="50.00000%"} Systematic Tests: Examples {#subsec_sys_effects} -------------------------- Finally, we briefly illustrate two examples chosen from the [*Systematics Suite*]{} (see Section \[subsec\_systematics\_suite\] and Table \[table\_systematics\_sims\_all\]), in order to highlight possible sources of systematics – among many – that can impact parameter constraints. The first case, classified as a numerical artifact but also carrying some theoretical implications, is related to the following aspect: the baseline BG cosmology of the [*Grid Suite*]{} does not contain any massive neutrinos, but only three massless active neutrinos as in the current standard model of particle physics, so that $N_{\rm eff}=3.046$. This choice is primarily motivated by the fact that we aim at quantifying and eventually isolating the effects of massive neutrinos on cosmic structures, with respect to a massless neutrino scenario. In Planck 2015 and 2018 cosmological parameter estimation, however, the baseline model adopted assumes a minimal-mass normal hierarchy for the neutrino masses, accurately approximated for current cosmological data as a single massive eigenstate with $M_{\nu} = 0.06~{\rm eV}$. This assumption is consistent with global fits to recent oscillation and other data, but of course is not the only possibility. In this respect, the massive neutrino cosmologies considered in the [*Sejong Suite*]{} – strictly speaking – refer to the degenerate neutrino hierarchy, while our baseline model neglects a minimal-mass normal hierarchy inferred from oscillation experiments. While this latter choice is well-motivated, at the level of parameter constraints, one may want to take into account the implications of our baseline model assumption, and assess how it propagates into the main cosmological observables. To this end, Figure \[fig\_systematics\_minimal\_neutrino\_mass\] shows ratios of total matter (left panel) and Ly$\alpha$ flux (right panel) power spectra (expressed in percentage) as a function of redshift (i.e., $z=2.0, 3.0, 4.0, 5.0$), indicated with different colors and line styles. Specifically, the $y$-axis displays the ratio between quantities evaluated in a baseline Planck-like model having a neutrino minimal-mass normal hierarchy of $M_{\nu} =0.06~{\rm eV}$ (simulation denoted as ‘BGM’ in Table \[table\_systematics\_sims\_all\]) versus our BG massless neutrino reference scenario, as a function of $k$. As is evident from the figure, in the Ly$\alpha$ forest regime of interest, deviations in terms of the total matter power spectrum are within $1\%$, while in terms of the 1D flux power spectrum, they remain within $0.3\%$. Therefore, one may want to account for this effect when presenting cosmological constraints. Another relevant source of potential systematics, also classified as a numerical artifact, is the starting redshift ($z_{\rm in}$) of the simulations – see e.g. Crocce et al. (2006). In most of our runs, we set initial conditions at $z_{\rm in}=33$ with 2LPT. This choice has been determined by taking into account a number of factors, in particular, the overall targeted accuracy, the computational cost in performing a large number of runs, and more importantly possible shot-noise effects introduced by the neutrino component. In fact, if the starting redshift is too high, the neutrino shot noise becomes too severe due to high thermal velocities related to the particle implementation (see Viel et al. 2010, as well as later studies from the same authors for further details). In previous works, we have extensively investigated the impact of this assumption. For analogous reasons, in Rossi et al. (2014) we started our runs at an even slightly lower redshift ($z_{\rm in}=30$) and demonstrated that our simulations are able to correctly recover the 1D Ly$\alpha$ flux power spectrum in the most relevant redshift range of interest for the Ly$\alpha$ forest ($2.2 \le z \le 4.2$, at scales up to $k=0.02$ \[km/s\]$^{-1}$), and proved those simulations to be very powerful in constraining massive neutrinos and dark radiation. Along these lines, Figure \[fig\_starting\_z\] shows the effects of systematics associated with the choice of the initial redshift. A run started at $z_{\rm in}=50$ with 2LPT and characterized by a $100 h^{-1}{\rm Mpc}$ box and $832^3$ particles per type in a massive neutrino cosmology ($\sum m_{\nu}=0.3~{\rm eV}$, indicated as NU03\_z50 in Table \[table\_systematics\_sims\_all\]), is confronted with a similar realization, having initial conditions set at $z_{\rm in}=33$ (simulation NU03\_z33 in Table \[table\_systematics\_sims\_all\]), in terms of 1D flux power spectrum statistics. The colored horizontal band represents the 0.5% level, with quantities expressed in power spectrum ratios; points at the same (final) redshift in the figure are slightly displaced along the $x$-axis, for ease of visibility. Error bars are $1-\sigma$ deviations computed from 100,000 simulated skewers extracted at different final $z$-intervals ($z=2.0, 3.0, 4.0, 5.0$, respectively). As inferred from the plot, in the range relevant for our constraints (for example, considering eBOSS data), the starting redshift effect is a sub-percent systematics, impacting the flux statistics at the $\simeq 0.5\%$ level, and thus always below the splicing uncertainty. At the stage of cosmological constraints, however, this effect can also be incorporated via a nuisance parameter, and eventually marginalized over. Summary and Conclusions: Novelties, Applications, and Outlook {#sec_conclusions} ============================================================= The Ly$\alpha$ forest has recently gained considerable attention as a unique tracer of the high-redshift cosmic web, complementary to lower-$z$ probes, with its statistical power greatly enhanced thanks to available data from the SDSS. In particular, the Ly$\alpha$ forest will play a leading role in unveiling the *dark sector* of the universe, as it is highly sensitive to neutrino masses, WDM, and additional dark radiation components such as sterile neutrinos – via significant attenuation effects on the matter and flux power spectra at small scales. Current and upcoming surveys (i.e., eBOSS, DESI) are or will be providing new exquisite high-quality data suitable for Ly$\alpha$ forest studies, potentially carrying novel discoveries along with the possibility of obtaining competitive cosmological parameter and [*dark sector*]{} constraints. Interpreting such statistically rich datasets demands equally high-quality numerical simulations: reliably modeling nonlinear evolution and the complex effects of baryons, neutrinos, and dark radiation at small scales as required by the Ly$\alpha$ forest is only possible via sophisticated high-resolution cosmological hydrodynamical simulations. Moreover, realistic numerical simulations able to reproduce observational surveys are also indispensable for controlling systematics that can spoil parameter constraints. A complete hydrodynamical treatment is mandatory to reach the precision that data are now beginning to show. And the most competitive bounds on neutrino and WDM masses will be obtained by including the key contribution of Ly$\alpha$ forest data; being able to rule out one of the neutrino hierarchies would have a relevant impact in particle physics, as its knowledge will complete the understanding of the neutrino sector and shed light onto leptogenesis, baryogenesis, and the origin of mass – with implications for neutrinoless double beta decay and double beta decay experiments. Within this context, and inspired by all of these reasons, we have carried out an extensive set of state-of-the-art high-resolution cosmological hydrodynamical simulations (over 300 runs) termed the *Sejong Suite*, primarily developed for modeling the Ly$\alpha$ forest in the redshift interval $5.0 \le z \le 2.0$. This paper is mainly intended as a presentation, technical description, and guide to the usage of the simulations and of the related post-processing products. Motivated by practical purposes, the entire suite has been organized into three main categories, targeting different scientific and technical aspects: (1) the [*Grid Suite*]{}, useful for cosmological parameter constraints especially regarding massive and sterile neutrinos and the [*dark sector*]{}; (2) the [*Supporting Suite*]{}, aimed at studying the detailed physical effects of exotic particles and dark radiation models, as well as their impact on the high-$z$ cosmic web; and (3) the [*Systematics Suite*]{}, meant to address several systematic effects, ranging from numerical challenges till parameter degeneracies. While the simulations of the [*Sejong Suite*]{} share the same philosophy as the runs developed in Rossi et al. (2014), there are a number of improvements and novelties at all levels in this new release related to [*technical*]{}, [*modeling*]{}, and [*innovative*]{} aspects. In particular, on the [*technical*]{} side, we devised a novel flexible pipeline able to produce a generic end-to-end simulation on a given high-performance supercomputing architecture. We ameliorated the time step integration and its accuracy, using a finer integration step. We customized Gadget-3 to our supercomputing architecture, resulting in a overall faster and more efficient performance of the code. We increased the resolution with respect to our previous simulations, thanks to the addition of more than $123$ million particles per type for our primary realizations (i.e., *Grid Suite*), corresponding to up to 369 million particles per simulation for our heaviest runs – equivalent of adding an extra simulation characterized by $~700^3$ particles/type to a given simulation set. In this way, although with additional enhancing techniques, we are able to reach an equivalent resolution up to $3 \times 3328^3 = 110$ billion particles in a $(100 h^{-1} {\rm Mpc})^3$ box size, corresponding to a grid resolution of $30h^{-1}{\rm kp}$ – ideal for meeting the observational requirements of eBOSS and DESI. On the [*modeling*]{} side, we expanded the parameter space for the [*Grid Suite*]{}, and tightened their variation range; we used a finer neutrino mass, introduced models with dark radiation and WDM, and simulated dark radiation scenarios without approximations. We also improved on the reionization history (i.e., better determination of the reionization redshift), which is well known to directly impact the parameter constraints. On the [*innovative*]{} side, the most significant novelty is the inclusion, for the first time, of extended mixed scenarios describing the combined effects of WDM, neutrinos, and dark radiation. These non-canonical models are quite interesting, particularly for constraining $N_{\rm eff}$ and WDM relic masses directly from Ly$\alpha$ forest data. In future releases of the [*Sejong Suite*]{}, we plan to expand around this framework and provide more realizations. In particular, the [*Grid Suite*]{} represents probably the ultimate word on splicing, as running larger-volume high-resolution hydrodynamical simulations able to meet the requirements of upcoming surveys is becoming progressively less prohibitive, in terms of computational costs. Therefore, abandoning splicing and interpolation techniques and adopting emulator-based strategies will be feasible, and we aim at extending the [*Sejong Suite*]{} in this direction as well. In addition to a thorough descriptions of the simulations, in this work we also presented a first analysis of the [*Sejong Suite*]{}, primarily focusing on the matter and flux statistics (Section \[sec\_sejong\_suite\_first\_results\]), and in particular, we showed that we are able to accurately reproduce the 1D flux power spectrum down to scales $k = 0.06~{\rm [km/s]^{-1}}$ as mapped by recent high-resolution quasar data. While primarily developed for Ly$\alpha$ forest studies, our high-resolution simulations and Ly$\alpha$ skewers may be useful for a broader variety of cosmological and astrophysical purposes. In addition to cosmological parameter constraints and the characterization of systematics, the [*Sejong Suite*]{} can in fact be helpful for novel methods that aim at painting baryonic physics into DM-only simulations, that intend to add small-scale physics to low-resolution larger-box simulations, or that attempt to accurately reproduce observables from a limited number of realizations in parameter space – as suggested by several recent studies. In this regard, there has been interesting progress toward efficient emulators (Bird et al. 2019; Giblin et al. 2019; Rogers et al. 2019; Van der Velden et al. 2019; Zhai et al. 2019), training sets, neural networks, and machine-learning techniques (Nadler et al. 2018; Rodr[í]{}guez et al. 2018; Mustafa et al. 2019; Ramanah et al. 2019; Shirasaki et al. 2019; Takhtaganov et al. 2019; Zamudio-Fernandez et al. 2019; Wibking et al. 2020), or novel approaches such as the Cosmological Evidence Modeling (Lange et al. 2019) or the Machine-assisted Semi-Simulation Model (Jo & Kim 2019). The common aspect between these methods is the reliance on some (even partial) information derived from high-resolution hydrodynamical realizations, and in this context, the [*Sejong Suite*]{} could be useful for calibrations purposes. Moreover, we also foresee numerous applications toward the galaxy-halo connection (e.g., Wechsler & Tinker 2018; Martizzi et al. 2019, 2020), the creation of mock galaxy catalogs with novel techniques (Balaguera-Antol[í]{}nez et al. 2019, 2020; Dai et al. 2019), as well as a number of interdisciplinary and interesting synergies with particle physics. All of the simulations carried out in this work are presented in Tables \[table\_grid\_sims\_base\]–\[table\_systematics\_sims\_all\], and the full list of products available from the [*Sejong Suite*]{} is summarized in Section \[subsec\_list\_available\_products\]. We plan to make them progressively available to the scientific community, and kindly ask you to refer to this publication if you use any of those products in your studies. This work is supported by the National Research Foundation of Korea (NRF) through Grants No. 2017R1E1A1A01077508 and No. 2020R1A2C1005655 funded by the Korean Ministry of Education, Science and Technology (MoEST), and by the faculty research fund of Sejong University. The numerical simulations presented in this work were performed using the Korea Institute of Science and Technology Information (KISTI) supercomputing infrastructure (Tachyon 2) under allocations KSC-2017-G2-0008 and KSC-2018-G3-0008, and post-processed with the KISTI KAT System (KISTI/TESLA ‘Skylake’ and ‘Bigmem’ architectures) under allocations KSC-2018-T1-0017, KSC-2018-T1-0033, and KSC-2018-T1-0061. We thank the KISTI supporting staff for technical assistance along the way, and Volker Springel for making Gadget-3 available. We also acknowledge extensive usage of our new computing resources (Xeon Silver 4114 master node and Xeon Gold 6126 computing node architecture) at Sejong University. Many thanks to the anonymous referee for insightful feedback and suggestions, and for empathizing with this challenging work. 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--- abstract: 'We extend the notion of belief function to the case where the underlying structure is no more the Boolean lattice of subsets of some universal set, but any lattice, which we will endow with a minimal set of properties according to our needs. We show that all classical constructions and definitions (e.g., mass allocation, commonality function, plausibility functions, necessity measures with nested focal elements, possibility distributions, Dempster rule of combination, decomposition w.r.t. simple support functions, etc.) remain valid in this general setting. Moreover, our proof of decomposition of belief functions into simple support functions is much simpler and general than the original one by Shafer.' author: - | Michel GRABISCH\ Université Paris I – Panthéon-Sorbonne\ email `[email protected]` bibliography: - '../BIB/fuzzy.bib' - '../BIB/grabisch.bib' - '../BIB/general.bib' title: Belief functions on lattices --- **Keywords:** belief function, lattice, plausibility, possibility, necessity Introduction ============ The theory of evidence, as established by Shafer [@sha76] after the work of Dempster [@dem67], and brought into a practically usable form by the works of Smets in particular [@sme90; @sme94], has become a popular tool in artificial intelligence for the representation of knowledge and making decision. In particular, many applications in classification have been done [@den95; @den00a]. The main advantage over more traditional models based on probability is that the model of Shafer allows for a proper representation of ignorance. On a mathematical point of view, belief functions, which are at the core of the theory of evidence, possess remarkable properties, in particular their links with the Möbius transform [@rot64] and the co-Möbius transform [@gra98c; @grla00a], called *commonality* by Shafer. Remarking that belief functions are non negative isotone functions defined on the Boolean lattice of subsets, one may ask if all these properties remain valid when more general lattices are considered. The aim of this paper is precisely to investigate this question, and we will show that amazingly they all remain valid. A first investigation of this question was done by Barthélemy [@bar00], and our work will complete his results. We are not aware of other similar works, except the one of Kramosil [@kra01], where belief functions are defined on Boolean lattices but take value in a partially ordered set, and the notion of bi-belief proposed by Grabisch and Labreuche [@grla03], where the underlying lattice is $3^n$. On an application point of view, one may ask about the usefulness of such a generalization, apart from its mathematical beauty. A general answer to this is that the objects we manipulate (events, logical propositions, etc.) may not form a Boolean lattice, i.e., distributive and complemented. Thus, a study on a weaker yet rich structure has its interest. Let us give some examples. - Case where the universal set $\Omega$ is the set of possible outcomes, states of nature, etc. In the classical case, all subsets of $\Omega$ (called events) are considered, but it may happen that some events are not observable or realizable, meaningful, etc. Then, the structure of the events is no more the Boolean lattice $2^\Omega$. - Case where the universal set $\Omega$ is the set of propositional variables, either true or false. As argued by Barthélemy [@bar00], in non-classical logics, the set of propositions need not be $2^\Omega$, and as we will see later, probability theory applies as far as the lattice induced by propositional calculus is distributive, and this covers intuitionistic logic and paraconsistent logic. If distributivity does not hold, then belief functions appear as a natural candidate, since as it will be shown, belief functions can live on any lattice. - Case where the universal set is the set of players/agents in some cooperative game or multiagent situation. Subsets of $\Omega$ are called coalitions, and most of the time, it happens that some coalitions are infeasible, .i.e., they cannot form, due to some inherent impossibility depending on the context. For example, in voting situations, clearly all coalitions of political parties cannot form. The same holds for agents or players in general where some incompatibilities exist between them. - Knowledge extraction and modeling: objects under study are often structured as lattices. For example, the popular Formal Concept Analysis of Ganter and Wille [@gawi99] build lattices of concepts, from a matrix of objects described by qualitative attributes. - Finally, in some cases, objects of interest are not subsets of some universal set. This is the case for example when one is interested into the collection of partitions of some set (again, this happens in game theory under the name “game in partition function form” [@thlu63], and also in knowledge extraction where the fundamental problem is to partition attributes), or when objects of interest are “bi-coalitions” like for bi-belief functions. A bi-coalition is a pair of subsets with empty intersection, and it may represent the set of criteria which are satisfied and the one which are not satisfied. The paper is organized as follows. Section \[sec:back\] recalls necessary material on lattices and classical belief functions. Section \[sec:bella\] gives the main results on belief defined over lattices, while the last one examine the case of necessity measures. Throughout the paper, we will deal with finite lattices. Background {#sec:back} ========== Lattices {#sec:latti} -------- We begin by recalling necessary material on lattices (a good introduction on lattices can be found in [@dapr90] and [@mon03]), in a finite setting. A *poset* is a set $P$ endowed with a partial order $\leq$ (reflexive, antisymmetric, transitive). A *lattice* $L$ is a poset such that for any $x,y\in L$ their least upper bound $x\vee y$ and greatest lower bound $x\wedge y$ always exist. For finite lattices, the greatest element of $L$ (denoted $\top$) and least element $\bot$ always exist. $x$ *covers* $y$ (denoted $x\succ y$) if $x> y$ and there is no $z$ such that $x>z>y$. Let $P$ be a poset, $Q\subseteq P$ is a *downset* if for any $y\in P$ such that $y\leq x$, $x\in Q$, then $y\in Q$. The set of all downsets of $P$ is denoted by $\mathcal{O}(P)$. A *linear lattice*, or *chain*, is such that $\leq$ is a total order. A chain $C$ in $L$ is *maximal* if no element $x\in L\setminus C$ can be added so that $C\cup\{x\}$ is still a chain. Lattices can be represented by their *Hasse diagram*, where nodes are elements of the lattice, and there is an edge between $x$ and $y$, with $x$ above $y$, if and only if $x\succ y$. Fig. \[fig:lat1\] shows three lattices. The middle and right ones are two different diagrams of the lattice of subsets of $\{1,2,3\}$ ordered by inclusion. (0,0)(3,4) (1,0)(3,2)(2,3)(0,1) (2,1)(3,2)(1,4)(0,3) (1,0)[0.1]{} (0,1)[0.1]{} (1,2)[0.1]{} (2,1)[0.1]{} (3,2)[0.1]{} (0,3)[0.1]{} (2,3)[0.1]{} (1,4)[0.1]{} (1,2)[$c$]{} (2,1)[$b$]{} (0,1)[$a$]{} (3,2)[$d$]{} (0,3)[$e$]{} (2,3)[$f$]{} (0,0)(2,4) (1,0)(0,1)(1,2)(2,1) (1,1)(0,2)(1,3)(2,2) (0,1)(0,2) (1,0)(1,1) (2,1)(2,2) (1,2)(1,3) (1,0)[0.1]{} (0,1)[0.1]{} (1,1)[0.1]{} (2,1)[0.1]{} (0,2)[0.1]{} (1,2)[0.1]{} (2,2)[0.1]{} (1,3)[0.1]{} (1,0)[$\emptyset$]{} (0,1)[$1$]{} (1,1)[$2$]{} (2,1)[$3$]{} (0,2)[$12$]{} (1,2)[$13$]{} (2,2)[$23$]{} (1,3)[$123$]{} (0,0)(2,4) (1,0)(0,1) (1,0)(1,1) (1,0)(2,1) (1,3)(0,2) (1,3)(1,2) (1,3)(2,2) (0,1)(1,2) (0,1)(2,2) (1,1)(0,2) (1,1)(2,2) (2,1)(0,2) (2,1)(1,2) (1,0)[0.1]{} (0,1)[0.1]{} (1,1)[0.1]{} (2,1)[0.1]{} (0,2)[0.1]{} (1,2)[0.1]{} (2,2)[0.1]{} (1,3)[0.1]{} (1,0)[$\emptyset$]{} (0,1)[$1$]{} (1,1)[$2$]{} (2,1)[$3$]{} (2,2)[$12$]{} (1,2)[$13$]{} (0,2)[$23$]{} (1,3)[$123$]{} Let $P,Q$ be two posets, and consider $f:P\rightarrow Q$. $f$ is *isotone* (resp. *antitone*) if $x\leq y$ implies $f(x)\leq f(y)$ (resp. $f(x)\geq f(y)$). $P$ and $Q$ are *isomorphic* (resp. *anti-isomorphic*), denoted by $P\cong Q$ (resp. $P\cong Q^\partial$), if it exists a bijection $f$ from $P$ to $Q$ such that $x\leq y\Leftrightarrow f(x)\leq f(y)$ (resp. $f(x)\geq f(y)$). Isomorphic posets have same Hasse diagrams, up to the labelling of elements. For any poset $(P,\leq$), one can consider its *dual* by inverting the order relation, which is denoted by $(P,\leq^\partial)$ (or simply $P^\partial$ if the order relation is not mentionned), i.e., $x\leq y$ if and only if $y\leq^\partial x$. *Autodual* posets are such that $P\cong P^\partial$ (i.e., they have the same Hasse diagram). The lattices of Fig. \[fig:lat1\] are all autodual, and Fig. \[fig:lat2\] shows their dual. (0,0)(3,4) (1,0)(3,2)(2,3)(0,1) (2,1)(3,2)(1,4)(0,3) (1,0)[0.1]{} (0,1)[0.1]{} (1,2)[0.1]{} (2,1)[0.1]{} (3,2)[0.1]{} (0,3)[0.1]{} (2,3)[0.1]{} (1,4)[0.1]{} (1,2)[$c$]{} (2,1)[$f$]{} (0,1)[$e$]{} (3,2)[$d$]{} (0,3)[$a$]{} (2,3)[$b$]{} (0,0)(2,4) (1,0)(0,1)(1,2)(2,1) (1,1)(0,2)(1,3)(2,2) (0,1)(0,2) (1,0)(1,1) (2,1)(2,2) (1,2)(1,3) (1,0)[0.1]{} (0,1)[0.1]{} (1,1)[0.1]{} (2,1)[0.1]{} (0,2)[0.1]{} (1,2)[0.1]{} (2,2)[0.1]{} (1,3)[0.1]{} (1,0)[$123$]{} (0,1)[$12$]{} (1,1)[$13$]{} (2,1)[$23$]{} (0,2)[$1$]{} (1,2)[$2$]{} (2,2)[$3$]{} (1,3)[$\emptyset$]{} (0,0)(2,4) (1,0)(0,1) (1,0)(1,1) (1,0)(2,1) (1,3)(0,2) (1,3)(1,2) (1,3)(2,2) (0,1)(1,2) (0,1)(2,2) (1,1)(0,2) (1,1)(2,2) (2,1)(0,2) (2,1)(1,2) (1,0)[0.1]{} (0,1)[0.1]{} (1,1)[0.1]{} (2,1)[0.1]{} (0,2)[0.1]{} (1,2)[0.1]{} (2,2)[0.1]{} (1,3)[0.1]{} (1,0)[$123$]{} (0,1)[$23$]{} (1,1)[$13$]{} (2,1)[$12$]{} (2,2)[$3$]{} (1,2)[$2$]{} (0,2)[$1$]{} (1,3)[$\emptyset$]{} A lattice $L$ is *lower semimodular* (resp. *upper semimodular*) if for all $x,y\in L$, $x\vee y\succ x$ and $x\vee y\succ y$ imply $x\succ x\wedge y$ and $y\succ x\wedge y$ (resp. $x\succ x\wedge y$ and $y\succ x\wedge y$ imply $x\vee y\succ x$ and $x\vee y\succ y$). A lattice being upper and lower semimodular is called *modular*. The lattice is *distributive* if $(x\vee y)\wedge z= (x\wedge z)\vee(y\wedge z)$ holds for all $x,y,z\in L$. (0,0)(4,4) (2,0)(0,2)(2,4)(4,2) (2,0)(2,4) (2,0)[0.1]{} (0,2)[0.1]{} (2,4)[0.1]{} (4,2)[0.1]{} (2,2)[0.1]{} (0,2)[$a$]{} (2,2)[$b$]{} (4,2)[$c$]{} (0,0)(4,4) (2,0)(0,1)(0,3)(2,4)(4,2) (2,0)[0.1]{} (0,1)[0.1]{} (0,3)[0.1]{} (2,4)[0.1]{} (4,2)[0.1]{} $(L,\leq)$ is said to be *lower (upper) locally distributive* if it is lower (upper) semimodular, and it does not contain a sublattice isomorphic to $M_3$. These are weaker conditions than distributivity, and if $L$ is both lower and upper locally distributive, then it is distributive. An element $j\in L$ is *join-irreducible* if $j=x\vee y$ implies either $j=x$ or $j=y$, i.e., it cannot be expressed as a supremum of other elements. Equivalently $j$ is join-irreducible if it covers only one element. Join-irreducible elements covering $\bot$ are called *atoms*, and the lattice is *atomistic* if all join-irreducible elements are atoms. The set of all join-irreducible elements of $L$ is denoted $\mathcal{J}(L)$. On Fig. \[fig:lat1\] and \[fig:lat2\], they are figured as black nodes. Similarly, *meet-irreducible elements* cannot be written as an infimum of other elements, and are such that they are covered by a single element. We denote by $\mathcal{M}(L)$ the set of meet-irreducible elements of $L$. *Co-atoms* are meet-irreducible elements covered by $\top$. For any $x\in L$, we say that $x$ *has a complement in $L$* if there exists $x'\in L$ such that $x\wedge x'=\bot$ and $x\vee x'=\top$. The complement is unique if the lattice is distributive. $L$ is said to be *complemented* if any element has a complement. On Fig. \[fig:lat1\] (left), no element has a complement, except top and bottom, while the two others are complemented lattices. *Boolean lattices* are distributive and complemented lattices, and in a finite setting, they are of the type $2^N$ for some set $N$, i.e. they are isomorphic to the lattice of subsets of some set, ordered by inclusion (see Fig. \[fig:lat1\] (middle,right)). Boolean lattices are atomistic, and atoms correspond to singletons, while co-atoms are of the form $N\setminus\{i\}$ for some $i\in N$. An important property is that in a lower locally distributive lattice, any element $x$ can be written as an irredundant supremum of join-irreducible elements in a unique way (this is called the *minimal decomposition* of $x$). We denote by $\eta^*(x)$ the set of join-irreducible elements in the minimal decomposition of $x$, and we denote by $\eta(x)$ the *normal decomposition* of $x$, defined as the set of join-irreducible elements smaller or equal to $x$, i.e., $\eta(x):=\{j\in \mathcal{J}(L)\mid j\leq x\}$. Hence $\eta^*(x)\subseteq \eta(x)$, and $$x=\bigvee_{j\in \eta^*(x)} j = \bigvee_{j\in \eta(x)} j.$$ Put differently, the mapping $\eta$ is an isomorphism of $L$ onto $\mathcal{O}(\mathcal{J}(L))$ (Birkhoff’s theorem). Likewise, any element in a upper locally distributive lattice can be written as a unique irredundant infimum of meet-irreducible elements. The decomposition are denoted by $\mu$ and $\mu^*$. Specifically, $\mu(x):=\{m\in\mathcal{M}(L)\mid m\geq x\}$, and $\displaystyle{x=\bigwedge_{m\in\mu(x)}m}$. The *height function* $h$ on $L$ gives the length of a longest chain from $\bot$ to any element in $L$. A lattice is *ranked* if $x\succ y$ implies $h(x)=h(y)+1$. A lattice is lower locally distributive if and only if it is ranked and the length of any maximal chain is $|\mathcal{J}(L)|$. The Möbius and co-Möbius transforms {#sec:mobi} ----------------------------------- We follow the general definition of Rota [@rot64] (see also [@bir67 p. 102]). Let $(L,\leq)$ be a poset which is locally finite (i.e., any interval is finite) having a bottom element. For any function $f$ on $(L,\leq)$, the *Möbius transform* of $f$ is the function $m:L\longrightarrow \mathbb{R}$ solution of the equation: $$\label{eq:mob} f(x) = \sum_{y\leq x}m(y).$$ This equation has always a unique solution, and the expression of $m$ is obtained through the Möbius function $\mu:L^2\rightarrow \mathbb{R}$ by: $$\label{eq:invm} m(x) = \sum_{y\leq x}\mu(y,x)f(y)$$ where $\mu$ is defined inductively by $$\label{eq:mu} \mu(x,y) = \left\{ \begin{array}{ll} 1, & \text{ if } x=y\\ -\sum_{x\leq t< y}\mu(x,t), & \text{ if } x< y\\ 0, & \text{ otherwise}. \end{array} \right.$$ Note that $\mu$ depends solely on $L$. The *co-Möbius transform* of $f$, denoted by $q$, is defined by [@gra98c; @grla00a]: $$\label{eq:com} q(x):=\sum_{y\geq x} m(y), \quad x\in L.$$ Belief functions and related concepts {#sec:beli} ------------------------------------- We recall only necessary definitions. For details, the reader is referred to, e.g., [@sme90; @sme94], or the monograph [@kra01a]. Let $\Omega$ be a finite space. A function $m:2^\Omega\rightarrow [0,1]$ is said to be a *mass allocation function* (or simply a *mass*) if $m(\emptyset)=0$ and $\sum_{A\subseteq \Omega}m(A)=1$. A subset $A\subseteq N$ is said to be a *focal element* if $m(A)>0$. A *belief function* on $\Omega$ is a function $\bel:2^\Omega\rightarrow [0,1]$ generated by a mass allocation function as follows: $$\label{eq:bel} \bel(A) := \sum_{B\subseteq A} m(B), \quad A\subseteq \Omega.$$ Note that $\bel(\emptyset)=0$ and $\bel(\Omega)=1$. One recognizes $m$ as being the Möbius transform of $\bel$ (apply Eq. (\[eq:mob\]) to $(L,\leq):=(2^\Omega,\subseteq)$). The inverse formula, obtained by using (\[eq:invm\]) and (\[eq:mu\]), is: $$\label{eq:invb} m(A) = \sum_{B\subseteq A} (-1)^{|A\setminus B|}\bel(B).$$ Given a mass allocation $m$, the *plausibility function* is defined by: $$\label{eq:plau} \pl(A):=\sum_{B\mid A\cap B\neq\emptyset} m(B) = 1-\bel(A^c), \quad A\subseteq \Omega.$$ Similarly, the *commonality function* is defined by: $$\label{eq:comm} q(A):=\sum_{B\supseteq A} m(B) , \quad A\subseteq \Omega.$$ It is the co-Möbius transform of $\bel$ (see (\[eq:com\])). Remark that $q(\emptyset)=1$. A *capacity* on $\Omega$ is a set function $v:2^\Omega\rightarrow [0,1]$ such that $v(\emptyset)=0$, $v(\Omega)=1$, and $A\subseteq B$ implies $v(A)\leq v(B)$ (*monotonicity*). Plausibility and belief functions are capacities. For any capacity $v$, its *conjugate* is defined by $\overline{v}(A):=1-v(A^c)$. Hence, plausibility functions are conjugate of belief functions (and vice versa). A capacity is *$k$-monotone* ($k\geq 2$) if for any family of $k$ subsets $A_1,\ldots,A_k$ of $\Omega$, it holds: $$v(\bigcup_{i\in K} A_i)\geq \sum_{I\subseteq K,I\neq\emptyset}(-1)^{|I|+1}v(\bigcap_{i\in I}A_i),$$ with $K:=\{1,\ldots,k\}$. A capacity is *totally monotone* if it is $k$-monotone for every $k\geq 2$. Shafer [@sha76] has shown that a capacity is totally monotone if and only if it is a belief function, hence there exists some mass allocation generating it. Given two mass allocations $m_1,m_2$, the *Dempster’s rule of combination* computes a combination of both masses into a single one: $$m(A) =:(m_1\oplus m_2)(A):= \sum_{B_1\cap B_2=A}m_1(B_1)m_2(B_2), \quad \forall A\subseteq \Omega, A\neq\emptyset,$$ and $m(\emptyset):=0$. Note that $m$ is no more a mass allocation in general, unless some normalization is carried out. It is well known that the Dempster rule of combination can be computed through the commonality functions much more easily. Specifically, calling $q,q_1,q_2$ the commonality functions associated to $m,m_1,m_2$, one has: $$\label{eq:drq} q(A) = q_1(A)q_2(A), \quad \forall A\subseteq \Omega.$$ A *simple support function focused on $A$* is a particular belief function $\bel_A$ whose mass allocation is: $$\label{eq:ssf} m_A(B):=\begin{cases} 1 - w_A, & \text{if } B=A\\ w_A, & \text{if } B=\Omega\\ 0, & \text{otherwise}. \end{cases}$$ with $0<w_A<1$. Smets [@sme95], using results of Shafer, has shown that any belief function such that $m(\Omega)\neq 0$ can be decomposed using only simple support functions as follows: $$\label{eq:decom} \bel=\bigoplus_{A\subseteq \Omega}\bel_A$$ with $$\label{eq:decom1} w_A=\prod_{B\supseteq A}q(B)^{(-1)^{|B\setminus A|+1}}, \quad\forall A\subseteq \Omega.$$ In the above decomposition, coefficients $w_A$ may be greater than 1. If this happens, the corresponding $\bel_A$ is no more a belief function. A *necessity function* or *necessity measure* is a belief function whose focal elements form a chain in $(2^\Omega,\subseteq)$, i.e., $A_1\subseteq A_2\subseteq\cdots\subseteq A_n$ (Dubois and Prade, [@dupr85b]). The characteristic property of necessity functions is that for any subsets $A,B$, $\N(A\cap B)=\min(\N(A),\N(B))$, where $\N$ denotes a necessity function. Conjugates of necessity functions are called *possibility functions*, denoted by $\Pi$, and are particular plausibility functions. It is easy to see that their characteristic property is that for any subsets $A,B$, $\Pi(A\cup B)=\max(\Pi(A),\Pi(B))$. This characteristic property implies that $\Pi$ is entirely determined by its value on singletons, i.e., $\Pi(A)=\max_{\omega\in A}\Pi(\{\omega\})$ for any $A\subseteq \Omega$. For this reason, $\pi(\omega):=\Pi(\{\omega\})$ is called the *possibility distribution* associated to $\Pi$. Note that necessarily there exists $\omega_0\in \Omega$ such that $\pi(\omega_0)=1$. Although this is generally not considered, one may define as well a *necessity distribution* $\nu(\omega):=\N(\Omega\setminus\omega)$, with the property that $\N(A)=\min_{\omega\in A^c}\nu(\omega)$. Let $\pi$ be a possibility distribution on $\Omega:=\{\omega_1,\ldots,\omega_n\}$, and assume that for some permutation $\sigma$ on $\{1.\ldots,n\}$, it holds $\pi(\omega_{\sigma(1)}) \leq\pi(\omega_{\sigma(2)})\leq \cdots\leq \pi(\omega_{\sigma(n)})=1$. Then it can be shown that the focal elements of the mass allocation associated to $\Pi$ are of the form $A_{\sigma(i)}:=\{\omega_{\sigma(i)},\ldots,\omega_{\sigma(n)}\}$, $i=1,\ldots, n$, and $m(A_{\sigma(i)})=\pi(\omega_{\sigma(i)}) - \pi(\omega_{\sigma(i-1)})$, with the convention $\pi(\omega_{\sigma(0)})=0$. Belief functions and capacities on lattices {#sec:bella} =========================================== Let $(L,\leq)$ be a finite lattice. A *capacity* on $L$ is a function $v:L\rightarrow [0,1]$ such that $v(\bot)=0$, $v(\top)=1$, and $x\leq y$ implies $v(x)\leq v(y)$ (isotonicity). To define the conjugate of a capacity, a natural way would be to write $\overline{v}(x):=1-v(x')$, where $x'$ is the complement of $x$. But this would impose that $L$ is complemented, which is very restrictive. For example, the lattice $3^n$ underlying bi-belief functions is not complemented. Moreover, if distributivity is imposed in addition, then only Boolean lattices are allowed, and we are back to the classical definition. We adopt a more general definition. A lattice $L$ is of *De Morgan type* if it exists a bijective mapping $n:L\rightarrow L$ such that for any $x,y\in L$ it holds $n(x\vee y)=n(x)\wedge n(y)$, and $n(\top)=\bot$. We call such a mapping a *$\vee$-negation*. The following is immediate. \[lem:nega\] Let $L$ be a De Morgan lattice, with $n$ a $\vee$-negation. Then: - $n(\bot)=\top$. - $n^{-1}(x\wedge y)=n^{-1}(x)\vee n^{-1}(y)$, for all $x,y\in L$ ($n^{-1}$ is called a *$\wedge$-negation*). - If $j$ is join-irreducible, then $n(j)$ is meet-irreducible, and if $m$ is meet-irreducible, then $n^{-1}(m)$ is join-irreducible. \(i) $n(x\vee \bot)=n(x) = n(x)\wedge n(\bot)$, for all $x\in L$, which implies $n(\bot)=\top$ because $n$ is a bijection. \(ii) Putting $x':=n(x)$ and $y':=n(y)$, we have $n^{-1}(x'\wedge y')=n^{-1}(n(x\vee y))=x\vee y= n^{-1}(x')\vee n^{-1}(y')$. \(iii) If $j$ is join-irreducible, $j=x\vee y$ implies that $j=x$ or $j=y$. Hence, $n(j)=n(x\vee y)=n(x)\wedge n(y)$ is either $n(x)$ or $n(y)$, which means that $n(j)$ is meet-irreducible. A complemented lattice with unique complement is of De Morgan type with $n(x):=x'$. If $L$ is isomorphic to its dual $L^\partial$, i.e. it is autodual, then it is of De Morgan type since it suffices to take for $n(x)$ the element in the Hasse diagram of $L^\partial$ which takes the place of $x$ in the Hasse diagram of $L$. In this case, we call $n$ a *horizontal symmetry*. In general, $n$ is not unique since there is no unique way to draw Hasse diagrams. Taking lattices of Fig. \[fig:lat1\] as examples, for the left one, we would have $n(a)=e$, for the middle one $n(12)=1$, and for the right one $n(12)=3$ (see Fig. \[fig:lat2\]). Since middle and right lattices are the same, this shows that several $n$ exist in general. Note that $n$ for the right lattice is nothing else than the usual complement. The following result shows that in fact the only De Morgan type lattices are those which are autodual. \[prop:gal\] A lattice $L$ is of De Morgan type if and only if it is autodual. We already know that if $L$ is autodual, then it is of De Morgan type. Conversely, assuming it is of De Morgan type, it suffices to show that $n$ is an anti-isomorphism. We already know that $n$ is a bijection. Taking $x\leq y$ implies that $x\vee y=y$, hence $n(x\vee y)=n(y)=n(x)\wedge n(y)$, which implies $n(y)\leq n(x)$. Conversely, $n(y)\leq n(x)$ implies $n(y)\wedge n(x)=n(y)=n(x\vee y)$, hence $x\vee y=y$ since $n$ is a bijection, so that $x\leq y$. In general, $n$ and $n^{-1}$ differ, that is, $n$ is not always *involutive*. Take for example the lattice $M_3$ of Fig. \[fig:M3\], and $n$ defined by $n(\top)=\bot$, $n(\bot)=\top$, $n(a)=b$, $n(b)=c$ and $n(c)=a$. Clearly, $n$ is a $\vee$-negation, but $n(n(a))=c\neq a$. The $\vee$-negation is involutive whenever $n$ is a horizontal symmetry on the Hasse diagram. If $n$ is involutive, it is simply called a *negation*. Let $L$ be an autodual lattice, and $n$ a $\vee$-negation on $L$. For any capacity $v$, its *$\vee$-conjugate* and *$\wedge$-conjugate* (w.r.t. $n$) are defined respectively by $$\begin{aligned} {}^\vee\overline{v}(x) &:=1-v(n(x))\\ {}^\wedge\overline{v}(x) &:=1-v(n^{-1}(x)),\end{aligned}$$ for any $x\in L$. If $n$ is a negation, then $\overline{v}(x):=1-v(n(x))$ is the *conjugate* of $v$. The following is immediate. \[lem:neg\] Let $L$ be an autodual lattice, and $n$ a $\vee$-negation on $L$. For any capacity $v$, it holds - ${}^\vee\overline{v}$ and ${}^\wedge\overline{v}$ are capacities on $L$. - ${}^\vee\overline{{}^\wedge\overline{v}} ={}^\wedge\overline{{}^\vee\overline{v}}=v$. \(i) ${}^\vee\overline{v}(\top)=1-v(n(\top))=1$, similarly for $\bot$. Isotonicity of ${}^\vee\overline{v}$ follows from antitonicity of $n$ and isotonicity of $v$. \(ii) ${}^\vee\overline{{}^\wedge\overline{v}}(x)=1-{}^\wedge\overline{v}(n(x)) = 1-(1-v(x))=v(x)$. The following definition of belief functions is in the spirit of the original one by Shafer. We used it also for defining bi-belief functions [@grla03]. A function $\bel:L\rightarrow [0,1]$ is called a *belief function* if $\bel(\top)=1$, $\bel(\bot)=0$, and its Möbius transform is non negative. Referring to (\[eq:mob\]), we recall that $$\label{eq:bell} \bel(x)=\sum_{y\leq x}m(y),\quad \forall x\in L.$$ Note that $\bel(\top)=1$ is equivalent to $\sum_{x\in L}m(x)=1$, and $\bel(\bot)=0$ is equivalent to $m(\bot)=0$. The inverse formula, giving $m$ in terms of $\bel$, has to be computed from (\[eq:mu\]), and depends only on the structure of $L$. Remark that $\bel$ is an isotone function by nonnegativity of $m$, and hence a capacity. Thanks to the definition of conjugation, if $L$ is autodual and $n$ is a $\vee$-negation, one can define *plausibility functions* as the $\vee$-conjugate of belief functions, which are again capacities. $k$-monotone functions {#sec:kmon} ---------------------- Barthélemy defines belief function as totally monotone functions. To detail this point, we define $k$-monotone functions. For $k\geq 2$, a function $f:L\rightarrow \mathbb{R}$ is said to be *$k$-monotone* (called *weakly $k$-monotone* by Barthélemy) if it satisfies, for any family of elements $x_1,\ldots,x_k\in L$: $$\label{eq:kmon} f(\bigvee_{i\in K}x_i) \geq \sum_{I\subseteq K, I\neq\emptyset}(-1)^{|I|+1} f(\bigwedge_{i\in I}x_i)$$ where $K:=\{1,\ldots,k\}$. A function is said to be *totally monotone* if it is $k$-monotone for all $k\geq 2$. One can prove that in fact, if $|L|=n$, total monotonicity is equivalent to $(n-2)$-monotonicity [@bar00]. For $k\geq 2$, a function is said to be a *$k$-valuation* if the inequality (\[eq:kmon\]) degenerates into an equality (called also *Poincaré’s inequality*). Similarly, a function is an *infinite valuation* or *total valuation* if it is a $k$-valuation for all $k\geq 2$. It is well known that monotone infinite valuations satisfying $f(\top)=1$ and $f(\bot)=0$ are probability measures. The following lemma, cited in [@bar00], summarizes well-known results from lattice theory (see Birkhoff [@bir67]). Let $L$ be a lattice. Then - $L$ is modular if and only if it admits a strictly monotone 2-valuation. - $L$ is distributive if and only if it is modular and every strictly monotone 2-valuation on $L$ is a 3-valuation. - $L$ is distributive if and only if it admits a strictly monotone 3-valuation. - $L$ is distributive if and only if it is modular and every strictly monotone 2-valuation on $L$ is an infinite valuation. Barthélemy showed in addition that any lattice admits a totally monotone function. In view of this result, Barthélemy defines belief functions as totally monotone function being monotone and satisfying $f(\top)=1$ and $f(\bot)=0$. In summary, a belief function can be defined on any lattice, while probability measures can live only on distributive lattices. The following proposition shows the relation between both definitions. Before, we state a result from [@bar00]. \[lem:2\] For any lattice $L$ and any function $m:L\rightarrow [0,1]$ such that $m(\bot)=0$ and $\sum_{x\in L}m(x)=1$, the function $f^m:L\rightarrow [0,1]$ defined by $f^m(x):=\sum_{y\leq x}m(y)$ is totally monotone and satisfies $f^m(\top)=1$ and $f^m(\bot)=0$. Any belief function is totally monotone. Let $\bel$ be a belief function, and $m$ its Möbius transform. We know that $m(\bot)=0$ and $\sum_{x\in L}m(x)=1$. Hence, by Lemma \[lem:2\], $\bel$ is totally monotone. A totally monotone function does not have necessarily a non negative Möbius function. Simple examples show that monotonicity is a necessary condition. The question to know whether monotonicity and total monotonicity imply non negativity of the Möbius function is still open. Properties of belief functions ------------------------------ A first result shown by Barthélemy shows that capacities collapse to belief functions when $L$ is linear [@bar00]. Any capacity on $L$ is a belief function if and only if $L$ is a linear lattice. In the sequel, we address the combination of belief functions and their decomposition in terms of simple support functions. We will see that classical results generalize. \[def:drc\] Let $\bel_1,\bel_2$ be two belief functions on $L$, with Möbius transforms $m_1,m_2$. The *Dempster’s rule of combination* of $\bel_1,\bel_2$ is defined through its Möbius transform $m$ by: $$m(x)=:(m_1\oplus m_2)(x):=\sum_{y_1\wedge y_2=x}m_1(y_1)m_2(y_2), \quad\forall x\in L.$$ Since $m$ defines unambiguously the belief function, we may write as well $\bel=\bel_1\oplus\bel_2$ to denote the combination. \[prop:comb\] Let $\bel_1,\bel_2$ be two belief functions on $L$, with co-Möbius transforms $q_1,q_2$, and consider their Dempster combination. Then, if $q$ denotes the co-Möbius transform of $\bel:= \bel_1\oplus\bel_2$, $$q(x)= q_1(x)q_2(x),\quad \forall x\in L.$$ We have: $$q(x) = \sum_{y\geq x}\sum_{y_1\wedge y_2=y}m_1(y_1)m_2(y_2) = \sum_{y_1\wedge y_2\geq x} m_1(y_1)m_2(y_2).$$ One can decompose the above sum since if $y_1\geq x$ and $y_2\geq x$, then $y_1\wedge y_2\geq x$ and reciprocally. Thus, $$q(x) = \sum_{y_1\geq x}m(y_1)\sum_{y_2\geq x}m(y_2) = q_1(x)q_2(x).$$ The above proposition generalizes (\[eq:drq\]), and gives a simple means to compute the Dempster combination. In Def. \[def:drc\], one may put as in the classical case $m(\bot)=0$. This does not affect the validity of Prop. \[prop:comb\], except for $x=\bot$. Indeed, by Prop. \[prop:comb\], one obtains $q(\bot)=1$, but $q(\bot)=\sum_{x\in L}m(x)<1$ in general if one puts $m(\bot)=0$ in Def. \[def:drc\]. Let $y\in L$. A *simple support function focused on $y$*, denoted by $y^w$, is a function on $L$ such that its Möbius transform satisfies: $$m(x) = \begin{cases} 1-w, & \text{if } x=y\\ w, & \text{if } x=\top\\ 0, & \text{otherwise. } \end{cases}$$ The decomposition of some belief function $\bel$ in terms of simple support functions is thus to write $\bel$ under the form: $$\label{eq:beldec} \bel(x) = \bigoplus_{y\in L} y^{w_y}(x).$$ The following result generalizes the decomposition in the classical case (see Sec. \[sec:beli\]). Let $\bel$ be a belief function such that its Möbius transform $m$ satisfies $m(\top)\neq 0$. The coefficients $w_y$ of the decomposition (\[eq:beldec\]) write $$w_y=\prod_{x\geq y}q(x)^{-\mu(x,y)}$$ where $\mu(x,y)$ is the Möbius function of $L$. We try to find $w_y$ such that $$\bel(x) = \bigoplus_{y\in L}y^{w_y}.$$ This expression can be written in terms of the co-Möbius transform: $$\label{eq:q} q(x)= \prod_{y\in L}q_y(x),\quad x\in L,$$ where $q_y$ is the co-Möbius transform of $y^{w_y}$: $$q_y(x)=\begin{cases} 1, & \text{if } x\leq y\\ w_y, & \text{otherwise.} \end{cases}$$ From (\[eq:q\]), we obtain: $$\begin{aligned} \log q(x) & = \sum_{y\in L}\log q_y(x) = \sum_{y\not\geq x}\log w_y \\ & = \sum_{y\in L}\log w_y - \sum_{y\geq x}\log w_y.\end{aligned}$$ On the other hand, $$q(\top) = \prod_{y\in L}q_y(\top) = \prod_{y\in L}w_y.$$ We supposed that $q(\top)=m(\top)\neq 0$, hence: $$\log q(x) = \log q(\top) - \sum_{y\geq x}\log w_y.$$ We set $Q(x):=\log q(x)$ and $W(y):=\log w_y$. The last equality becomes: $$Q(x) = Q(\top)-\sum_{y\geq x}W(y).$$ If we define $Q'(x) = Q(\top) - Q(x)$, we finally obtain: $$Q'(x) = \sum_{y\geq x} W(y).$$ We recognize here the equation defining the Möbius transform of $Q'$, up to an inversion of the order (dual order)(see (\[eq:mob\])). Hence, using (\[eq:invm\]): $$W(y)=\sum_{x\geq y}\mu(x,y)Q'(x)$$ with $\mu$ defined by (\[eq:mu\]). Rewriting this with original notation, we obtain: $$\log w_y=\sum_{x\geq y}\mu(x,y)[\log q(\top) - \log q(x)].$$ Remarking that $\sum_{x\geq y}\mu(x,y)\log q(\top)$ is zero, since it corresponds to the Möbius transform of a constant function, we finally get: $$w_y=\prod_{x\geq y} q(x)^{-\mu(x,y)}.$$ Note that the above proof is much shorter and general than the original one by Shafer [@sha76]. As in the classical case, these coefficients may be strictly greater than 1, hence corresponding simple support functions have negative Möbius transform and are no more belief functions. Necessity functions {#sec:nece} =================== A function $\N:L\rightarrow[0,1]$ is called a *necessity function* if it satisfies $\N(x\wedge y)=\min(\N(x),\N(y))$, for all $x,y\in L$, and $\N(\bot)=0$, $\N(\top)=1$. The following result is due to Barthélemy [@bar00]. $\N$ is a necessity function if and only if it is belief function whose Möbius transform $m$ is such that its focal elements form a chain in $L$. We define possibility functions as $\vee$-conjugates of necessity functions. Let $L$ be an autodual lattice, and $n$ a $\vee$-negation on $L$. For any necessity function $\N$ on $L$, its $\vee$-conjugate is called a *possibility function*. Let $\Pi$ be a possibility function. Then ${}^\wedge\overline{\Pi}$ is its corresponding necessity function by Lemma \[lem:neg\] (ii). Let $L$ be an autodual lattice, and $n$ a $\vee$-negation on $L$. The mapping $\Pi:L\rightarrow [0,1]$ is a possibility function if and only if $$\label{eq:pi} \Pi(x\vee y) = \max(\Pi(x),\Pi(y)), \quad \forall x,y\in L.$$ Let $\Pi$ be a possibility function being the $\vee$-conjugate of some necessity function $N$. Then: $$\begin{aligned} \Pi(x\vee y) & = 1-\N(n(x\vee y)) = 1- \N(n(x)\wedge n(y))\\ & = 1- \min(\N(n(x)),\N(n(y))) = \max(1-\N(n(x)),1-\N(n(y)))\\ & = \max(\Pi(x),\Pi(y)).\end{aligned}$$ Conversely, let $\Pi$ satisfy (\[eq:pi\]) and consider its $\wedge$-conjugate ${}^\wedge\overline{\Pi}$. We have: $$\begin{aligned} {}^\wedge\overline{\Pi}(x\wedge y) & = 1-\Pi(n^{-1}(x\wedge y))= 1-\Pi(n^{-1}(x)\vee n^{-1}(y))\\ & = 1-\max(\Pi(n^{-1}(x)),\Pi(n^{-1}(y)))\\ & = \min({}^\wedge\overline{\Pi}(x),{}^\wedge\overline{\Pi}(y)).\end{aligned}$$ Hence ${}^\wedge\overline{\Pi}$ is a necessity function, which implies that $\Pi$ is a possibility function since ${}^\vee\overline{{}^\wedge\overline{\Pi}}=\Pi$ by Lemma \[lem:neg\] (ii). The next topic we address concerns distributions. Since we need the property of decomposition of elements into supremum of join-irreducible elements, we impose that $L$ is lower locally distributive. Since $L$ has to be autodual, then it is also upper locally distributive, and so it is distributive. We propose the following definition. Let $L$ be an autodual distributive lattice, some $\vee$-negation $n$ on $L$, and $\N$ a necessity function. The *possibility distribution* $\pi:\mathcal{J}(L)\rightarrow [0,1]$ associated to $\N$ is defined by $\pi(j):=\Pi(\{j\})$, $j\in \mathcal{J}(L)$, with $\Pi$ the possibility function which is $\vee$-conjugate of $\N$. The *necessity distribution* $\nu:\mathcal{M}(L)\rightarrow [0,1]$ associated to $\N$ is defined by $\nu(m):=\N(\{m\})$, $m\in\mathcal{M}(L)$. Then, the value of $\Pi$ and $\N$ at any $x\in L$ can be computed as follows: $$\Pi(x) = \max(\pi(j) \mid j\in\eta^*(x)), \quad \N(x) = \min(\nu(m)\mid m\in\mu^*(x)).$$ Remark that due to isotonicity of $\Pi$ and $\N$, and hence of $\pi$ and $\nu$, one can replace as well $\eta^*,\mu^*$ by $\eta,\mu$. The above formulas are well-defined since the decomposition is unique for distributive lattices. Lastly, remark that necessarily there exists $j_0\in\mathcal{J}(L)$ such that $\pi(j_0)=1$, and $m_0\in\mathcal{M}(L)$ such that $\nu(m_0)=0$, since $\Pi(\top)=1$ and $\N(\bot)=0$. $\pi$ and $\nu$ are related through conjugation since $n$ maps join-irreducible elements to meet-irreducible elements and vice-versa for $n^{-1}$ (see Lemma \[lem:nega\] (iii)). Hence, for $j\in\mathcal{J}(L)$ and $m\in\mathcal{M}(L)$: $$\begin{aligned} \pi(j) & =1 - \N(n(j)) = 1 - \nu(m_j)\\ \nu(m) & =1 - \Pi(n^{-1}(m)) = 1 - \pi(j_m),\end{aligned}$$ where $m_j:=n(j)$, and $j_m:=n^{-1}(m)$. Given a mass allocation defining some necessity function, it is easy to derive the corresponding possibility distribution. The converse problem, i.e., given a possibility distribution, find (if possible) the corresponding chain of focal elements and mass allocation giving rise to this possibility distribution, is less simple. Interestingly enough, this problem has always a unique solution, which is very close to the classical case. \[th:2\] Let $L$ be autodual, distributive, and $n$ be a $\vee$-negation on $L$. Let $\pi$ be a possibility distribution, and assume that the join-irreducible elements of $L$ are numbered such that $\pi(j_1)<\cdots <\pi(j_n)=1$. Then there is a unique maximal chain of focal elements generating $\pi$, given by the following procedure: > Going from $j_n$ to $j_1$, at each step $k=n,n-1,\ldots,1$, select the unique join-irreducible element $\iota_k$ such that: $$\label{eq:proc} > \iota_k\not\in \eta(n(j_k)),\quad \iota_k\in\bigcap_{l=1}^{k-1}\eta(n(j_l)).$$ Then the maximal chain is defined by $C_\pi:=\{\iota_n, > \iota_n\vee\iota_{n-1},\ldots, \iota_n\vee\cdots\vee\iota_2,\top\}$, and $$\label{eq:mobpi} > m(\iota_n\vee\iota_{n-1}\vee\cdots\vee\iota_k)=\pi(j_k)-\pi(j_{k-1}),\quad k=1,\ldots,n,$$ with $\pi(j_0):=0$. Moreover, at each step $k$, it is equivalent to choose $\iota_k$ as the smallest in $\eta(n(j_{k-1}))\setminus \eta(n(j_k))$. For ease of notation, denote $n(j_k)$ by $m_k$ (meet-irreducible). We first show that such a procedure can always work and leads to a unique solution for $C_\pi$. Assume that the poset $\mathcal{J}(L)$ has $q$ connected components $J_1,\ldots,J_q$. By definition, $j_n$ is one of the maximal elements of one of the connected components, say $J_{q_0}$. Clearly, $\bigwedge_{k=1}^n m_k=n(\bigvee_{k=1}^n j_k)=n(\top)=\bot$. But $\bigvee_{k=1}^{n-1} j_k\neq\top$, otherwise $\big(\bigcup_{l=1,\ldots,q,l\neq q_0}J_l\big)\cup J'_{q_0}$, where $J'_{q_0}$ is a maximal downset of $J_{q_0}\setminus \{j_n\}$, would be another downset corresponding to $\top$, which is impossible since $L$ is distributive (Birkhoff’s theorem). This implies that there exists $\iota_n\in \bigcap_{k=1}^{n-1}\eta(m_k)$, and $\iota_n\not\in \eta(m_n)$. Let us show that $\iota_n$ is unique. Since $L$ is distributive, it is ranked and any maximal chain has length $|\mathcal{J}(L)|=n$. Hence, $\bigvee_{k=1}^{n-1}j_k$ has height $n-1$ (it is a co-atom), and $\bigwedge_{k=1}^{n-1}m_k$ is an atom. Therefore, $\bigcap_{k=1}^{n-1}\eta(m_k)$ is a singleton. For $\iota_{n-1}$ and subsequent ones, we apply the same reasoning on the lattice $\mathcal{O}(\mathcal{J}(L)\setminus\{\iota_n\})$, then on $\mathcal{O}(\mathcal{J}(L)\setminus\{\iota_n,\iota_{n-1}\})$, etc., instead of $L=\mathcal{O}(\mathcal{J}(L))$. Hence, there will be $n$ steps, and at each step one join-irreducible element is chosen in a unique way. We prove now that the sequence $\{\iota_n, \iota_n\vee\iota_{n-1},\ldots, \iota_n\vee\cdots\vee\iota_2,\top\}$ is a maximal chain, denoted $C_\pi$. It suffices to prove that $\iota_n\vee\iota_{n-1}\vee\cdots\vee\iota_k\succ \iota_n\vee\iota_{n-1}\vee\cdots\vee\iota_{k+1}$, $k=1,\ldots,n-1$. The fact that the former is greater or equal to the latter is obvious, hence $C_\pi$ is a chain. To prove that it is maximal, we have to show that equality cannot occur among any two subsequent elements. To see this, observe that at each step $k$: $$\label{eq:proof} \iota_k\not\leq m_k,\quad \iota_k\leq m_{k-1},\iota_k\leq m_{k-2},\ldots,\iota_k\leq m_1.$$ Hence $\iota_{k-1}\not\leq \iota_k$, otherwise $\iota_{k-1}\leq m_{k-1}$ would hold, a contradiction. Hence, the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ is non decreasing, and equality cannot occur. Let us prove that it suffices to choose $\iota_k$ as the smallest in $\eta(n(j_{k-1})\setminus \eta(n(j_k))$. If at step $k$, a smallest $\iota_k$ is not chosen in $\eta(n(j_{k-1})\setminus \eta(n(j_k))$, it will be taken after, and the sequence $\iota_n,\iota_{n-1},\ldots,\iota_1$ will be no more non decreasing, a contradiction. It remains to prove that $\pi$ is strictly increasing and to verify the expression of $m$. Let us prove by induction that $$\label{eq:pi1} \pi(j_k)=1-m(\iota_n)-m(\iota_n\vee\iota_{n-1}) - \cdots- m(\iota_n\vee\cdots\vee \iota_{k+1}),\quad k=n,\ldots,1.$$ We show it for $k=n$. We have $$\pi(j_n)=1-\nu(m_n) = 1-\sum_{\substack{x\leq m_n\\x\in C_\pi}}m(x).$$ Since $\iota_n\not\in\eta(m_n)$, no $x$ in $C_\pi$ can be smaller than $m_n$. Hence $\pi(j_n)=1$. Let us assume (\[eq:pi1\]) is true from $n$ up to some $k$, and prove it is still true for $k-1$. Using (\[eq:proof\]), we have: $$\begin{aligned} \pi(j_{k-1}) &=1-\nu(m_{k-1}) = 1-\sum_{\substack{x\leq m_{k-1}\\x\in C_\pi}}m(x) \\ & = 1 - \sum_{\substack{x\leq m_{k}\\x\in C_\pi}}m(x) - m(\iota_n\vee\cdots\vee\iota_k) = \pi(j_k)-m(\iota_n\vee\cdots\vee\iota_k),\end{aligned}$$ which proves (\[eq:pi1\]). Lastly, remark that the linear system of $n$ equations (\[eq:pi1\]) is triangular, with no zero on the diagonal. Hence it has a unique solution, which is easily seen to be (\[eq:mobpi\]). As illustration of the theorem, we give an example. Let us consider the distributive autodual lattice given on Fig. \[fig:lat3\]. (0,0)(5,5) (0,1)(0,2) (2,1)(1,2)(2,3)(3,2) (0,1)[0.1]{} (0,2)[0.1]{} (2,1)[0.1]{} (1,2)[0.1]{} (2,3)[0.1]{} (3,2)[0.1]{} (0,1)[$a$]{} (0,2)[$b$]{} (2,1)[$c$]{} (1,2)[$d$]{} (3,2)[$e$]{} (2,3)[$f$]{} (0,0)(9,6) (7,0)(4,0.5)(4,1.5)(1,2)(2,3)(1,4)(1,5) (1,2)(0,3)(1,4)(2,3) (4,1.5)(3,2.5)(4,3.5)(5,2.5) (7,1)(6,2)(7,3)(8,2) (1,1)(1,2) (4,0.5)(4,1.5) (7,0)(7,1) (1,4)(1,5) (4,3.5)(4,4.5) (7,3)(7,4) (1,1)(7,0) (1,2)(7,1) (0,3)(6,2) (2,3)(8,2) (1,4)(7,3) (1,5)(7,4) (1,1)[0.1]{} (4,0.5)[0.1]{} (7,0)[0.1]{} (1,2)[0.1]{} (4,1.5)[0.1]{} (7,1)[0.1]{} (0,3)[0.1]{} (3,2.5)[0.1]{} (6,2)[0.1]{} (2,3)[0.1]{} (5,2.5)[0.1]{} (8,2)[0.1]{} (1,4)[0.1]{} (4,3.5)[0.1]{} (7,3)[0.1]{} (1,5)[0.1]{} (4,4.5)[0.1]{} (7,4)[0.1]{} (4,0.5)[$a$]{} (1,1)[$b$]{} (7,1)[$c$]{} (6,2)[$d$]{} (8,2)[$e$]{} (7,4)[$f$]{} (4,4.5)[$\alpha$]{} (1,4)[$\gamma$]{} (0,3)[$\delta$]{} (2,3)[$\epsilon$]{} Join-irreducible elements are $a,b,c,d,e,f$, while meet-irreducible ones are $\alpha,b,\gamma,\delta,\epsilon,f$. We propose as $\vee$-negation the following: $x$ $n(x)$ $x$ $n(x)$ ----- ---------- ----- ------------ $a$ $\alpha$ $d$ $\epsilon$ $b$ $f$ $e$ $\delta$ $c$ $\gamma$ $f$ $b$ Let us consider a possibility distribution satisfying $$\pi(c)<\pi(d)<\pi(e)<\pi(a)<\pi(f)<\pi(b)=1.$$ (observe that the sequence $c,d,e,a,f,b$ is non decreasing, as requested). We apply the procedure of Th. \[th:2\]. For $b$, we have $n(b)=f=c\vee d\vee e\vee f$, and for $f$, we have $n(f)=b=a\vee b$. Hence the first join-irreducible element of the sequence, $\iota_6$, is $a$ (not in $\eta(f)$, and minimal in $\eta(b)$). Table \[tab:1\] summarizes all the steps. step $k$ $x$ $n(x)$ $\eta(n(x))$ $\iota_k$ chain ---------- ----- ------------ -------------- ----------- ----------------------------- 6 $b$ $f$ $c,d,e,f$ $a$ $a$ 5 $f$ $b$ $a,b$ $c$ $a\vee c$ 4 $a$ $\alpha$ $a,c,d,e,f$ $b$ $a\vee c\vee b$ 3 $e$ $\delta$ $a,b,c,d$ $e$ $a\vee c\vee b\vee e$ 2 $d$ $\epsilon$ $a,b,c,e$ $d$ $a\vee c\vee b\vee e\vee d$ 1 $c$ $\gamma$ $a,b,c,d,e$ $f$ $\top$ : Computation of $C_\pi$[]{data-label="tab:1"} The maximal chain is in gray on Fig. \[fig:lat3\]. We deduce that: $$\begin{aligned} \pi(b) &=1\\ \pi(f) &=1-m(a)\\ \pi(a) &=1-m(a) -m(a\vee c)\\ \pi(e) &=1-m(a) -m(a\vee c) - m(a\vee c\vee b)\\ \pi(d) &=1-m(a) -m(a\vee c) - m(a\vee c\vee b) - m(a\vee c\vee b\vee e)\\ \pi(c) &=1-m(a) -m(a\vee c) - m(a\vee c\vee b) - m(a\vee c\vee b\vee e) - m(a\vee c\vee b\vee e\vee d)\end{aligned}$$ from which we deduce $$\begin{aligned} m(a) &= \pi(b)-\pi(f)\\ m(a\vee c ) &= \pi(f) -\pi(a)\\ m(a\vee c\vee b) &= \pi(a)-\pi(e)\\ m(a\vee c\vee b\vee e) &= \pi(e) -\pi(d)\\ m(a\vee c\vee b\vee e\vee d) &= \pi(d) -\pi(c)\\ \end{aligned}$$ and $m(\top) = 1-m(a) -m(a\vee c) - m(a\vee c\vee b) - m(a\vee c\vee b\vee e) - m(a\vee c\vee b\vee e\vee d)= \pi(c)$. Acknowledgment ============== The author addresses all his thanks to Bruno Leclerc and Bernard Monjardet for fruitful discussions on negations.

--- abstract: 'We resolve a basic problem on subspace distances that often arises in applications: How can the usual Grassmann distance between equidimensional subspaces be extended to subspaces of different dimensions? We show that a natural solution is given by the distance of a point to a Schubert variety within the Grassmannian. This distance reduces to the Grassmann distance when the subspaces are equidimensional and does not depend on any embedding into a larger ambient space. Furthermore, it has a concrete expression involving principal angles, and is efficiently computable in numerically stable ways. Our results are largely independent of the Grassmann distance — if desired, it may be substituted by any other common distances between subspaces. Our approach depends on a concrete algebraic geometric view of the Grassmannian that parallels the differential geometric perspective that is well-established in applied and computational mathematics.' address: - 'Department of Mathematics, University of Chicago, Chicago, IL 60637.' - 'Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL 60637.' author: - Ke Ye - 'Lek-Heng Lim' title: Schubert varieties and distances between subspaces of different dimensions --- Introduction {#sec:motivation} ============ Biological data (e.g. gene expression levels, metabolomic profile), image data (e.g.  <span style="font-variant:small-caps;">mri</span> tractographs, movie clips), text data (e.g. blogs, tweets), etc., often come in the form of a set of feature vectors $a_1,\dots,a_m \in \mathbb{R}^d$ and can be conveniently represented by a matrix $A \in \mathbb{R}^{m \times d}$ (e.g. gene-microarray matrices of gene expression levels, frame-pixel matrices of grey scale values, term-document matrices of term frequencies-inverse document frequencies). In modern applications, it is often the case that one will encounter an exceedingly large sample size $m$ (massive) or an exceedingly large number of variables $d$ (high-dimensional) or both. The raw data $A$ is usually less interesting and informative than the spaces it defines, e.g. its row and column spaces or its principal subspaces. Moreoever, it often happens that $A$ can be well-approximated by a subspace $\mathbf{A} \in \operatorname{Gr}(k,n)$ where $k \ll m$ and $n \ll d$. The process of getting from $A$ to $\mathbf{A}$ is well-studied, e.g. randomly sample a subset of representative landmarks or compute principal components. Subspace-valued data appears in a wide range of applications: computer vision [@MYDF; @VidMaSas2005], bioinformatics [@HH], machine learning [@HRS; @LZ], communication [@LHS; @ZT], coding theory [@AC; @BN; @CHS; @DHST], statistical classification [@HamLee2008], and system identification [@MYDF]. In computational mathematics, subspaces arise in the form of Krylov subspaces [@LS] and their variants [@C], as subspaces of structured matrices (e.g. Toeplitz, Hankel, banded), and in recent developments such as compressive sensing (e.g. Grassmannian dictionaries [@SH], online matrix completion [@BNR]). One of the most basic problems with subspaces is to define a notion of separation between them. The solution is well-known for subspaces of the same dimension $k$ in $\mathbb{R}^n$. These are points on the Grassmannian $\operatorname{Gr}(k,n)$, a Riemannian manifold, and the geodesic distance between them gives us an intrinsic distance. The *Grassmann distance* is independent of the choice of coordinates and can be readily related to principal angles and thus computed via the singular value decomposition (<span style="font-variant:small-caps;">svd</span>): For subspaces $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$, form matrices $A, B \in \mathbb{R}^{n \times k}$ whose columns are their respective orthonormal bases, then $$\label{eq:grassdist} d(\mathbf{A},\mathbf{B}) = \Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2},$$ where $\theta_i = \cos^{-1} \bigl(\sigma_i(A^\mathsf{T} B)\bigr)$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$. This is the geodesic distance on the Grassmannian viewed as a Riemannian manifold. There are many other common distances defined on Grassmannians — Asimov, Binet–Cauchy, chordal, Fubini–Study, Martin, Procrustes, projection, spectral (see Table \[tab:distances\]). What if the subspaces are of different dimensions? In fact, if one examines the aforementioned applications, one invariably finds that the most general settings for each of them would fall under this situation. The restriction to equidimensional subspaces thus somewhat limits the utility of these applications. For example, the principal subspaces of two matrices $A$ and $B$ for a given noise level would typically be of different dimensions, since there is no reason to expect the number of singular values of $A$ above a given threshold to be the same as that of $B$. As such one may also find many applications that involve distances between subspaces of different dimensions: numerical linear algebra [@BES; @SS], information retrieval [@ref2a; @text], facial recognition [@ref2b; @face], image classification [@ref2a; @ref2b], motion segmentation [@motion; @ref2c; @ref2d], <span style="font-variant:small-caps;">eeg</span> signal analysis [@eeg], mechanical engineering [@mech], economics [@econ], network analysis [@network], blog spam detection [@blog], and decoding colored barcodes [@bar]. These applications are all based on two existing proposals for a distance between subspaces of different dimensions: The *containment gap* [@Kato pp. 197–199] and the *symmetric directional distance* [@SWF; @WWF]. They are however somewhat ad hoc and bear little relation to the natural geometry of subspaces. Also, it is not clear what they are suppose to measure and neither restricts to the Grassmann distance when the subspaces are of the same dimension. Our main objective is to show that there is an alternative definition that does generalize the Grassmann distance but our work will also shed light on these two distances. Main Contributions ------------------ Our main result (see Theorem \[thm1\]) can be stated in simple linear algebraic terms: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$, assuming $k< l$ without loss of generality, the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives the distance between $\mathbf{A}$ and $\mathbf{B}$. Taking an algebraic geometric point-of-view: 1. \[quote\] *The distance between subspaces of different dimensions is the distance between a point and a certain Schubert variety within the Grassmannian.* This distance has the following properties, established in Section \[sec:main\]: 1. readily computable via <span style="font-variant:small-caps;">svd</span>; 2. restricts to the usual Grassmann distance for subspaces of the same dimension; 3. independent of the choice of local coordinates; 4. independent of the dimension of the ambient space (i.e., $n$); 5. \[other\] may be defined in conjunction with other common distances in Table \[tab:distances\]. We will see in Section \[sec:existing\] that the two existing notions of distance between subspaces of different dimensions are special cases of . Evidently, the word ‘distance’ in ($\ast$) is used in the sense of a distance of a point to a set. For example, if a subspace is contained in another, then the distance between them is zero, even if they are distinct subspaces. Thus the distance in ($\ast$) is not a metric[^1]. In Section \[sec:metric\], we define a metric on the set of subspaces of all dimensions using an analogue of our main result: Given any two subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with $k< l$, the distance from $\mathbf{A}$ to the furthest $k$-dimensional subspace contained in $\mathbf{B}$ *equals* the distance from $\mathbf{B}$ to the furthest $l$-dimensional subspace that contains $\mathbf{A}$. Their common value gives a metric between $\mathbf{A}$ and $\mathbf{B}$. The most interesting metrics for subspaces of different dimensions can be found in Table \[tab:metrics\]. In Section \[sec:volume\], we obtain a volumetric analogue of our main result: Given two arbitrary subspaces in $\mathbb{R}^n$, $\mathbf{A}$ of dimension $k$ and $\mathbf{B}$ of dimension $l$ with $k < l$, we show that the probability a random $l$-dimensional subspace contains $\mathbf{A}$ *equals* the probability a random $k$-dimensional subspace is contained in $\mathbf{B}$. The far-reaching work [@EAS] popularized the basic *differential geometry* of Stiefel and Grassmannian manifolds by casting the discussions concretely in terms of matrices. Subsequent works, notably [@AMSbook; @AMS; @AMSV], have further enriched this concrete matrix-based approach. A secondary objective of our article is to do the same for the basic *algebraic geometry* of Grassmannians. In particular, we introduce some of the objects in Table \[tab:objects\] to an applied and computational mathematics readership. The proofs of our main results essentially use only the <span style="font-variant:small-caps;">svd</span>. Everything else is explained within the article and accessible to anyone willing to accept a small handful of unfamiliar terminologies and facts on faith. -------------------------------- --------------------------------------------- ------------------------------------------------------------------------------------------------------------- ------------------- *Grassmannian* $\operatorname{Gr}(k,n)$ models $k$-dimensional subspaces in $\mathbb{R}^n$ §\[sec:Grass\] *Infinite Grassmannian* $\operatorname{Gr}(k,\infty)$ models $k$-dimensional subspaces regardless of ambient space §\[sec:infty\] *Doubly-infinite Grassmannian* $\operatorname{Gr}(\infty,\infty)$ models subspaces of all dimensions regardless of ambient space §\[sec:dinfty\] *Flag variety* $\operatorname{Flag}(k_1,\dots,k_m,n)$ models nested sequences of subspaces in $\mathbb{R}^n$; $\operatorname{Flag}(k,n) = \operatorname{Gr}(k,n)$ §\[sec:schubert\] *Schubert variety* $\Omega(\mathbf{X}_1,\dots,\mathbf{X}_m,n)$ ‘linearly constrained’ subset of $\operatorname{Gr}(k,n)$ §\[sec:schubert\] -------------------------------- --------------------------------------------- ------------------------------------------------------------------------------------------------------------- ------------------- Grassmannian of linear subspaces {#sec:Grass} ================================ We will selectively review some basic properties of the Grassmannian. The differential geometric perspectives are drawn from [@Husemoller; @MS], the more concrete matrix-theoretic view from [@AMS; @EAS; @Wong], and the computational aspects from [@GVL]. We fix the ambient space $\mathbb{R}^n$. A *$k$-plane* is a $k$-dimensional subspace of $\mathbb{R}^n$. A *$k$-frame* is an ordered orthonormal basis of a $k$-plane, regarded as an $n \times k$ matrix whose columns $a_1,\dots, a_k$ are the orthonormal basis vectors. A *flag* is a strictly increasing sequence of nested subspaces, $\mathbf{X}_0\subset \mathbf{X}_1\subset \cdots \subset \mathbf{X}_m\subset \mathbb{R}^n$; it is *complete* if $m=n$. We write $\operatorname{Gr}(k,n)$ for the *Grassmannian* of $k$-planes in $\mathbb{R}^n$, $\operatorname{V}(k,n)$ for the *Stiefel manifold* of orthonormal $k$-frames, and $\operatorname{O}(n) \coloneqq \operatorname{V}(n,n)$ for the *orthogonal group* of $n \times n$ orthogonal matrices. $\operatorname{V}(k,n)$ may be regarded as a homogeneous space, $$\operatorname{V}(k,n) \cong \operatorname{O}(n)/\operatorname{O}(n-k),$$ or more concretely as the set of $n \times k$ matrices with orthonormal columns. There is a *right action* of $\operatorname{O}(k)$ on $\operatorname{V}(k,n)$: For $Q\in \operatorname{O}(k)$ and $A \in \operatorname{V}(k,n)$, the action yields $AQ \in \operatorname{V}(k,n)$ and the resulting homogeneous space is $\operatorname{Gr}(k,n)$, i.e., $$\label{eq:homo} \operatorname{Gr}(k,n) \cong \operatorname{V}(k,n)/\operatorname{O}(k) \cong \operatorname{O}(n)/\bigl(\operatorname{O}(n-k) \times \operatorname{O}(k)\bigr).$$ In this picture, a subspace $\mathbf{A} \in \operatorname{Gr}(k,n)$ is identified with an equivalence class comprising all its $k$-frames $\{ AQ \in \operatorname{V}(k,n): Q \in \operatorname{O}(k)\}$. Note that $\operatorname{span}(AQ) = \operatorname{span}(A)$ for all $Q \in \operatorname{O}(k)$. There is a *left action* of $\operatorname{O}(n)$ on $\operatorname{Gr}(k,n)$: For any $Q \in \operatorname{O}(n)$ and $\mathbf{A} = \operatorname{span}(A) \in \operatorname{Gr}(k,n)$ where $A$ is a $k$-frame of $\mathbf{A}$, the action yields $$\label{eq:action} Q \cdot \mathbf{A}\coloneqq \operatorname{span}(QA) \in \operatorname{Gr}(k,n).$$ This action is transitive as any $k$-plane can be rotated onto any other $k$-plane by some $Q \in \operatorname{O}(n)$. A $k$-plane $\mathbf{A} \in \operatorname{Gr}(k,n)$ will be denoted in boldface; the corresponding italized letter $A =[a_1,\dots,a_k] \in \operatorname{V}(k,n)$ will denote a $k$-frame of $\mathbf{A}$. $\operatorname{Gr}(k,n)$ and $\operatorname{V}(k,n)$ are smooth manifolds of dimensions $k(n-k)$ and $nk-k(k+1)/2$ respectively. As a set of $n\times k$ matrices, $\operatorname{V}(k,n)$ is a submanifold of $\mathbb{R}^{n\times k}$ and inherits a Riemannian metric from the Euclidean metric on $\mathbb{R}^{n\times k}$, i.e., given $A=[a_1,\dots, a_k]$ and $B=[b_1,\dots, b_k]$ in $T_X \operatorname{V}(k,n)$, the tangent space at $X \in \operatorname{V}(k,n)$, the Riemannian metric $g$ is defined by $g_X(A,B)=\sum_{i=1}^k a_i^\mathsf{T} b_i= \operatorname{tr}(A^\mathsf{T}B)$. As $g$ is invariant under the action of $\operatorname{O}(k)$, it descends to a Riemannian metric on $\operatorname{Gr}(k,n)$ and in turn induces a geodesic distance on $\operatorname{Gr}(k,n)$ which we define below. Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ respectively. Let $r \coloneqq \min (k,l)$. The $i$th *principal vectors* $(p_{i},q_{i})$, $i=1,\dots,r$, are defined recursively as solutions to the optimization problem$$\begin{tabular} [c]{rcl}maximize & & $p^{\mathsf{T}}q$\\ subject to & & $p\in \mathbf{A},\; p^{\mathsf{T}}p_{1}=\dots=p^{\mathsf{T}}p_{i-1}=0,\;\lVert p\rVert=1,$\\ & & $q\in \mathbf{B}, \; q^{\mathsf{T}}q_{1}=\dots=q^{\mathsf{T}}q_{i-1}=0,\;\lVert q\rVert=1,$\end{tabular}$$ for $i=1,\dots,r$. The *principal angles* are then defined by$$\cos\theta_{i}=p_{i}^\mathsf{T}q_{i}, \quad i = 1,\dots,r.$$ Clearly $0\le \theta_{1}\leq\dots\leq\theta_r \le \pi/2$. We will let $\theta_i(\mathbf{A},\mathbf{B})$ denote the $i$th principal angle between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. Principal vectors and principal angles may be readily computed using <span style="font-variant:small-caps;">qr</span> and <span style="font-variant:small-caps;">svd</span> [@BG; @GVL]. Let $A = [a_1,\dots,a_k]$ and $B =[ b_1,\dots, b_l]$ be orthonormal bases and let $$\label{eq:svd} A^\mathsf{T} B = U\Sigma V^\mathsf{T}$$ be the full <span style="font-variant:small-caps;">svd</span> of $A^\mathsf{T} B$, i.e., $U \in \operatorname{O}(k)$, $V \in \operatorname{O}(l)$, $\Sigma = \left[\begin{smallmatrix}\Sigma_1 & 0 \\ 0 & 0 \end{smallmatrix}\right]\in \mathbb{R}^{k \times l}$ with $\Sigma_1 = \operatorname{diag}(\sigma_1,\dots,\sigma_r) \in \mathbb{R}^{r \times r}$ where $\sigma_1 \ge \dots \ge \sigma_{r} \ge 0$. The principal angles $\theta_{1}\leq\dots\leq\theta_{r}$ are given by $$\label{eq:angles} \theta_i = \cos^{-1} \sigma_i, \quad i = 1,\dots,r.$$ It is customary to write $A^\mathsf{T} B = U (\cos\Theta) V^\mathsf{T}$, where $\Theta = \operatorname{diag}(\theta_1,\dots,\theta_{r},1,\dots,1) \in \mathbb{R}^{k \times l}$ and $\Theta_1 = \operatorname{diag}(\theta_1,\dots,\theta_{r}) \in \mathbb{R}^{r \times r}$. Consider the column vectors, $$AU = [p_1,\dots,p_k], \quad BV = [q_1,\dots,q_l].$$ The principal vectors are given by $(p_1,q_1),\dots,(p_r, q_r)$. Strictly speaking, principal vectors come in pairs but we will also call the vectors $p_{r+1},\dots,p_k$ (if $r = l < k$) or $q_{r+1},\dots,q_l$ (if $r = k < l$) principal vectors for lack of a better term. We will use the following fact from [@GVL Theorem 6.4.2]. \[prop:angles\] Let $r = \min (k,l)$ and $\theta_1,\dots,\theta_{r}$ and $(p_1,q_1),\dots,(p_{r}, q_{r})$ be the principal angles and principal vectors between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ respectively. If $m < r$ is such that $1 = \cos \theta_1 = \dots = \cos \theta_m > \cos \theta_{m+1}$, then $$\mathbf{A} \cap \mathbf{B} = \operatorname{span} \{p_1,\dots, p_m \} = \operatorname{span} \{q_1,\dots, q_m \}.$$ If $k = l$, the geodesic distance between $\mathbf{A}$ and $\mathbf{B}$ in $\operatorname{Gr}(k,n)$ is called the *Grassmann distance* and is given by $$\label{eq:grassdist1} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2} = \lVert\cos^{-1}\Sigma\rVert_{F}.$$ An explicit expression for the geodesic [@AMS] connecting $\mathbf{A}$ to $\mathbf{B}$ on $\operatorname{Gr}(k,n)$ that minimizes the Grassmann distance is given by $\gamma : [0,1] \to \operatorname{Gr}(k,n)$, $$\label{eq:geodesic} \gamma(t) = \operatorname{span}( AU\cos t\Theta+ Q\sin t \Theta )$$ where $M= Q(\tan\Theta) U^{\mathsf{T}}$ is a condensed <span style="font-variant:small-caps;">svd</span> of the matrix $$M \coloneqq (I - AA^\mathsf{T})B(A^\mathsf{T} B)^{-1} \in \mathbb{R}^{n \times k}$$ and where $U \in \operatorname{O}(k)$ and $\Theta=\operatorname{diag}(\theta_1,\dots,\theta_k)\in \mathbb{R}^{k\times k}$ are as in and . Note that if $\cos\Theta=\Sigma$, then $\tan\Theta=(\Sigma^{-2}-I)^{1/2}$. Also, $\gamma(0) =\mathbf{A}$ and $\gamma(1) = \mathbf{B}$. Aside from the Grassmann distance, there are many well-known distances between subspaces [@BN; @DD; @DHST; @EAS; @HamLee2008]. We present some of these in Table \[tab:distances\]. -------------- --------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- *Principal angles* *Orthonormal bases* Asimov $d^{\alpha}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \theta_k$ $\cos^{-1} \lVert A^\mathsf{T} B \rVert_2$ Binet–Cauchy $d^{\beta}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(1 - \prod\nolimits_{i=1}^k\cos^2\theta_i\right)^{1/2}$ $(1 - (\det A^\mathsf{T} B)^2)^{1/2}$ Chordal $d^{\kappa}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(\sum\nolimits_{i=1}^k\sin^2\theta_i\right)^{1/2}$ $\frac{1}{\sqrt{2}} \lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_F$ Fubini–Study $d^{\phi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \cos^{-1} \left(\prod\nolimits_{i=1}^k\cos \theta_i\right)$ $\cos^{-1} \lvert \det A^\mathsf{T} B \rvert$ Martin $d^{\mu}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left( \log \prod\nolimits_{i=1}^k 1/\cos^2 \theta_i \right)^{1/2}$ $(-2 \log \det A^\mathsf{T} B)^{1/2}$ Procrustes $d^{\rho}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\left(\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\right)^{1/2}$ $\lVert AU - BV \rVert_F$ Projection $d^{\pi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \sin \theta_k$ $\lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_2$ Spectral $d^{\sigma}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\sin (\theta_k/2)$ $\lVert AU - BV \rVert_2$ -------------- --------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------- The value $\sin \theta_1$ is sometimes called the *max correlation distance* [@HamLee2008] or spectral distance [@DHST] but it is not a distance in the sense of a metric (can be zero for a pair of distinct subspaces) and thus not listed. The spectral distance $d^{\sigma}_{\operatorname{Gr}(k,n)}$ is also called chordal $2$-norm distance [@BN]. For each distance in Table \[tab:distances\] defined for equidimensional $\mathbf{A}$ and $\mathbf{B}$, Theorem \[thm:othermetrics\] provides a corresponding version for when $\dim \mathbf{A} \ne \dim \mathbf{B}$. The fact that all these distances in Table \[tab:distances\] depend on the principal angles is not a coincidence — the result [@Wong Theorem 3] implies the following. \[thm:wong\] Any notion of distance between $k$-dimensional subspaces in $\mathbb{R}^n$ that depends only on the relative positions of the subspaces, i.e., invariant under any rotation in $\operatorname{O}(n)$, must be a function of their principal angles. To be more specific, if a distance $d : \operatorname{Gr}(k,n) \times \operatorname{Gr}(k,n) \to [0, \infty)$ satisfies $$d(Q \cdot \mathbf{A}, Q \cdot \mathbf{B}) =d(\mathbf{A}, \mathbf{B}),$$ for all $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$ and all $Q \in \operatorname{O}(n)$, where the action is as defined in , then $d$ must be a function of $\theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$. We will next introduce the *infinite Grassmannian* $\operatorname{Gr}(k,\infty)$ to show that these distances between subspaces are independent of the dimension of their ambient space. The Infinite Grassmannian {#sec:infty} ========================= One way of defining a distance between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$ where $k \ne l$ is to first isometrically embed $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(l,n)$ into an ambient Riemannian manifold and then define the distance between $\mathbf{A}$ and $\mathbf{B}$ as their distance in the ambient space. This approach is taken in [@CHS; @SLLM], via an isometric embedding of $\operatorname{Gr}(0,n), \operatorname{Gr}(1,n), \dots, \operatorname{Gr}(n,n )$ into a sphere of dimension $(n-1)(n+2)/2$. Such a distance suffers from two shortcomings: It is not intrinsic to the Grassmannian and it depends on both the embedding and the ambient space. The distance that we propose in Section \[sec:main\] will depend only on the intrinsic distance of the Grassmannian and is independent of $n$, i.e., a $k$-plane $\mathbf{A}$ and an $l$-plane $\mathbf{B}$ in $\mathbb{R}^n$ will have the same distance if we regard them as subspaces in $\mathbb{R}^m$ for any $m \ge \min(k,l)$. We will first establish this for the special case $k = l$. Consider the inclusion map $\iota_n:\mathbb{R}^n\to \mathbb{R}^{n+1}$, $\iota_n(x_1,\dots, x_n)=(x_1,\dots, x_n,0)$. It is easy to see that $\iota_n$ induces a natural inclusion of $\operatorname{Gr}(k,n)$ into $\operatorname{Gr}(k,n+1)$ which we will also denote by $\iota_n$. For any $m > n$, composition of successive natural inclusions gives the inclusion $\iota_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(k,m)$, where $\iota_{nm} \coloneqq \iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1}$. To be more concrete, if $A \in \mathbb{R}^{n \times k}$ has orthonormal columns, then $$\label{eq:iota} \iota_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(k,m), \qquad \operatorname{span} (A) \mapsto \operatorname{span} \left( \begin{bmatrix}A \\ 0 \end{bmatrix} \right),$$ where the zero block matrix is $(m-n) \times k$ so that $\left[ \begin{smallmatrix}A \\ 0 \end{smallmatrix} \right] \in \mathbb{R}^{m \times k}$. For a fixed $k$, the family of Grassmannians $\{ \operatorname{Gr}(k,n): n\in \mathbb{N}, \; n \ge k \}$ together with the inclusion maps $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ for $m > n$ form a direct system. The *infinite Grassmannian* of $k$-planes is defined to be the direct limit of this system in the category of topological spaces and denoted by $$\operatorname{Gr}(k,\infty)\coloneqq \varinjlim\operatorname{Gr}(k,n).$$ Those unfamiliar with the notion of direct limits may simply take $$\operatorname{Gr}(k, \infty) = \bigcup\nolimits_{n=k}^\infty \operatorname{Gr}(k,n),$$ where we regard $\operatorname{Gr}(k,n) \subset \operatorname{Gr}(k,n+1)$ by identifying $\operatorname{Gr}(k,n)$ with $\iota_n\bigl(\operatorname{Gr}(k,n)\bigr)$. With this identification, we no longer need to distinguish between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and its image $\iota_{n}(\mathbf{A})\in \operatorname{Gr}(k,n+1)$ and may regard $\mathbf{A} \in \operatorname{Gr}(k,m)$ for all $m > n$. We now define a distance $d_{\operatorname{Gr}(k, \infty)}$ on $\operatorname{Gr}(k,\infty)$ that is consistent with the Grassmann distance on $\operatorname{Gr}(k,n)$ for all $n$ sufficiently large. \[lem:infty\] The natural inclusion $\iota_n:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,n+1)$ is isometric, i.e., $$\label{eq:iso} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,n+1)}\bigl(\iota_n(\mathbf{A}),\iota_n(\mathbf{B})\bigr).$$ Repeated applications of yields $$\label{eq:iso1} d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,m)}\bigl(\iota_{nm}(\mathbf{A}),\iota_{nm}(\mathbf{B})\bigr)$$ for all $m > n$ and if we identify $\operatorname{Gr}(k,n)$ with $\iota_n\bigl(\operatorname{Gr}(k,n)\bigr)$, we may rewrite as $$d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(k,m)}(\mathbf{A},\mathbf{B})$$ for all $m > n$. If $a\in \mathbb{R}^n$, we write $\hat{a} = \left[\begin{smallmatrix} a \\ 0 \end{smallmatrix} \right]\in \mathbb{R}^{n+1}$. Let $A = [a_1,\dots, a_k]$ and $B = [b_1,\dots,b_k]$ be any orthonormal bases of $\mathbf{A}$ and $\mathbf{B}$ respectively. By the definition of $\iota_n$, $\iota_n(\mathbf{A})$ is the subspace in $\mathbb{R}^{n+1}$ spanned by an orthonormal basis that we will denote by $\iota_n(A) \coloneqq [\hat{a}_1,\dots, \hat{a}_k] \in \mathbb{R}^{(n+1) \times k}$. Hence we have $$\iota_n(A)^\mathsf{T} \iota_n(B) = \begin{bmatrix} A^\mathsf{T} B \\ 0 \end{bmatrix}.$$ By the expression for Grassmann distance in , we see that must hold. Since the inclusion of $\operatorname{Gr}(k,n)$ in $\operatorname{Gr}(k,n+1)$ is isometric, a geodesic in $\operatorname{Gr}(k,n)$ remains a geodesic in $\operatorname{Gr}(k,n+1)$. Given $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(k, \infty)$, there must exist some $n$ sufficiently large so that both $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(k, n)$ and in which case we define the distance between $\mathbf{A}$ and $\mathbf{B}$ in $ \operatorname{Gr}(k, \infty)$ to be $$d_{\operatorname{Gr}(k, \infty)}(\mathbf{A},\mathbf{B}) \coloneqq d_{\operatorname{Gr}(k, n)}(\mathbf{A},\mathbf{B}).$$ By Lemma \[lem:infty\], this value is independent of our choice of $n$ and is the same for all $m \ge n$. In particular, $d_{\operatorname{Gr}(k, \infty)}$ is well-defined and yields a distance on $\operatorname{Gr}(k,\infty)$. We summarize these observations below. \[cor:infty\] The Grassmann distance between two $k$-planes in $\operatorname{Gr}(k,n)$ is the geodesic distance in $\operatorname{Gr}(k,\infty)$ and is therefore independent of $n$. Also, the expression for a distance minimizing geodesic in $\operatorname{Gr}(k,n)$ extends to $\operatorname{Gr}(k,\infty)$. Lemma \[lem:infty\] also holds for other distances on $\operatorname{Gr}(k,n)$ in Table \[tab:distances\], allowing us to define them on $\operatorname{Gr}(k, \infty)$. \[lem:inclusion\] For all $m > n$, the inclusion $\iota_{nm}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(k,m)$ is isometric when $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(k,m)$ are both equipped with the same distance in Table \[tab:distances\], i.e., $$d^*_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=d^*_{\operatorname{Gr}(k,m)}(\iota_{nm}(\mathbf{A}),\iota_{nm}\bigl(\mathbf{B})\bigr),$$ $* = \alpha, \beta, \kappa, \phi, \mu, \rho, \pi, \sigma$. Consequently $d^{\ast}_{\operatorname{Gr}(k,\infty)}$ is well-defined. $d^{\ast}_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})$ and $d^{\ast}_{\operatorname{Gr}(k,n+1)}\bigl(\iota_n(\mathbf{A}),\iota_n\bigl(\mathbf{B})\bigr)$ depend only on the principal angles between $\mathbf{A}$ and $\mathbf{B}$, so the distance remains unchanged under $\iota_n$. Repeated applications to $\iota_n \circ \iota_{n+1} \circ \dots \circ \iota_{m-1} = \iota_{nm}$ yield the required isometry. Distances between subspaces of different dimensions {#sec:main} =================================================== We now address our main problem. The proposed notion of distance will be that of a point $x \in X$ to a set $S \subset X$ in a metric space $(X,d)$. Recall that this is defined by $d(x, S) \coloneqq \inf\{d(x,y) : y \in S \}$. For us, $X$ is a Grassmannian, therefore compact, and so $d(x,S)$ is finite. Also, $S$ will be a closed subset and so we write $\min$ instead of $\inf$. We will introduce two possible candidates for $S$. \[def:Omega\] Let $k,l, n \in \mathbb{N}$ be such that $k\le l \le n$. For any $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in \operatorname{Gr}(l,n)$, we define the subsets $$\Omega_{+}(\mathbf{A})\coloneqq\bigl\{\mathbf{X}\in \operatorname{Gr}(l,n) : \mathbf{A}\subseteq \mathbf{X}\bigr\},\quad \Omega_{-}(\mathbf{B})\coloneqq\bigl\{\mathbf{Y}\in \operatorname{Gr}(k,n) : \mathbf{Y} \subseteq \mathbf{B}\bigr\}.$$ We will call $\Omega_{+}(\mathbf{A})$ the *Schubert variety of $l$-planes containing $\mathbf{A}$* and $\Omega_{-}(\mathbf{B})$ the *Schubert variety of $k$-planes contained in $\mathbf{B}$*. As we will see in Section \[sec:schubert\], $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are indeed Schubert varieties and therefore closed subsets of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. Furthermore, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are uniquely determined by $\mathbf{A}$ and $\mathbf{B}$ (see Proposition \[prop:Omega1\]) and may be regarded as ‘sub-Grassmannians’ of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively (see Proposition \[prop:OmegaGrass\]). How could one define the distance between a subspace $\mathbf{A}$ of dimension $k$ and a subspace $\mathbf{B}$ of dimension $l$ in $\mathbb{R}^n$ when $k \ne l$? We may assume $k < l\le n$ without loss of generality. In which case a very natural solution is to define the required distance $\delta(\mathbf{A}, \mathbf{B}) $ as that between the $k$-plane $\mathbf{A}$ and the closest $k$-plane $\mathbf{Y}$ contained in $\mathbf{B}$, measured within $\operatorname{Gr}(k,n)$. In other words, we want the Grassmann distance from $\mathbf{A}$ to the closed subset $\Omega_{-}(\mathbf{B})$, $$\label{eq:minus} \delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) = \min\bigl\{ d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{Y}) : \mathbf{Y}\in \Omega_{-}(\mathbf{B})\bigr\}.$$ This has the advantage of being intrinsic — the distance $\delta(\mathbf{A}, \mathbf{B}) $ is measured in $d_{\operatorname{Gr}(k,n)}$ and is defined wholly within $\operatorname{Gr}(k,n)$ without any embedding of $\operatorname{Gr}(k,n)$ into an arbitrary ambient space. Furthermore, by the property of $d_{\operatorname{Gr}(k,n)}$ in Corollary \[cor:infty\], $\delta(\mathbf{A}, \mathbf{B}) $ does not depend on $n$ and takes the same value for any $m \ge n$. We illustrate this in Figure \[fig:sphere\]: The sphere is intended to be a depiction of $\operatorname{Gr}(1,3)$ though to be accurate antipodal points on the sphere should be identified. (-4,0) circle\[radius=0.05\]; at (-4.08,0) [$\mathbf{A}$]{}; (0,4) arc (90:270:2cm and 4cm); (0,4) arc (90:-90:2cm and 4cm); (0,0) circle (4cm); (0,0) circle (4cm); at (3,-3) [$\operatorname{Gr}(1,3)$]{}; at (0,2.5) [$\Omega_{-}(\mathbf{B})$]{}; at (-3,-1.25) [$\gamma$]{}; (-4,0) arc (180:243:4cm and 2cm); (-1.8,-1.8) circle\[radius=0.05\]; at (-2,-1.9) [$\mathbf{X}$]{}; However, it is equally natural to define $\delta(\mathbf{A}, \mathbf{B}) $ as the distance between the $l$-plane $\mathbf{B}$ and the closest $l$-plane $\mathbf{Y}$ containing $\mathbf{A}$, measured within $\operatorname{Gr}(l,n)$. In other words, we could have instead defined it as the Grassmann distance from $\mathbf{B}$ to the closed subset $\Omega_{+}(\mathbf{A})$, $$\label{eq:plus} \delta(\mathbf{A}, \mathbf{B}) \coloneqq d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr) = \min\bigl\{ d_{\operatorname{Gr}(l,n)}(\mathbf{B},\mathbf{X}) : \mathbf{X}\in \Omega_{+}(\mathbf{A})\bigr\}.$$ It will have the same desirable features as the one in except that the distance is now measured in $d_{\operatorname{Gr}(l,n)}$ and within $\operatorname{Gr}(l,n)$. It turns out that the two values in and are equal, allowing us to define $\delta(\mathbf{A}, \mathbf{B})$ as their common value. We will establish this equality and the properties of $\delta(\mathbf{A}, \mathbf{B})$ in the remainder of this section. The results are summarized in Theorem \[thm1\]. Our proof is constructive: In addition to showing the equality of and , it shows how one may explicitly find the closest points on Schubert varieties $\mathbf{X} \in \Omega_{-}(\mathbf{B})$ and $\mathbf{Y} \in \Omega_{+}(\mathbf{A})$ to any given point in the respective Grassmannians. \[thm1\] Let $\mathbf{A}$ be a subspace of dimension $k$ and $\mathbf{B}$ be a subspace of dimension $l$ in $\mathbb{R}^n$. Suppose $k \le l \le n$. Then $$\label{eq:equal} d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}\bigl(\mathbf{B})\bigr) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}\bigl(\mathbf{A})\bigr).$$ Their common value defines a distance $\delta(\mathbf{A}, \mathbf{B}) $ between the two subspaces with the following properties: 1. \[indp\] $\delta(\mathbf{A}, \mathbf{B}) $ is independent of the dimension of the ambient space $n$ and is the same for all $n \ge l+1$; 2. \[reduce\] $\delta(\mathbf{A}, \mathbf{B}) $ reduces to the Grassmann distance between $\mathbf{A}$ and $\mathbf{B}$ when $k = l$; 3. \[explicit\] $\delta(\mathbf{A}, \mathbf{B}) $ may be computed explicitly as $$\label{eq:grassdist2} \delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^{\min \{k,l\}}\theta_i(\mathbf{A},\mathbf{B})^2 \Bigr)^{1/2}$$ where $\theta_i(\mathbf{A},\mathbf{B})$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$, $i =1,\dots,\min(k,l)$. Rewriting as $$\min_{\mathbf{X}\in \Omega_{+}(\mathbf{A})}d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) = \min_{\mathbf{Y} \in \Omega_{-}(\mathbf{B})} d_{\operatorname{Gr}(k,n)}(\mathbf{Y},\mathbf{A}),$$ the equation says that the distance from $\mathbf{B}$ to the nearest $l$-dimensional subspace that contains $\mathbf{A}$ equals the distance from $\mathbf{A}$ to the nearest $k$-dimensional subspace contained in $\mathbf{B}$. This relation has several parallels. We will see that: 1. the Grassmann distance may be replaced by any of the distances in Table \[tab:distances\] (see Theorem \[thm:othermetrics\]); 2. ‘nearest’ may be replaced by ‘furthest’ and ‘min’ above replaced by ‘max’ when $n$ is sufficiently large (see Proposition \[prop:prob4\]); 3. ‘distance’ may be replaced by ‘volume’ with respect to the intrinsic uniform probability density on the Grassmannian (see Section \[sec:volume\]). We will prove Theorem \[thm1\] by way of the next two lemmas. \[lem:ineq1\] Let $k \le l \le n$ be positive integers. Let $\delta :\operatorname{Gr}(k,n)\times \operatorname{Gr}(l,n)\to [0,\infty)$ be the function defined by $$\delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k\theta_i^2\Bigr)^{1/2}$$ where $\theta_i \coloneqq \theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$. Then $$\delta(\mathbf{A},\mathbf{B})\ge d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr).$$ It suffices to find an $\mathbf{X}\in \Omega_{+}(\mathbf{A})$ such that $\delta(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})$. Let $(p_1,q_1),\dots, (p_k,q_k)$ be the principal vectors between $\mathbf{A}$ and $\mathbf{B}$. We will extend $q_1,\dots,q_k$ into an orthonormal basis of $\mathbf{B}$ by appending appropriate orthonormal vectors $q_{k+1},\dots,q_l$. The principal angles are given by $\theta_i=\cos^{-1} p_i^\mathsf{T} q_i$, $\lVert p_i \rVert =\lVert q_i \rVert=1$. If we take $\mathbf{X} \in \operatorname{Gr}(l,n)$ to be the subspace spanned by $p_1,\dots,p_k,q_{k+1},\dots, q_l$, then $$\begin{aligned} \label{eq:ineq1} d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) &= [(\cos^{-1} p_1^\mathsf{T} q_1)^2 + \dots + (\cos^{-1} p_k^\mathsf{T} q_k)^2 \nonumber \\ &\qquad + (\cos^{-1} q_{k+1}^\mathsf{T} q_{k+1})^2 + \dots + (\cos^{-1} q_l^\mathsf{T} q_l)^2 ]^{1/2}\\ &= [ \theta_1^2 + \dots + \theta_k^2 + 0^2 + \dots + 0^2]^{1/2} = \delta(\mathbf{A},\mathbf{B}). \nonumber \end{aligned}$$ We state the following well-known fact [@Horn Corollary 3.1.3] for easy reference and deduce a corollary that will be useful for Lemma \[lem:ineq2\]. \[prop:compare\] Let $k\le l\le n$ be positive integers. Suppose $B \in \mathbb{R}^{n\times l}$ and $B_k \in \mathbb{R}^{n\times k}$ is a submatrix obtained by removing any $l-k$ columns from $B$. Then the $i$th singular values satisfy $\sigma_i(B_k)\le \sigma_i(B)$ for $i=1,\dots, k$. \[cor:compare\] Let $B$ and $B_k$ be as in Proposition \[prop:compare\] and $\mathbf{B}$ and $\mathbf{B}_k$ be subspaces of $\mathbb{R}^n$ spanned by the column vectors of $B$ and $B_k$ respectively. Then for any subspace $\mathbf{A}$ of $\mathbb{R}^n$, the principal angles between the respective subspaces satisfy $$\theta_i(\mathbf{A},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{B}_k)$$ for $i = 1,\dots,\min(\dim \mathbf{A}, \dim \mathbf{B}_k)$. By appropriate orthogonalization if necessary, we may assume that $B$ and its submatrix $B_k$ are orthonormal bases of $\mathbf{B}$ and $\mathbf{B}_k$. Let $A$ be an orthonormal basis of $\mathbf{A}$. Then $\sigma_i(A^\mathsf{T}B)$ and $\sigma_i(A^\mathsf{T}B_k)$ take values in $[0,1]$. Since $\theta_i(\mathbf{A},\mathbf{B})= \cos^{-1}(\sigma_i\bigl(A^\mathsf{T}B)\bigr)$ and $\cos^{-1}$ is monotone decreasing in $[0,1]$, the result follows from $\sigma_i(A^\mathsf{T}B)\ge \sigma_i(A^\mathsf{T}B_k)$, by Proposition \[prop:compare\] applied to the submatrix $A^\mathsf{T}B_k$ of $A^\mathsf{T}B$. \[lem:ineq2\] Let $\mathbf{A}$, $\mathbf{B}$ be as in Lemma \[lem:ineq1\]. Then $\delta(\mathbf{A}, \mathbf{B})\le d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)$. Let $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$. Then $\mathbf{Y}$ is a $k$-dimensional subspace contained in $\mathbf{B}$ and in the notation of Corollary \[cor:compare\], we may write $\mathbf{Y} = \mathbf{B}_{k}$. By the same corollary we get $\theta_i(\mathbf{A},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{Y})$ for $i = 1, \dots, k$. Hence $$\label{eq:ineq2} \delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^{k}\theta_i(\mathbf{A},\mathbf{B})^2\Bigr)^{1/2} \le \Bigl(\sum\nolimits_{i=1}^{k}\theta_i(\mathbf{A},\mathbf{Y})^2\Bigr)^{1/2}=d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{Y}).$$ The desired inequality follows since this holds for arbitrary $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$. Recall that Grassmannians satisfy an isomorphism $$\operatorname{Gr}(k,n)\cong \operatorname{Gr}(n-k,n)$$ that takes a $k$-plane $\mathbf{Y}$ to the $(n-k)$-plane $\mathbf{Y}^{\perp}$ of linear forms vanishing on $\mathbf{Y}$. It is easy to see that this isomorphism is an isometry. Using this isometric isomorphism, together with Lemma \[lem:ineq1\] and Lemma \[lem:ineq2\], we can immediately deduce that $$\delta(\mathbf{A}, \mathbf{B}) \le d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}\bigl(\mathbf{B})\bigr)=d_{\operatorname{Gr}(n-k,n)}\bigl(\mathbf{A^{\perp}},\Omega_{+}\bigl(\mathbf{B}^{\perp})\bigr)\le \delta(\mathbf{A}^{\perp},\mathbf{B}^{\perp}).$$ On the other hand, by results in [@Knyazev], we have $\delta(\mathbf{A}, \mathbf{B}) = \delta(\mathbf{A}^{\perp},\mathbf{B}^{\perp})$ and hence $$\delta(\mathbf{A}, \mathbf{B}) = d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr).$$ Similarly we can obtain $$\delta(\mathbf{A}, \mathbf{B}) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr).$$ Hence we have the required equalities and in Theorem \[thm1\]. Property  is obvious from and Property  follows from Lemma \[lem:infty\]. The proof of Lemma \[lem:ineq1\] provides a simple way to find a point $\mathbf{X} \in \Omega_{+}(\mathbf{A})$ that realizes the distance $d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)=\delta(\mathbf{A},\mathbf{B})$. Similarly we may explicitly determine a point $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$ that realizes the distance $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)=\delta(\mathbf{A},\mathbf{B})$. One might wonder whether or not Theorem \[thm1\] still holds if we replace $d_{\operatorname{Gr}(k,n)}$ by other distance functions described in Table \[tab:distances\]. The answer is yes. \[thm:othermetrics\] Let $k \le l \le n$. Let $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B}\in\operatorname{Gr}(l,n)$. Then $$d^*_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A}, \Omega_{-}(\mathbf{B})\bigr) = d^*_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B}, \Omega_{+}(\mathbf{A})\bigr),$$ for $* = \alpha, \beta, \kappa, \phi, \mu, \rho, \pi, \sigma$. Their common value $\delta^*(\mathbf{A},\mathbf{B})$ is given by: $$\begin{aligned} \delta^{\alpha}(\mathbf{A}, \mathbf{B}) &= \theta_k, & \delta^{\beta}(\mathbf{A},\mathbf{B}) &= \Bigl(1 - \prod\nolimits_{i=1}^k\cos^2\theta_i\Bigr)^{1/2},\\ \delta^{\kappa}(\mathbf{A}, \mathbf{B}) &= \Bigl(\sum\nolimits_{i=1}^k\sin^2\theta_i\Bigr)^{1/2}, & \delta^{\phi}(\mathbf{A}, \mathbf{B}) &= \cos^{-1}\bigl(\prod\nolimits_{i=1}^k\cos \theta_i\Bigr),\\ \delta^{\mu}(\mathbf{A}, \mathbf{B}) &= \Bigl( \log \prod\nolimits_{i=1}^k\frac{1}{\cos^2 \theta_i}\Bigr)^{1/2}, & \delta^{\rho}(\mathbf{A}, \mathbf{B}) &= \Bigl(2\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\Bigr)^{1/2},\\ \delta^{\pi}(\mathbf{A}, \mathbf{B}) &= \sin \theta_k, & \delta^{\sigma}(\mathbf{A}, \mathbf{B}) &= 2\sin (\theta_k/2),\end{aligned}$$ or more generally with $\min(k,l)$ in place of the index $k$ when we do not require $k \le l$. This follows from observing that our proof of Theorem \[thm1\] only involves principal angles between $\mathbf{A}$ and $\mathbf{B}$ and the diffeomorphism between $\operatorname{Gr}(k,n)$ and $\operatorname{Gr}(n-k,n)$ remains an isometry under these distances. In particular, both and still hold with any of these distances in place of the Grassmann distance. We will see in Section \[sec:existing\] that the projection distance $\delta^\pi$ in Theorem \[thm:othermetrics\] is equivalent to the containment gap, a measure of distance between subspaces of different dimensions originally proposed in operator theory [@Kato]. Grassmannian of subspaces of all dimensions {#sec:dinfty} =========================================== We view the equality of $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr)$ and $d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)$ as the strongest evidence that their common value $\delta(\mathbf{A}, \mathbf{B})$ provides the most natural notion of distance between subspaces of different dimensions. As we pointed out earlier, $\delta$ is a distance in the sense of a distance from a point to a set, but not a distance in the sense of a metric on the set of all subspaces of all dimensions. For instance, $\delta$ does not satisfy the separation property: $\delta(\mathbf{A}, \mathbf{B}) =0$ for any $\mathbf{A} \subsetneq \mathbf{B}$. In fact, it is easy to observe the following. \[lem:zero\] Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. Then $\delta(\mathbf{A},\mathbf{B}) = 0$ iff $\mathbf{A} \subseteq \mathbf{B}$ or $\mathbf{B} \subseteq \mathbf{A}$. $\delta$ also does not satisfy the triangle inequality: For a line $\mathbf{L}$ not contained in a subspace $\mathbf{A}$, the triangle inequality, if true, would imply $$\begin{aligned} \delta(\mathbf{L},\mathbf{A})&=\delta(\mathbf{L},\mathbf{A})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{B}),\\ \delta(\mathbf{L},\mathbf{B})&=\delta(\mathbf{L},\mathbf{B})+\delta(\mathbf{A},\mathbf{B})\ge \delta(\mathbf{L},\mathbf{A}),\end{aligned}$$ giving $\delta(\mathbf{L},\mathbf{A})=\delta(\mathbf{L},\mathbf{B})$ for any subspace $\mathbf{B}$, which is evidently false by Lemma \[lem:zero\] (e.g. take $\mathbf{B} = \mathbf{A} \oplus \mathbf{L}$). These observations also apply verbatim to all the other similarly-defined distances $\delta^*$ in Theorem \[thm:othermetrics\], i.e., none of them are metrics. The set of all subspaces of all dimensions is parameterized by $\operatorname{Gr}(\infty, \infty)$, the *doubly infinite Grassmannian* [@FH], which may be viewed informally as the disjoint union of all $k$-dimensional subspaces[^2] over all $k \in \mathbb{N}$, $$\operatorname{Gr}(\infty, \infty)=\coprod\nolimits_{k=1}^{\infty} \operatorname{Gr}(k,\infty).$$ To define a metric between any pair of subspaces of arbitrary dimensions is to define one on $\operatorname{Gr}(\infty,\infty)$. It is easy to define metrics on $\operatorname{Gr}(\infty,\infty)$ that bear little relation to the geometry of Grassmannian but we will propose one in Section \[sec:metric\] that is consistent with $\delta$ and with $d_{\operatorname{Gr}(k,n)}$ for all $k \le n$. We will require the formal definition of $\operatorname{Gr}(\infty, \infty)$, namely, it is the direct limit of the direct system of Grassmannians $\{ \operatorname{Gr}(k,n) : (k,n) \in \mathbb{N} \times \mathbb{N} \}$ with inclusion maps $\iota^{kl}_{nm} : \operatorname{Gr}(k,n) \to \operatorname{Gr}(l,m)$ for all $k \le l$ and $n \le m$ such that $l-k\le m-n$. For $A \in \mathbb{R}^{n \times k}$ with orthonormal columns, the embedding is given by $$\label{eq:epsilon} \iota^{kl}_{nm} :\operatorname{Gr}(k,n) \to \operatorname{Gr}(l,m), \qquad \operatorname{span} (A) \mapsto \operatorname{span} \left( \begin{bmatrix} A & 0\\ 0 & 0\\ 0 & I_{l-k} \end{bmatrix}\right),$$ where $I_{l-k} \in \mathbb{R}^{(l -k) \times (l -k)}$ is an identity matrix and we have $(m-n) - (l -k)$ zero rows in the middle so that the $3 \times 2$ block matrix is in $\mathbb{R}^{m \times l}$. Note that for a fixed $k$, $\iota^{kk}_{nm} $ reduces to $\iota_{nm}$ in . Since our distance $\delta(\mathbf{A},\mathbf{B})$ is defined for subspaces $\mathbf{A}$ and $\mathbf{B}$ of all dimensions, it defines a function $\delta:\operatorname{Gr}(\infty,\infty)\times \operatorname{Gr}(\infty,\infty)\to \mathbb{R}$ that is a *premetric* on $\operatorname{Gr}(\infty,\infty)$, i.e., $\delta(\mathbf{A},\mathbf{B})\ge 0$ and $\delta(\mathbf{A},\mathbf{A})=0$ for all $\mathbf{A},\mathbf{B} \in \operatorname{Gr}(\infty,\infty)$. This in turn defines a topology $\tau $ on $\operatorname{Gr}(\infty,\infty)$ in a standard way: The $\varepsilon$-ball centered at $\mathbf{A}$ is $$B_\varepsilon(\mathbf{A})\coloneqq\{ \mathbf{X}\in \operatorname{Gr}(\infty,\infty) : \delta(\mathbf{A},\mathbf{X})< \varepsilon \},$$ and $U \subseteq \operatorname{Gr}(\infty,\infty)$ is defined to be open if for any $\mathbf{A}\in U$, there is an $\varepsilon$-ball $B_\varepsilon(\mathbf{A}) \subseteq U$. The topology $\tau$ is consistent with the usual topology of Grassmannians (but it is not the disjoint union topology). If we restrict $\tau$ to $\operatorname{Gr}(k,\infty)$, then the subspace topology is the same as the topology induced by the metric $d_{\operatorname{Gr}(k,\infty)}$ on $\operatorname{Gr}(k,\infty)$ as defined in Section \[sec:infty\]. Nevertheless this apparently natural topology on $\operatorname{Gr}(\infty,\infty)$ turns out to be a strange one. \[prop:tau\] The topology $\tau$ on $\operatorname{Gr}(\infty,\infty)$ is non-Hausdorff and therefore non-metrizable. $\tau $ is not Hausdorff since it is not possible to separate $\mathbf{A}\subsetneq \mathbf{B}$ by open subsets, as we saw in Lemma \[lem:zero\]. Metrizable spaces are necessarily Hausdorff. Even though $\tau$ restricts to the metric space topology on $\operatorname{Gr}(k,\infty)$ induced by the Grassmann distance $d_{\operatorname{Gr}(k,\infty)}$ for every $k \in \mathbb{N}$, it is not itself a metric space topology. We view this as a consequence of a more general phenomenon, namely, the category $\boldsymbol{\mathsf{Met}}$ of metric spaces (objects) and continuous contractions (morphisms) has no coproduct, i.e., given a collection of metric spaces, there is in general no metric space that will behave like the disjoint union of the collection of metric spaces. To see this, take metric spaces $(X_1,d_1)$ and $(X_2,d_2)$ where $X_1 = \{ x_1\}$, $X_2 = \{x_2\}$. Suppose a coproduct $(X,d)$ of $(X_1,d_1)$ and $(X_2,d_2)$ exists. Let $Y=\{y_1,y_2\}$ and let $d_Y$ be the metric on $Y$ induced by $d_Y(y_1,y_2)=2 d(x_1,x_2)\ne 0$. Now define $\varphi_i: X_i\to Y$ by $\varphi_i(x_i)=y_i$, $i=1,2$. One sees that no morphism $\varphi:X\to Y$ in $\boldsymbol{\mathsf{Met}}$ is compatible with $\varphi_1$ and $\varphi_2$, contradicting the assumption that $X$ is the coproduct of $X_1$ and $X_2$. If we instead look at the category of metric spaces with continuous or uniformly continuous maps as morphisms, then coproducts always exist [@Helemskii]. In Section \[sec:metric\], we will relax our requirement and construct a metric $d_{\operatorname{Gr}(\infty,\infty)}$ on $\operatorname{Gr}(\infty,\infty)$ that restricts to $d_{\operatorname{Gr}(k,\infty)}$ for all $k \in \mathbb{N}$ but without requiring that it comes from a coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)}) : k \in \mathbb{N}\}$ in $\boldsymbol{\mathsf{Met}}$. Metrics for subspaces of all dimensions {#sec:metric} ======================================= We will describe a simple recipe for turning the distances $\delta^*$ in Theorem \[thm:othermetrics\] into metrics on $\operatorname{Gr}(\infty,\infty)$. Suppose $k \le l$ and we have $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. In this case there are $k$ principal angles between $\mathbf{A}$ and $\mathbf{B}$, $\theta_1,\dots, \theta_k$, as defined in . First we will set $\theta_{k+1} = \dots = \theta_l = \pi/2$. Then we take the Grassmann distance $\delta$ or any of the distances $\delta^*$ in Theorem \[thm:othermetrics\], replace the index $k$ by $l$, and call the resulting expressions $d_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B})$ (for Grassmann distance) and $d^*_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B})$ (for other distances) respectively. When $n$ is sufficiently large, setting $\theta_{k+1},\dots,\theta_l$ all equal to $\pi/2$ is equivalent to completing $\mathbf{A}$ to an $l$ dimensional subspace of $\mathbb{R}^n$, by adding $l-k$ vectors orthonormal to the subspace $\mathbf{B}$. Hence the distance between $\mathbf{A}$ and $\mathbf{B}$ is defined by the distance function on the Grassmannian $\operatorname{Gr}(l,n)$. We show in Proposition \[prop:metric\] that these expressions will indeed define metrics on $\operatorname{Gr}(\infty,\infty)$. Applying the above recipe to the Grassmann, chordal, and Procrustes distances yield the *Grassmann*, *chordal*, and *Procrustes metrics* on $\operatorname{Gr}(\infty,\infty)$ given in Table \[tab:metrics\]. ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Grassmann metric $d_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B}) = \Bigl(\lvert k - l \rvert\pi^2/4 + \sum\nolimits_{i=1}^{\min(k,l)}\theta_i^2\Bigr)^{1/2}$ Chordal metric $d_{\operatorname{Gr}(\infty, \infty)}^{\kappa}(\mathbf{A}, \mathbf{B}) = \Bigl(\lvert k - l \rvert + \sum\nolimits_{i=1}^{\min(k,l)} \sin^2\theta_i\Bigr)^{1/2}$ Procrustes metric $d_{\operatorname{Gr}(\infty, \infty)}^{\rho}(\mathbf{A}, \mathbf{B}) =\Bigl(\lvert k - l \rvert + 2\sum\nolimits_{i=1}^{\min(k,l)}\sin ^2(\theta_i/2)\Bigr)^{1/2}$ ------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Evidently the metrics in Table \[tab:metrics\] are all of the form $$\label{eq:rms} d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = \sqrt{\delta^*(\mathbf{A}, \mathbf{B})^2 + c_*^2 \epsilon(\mathbf{A},\mathbf{B})^2},$$ where $\epsilon(\mathbf{A},\mathbf{B}) \coloneqq \lvert \dim \mathbf{A} - \dim \mathbf{B} \rvert^{1/2}$. On the other hand, applying the above recipe to other distances in Table \[tab:distances\] yield the *Asimov*, *Binet–Cauchy*, *Fubini–Study*, *Martin*, *projection*, and *spectral metrics* on $\operatorname{Gr}(\infty,\infty)$ given by $$\label{eq:indicator} d_{\operatorname{Gr}(\infty, \infty)}^{*}(\mathbf{A}, \mathbf{B}) \\ = \begin{cases} d_{\operatorname{Gr}(k,\infty)}^* (\mathbf{A}, \mathbf{B}) & \text{if } \dim \mathbf{A} = \dim \mathbf{B} = k,\\ c_* & \text{if } \dim \mathbf{A} \ne \dim \mathbf{B}, \end{cases}$$ for $ * = \alpha, \beta, \phi, \mu, \pi, \sigma$, respectively. The constants $c_* > 0$ can be seen to be $$c = c_\alpha = \pi/2, \quad c_\sigma = \sqrt{2}, \quad c_\mu = \infty,\quad c_\beta = c_\phi = c_\pi = c_\kappa = c_\rho = 1.$$ In all cases, for subspaces $\mathbf{A}$ and $\mathbf{B}$ of equal dimension $k$, these metrics on $\operatorname{Gr}(\infty,\infty)$ restrict to the corresponding ones on $\operatorname{Gr}(k,\infty)$, i.e., $$d^*_{\operatorname{Gr}(\infty, \infty)}(\mathbf{A}, \mathbf{B}) = d^*_{\operatorname{Gr}(k, \infty)}(\mathbf{A}, \mathbf{B}),$$ where the latter is as described in Corollary \[cor:infty\] and Lemma \[lem:inclusion\]. These metrics on $\operatorname{Gr}(\infty,\infty)$ are the amalgamation of two pieces of information, the distance $\delta^*(\mathbf{A},\mathbf{B})$ and the difference in dimensions $\lvert \dim \mathbf{A} - \dim \mathbf{B} \rvert$, either via a root mean square or an indicator function. The Grassmann metric has a natural interpretation (see Proposition \[prop:prob4\]): > *$d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B})$ is the distance from $\mathbf{B}$ to the furthest $l$-dimensional subspace that contains $\mathbf{A}$, which equals the distance from $\mathbf{A}$ to the furthest $k$-dimensional subspace contained in $\mathbf{B}$.* The chordal metric in Table \[tab:metrics\] is equivalent to the symmetric directional distance, a metric on subspaces of different dimensions [@SWF; @WWF] popular in machine learning [@bar; @ref2a; @motion; @ref2b; @eeg; @mech; @blog; @ref2c; @econ; @network; @face; @ref2d; @text] (see Section \[sec:existing\]). \[prop:metric\] The expressions in Table \[tab:metrics\] and are metrics on $\operatorname{Gr}(\infty,\infty)$. It is trivial to see that the expression defined in yields a metric on $\operatorname{Gr}(\infty,\infty)$ for $ * = \alpha, \beta, \mu, \pi, \sigma, \phi$, and so we just need to check the remaining three cases that take the form in . Of the four defining properties of a metric, only the triangle inequality is not immediately clear from . Let $k = \dim \mathbf{A}$, $l = \dim \mathbf{B}$, and $m = \dim \mathbf{C}$. We may assume <span style="font-variant:small-caps;">wlog</span> that $k\le l \le m \le n$ where $n$ is chosen sufficiently large so that $\mathbf{A}, \mathbf{B}, \mathbf{C}$ are subspaces in $\mathbb{R}^n$. Let $A \in \mathbb{R}^{n\times k}$, $B \in \mathbb{R}^{n \times l}$, $C \in \mathbb{R}^{n \times m}$ be matrices whose columns are orthonormal bases of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ respectively. Consider the following $(n+m-k)\times m$ matrices: $$A'=\begin{bmatrix} A & 0\\ 0 & I_{m-k} \end{bmatrix},\quad B'=\begin{bmatrix} B & 0\\ 0 & 0\\ 0 & I_{m-l} \end{bmatrix},\quad C'=\begin{bmatrix} C\\ 0 \end{bmatrix}.$$ and set $\mathbf{A}' = \operatorname{span}(A')$, $\mathbf{B}' = \operatorname{span}(B')$, $\mathbf{C}' = \operatorname{span}(C')$; note that these are just $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ embedded in $\operatorname{Graff}(m,n+m-k)$ via . The expressions in Table \[tab:metrics\] satisfy $$\begin{aligned} d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{A}',\mathbf{B}'),\\ d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{B},\mathbf{C}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{B}',\mathbf{C}'),\\ d^*_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{C}) = d^*_{\operatorname{Gr}(m,n+m-k)}(\mathbf{A}',\mathbf{C}').\end{aligned}$$ Since $\mathbf{A}',\mathbf{B}',\mathbf{C}' \in \operatorname{Gr}(m,n+m-k)$, the triangle inequality for $d^*_{\operatorname{Gr}(m,n+m-k)}$ immediately yields the triangle inequality for $d^*_{\operatorname{Gr}(\infty,\infty)}$. The proof shows that for any $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ where $k \le l \le n$, $$d^*_{\operatorname{Gr}(\infty,\infty)}(\mathbf{A},\mathbf{B}) =d^*_{\operatorname{Gr}(l,n+l-k)}\bigl(\iota^{k,l}_{n,n+l-k}(\mathbf{A}),\iota^{l,l}_{n,n+l-k}(\mathbf{B})\bigr).$$ The embeddings $\iota^{k,l}_{n,n+l-k}:\operatorname{Gr}(k,n)\to \operatorname{Gr}(l,n+l-k)$ and $\iota^{l,l}_{n,n+l-k}:\operatorname{Gr}(l,n)\to \operatorname{Gr}(l,n+l-k)$ are as defined in and are isometric for all $ k\le l \le n$. \[prop:prob4\] Let $k\le l \le n/2$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B}\in \operatorname{Gr}(l,n)$. Then $$\label{eq:equal2} \max_{\mathbf{X}\in \Omega_{+}(\mathbf{A})}d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) = \max_{\mathbf{Y} \in \Omega_{-}(\mathbf{B})} d_{\operatorname{Gr}(k,n)}(\mathbf{Y},\mathbf{A}) = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}),$$ i.e., $d_{\operatorname{Gr}(\infty,\infty)}$ is the distance between *furthest* subspaces. We assume <span style="font-variant:small-caps;">wlog</span> that $\mathbf{A}\cap \mathbf{B}=\{0 \}$ by Proposition \[prop:angles\]. Since $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})=\delta(\mathbf{X},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^l \theta_i(\mathbf{X},\mathbf{B})^2\Bigr)^{1/2},$$ and by Corollary \[cor:compare\], $\theta_i(\mathbf{X},\mathbf{B})\le \theta_i(\mathbf{A},\mathbf{B})$, $i =1,\dots,k$, we obtain $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})\le \Bigl(\delta(\mathbf{A},\mathbf{B})^2+\sum\nolimits_{i=k+1}^l\theta_i(\mathbf{X},\mathbf{B})^2\Bigr)^{1/2}.$$ Let $(a_1,b_1),\dots, (a_k, b_k)$ be the principal vectors between $\mathbf{A}$ and $\mathbf{B}$. We extend $b_1,\dots, b_k$ to obtain an orthonormal basis $b_1,\dots, b_k,b_{k+1},\dots, b_l$ of $\mathbf{B}$. Let $\mathbf{X}\cap\mathbf{A}^\perp$ be the orthogonal complement of $\mathbf{A}$ in $\mathbf{X}$ and let $\mathbf{B}_0 \coloneqq\operatorname{span}\{b_{k+1},\dots, b_l\}$. Then $$\Bigl(\sum\nolimits_{i=k+1}^l\theta_i(\mathbf{X},\mathbf{B})^2 \Bigr)^{1/2} = \delta(\mathbf{X}\cap\mathbf{A}^\perp,\mathbf{B}_0),$$ and the last inequality becomes $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B}) \le \sqrt{\delta(\mathbf{A},\mathbf{B})^2+\delta(\mathbf{X}\cap\mathbf{A}^\perp,\mathbf{B}_0)^2}.$$ If $n\ge 2l$, then there exist $l-k$ vectors $c_1,\dots,c_{l-k}$ orthogonal to $\mathbf{A}$ and $\mathbf{B}$ simultaneously. Choosing $\mathbf{X} = \operatorname{span}\{a_1,\dots,a_k,c_1,\dots,c_{l-k}\}$, we attain the required maximum: $$d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})=\sqrt{\delta(\mathbf{A},\mathbf{B})^2+(l-k)\pi^2/4} = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}).$$ The second equality in follows from $d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{A},\mathbf{B}) = d_{\operatorname{Gr}(\infty,\infty)} (\mathbf{B},\mathbf{A})$, given that $d_{\operatorname{Gr}(\infty,\infty)}$ is a metric by Proposition \[prop:metric\]. The existence of the metrics $d^*_{\operatorname{Gr}(\infty,\infty)}$ as defined in and does not contradict our earlier discussion about the general nonexistence of coproduct in $\boldsymbol{\mathsf{Met}}$ as these metrics do not respect continuous *contractions*. Take the Grassmann metric on $\operatorname{Gr}(\infty,\infty)$ for instance. $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is an object of the category $\boldsymbol{\mathsf{Met}}$ but it is *not* the coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$. Indeed, let $Y = \{y_1, y_2\}$ with metric defined by $d_Y(y_1, y_2) = 1$. Consider a family of maps $f_k:\operatorname{Gr}(k,\infty)\to Y$, $$f_k(\mathbf{A})= \begin{cases} y_1&\text{if }k=2,\\ y_2&\text{otherwise}. \end{cases}$$ Then $f_k$ is a continuous contraction between $\operatorname{Gr}(k,\infty)$ and $Y$. So $\{f_k :k \in \mathbb{N}\}$ is a family of morphisms in $\boldsymbol{\mathsf{Met}}$ compatible with $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$. If $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is the coproduct of this family, then there must be a continuous contraction $f:\operatorname{Gr}(\infty,\infty)\to Y$ such that $f\circ \iota_k=f_k$ with $\iota_k$ being the natural inclusion of $\operatorname{Gr}(k,\infty)$ into $\operatorname{Gr}(\infty,\infty)$. But taking $\mathbf{A}\in \operatorname{Gr}(2,\infty)$ and $\mathbf{B}\in \operatorname{Gr}(3,\infty)$, we see that $$d_{\operatorname{Gr}(\infty,\infty)}(\mathbf{A},\mathbf{B})\ge \frac{\pi}{2} > 1=d_Y\bigl(f(\mathbf{A}),f(\mathbf{B})\bigr),$$ contradicting the surmise that $f$ is a contraction. Similarly, one may show that $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)}^*)$ is not a coproduct in $\boldsymbol{\mathsf{Met}}$ for any $* = \alpha, \beta, \kappa, \mu, \pi, \rho, \sigma, \phi$. $(\operatorname{Gr}(\infty,\infty),d_{\operatorname{Gr}(\infty,\infty)})$ is also not the coproduct of $\{(\operatorname{Gr}(k,\infty), d_{\operatorname{Gr}(k,\infty)} ) : k \in \mathbb{N}\}$ in the category of metric spaces with continuous (or uniformly continuous) maps as morphisms. The coproduct in this category is simply $\operatorname{Gr}(\infty,\infty)$ with the metric induced by the disjoint union topology, which is too fine (in the sense of topology) to be interesting. In particular, such a metric is unrelated to the distance $\delta$. Comparison with existing works {#sec:existing} ============================== There are two existing proposals for a distance between subspaces of different dimensions — the containment gap and the symmetric directional distance. These turn out to be special cases of our distance in Section \[sec:main\] and our metric in Section \[sec:metric\]. Let $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. The *containment gap* is defined as $$\gamma(\mathbf{A}, \mathbf{B}) \coloneqq \max_{a\in \mathbf{A}} \; \min_{b\in \mathbf{B}} \frac{\lVert a - b \rVert}{\lVert a \rVert}.$$ This was proposed in [@Kato pp. 197–199] and used in numerical linear algebra [@SS] for measuring separation between Krylov subspaces [@BES]. It is equivalent to our *projection distance* $\delta^\pi$ in Theorem \[thm:othermetrics\]. It was observed in [@BES p. 495] that $$\gamma(\mathbf{A}, \mathbf{B}) = \sin\bigl( \theta_k(\mathbf{A},\mathbf{Y}) \bigr)$$ where $\mathbf{Y} \in \Omega_{-}(\mathbf{B})$ is nearest to $\mathbf{A}$ in the projection distance $d^\pi_{\operatorname{Gr}(k,n)}$. By Theorem \[thm:othermetrics\], we deduce that it can also be realized as $$\gamma(\mathbf{A}, \mathbf{B}) = \sin\bigl( \theta_l(\mathbf{B},\mathbf{X}) \bigr)$$ where $\mathbf{X} \in \Omega_{+}(\mathbf{A})$ is nearest to $\mathbf{B}$ in the projection distance $d^\pi_{\operatorname{Gr}(l,n)}$, a fact about the containment gap that had not been observed before. Indeed, by Theorem \[thm:othermetrics\], we get $$\gamma(\mathbf{A}, \mathbf{B}) = \delta^\pi (\mathbf{A}, \mathbf{B})$$ for all $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$. The *symmetric directional distance* is defined as $$\label{eq:sdd} d_{\Delta}(\mathbf{A},\mathbf{B}) \coloneqq \Bigl( \max(k,l) - \sum\nolimits_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^2 \Bigr)^{1/2}$$ where $A = [a_1,\dots,a_k]$ and $B = [b_1,\dots,b_l]$ are the respective orthonormal bases. This was proposed in [@SWF; @WWF], and is widely used [@bar; @ref2a; @motion; @ref2b; @eeg; @mech; @blog; @ref2c; @econ; @network; @face; @ref2d; @text]. The definition is equivalent to our *chordal metric* $d^\kappa_{\operatorname{Gr}(\infty,\infty)}$ in Table \[tab:metrics\], $$d_{\operatorname{Gr}(\infty,\infty)}^\kappa (\mathbf{A},\mathbf{B})^2 = \lvert k - l \rvert + \sum_{i=1}^{\min(k,l)} \sin^2 \theta_i = \max(k,l) - \sum_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^{2} =d_\Delta (\mathbf{A}, \mathbf{B})^2,$$ since $ \lvert k - l \rvert = \max(k,l) - \min(k,l)$, and $$\sum\nolimits_{i,j=1}^{k,l} (a_i^\mathsf{T} b_j)^{2} = \lVert A^\mathsf{T} B \rVert_F^2 = \sum\nolimits_{i=1}^{\min(k,l)} \cos^2 \theta_i = \min(k,l) - \sum\nolimits_{i=1}^{\min(k,l)} \sin^2 \theta_i .$$ Geometry of $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ {#sec:schubert} ================================================================= Up to this point, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$, as defined in Definition \[def:Omega\], are treated as mere subsets of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. We will see that $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ have rich geometric properties. Firstly, we will show that they are Schubert varieties, justifying their names. \[def:Schubert\] Let $\mathbf{X}_1\subset \mathbf{X}_2 \subset \dots \subset \mathbf{X}_k$ be a fixed flag in $\mathbb{R}^n$. The *Schubert variety* $\Omega(\mathbf{X}_1, \dots, \mathbf{X}_k , n)$ is the set of $k$-planes $\mathbf{Y}$ satisfying the Schubert conditions $\dim (\mathbf{Y}\cap \mathbf{X}_i)\ge i$, $i=1,\dots, k$, i.e., $$\Omega(\mathbf{X}_1, \dots, \mathbf{X}_k , n) = \{\mathbf{Y} \in \operatorname{Gr}(k,n): \dim (\mathbf{Y}\cap \mathbf{X}_i)\ge i, \; i=1,\dots, k \}.$$ Let $0 \eqqcolon k_0 < k_1<\cdots < k_{m+1} \coloneqq n$ be a sequence of increasing nonnegative integers. The associated *flag variety* is the set of flags satisfying the condition $\dim \mathbf{X}_i = k_i$, $i=0, 1,\dots, m+1$. We denote it by $\operatorname{Flag}(k_1,\dots, k_m, n)$, i.e., $$\{(\mathbf{X}_1,\dots,\mathbf{X}_{m}) \in \operatorname{Gr}(k_1,n) \times \dots \times \operatorname{Gr}(k_m,n) : \mathbf{X}_i\subset \mathbf{X}_{i+1}, \; i =1,\dots,m \}.$$ Observe that a Schubert variety depends on a specific increasing sequence of subspaces whereas a flag variety depends only on an increasing sequence of dimensions (of subspaces). Flag varieties may be viewed as a generalization of Grassmannians since if $m=1$, then $\operatorname{Flag}(k,n) = \operatorname{Gr}(k,n)$. Like Grassmannians, $\operatorname{Flag}(k_1,\dots, k_m, n)$ is a smooth manifold and sometimes called a *flag manifold*. The parallel goes further, $\operatorname{Flag}(k_1,\dots, k_m, n) $ is a homogeneous space, $$\label{eq:flag} \operatorname{Flag}(k_1,\dots, k_m, n) \cong \operatorname{O}(n)/\bigl(\operatorname{O}(d_1) \times \dots \times \operatorname{O}(d_{m+1})\bigr)$$ where $d_i = k_{i} - k_{i-1}$ for $i =1,\dots, m+1$, generalizing . Let $\mathbf{A}\in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ with $k\le l$. Then $$\Omega_{+}(\mathbf{A}) = \Omega(\mathbf{A}_1, \dots, \mathbf{A}_l , n),\qquad \Omega_{-}(\mathbf{B}) = \Omega(\mathbf{B}_1, \dots, \mathbf{B}_{k} , n),$$ are Schubert varieties in $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively with the flags $$\begin{gathered} \{0\} \eqqcolon \mathbf{A}_0 \subset \mathbf{A}_1\subset \dots \subset \mathbf{A}_k\coloneqq\mathbf{A}\subset \mathbf{A}_{k+1}\dots \subset \mathbf{A}_{l}, \\ \{0\} \eqqcolon \mathbf{B}_0 \subset \mathbf{B}_1 \subset \dots \subset \mathbf{B}_{k} \coloneqq\mathbf{B}.\end{gathered}$$ where $\mathbf{A}_{k+i}$ is a subspace of $\mathbb{R}^n$ containing $\mathbf{A}$ of dimension $n-l+(k+i)$ for $1\le i\le l-k$. The isomorphism $\operatorname{Gr}(l,n) \cong \operatorname{Gr}(n-l,n)$ (resp. $\operatorname{Gr}(k,n) \cong \operatorname{Gr}(n-k,n)$) that sends $\mathbf{X}$ to $\mathbf{X}^{\perp}$ takes $\Omega_{+}(\mathbf{A})$ to $\Omega_{-}(\mathbf{A}^{\perp})$ (resp. $\Omega_{-}(\mathbf{B})$ to $\Omega_{+}(\mathbf{B}^{\perp})$). Thus $\Omega_{+}(\mathbf{A})$ (resp. $\Omega_{-}(\mathbf{B})$) may also be viewed as Schubert varieties in $\operatorname{Gr}(n-l,n)$ (resp. $\operatorname{Gr}(n-k,n)$). More importantly, this observation implies that $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$, despite superficial difference in their definitions, are essentially the same type of objects. \[prop:Omega\] For any $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$, we have $$\Omega_{+}(\mathbf{A}) \cong \Omega_{-}(\mathbf{A}^{\perp}) \quad\text{and}\quad \Omega_{-}(\mathbf{B}) \cong \Omega_{+}(\mathbf{B}^{\perp}).$$ Also, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are uniquely determined by $\mathbf{A}$ and $\mathbf{B}$ respectively. \[prop:Omega1\] Let $\mathbf{A}, \mathbf{A}' \in \operatorname{Gr}(k,n)$ and $\mathbf{B},\mathbf{B}' \in \operatorname{Gr}(l,n)$. Then $$\begin{gathered} \Omega_{+}(\mathbf{A}) =\Omega_{+}(\mathbf{A'})\quad \text{if and only if} \quad \mathbf{A} = \mathbf{A}',\\ \Omega_{-}(\mathbf{B}) =\Omega_{-}(\mathbf{B}')\quad \text{if and only if} \quad \mathbf{B} = \mathbf{B}'.\end{gathered}$$ Suppose $\Omega_{+}(\mathbf{A})=\Omega_{+}(\mathbf{A'})$. Observe that the intersection of all $l$-planes containing $\mathbf{A}$ is exactly $\mathbf{A}$ and ditto for $\mathbf{A}'$. So $$\mathbf{A} = \bigcap\nolimits_{\mathbf{X} \in\Omega_{+}(\mathbf{A})} \mathbf{X} = \bigcap\nolimits_{\mathbf{X} \in\Omega_{+}(\mathbf{A}')} \mathbf{X} = \mathbf{A}'.$$ The converse is obvious. The statement for $\Omega_{-}$ then follows from Proposition \[prop:Omega\]. This observation allows us to treat subspaces of different dimensions on the same footing by regarding them as *subsets* in the same Grassmannian. If we have a collection of subspaces of dimensions $k\le k_1 < k_2 < \dots < k_m \le l$, the injective map $\mathbf{A} \mapsto \Omega_{+}(\mathbf{A})$ takes all of them into distinct subsets of $\operatorname{Gr}(l,n)$. Alternatively, the injective map $\mathbf{B} \mapsto \Omega_{-}(\mathbf{B})$ takes all of them into distinct subsets of $\operatorname{Gr}(k,n)$. The resemblance between $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ in Proposition \[prop:Omega\] goes further — we may view them as ‘sub-Grassmannians’. \[prop:OmegaGrass\] Let $k \le l \le n$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B} \in \operatorname{Gr}(l,n)$. Then $$\Omega_{+}(\mathbf{A}) \cong \operatorname{Gr}(l-k,n-k), \quad \Omega_{-}(\mathbf{B}) \cong \operatorname{Gr}(k,l),$$ isomorphic as algebraic varieties and diffeomorphic as smooth manifolds. Thus $$\dim \Omega_{+}(\mathbf{A})=(n-l)(l-k),\quad \dim \Omega_{-}(\mathbf{B}) = k(l-k).$$ The first isomorphism is the quotient map $\varphi: \Omega_{+}(\mathbf{A})\to \operatorname{Gr}_{l-k}(\mathbb{R}^n/\mathbf{A})$, $\mathbf{X} \mapsto \mathbf{X}/\mathbf{A} \subseteq \mathbb{R}^n/\mathbf{A}$, composed with the isomorphism $ \operatorname{Gr}_{l-k}(\mathbb{R}^n/\mathbf{A}) \cong \operatorname{Gr}(l-k,n-k)$. The second isomorphism is obtained by regarding a $k$-dimensional subspace $\mathbf{Y}$ of $\mathbb{R}^n$ in $\Omega_{-}(\mathbf{B}) $ as a $k$-dimensional subspace of $\mathbf{B}$, i.e., $\Omega_{-}(\mathbf{B}) = \operatorname{Gr}_k(\mathbf{B}) \cong \operatorname{Gr}(k,l)$. That $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are Grassmannians allows us to infer the following: 1. \[top\] as topological spaces, they are compact and path-connected; 2. as algebraic varieties, they are irreducible and nonsingular; 3. \[diff\] as differential manifolds, they are smooth and any two points on them can be connected by a length-minimizing geodesic. The topology in refers to the metric space topology, not Zariski topology. A consequence of compactness is that the distance $d_{\operatorname{Gr}(k,n)}\bigl(\mathbf{A},\Omega_{-}(\mathbf{B})\bigr) = d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr)$ can be attained by points in $\Omega_{-}(\mathbf{B})$ and $\Omega_{+}(\mathbf{A})$ respectively. We constructed these closest points explicitly when we proved Theorem \[thm1\]. Many more topological and geometric properties of $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ follow from Proposition \[prop:OmegaGrass\] as they inherit everything that we know about Grassmannians (e.g. coordinate ring, cohomology ring, Plücker relations, etc.); in particular, $\Omega_{+}(\mathbf{A})$ and $\Omega_{-}(\mathbf{B})$ are also flag varieties. The last property in requires a proof. The length-minimizing geodesic is not unique and so $\Omega_+(\mathbf{A})$ and $\Omega_-(\mathbf{B})$ are not *geodesically convex* [@Nicolaescu Definition 4.1.35]. Any two points in $\Omega_{-}(\mathbf{B})$ (resp. $\Omega_{+}(\mathbf{A})$) can be connected by a length-minimizing geodesic in $\operatorname{Gr}(k,n)$ (resp. $\operatorname{Gr}(l,n)$). By Proposition \[prop:Omega\], it suffices to show that any two points in $\Omega_{-}(\mathbf{B})$ can be connected by a geodesic curve in $\Omega_{-}(\mathbf{B})$. By Proposition \[prop:OmegaGrass\], $\Omega_{-}(\mathbf{B})$ is the image of $\operatorname{Gr}(k,l)$ embedded isometrically in $\operatorname{Gr}(k,n)$. So by Lemma \[lem:infty\], for any $\mathbf{X}_1,\mathbf{X}_2 \in \operatorname{Gr}(k,l)$, $d_{\operatorname{Gr}(k,n)}(\mathbf{X}_1,\mathbf{X}_2)= d_{\operatorname{Gr}(k,l)}(\mathbf{X}_1,\mathbf{X}_2) = d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$, where the last is the geodesic distance in $\Omega_{-}(\mathbf{B})$. Hence if $d_{\Omega_{-}(\mathbf{B})}(\mathbf{X}_1,\mathbf{X}_2)$ is realized by a geodesic curve $\gamma$ in $\Omega_{-}(\mathbf{B})$, then $\gamma$ must also be a geodesic curve in $\operatorname{Gr}(k,n)$. We have represented $\operatorname{Gr}(k,n)$ as a set of *equivalence classes* of matrices but it may also be represented as a set of *actual matrices* [@Nicolaescu Example 1.2.20], namely, idempotent symmetric matrices of trace $k$: $$\operatorname{Gr}(k,n)\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=k \}.$$ The isomorphism maps each subspace $\mathbf{A}\in \operatorname{Gr}(k,n)$ to $P_{\mathbf{A}} \in \mathbb{R}^{n \times n}$, the unique orthogonal projection onto $\mathbf{A}$, and its inverse takes an orthogonal projection $P$ to the subspace $\operatorname{im}( P) \in \operatorname{Gr}(k,n)$. $P$ is an orthogonal projection iff it is symmetric and idempotent, i.e., $P^{\mathsf{T}} = P^2 = P$. The eigenvalues of an orthogonal projection onto a subspace of dimension $k$ are $1$’s and $0$’s with multiplicities $k$ and $n-k$, so $\operatorname{tr}(P) = k$ is equivalent to $\operatorname{rank}(P) = k$, ensuring $\operatorname{im}(P)$ has dimension $k$. In this representation, $$\begin{aligned} \Omega_{+}(\mathbf{A}) &\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=l, \; \operatorname{im}(A) \subseteq \operatorname{im}(P) \},\\ \Omega_{-}(\mathbf{B}) &\cong \{ P \in \mathbb{R}^{n \times n} : P^{\mathsf{T}} = P^2 = P, \; \operatorname{tr}(P)=k, \; \operatorname{im}(P) \subseteq \operatorname{im}(B) \},\end{aligned}$$ allowing us to treat $\operatorname{Gr}(k,n)$, $\operatorname{Gr}(l,n)$, $\Omega_{+}(\mathbf{A})$, $\Omega_{-}(\mathbf{B})$ all as subvarieties of $\mathbb{R}^{n \times n}$. Probability density on the Grassmannian {#sec:volume} ======================================= We determine the relative volumes of $\Omega_{+}(\mathbf{A})$, $\Omega_{-}(\mathbf{B})$ and prove a volumetric analogue of in Theorem \[thm1\]: > *Given $k$-dimensional subspace $\mathbf{A}$ and $l$-dimensional subspace $\mathbf{B}$ in $\mathbb{R}^n$, the probability that a randomly chosen $l$-dimensional subspace in $\mathbb{R}^n$ contains $\mathbf{A}$ equals the probability that a randomly chosen $k$-dimensional subspace in $\mathbb{R}^n$ is contained in $\mathbf{B}$.* Every Riemannian metric on a Riemannian manifold yields a *volume density* that is consistent with the metric [@Nicolaescu Example 3.4.2]. The Riemannian metric[^3] on $\operatorname{Gr}(k,n)$ that gives us the Grassmann distance in and the geodesic in also gives a density $d\gamma_{k,n}$ on $\operatorname{Gr}(k,n)$. The volume of $\operatorname{Gr}(k,n)$ is then $$\label{eq:volume} \operatorname{Vol}\bigl(\operatorname{Gr}(k,n)\bigr)=\int_{\operatorname{Gr}(k,n)} |d\gamma_{k,n}| =\binom{n}{k}\frac{\prod_{j=1}^{n}{\omega_j}}{\bigl(\prod_{j=1}^k\omega_j \bigr)\bigl(\prod_{j=1}^{n-k}\omega_j\bigr)},$$ where $\omega_m \coloneqq \pi^{m/2}/\Gamma(1+m/2)$, volume of the unit ball in $\mathbb{R}^m$ [@Nicolaescu Proposition 9.1.12]. The normalized density $d\mu_{k,n} \coloneqq \operatorname{Vol}\bigl(\operatorname{Gr}(k,n)\bigr)^{-1} \lvert d\gamma_{k,n}\rvert$ defines a natural *uniform probability density* on $\operatorname{Gr}(k,n)$. With respect to this, the probability of landing on $\Omega_{+}(\mathbf{A})$ in $\operatorname{Gr}(l,n)$ equals the probability of landing on $\Omega_{-}(\mathbf{B})$ in $\operatorname{Gr}(k,n)$. \[cor:volume\] Let $k \le l \le n$ and $\mathbf{A} \in \operatorname{Gr}(k,n)$, $\mathbf{B} \in \operatorname{Gr}(l,n)$. The relative volumes of $\Omega_{+}(\mathbf{A})$ in $\operatorname{Gr}(l,n)$ and $\Omega_{-}(\mathbf{B})$ in $\operatorname{Gr}(k,n)$ are equal and their common value depends only on $k,l,n$, $$\mu_{l,n}\bigl(\Omega_{+}(\mathbf{A})\bigr) = \mu_{k,n}\bigl(\Omega_{-}(\mathbf{B})\bigr) = \frac{l!(n-k)!\prod_{j=l-k+1}^{l}{\omega_j}}{n!(l-k)!\prod_{j=n-k+1}^{n}\omega_j}.$$ By Proposition \[prop:OmegaGrass\], $\Omega_{+}(\mathbf{A})$ is isometric to $\operatorname{Gr}(n-l,n-k)$ and $\Omega_{-}(\mathbf{B})$ is isometric to $\operatorname{Gr}(k,l)$, so by their volumes are $$\binom{n-k}{n-l}\frac{\prod_{j=1}^{n-k}{\omega_j}}{\bigl(\prod_{j=1}^{n-l}\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)}, \qquad \binom{l}{k}\frac{\prod_{j=1}^{l}{\omega_j}}{\bigl(\prod_{j=1}^k\omega_j\bigr)\bigl(\prod_{j=1}^{l-k}\omega_j)}$$ respectively. Now divide by the volumes of $\operatorname{Gr}(l,n)$ and $\operatorname{Gr}(k,n)$ respectively. By definition, relative volume depends on the volume of ambient space and the dependence on $n$ is expected, a slight departure from Theorem \[thm1\]. Conclusions =========== We provided what we hope is a thorough study of subspace distances, a topic of wide-ranging interest. We investigated the topic from different angles and filled in the most glaring gap in our existing knowledge — defining distances and metrics for inequidimensional subspaces. We also developed simple geometric models for subspaces of all dimensions and enriched the existing differential geometric view of Grassmannians with algebraic geometric perspectives. We expect these to be of independent interest to applied and computational mathematicians. Most of the topics discussed in this article have been extended to affine subspaces in [@LWY]. Acknowledgment {#acknowledgment .unnumbered} -------------- We are very grateful to the two anonymous referees for their invaluable suggestions, both mathematical and stylistic. 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--- abstract: | We consider a stochastic process in which independent identically distributed random matrices are multiplied and where the Lyapunov exponent of the product is positive. We continue multiplying the random matrices as long as the norm, $\epsilon$, of the product is *less* than unity. If the norm is greater than unity we reset the matrix to a multiple of the identity and then continue the multiplication. We address the problem of determining the probability density function of the norm, $P_\epsilon$. We argue that, in the limit as $\epsilon\to 0$, $P_\epsilon\sim (\ln (1/\epsilon))^\mu \epsilon^\gamma$, where $\mu $ and $\gamma$ are two real parameters. Our motivation for analysing this *matrix contraction process* is that it serves as a model for describing the fine-structure of strange attractors, where a dense concentration of trajectories results from the differential of the flow being contracting in some region. We exhibit a matrix-product model for the differential of the flow in a random velocity field, and show that there is a phase transition, with the parameter $\mu$ changing abruptly from $\mu=0$ to $\mu=-\frac{3}{2}$ as a parameter of the flow field model is varied. address: 'Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, England.' author: - Michael Wilkinson and John Grant title: A Matrix Contraction Process --- Introduction {#sec: 1} ============ Consider a random multiplicative process obtained from a sequence $a_1,a_2,a_3,\cdots$ of real positive independent identically distributed random variables, each having a finite probability of being less than unity and of being greater than unity. Let the value, $\epsilon$, of the process after $N$ steps be given by the random variable $$\label{eq: 1.1} \epsilon= \epsilon_0 \left(\prod_{k=1}^N a_k \right)$$ where $\epsilon_0<1$ is a positive scalar constant, providing this value is *less* than unity. If this product exceeds unity we reset $\epsilon$ to the value $\epsilon_0$, and reset $N$ to 0. This process has one of two possible types of behaviour, depending on the probability distribution of the $a_k$. The value of $\epsilon$ given by equation (\[eq: 1.1\]) may tend to decrease as $N\to \infty$. The other possibility is that the product tends to increase and the process is repeatedly reset. We are interested in this latter case as it generates a statistically stationary sequence of values of $\epsilon$, and we can consider the probability density function (PDF), $P_\epsilon$ (throughout we use $P_X$ to denote the PDF of a random variable $X$, and $\langle X\rangle$ to denote its expectation value). The distribution of $\epsilon$ is most easily understood in terms of the random variable, $Z=\ln\,\epsilon$ which is a *sum* of independent random numbers, $z_k=\ln\,a_k$, each having a positive mean value. In the limit as $Z\to -\infty$ a master equation for $Z$ becomes independent of $Z$, unless the probability distribution of the $z_k$ has heavy tails’. This symmetry is respected by choosing a PDF of the form $P_Z=\exp(\alpha Z)$, for some coefficient $\alpha$ which must be positive to give a normalisable probability density. So, for small $\epsilon$, the PDF $P_\epsilon$, is given by a power-law of the form $$\label{eq: 1.2} P_\epsilon \sim \ \epsilon^\gamma$$ where $\gamma=\alpha-1$. This argument showing that this *scalar contraction process* has a power-law distribution for $\epsilon$ was previously presented in [@Gui+16]. It is explained in [@Gui+16] that the scalar contraction process has applications in dynamical systems theory, because the differential of a dynamical map is, by the chain rule, a product of the differentials for each iteration. In the case of a system with one degree of freedom, involving a chaotic map of one variable, it is reasonable to model this differential as a product of independent random numbers, analogous to equation (\[eq: 1.1\]). The resetting process addresses what happens when the separation of trajectories is no longer small and the linearisation approximation fails. Since we are concerned with small separations, we ignore the dynamics while the separation becomes large and reset the process when the separation of trajectories becomes small again. For a system with more than one degree of freedom the differential of the dynamical map is described by a product of *stability matrices*, rather than a product of scalars [@Ott02]. In this paper we generalise from the scalar case to consider a *matrix contraction process* involving a product of $M\times M$ square matrices: $$\label{eq: 1.3} {\bf A} = \frac{\epsilon_0}{\sqrt{M}} {\bf a}_N{\bf a}_{N-1}\ldots {\bf a}_2 {\bf a}_1$$ where the ${\bf a}_i$ are independent, identically distributed random matrices. When $\epsilon$, the (Frobenius) norm of [**A**]{}, defined by $$\label{eq: 1.4} \epsilon=\sqrt{{\rm tr}\left({\bf A}^{\rm T}{\bf A}\right)},$$ equals or exceeds unity we reinitialise ${\bf A}$ to $\epsilon_0/\sqrt{M}$ times the identity matrix, so that the norm is $\epsilon_0$, and continue the iteration. This generates a sequence of values of $\epsilon$ characterized by a PDF $P_\epsilon$. Analogy with the scalar contraction process suggests that $P_\epsilon$ is a power-law, however we claim that for these matrix-valued contraction processes a more general form may be required to describe the PDF as $\epsilon \to 0$: $$\label{eq: 1.5} P_\epsilon\sim \left(\ln\,\frac{1}{\epsilon} \right)^\mu \epsilon^\gamma$$ where $\mu $ is another parameter. Our objective is to justify this assertion and to show how $\gamma $ and $\mu$ can be determined. We discuss this in detail for a specific class of models, but the construction can be generalised. We find that $\gamma$ depends continuously on parameters of the model, but that $\mu$ is constant over intervals of the parameters, exhibiting discontinuous jumps. In some cases we find $\mu=0$ so that $P_\epsilon$ is a power-law. The matrix contraction process has a natural application to describing the structure of strange attractors. In particular, it is possible to relate the distribution of $\epsilon$ to fractal dimensions. We will consider these connections in detail in a companion paper [@Wil+17], where we characterise the structure of compact constellations of phase points and discuss their relation to Renyi dimensions of the attractor. In order to simplify the discussion, we restrict ourselves to a model where ${\bf A}$ is a continuous function of some time variable, $t$, and the factors in equation (\[eq: 1.3\]) are close to the identity so that we may write $$\label{eq: 1.6} {\bf A}(t)=\frac{\epsilon_0}{\sqrt{M}}({\bf I}+\delta {\bf a}_n)({\bf I}+\delta {\bf a}_{n-1})\cdots ({\bf I}+\delta {\bf a}_2)({\bf I}+\delta {\bf a}_1)$$ where $n \equiv {\rm Int}(t/ \delta t)$, with $\delta t$ a small increment of time. The elements of the $\delta {\bf a}_n$ are assumed to have the following statistics $$\label{eq: 1.7} \langle (\delta {\bf a}_n)_{ij}\rangle=0 \ ,\ \ \ \langle (\delta {\bf a}_n)_{ij}(\delta {\bf a}_m)_{kl}\rangle=2{\cal D}_{ijkl}\ \delta _{nm}\delta t$$ so that the evolution of ${\bf A}(t)$ is characterised by a diffusive process, with diffusion coefficients ${\cal D}_{ijkl}$. We consider the simplest case where ${\bf A}(t)$ is a $2\times 2$ matrix. The initial value of ${\bf A}(t)$, and the value it takes whenever it is reset, is a scalar multiple of the $2 \times 2$ identity matrix, equal to $\frac{1}{\sqrt {2}}\epsilon_0 {\bf I}$, where $\epsilon_0$ is the initial value of the matrix norm and $0< \epsilon_0 < 1$. Evolution of the process ${\bf A}(t)$ {#sec: 2} ===================================== The analysis of the evolution of ${\bf A}(t)$ may be simplified by using the singular value decomposition (SVD) of ${\bf A}(t)$. This can be written in the form (see [@Horn13]) $$\label{eq: 2.1} {\bf A}(t)={\bf R}_1\,{\bf \Lambda}\,{\bf R}_2$$ where ${\bf R}_1 \equiv {\bf R}(\theta_1)$ and ${\bf R}_2 \equiv {\bf R}(\theta_2)$ are rotation matrices with $$\label{eq: 2.1a} {\bf R}(\theta)=\left(\begin{array}{cc} \cos \theta & \sin \theta \cr -\sin \theta & \cos \theta \end{array}\right) \ , \$$ and ${\bf \Lambda}$ is a diagonal matrix whose entires are the singular values $\lambda_1$ and $\lambda_2$: $$\label{eq: 2.1b} {\bf \Lambda}= \left( \begin{array}{cc} \lambda_1 & 0 \cr 0 & \lambda_2 \end{array} \right)\ .$$ The rotation angles and singular values depend on $t$. Consider the process at times $n \delta t$ and $(n+1) \delta t$. Suppose that, in the time interval, the increments in matrices ${\bf R}_1$, ${\bf R}_2$ and ${\bf \Lambda}$ are, respectively, $\delta {\bf R}_1$, $\delta {\bf R}_2$, and $\delta{\bf \Lambda}$ (we have suppressed the arguments of the matrices for clarity). Then writing ${\bf A}_n$ for ${\bf A}(n \delta t)$ we have $$\label{eq: 2.1c} {\bf A}_{n+1} = {\bf a}_{n+1}{\bf A}_{n}= \left({\bf I} + \delta {\bf a}_{n+1} \right){\bf A}_{n}$$ and the SVD of ${\bf A}_{n+1}$ can be written as $$\label{eq: 2.1d} {\bf A}_{n+1} = \left( {\bf R}_1 + \delta {\bf R}_1 \right ) \left( {\bf \Lambda}+ \delta{\bf \Lambda} \right) \left( {\bf R}_2 + \delta {\bf R}_2 \right) \ .$$ Since ${\bf A}_{n}={\bf R}_1{\bf \Lambda}{\bf R}_2 $, expanding and comparing these expressions for ${\bf A}_{n+1}$ gives $$\begin{aligned} \label{eq: 2.1e} \delta{\bf a}_{n+1} {\bf A}_{n} & = & {\bf R}_1 {\bf \Lambda} \delta{\bf R}_2 + {\bf R}_1 \delta{\bf \Lambda} {\bf R}_2 + \delta{\bf R}_1 {\bf \Lambda} {\bf R}_2 \nonumber \\ % & +& {\bf R}_1 \delta{\bf \Lambda} \delta{\bf R}_2 + \delta {\bf R}_1 {\bf \Lambda} \delta{\bf R}_2 + \delta{\bf R}_1 \delta {\bf \Lambda} {\bf R}_2 \nonumber \\ % & +& \delta{\bf R}_1 \delta {\bf \Lambda} \delta{\bf R}_2 .\end{aligned}$$ Pre-multiplying this equation by ${\bf R}_{1 }^{-1}$ and postmultiplying it by $ {\bf R}_2^{-1}{\bf \Lambda}^{-1}$ gives $$\begin{aligned} \label{eq: 2.1f} \delta \tilde{{\bf a}}_{n+1} & =& {\bf \Lambda} \delta{\bf R}_2 {\bf R}_2^{-1} {\bf \Lambda}^{-1} + \delta{\bf \Lambda}{\bf \Lambda}^{-1} + {\bf R}_1^{-1} \delta{\bf R}_1 \nonumber \\ & + & \delta{\bf \Lambda} \delta{\bf R}_2 {\bf R}_2^{-1} {\bf \Lambda}^{-1} + {\bf R}_1^{-1} \delta{\bf R}_1 {\bf \Lambda} \delta{\bf R}_2 {\bf R}_2^{-1} {\bf \Lambda}^{-1} + {\bf R}_1^{-1} \delta {\bf R}_1 \delta {\bf \Lambda}{\bf \Lambda}^{-1} \nonumber \\ % & +& {\bf R}_1^{-1} \delta{\bf R}_1 \delta {\bf \Lambda} \delta{\bf R}_2 {\bf R}_2^{-1}{\bf \Lambda}^{-1} \ ,\end{aligned}$$ where $$\label{eq: 2.1x} \delta \tilde{{\bf a}}_{n+1} \equiv {{\bf R}_1}^{-1} \delta {\bf a}_{n+1} {\bf R}_1 \ .$$ Now, in terms of the increments in the singular values, $\delta \lambda_i$, we have $$\label{eq: 2.1g} \delta {\bf \Lambda} {\bf \Lambda}^{-1} =\left( \begin{array}{cc} \frac{\delta \lambda_1}{\lambda_1} & 0 \cr 0 & \frac{\delta \lambda_2}{\lambda_2} \end{array} \right) = {\bf \Lambda}^{-1} \delta {\bf \Lambda}$$ and, to second order in the $\delta \theta_i$, $$\label{eq: 2.1h} {\bf R}_{i}^{-1} \delta {\bf R}_{i}= \left( \begin{array}{cc} -\frac{\delta \theta_i^{2}}{2} & \delta \theta_i \cr -\delta \theta_i & -\frac{\delta \theta_i^{2}}{2} \cr \end{array} \right) =\delta {\bf R}_{i} {\bf R}_{i}^{-1} \ .$$ Therefore, to the second order of increments, equation (\[eq: 2.1f\]) gives $$\label{eq: 2.1i} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ \delta \widetilde{{\bf a}}_{n+1} = \left( \begin{array}{cc} \frac{\delta \lambda_1}{\lambda_1} - \frac{1}{2} \left(\delta \theta_1^{2} + \delta \theta_2^{2} \right) - \left( \frac{\lambda_2}{\lambda_1} \right) \delta \theta_1 \delta \theta_2 \ , & \left(1+\frac{\delta \lambda_2}{\lambda_2} \right)\delta \theta_1 + \left( \frac{\lambda_1}{\lambda_2} \right) \left(1+\frac{\delta \lambda_1}{\lambda_1} \right) \delta \theta_2 \ , \cr -\left( 1+\frac{\delta \lambda_1}{\lambda_1} \right)\delta \theta_1 -\left( \frac{\lambda_2}{\lambda_1} \right) \left( 1+\frac{\delta \lambda_2}{\lambda_2} \right) \delta \theta_2\ , & \frac{\delta \lambda_2}{\lambda_2} - \frac{1}{2} \left( \delta \theta_1^{2} + \delta \theta_2^{2} \right) - \left( \frac{\lambda_1}{\lambda_2} \right) \delta \theta_1 \delta \theta_2 \cr \end{array} \right)$$ In the limit as $\delta t \to 0$, equation (\[eq: 2.1i\]) reduces to a system of coupled stochastic differential equations (SDEs) for the singular values and rotation angles: $$\begin{aligned} \label{eq: 2.1j} \frac{\rm{d} \lambda_1}{\lambda_1} &=& \rm{d}{\tilde a}_{11} + \frac{1}{2} \left(\rm{d} \theta_1^{2} + \rm{d} \theta_2^{2} \right) + \nu \rm{d} \theta_1\rm{d} \theta_2 \\ \cr \label{eq: 2.1k} \frac{\rm{d} \lambda_2}{\lambda_2} &=& \rm{d}{\tilde a}_{22} + \frac{1}{2} \left(\rm{d} \theta_1^{2} + \rm{d} \theta_2^{2} \right) + \left( \frac{1}{\nu} \right) \rm{d} \theta_1\rm{d} \theta_2 \\ % \label{eq: 2.1l} \rm{d} \theta_1 &=& - \frac{ \frac{1}{\nu}\left(1+\frac{\rm{d} \lambda_1}{\lambda_1} \right) \rm{d}{\tilde a}_{21}+{\nu}\left(1 + \frac{\rm{d} \lambda_2}{\lambda_2}\right)\rm{d}{\tilde a}_{12} } {\frac{1}{\nu}\left(1+\frac{\rm{d} \lambda_1}{\lambda_1} \right)^2 - {\nu} \left(1 + \frac{\rm{d} \lambda_2}{\lambda_2} \right)^2 } \\ \label{eq: 2.1m} % \rm{d} \theta_2 &=& \frac{ \left(1 + \frac{\rm{d} \lambda_1}{\lambda_1}\right)\rm{d}{\tilde a}_{12} +\left(1+\frac{\rm{d} \lambda_2}{\lambda_2} \right) \rm{d}{\tilde a}_{21}} {\frac{1}{\nu} \left(1 + \frac{\rm{d} \lambda_1}{\lambda_1}\right)^2-\nu \left(1+\frac{\rm{d} \lambda_2}{\lambda_2} \right)^2 }\end{aligned}$$ where $ \nu \equiv \lambda_2/\lambda_1 $ and we have suppressed the time subscript on the $ \rm{d}{\tilde a}_{ij} $ (the remaining subscripts denote the row and column position of the matrix element). It is convenient to replace the singular values with logarithmic variables ${Z_i = \rm ln}\lambda_i$: we have $$\label{eq: 2.2} {\rm d}Z_i = \frac{{\rm d} \lambda_i}{\lambda_i}-\frac{1}{2} \left( \frac{{\rm d} \lambda_i}{\lambda_i} \right)^2$$ then, retaining only terms upto the second order in small increments, we obtain the following set of SDEs for the increments of the $Z_i$ and $\theta_i$ in terms of the matrix elements, ${\rm d}{\tilde a}_{ij}$. Defining $$\label{eq: 2.6b} \alpha \equiv \frac{\nu}{\left( \nu^2-1 \right)}$$ we find: $$\begin{aligned} \label{eq: 2.3} \rm{d} Z_1&=& \rm{d}\tilde{a}_{11} -\frac{1}{2}\rm{d}\tilde{a}_{11}^2 -\frac{\alpha}{2}\left[\frac{1}{\nu} \rm{d}\tilde{a}_{21}^2 +2\nu\rm{d}\tilde{a}_{12}\rm{d}\tilde{a}_{21} +\nu \rm{d}\tilde{a}_{12}^2\right] \\ % \label{eq: 2.4} \rm{d}Z_2 &=& \rm{d}\tilde{a}_{22} -\frac{1}{2}\rm{d}\tilde{a}_{22}^2 +\frac{\alpha}{2}\left[\nu\rm{d}\tilde{a}_{12}^2 +\frac{2}{\nu}\rm{d}\tilde{a}_{12}\rm{d}\tilde{a}_{21} +\frac{1}{\nu}\rm{d}\tilde{a}_{21}^2\right]\end{aligned}$$ and $$\begin{aligned} \label{eq: 2.5} \rm{d}\theta_1 &=& \alpha\left[\nu \rm{d}\tilde{a}_{12}+\frac{1}{\nu}\rm{d}\tilde{a}_{21}\right] +\alpha^2\rm{d}\tilde{a}_{11}\left[2\rm{d}\tilde{a}_{12} +\frac{\nu^2+1}{\nu^2}\rm{d}\tilde{a}_{21}\right] \nonumber \\ % &&-\alpha^2\rm{d}\tilde{a}_{22}\left[2\rm{d}\tilde{a}_{21} +(\nu^2+1)\rm{d}\tilde{a}_{12}\right] \\ % \label{eq: 2.6} \rm{d}\theta_2 &=& -\alpha \left[\rm{d}\tilde{a}_{12}+\rm{d}\tilde{a}_{21}\right] -\frac{\alpha^2}{\nu}\rm{d}\tilde{a}_{11}\left[2\rm{d}\tilde{a}_{12} +(\nu^2+1)\rm{d}\tilde{a}_{21}\right] \nonumber \\ % &&+\alpha^2\rm{d}\tilde{a}_{22}\left[2\nu \rm{d}\tilde{a}_{12} +\frac{(\nu^2+1)}{\nu} \rm{d}\tilde{a}_{21}\right] \ .\end{aligned}$$ These equations can now be used to produce a Fokker-Planck equation for the joint probability density of the variables, $Z_i$ and $\theta_i$. The same Fokker-Planck equation arises if we replace the second-order terms by their mean values so we simplify the equations by taking expectation values of the second-order terms. At this stage there is nothing to distinguish between $\lambda_1$ and $\lambda_2$. However we do expect that both $\lambda_1$ and $\lambda_2$ have non-zero and distinct Lyapunov exponents, so that either $\nu\to 0$ or $\nu\to\infty$ as $t\to \infty$, with probability unity. Which case occurs is random and equiprobable. Let us assume that symmetry breaks so that $\nu\to 0$ in the long time limit. Accordingly, we consider this limit (noting that $\alpha/\nu \to -1$ as $\nu\to 0$). We obtain the following Langevin equations: $$\begin{aligned} \label{eq: 2.7} \rm{d} Z_1=& \rm{d}\tilde{a}_{11} -\frac{1}{2}\langle \rm{d}\tilde{a}_{11}^2\rangle +\frac{1}{2}\langle\rm{d}\tilde{a}_{21}^2\rangle \\ \cr \label{eq: 2.8} \rm{d}Z_2 =&\rm{d}\tilde{a}_{22} -\frac{1}{2}\langle \rm{d}\tilde{a}_{22}^2\rangle -\langle \rm{d}\tilde{a}_{12}\rm{d}\tilde{a}_{21}\rangle -\frac{1}{2}\langle \rm{d}\tilde{a}_{21}^2\rangle \\ \cr \label{eq: 2.9} \rm{d}\theta_1 =&-\rm{d}\tilde{a}_{21} +\langle \rm{d}\tilde{a}_{11}\rm{d}\tilde{a}_{21}\rangle \\ \cr \label{eq: 2.10} \rm{d}\theta_2 =& 0 \ .\end{aligned}$$ Note that $\theta_2$ freezes as $t\to \infty$. This is to be expected because the direction along which the norm is most rapidly increasing is expected to approach a limit as $t\to \infty$. Equations (\[eq: 2.7\]) and (\[eq: 2.8\]) are independent of the variables $Z_i$. These quantities therefore have a diffusive evolution at long times and we have $$\label{eq: 2.11} \langle Z_i(t)\rangle =v_i t \ ,\ \ \ \langle (Z_i-v_it)(Z_j-v_jt)\rangle =2{\cal D}_{ij}t \ .$$ The joint probability density of the $Z_i$ after time $t$ is therefore determined using a Green’s function $$\label{eq: 2.12} G(Z_1,Z_2,t)=\frac{1}{4\pi\sqrt{{\rm det}({\bf D})}t} \exp\left[-S(\mbox{\boldmath$Z$},t)\right]$$ where $$\label{eq: 2.13} S(\mbox{\boldmath$Z$},t)= \frac{1}{4t}(\mbox{\boldmath$Z$}-\mbox{\boldmath$v$}t) \cdot{\bf D}^{-1} (\mbox{\boldmath$Z$}-\mbox{\boldmath$v$}t) \ .$$ Because we have assumed that $\nu \to 0$, these equations are valid provided that $Z_1-Z_2$ is sufficiently large. Matrix contraction {#sec: 3} ================== Principles of calculation {#sec: 3.1} ------------------------- Our objective is to understand the distribution of the norm, $\epsilon$, for the matrix contraction process. In terms of the singular values and the logarithmic variables the norm of the matrix ${\bf A}(t)$ is $$\label{eq: 3.1} \epsilon=\sqrt{\lambda_1^2+\lambda_2^2}=\sqrt{(\exp(2Z_1)+\exp(2Z_2}) \ .$$ The PDF of $\epsilon$ may therefore be obtained from the joint PDF, $P_{(Z_1,Z_2)}$, of $Z_1$ and $Z_2$. In section \[sec: 2\] we showed that $\mbox{\boldmath$Z$}=(Z_1,Z_2)$ undergoes a diffusive process with drift. The matrix contraction process may therefore be represented by a point, or a notional particle, in the $(Z_1,Z_2)$ plane which undergoes advective diffusion, with drift velocity $\mbox{\boldmath$v$}$ and diffusion tensor ${\bf D}$. This process is illustrated schematically in figure \[fig: 1\]. Since we have ordered the singular values so that $Z_1\ge Z_2$, the line $Z_1=Z_2$ is a reflecting boundary. The resetting process occurs when the norm of the matrix is equal to unity, which is represented by a contour $\rm{B}$. The contour ${\rm B}$ is thereforean absorbing boundary. When the representative point reaches this absorbing boundary, $\bf A(t)$ is re-set to $\frac{1}{\sqrt{2}}\epsilon_0 {\mathbf{I}}$ and the representative diffusing particle’ is re-introduced at the source at the point ${\rm S}=\left( {\rm ln} (\epsilon_0/\sqrt{2}),{\rm ln}(\epsilon_0/\sqrt{2}) \right)$. The process may therefore be modelled by an ensemble of particles’, which diffuse and drift in a wedge-like domain of the $(Z_1,Z_2)$ plane, being reflected at one edge and absorbed at the other. The loss of particles by absorption on ${\rm B}$ is balanced by particles being injected at ${\rm S}$. [figure1.eps]{} (100,83)[$Z_1$]{} (84,98)[$Z_2$]{} (98,95)[$y$]{} (100,70)[$x$]{} (46,86)[$(X,0)$]{} (72,74)[${\rm S}$]{} (-4,81)[${\rm B}$]{} (-4,52)[${\rm C}$]{} (55,30)[$\mbox{\boldmath$Z$}^{\ast}$]{} (90,29)[$\mbox{\boldmath$v$}$]{} (87,15)[${\bf D}$]{} (35,15)[$\mbox{\boldmath$u$}^{\ast}$]{} Consider how to determine the probability density $P_{(Z_1,Z_2)}$ for the representative point to reach $(Z_1,Z_2)$. Let $p(Z_1,Z_2,t,\Delta t)$ be the probability density for the point to be at $(Z_1,Z_2)$ at time $t$ after it is emiitted from S, and to be absorbed on B after time $t+\Delta t$. The probability density $P_{(Z_1,Z_2)}$ is obtained by integrating over the times: $$\label{eq: 3.1a} P_{(Z_1,Z_2)}=\int_0^\infty {\rm d}t\int_0^\infty {\rm d}\Delta t\ p(Z_1,Z_2,t,\Delta t)$$ Because the diffusion process is Markovian, we can write $p$ as a product of two functions: $p(Z_1,Z_2,t,\Delta t)=G(Z_1,Z_2,t)p_2(Z_1,Z_2,\Delta t)$, where $G(Z_1,Z_2,t)$ is the Green’s function (probability density to reach $(Z_1,Z_2)$ after time $t$), and $p_2(Z_1,Z_2,\Delta t)$ is the probability density for a particle released at $(Z_1,Z_2)$ to be absorbed on B after time $\Delta t$. From this definition of $p_2$, we have $$\label{eq: 3.1b} \int_0^\infty {\rm d}\Delta t\ p_2(Z_1,Z_2,\Delta t)=1$$ so that $$\label{eq: 3.2} P_{(Z_1,Z_2)}=\int_0^\infty {\rm d}t\ G(Z_1,Z_2,t) \ .$$ Strictly speaking, $G(Z_1,Z_2,t)$ should be the Green’s function for reaching $(Z_1,Z_2)$ after time $t$ by a path that does not cross B. However, when we determine the probability density of $\epsilon$, we find that we require $P_{(Z_1,Z_2)}$ for positions which are not close to the absorbing boundary B. In this case we can use the Green’s function for the non-absorbing boundary. In cases where the required values of ${(Z_1,Z_2)}$ are not close to the reflecting line, we can use equation (\[eq: 2.12\]) to approximate the Green’s function. Following the approach used in [@Gra+15], we use the Laplace principle to estimate the integral in equation (\[eq: 3.2\]). At the time $t^\ast$ when the propagator is maximal, $\partial S/\partial t(\mbox{\boldmath$Z$},t^\ast)=0$, so that $t^\ast$ is given by $$\label{eq: 3.2a} t^\ast= \sqrt{\frac {\mbox{\boldmath$Z$}\cdot {\bf D}^{-1}\mbox{\boldmath$Z$}} {\mbox{\boldmath$v$}\cdot {\bf D}^{-1}\mbox{\boldmath$v$}} }$$ and, therefore $$\label{eq: 3.3} P_{(Z_1,Z_2)}\sim \exp\left[-\Phi(\mbox{\boldmath$Z$})\right]$$ where $$\label{eq: 3.4} \Phi (\mbox{\boldmath$Z$})=S(\mbox{\boldmath$Z$},t^\ast) =\frac{1}{2}\left[\sqrt{\mbox{\boldmath$Z$}\ \cdot{\bf D}^{-1}\mbox{\boldmath$Z$}}\sqrt{\mbox{\boldmath$v$}\ \cdot{\bf D}^{-1}\mbox{\boldmath$v$}} -\mbox{\boldmath$Z$}\cdot{\bf D}^{-1}\mbox{\boldmath$v$} \right] \ .$$ The function $\Phi (Z_1,Z_2)$ may be interpreted as the height, above the $(Z_1,Z_2)$ plane, of a tilted conical surface which touches this plane along the ray $\mbox{\boldmath$Z$}=\lambda \mbox{\boldmath$v$}$, where $\lambda$ is a positive real parameter. Since this cone touches the plane in the direction of the drift vector $\mbox{\boldmath$v$}$, downwind’ of the origin, along any other ray through the origin $\Phi (Z_1,Z_2)$ must increase linearly with distance from the origin. Therefore, asymptotically, $P_{(Z_1,Z_2)}$ decays exponentially with distance from the origin along such a ray. We remark that this construction for $P_{(Z_1,Z_2)}$ is similar in structure to the result obtained in [@Wil+15]. Recall that we wish to determine the probability density for the matrix norm, $\epsilon$, to reach a very small value. Accordingly, we consider the form of lines of constant $\epsilon$ in the $(Z_1,Z_2)$ plane. These lines are determined from equation (\[eq: 3.1\]), together with the condition that $Z_1\ge Z_2$. The contour ${\rm C}$ in figure \[fig: 1\] is one such line. It is clear from the form of (\[eq: 3.1\]) that the contours of constant $\epsilon$ are asymptotic to the vertical line $Z_1=\ln\,\epsilon$. The probability density to reach $\epsilon$ can therefore be obtained from that of $X={\rm ln}\ \epsilon$, which can be estimated using the expression (\[eq: 3.3\]) on the segment of the asymptote lying on or below the line $Z_1=Z_2$. The PDF of $X$ is obtained by applying Laplace’s principle again, so that we determine the value $\mbox{\boldmath$Z$}^\ast$ that minimises $\Phi(Z_1,Z_2)$ on the line $(Z_1,Z_2)=(X,X-Y)$, with $0 \le Y<\infty$. Because $\Phi(Z_1,Z_2)$ increases linearly along any ray, we find that the saddle point $\mbox{\boldmath$Z$}^{\ast}$, at which $\Phi (\mbox{\boldmath$Z$})$ has a minimum value, lies along the *direction*, $\mbox{\boldmath$u$}^\ast$ which minimises (\[eq: 3.4\]) on the line $(Z_1,Z_2)=-(1,\eta)$, with $\eta\ge 1$. If the minimum lies at $\eta^\ast$, we have ${\bm u}^\ast=-(1,\eta^\ast)$. Since $\mbox{\boldmath$Z$}^\ast = |X| \mbox{\boldmath$u$}^\ast$, the probability density of $X$ is therefore of the form $$\label{eq: 3.6} P_X \sim \exp[-\Phi(\mbox{\boldmath$u$}^\ast)|X|] \ .$$ This is consistent with $P_\epsilon$ having a power-law distribution, and suggests that the exponent $\gamma$ in (\[eq: 1.5\]) is $$\label{eq: 3.7} \gamma=[\Phi(\mbox{\boldmath$u$}^\ast)-1] \ .$$ In the discussion above it has been assumed that the saddle point $\mbox{\boldmath$Z$}^{\ast}$, lies below the line $Z_1=Z_2$, so that $\eta^\ast>1$, as illustrated schematically in figure \[fig: 1\]. We refer to this case as the *non-degenerate case*. However, the minimum of $\Phi(Z_1,Z_2)$ along a line of constant $Z_1$ may occur at a *physically inaccesible* point, i.e. one for which $Z_2 \geq Z_1$ or, equivalently, $\eta^\ast \leq 1$. In this case, which we refer to as the *degenerate* case, the discussion in section (\[sec: 3\]) must be replaced by a consideration of what happens in the neighbourhood of the boundary point $(Z_1=X,Z_2=X)$. This degenerate case is considered in section \[sec: 4\]. Estimate for the pre-exponential factor {#sec: 3.2} --------------------------------------- We have argued that $P_X\sim \exp(\alpha |X|)$, with $\alpha =-\Phi(\mbox{\boldmath$u$}^\ast)$, where $\mbox{\boldmath$u$}^\ast$ minimises $\Phi$ along the line $Z_1=X$. The exponential form of $P_X$ is consistent with a power-law, $P_\epsilon\sim \epsilon^\gamma$, with $\gamma=-\left(\alpha+1\right)$, but it is necessary to examine the pre-exponential factor to determine the true asymptotic form of $P_\epsilon$ as $\epsilon\to 0$. To this end, we consider in more detail the Laplace estimates of the integrals. First consider the integration over $t$: the Laplace method applied to (\[eq: 3.2\]) yields $$\label{eq: 3.8} P_{(Z_1,Z_2)}\sim \frac{\exp[-\Phi(\mbox{\boldmath$Z$})]} {t^\ast\sqrt{2\pi{\rm det}({\bf D})}}\left(\frac{\partial^2S}{\partial t^2} (\mbox{\boldmath$Z$},t^\ast)\right)^{-1/2} \ .$$ The stationary point is at $t^{\ast}$, given by equation (\[eq: 3.2a\]) and we find $$\label{eq: 3.10} \frac{\partial^2 S}{\partial t^2}(\mbox{\boldmath$Z$},t^\ast) =\frac{(\mbox{\boldmath$v$}\cdot {\bf D}^{-1}\mbox{\boldmath$v$})^{3/2}} {2(\mbox{\boldmath$Z$}\cdot {\bf D}^{-1}\mbox{\boldmath$Z$})^{1/2}} \ .$$ It follows that $$\label{eq: 3.11} P_{(Z_1,Z_2)}\sim K (\mbox{\boldmath$Z$}\cdot {\bf D}^{-1}\mbox{\boldmath$Z$})^{-1/4} \exp[-\Phi(\mbox{\boldmath$Z$})]$$ where $K$ is independent of $\mbox{\boldmath$Z$}$. Now, in order to determine $P_X$ we integrate (\[eq: 3.11\]) down the asymptote $Z_1=X$ from the point $\left(X,X\right)$, i.e. over $Z_2$ with $Z_2< X$, using the Laplace approximation for the second time. We are concerned only with the way in which $P_X$ depends on $Z_1$, and not on the precise form of any coefficients. We can therefore avoid detailed calculation of coefficients by using power-counting arguments. The expression for the second derivative of $\Phi(Z_1,Z_2)$ with respect to $Z_2$ contains terms proportional to $({\bm Z}\cdot {\bm D}^{-1}{\bm Z})^{-1/2}$, and other terms which scale as $Z_1^{-1}$. Performing the Gaussian integral therefore introduces a factor $Z_1^{1/2}$ which cancels the $Z_1$ dependence of the factor $({\bm Z}\cdot {\bm D}^{-1}{\bm Z})^{-1/4}$ in (\[eq: 3.11\]). At leading order, there is therefore no overall $Z_1$ dependence in the coefficient of the exponential term in the expression for $P_X$ and we conclude that $P_X\sim \exp[-\Phi(\mbox{\boldmath$u$}^\ast) |X|]$. Therefore in the non-degenerate case the probability density $P_\epsilon$ is of the form (\[eq: 1.5\]) with $$\label{eq: 3.12} \gamma=[\Phi(\mbox{\boldmath$u$}^\ast)-1] \ ,\ \ \ \mu=0 \ .$$ Treating the degenerate case {#sec: 4} ============================ If there is no stationary point of $\Phi (\mbox{\boldmath$Z$})$ with $Z_2<Z_1$, then the discussion in section \[sec: 3.2\] does not apply. We now consider what happens in this degenerate case. Making a coordinate transformation {#sec: 4.1} ---------------------------------- We have seen that the matrix multiplication process in our model corresponds to the diffusive evolution of the variables $Z_1$ and $Z_2$. From equations (\[eq: 1.7\]) it follows that the diffusion tensor is a symmetric $2\times 2$ matrix with equal diagonal elements, it can therefore be diagonalised by rotating the coordinate axes through $\pi/4$. We therefore find it convenient to consider the advective diffusion process in the coordinate system $x=(Z_1-Z_2)$ and $y=(Z_1+Z_2)$, where the diffusion tensor is diagonal (see figure \[fig: 1\]). In the $x$, $y$ coordinate system, the source is at $\left( 0, {\rm ln} \left( \epsilon_0^2/2 \right ) \right) $, the $y$ axis is a reflecting barrier and the absorbing boundary is approximated by the line $y=-x$. The equations of motion, (\[eq: 2.5\]), transform into $$\begin{aligned} \label{eq: 4.1.1} {\rm d}x&=&\left({\rm d}\tilde{a}_{11}-{\rm d}\tilde{a}_{22}\right) + \frac{1}{2}\left(\langle{\rm d}\tilde a_{22}^2 \rangle- \langle {\rm d} \tilde a_{11}^2 \rangle \right) \nonumber \\ % && +\frac{\nu}{1-\nu^2}\left[\nu \langle {\rm d}\tilde{a}_{12}^2 \rangle +\frac{1}{\nu} \langle {\rm d}a_{21}^2 \rangle + \left(\nu+\frac{1}{\nu}\right) \langle {\rm d}\tilde{a}_{12}{\rm d}\tilde{a}_{21} \rangle \right] \nonumber \\ % {\rm d}y &=&\left({\rm d}\tilde{a}_{11}+{\rm d}\tilde{a}_{22}\right)- \frac{1}{2}\left( \langle{\rm d}\tilde{a}_{11}^2 \rangle+ \langle{\rm d}\tilde{a}_{22}^2 \rangle \right) - \langle {\rm d}\tilde{a}_{12}{\rm d}\tilde{a}_{21} \rangle \ .\end{aligned}$$ From these it follows that the drift velocity in the $y$ direction, $v_y$, is a constant and the drift velocity in the $x$ direction, $v_x$, is a function of $\nu=\exp(-x)$ and therefore of $x$. Also, the diffusion tensor for the fluctuations of $x$ and $y$ is diagonal, with diffusion coefficients $D_x$, $D_y$, which are independent of $x$ and $y$. Therefore the dynamics in the $y$ direction is simple: diffusion with a constant drift velocity. The dynamics in the $x$ direction is more complex: diffusion with a drift velocity which is a function of $x$. The motion in the $x$ and $y$ directions are independent. In section \[sec: 4.2\] we consider motion in the $x$ direction alone, in order to describe the effect of the reflecting boundary at $x=0$. In section \[sec: 4.3\] we combine the results for the $x$ and $y$ motions to model the distribution of $\ln \epsilon$. One-dimensional diffusion with reflecting wall {#sec: 4.2} ---------------------------------------------- The $x$-coordinate of the particle representing the matrix contraction process undergoes diffusion with a constant diffusion coefficient $D_x$, and drift with a position-dependent drift velocity, which is a function of $x$ alone. This velocity, $v_x(x)$, approaches a positive constant value, $v_0$, as $x\to\infty$, and it approaches $\infty$ as $x\to 0$, so that $x=0$ is a reflective barrier. We shall assume that the particle is released at $x=x_0$ at time $t=0$ and we wish to determine the distribution $P_x(x,t)$ at later times. The probability density satisfies $$\label{eq: 4.2.1} \frac{\partial P_x}{\partial t}=D_x\frac{\partial^2P_x}{\partial x^2} -\frac{\partial}{\partial x}[v_x(x)P_x] \ .$$ This equation can be transformed to a Hermitean form by writing $$\label{eq: 4.2.2} P_x=\exp[\chi(x)]\psi(x,t) \ ,\ \ \ \chi(x)=\frac{1}{2D_x}\int^x{\rm d}x'\ v_x(x') \ .$$ The function $\psi(x,t)$ satisfies a Schrödinger-like equation, with a Hermitian operator $\hat{\cal H}$: $$\label{eq: 4.2.3} \frac{\partial\psi}{\partial t}=D_x\frac{\partial^2\psi}{\partial x^2}-V(x)\psi \equiv-\hat{\cal H}\psi$$ where $$\label{eq: 4.2.4} V(x)=\frac{1}{2}v_x^{\prime}(x)+\frac{[v_x(x)]^2}{4D_x} \ .$$ By introducing a nominal absorbing barrier at x=L we can develop the solution to equation (\[eq: 4.2.1\]) in the finite interval \[0,L\] as an infinite series of orthonormal eigenfunctions of the Hermitean operator $\hat{\cal H}$; we then choose $L$ sufficiently large that its value may be assumed to have no influence. The details of this approach may be found in section 7.1 of [@Mah09]; we merely quote the result: $$\label{eq: 4.2.11} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ P_x(x,t)=\frac{2}{L}\exp\left(\frac{v_0(x-x_0)}{2D_x}-\frac{v_0^2t}{4D_x}\right) \sum_{n=1}^\infty\exp\left(-\frac{n^2\pi^2D_xt}{L^2}\right) \sin\left(\frac{n\pi x}{L}\right)\sin\left(\frac{n\pi x_0}{L}\right) \ .$$ Approximating the sum by an integral gives $$\begin{aligned} \label{eq: 4.2.12} P_x(x,t)&=&\frac{1}{L}\exp\left(\frac{v_0(x-x_0)}{2D_x}-\frac{v_0^2t}{4D_x}\right) \int_0^\infty {\rm d}n\ \exp\left(-\frac{n^2\pi^2D_xt}{L^2}\right) \nonumber \\ % &\times& \left[\cos\left(\frac{n\pi (x-x_0)}{L}\right)-\cos\left(\frac{n\pi (x+x_0)}{L}\right)\right] \ .\end{aligned}$$ Using the standard integral $$\label{eq: 4.2.13} \int_{-\infty}^\infty{\rm d}x\ \exp(-\alpha x^2)\cos(kx)=\sqrt{\frac{\pi}{\alpha}}\exp(-k^2/4\alpha)$$ we obtain the solution $$\begin{aligned} \label{eq: 4.2.14} P_x(x,t)&=&\frac{1}{\sqrt{4\pi D_xt}}\exp\left(\frac{v_0(x-x_0)}{2D_x}-\frac{v_0^2t}{4D_x}\right) \nonumber \\ % &&\times \left[\exp\left(-\frac{(x-x_0)^2}{4D_xt}\right) -\exp\left(-\frac{(x+x_0)^2}{4D_xt}\right)\right] \ .\end{aligned}$$ In the limit where $x_0\to 0$, we have $$\label{eq: 4.2.15} P_x(x,t)\sim \frac{x_0}{\sqrt{4\pi D_x^3}}\frac{x}{t^{3/2}}\exp\left(\frac{v_0x}{2D_x}\right) \exp\left(-\frac{x^2}{4D_xt}\right) \exp\left(-\frac{v_0^2t}{4D_x}\right) \ . %$$ Implications for degenerate case {#sec: 4.3} -------------------------------- We are interested in the probability density for the representative particle to reach the line in $(Z_1,Z_2)$ space corresponding to a given value of $X=\ln \epsilon$. In terms of the variables $x=(Z_1-Z_2)$ and $y=(Z_1+Z_2)$ we have $$\label{eq: 4.3.0} X\equiv g(x,y)=\ln \sqrt{\exp(y+x)+\exp(y-x)}=\frac{y}{2}+\frac{1}{2}\ln \left(2\cosh\, x\right)$$ The probability density for $X$ is $$\begin{aligned} \label{eq: 3.4.01} P_X&=&\int_0^\infty{\rm d}x\int_0^\infty{\rm d}y\ \delta \left(X-g(x,y)\right)\,P_{(x,y)}(x,y) \nonumber \\ % &=&2\int_0^\infty{\rm d}x\int_0^\infty{\rm d}y\ \delta \left(y-2X+\ln (2\cosh\,x)\right)\,P_{(x,y)}(x,y) \nonumber \\ % &=&2\int_0^\infty{\rm d}x\ P_{(x,y)}(x,2X-\ln(2\cosh\,x)) \ .\end{aligned}$$ Because the diffusive motions in the $x$ and $y$ coordinates are independent, the probability density to reach $(x,y)$ after time $t$ is expressed as a product: $$\label{eq: 4.3.2} G(x,y,t)=P_x(x,t)P_y(y,t) \ .$$ The probability density $P_y(y,t)$ is that for a simple advection-diffusion process: $$\label{eq: 4.3.3} P_y(y,t)=\frac{1}{\sqrt{4\pi D_y t}}\exp\left[-\frac{(y-v_yt)^2}{4D_yt}\right] \ .$$ The probability density $P_x(x,t)$, which must take account of the fact that $x=0$ is a reflecting barrier, is given by equation (\[eq: 4.2.15\]). Therefore $$\begin{aligned} \label{eq: 4.3.4} P_X&=&2\int_0^\infty {\rm d}x\int_0^\infty{\rm d}t\ P_x(x,t)P_y(2X-\ln(2\cosh\,x),t) \nonumber \\ % &=&\frac{x_0}{2\pi D_x\sqrt{D_xD_y}}\exp\left(\frac{X v_y}{D_y}\right) \int_0^\infty {\rm d}x\ x \exp\left(\frac{v_0x}{2D_x}-\frac{\ln(2\cosh\, x)v_y}{2D_y}\right) \nonumber \\ % &&\times I(x,2X-\ln(2\cosh\, x))\end{aligned}$$ with $$\label{eq: 4.3.5} I(x,y)=\int_0^\infty {\rm d}t\ \frac{1}{t^2}\exp\left[-\left({\cal A}t+\frac{{\cal B}(x,y)}{t}\right)\right]$$ where $$\label{eq: 4.3.5a} {\cal A}=\frac{v_0^2}{4D_x}+\frac{v_y^2}{4D_y} ,\ \ \ \ {\cal B}(x,y)=\frac{x^2}{4D_x} +\frac{y^2}{4D_y} \ .$$ Using the Laplace method to approximate the integral $I(x,y)$, we find that the exponent in the integrand is minimised at time $$\label{eq: 4.3.6} t^\ast=\sqrt{\frac{{\cal B}}{{\cal A}}}$$ and the Laplace estimate for this integral is $$\label{eq: 4.3.7} I(x,y)=\sqrt{\frac{\pi}{2}} \frac{{\cal A}^{1/4}}{{\cal B}^{3/4}} \exp[-2\sqrt{\cal AB}] \ .$$ We are concerned with determining the leading order behaviour as $|X|\to \infty$. The Laplace approximation for $I(x,2X-\ln (2\cosh\,x)))$ is valid when ${\cal B}$ is sufficiently large, that is when $|X|$ is large. This is $$\begin{aligned} \label{eq: 4.3.7b} \lefteqn {I(x,2X-\ln (2\cosh\, x))} \nonumber \\ & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ \sim \sqrt{\frac{\pi}{2}}{{\cal A}}^{1/4}\left(\frac{X^2}{D_y}\right)^{-3/4} \times \exp\left[-2\sqrt{{\cal A}} \sqrt{\frac{X^2}{D_y}-\frac{X\ln(2\cosh\,x)}{D_y} +\frac{[\ln(2\cosh\,x)]^2}{4D_y}+\frac{x^2}{4D_x} }\right] \nonumber \\ % & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ \sim \sqrt{\frac{\pi}{2}}{{\cal A}}^{1/4}D_y^{3/4}|X|^{-3/2} \times\exp\left[-2\sqrt{{\cal A}}\frac{|X|}{\sqrt{D_y}} \sqrt{1-\frac{\ln(2\cosh\,x)}{X}+O( X^{-2} ) }\right] \nonumber \\ % & & \!\!\!\!\!\!\!\!\!\ \sim {\cal C}|X|^{-3/2} \times\exp\left[-2\sqrt{\frac{{\cal A}}{D_y}} \left(|X| + \frac{1}{2}\ln(2\cosh\,x)\right) \right]\end{aligned}$$ where ${\cal C}$ is independent of $X$. From equations (\[eq: 4.3.4\]) and (\[eq: 4.3.7\]) we see that, to leading order as $X\to \infty$, the asymptotic behaviour of $P_X$ is $$\label{eq: 4.3.9} P_X\sim |X|^{-3/2}\exp[-\Lambda |X|] \ ,\ \ \ \Lambda \equiv \sqrt{\frac{v_0^2}{D_xD_y}+\frac{v_y^2}{D_y^2}} + \frac{v_y}{D_y} \ ,$$ providing that the following integral is finite: $$\label{eq: 4.3.10} J=\int_0^\infty {\rm d}x\ x \exp\left[ \frac{v_0x}{2D_x}-\frac{v_y}{2D_y}\ln (2\cosh\,x)-\sqrt{\frac{{\cal A}}{D_y}\ln(2\cosh\,x)} \right] \ .$$ This integral converges provided $$\label{eq: 4.3.10a} \sqrt{\frac{v_0^2}{D_xD_y}+\frac{v_y^2}{D_y^2}}-\frac{v_0}{D_y}+\frac{v_y}{D_y}>0 \ .$$ This condition can be shown to be equivalent to the degeneracy condition, namely that at the stationary point $(Z_1^{\ast}, Z_2^{\ast})$ of $\Phi(Z_1,Z_2)$, $Z_2^{\ast} > Z_1^{\ast} $. Because, in the $(x,y)$ coordinate system, ${\bf D}$ is diagonal and the function $\Phi(x,y)$ has the form $$\label{eqn: 4.3.11} \Phi(x,y)=\frac{1}{2} \left[ \sqrt{ \frac{v_0^2}{D_x}+\frac{v_y^2}{D_y} } \sqrt{\frac{x^2}{D_x}+\frac{y^2}{D_y}} - \left( \frac{v_0}{D_x}+\frac{v_y}{D_y} \right) \right]$$ Therefore, requiring that the minimum of $\Phi(x,y)$, on the line $y=2X-x$, is degenerate, and so lies at a negative value of $x$, gives (\[eq: 4.3.10a\]). Note that equation (\[eq: 4.3.9\]) shows that in the degenerate case $$\label{eq: 4.3.12} P_\epsilon \sim \left(\ln\,\frac{1}{\epsilon} \right)^\mu \epsilon^\gamma$$ with $\mu=-3/2$, and $\gamma=\Lambda-1$. Advective flow model {#sec: 5} ==================== Description of the model {#sec: 5.1} ------------------------ As a specific example, consider the matrix representing the differential of a random flow. Suppose that the velocity field of the flow, ${\bm u}\left({\bm x}(t),t\right)$, is defined by a random potential $\phi\left({\bm x}(t), t\right)$ and a random stream function $\mbox{\bm $\psi $}\left({\bm x}(t), t\right)$ such that $$\label{eq: 5.1} \mbox{\boldmath$u$} =\mbox{\boldmath$\nabla$}\wedge \mbox{\boldmath$\psi$} +\beta \mbox{\boldmath$\nabla$}\phi \ ,$$ where $\beta$ is a compressibility parameter. We assume that the fields $\mbox{\boldmath $\psi $}$ and $\phi$ are independent, have zero mean values and the same correlation function. We consider the case where the there are no temporal correlations, so that the position of a particle advected by the fluid, ${\bm x}(t)$, is a vector-valued random process satisfying $$\label{eq: 5.2} \mbox{\boldmath$x$}\left( t +\delta t \right) = \mbox{\boldmath$x$} \left( t \right) + \mbox{\boldmath$u$} \left(\mbox{\boldmath$x$} \left( t \right), t\right) \sqrt{\delta t}$$ and to leading order the separation $\delta \bm{r}=\bm{x}_1-\bm{x}_0$ of two nearby particles can be written as $$\label{eq: 5.4} \delta \mbox{\boldmath$r$} \left( t +\delta t \right) \equiv \mbox{\boldmath$a$}(\mbox{\boldmath$x$}_0 \left( t \right),t) \delta \mbox{\boldmath$r$} \left( t \right) = \left[ \mbox{\boldmath$I$} + \delta \mbox{\boldmath$a$} \left(\mbox{\boldmath$x$}_0\left( t \right),t \right) \right] \delta \mbox{\boldmath$r$} \left( t \right)$$ where $$\label{eq: 5.5} \left[ \delta \mbox{\boldmath$a$} \left(\mbox{\boldmath$x$}_0\left( t \right),t \right) \right]_{ij} = \left(\frac{\partial u_i \left(\mbox{\boldmath$x$}_{0}(t), t\right)}{\partial x_j} \right) \sqrt{\delta t}.$$ The elements of the matrix $ \delta {\bm a}$ are random variables constructed from the second derivatives of the velocity field potentials evaluated at time $t$ and position ${\bm x}_0(t)$ : $$\label{eq: 5.6} \delta {\bf a}=\left(\begin{array}{cc} \psi_{xy}+\beta \phi_{xx} & -\psi_{yy}+\beta \phi_{xy}\cr \psi_{xx}+\beta \phi_{xy} & -\psi_{xy}+\beta \phi_{yy} \end{array}\right)\sqrt{\delta t}.$$ All the derivatives have mean value zero, and are normalized so that their non-zero covariances are: $$\label{eq: 5.7a} \langle \psi_{xx}^2\rangle = \langle\psi_{yy}^2\rangle =3 \ \ , \ \ \ \ \langle \psi_{xx}\psi_{yy}\rangle = \langle \psi_{xy}^2\rangle =1\ ,$$ and similarly for derivatives of $\phi$. Also, since the model is rotationally invariant, and since $\delta \tilde{{\bf a}}_{n}$ , defined by (\[eq: 2.1x\]) is a rotational transformation of $\delta{\bf a}_{n}$ the elements of $\delta \tilde{{\bf a}}_{n}$ have the same statistics as those of $\delta{\bf a}_{n}$. From equations (\[eq: 5.6\]) and (\[eq: 5.7a\]), using equations (\[eq: 2.7\]), (\[eq: 2.8\]) and (\[eq: 2.11\]), it follows that the drift velocity and diffusion tensor for this model are, respectively: $$\label{eq: 5.7b} {\bm v}=\left(\begin{array}{cc} 1-\beta^2 \cr -(1+3\beta^2) \end{array}\right)$$ $$\label{eq: 5.7c} {\bf D}=\frac{1}{2}\left(\begin{array}{cc} 1+3\beta^2, & \beta^2 - 1 \cr \beta^2 - 1, & 1+3\beta^2 \end{array}\right).$$ Theoretical predictions for $\gamma$ {#sec: 5.2} ------------------------------------ Using the above expressions for ${\bm v}$ and ${\bf D}$ to compute the function $\Phi(Z_1,Z_2)$ using equation (\[eq: 3.4\]) gives $$\label{eq: 5.10} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\ \Phi(Z_1,Z_2)=\frac{1}{2\beta}\sqrt{\frac{1+3\beta^2}{2(1+\beta^2)}} \sqrt{(1+3\beta^2)(Z_1^2+Z_2^2)+2(1-\beta^2)Z_1Z_2}+Z_2 \ .$$ Considering the discussion immediately preceding equation (\[eq: 3.6\]), the critical point is $$\label{eq: 5.10a} {\bm Z}^\ast=|X|{\bm u}^\ast=|X|\left(\begin{array}{cc} -1 \cr -\eta^\ast \end{array}\right)$$ where $f^{\prime}(\eta^\ast)=0$ with $f(\eta) \equiv \Phi(-1,-\eta)$. Therefore, in terms of the compressibility parameter, $\beta$, we have $$\label{eq: 5.10b} \eta^\ast=\frac{7 \beta^4+10\beta^2-1}{(1+3\beta^2)(1-\beta^2)} \ .$$ Noting that $$\label{eq: 5.10c} (\eta^\ast-1)=\frac{2(1+\beta^2)(5 \beta^2-1)}{(1+3\beta^2)(1-\beta^2)}$$ if $1 > \beta > 1/\sqrt{5}$ then we must have $\eta^\ast > 1$ so that $Z_2^\ast < X=Z_1^\ast$ and the critical point is non-degenerate. However, if $0 \le \beta < 1/\sqrt{5}$ then $\eta^\ast < 1$ so that $Z_2^\ast > X=Z_1^\ast$, in which case the critical point is degenerate. In the non-degenerate case, we find that $\mbox{$\Phi({\bm u}^\ast)=f(\eta^\ast)= -2(\beta^2-1)/(1+3\beta^2)$}$ and, from equation (\[eq: 3.7\]), $$\label{eq: 5.10d} \gamma= -\frac{2(1+\beta^2)}{(1+3\beta^2)} -1\ .$$ In the degenerate case, setting $\eta^\ast=1$ so that $Z_1^\ast=Z_2^\ast=X$, and noting that, since $X<0$, $\sqrt{Z_1^\ast Z_2^\ast}=|X|=-X$, gives $\Phi({\bm u}^\ast)=\sqrt{(1+3\beta^2)/(2\beta^2)}-1$ so that $$\label{eq: 5.10e} \gamma= \frac{1}{\beta} \sqrt{\frac{1+3\beta^2}{2}}-2 \ .$$ This expression is suspect, however, because the derivation of (\[eq: 3.7\]) depends on the assumption that $Z_1>Z_2$, which is violated by the critical point condition. Exact equations for evolution of singular values {#sec: 5.3} ------------------------------------------------ In order to understand the degenerate case in more detail, we need to consider a more refined treatment that does not assume $Z_1\gg Z_2$. Using the statistics for the increments ${\rm d}\tilde{a}_{ij}$, obtained from (\[eq: 5.6\]) and (\[eq: 5.7a\]), the exact equations of motion in the the $(x,y)$ coordinate system, equations (\[eq: 4.1.1\]), become: $$\begin{aligned} \label{eq: 5.12} {\rm d}x&=&\left({\rm d}\tilde{a}_{11}-{\rm d}\tilde{a}_{22}\right) +2(1+\beta^2)\frac{1+\nu^2}{1-\nu^2}{\rm d}t \nonumber \\ % {\rm d}y&=&\left({\rm d}\tilde{a}_{11}+{\rm d}\tilde{a}_{22}\right) -4\beta^2{\rm d}t\end{aligned}$$ and the second moments of the increments are therefore $$\label{eq: 5.13} \langle {\rm d}x^2\rangle = 4(1+\beta^2){\rm d}t, \ \ \ \langle {\rm d}y^2\rangle = 8\beta^2{\rm d}t, \ \ \ \langle {\rm d}x{\rm d}y\rangle = 0 \ .$$ Noting that $\nu=\lambda_2/\lambda_1=\exp(-x)$ we see that $x$ and $y$ make independent diffusive motions, with the following drift velocity and diffusion tensor: $$\label{eq: 5.18} {\bm v}=\left(\begin{array}{cc} 2(1+\beta^2){\rm coth}(x)\cr -4\beta^2 \end{array}\right) \ , \ \ \ % {\bf D}=\left(\begin{array}{cc} 2(1+\beta^2) & 0 \cr 0 & 4\beta^2 \end{array}\right).$$ These results are consistent with the remarks regarding $v_y$ and $v_x$ in sections (\[sec: 4.1\]) and (\[sec: 4.2\]), respectively: $v_y$ is constant and, since $v_x(x)=2(1+\beta^2) {\rm coth}(x)$, we have $v_0=2(1+\beta^2)$ and $v_x \to \infty$ as $x \to 0$. Ratio of singular values {#sec: 6} ======================== Thus far we have considered the PDF of the norm, $\epsilon$, of the matrix product in the limit as $\epsilon \to 0$, showing that it is a power-law, with an additional logarithmic correction in the degenerate case. We can also consider conditional probabilities. One interesting example is the distribution of the ratio of singular values $\lambda_2/\lambda_1$, for a given value of $\epsilon$. Equivalently, we can consider the PDF of $x =\Delta Z = Z_1-Z_2 = \ln (\lambda_1/\lambda_2)$, for a given value of $X =\ln\,\epsilon$, this will be denoted $P_{\Delta Z\vert X}$. In the non-degenerate case it is clear from the discussion in section \[sec: 3\] that the ratio $\lambda_2/\lambda_1$ approaches zero as $\epsilon\to 0$. In the degenerate case, however, the values of $Z_1$ and $Z_2$ are comparable and the distribution of $\Delta Z$ is non-trivial. Accordingly, we concentrate on $P_{\Delta Z\vert X}$ for the degenerate case. To simplify the discussion we only give explicit formulae for the case where $\vert \Delta Z/X \vert \ll 1$. The joint distribution of $\mbox{\boldmath$Z$}=(Z_1,Z_2)$ takes the form given by equation (\[eq: 3.11\]), namely $P_{\bm Z}(\mbox{\boldmath$Z$}) \sim (\mbox{\boldmath$Z$}\cdot{\bf D}^{-1}\mbox{\boldmath$Z$})^{-1/4} \exp[-\Phi(\mbox{\boldmath$Z$})]$. We are interested in the distribution $$\label{eq: 6.2} P_{\Delta Z\vert X}=\frac{P_{\bm Z}(X,X-\Delta Z)}{\int_0^\infty {\rm d}\Delta Z\ P_{\bm Z}(X,X-\Delta Z)} \ .$$ Provided $\vert \Delta Z/X \vert \ll 1$ the dependence of the pre-exponential factor in (\[eq: 3.11\]) on $\Delta Z$ can be neglected and we have $$\label{eq: 6.3} P_{\Delta Z\vert X}={\cal K}\exp[\Phi(X,X)-\Phi(X,X-\Delta Z)]$$ where ${\cal K}$ is a normalisation factor. Equation (\[eq: 6.3\]) is based upon the assumption that the $Z_i$ undergo diffusion with a constant drift velocity. This assumption ceases to be valid close to the reflecting boundary, $x=0$ where $\Delta Z$ is very small and the drift velocity is given by (\[eq: 5.18\]). We are interested in finding a solution which matches (\[eq: 4.2.15\]) when $x \gg 1$ but which obeys the correct Fokker-Planck equation, namely $$\label{eq: 6.4} \frac{1}{2(1+\beta^2)}\partial_t P=\partial_x^2 P -\partial_x\left[\frac{1}{{\rm tanh}(x)}P\right]$$ Making the same transformation to Hermitean form as equations (\[eq: 4.2.2\])-(\[eq: 4.2.4\]), and setting $\psi(x,t)=\phi(x)\exp[-v_0^2t/4D_x]$ in order to find a solution which matches (\[eq: 4.2.15\]), we find that $\phi(x)$ satisfies: $$\label{eq: 6.8} \phi ^{\prime \prime} =-\frac{1}{4\,{\rm sinh}^2(x)}\phi$$ We require a solution $\phi(x)$ which approaches a constant as $x\to \infty$ and which approaches zero as $x\to 0$. Close to $x=0$ the differential equation is approximated by $\phi ^{\prime \prime} = -\phi/4x^2 $. This has general solution $\phi(x)=\sqrt{x} (a+b \ln(x) )$, where $a$ and $b$ are arbitrary constants. Since $\phi(x) \to 0$ as $x \to 0$ we conclude that $ \phi(x) \sim \sqrt{x}$ as $x \to 0$. From equation (\[eq: 4.2.2\]) with $v_x=D_x \coth(x)$ we have $\chi(x) = \ln(\sqrt{\sinh(x)})$ and, therefore, $\exp(\chi(x) ) \to \sqrt{x}$ as $x \to 0$. Hence, from equations (\[eq: 4.2.2\]) and (\[eq: 6.2\]) we conclude that $P_{\Delta Z\vert X}$ is of the form $$\label{eq: 6.9} P_{\Delta Z\vert X}=F(\Delta Z)\exp[\Phi(X,X)-\Phi(X,X-\Delta Z)]$$ where $F(\Delta Z)\sim \Delta Z$ for $\Delta Z\ll 1$, but where $F(\Delta Z)$ approaches a constant as $\Delta Z\to \infty$. Numerical investigations of the advective flow model {#sec: 7} ==================================================== Our analysis of the matrix contraction process has led us to consider a diffusive model for the evolution of the singular values. The predictions of this model are a consequence of the fact that the diffusion process has an unusual combination of reflecting and absorbing boundary conditions. Because the problem is too complex for a rigorous analysis to be practicable, we have tested the predictions by means of numerical simulations. \[fig: 2\] The results of direct simulations of the matrix contraction process, using the random advection model discussed in section \[sec: 5\] to generate the ensemble of random matrices, are given in plots (a) and (b) of figure 2. In plots (c) and (d), which deal with the degenerate case, we show the results obtained using the exact equations of motion for the singular values, as given in section (\[sec: 5.3\]) (having first verified that these equations produce identical results to the direct simulation approach). These results show that the predictions for $\gamma$ are correct for both the non-degenerate case ($1 > \beta > 1/\sqrt{5}$) and the degenerate case ($0 < \beta < 1/\sqrt{5}$). Figure \[fig: 7.3\] shows the result of a simulation of the conditional PDF $P_{{\Delta Z} \vert {\rm X}}$, where $\Delta Z=Z_1-Z_2 $ and $ {\rm X}= {\rm ln} (\epsilon)$, for the degenerate case where $\beta=0.3$, with $\epsilon_0=0.01$ and ${\rm X} = -10$. The plot also shows segments of two fitted curves: the straight line $ P_{{\Delta Z} \vert {\rm X}}=K_1 {\Delta Z}$ where $K_1$ is a constant (shown in black) and the exponential tail $P_{{\Delta Z} \vert {\rm X}}=K_2 \exp{[\Phi(X,X)-\Phi(X,X-{\Delta Z})]}$ where $K_2$ is a constant (shown in blue). These demonstrate that the form of $P_{{\Delta Z}|{\rm X}}$ is as given in equation (\[eq: 6.9\]). {width="65.00000%"} We remark that it is not actually necessary to compute the velocity field to simulate the matrices $\delta {\bf a}$. Defining the vectors ${\bm C} = (C_1,C_2,C_3)=(\psi_{xx},\psi_{yy},\psi_{xy})$ and ${\bm D}=(D_1,D_2,D_3)=(\phi_{xx},\phi_{yy},\phi_{xy})$ then the covariance matrix of ${\bm C}$ and of ${\bm D}$ is $$\label{eq: 7.2.1} {\bm K}=\langle C_iC_j\rangle= \left(\begin{array}{ccc} 3 & 1 & 0 \cr 1 & 3 & 0 \cr 0 & 0 & 1 \cr \end{array}\right) .$$ Therefore if ${\bm \xi}= (\xi_1,\xi_2,\xi_3)$ and ${\bm \eta}=(\eta_1,\eta_2,\eta_3)$ are each vectors whose elements are uncorrelated Gaussian random variables with zero mean and unit variance, we may write ${\bm C} ={\bm K}^{1/2} {\bm \xi}$, and ${\bm D} ={\bm K}^{1/2} {\bm \eta} $, where $$\label{eq: 7.2.2} {\bf K}^{1/2}= \frac{1}{\sqrt{2}} \left(\begin{array}{ccc} \sqrt{2}+1 & \sqrt{2}-1 & 0 \cr \sqrt{2}-1 & \sqrt{2}+1 & 0 \cr 0 & 0 & \sqrt{2} \end{array}\right)$$ We used this approach to simulate the stability matrices of the random flow model. Conclusions {#sec: 8} =========== We have investigated a process which occurs naturally in models for chaotic dynamical systems, considering the distribution of small values of the norm of the product of random matrices, and resetting the process to the identity matrix whenever the norm ceases to be small. We considered a model involving a product of matrices which are close to the identity, and which have diffusive fluctuations. For the scalar version of this problem, the distribution of $\epsilon$ is always a power-law. For the matrix contraction process, we find that the distribution is of the form (\[eq: 1.5\]) when the matrix has diffusive fluctuations, with two possible values of $\mu$ ($0$ or $-3/2$). It would be of interest to know about the distribution of $\epsilon$ for more general classes of matrix. [*Acknowledgements*]{}. We are grateful for the hospitality of the Department of Physics at the University of Auckland, where much of this paper was written. 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--- address: | $^{1}$ Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan; [email protected]\ $^{2}$ Department of Physics and Sezione INFN Bari, 70126 Bari, Italy; [email protected]\ $^{3}$ Istituto Applicazioni Calcolo, CNR, 70126 Bari, Italy; [email protected]\ $^{4}$ Departmento of Physics and Astronomy and Sezione INFN, Università di Padova, 35131 Padova, Italy; [email protected]\ $^{5}$ Center for Life Nano Science, IIT, 00161 Roma, Italy; [email protected] --- Introduction ============ Liquid crystals (LC) are fluids often made up of rod-like molecules that can arrange in a variety of equilibrium phases, depending upon their geometry and reciprocal interaction [@deGennes; @Chandra]. Of particular relevance is the cholesteric phase where the local molecular alignment is captured by a unit magnitude director field ${\bf n}$ displaying a natural twist deformation in the direction perpendicular to the molecules. Such liquid crystals can be found in a wide range of biological systems, such as bacterial flagella [@berg] and DNA molecules in solution [@watson], and have found increasing application in modern display devices, in which the cholesteric phase results from mixing a nematic liquid crystal (such as E7) with a chiral dopant. In recent years much interest has been focused on the realization of liquid crystal - particles composites, a new exciting class of soft material with tunable elastic and electro-optic properties, in which either colloidal particles or droplets of standard fluids are dispersed in a LC phase [@musevic]. The presence of these inclusions induces a disturbance in the director orientation which typically aligns either tangentially or normally to the surface of the particles. This leads to the formation of a variety of topological defects that mediate long-range anisotropic particle-particle interactions and stabilize novel ordered structures, such as chain [@loudet; @poulin] and defect glass [@wood] in nematics or colloidal crystals [@ravnik] and isolated clusters [@lintuvuori] in cholesterics, with potential use in photonics [@smalyukh] and in new devices [@chari]. Recent theoretical and experimental works have investigated the typical defect structures observed at equilibrium when particles are dispersed in cholesteric liquid crystals (CLC), and have shown that these can be controlled by tuning the ratio between the particle size $R$ (typically its radius) and the cholesteric pitch $p$. For instance, if strong perpendicular (homeotropic) anchoring is set on the surface of a spherical colloid, one observes either a planar Saturn ring (a circular defect line of half-integer charge located around the equator of the particle) if $p >> R$, or a twisted Saturn rings, wrapping around the particle, if $p<R$ [@lintuvuori2; @lintuvuori3; @review; @musevic2]. If, on the other hand, the anchoring is tangential to the surface, the defect pattern ranges from the “boojum” (two surface defects of integer charge located on opposite sides) if $R/p$ is small, to its twisted version if $R/p$ is large enough [@review]. A further complication arises when, in place of a solid particle, one considers a deformable liquid droplet. Here the strength of the interface anchoring relative to the elasticity of the LC becomes a crucial parameter to control the droplet shape (in addition to the director pattern near the surface). For instance, one observes either a spherical shape or a nutshell-like structure if the anchoring strength is respectively weak or strong [@Soft_Matter]. So far most of the studies have focused on the equilibrium properties of these inclusions [@review] and much less is known when these systems are subject to external perturbations, such as electric and flow fields. Liquid crystal have long been known as systems that exhibit a rich dynamical response such as shear banding [@olmsted; @olmsted2; @orlandini] and molecular tumbling [@deGennes; @tsuji; @tsuji2] when subject to shear flow [@deGennes]. This is due to the complex coupling between hydrodynamics and director field (known as backflow), a feature that is even more relevant in a CLC where the inherent three-dimensional twisted arrangement of the director field can give rise to striking effects, such as a significant increase in viscosity when subject to a shear flow in the direction of its helix (*permeation*) [@helfrich; @hongla; @marenduzzo]. Recent simulations on colloidal dispersions in CLC report, for instance, a violation of the Stokes law when a single particle is dragged parallel to the cholesteric helix  [@lintuvuori2], or an induced rotation, either continuous or stepwise, when two particles, forming a dimer, are pulled through the cholesteric phase with a constant force along the helical axis [@juho_dimer]. Here we present a preliminary study on the effect that an imposed shear flow has on the rheology of an inverted cholesteric emulsion, described as a single isotropic (liquid crystal) droplet (in which molecules are randomly oriented) surrounded by a cholesteric liquid crystal. The study has been carried out by using a lattice Boltzmann approach [@succi], a method which solves numerically the Beris-Edwards equations of cholesteric hydrodynamics [@Beris-Edwards] coexisting with an isotropic phase and already tested for an inverted nematic emulsion in the presence of an electric field [@sulaiman] or a shear flow [@Soft_Matter]. The main strength of the algorithm is that backflow effects are automatically included and the dynamics of topological defects can be easily tracked. By varying the shear rate and the ratio between the elastic energy scales of the cholesteric in the bulk and at the droplet interface, we show that the presence of a liquid droplet strongly reduces the secondary flow usually observed in a pure cholesteric phase. Such effect has been found more pronounced when strong perpendicular anchoring is set on the droplet interface, and is also confirmed when the chirality degree of the cholesteric is increased, even though in this case the system may temporarily enter the blue phase for high shear rates [@mermin]. The paper is organized as follows. In Section 2 we describe the numerical model of a 2D droplet of isotropic fluid suspended in a CLC, in particular its equilibrium phase behavior, encoded by a Landau-de Gennes free energy, and its hydrodynamics, captured by the Beris-Edwards equations of motion for CLCs coupled to Cahn-Hilliard dynamics. In Section 3 we first discuss the equilibrium properties of a cholesteric sample and then those of a single droplet of Newtonian fluid dispersed into it, both in the absence of anchoring and with homeotropic anchoring. We next present the main results of the paper, namely the effect that a symmetric shear flow has on the droplet-CLC system when different values of interface anchoring and chirality are considered. Finally the last section is dedicated to discuss the results and to conclusions. Model and methods ================= We consider an isolated droplet of Newtonian isotropic fluid dispersed in a medium of cholesteric liquid crystal. This setup is usually referred as [*inverted*]{} cholesteric emulsion [@poulin; @weitz_PRE; @Soft_Matter] to be distinguished from a [*direct*]{} emulsion, where a liquid crystal droplet is immersed in an isotropic fluid. The hydrodynamics of such system can be described by using an extended version of the Beris-Edwards theory [@Beris-Edwards] for chiral fluids, already adopted in previous works on liquid crystals [@cates_soft; @tiribocchi_soft; @tiribocchi_prl; @tiribocchi_soft2]. Here we briefly recap the model. The equilibrium properties are captured by a Landau-de Gennes [@deGennes] free-energy functional ${\cal F}=\int_V{fdV}+\int_S f_WdS$, where the free-energy density $f$ is given by the sum of the following terms $$\label{free_energy} f= f_{bf} (\phi) + f_{lc} (\phi, Q) + f_{int} (\phi, Q),$$ while the second contribution $f_W(Q)$ is added in the presence of a bounding surface. Here $\phi({\bf r},t)$ is a scalar order parameter related to the concentration of the cholesteric phase relative to the isotropic phase, while ${\bf Q}({\bf r},t)$ is a tensor order parameter that, within the Beris-Edwards theory [@Beris-Edwards], describes the cholesteric phase. It is defined as $Q_{\alpha\beta}=q(\hat{n}_{\alpha}\hat{n}_{\beta}-1/3\delta_{\alpha\beta})$, where $\hat{n}({\bf r},t)$ is the director field[^1] describing the local orientation of the molecules (in the uniaxial approximation $\textbf{n}=-\textbf{n}$), and $q$ is the local degree of order, proportional to the largest eigenvalue of ${\bf Q}$ ($0\le q\le 2/3$). The first term of Eq. \[free\_energy\] describes the bulk properties of the mixture and is given by $$\label{free_binary} f_{bf}(\phi)=\frac{a}{4}\phi^2(\phi-\phi_0)^2+\frac{K_{bf}}{2}|\nabla\phi|^2,$$ where $a$ and $K_{bf}$ are two positive phenomenological constants controlling the interface width $\Delta$ of the droplet, which, for a binary fluid without liquid crystal, goes as $\Delta\sim\sqrt{K_{bf}/a}$. Eq. \[free\_binary\] is borrowed from binary fluid mixtures and enables the formation of two phases: The isotropic one (inside the droplet where $\phi\simeq 0$) and the cholesteric one (outside the droplet where $\phi\simeq\phi_0$), separated by an interface whose energetic cost is gauged by the gradient term. The second term is the cholesteric liquid crystal free-energy density given by $$\begin{aligned} \label{free_bulk_chol} f_{lc}(\phi, Q)&=&A_0\left[\frac{1}{2}\left(1-\frac{\zeta (\phi)}{3}\right)Q_{\alpha\beta}^2-\frac{\zeta (\phi)}{3}Q_{\alpha\beta}Q_{\beta\gamma}Q_{\gamma\alpha}+\frac{\zeta (\phi)}{4}(Q_{\alpha\beta}^2)^2\right]\nonumber\\ &&+\frac{K}{2}(\partial_{\beta}Q_{\alpha\beta})^2+\frac{K}{2}(\epsilon_{\alpha\delta\gamma}\partial_{\delta}Q_{\gamma\beta}+2q_0Q_{\alpha\beta})^2.\end{aligned}$$ The terms multiplied by the positive constant $A_0$ stem from a truncated polynomial expansion up to the fourth order in ${\bf Q}$ [@deGennes], and describe the bulk properties of an uniaxial nematic liquid crystal with an isotropic-to-nematic transition at $\zeta(\phi)=\zeta_c=2.7$. Here $\zeta$ plays the role of an effective temperature. For a nematogen without chirality ($q_0=0$), the phase is isotropic if $\zeta(\phi)<\zeta_c=2.7$, otherwise it is cholesteric. By following previous studies [@sulaiman; @Soft_Matter; @Fadda] we set $\zeta(\phi)=\zeta_0+\zeta_s\phi$, where $\zeta_{0}$ and $\zeta_{s}$ control the boundary of the coexistence region. The remaining terms of Eq. \[free\_bulk\_chol\], multiplied by the constant $K$, take into account the elastic energy due to the local deformations of the cholesteric arrangement and enter the free energy through first order gradient contributions, except a gradient-free term included to have a positive elastic free energy. The parameter $q_0=2\pi/p_0$ determines the pitch length $p_0$ of the cholesteric and $\epsilon_{\alpha\delta\gamma}$ is the Levi-Civita antisymmetric tensor. Here we consider the “one elastic constant” approximation, an approach usually adopted when investigating liquid crystals as it considerably simplifies theoretical calculations [@deGennes]. The energetic cost due to the anchoring of the liquid crystal at the droplet interface is included through $$f_{int}(\phi,Q)=L(\partial_{\alpha}\phi)Q_{\alpha\beta}(\partial_{\beta}\phi),$$ where the constant $L$ controls the strength of the anchoring. If it is negative the liquid crystals is homeotropic (perpendicular) to the interface, whereas if positive the liquid crystal lies tangentially. Finally, if confining walls are included, a further term needs to be considered in the free energy functional. For homeotropic anchoring, which is the case considered here, one has $$f_W(Q)=\frac{1}{2}W(Q_{\alpha\beta}-Q^0_{\alpha\beta})^2,$$ where $W$ controls the strength of the anchoring at the walls and $Q_{\alpha\beta}^{0}$ sets the preferred configuration of the tensor order parameter at the surface. The thermodynamic state of our mixture is specified by two dimensionless quantities, the reduced temperature and the reduced chirality, given by $$\begin{aligned} \tau&=&\frac{27(1-\zeta(\phi_0)/3)}{\zeta(\phi_0)},\label{red_temp}\\ \kappa&=&\sqrt{\frac{108Kq_0^2}{A_0\zeta(\phi_0)}}.\label{chiral}\end{aligned}$$ The former multiplies the quadratic terms of the dimensionless bulk free energy and vanishes at the spinodal point of a nematic, while the latter multiplies the gradient terms and gauges the amount of twist accumulated in the system [@mermin]. Such parameters have been used, for instance, in Refs. [@dupuis; @henrich2] for the numerical calculation of the phase diagram of cholesteric liquid crystals. Note, in particular, that according to Eq. \[chiral\] the knowledge of $q_0$ is not a sufficient information (except if $q_0=0$) to correctly assess the value of the chirality (hence the phase of the liquid crystal), as this is also affected by other thermodynamic parameters ($A$, $K$ and $\zeta$). The dynamics of the system is governed by a set of balance equations, the first of which is the equation of the tensor order parameter ${\bf Q}$ $$\label{Q_eq} (\partial_t+{\bf u}\cdot\nabla){\bf Q}-{\bf S}({\bf W},{\bf Q})=\Gamma {\bf H},$$ where ${\bf u}$ is the velocity of the fluid and the term on the left hand side is the material derivative, describing the rate of change of ${\bf Q}$ advected by the flow. The derivative includes the tensor ${\bf S}({\bf W},{\bf Q})$ since the order parameter can be rotated and stretched by local velocity gradients $W_{\alpha\beta}=\partial_{\beta}u_{\alpha}$, and is given by $$\label{eq_S} {\bf S}({\bf W},{\bf Q})=(\xi{\bf D}+{\bf \Omega})({\bf Q}+{\bf I}/3)+({\bf Q}+{\bf I}/3)(\xi{\bf D}-{\bf \Omega})-2\xi({\bf Q}+{\bf I}/3)Tr({\bf Q}{\bf W}).$$ Here $\textbf{D}=(\textbf{W}+\textbf{W}^{T})/2$ and ${\bf \Omega}=(\textbf{W}-\textbf{W}^{T})/2$ are the symmetric and antisymmetric parts of the velocity gradient tensor, $Tr$ denotes the tensorial trace and $\textbf{I}$ is the unit matrix. The constant $\xi$ depends upon the molecular details of the liquid crystal, and controls the dynamics of the director field under shear flow. Indeed after imposing a homogeneous shear on a nematic liquid crystal, at steady state the director will align along the flow gradient at an angle $\theta$ fulfilling the relation $\xi\cos(2\theta)=(3q)/(2+q)$. Real solutions are obtained when $\xi\geq 0.6$. Throughout our simulations we have set $\xi=0.6$. Finally, on the right hand side of Eq. \[Q\_eq\], $\Gamma$ is the collective rotational diffusion constant and ${\bf H}$ is the molecular field given by $${\bf H}=-\frac{\delta {\cal F}}{\delta {\bf Q}}+\frac{{\bf I}}{3}Tr\frac{\delta {\cal F}}{\delta {\bf Q}}.\label{H_eq}$$ The time evolution of the concentration field $\phi({\bf r},t)$ is governed by a convection-diffusion equation $$\partial_{t}\phi+ \partial_{\alpha}(\phi u_{\alpha})=\nabla \left (M \nabla\frac{\delta {\cal F}}{\delta \phi}\right),\label{phi_eq}$$ where $M$ is the mobility and $\mu=\delta {\cal F}/\delta\phi$ is the chemical potential, while the force balance is ensured by the incompressible Navier-Stokes equation, $$\begin{aligned} \nabla\cdot{\bf u}&=&0,\label{cont}\\ \rho(\partial_t+u_{\beta}\partial_{\beta})u_{\alpha}&=&\partial_{\beta}\sigma_{\alpha\beta}^{total},\label{nav_stok}\end{aligned}$$ where the total stress tensor $\sigma^{total}_{\alpha\beta}$ is the sum of four contributions $$\sigma^{total}_{\alpha\beta}=-\rho T+\sigma^{visc}_{\alpha\beta}+\sigma^{lc}_{\alpha\beta}+\sigma^{s}_{\alpha\beta}.$$ The first term $\rho T$ is the ideal background pressure and $T$ is the temperature. The second one is the viscous stress tensor $$\sigma^{visc}_{\alpha\beta}=\eta(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}),$$ where $\eta$ is the isotropic shear viscosity. The third one is the elastic stress due to the liquid crystalline order $$\begin{aligned} \label{lc_stress} \sigma^{lc}_{\alpha\beta}&=&\frac{K}{2}(\nabla\textbf{Q})^2\delta_{\alpha\beta}-\xi H_{\alpha\gamma}\left ( Q_{\gamma\beta}+\frac{1}{3}\delta_{\gamma\beta} \right )-\xi\left ( Q_{\alpha\gamma}+\frac{1}{3}\delta_{\alpha\gamma} \right )H_{\gamma\beta}\nonumber\\ &&+2\xi\left ( Q_{\alpha\beta}-\frac{1}{3}\delta_{\alpha\beta} \right )Q_{\gamma\mu}H_{\gamma\mu}+Q_{\alpha\nu}H_{\nu\beta}-H_{\alpha\nu}Q_{\nu\beta},\end{aligned}$$ and the last term is the interfacial stress between the isotropic and liquid crystal phase $$\label{int_stress} \sigma_{\alpha\beta}^{s}=-\left(\frac{\delta{\cal F}}{\delta\phi}\phi-{\cal F}\right)\delta_{\alpha\beta}-\frac{\delta{\cal F}}{\delta(\partial_{\beta}\phi)}\partial_{\alpha}\phi-\frac{\delta{\cal F}}{\delta(\partial_{\beta}Q_{\gamma\mu})}\partial_{\alpha}Q_{\gamma\mu}.$$ The total isotropic pressure of the system is the sum of the ideal background pressure $\rho T$ plus the isotropic term of Eq. \[lc\_stress\] (constant in our simulations, except nearby the defects and close to the droplet interface where it augments) and of Eq. \[int\_stress\]. Both $\sigma^{lc}_{\alpha\beta}$ and $\sigma_{\alpha\beta}^{s}$ take into account the non-Newtonian fluid effects. In fact, if ${\bf Q}=0$, they reduce, respectively, to the scalar pressure and to the interfacial stress of a binary Newtonian fluid mixture. Equations (\[Q\_eq\]) (\[phi\_eq\]) (\[cont\]) (\[nav\_stok\]) are solved numerically by means of a hybrid lattice Boltzmann method [@henrich; @Fadda], which uses a combination of a standard lattice Boltzmann approach to solve the Navier-Stokes and the continuity equation, and a finite-difference scheme to solve Eqs. (\[Q\_eq\]) and (\[phi\_eq\]). All simulations are performed on a quasi two-dimensional rectangular box of size $L_{x}=1 \times L_{y}=300 \times L_{z}=100$ in which the cholesteric liquid crystal and the droplet are embedded. We choose a longer size along the $y$-direction in order to minimize the interference between periodic images of the droplet moving with the flow when a shear flow is applied. The droplet, in particular, is modeled as a circular isotropic (liquid-crystalline) region with radius $R=32$ lattice sites, initially placed at the centre of the sample and surrounded by the liquid crystal phase. This quasi-2d setup is chosen to allow an out-of-plane component of the vector fields along the $x$-direction that accommodates the three-dimensional twisted arrangement of the cholesteric, and, in the presence of a shear flow, captures the secondary flow appearing perpendicular to the direction of the shearing, along the $x$-direction. The entire system is periodic along the $y$-direction and is sandwiched between two flat walls parallel to the $xy$-plane and lying at $z=0$ and $z=L_z$. The walls can be either at rest or moving along the $y$-direction with velocity $-u_w$ and $u_w$ (with $u_w>0)$ at $z=0$ and $z=L_z$, respectively. There we impose neutral wetting conditions (i.e. ${\bf a}\cdot\nabla\phi=0$, where ${\bf a}$ is a unit vector perpendicular to the wall) for the concentration, no-slip conditions (the fluid moves with the same velocity of the walls) for the fluid velocity and homeotropic conditions (i.e. the director is perpendicular to the walls) for the liquid crystal order parameter. Initial conditions of $\phi$ and ${\bf Q}$ in the bulk are as follows. They are both set to zero inside the droplet, while outside $\phi$ is fixed to a constant value ($\phi_{0}\simeq 2$) and the components of $\textbf{Q}$ are given by $$\begin{aligned} Q_{xx}&=& (c_0-c_1/2)\cos(2q_0y) + c_1/2,\\ Q_{yy}&=& -2c_1,\\ Q_{xz}&=& -(c_0-c_1/2)\sin(2q_0y),\\ Q_{xy}&=&Q_{yz}=0,\end{aligned}$$ where $c_0=0.546$, $c_1=0.272$, to accommodate a cholesteric liquid crystal with helical axis parallel to the $y$-direction. Such components stem from assuming that the ground state of the director field is given by $\hat{\bf n}(y)=\cos(q_0y){\bf e}_x+\sin(q_0y){\bf e}_z$, where ${\bf e}_{\alpha}$ is a unit vector oriented along the Cartesian axis. The parameter $q_0 = 2\pi/p_0$ controls the number $N$ of $\pi$ twists that the liquid crystal displays in a cell of length $L_y$. Indeed one can define $p_0$ in terms of the linear size $L_y$, as $p_0=2L_y/N$, to compare the pitch length with the cell size. Starting from these initial conditions, the system is first let to relax to its equilibrium state and afterwards both walls are sheared along opposite directions at constant speed. Typical parameters used in our simulations are $a=7 \times 10^{-2}$, $K_{bf}=0.14$, $M=5 \times 10^{-2}$, $\Gamma=1$, $A_{0}=0.1$, $K=0.1$, $\eta=0.38$, $L=-0.04$ and $W=0.4$. Finally one has to choose the values of $\tau$ and $\kappa$ in order to have the cholesteric phase outside the droplet [@dupuis; @henrich2]. We set $\tau=0$ (i.e. $\zeta(\phi_0)=3$) and $q_0$ (which controls the chirality $\kappa$, see Eq. \[chiral\]) equal to $\pi/150$ or $\pi/75$, in order to have $N=2$ or $N=4$ on a lattice of linear size $L_y=300$. If, for instance, $q_0=\pi/75$, one has $\kappa~\simeq 0.25$, well inside the cholesteric phase according to the phase diagram of Ref. [@dupuis; @henrich2]. In order to map such numerical values to typical physical ones, we consider a droplet of size $1-10\mu$$m$ immersed in a cholesteric liquid crystal of elastic constant roughly $10$pN and viscosity of $1$ Poise. This corresponds to a lattice space $\Delta x=10^{-7}$m and the time step is $\Delta t=10^{-8}$s (they are both equal to one in our simulations). Finally the anchoring coefficient $L$ correspond to values $10^{-6}$J$m^{-2}$ [@sulaiman; @poon]. Results ======= We investigate the physics of two different systems, one in which the sole cholesteric liquid crystal is sandwiched between flat walls and the other one in which an isotropic droplet is dispersed inside it. We initially study their equilibrium properties, in particular when homeotropic anchoring of the director is set both at the walls and at the droplet interface, and for two different values of the cholesteric pitch. Afterwards the dynamic response under different shear rates (only in the flow aligning regime) and interface anchoring is studied. Equilibrium properties ---------------------- We first consider a CLC initialized in a quasi two-dimensional rectangular box of size $L_{y}=300 \times L_{z}=100$ with walls at rest. In Fig. \[fig1\](a) and (d) we show the equilibrium configurations of the director field (obtained after $t~\simeq 10^5$ timesteps, when the free-energy is at its minimum value) of the cholesteric phase for $N=2$ (a) and $N=4$ (d) and with strong homeotropic anchoring on both walls. In both cases the cholesteric axis lies along the $y$-direction. One can clearly distinguish regions where the director displays a local nematic-like order (with orientation almost entirely along the $z$-direction, forced by the anchoring at the walls) from regions where it flips in the third direction (i.e. along the $x$-axis). Due to the interplay between the orientation of the cholesteric phase near the walls and the imposed perpendicular anchoring, topological defects emerge. Although in cholesterics four different types of defects may occur [@Fadda; @kleman; @kleman2], here we observe $\lambda^{\pm m}$ and $\tau^{\pm m}$ disclinations defects ($m$ is the topological charge, an integer or semi-integer number). While in $\lambda$ defects the director field is continuous (it is the local cholesteric pitch axis, winding around the defect, that is discontinuous) and flips into the third dimension at the $\lambda$ line, $\tau$ defects are true disclinations as the director drops down at the defect core. Note that in Fig. \[fig1\](a) and (d) those pinned at the walls are $\tau$ disclinations (of charge $-1/2$) which sustain a $\lambda^{+1}$-charge region, where the director field flips into the third dimension. Changing the chirality also affects their position; indeed, when $N=4$, $\tau$ disclinations appear closer to the boundaries and $\lambda$-charge regions are longer and narrower than those observed when $N=2$. It important to stress that, while it is easy to track numerically the location of $\tau$ disclination defects (for instance by computing the local orientational order at each point), it is more difficult to resolve the correct position of $\lambda$ defects, as the radius of their core is usually comparable with the helix pitch $p$, and only a limited number of lattice points can be used to calculate its extension. This is why we prefer indicating regions containing $\lambda$ defects (either of charge $\pm 1$ or couple of charge $\pm 1/2$ ) also as $\lambda$-charge regions. The presence of an isotropic droplet in the centre of the cell substantially modifies the equilibrium configuration of the cholesteric. In the absence (or for very weak values) of interface anchoring and with $N=2$ (Fig. \[fig1\](b)), the droplet removes the $\lambda$-charge region and the two $\tau$ disclinations in the centre of the box leaving the rest of the cholesteric almost unaltered. If, on the other hand $N=4$, the two $\tau$ disclinations in the centre of the box survive (as they are closer to the walls), and the $\lambda$-charge region is simply covered by the droplet (Fig. \[fig1\](e)). If strong homeotropic anchoring is imposed at the droplet interface (Fig. \[fig1\](c)-(f)), two fully in-plane topological defects of charge $-1/2$ emerge on opposite sides of the droplet and located along the equatorial line. Such strong anchoring importantly alters the liquid crystal orientation in the surroundings of the droplet, favoring the formation of a pronounced splay deformation. A suitable quantity to measure the strength of the interface anchoring relative to the bulk elastic deformation is the dimensionless number $\Lambda={\cal F}_{int}R/\Sigma K$, where ${\cal F}_{int}=\int_V f_{int}dV$ and $\Sigma$ is the perimeter of the droplet. For a droplet of size $1-10\mu$m, $\Lambda$ usually ranges from $10^{-2}$ to $10^3$. If $\Lambda\ll 1$ the anchoring is weak, and no large deformations are observed in the surroundings of the droplet (as in Fig. \[fig1\](b)-(e), where $\Lambda\simeq 0$), whereas if $\Lambda\sim 1$ the anchoring is considered strong and large distortions of director with topological defects are found (see Fig. \[fig1\](c)-(f), where $\Lambda\simeq 2.88$). {width="100.00000%"} Cholesteric liquid crystal under shear -------------------------------------- By starting from the equilibrated configurations previously described, we now study the rheology of such systems when a symmetric shear flow is imposed. We move top and bottom walls along the $y$-axis with velocity $u_w$ (top) and $-u_w$ (bottom). This sets a shear rate $\dot{\gamma} = 2 u_w/L_z$ (measured in $\Delta t^{-1}$ in simulation units) and a shear flow along the $y$-direction. We first consider the homogeneous case in which no droplets are present. In Fig. \[fig2\](a)-(c) we show the steady state configurations of the director field for $N=2$ and $N=4$ and for a relatively weak shear rate ($\dot{\gamma}=6\times 10^{-5}$, i.e. $u_w=0.003$). With respect to the equilibrium state, the cholesteric layers are tilted along the direction imposed by the shear flow, while $\tau$ disclinations, still pinned at the walls, sustain $S$-shaped $\lambda$-charge regions. The primary flow $u_y$ (measured at $y=L_y/2$) is approximately Newtonian for $N=2$ while it gets weakly sigmoidal $N=4$ (Fig. \[fig2\]b), an effect due to the larger resistance to overcome elastic deformations encountered by the flow when higher values of $q_0$ are considered. Note that, as reported in Ref. [@marenduzzo], a more flattened velocity profile (in which $u_y$ is zero almost everywhere along the $z$-direction) could be achieved if larger values of $L_z$ and shorter values of $L_y$ are considered. As expected, unlike the case of a binary fluid mixture without liquid crystal (i.e. when ${\bf Q}=0$), a secondary flow $u_x$ emerges due to the action of the director field on the velocity field along the $x$-direction (Fig. \[fig2\](d)), stronger in the center of the sample (where it attains its larger value) but smaller than the primary flow, and zero near the walls. Its negative sign is due to the handedness of the initial configuration of the liquid crystal. Note that the presence of a secondary flow is a signature of the permeation, i.e. a significant increase in the viscosity (calculated as $\eta_{yz}=\sigma_{yz}^{total}/\partial_zu_y$ [@marenduzzo]) of the cholesteric liquid crystal observed when it is subject to a flow in the direction of its helix (i.e. perpendicular to the director field) [@deGennes; @helfrich; @lubensky; @marenduzzo]. Our simulations show, for instance, that, if $N=4$, $\eta_{yz}$ attains a maximum at $\sim 0.65$ for $\dot{\gamma}=6\times 10^{-5}$ (in the centre of the sample) and at $\sim 1.4$ for $\dot{\gamma}=8\times 10^{-5}$. Such values are higher than the isotropic viscosity $\eta=0.38$. {width="100.00000%"} A well-established result of the rheology of liquid crystals is that if the shear-rate is increased, the temperature at which the cholesteric-nematic transition occurs actually diminishes [@olmsted; @marenduzzo]. While we confirm this result, we also find that the transition depends upon the pitch length, namely by $N$. In order to better appreciate it, in Fig. \[fig3\] we show two dynamic configurations of the director field and of the corresponding velocity field for $N=4$ and $\dot{\gamma}=2\times 10^{-4}$ ($u_w=0.01$), in a full three dimensional box of size $L_x=5$, $L_y=300$, $L_z=100$. While if $N=2$ (and for the same shear rate) the cholesteric is already in the nematic state (not shown), if $N=4$ the system exhibits a more complex dynamics, in which double-twist cylinders (one of them is highlighted with a black dotted square in Fig. \[fig3\]a) periodically form and get destroyed by the fluid flow. Afterwards the cholesteric arrangement is temporarily restored (Fig. \[fig3\]c) and disrupted again. The presence of double twist cylinders indicate that, within a range of values of shear-rate, the cholesteric-nematic transition is replaced by the cholesteric-blue phase one for high values of $N$ (namely for a higher chirality). Interestingly, during the blue-phase-like regime topological defects at the walls acquire a positive charge (they are now actually $\tau^{1/2}$ disclinations, indicated by red cylinders in Fig. \[fig3\]a) while the topological charge of the $\lambda$ region (highlighted by a green circle in Fig. \[fig3\]a) switches to $-1$. The velocity field shows large vortices in the surroundings of such defects (Fig. \[fig3\]b) while far from them it has the typical structure of sheared systems (such as in Fig. \[fig3\]d), i.e. a bidirectional flow, strong near the walls and much weaker in the middle of the cell. {width="100.00000%"} Isotropic droplet in cholesteric phase under shear -------------------------------------------------- We now describe the rheological properties of a system in which an isotropic droplet is included in the cholesteric phase. We consider low/moderate values of shear rate, comparable to those used for the cholesteric cell without droplet. This also ensures that the system remains firmly in the cholesteric phase and finite size effects due to periodic boundary conditions and artificial interactions between droplet mirror images should also be minimized. Two dimensionless numbers useful to quantify the effects of the shear flow on the droplet are the Capillary number $Ca$ and the deformation parameter $D$. The former is defined as $Ca=\frac{R\dot{\gamma}\eta\Sigma}{{\cal F}_{K_{bf}}}$ (where $F_{K_{bf}}=\int_V dV (K_{bf}/2)|\nabla\phi|^2$) and measures the strength of viscous forces relative to the surface tension mediated by the constant $K_{bf}$, while the latter is defined as $D=\frac{a-b}{a+b}$ [@taylor] (where $a$ and $b$ are the major and the minor axis of an elliptical droplet) and quantifies the droplet deformation under shear. If low or moderate shear flows are considered, $Ca$ is expected to be $\sim 0.01$ while $D$ between $0$ (no deformation) and $0.1$ (weak/moderate deformation). {width="100.00000%"} We first consider the case with $N=2$. In Fig. \[fig4\](a)-(d) we show two configurations of the director field for $\Lambda\sim 0$ (a) and $\Lambda\sim 1$ (d) under shear flow with $\dot{\gamma}=6\times 10^{-5}$ ($u_w=0.003$). The starting equilibrium states are those shown in Fig \[fig1\](b) and (c), respectively. While if $\Lambda\sim 0$ the director field in the bulk of the cell is almost everywhere tilted along the direction imposed by the shear flow (except where it flips in the $x$-direction), with $\Lambda\sim 1$ this occurs only far from the droplet as the two defects near the droplet interface foster the formation of large splay distortions. Note that such defects, although still on opposite sides of the droplet as in Fig. \[fig1\](c), have been dragged by the fluid flow around the interface in a clockwise direction. While in this case the droplet has slightly moved from its initial position rightwards and then has got stuck in the steady state with $y_{cm}$ not changing, if $\Lambda\sim 0$ it has acquired a persistent unidirectional motion rightwards for the lower shear rate (see Fig. \[fig4\](b), where the $y$-position of the center of mass of the droplet is plotted). When increasing the shear rate, $y_{cm}$ reaches a steady value. This unidirectional motion occurs as the center of mass of the droplet shifts slightly upwards with respect to its initial position $z=L_z/2$, enough to move the droplet where the flow (in the upper part of the lattice) can drag it rightwards. The symmetric case, in which the droplet is dragged leftwards, would occur if its barycenter were shifted downwards. The deformation $D$ (see Fig. \[fig4\](e)) is only weakly affected by the value of $\Lambda$, while the droplet becomes more elliptical when increasing the shear rate. More important, instead, is the effect that the inclusion of a droplet has on the fluid flow profile. Unlike the case of the pure cholesteric cell, here the primary flow ($u_y$) flattens at zero at the center of the sample with an effective shear rate smaller than the imposed one, and the secondary flow ($u_x$) is almost one order of magnitude lower than that of the cholesteric cell. However, while it increases by augmenting $Ca$ if $\Lambda\sim 0$, it is strongly reduced if $\Lambda\sim 1$. The former occurs as, despite an increase in the shear rate, the liquid crystal is still firmly in the cholesteric phase where the secondary flow is expected to be relevant. The latter instead is due to the strong anchoring of the director at the droplet interface and at the walls, a constraint that forces the director to remain fully in the two-dimensional $yz$-plane. We now briefly turn to discuss the physics observed when $N=4$. While for low/moderate shear rate the droplet undergoes weak deformations, overall analogous to those shown for $N=2$, for higher values it is generally more difficult to achieve a stationary state as the large number of topological defects favors the formation of intense flows which significantly affect the liquid crystal orientation (through the backflow effect). Despite this limitation, it is interesting to track the secondary flow by varying the shear rate (even out of the cholesteric phase) and the interface anchoring strength in order to assess how the rheological response is affected by the chirality. In Fig. \[fig5\] we show the secondary flow profiles taken at three different positions (namely at $y=L_y/4, L_y/2$, and $3L_y/4$) at late times, to give a uniform view of the behavior even in regions where the droplet is absent. As long as $\Lambda$ and $Ca$ are relatively weak ($\Lambda\sim 0$, $Ca\sim 0.016$), $u_x$ attains values comparable with those observed in a sample of cholesteric and almost one order of magnitude higher than those obtained for $N=2$, suggesting that increasing the chirality essentially favors a non-Newtonian response of the fluid. In these cases the droplet achieves a stationary value for $y_{cm}$ (without any appreciable motion) with a deformation overall comparable with that seen for $N=2$. Interestingly though, increasing both shear rate and interface anchoring strength leads to a drastic reduction of the secondary flow as, on one hand, the cholesteric order is gradually destroyed in favor of a nematic-like one (where there is no secondary flow) and, on the other hand, the strong interface anchoring impedes the director field to acquire an out-of-plane component (as seen when $N=2$). {width="100.00000%"} Conclusions =========== To summarize, we have studied, by numerical simulations, the rheological response of an inverted cholesteric droplet sandwiched between two planar walls under a symmetric shear flow. We have shown that the dynamics is affected by the shear rate, the strength of the interface anchoring and the chirality of the cholesteric. In particular the presence of a relatively intense secondary flow (emerging out of the plane of the cholesteric) suggests that the system is essentially non-Newtonian, an effect generally weakened if an isotropic droplet is included and further reduced if strong interface anchoring is imposed. If the droplet is absent, a moderate shear flow drives the sample towards a steady state in which the director field is almost everywhere tilted along the direction imposed by the shear itself, except in regions where $S$-like $\lambda$-charge region are sustained by $\tau$ disclinations pinned at the walls. Importantly, increasing the chirality favors a non-Newtonian response of the cholesteric witnessed by a sustained secondary flow. On the other hand, augmenting the shear-rate determines a decrease of the transition temperature, which leads the system either into the nematic state or, if the chirality is higher, into a blue-phase-like state. The dynamics is significantly different if an isotropic droplet is embedded in the sample. If the interface anchoring is weak, we generally find a reduction of the secondary flow with respect to the droplet-free cholesteric sample, an effect even more pronounced if the interface anchoring is strong. We ascribe the latter effect to the resistance encountered by the flow to overcome a larger elastic deformation of the director field, in particular near the droplet interface where two fully in-plane topological defects form. These results are overall confirmed when the chirality is increased, although here an intense secondary flow, comparable to that observed in the droplet-free sample, is found for low shear-rate and weak anchoring. This highlights the fact that the rheological response displayed by an inverted cholesteric droplet has a complex landscape where a key role is played by at least three quantities: Shear rate, elasticity, and chirality. Our results represent a first step on the study of the rheology of an inverted cholesteric droplet, and are of possible interest for designing CLC based devices built from an emulsion. This opens up several directions for future research. It would be worth investigating the case in which tangential anchoring is considered (both at the droplet interface and at the walls) in a cholesteric sample whose axis is perpendicular to the walls. Besides yielding to the formation of different topological defects (such as a twisted Saturn ring in 3D), such configuration is expected to display a rich dynamical behavior like that observed with homeotropic interface anchoring which strongly depends upon the cholesteric pitch (or the chirality). This can pave the way to extend the study to include several droplets, either in a monodisperse setup or in the more intriguing polydisperse one where different droplet interface anchoring sets may be considered. One can envisage, for instance, the design of a new soft material made up of highly-packed isotropic droplets whose resistance to deformation could be sustained by the liquid crystal dispersed in between. Finally, although three-dimensional simulations can be computationally demanding, it would be surely of great interest to investigate the physics of more realistic systems in order, for instance, to minimize finite size effects. However we note that quasi two-dimensional liquid crystal devices could be experimentally realized, such as that described in Ref. [@cluzeau], where a smectic-C film surrounding droplets, obtained by nucleation, is proposed. [99]{} de Gennes, P. G. and Prost, J. In [*The Physics of Liquid Crystals*]{}; Clarendon Press Oxford; 1993. Chandrasekhar, S. In [*Liquid Crystals*]{}; Cambridge University Press; 1977. Berg, H. C. and Anderson, R. A., Bacteria Swim by Rotating their Flagellar Filaments. [*Nature*]{} [**1973**]{}, [*245*]{} 380. Watson, J. 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--- abstract: 'In a previous paper, s and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the  is isomorphic to $\BZ A_{\infty}$.' address: 'Danish National Library of Science and Medicine, Nørre Allé 49, 2200 København N, DK–Denmark' author: - Peter Jørgensen title: 'The Auslander-Reiten quiver of a Poincaré duality space' --- Introduction {#sec:introduction} ============ In [@PJARtq], the concepts of s and s from the representation theory of Artin algebras were introduced into algebraic topology. The main theorem was that s exist precisely over Poincaré duality spaces. On the other hand, the only concrete s computed in [@PJARtq] were those over spheres. An ad hoc computation showed that the  over $S^d$, the $d$-dimensional sphere, consists of $d-1$ components, each isomorphic to $\BZ A_{\infty}$. The purpose of this paper is to show that this result was no accident. Namely, let $k$ be a field of characteristic zero, and let $\Space$ be a simply connected topological space which has Poincaré duality of dimension $d \geq 2$ over $k$, that is, satisfies $\Hom_k(\H^{\ast}(\Space;k),k) \cong \H^{d-\ast}(\Space;k)$, where $\HXk$ is singular cohomology of $X$ with coefficients in $k$. The main result here is the following. \[thm:topology\] Let $\Component$ be a component of the  of the category $\DcXk$. Then $\Component$ is isomorphic to $\BZ A_{\infty}$. Here $\CXk$ is the singular cochain Differential Graded Algebra of $\Space$ with coefficients in $k$, and if $\D(\CXk)$ is the derived category of Differential Graded left-modules over $\CXk$, then $\DcXk$ denotes the full subcategory of  objects, that is, objects $M$ for which the functor $\Hom(M,-)$ preserves set indexed coproducts. The ingredients in the proof are standard. First the Riedtmann structure theorem is invoked to show that the component $\Component$ has the form $\BZ \Tree/\Group$ for a directed tree $\Tree$ and an admissible group of automorphisms $\Group \subseteq \Aut(\BZ \Tree)$ (section \[sec:Riedtmann\]). Then a certain function $\AddFunct$ is constructed on $\Component$ (section \[sec:AddFunct\]). This induces an unbounded additive function $\AddFunctTree$ on $\Tree$ forcing its underlying graph to be $A_{\infty}$, and finally an elementary argument shows that $\Group$ acts trivially (section \[sec:shape\]). Hence $\Component$ is $\BZ A_{\infty}$. In fact, I will prove a more general result than theorem \[thm:topology\]. Namely, I will not restrict myself to Differential Graded Algebras (DGAs) of the form $\CXk$, but will work over an abstract DGA denoted $R$. \[set:blanket\] Throughout, $R$ denotes a DGA over the field $k$ which satisfies the following. 1. $R$ is a cochain DGA, that is, $R^i = 0$ for $i < 0$. 2. $R^0 = k$. 3. $R^1 = 0$. 4. $\dim_k R < \infty$. $\Box$ This setup is the same as in [@PJARtq sec. 3], so I can use the results about $R$ proved in [@PJARtq]. For brief introductions to s, s, DGAs, Differential Graded modules (s), and the way they interact, I refer the reader to [@PJARtq]. My notation is mostly standard and identical to the notation of [@PJARtq]; however, I do want to recapitulate a few ubiquitous items. By $\D(R)$ is denoted the derived category of s, and by $\DcR$ the full subcategory of  objects. By $R^{\opp}$ is denoted the opposite DGA of $R$ with multiplication $$r \stackrel{\opp}{\cdot} s = (-1)^{|r||s|}sr.$$ s are identified with s, and so $\D(R^{\opp})$ is identified with the derived category of s, and $\DcRo$ with its full subcategory of  objects. The s $$k = R/R^{\geq 1} \; \; \mbox{and} \; \; \dual\!R = \Hom_k(R,k)$$ are used frequently throughout. When I wish to emphasize that I am viewing some $M$ as either a  or a , I do so with subscripts, writing either ${}_{R}M$ or $M_R$. Applying the Riedtmann structure theorem {#sec:Riedtmann} ======================================== This section will apply the Riedtmann structure theorem to show under some conditions that a component $\Component$ of the  of $\DcR$ has the form $\BZ \Tree/\Group$ for a directed tree $\Tree$ and an admissible group of automorphisms $\Group \subseteq \Aut(\BZ \Tree)$. Let me start by recalling some facts from [@PJARtq]. The triangulated category $\DcR$ does not necessarily have s, but if it does, then by [@PJARtq prop. 4.4(ii)] its  is $$\label{equ:tau} \tau(-) = \Sigma^{-1}(\dual\!R \LTensor_R -).$$ Here $\Sigma$ denotes suspension of s. When $\DcR$ has s, $\tau$ induces a map from the  of $\DcR$ to itself which is also referred to as the  and denoted $\tau$. The  of $\DcR$ is then a stable translation quiver with translation $\tau$; see [@PJARtq sec. 2]. This is even true in the strong sense that $\tau$ is an automorphism of the , under the extra assumption that $\DcRo$ also has s. This last fact was not mentioned in [@PJARtq], but is proved in the following lemma. \[lem:tau\_automorphism\] Suppose that $\DcR$ and $\DcRo$ have s. 1. The functor $\Sigma^{-1}(\dual\!R \LTensor_R -)$ is an auto-equivalence of $\DcR$, with quasi-inverse $\Sigma(P \LTensor_R -)$ for a suitable  $P$. 2. The  $\tau$ is an automorphism of the  of $\DcR$. 3. For each integer $p \not= 0$, the map $\tau^p$ is without fixed points in the  of $\DcR$. \(i) Both $\DcR$ and $\DcRo$ have s, so it follows from [@PJARtq cor. 5.2 and thm. 5.1 and its proof] that there are isomorphisms ${}_{R}(\dual\!R) \cong {}_{R}(\Sigma^d R)$ in $\DcR$ and $(\dual\!R)_R \cong (\Sigma^d R)_R$ in $\DcRo$. (Note that I do not know ${}_{R}(\dual\!R)_R \cong {}_{R}(\Sigma^d R)_R$!) This makes it easy to check that with $P$ equal to $\RHom_R(\dual\!R,R)$, the functors $\Sigma^{-1}(\dual\!R \LTensor_R -)$ and $\Sigma(P \LTensor_R -)$ are quasi-inverse endofunctors on $\DcR$. \(ii) This clearly follows from (i). \(iii) Let $M$ be an indecomposable object of $\DcR$ with vertex $[M]$ in the  of $\DcR$. If $\tau^p$ had the fixed point $[M]$, then $p$ applications of equation would give $$M \cong \Sigma^{-p}((\dual\!R) \LTensor_R \cdots \LTensor_R (\dual\!R) \LTensor_R M).$$ But this is impossible, as one proves for instance by checking $$\begin{aligned} & \inf \{\, i \,|\, \H^i(M) \not= 0 \,\} \\ & \;\;\;\;\;\;\;\;\;\; \not= \inf \{\, i \,|\, \H^i(\Sigma^{-p}((\dual\!R) \LTensor_R \cdots \LTensor_R (\dual\!R) \LTensor_R M)) \not= 0 \,\},\end{aligned}$$ for which the general formula $$\begin{aligned} & \inf \{\, i \,|\, \H^i(N \LTensor_R M) \not= 0 \,\} \\ & \;\;\;\;\;\;\;\;\;\; = \inf \{\, i \,|\, \H^i(N) \not= 0 \,\} + \inf \{\, i \,|\, \H^i(M) \not= 0 \,\}\end{aligned}$$ is handy. A loop in a quiver is an arrow that starts and ends at the same vertex. \[lem:no\_loops\] Suppose that $\DcR$ has s. In this case, the  of $\DcR$ has no loops. Suppose that there were an indecomposable object $M$ of $\DcR$ whose vertex $[M]$ in the  of $\DcR$ had a loop $[M] \longrightarrow [M]$. The arrow $[M] \longrightarrow [M]$ would mean that there existed an irreducible morphism $M \stackrel{\mu}{\longrightarrow} M$ in $\DcR$. Such a morphism would be non-invertible, so it would be in the radical of the local artinian ring $\End_{\Dsmall^{\operatorname{c}}(R)}(M)$ (cf.[@PJARtq lem. 3.6]). Hence some power $\mu^n$ would be zero. However, it is not hard to mimick the proof of [@ARSbook lem.VII.2.5] to show that if $M$ is indecomposable and $M \stackrel{\mu}{\longrightarrow} M$ is irreducible, and the composition $M \stackrel{\mu}{\longrightarrow} \cdots \stackrel{\mu}{\longrightarrow} M$ is zero, then $[M] = \tau[M]$ holds, and this is false by lemma \[lem:tau\_automorphism\](iii). In the following lemma and the rest of the paper, $\Component$ is a component of the  viewed as a stable translation quiver, so $\Component$ is closed under the  $\tau$ and its inverse. This implies that $\Component$ is a connected stable translation quiver with translation the restriction of $\tau$. For the lemma’s terminology of directed trees, stable translation quivers of the form $\BZ \Tree$, and admissible groups of automorphisms, see [@Bensonbook sec. 4.15]. \[lem:Riedtmann\] Suppose that $\DcR$ and $\DcRo$ have s. Let $\Component$ be a component of the  of $\DcR$. Then there exists a directed tree $\Tree$ and an admissible group of automorphisms $\Group \subseteq \Aut(\BZ \Tree)$ so that $\Component$ is isomorphic to $\BZ \Tree/\Group$ as a stable translation quiver. Lemma \[lem:tau\_automorphism\](ii) implies that the restriction of $\tau$ to $\Component$ is an automorphism of $\Component$, and lemma \[lem:no\_loops\] implies that $\Component$ has no loops. Also, by construction, the  has no multiple arrows, so neither has $\Component$. Hence $\Component$ satisfies the conditions of the Riedtmann structure theorem, [@Bensonbook thm. 4.15.6], and the conclusion of the lemma follows. Labelling the {#sec:labelling} ============== This section constructs a labelling of the  of $\DcR$, that is, a pair of positive integers $(a_{m \rightarrow n},b_{m \rightarrow n})$ for each arrow $m \longrightarrow n$ in the quiver. The construction is classical, works in good cases such as when $\DcR$ and $\DcRo$ both have s, and goes as follows. Let $M$ and $N$ be indecomposable objects of $\DcR$ for which there is an arrow $[M] \longrightarrow [N]$ in the , and let $$M \longrightarrow X \longrightarrow \tau^{-1}M \longrightarrow \; \; \mbox{and} \; \; \tau N \longrightarrow Y \longrightarrow N \longrightarrow$$ be s in $\DcR$. Here $\tau^{-1}M$ makes sense because of lemma \[lem:tau\_automorphism\]. Note that the triangles are determined up to isomorphism by [@HapDerCat prop. 3.5(i)]. The arrow $[M] \longrightarrow [N]$ means that there exists an irreducible morphism $M \longrightarrow N$ in $\DcR$, and by [@HapDerCat prop. 3.5] this means that $M$ is a direct summand of $Y$ and that $N$ is a direct summand of $X$. Let $a$ be the multiplicity of $M$ as a direct summand of $Y$ and let $b$ be the multiplicity of $N$ as a direct summand of $X$. These numbers are well-defined because $\DcR$ is a Krull-Schmidt category by [@HapDerCat par.3.1] and [@PJARtq lem. 3.6]. Now define the labelling by equipping $[M] \longrightarrow [N]$ with the pair of positive integers $(a,b)$. \[lem:symmetry\] Suppose that $\DcR$ and $\DcRo$ have s. The above labelling of the  of $\DcR$ satisfies the following. 1. $a_{\tau[M] \rightarrow [N]} = b_{[N] \rightarrow [M]}$. 2. $b_{\tau[M] \rightarrow [N]} = a_{[N] \rightarrow [M]}$. 3. $(a_{\tau^p[M] \rightarrow \tau^p[N]}, b_{\tau^p[M] \rightarrow \tau^p[N]}) = (a_{[M] \rightarrow [N]},b_{[M] \rightarrow [N]})$ for each integer $p$. \(i) Let $$\label{equ:AR_triangle} \tau N \longrightarrow Y \longrightarrow N \longrightarrow$$ be an . Then $a_{\tau[M] \rightarrow [N]}$ is defined as the multiplicity of $\tau M$ as a direct summand of $Y$. Now, $\tau$ is given by the formula $\Sigma^{-1}(\dual\!R \LTensor_R -)$ which also defines an auto-equivalence of $\DcR$ by lemma \[lem:tau\_automorphism\](i). A quasi-inverse auto-equivalence is given by $\Sigma(P \LTensor_R -)$, and applying this to the gives an   $$N \longrightarrow \Sigma(P \LTensor_R Y) \longrightarrow \tau^{-1}N \longrightarrow.$$ Then $b_{[N] \rightarrow [M]}$ is defined as the multiplicity of $M$ as a direct summand of $\Sigma(P \LTensor_R Y)$. Applying $\Sigma^{-1}(\dual\!R \LTensor_R -)$, this multiplicity equals the multiplicity of $\Sigma^{-1}(\dual\!R \LTensor_R M)$ as a direct summand of $\Sigma^{-1}(\dual\!R \LTensor_R \Sigma(P \LTensor_R Y))$, that is, the multiplicity of $\tau M$ as a direct summand of $Y$. And this again equals $a_{\tau[M] \rightarrow [N]}$ by the above. \(ii) and (iii) are proved by similar means. The function $\AddFunct$ {#sec:AddFunct} ======================== Let $$\AddFunct(M) = \dim_k \Ext_R(M,k) = \dim_k \H(\RHom_R(M,k))$$ for $M$ in $\D(R)$. Observe that $\AddFunct$ is constant on each isomorphism class in $\DcR$, so induces a well-defined function on the vertices $[M]$ of the  of $\DcR$. I also denote the induced function by $\AddFunct$, so $\AddFunct([M]) = \AddFunct(M)$. The purpose of this section is to study $\AddFunct$. \[lem:AddFunct\] Suppose that $\DcR$ and $\DcRo$ have s and that the ${}_{R}k$ is not in $\DcR$. Let $N$ be an indecomposable object of $\DcR$ with vertex $[N]$ in the  of $\DcR$. 1. If $\tau N \longrightarrow Y \longrightarrow N \longrightarrow$ is an  in $\DcR$, then $\AddFunct(\tau N) + \AddFunct(N) - \AddFunct(Y) = 0$. 2. $\AddFunct(\tau[N]) + \AddFunct([N]) - \sum_{[M] \rightarrow [N]}a_{[M] \rightarrow [N]}\AddFunct([M]) = 0$, where the sum is over all arrows in the  of $\DcR$ which end in $[N]$. 3. $\AddFunct(\tau[N]) = \AddFunct([N])$. \(i) The  $\tau N \longrightarrow Y \longrightarrow N \longrightarrow$ gives a long exact sequence consisting of pieces $\Ext_R^i(N,k) \longrightarrow \Ext_R^i(Y,k) \longrightarrow \Ext_R^i(\tau N,k) \stackrel{\delta}{\longrightarrow}$. Part (i) of the lemma will follow if the connecting maps $\delta$ are zero. And they are: By [@PJARtq lem. 4.2], the  is also an in $\DfR$, the full subcategory of $\D(R)$ consisting of $M$’s with $\dim_k \H\!M < \infty$. Moreover, no morphism $\Sigma^{-i}(\tau N) \longrightarrow {}_{R}k$ is a section, for any such section would clearly have to be an isomorphism, but $\Sigma^{-i}(\tau N)$ is in $\DcR$ and ${}_{R}k$ is not. The definition of  now says that the composition $\Sigma^{-i-1}N \longrightarrow \Sigma^{-i}(\tau N) \longrightarrow {}_{R}k$ is zero, where the first arrow is a suspension of the connecting morphism from the . In other words, the map $$\Hom_{\Dsmall(R)}(\Sigma^{-i}(\tau N),k) \longrightarrow \Hom_{\Dsmall(R)}(\Sigma^{-i-1}N,k)$$ is zero. But this map equals $$\Ext_R^i(\tau N,k) \stackrel{\delta}{\longrightarrow} \Ext_R^{i+1}(N,k).$$ \(ii) Consider again $\tau N \longrightarrow Y \longrightarrow N \longrightarrow$, the  from (i). From [@HapDerCat prop.3.5] follows that $Y$ is a coproduct of copies of those indecomposable objects $M$ of $\DcR$ for which there are irreducible morphisms $M \longrightarrow N$. That is, $Y$ is a coproduct of copies of those indecomposable objects $M$ of $\DcR$ for which there are arrows $[M] \longrightarrow [N]$ in the  of $\DcR$. And by the definition at the beginning of section \[sec:labelling\], the multiplicity of $M$ as a direct summand of $Y$ is $a_{[M] \rightarrow [N]}$. This clearly implies $$\AddFunct(Y) = \sum_{[M] \rightarrow [N]}a_{[M] \rightarrow [N]}\AddFunct([M]),$$ and then (ii) follows from (i). \(iii) Since both $\DcR$ and $\DcRo$ have s, it follows from [@PJARtq cor. 5.2 and thm. 5.1 and its proof] that there is an isomorphism ${}_{R}(\dual\!R) \cong {}_{R}(\Sigma^d R)$ in $\DcR$. So as $k$-vector spaces, $$\H(\RHom_R(\dual\!R,k)) \cong \H(\RHom_R(\Sigma^d R,k)) \cong \Sigma^{-d}k,$$ which easily implies $\RHom_R(\dual\!R,k) \cong \Sigma^{-d}k$ in $\D(R)$. This again gives (a) in $$\begin{aligned} \RHom_R(\tau N,k) & = \RHom_R(\Sigma^{-1}(\dual\!R \LTensor_R N),k) \\ & \cong \Sigma \RHom_R(\dual\!R \LTensor_R N,k) \\ & \cong \Sigma \RHom_R(N,\RHom_R(\dual\!R,k)) \\ & \stackrel{\rm (a)}{\cong} \Sigma \RHom_R(N,\Sigma^{-d}k) \\ & \cong \Sigma^{1-d} \RHom_R(N,k),\end{aligned}$$ and taking cohomology and $k$-dimension shows $\AddFunct(\tau[N]) = \AddFunct([N])$. The following results leave $\AddFunct$ for a while, but return to it in corollary \[cor:AddFunct\_unbounded\]. There is an abelian category $\sC(R)$ whose objects are s and whose morphisms are homomorphisms of s, that is, homomorphisms of s which are compatible with the differentials. The following Harada-Sai lemma for $\sC(R)$ is proved just like [@Bensonbook lem. 4.14.1]. \[lem:composition\] Let $F_0, \ldots, F_{2^p-1}$ be indecomposable objects of $\sC(R)$ with $\dim_k F_i \leq p$ for each $i$, and let $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex] F_{2^p-1} & \rTo^{\varphi_{2^p-1}} & F_{2^p-2} & \rTo^{\varphi_{2^p-2}} & \cdots & \rTo^{\varphi_1} & F_0 \\ \end{diagram}$$ be non-isomorphisms in $\sC(R)$. Then $\varphi_1 \circ \cdots \circ \varphi_{2^p-1} = 0$. \[cor:composition\] Let $M_0, \ldots, M_{2^p-1}$ be indecomposable objects of $\DcR$ with $\AddFunct(M_i) \leq \frac{p}{\dim_k R}$ for each $i$, and let $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex] M_{2^p-1} & \rTo^{\mu_{2^p-1}} & M_{2^p-2} & \rTo^{\mu_{2^p-2}} & \cdots & \rTo^{\mu_1} & M_0 \\ \end{diagram}$$ be non-isomorphisms in $\DcR$. Then $\mu_1 \circ \cdots \circ \mu_{2^p-1} = 0$. Pick minimal semi-free resolutions $F_i \stackrel{\simeq}{\longrightarrow} M_i$ (minimality means that the differential $\partial_{F_i}$ takes values inside $R^{\geq 1} \cdot F_i$, see [@PJARtq lem. 3.3]). Then the morphisms $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] M_{2^p-1} & \rTo^{\mu_{2^p-1}} & M_{2^p-2} & \rTo^{\mu_{2^p-2}} & \cdots & \rTo^{\mu_1} & M_0 \\ \end{diagram} $ in $\DcR$ are represented by morphisms $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] F_{2^p-1} & \rTo^{\varphi_{2^p-1}} & F_{2^p-2} & \rTo^{\varphi_{2^p-2}} & \cdots & \rTo^{\varphi_1} & F_0 \\ \end{diagram} $ in $\sC(R)$. If $\varphi_i$ were an isomorphism in $\sC(R)$, then $\mu_i$ would be an isomorphism in $\DcR$, so each $\varphi_i$ is a non-isomorphism in $\sC(R)$. Also, since $F_i$ is minimal, it is easy to see that if $F_i$ decomposed non-trivially in $\sC(R)$, then $M_i$ would decompose non-trivially in $\DcR$, so each $F_i$ is indecomposable in $\sC(R)$. Now $$\begin{aligned} \nonumber \Ext_R(M_i,k) & = \H(\RHom_R(M_i,k)) \cong \H(\Hom_R(F_i,k)) \\ \label{equ:Ext} & \stackrel{\rm (a)}{\cong} \Hom_R(F_i,k)^{\natural} = \Hom_{R^{\natural}}(F_i^{\natural},k^{\natural}).\end{aligned}$$ Here $\natural$ indicates the functor which forgets differentials, and therefore sends DGAs to graded algebras and s to graded modules. The isomorphism (a) holds because $F_i$ is minimal, whence the differential of the complex $\Hom_R(F_i,k)$ is zero, so taking cohomology amounts simply to forgetting the differential. Taking $k$-dimensions in equation shows that $\AddFunct(M_i)$ equals the $k$-dimension of $\Hom_{R^{\natural}}(F_i^{\natural},k^{\natural})$. However, since $F_i$ is semi-free I have $$F_i^{\natural} \cong \bigoplus_j \Sigma^{\sigma_j}(R^{\natural})$$ for certain integers $\sigma_j$, and the $k$-dimension of $\Hom_{R^{\natural}}(F_i^{\natural},k^{\natural})$ clearly equals the number of summands $\Sigma^{\sigma_j}(R^{\natural})$ in $F_i^{\natural}$. All in all, $\AddFunct(M_i)$ equals the number of summands $\Sigma^{\sigma_j}(R^{\natural})$ in $F_i^{\natural}$ which gives (b) in $$\begin{aligned} \dim_k F_i & = \dim_k F_i^{\natural} \stackrel{\rm (b)}{=} \AddFunct(M_i) \dim_k R^{\natural} \\ & = \AddFunct(M_i) \dim_k R \leq \frac{p}{\dim_k R} \dim_k R = p.\end{aligned}$$ But now lemma \[lem:composition\] gives $\varphi_1 \circ \cdots \circ \varphi_{2^p - 1} = 0$ which clearly implies $\mu_i \circ \cdots \circ \mu_{2^p - 1} = 0$. \[lem:non\_zero\_composition\] Suppose that $\DcR$ has s and that ${}_{R}k$ is not in $\DcR$. Given an indecomposable object $M_0$ of $\DcR$ and an integer $q \geq 0$, there exist indecomposable objects and irreducible morphisms $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex] M_q & \rTo^{\mu_q} & M_{q-1} & \rTo^{\mu_{q-1}} & \cdots & \rTo^{\mu_1} & M_0 \\ \end{diagram}$$ in $\DcR$ with $\mu_1 \circ \cdots \circ \mu_q \not= 0$. If $M$ is a non-zero object in $\DfR$, the full subcategory of $\D(R)$ consisting of $M$’s with $\dim_k \H\!M < \infty$, then the $k$-dual $\dual\!M = \Hom_k(M,k)$ is non-zero in $\DfRo$, so the minimal semi-free resolution $F$ of $\dual\!M$ is non-trivial. Hence $\RHom_{R^{\opp}}(\dual\!M,k) \cong \Hom_{R^{\opp}}(F,k)$ has non-zero cohomology, so by dualization the same holds for $$\RHom_R(\dual\!k,\dual\!\dual\!M) \cong \RHom_R(k,M).$$ So there exists an $i$ so that $\H^{-i}(\RHom_R(k,M)) \cong \Hom_{\Dsmall(R)}(\Sigma^i k,M)$ is non-zero; that is, $$\begin{aligned} \nonumber & \mbox{For $M$ non-zero in $\DfR$, there exists} & \\ \label{equ:non_zero_morphism} & \mbox{a non-zero morphism ${}_{R}(\Sigma^i k) \longrightarrow M$ in $\DfR$.} &\end{aligned}$$ Moreover, since $M$ is non-zero, each retraction ${}_{R}(\Sigma^i k) \longrightarrow M$ in $\DfR$ is clearly an isomorphism. If $M$ is in $\DcR$, then there is no such isomorphism because ${}_{R}(\Sigma^i k)$ is not in $\DcR$, so $$\begin{aligned} \nonumber & \mbox{For $M$ non-zero in $\DcR$, no} & \\ \label{equ:non_retraction} & \mbox{morphism ${}_{R}(\Sigma^i k) \longrightarrow M$ is a retraction in $\DfR$.} &\end{aligned}$$ Now for the proof proper. In fact, I shall prove slightly more than claimed, namely, there exists $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex] {}_{R}(\Sigma^i k) & \rTo^{\kappa_q} & M_q & \rTo^{\mu_q} & M_{q-1} & \rTo^{\mu_{q-1}} & \cdots & \rTo^{\mu_1} & M_0 \\ \end{diagram}$$ with the $M_i$ indecomposable and the $\mu_i$ irreducible in $\DcR$, with $\kappa_q$ not a retraction in $\DfR$, and with $\mu_1 \circ \cdots \circ \mu_q \circ \kappa_q \not= 0$. I do so by induction on $q$. For $q = 0$, existence of $\kappa_0$ holds by equations and . For $q \geq 1$, the induction gives data $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex,midshaft] {}_{R}(\Sigma^i k) & \rTo^{\kappa_{q-1}} & M_{q-1} & \rTo^{\mu_{q-1}} & & \cdots & \rTo^{\mu_1} & M_0. \\ \end{diagram}$$ Let $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] \tau M_{q-1} & \rTo & X_q & \rTo^{\mu_q^{\prime}} & M_{q-1} & \rTo \\ \end{diagram} $ be an  in $\DcR$. By [@PJARtq lem. 4.2] this is also an  in $\DfR$, so since $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] {}_{R}(\Sigma^i k) & \rTo^{\kappa_{q-1}} & M_{q-1} \\ \end{diagram} $ is not a retraction in $\DfR$, it factors through $X_q$ as $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] {}_{R}(\Sigma^i k) & \rTo^{\kappa_q^{\prime}} & X_q & \rTo^{\mu_q^{\prime}} & M_{q-1}, \\ \end{diagram} $ and so $\mu_1 \circ \cdots \circ \mu_{q-1} \circ \mu_q^{\prime} \circ \kappa_q^{\prime} = \mu_1 \circ \cdots \circ \mu_{q-1} \circ \kappa_{q-1}$ is non-zero. Splitting $X_q$ into indecomposable summands and $\kappa_q^{\prime}$ and $\mu_q^{\prime}$ into components gives that there exists $ \begin{diagram}[labelstyle=\scriptstyle,width=3ex] {}_{R}(\Sigma^i k) & \rTo^{\kappa_q} & M_q & \rTo^{\mu_q} & M_{q-1} \\ \end{diagram} $ with $\mu_1 \circ \cdots \circ \mu_{q-1} \circ \mu_q \circ \kappa_q \not= 0$. Here $M_q$ is an indecomposable summand of $X_q$ so is in $\DcR$, so equation gives that $\kappa_q$ is not a retraction in $\DfR$. And $\mu_q$ is a component of $\mu_q^{\prime}$, so $\mu_q$ is irreducible in $\DcR$ by [@HapDerCat prop. 3.5]. I now return to the function $\AddFunct$. \[cor:AddFunct\_unbounded\] Suppose that $\DcR$ and $\DcRo$ have s and that ${}_{R}k$ is not in $\DcR$. Let $\Component$ be a component of the  of $\DcR$. Then $\AddFunct$ is unbounded on $\Component$. Let $M_0$ be an indecomposable object of $\DcR$ for which $[M_0]$ is a vertex of $C$. Lemma \[lem:non\_zero\_composition\] says that there exist arbitrarily long sequences of indecomposable objects and irreducible morphisms $$\begin{diagram}[labelstyle=\scriptstyle,width=3ex] M_q & \rTo^{\mu_q} & M_{q-1} & \rTo^{\mu_{q-1}} & \cdots & \rTo^{\mu_1} & M_0 \\ \end{diagram}$$ in $\DcR$ with $\mu_1 \circ \cdots \circ \mu_q \not= 0$. Each $[M_i]$ is clearly a vertex of $\Component$. So corollary \[cor:composition\] implies that $\AddFunct$ cannot be bounded on $\Component$. Structure of the {#sec:shape} ================= This section combines the material of previous sections to prove the main result of this paper, theorem \[thm:topology\]. By lemma \[lem:Riedtmann\], if $\DcR$ and $\DcRo$ have s, then each component $\Component$ of the  of $\DcR$ is of the form $\BZ \Tree/\Group$, where $\Tree$ is a directed tree and $\Group \subseteq \Aut(\BZ \Tree)$ is an admissible group of automorphisms. Recall from [@Bensonbook sec. 4.15] that the vertices of $\BZ \Tree$ are the pairs $(p,t)$ where $p$ is an integer and $t$ is a vertex of $\Tree$, and that the vertices of $\BZ \Tree/\Group$ are the orbits $\Group(p,t)$ under the group $\Group$. By identifying $\BZ \Tree/\Group$ with $\Component$, I shall also view the $\Group(p,t)$’s as vertices of $\Component$, so the following makes sense. First, the start of section \[sec:labelling\] constructs a labelling of the , and so in particular a labelling of $\Component$. Now, if there is an arrow $t \longrightarrow u$ in $\Tree$, then there is an arrow $\Group(0,t) \longrightarrow \Group(0,u)$ in $\BZ \Tree/\Group$, that is, an arrow $\Group(0,t) \longrightarrow \Group(0,u)$ in $\Component$. Hence the labelling of $\Component$ induces a labelling of $\Tree$ by $$\label{equ:Tree_labelling} (a_{t \rightarrow u},b_{t \rightarrow u}) = (a_{\Group(0,t) \rightarrow \Group(0,u)}, b_{\Group(0,t) \rightarrow \Group(0,u)}).$$ Secondly, the start of section \[sec:AddFunct\] defines a function $\AddFunct$ on the vertices of $\Component$. That is, $\AddFunct$ is defined on the $\Group(p,t)$’s, and so induces a function $\AddFunctTree$ on $\Tree$ by $$\label{equ:AddFunctTree} \AddFunctTree(t) = \AddFunct(\Group(0,t)).$$ \[lem:induction\] Suppose that $\DcR$ and $\DcRo$ have s. 1. The function $\AddFunctTree$ is additive with respect to the labelling of $\Tree$ given by equation . 2. The function $\AddFunctTree$ is unbounded on $\Tree$. To explain part (i), let me recall from [@Bensonbook sec. 4.5] that when $\Tree$ is a labelled directed tree, a function $\AddFunctTree$ on the vertices of $\Tree$ is called additive if it satisfies $$\label{equ:f_additive} 2\AddFunctTree(t) - \sum_{s \rightarrow t}a_{s \rightarrow t}\AddFunctTree(s) - \sum_{t \rightarrow u}b_{t \rightarrow u}\AddFunctTree(u) = 0$$ for each vertex $t$. \[Proof of Lemma \[lem:induction\]\] (i) To show that the left hand side of equation is zero, let me rewrite it as $$\begin{aligned} \nonumber 2\AddFunct(\Group(0,t)) & - \sum_{s \rightarrow t} a_{\Group(0,s) \rightarrow \Group(0,t)}\AddFunct(\Group(0,s)) \\ \label{equ:form_i} & - \sum_{t \rightarrow u} b_{\Group(0,t) \rightarrow \Group(0,u)}\AddFunct(\Group(0,u)),\end{aligned}$$ using equations and . Recall from [@Bensonbook sec. 4.15] that the translation of $\BZ \Tree/\Group$ is given by $\tau(\Group(p,t)) = \Group(p+1,t)$. Identifying $\BZ \Tree/\Group$ with $\Component$, lemma \[lem:symmetry\](i) gives $b_{\Group(0,t) \longrightarrow \Group(0,u)} = a_{\tau(\Group(0,u)) \longrightarrow \Group(0,t)} = a_{\Group(1,u) \longrightarrow \Group(0,t)}$, and lemma \[lem:AddFunct\](iii) gives $\AddFunct(\Group(0,v)) = \AddFunct(\tau(\Group(0,v))) = \AddFunct(\Group(1,v))$. Hence is $$\begin{aligned} \nonumber \AddFunct(\tau \Group(0,t)) + \AddFunct(\Group(0,t)) & - \sum_{s \rightarrow t} a_{\Group(0,s) \rightarrow \Group(0,t)}\AddFunct(\Group(0,s)) \\ \label{equ:form_ii} & - \sum_{t \rightarrow u} a_{\Group(1,u) \rightarrow \Group(0,t)}\AddFunct(\Group(1,u)).\end{aligned}$$ To rewrite further, recall also from [@Bensonbook sec. 4.15] that the arrows in $\BZ \Tree$ which end in $(0,t)$ are obtained as follows: There is an arrow $(0,s) \longrightarrow (0,t)$ in $\BZ \Tree$ for each arrow $s \longrightarrow t$ in $\Tree$, and there is an arrow $(1,u) \longrightarrow (0,t)$ in $\BZ \Tree$ for each arrow $t \longrightarrow u$ in $\Tree$. Moreover, the canonical map $\BZ \Tree \longrightarrow \BZ \Tree/\Group$ is a so-called covering, see [@Bensonbook p. 156], so the map which sends the arrow $m \longrightarrow (0,t)$ to the arrow $\Group m \longrightarrow \Group(0,t)$ is a bijection between the arrows in $\BZ \Tree$ which end in $(0,t)$ and the arrows in $\BZ \Tree/\Group$ which end in $\Group(0,t)$. All this implies that taken together, the two sums in contain exactly one summand for each arrow in $\BZ \Tree/\Group$ which ends in $\Group(0,t)$, so is $$\AddFunct(\tau \Group(0,t)) + \AddFunct(\Group(0,t)) - \sum_{m \rightarrow \Group(0,t)} a_{m \rightarrow \Group(0,t)}\AddFunct(m).$$ Identifying $\BZ \Tree/\Group$ with $\Component$, this is zero by lemma \[lem:AddFunct\](ii). \(ii) I have $$\AddFunct(\Group(p,t)) = \AddFunct(\tau^p(\Group(0,t))) \stackrel{\rm (a)}{=} \AddFunct(\Group(0,t)) = \AddFunctTree(t),$$ where (a) is by lemma \[lem:AddFunct\](iii). So if $\AddFunctTree$ were bounded on $\Tree$, then $\AddFunct$ would be bounded on $C$ which it is not by corollary \[cor:AddFunct\_unbounded\]. The following is my abstract main result, which has theorem \[thm:topology\] as an easy consequence. I emphasize that as above, $\Component$ is a component of the viewed as a stable translation quiver, so $\Component$ is closed under the  $\tau$ and its inverse, and is a connected stable translation quiver with translation the restriction of $\tau$. \[thm:main\] Suppose that $\DcR$ and $\DcRo$ have s and that ${}_{R}k$ is not in $\DcR$. Let $\Component$ be a component of the  of $\DcR$. Then $\Component$ is isomorphic to $\BZ A_{\infty}$ as a stable translation quiver. Lemma \[lem:Riedtmann\] says that $\Component$ is of the form $\BZ \Tree/\Group$. Lemma \[lem:induction\] gives an additive function $\AddFunctTree$ on $\Tree$, and shows that $\AddFunctTree$ is unbounded. Consequently, the underlying graph of $\Tree$ must be $A_{\infty}$ by [@Bensonbook thm. 4.5.8(iv)]. So $\Component$ is $\BZ A_{\infty}/\Group$. (Note that when the underlying graph of $\Tree$ is $A_{\infty}$, the quiver $\BZ \Tree$ is determined up to isomorphism, independently of the directions of the arrows in $\Tree$. So it makes sense to denote $\BZ \Tree$ by $\BZ A_{\infty}$.) Now, $\Group$ must send vertices at the end of $\BZ A_{\infty}$ to other such vertices. If $\Group$ acted non-trivially on the vertices at the end of $\BZ A_{\infty}$, then there would exist a $g$ in $\Group$ and a vertex $m$ at the end of $\BZ A_{\infty}$ so that $gm$ was a different vertex at the end of $\BZ A_{\infty}$. As the other vertices at the end of $\BZ A_{\infty}$ have the form $\tau^p m$ for $p \not= 0$, this would mean $gm = \tau^p m$ for some $p \not= 0$. Then $m$ and $\tau^p m$ would get identified in $\BZ A_{\infty}/\Group$, so $\Group m$ would be a fixed point of $\tau^p$ in $\BZ A_{\infty}/\Group$, that is, a fixed point of $\tau^p$ in $\Component$ contradicting lemma \[lem:tau\_automorphism\](iii). So $\Group$ acts trivially on the vertices at the end of $\BZ A_{\infty}$, and it is easy to see that this forces $\Group$ to act trivially on all of $\BZ A_{\infty}$. So $\BZ A_{\infty}/\Group$ is just $\BZ A_{\infty}$, and the theorem follows. \[Proof of Theorem \[thm:topology\]\] This proof uses a bit of rational homotopy theory for which my source is [@FHTbook]. By [@FHTbook exam. 6, p. 146] I have that $\CXk$ is equivalent by a series of quasi-isomorphisms to a DGA, $R$, which satisfies setup \[set:blanket\]. Hence the various derived categories of $\CXk$ are equivalent to those of $R$ by [@KrizMayAst thm. III.4.2], so theorem \[thm:main\] can be applied to $\CXk$. To get the desired conclusion, that each component $\Component$ of the  of $\DcXk$ is isomorphic to $\BZ A_{\infty}$, I must show that the premises of theorem \[thm:main\] hold for $\CXk$. So I must show that $\Dc(\CXk)$ and $\Dc(\CXk^{\opp})$ have s, and that ${}_{\CXk}k$ is not in $\Dc(\CXk)$. The first two claims hold by the main result of [@PJARtq], theorem 6.3, because $\Space$ has Poincaré duality over $k$. The last claim can be seen as follows. By assumption, $\Space$ has Poincaré duality of dimension $d \geq 2$ over $k$, so I have $\H^{\geq 2}(\Space;k) \not= 0$. Hence the minimal Sullivan model $\Lambda V$ of $\CXk$ has $V^{\geq 2} \not= 0$; see [@FHTbook chp. 12]. But $V$ is $\Hom_{\BZ}(\pi_{\ast}\Space,k)$ by [@FHTbook chp. 15], where $\pi_{\ast}\Space$ is the sequence of homotopy groups of $\Space$, so as $k$ has characteristic zero, $\pi_{\geq 2}\Space$ cannot be torsion, so $\BQ \otimes_{\BZ} \pi_{\geq 2}\Space \not= 0$. Since [@FHTbook p. 434] gives $\dim_{\BQ}(\BQ \otimes_{\BZ} \pi_{\rm odd}\Space) - \dim_{\BQ} (\BQ \otimes_{\BZ} \pi_{\rm even}\Space) \geq 0$, there follows $\BQ \otimes_{\BZ} \pi_{\rm odd}\Space \not= 0$. This implies that the denominator is non-trivial in the formula [@FHTbook (33.7)] from which follows $\dim_{\BQ} \H^{\ast}(\Omega \Space;\BQ) = \infty$, where $\Omega \Space$ is the Moore loop space of $\Space$. So also $$\label{equ:H_Omega_X_infinite} \dim_k \H^{\ast}(\Omega \Space;k) = \infty.$$ Now consider the Moore path space fibration $\Omega \Space \longrightarrow P\Space \longrightarrow \Space$ from [@FHTbook exam. 1, p. 29]. The Moore path space $P\Space$ is contractible, and this implies that the  $\C^{\ast}(P\Space;k)$ is quasi-isomorphic to ${}_{\CXk}k$. Inserting the fibration into the Eilenberg-Moore theorem [@FHTbook thm.7.5] therefore gives a quasi-isomorphism $\C^{\ast}(\Omega \Space;k) \simeq k \LTensor_{\CXk} k$, and taking cohomology shows $$\H^{\ast}(\Omega \Space;k) \cong \Tor_{-\ast}^{\CXk}(k,k).$$ Combining with equation gives $$\dim_k \Tor^{\CXk}(k,k) = \infty.$$ So ${}_{\CXk}k$ cannot be finitely built from ${}_{\CXk}\CXk$ whence ${}_{\CXk}k$ is not in $\Dc(\CXk)$ by [@PJARtq lem. 3.2]. [9]{} M. Auslander, I. Reiten, and S. Smalø, “Representation theory of Artin algebras”, Cambridge Stud. Adv.Math., Vol. 36, Cambridge University Press, Cambridge, 1997, paperback edition. D. Benson, “Representations and cohomology, I”, Cambridge Stud. Adv. Math., Vol. 30, Cambridge University Press, Cambridge, 1998, paperback edition. Y. Félix, S. Halperin, and J.-C. Thomas, “Rational Homotopy Theory”, Grad. Texts in Math., Vol. 205, Springer, Berlin, 2001. D. Happel, [*On the derived category of a finite-dimensional algebra*]{}, Comment. Math. Helv. [**62**]{} (1987), 339–389. P. Jørgensen, Auslander-Reiten triangles and quivers over topological spaces, preprint (2002). [math.AT/0304079]{}. I. Kriz and J. P. May, “Operads, algebras, modules and motives”, Astérisque [**233**]{} (1995).

--- author: - 'N. Huélamo' - 'S. Lacour' - 'P. Tuthill' - 'M. Ireland' - 'A. Kraus' - 'G. Chauvin' bibliography: - '16395.bib' date: 'Received; accepted' title: 'A companion candidate in the gap of the T Cha transitional disk[^1]' --- [T Cha is a young star surrounded by a cold disk. The presence of a gap within its disk, inferred from fitting to the spectral energy distribution, has suggested on-going planetary formation.]{} [The aim of this work is to look for very low-mass companions within the disk gap of T Cha.]{} [We observed T Cha in $L'$ and $K_s$ with NAOS-CONICA, the adaptive optics system at the VLT, using sparse aperture masking.]{} [We detected a source in the $L'$ data at a separation of 62$\pm$7mas, position angle of $\sim$78$\pm$1 degrees, and a contrast of $\Delta L'\, =\, 5.1\pm0.2$mag. The object is not detected in the $K_s$ band data, which show a 3-$\sigma$ contrast limit of 5.2mag at the position of the detected $L'$ source. For a distance of 108pc, the detected companion candidate is located at 6.7AU from the primary, well within the disk gap. If T Cha and the companion candidate are bound, the comparison of the $L'$ and $K_s$ photometry with evolutionary tracks shows that the photometry is inconsistent with any unextincted photosphere at the age and distance of T Cha. The detected object shows a very red $K_s-L'$ color for which a possible explanation would be a significant amount of dust around it. This would imply that the companion candidate is young, which would strengthen the case for a physical companion, and moreover that the object would be in the substellar regime, according to the $K_s$ upper limit. Another exciting possibility would be that this companion is a recently formed planet within the disk. Additional observations are mandatory to confirm that the object is bound and to properly characterize it.]{} Introduction ============ In recent years, a large number of disks characterized by a lack of significant mid-infrared (IR) emission and a rise into the far-IR have been detected [e.g. @Brown2007; @Merin2010]. These are the so-called ‘transitional disks’, and they are thought to be in an intermediate evolutionary state between primordial Class II protoplanetary disks and Class III debris disks. The lack of mid-IR excess in cold disks has been interpreted as a sign of dust clearing, which can result in gaps or holes within the disk. These gaps and holes can be created by several mechanisms, such as a close stellar companion, disk photoevaporation, grain growth or a planet formed within the disk. A planet forming within the disk is expected to generate a gap while the dust and gas is accreted onto its surface, sweeping out the orbital region [e.g. @Lubow1999]. In this work, we present high angular resolution deep IR observations of T Cha, a young star with a cold disk. Its spectral energy distribution (SED) shows a small IR excess between 1–10$\mu$m and a very steep rise between 10–30$\mu$m. The SED has only been successfully modeled by including a gap from 0.2 to 15 AU [@Brown2007; @Schisano2009]. In fact, an inner dusty disk has recently been detected by @Olofsson2011. Because one of the possibilities is that the gap has been cleared by a very low-mass object, we obtained adaptive optics (AO) sparse aperture masking (SAM) observations of T Cha aimed at detecting faint companions within the disk gap. The target: T Cha ================= T Cha is a high probability member of the young $\epsilon$ Cha association [@Torres2008]. It is a G8-type star with a mass of $\sim$ 1.5M$_{\odot}$, classified as a weak-lined T Tauri star based on the H$_\alpha$ equivalent width from single epoch spectroscopy [@Alcala1993]. Subsequent photometric and spectroscopic monitoring has indicated a strong variability of this line, which shows significant changes in its equivalent width, intensity, and profile [@GregorioHetem1992; @Alcala1993; @Schisano2009]. If the line is related to accretion episodes, then the average accretion rate is $\dot{M}$ = 4$\times$ 10$^{-9}\,M_{\odot}/yr$ [@Schisano2009]. T Cha shows variable circumstellar extinction with a most frequent value of $A_V$=1.7mag according to @Schisano2009. The authors derive a disk extinction law characterized by $R_V=5.5$, which suggests the presence of large dust grains within the disk. The age of the source is variously estimated to be between 2–10Myr according to different methods [@Fernandez2008]. A complete study of the $\epsilon$ Cha association by @Torres2008 provides an average age of 6Myr, while @daSilva2009 estimate a slightly older age (between 5-10Myr) based on the lithium content of the $\epsilon$ Cha members. Finally, a dynamical evolution study of the $\eta$ Cha cluster, which probably belongs to the $\epsilon$ Cha association, provides an age of 6.7Myr [@Ortega2009]. For the purpose of this paper, we adopt an age of 7Myr. The distance to the source, based on the Hipparcos parallax, is 66pc$\pm$15pc. A more reliable value of 100pc was obtained using proper-motion studies [@Frink1998; @Terranegra1999]. @Torres2008 provided a kinematical distance of 109pc for T Cha, and an average value of 108$\pm$9pc for the whole association. We adopted the latter value for this paper. Finally, previous works based on radial velocity (RV) and direct imaging and coronographic techniques have not reported the presence of any (stellar or very low-mass) companion around T Cha [@Schisano2009; @Chauvin2010; @Vicente2011]. The SAM observations allow us to fill the gap between between RV and direct imaging observations. Observations and data reduction =============================== The observations presented here were obtained with NAOS-CONICA (NACO), the AO system at the Very Large Telescope (VLT), and SAM [@Tuthill2010] in two different campaigns. The [*L’*]{} observations were obtained in March 2010 under excellent atmospheric conditions (average coherent time of $\tau_0$=8ms, and average seeing of 06), while the $K_s$ data were obtained in July 2010 under moderate atmospheric conditions ($\tau_0$=4ms, and seeing of 10). In March 2010, T Cha was observed with the L27 objective, the seven-hole mask and the $L'$ filter ($\lambda_c$= 3.80$\mu$m, $\Delta\lambda$=0.62 $\mu$m). The target and a calibrator star (HD102260) were observed during 10 minutes each. We repeated the sequence star+calibrator nine times, integrating a total of 48 minutes on-source. The observational procedure included a dithering pattern that placed the target in the four quadrants of the detector. We acquired datacubes of 100 frames of 0.4sec integration time in each offset position. The plate scale, 27.10$\pm$0.10mas/pix, and true north orientation of the detector, -0.48$\pm$0.25 degrees, were derived using the astrometric calibrator $\theta$ Ori$^1$ C observed in April 2010. For the $K_s$-band data we used the S27 objective and the same strategy, but integrating in datacubes of 100 frames with 0.5sec of individual exposure time. We spent a total of 20 minutes on-source, and we used two stars, HD102260 and HD101251, as calibrators. All data were reduced using a custom pipeline detailed in Lacour et al. (in preparation). In brief, each frame is flat-fielded, dark-subtracted, and bad-pixel-corrected. The complex amplitudes of each one of the $7\times6/2=21$ fringe spatial frequencies were then used to calculate the bispectrum, from which the argument is taken to derive the closure phase. Lastly, the closure phases were fitted to a model of a binary source. ![Results from the $L'$ SAM observations: a companion candidate is detected at $61.8\pm 7.4$mas from the central source with a flux ratio of 0.92$\pm$0.20% in [*L’*]{}. [*Upper left panel:*]{} $\chi^2$ as a function of the position of companion candidate (degrees of freedom = 314). The black circle corresponds to the imaging resolution of the telescope ($1.22\lambda/D$). [*Upper right panel:*]{} best fit of the model of the companion (solid line) overplotted on the deconvolved phase. [*Lower panel:*]{} Orientation of companion detections made using each of the nine individual data files plotted in raw detector coordinates. Spurious structures should appear fixed, while real features will rotate with the sky, as illustrated by the overplotted solid line that depicts the expected orientation for an object with a sky position angle of 78 degrees.[]{data-label="fig1"}](16395fg1c.eps) Main results ============ $L'$ detection {#detect} -------------- The three free parameters of the fit are the flux ratio, the separation, and the position angle of the companion candidate. The upper left panel of Fig. \[fig1\] depicts the minimum $\chi^2$ as a function of position angle and separation. For an arbitrary fit, $\chi^2$ is high (reduced $\chi^2$ of $\approx 9$), but the map shows a clear minimum for a companion to the west of the star. The phase corresponding to the best-fit model is shown in the right panel of the same figure. It consists of a sinusoidal curve with a specific angular direction, and a period of half the resolution of the 8.2 meter telescope. In the same figure we plotted the deconvolved phases from the measured closure phases that were projected onto the orientation of the best-fit binary. The best-fitting companion parameters for the L-band data are a separation of $61.8 \pm 7.4$mas, a position angle of $78.5\pm 1.2$ degrees, and a fractional flux with respect to the central object of $0.92 \pm 0.20$%. To confirm the validity of the detection, each one of the nine star+calibrator data pairs was also analyzed separately. Because the observations were taken in ‘pupil tracking mode’, all optical and electronic aberrations should remain at frozen orientation on the detector, while a real structure on the sky will rotate with the azimuth pointing of the telescope (close to the sidereal rate). This expected rotation of the detection is illustrated in Fig. \[fig1\], which strongly argues against an instrumental artifact. The detection error bars reported above are 1-$\sigma$, but owing to the low separation, there is a strong degeneracy between separation and flux ratio. This is highlighted in the contours shown in Fig. \[fig2\]. The limits of the 3 $\sigma$ contours correspond to a spread of parameters between flux ratio of 9% at 26mas, and 0.6% at 80mas. $K_s$ upper limit ----------------- We did not detect any source around T Cha in the $K_s$-band data. Figure \[fig3\] shows the 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$ sensitivity curves as a function of the separation to the central source. The data analysis shows that we can rule out a companion between 40 and 62mas, with contrast ratios varying between 1.3% and 0.83%, respectively, at 99% confidence. ![Error contours as a function of separation and flux ratio in the [*L’*]{} filter. The contour levels correspond to 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$. The upper horizontal labels provide the separation in astronomical units assuming a distance of 108pc.[]{data-label="fig2"}](16395fg2.eps) Physical parameters ------------------- According to Section \[detect\], we have detected a companion candidate at a separation and position angle of 61.8$\pm$7.4mas and 78.5$\pm$1.2 degrees, respectively, and a flux ratio of 0.0092$\pm$0.0020. Assuming a distance of 108$\pm$9pc, the separation between the star and the companion candidate is $\sim$6.7$\pm$1.0AU. This lies well within the disk gap of T Cha, according to the @Brown2007 [0.2-15AU] and @Olofsson2011 [0.17-7.5AU] disk models. The flux ratio between the primary and the companion translates into a difference of magnitude of $\Delta L'\, =\, 5.1\pm0.2$mag. Because our observational methods are not optimized for photometry, we instead rely on $L$-band magnitudes for T Cha from the literature. The reported Johnson L-band ($\lambda_c$=3.45$\mu$m, $\Delta\lambda$=0.472$\mu$m) brightness of T Cha is 5.86$\pm$0.02mag [@Alcala1993], while the IRAC channel-1 (3.55$\mu$m, $\Delta\lambda$=0.75$\mu$m) brightness is 5.74$\pm$0.06mag. For the purpose of this paper, we assume an $L'$ magnitude of 5.8$\pm$0.1mag for the primary. Our measured contrast ratio then implies $L'\sim$10.9$\pm$0.2mag for the companion candidate. We can provide a 3-$\sigma$ limit of $\Delta K_s$$>$5.2mag at the separation and position angle of the $L'$ detection for the $K_s$-band data. Because T Cha itself shows $K_s$=6.95$\pm$0.02mag [@2MASS], the limit to the companion candidate brightness would be $K_s >$12.15mag. The extinction curve derived for T Cha using optical data seems significantly flatter than that from the interstellar medium [@Schisano2009]. However, because the disk extinction curve is not well known in the infrared regime, we corrected the observed magnitudes as a first approximation using the @Mathis1990 extinction curve with $R_V$=5, and assuming $A_V$=1.7mag. We obtained $A_{K_s}$ $\sim$ 0.2mag and $A_{L'}$ $\sim$ 0.1mag, which results in magnitudes of $L'=10.8\pm0.2$ and $K_s$ $>$11.95mag. ![Contrast in the $K_s$-band data. The solid red lines represent the 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$ upper limits at different separations from the central source. We estimate a 3-$\sigma$ upper limit of 0.83% at 62mas, that is, at the position of the $L'$ detected source.[]{data-label="fig3"}](16395fg3.eps) Discussion ========== We have detected a source within the disk gap of T Cha. We will now discuss the possible nature of this object, keeping in mind that a detection at a single epoch and waveband provides limited information. If we assume the two objects are co-moving and at a distance of 108$\pm$9pc, the companion candidate would show absolute magnitudes of $M_{L'}$=5.6$\pm$0.5mag, and $M_{K_s}>$6.8mag. We used these values to place the object in two magnitude-color diagrams (Fig. \[fig4\]). Assuming they are coeval, and assuming an age for the system of 7$\pm$2Myr, both diagrams show that the observed properties are inconsistent with any unextincted photosphere at the age and distance of T Cha, according to the NextGen [@Baraffe1998] and DUSTY [@Chabrier2000] models.[^2] The companion candidate shows a very red $K_s-L'$ color. A possible explanation for this $K_s-L'$ excess would be a significant amount of dust around the object. If this scenario is correct, it would imply that the object is young, which would strengthen the case for a physical companion, and moreover according to Fig. \[fig4\]b, the $K_s$ upper limit would place it in the substellar (brown dwarf) regime. Brown dwarf (BD) companions to stars are rare, as we know from radial velocity studies of samples of nearby stars [e.g. @Grether2006], giving rise to the so-called ”Brown Dwarf Desert”. Although this could be interpreted as evidence against a BD for the T Cha companion, the separation of 6.7AU places it near to the known shores of this desert (longer period companions are not well studied). Indeed, @Kraus2011 claim that intermediate separation ranges (5–50AU) show no evidence for this desert, which makes T Cha system interesting whether or not the companion mass lies above or below the BD cutoff. Because disk gaps can be the result of dust clearing owing to planet formation, we also investigated if the companion candidate could be a recently formed planet within the disk. The T Cha system shows properties that are consistent with this scenario. First, the object is detected well within the disk gap. The total disk mass derived by @Olofsson2011 is 1.74$\pm$0.25$\times$10$^{-2}$M$_{\odot}$, while the average accretion rate is 4$\times10^{-9}$$M_{\odot}/yr$. These properties seem consistent with a planet-forming disk according to @AlexArmi2007, keeping in mind that both measurements can be affected by large uncertainties. If this is the case, the evolutionary models used here are not well suited to derive the mass of planetary objects, because we are probably observing the planet at the initial formation phase, when the brightness depends only on the accretion history and accretion rate. Indeed, one of the biggest advantages of observing transitional disks is that recently formed planets are probably still accreting material and therefore should be in their brightest evolutionary phase. Additional observations are needed to shed light on the nature of this exciting object, the first potential substellar object detected within the gap of a transitional disk. In particular, observations that detect this object at other wavelengths or determine the disk position angle and inclination would be most useful. Conclusions =========== We have observed T Cha with NACO/SAM in two filters, $L'$ and $K_s$. Our main results can be summarized as follows: - We detected a faint companion around T Cha at $62\pm7.4$mas of separation, position angle of 78.0$\pm$1.2 degrees, and contrast ratio of $\Delta\,L'=5.04\pm0.2$mag. We did not detect any source in the $K_s$-band. The 3-$\sigma$ contrast ratio at a separation of 62mas is 0.82%, that is $\Delta K_s$$>$5.2mag. - If T Cha and the detected object are bound, and assuming a distance of 108pc, the faint companion lies at 6.7AU, that is, within the disk gap of the central source. - If T Cha and its companion are bound and coeval, the infrared magnitude-color diagrams show that the object photometry is inconsistent with any unextincted photosphere at the age and distance of T Cha when compared with the evolutionary tracks. - The companion candidate displays a strong $Ks-L'$ excess, which could be explained by a significant amount of dust around it. This scenario would strengthen the case of a physical companion and, according to the $K_s$ upper limit, it would place the object in the substellar regime. - The overall properties of the T Cha system also suggest that the newly detected source might be a recently formed planet within the disk. In this case, suitable planetary formation models are needed to derive its physical properties. ![Magnitude-color diagrams with the T Cha companion candidate. [**Left:**]{} The solid lines show NextGEN models for 5,7, and 10Myr, while the shaded area represents the $K_s-L'$ limit. We provide masses (in M$_{Jup}$) for an age of 7Myr. [**Right:**]{} The solid lines correspond to the NextGEN (blue) and DUSTY (black) models for 5,7, and 10Myr, and the masses are derived for an age of 7 Myr. The shaded area shows the possible location of the companion candidate, according to the derived $M_{K_s}$ and $K_s-L'$ limits.[]{data-label="fig4"}](16395fg4.eps) Second epoch observations in different photometric bands are mandatory to confirm if the object is bound, and to properly characterize it. This research has been funded by Spanish grants MEC/ESP2007-65475-C02-02, MEC/Consolider-CSD2006-0070, and CAM/PRICIT-S2009ESP-1496. We thank the Paranal staff, in particular A. Bayo and F. Selman, for their support during the observations. NH is indebted to D.Stamatellos, H. Bouy, J. Olofsson, and B. Merín for useful discussions. [^1]: Based on observations obtained at the European Southern Observatory using the Very Large Telescope in Cerro Paranal, Chile, under program 84.C-0755(A) [^2]: We have used the latest NextGEN and DUSTY models [@allard2010] convolved with the NACO filters.

--- abstract: | Differential cross sections and beam asymmetries for coherent $\pi^{\circ}$ photoproduction from $^4$He in the $\Delta$ energy-range have been measured with high statistical and systematic precisions using both decay photons for identifying the process. The experiment was performed at the MAinz MIcrotron using the TAPS photon spectrometer and the Glasgow/Mainz tagged photon facility. The differential cross sections are in excellent agreement with predictions based on the DWIA if an appropriate parametrization of the $\Delta$-nuclear interaction is applied. The beam asymmetries are interpreted in terms of degrees of linear polarization of collimated coherent bremsstrahlung. The expected increase of the degree of linear polarization with decreasing collimation angle is confirmed. Agreement with calculations is obtained on a few-percent level of precision in the maxima of the coherent peaks. PACS classification: 25.20.Lj; 24.70+s; 29.27.Hj address: - 'II. Physikalisches Institut, Universität Göttingen, Bunsenstrasse 7-9, D-37073 Göttingen, Germany' - 'Institut für Kernphysik, Universität Mainz, D-55099 Mainz, Germany' - 'Department of Physics and Astronomy, University of Glasgow, Glasgow G128QQ, UK' - 'II. Physikalisches Institut, Universität Giessen, D-35392 Giessen, Germany' - 'Institut für Kernphysik, Universität Mainz, D-55099 Mainz, Germany and Department of Physics, National Taiwan University, Taipei 10617, Taiwan' - 'Physikalisches Institut, Universität Tübingen, D-72076 Tübingen, Germany' author: - 'F. Rambo' - 'P. Achenbach' - 'J. Ahrens' - 'H.-J. Arends' - 'R. Beck' - 'S. J. Hall' - 'V. Hejny' - 'P. Jennewein' - 'S.S. Kamalov' - 'M. Kotulla' - 'B. Krusche' - 'V. Kuhr' - 'R. Leukel' - 'V. Metag' - 'R. Novotny' - 'V. Olmos de Léon' - 'A. Schmidt' - 'M. Schumacher' - 'U. Siodlaczek' - 'F. Smend' - 'H. Ströher' - 'J. Weiss' - 'F. Wissmann' - 'M. Wolf' title: 'Coherent $\pi^{\circ}$ photoproduction from $^4$He' --- , , , , , , , , , , , , , , , , , , , , , , , Nuclear Reactions $^4$He($\gamma,\pi^{\circ}$)$^4$He, $E_{\gamma}=200-400$ MeV, tagged coherent bremsstrahlung beam. Measured total and differential cross sections. New theoretical calculations taking into account in-medium modification of the $\Delta$ resonance parameters lead to excellent agreement with the experimental data. Introduction ============ Coherent photoproduction of $\pi^{\circ}$ mesons from nuclei is an ideal tool to study fundamental properties of pion-nucleus interactions. Another important aspect in the study of this reaction is the unique opportunity to get information about modifications of the $\Delta(1232)$-resonance characteristics in the nuclear medium. The nucleus $^4$He is especially suited for studies of this type because it combines the well known structure of a few-body system with the large binding energy of a complex nucleus. Furthermore, because of the high excitation energy of 20 MeV of the first excited level it is comparatively easy to separate coherent from incoherent $\pi^{\circ}$ photoproduction processes. The advantage of coherent $\pi^{\circ}$ photoproduction as compared to, e.g., the production of charged mesons is that the nucleus is identical in the initial and final states. This eliminates complicated nuclear rearrangement processes and makes a clear-cut theoretical interpretation possible. Furthermore, the coherent $\pi^{\circ}$ photoproduction process is uniformly distributed over the whole nuclear volume. This leads to a substantial enhancement of the cross section in contrast to, e.g., $\pi^+$ photoproduction. In spite of the fundamental importance of this process the available experimental data were very scarce. There were only four papers available [@Sta69; @Lef70; @Tieger84; @Ananin85] providing few and partly controversial differential cross sections. The reason for this rather unsatisfactory experimental situation was that the experiments were extremely difficult with the previously available photon beams and photon detectors. A major step forward came with the advent of $cw$ electron accelerators [@Her76; @Her81; @Her90] and of broad-band tagging facilities [@Ant91; @hall96] as they are now available, e.g., at MAMI in Mainz. A further large progress came with the availability of high energy-resolution and large angular-acceptance photon detectors. The present experiment is a continuation of our previous research on coherent $\pi^\circ$ photoproduction on $^4$He [@kraus97] performed at MAMI (Mainz) [@Her76; @Her81; @Her90; @Ant91; @hall96] which were mainly carried out to measure the degree of linear polarization of coherent bremsstrahlung produced via a diamond radiator [@lohmann94]. In these previous investigations [@kraus97] the 48 cm Ø$\times$ 64 cm Mainz NaI(Tl) detector [@wiss94] has been used as a high detection-efficiency and high energy-resolution (1.5%) spectrometer for the high-energy photon emitted in an asymmetric $\pi^\circ$ decay. The experiment [@kraus97] had been carried out for two different collimation angles (half of the angular opening) of $\theta_{coll}$=0.7 mrad and $\theta_{coll}$=0.5 mrad of the photon beam, corresponding to $\theta_{coll}=1.17 \times \theta_{BH}$ and $\theta_{coll}=0.83 \times \theta_{BH}$ where $\theta_{BH}$=mc$^2$/E (m, E = electron mass and energy, respectively) is the characteristic (Bethe-Heitler) angle. In this range of collimation angles a pronounced dependence of the degree of linear polarization on the collimation angle $\theta_{coll}$ is expected and this effect was clearly observed for the first time in that experiment [@kraus97]. In parallel to the experiment a program was started to precisely calculate the degree of linear polarization including all geometrical effects like the parameters of the electron beam and the collimation of the photon beam [@rambo98]. As a result of these investigations [@kraus97; @rambo98] an agreement between measured and calculated degrees of linear polarization on a ${\stackrel{>}{\sim}}$ 3% level of precision was obtained. Differing from our previous experiment [@kraus97] where one decay photon has been used to identify a $\pi^{\circ}$ event, our present experiment carried out with the TAPS setup [@nov87; @nov91; @nov96] at MAMI makes use of both decay photons and a much larger angular acceptance. Therefore, this latter method permits to measure the degree of linear polarization of collimated photons with higher systematic and statistical precisions than it was possible before [@kraus97; @rambo98]. Furthermore, it was possible to evaluate differential cross sections for coherent $\pi^{\circ}$ photoproduction in wide energy and angular ranges. After completion of the present paper a similar experiment performed by Bellini et al. [@belli99] came to our attention. It used the LEGS tagged polarized photon beam and covers a similar, but smaller range of photon energies with a smaller number of pion production angles. Using one large $48 cm \oslash \times 48 cm $ NaI(Tl) detector for the high-energy photon and an array of smaller NaI(Tl) detectors for the low-energy photon of the asymmetric $\pi^\circ$ decay it was possible to single out the coherent ${\vec \gamma} + ^4{}$He $\to \pi^\circ + ^4{}$He production channel. The present and the previous data [@belli99] are in a general reasonable agreement with each other except for some deviations especially at the highest energies accessible in [@belli99], as will be shown later. Experiment ========== The experiments have been carried out using the BaF$_2$ multi-detector array TAPS [@nov87; @nov91; @nov96] at the tagged-photon facility of the A2-collaboration [@Ant91; @hall96] installed at the microtron MAMI in Mainz [@Her76; @Her81; @Her90]. Using a diamond crystal as radiator and positioning a collimator at a distance of 2500 mm downstream from the radiator the tagged-photon facility provides linearly polarized photons with degrees of linear polarization up to 50 $-$ 60% [@kraus97]. The accelerator was operated at its nominal maximum energy of 855 MeV. The intensity of the photon flux is limited by the tagging technique which does not allow electron rates in a tagger channel ($\Delta$ E $\approx$ 2 MeV) larger than 10$^6$ s$^{-1}$. Since the electron rates are strongly increasing with decreasing energies of the bremsstrahlung photons the useful photon flux can be increased by switching off unused tagger channels at low photon energies. In the present experiment the orientation of the diamond crystal was chosen such that the position of the discontinuity at the high-energy side of the coherent-bremsstrahlung spectrum produced by the most prominent reciprocal lattice vector $[02\overline{2}]$ was located close to 360 MeV. This means that significant linear polarization appeared within the energy range between 200 MeV and 360 MeV. Therefore, the tagging channels corresponding to photon energies below 200 MeV were switched off for the purpose mentioned above. The setup of the experiment is shown in Fig. \[fig1\]. TAPS consists of 384 BaF$_2$ modules having 25 cm length and hexagonal cross section (inner diameter 5.9 cm), mounted in six blocks of 64 [@Krusche95; @hejny98]. Each module is equipped with its individual veto detector in front to identify incoming charged particles. The blocks were arranged in a horizontal plane in a distance of 57 cm from the target center. Their central axes were located in the scattering plane at polar angles of -149$^\circ$, -99$^\circ$, -49$^\circ$, 50$^\circ$, 100$^\circ$ and 151$^\circ$, respectively. Additional 120 BaF$_2$ modules were mounted in the forward direction symmetrically around the photon beam to build a hexagonal forward wall (FW) which covered a range of polar angles from 2$^\circ$ to 19$^\circ$. The FW modules are of the phoswich type with a plastic detector mounted directly onto the crystal. The light output from both parts is viewed by the same photomultiplier tube. This gives an excellent identification of charged particles in addition to that via the pulse-shape discrimination provided by BaF$_2$. The liquid helium target had a diameter of 3 cm and a length of 10 cm, corresponding to about 1.86$\times$10$^{23}$nuclei/cm$^2$. The target cell was made of Kapton and was mounted in a large (Ø90cm), evacuated scattering chamber of carbon fiber to minimize scattering, conversion and energy loss between target and detector. For automatic refilling, the Kapton cell was connected to a 5 liter $^4$He Dewar vessel contained in a lq. N$_2$ shield above the scattering chamber. Several layers of superisolation foil were used to protect the target from thermal radiation. With these precautions an uninterrupted measurement was possible for 9 h. Three different collimators having diameters of 3 mm, 4 mm and 6 mm were used to study the effect of collimation on the degree of linear polarization. In order to obtain approximately the same statistical precision in the three measurements, the total beam time of 125 h on the target was subdivided into three periods of 57, 43 and 25 h, respectively. Data analysis and results ========================= The two-photon decay was used to identify neutral pions. In the evaluation of data, events were selected which contained at least two coincident photons, being unambiguously identified using pulse-shape analysis, information from the charged-particle veto detectors, and time-of-flight analysis. As a first step to identify $\pi^\circ$ mesons from coincident (within a time window of 0.7 ns) photon pairs in two different TAPS blocks, spectra of invariant masses $m_{\gamma\gamma}$ were constructed where the invariant mass is given by $$m_{\gamma\gamma}=\sqrt{2E_1E_2(1-\cos\Phi)}, \label{meff}$$ with $E_1$ and $E_2$ being the photon energies and $\Phi$ the angle between the two photons. Since most of the two-photon events are due to $\pi^\circ$ or $\eta$ decays, the spectrum of invariant masses contains two pronounced peaks in addition to a small combinatoric background from events with more than two photons in the time window. These peaks are slightly asymmetric because of the asymmetric response function of the BaF$_2$ detectors [@Gabler93]. To take care of this, also the window of accepted events around the invariant mass of 135 MeV had to be chosen asymmetric, i. e. from 90 to 160 MeV. Because of the high angular resolution of TAPS the opening angle $\Phi$ between two decay photons has been used to separate coherent from incoherent $\pi^\circ$ photoproduction processes. For this purpose use was made of the relation $$\Phi_{min} = 2\arccos \left( \frac{\sqrt{(E_{\pi^{\circ}}^{lab})^2 - m_{\pi^{\circ}}^2}}{E_{\pi^{\circ}}^{lab}} \right) \label{Phimin}$$ between the minimum opening angle $\Phi_{min}$ which corresponds to the symmetric two-photon decay and the $\pi^{\circ}$ energy $E_{\pi^{\circ}}^{lab}$ in the laboratory. Since all incoherent processes lead to a breakup of the helium nucleus, the respective $\pi^\circ$ mesons have at least 20 MeV less energy than the coherently produced ones for a given and, in the following discussion, fixed primary photon energy. Correspondingly, the opening angles $\Phi$ from incoherent processes are shifted to distinctly higher values (typically by 3 times the angular resolution). This shift defines, via Eq. (\[Phimin\]), a range of opening angles $\Phi$ which only contains coherently produced $\pi^\circ$ mesons. For the verification of the effectiveness of the procedure of separating coherent from incoherent events described above, use has been made of an analysis of the “missing energy” of the event which makes an explicit use of the energy information provided by the tagger. Assuming coherent $\pi^\circ$ photoproduction the $cm$ energy of the $\pi^\circ$ meson can be calculated via $$E^*_{\pi^{\circ}}(E_\gamma)= \frac{s+m_{\pi^{\circ}}^2-m^2_{He}}{2\sqrt{s}} \label{miss1}$$ from the energy $E_{\gamma}$ of the primary photon, where ${\sqrt s}=\sqrt{2E_\gamma m_{He}+m^2_{He}}$ is the total energy in the $cm$ system and $m_{He}$ the mass of the He nucleus. On the other hand the same energy can be obtained via Lorentz transformation from the measured energies $E_1$ and $E_2$ and emission angles $\Theta_1$ and $\Theta_2$ of the decay photons in the laboratory system via $$E^*_{\pi^{\circ}}(\gamma_1,\gamma_2)= \gamma(E_1+E_2-\beta(E_1\cos\Theta_1+E_2\cos\Theta_2)), \label{miss2}$$ with $\beta=E_\gamma/(E_\gamma+m_{He})$ and $\gamma=\sqrt{1-\beta^2}$. The difference of the two energies $$\Delta E_{\pi^{\circ}} = E^*_{\pi^{\circ}}(\gamma_1,\gamma_2)-E^*_{\pi^{\circ}}(E_\gamma) \label{missing}$$ is the “missing energy” which should be zero in case of coherent $\pi^\circ$ photoproduction for an infinitely high precision of the experimental quantities. The verification of the effectiveness of the procedure of separating coherent from incoherent processes is illustrated in Fig. \[fig2\], where numbers of events are shown as a function of the missing energy $\Delta E_{\pi^{\circ}}$ for the four primary photon energies of $E_\gamma$= 224, 294, 320 and 366 MeV. The upper curves contain the numbers of events without cuts in the spectra of opening angles $\Phi$ as described in the preceding paragraph. These curves contain a peak structure around zero missing energy which is mainly due to coherent $\pi^\circ$ photoproduction and a tail at the negative-energy side which is due to incoherent $\pi^{\circ}$ photoproduction. The two lower curves are (i) experimental numbers of events after the cuts in the spectra of opening angles $\Phi$ are made, and (ii) completely analogously simulated [@Geant94] numbers of events where the simulation is carried out under the premise of coherent $\pi^\circ$ photoproduction. These two lower curves are adjusted to each other to give the same area. This, apparently, leads to a very good overall agreement of the two distributions, thus proving that indeed the respective experimental data are due to coherent $\pi^\circ$ photoproduction only. A further quantity which is obtained from the simulation [@Geant94] is the detection efficiency of the detector which relates numbers of events to differential cross sections. Using this detection efficiency the experimental differential cross sections shown in Figs. \[fig3\] and \[fig4\] and in Table \[sigdif\] are obtained. The errors given are statistical errors only. Table \[sigtot\] shows the total cross sections. The total systematic error of the cross section is estimated to be 7%. It stems from the following contributions:\ (i) The detection efficiency of the TAPS apparatus: 3%,\ (ii) the analysis cuts in the pulse-shape and time-of-flight spectra: 5 %,\ (iii) the tagging efficiency for coherent bremsstrahlung which is more strongly influenced by beam instabilities than the one for incoherent bremsstrahlung: 3%,\ (iv) dead-time corrections of the photon flux: 1.5 % , and\ (v) target thickness: 2%. As is illustrated by the reduction of the numbers of experimental events shown in the peaks around zero missing energy of Fig. \[fig2\], the cuts on the opening angle $\Phi$ have the unavoidable side effect of decreasing the acceptance of the TAPS detector. In the angular ranges between the TAPS blocks the acceptance almost drops to zero. This leads to gaps in the experimental differential cross sections close to the maxima of the angular distributions, as can be seen in part of the experimental data shown in Figs. \[fig3\] and \[fig4\]. Fortunately, the differential cross sections are smooth functions of polar angle so that the gaps may be safely bridged by interpolation. The data analysis described so far averages over the two directions (vertical and horizontal) of linear polarization so that differential cross sections for unpolarized photons are obtained. In a further step of analysis use has been made of the linear polarization provided by coherent bremsstrahlung and of the properties of coherent $\pi^\circ$ photoproduction on $^4$He as a polarimeter for measuring the degree of linear polarization on an absolute scale. These latter properties are provided by the fact that both the nucleus and the $\pi^\circ$ meson have no spins, so that $\pi^\circ$ mesons are emitted exclusively as $p$ waves through $M1$ excitation of the $\Delta$ resonance, a channel which is already strongly favored for the free nucleon. As a consequence, the degree of linear polarization of the photon beam is completely transferred to the azimuthal asymmetry of the $\pi^\circ$ mesons. In this case the general expression for the differential cross section in the $cm$ frame can be given by $$\frac{d\sigma}{d\Omega}=| (\vec{\epsilon} \times \hat{\vec{k}})\cdot \hat{\vec{q}}|^2 |F(\vec{q},\vec{k})|^2, \label{lin}$$ where $\vec{\epsilon}$ denotes the polarization vector, $\hat{\vec{k}}$ the direction of the photon and $\hat{\vec{q}}$ the direction of the pion. The function $F(\vec{q},\vec{k})$ is a polarization-independent normalization factor. The azimuthal acceptance of TAPS is limited to angular intervals of $\Delta\varphi^{max}=\pm 25^\circ$ width around $\varphi=0^\circ$ and $\varphi=180^\circ$, i.e., above and below the horizontal reaction plane on both sides of the photon beam. In order to make use of the full sizes of these intervals without losing information, the azimuthal distributions $N_V(\varphi)$ and $N_H(\varphi)$ of coherent $\pi^\circ$ events were determined separately for each tagger channel. These two quantities $N_V(\varphi)$ and $N_H(\varphi)$ are the numbers of events from coherent $\pi^\circ$ photoproduction as functions of the azimuthal angle $\varphi$ of the $\pi^\circ$ meson with the electric vector being preferentially vertical (vertical polarization V) and horizontal (horizontal polarization H), respectively, normalized to the same numbers of primary photons. The angular characteristic given by Eq. (\[lin\]) predicts that the numbers of events $N_V(\varphi)$ are larger than the numbers of events $N_H(\varphi)$ with the intensity ratio being determined by the degree of linear polarization (in case of complete linear polarization $N_H(\varphi)$ would be zero). The degree of linear polarization was evaluated via $$P(E_\gamma) = \frac{1}{\cos 2\varphi} \frac{N_V(\varphi) - N_H(\varphi)}{N_V(\varphi) + N_H(\varphi)}~~ \label{phi-dep}$$ using angular bins of $\Delta\varphi^{bin}= 3^\circ$ widths. As expected, Eq. (\[phi-dep\]) neither contains the efficiency of the detector nor the absolute value of the photon flux. This fact to a large extent removes the systematic errors discussed above in connection with the differential cross sections. The factor $1 / \cos{2\varphi}$ corrects for the azimuthal distribution of the $\pi^\circ$ mesons and is equal to 1 in case the $\pi^\circ$ meson is emitted exactly in the horizontal plane. The final result for the degree of linear polarization in each tagger channel is obtained by averaging the results obtained from the expression on the r.h.s. of Eq. (\[phi-dep\]) over the angular bins $\Delta\varphi^{bin}$ between $\Delta\varphi^{max}=- 25^\circ$ and $\Delta\varphi^{max}=+ 25^\circ$. Discussion ========== Cross sections for coherent $\pi^{\circ}$ photoproduction --------------------------------------------------------- The differential cross sections obtained in the present experiment for photon energies between 200 MeV and 400 MeV are compared with previous experimental data [@Sta69; @Lef70; @Tieger84; @Ananin85; @belli99] and with theoretical predictions [@drechsel99]. Apparently, the present data are of considerable superiority to the previous ones, both with respect to self-consistency and with respect to completeness. Where available, the BNL data [@belli99] are in a general reasonable agreement with the present ones except for some deviations especially at the highest energies accessible in [@belli99]. The experimental data are compared with predictions [@drechsel99] based on an extended version of the distorted wave impulse approximation (DWIA) [@Chu87; @Gmi87; @Kamalov85; @Kamalov97]. In this approach the elementary process is described by the recently developed unitary isobar model [@drechsel99a]. Medium effects are considered by introducing a phenomenological $\Delta$ self-energy. This allows to incorporate these effects in a self-consistent way with regard to both the “bare” $\gamma N\Delta$ vertices and $\Delta$ excitations due to pion rescattering. Without going into details here, the following points should be discussed: In Figs. \[fig3\] and \[fig4\] where the differential cross sections are shown, the dashed curves represent the results of the distorted wave impulse-approximation (DWIA). These predictions apparently are not in agreement with the experimental data especially for energies not too far away from the maximum of the $\Delta$ resonance. The inclusion of $\Delta$-nuclear interaction effects (or $\Delta$ self energy as introduced in [@drechsel99]) in the free $\Delta$ propagator leads to an almost perfect agreement between data and predictions showing that medium modifications of the main resonance characteristics (width and position) supposedly take care of that part of the reaction mechanism which is missing in the DWIA. To give a quantitative information about the $\Delta$-nuclear interaction we quote from ref. [@drechsel99] the $\Delta$ self-energy at E$_\gamma$ = 290 MeV, being V $= 19 - i 33$ MeV which corresponds to an increase of the $\Delta$ resonance mass and width by 19 and 66 MeV, respectively. This result is in reasonable agreement with the results obtained in pion-nuclear scattering. It is interesting to note that by using the same parameters for the $\Delta$ self energy also the data for coherent $\pi^\circ$ photoproduction on $^{12}$C [@Arends83; @Schmitz96] can be described. This supports our conclusion about the general validity of the theoretical approach suggested in Ref. [@drechsel99]. Fig. \[fig5\] shows the experimental total cross section for coherent $\pi^\circ$ photoproduction obtained in the present experiment together with predictions [@drechsel99]. As a dotted line this figure also contains the plane-wave impulse approximation (PWIA) which was omitted in Figs. \[fig3\] and \[fig4\] because of its large deviations from the experimental data. The comparison of the PWIA with the DWIA (dashed curve) shows that the distortion of the pion wave-function through a pion-nucleus optical potential takes care of the largest part of the discrepancy between experiment and prediction. The final adjustment, then, is achieved by the introduction of the medium effects in the $\Delta$ propagator [@drechsel99]. Degrees of linear polarization ------------------------------ The beam asymmetry measurements have been used to calculate the degree of linear polarization of collimated coherent bremsstrahlung. The experimental results are shown in Fig. \[fig6\] together with predictions, the latter ones calculated on an absolute scale by the method described in detail in [@rambo98]. Above 300 MeV, i.e. close to the maximum polarization, the calculation fits the measured data precisely, whereas below 300 MeV the calculation underestimates the measured degree of linear polarization by up to 0.03 (absolute value). This effect appears to increase with decreasing collimation angle and is especially pronounced for $\theta_{coll}=$ 0.6 mrad. Here we find that the predictions systematically underestimate the data in the whole range between 200 and 300 MeV. This effect had already been observed in our previous investigation [@kraus97] but was to some extent masked by the lower statistical precision achieved in that experiment for the smallest collimation angle. An elaborate investigation carried out in the present work showed that this systematic deviation cannot be explained by geometrical imperfections of the collimation system, i.e. it can neither be explained by a slightly decentral placement of the collimator nor by effects related with the beam properties. It is desirable to clear up the physical origin of the remaining discrepancies between predictions and experimental data for the degree of linear polarization. Nevertheless, from the point of view of the application of collimated coherent bremsstrahlung as a source of linearly polarized photons the achieved precision is quite satisfactory. The argument for this is that in the region of large linear polarization, which is the most important one for practical applications, the agreement between experiment and calculation is perfect. The measurement of linear polarization may be improved by on-line monitoring of the intensity spectrum of coherent bremsstrahlung and, furthermore, by developing reliable methods to convert the intensity spectrum directly into that of linear polarization. Summary and conclusion ====================== Differential cross sections for coherent $\pi^\circ$ photoproduction on $^4$He in the energy range between 200 MeV and 400 MeV were measured with very high statistical and systematic precisions. The experimental data are compared with predictions obtained on the basis of the DWIA. These predictions are in perfect agreement with the experimental data if appropriate medium modifications of the $\Delta(1232)$-resonance width and position are taken into account through a semiempirical procedure. 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Schmitz, Ph.D. thesis, University of Mainz, unpublished (1996). $\vartheta_{cm}$ 201-210 211-222 223-234 235-246 247-258 259-270 ------------------ ------------ ------------ ------------ ------------ ------------ ------------ 7 0.1 (0.1) 0.3 (0.1) 0.4 (0.1) 0.4 (0.2) 1.8 (0.3) 1.3 (0.3) 12 2.1 (0.1) 2.9 (0.1) 3.6 (0.2) 5.4 (0.3) 6.2 (0.3) 7.5 (0.3) 17 4.2 (0.2) 5.5 (0.2) 6.9 (0.2) 9.2 (0.3) 11.2 (0.3) 13.2 (0.3) 22 6.9 (0.2) 8.3 (0.2) 10.6 (0.2) 14.3 (0.3) 16.8 (0.3) 20.3 (0.3) 27 9.2 (0.3) 11.5 (0.3) 14.0 (0.3) 18.6 (0.4) 22.4 (0.4) 26.6 (0.4) 32 11.4 (0.3) 13.3 (0.3) 15.8 (0.4) 22.5 (0.4) 28.7 (0.4) 33.2 (0.4) 37 12.7 (0.4) 15.9 (0.4) 18.9 (0.4) 26.4 (0.5) 33.2 (0.5) 39.7 (0.6) 42 15.0 (0.4) 17.7 (0.4) 23.6 (0.5) 29.9 (0.6) 37.6 (0.7) 43.2 (0.7) 47 15.9 (0.4) – – – – – 62 – – 31.5 (0.7) 41.4 (0.8) 48.0 (0.8) 54.4 (0.8) 67 – 24.7 (0.7) 34.8 (0.6) 40.7 (0.5) 46.2 (0.5) 49.2 (0.5) 72 19.5 (0.6) 25.7 (0.6) 32.5 (0.5) 39.0 (0.5) 42.7 (0.4) 45.3 (0.4) 77 20.4 (0.5) 24.4 (0.4) 30.2 (0.4) 36.5 (0.4) 39.5 (0.4) 41.0 (0.4) 82 19.2 (0.4) 23.8 (0.4) 27.5 (0.4) 32.7 (0.4) 35.2 (0.4) 35.8 0.3) 87 17.6 (0.3) 20.1 (0.3) 24.9 (0.3) 29.1 (0.4) 30.8 (0.4) 30.3 (0.3) 92 17.0 (0.3) 19.1 (0.3) 22.0 (0.3) 26.0 (0.4) 27.4 (0.4) 26.6 (0.4) 97 14.9 (0.3) 16.7 (0.3) 19.3 (0.3) 21.9 (0.4) 22.5 (0.4) 21.7 (0.5) 102 13.3 (0.2) 14.1 (0.2) 15.6 (0.3) 17.1 (0.3) 16.4 (0.4) 15.4 (0.5) 107 11.9 (0.2) 12.7 (0.2) 13.0 (0.3) 14.3 (0.3) 12.6 (0.4) 12.0 (0.4) 112 10.3 (0.2) 11.1 (0.2) 11.8 (0.3) 11.9 (0.3) 10.2 (0.3) 8.9 (0.3) 117 8.7 (0.2) 8.8 (0.2) 9.2 (0.2) 9.6 (0.2) 8.2 (0.2) 6.3 (0.2) 122 7.0 (0.2) 7.8 (0.2) 7.4 (0.2) 7.0 (0.2) 6.6 (0.2) 4.8 (0.2) 127 6.1 (0.3) 6.0 (0.2) 6.1 (0.3) 6.4 (0.2) 5.0 (0.2) 3.6 (0.2) 132 5.1 (0.3) 5.1 (0.2) 5.5 (0.3) 4.9 (0.3) 3.7 (0.2) 2.5 (0.1) 137 3.3 (0.2) 3.9 (0.2) 3.1 (0.3) 3.2 (0.3) 2.9 (0.2) 1.9 (0.2) 142 3.1 (0.2) 3.5 (0.2) 2.8 (0.4) 0.8 (0.3) 1.0 (0.2) 0.5 (0.2) 147 2.8 (0.2) 2.3 (0.2) 1.5 (0.3) – – – : Unpolarized differential cross sections (in the $cm$ system) for coherent $\pi^{\circ}$ photoproduction on $^4$He in units of $\mu$b/sr. First column: $\vartheta_{cm}$ production angle in degrees. Following columns: cross sections with 1$\sigma$ statistical errors for the intervals of photon laboratory energy (in MeV) given on top of each column. \[sigdif\] $\vartheta_{cm}$ 271-282 283-294 295-308 309-318 319-330 331-342 ------------------ ------------ ------------ ------------ ------------ ------------ ------------ 7 2.2 (0.3) 3.4 (0.3) 2.9 (0.2) 4.5 (0.3) 4.2 (0.3) 2.5 (0.3) 12 8.8 (0.3) 10.3 (0.3) 10.3 (0.3) 11.9 (0.4) 11.6 (0.3) 10.9 (0.3) 17 15.5 (0.3) 16.7 (0.3) 19.0 (0.3) 20.2 (0.4) 20.5 (0.3) 20.5 (0.3) 22 23.0 (0.4) 24.7 (0.4) 27.3 (0.4) 29.2 (0.5) 29.3 (0.4) 30.7 (0.4) 27 30.0 (0.4) 32.6 (0.4) 35.6 (0.4) 38.4 (0.5) 37.9 (0.4) 37.4 (0.4) 32 36.9 (0.4) 40.3 (0.4) 42.7 (0.4) 46.3 (0.6) 44.6 (0.4) 43.8 (0.4) 37 44.1 (0.5) 45.9 (0.5) 49.3 (0.5) 50.5 (0.6) 50.0 (0.5) 47.8 (0.5) 42 47.7 (0.7) 50.6 (0.6) 53.0 (0.6) 54.4 (0.8) 51.9 (0.6) 50.0 (0.5) 47 52.2 (1.2) 52.6 (1.0) 54.2 (0.9) 53.5 (1.0) 51.7 (0.7) 48.5 (0.7) 51 51.4 (3.3) 49.2 (2.4) 55.4 (2.0) 55.8 (1.9) 49.6 (1.2) 45.9 (1.1) 58 50.0 (1.6) 51.3 (1.4) 51.2 (1.4) 52.3 (1.4) 48.8 (1.0) 40.7 (0.8) 62 52.9 (0.7) 49.7 (0.6) 50.7 (0.6) 44.9 (0.7) 41.0 (0.5) 33.4 (0.4) 67 49.3 (0.5) 46.2 (0.4) 44.2 (0.4) 39.0 (0.5) 33.9 (0.3) 28.3 (0.3) 72 44.0 (0.4) 41.0 (0.4) 37.8 (0.3) 32.9 (0.4) 27.9 (0.3) 22.0 (0.2) 77 39.4 (0.3) 34.4 (0.3) 31.0 (0.3) 25.6 (0.3) 20.8 (0.2) 16.4 (0.2) 82 33.9 (0.3) 29.3 (0.3) 24.7 (0.2) 20.6 (0.3) 15.5 (0.2) 12.1 (0.2) 87 28.2 (0.3) 23.9 (0.3) 19.0 (0.2) 15.0 (0.3) 10.9 (0.2) 8.8 (0.1) 92 24.3 (0.4) 19.0 (0.3) 14.4 (0.2) 10.8 (0.3) 7.8 (0.2) 6.0 (0.1) 97 20.1 (0.5) 17.0 (0.4) 10.8 (0.3) 8.0 (0.3) 5.2 (0.2) 4.3 (0.1) 101 12.1 (0.6) 10.8 (0.8) 7.1 (0.5) 4.9 (0.4) 3.7 (0.2) 2.1 (0.2) 107 7.5 (0.5) 4.7 (0.5) 1.6 (0.3) 3.0 (0.4) 1.9 (0.2) 1.4 (0.2) 112 6.4 (0.3) 4.3 (0.2) 2.6 (0.2) 2.1 (0.2) 1.7 (0.1) 1.6 (0.1) 117 4.1 (0.2) 2.7 (0.1) 2.0 (0.1) 1.8 (0.1) 1.5 (0.1) 1.3 (0.1) 122 3.2 (0.1) 2.0 (0.1) 1.6 (0.1) 1.3 (0.1) 1.4 (0.1) 1.3 (0.1) 127 2.2 (0.1) 1.3 (0.1) 1.3 (0.1) 1.3 (0.1) 1.2 (0.1) 1.2 (0.1) 132 1.4 (0.1) 0.9 (0.1) 1.0 (0.1) 1.1 (0.1) 1.1 (0.1) 1.0 (0.1) 137 1.0 (0.1) 0.8 (0.1) 0.9 (0.1) 1.1 (0.1) 1.1 (0.1) 0.9 (0.1) : (continued) $\vartheta_{cm}$ 343-355 356-366 367-378 379-390 391-401 ------------------ ------------ ------------ ------------ ------------ ------------ 8 2.4 (0.3) 2.0 (0.4) – 9.9 (3.5) – 12 9.1 (0.3) 10.7 (0.4) 5.2 (0.6) 7.4 (0.8) 8.4 (1.5) 17 19.9 (0.3) 21.4 (0.4) 18.2 (0.6) 18.0 (0.7) 17.0 (0.7) 22 28.9 (0.4) 30.9 (0.4) 27.4 (0.6) 26.7 (0.6) 24.2 (0.6) 27 36.1 (0.4) 37.7 (0.4) 34.1 (0.6) 31.2 (0.5) 29.2 (0.5) 32 40.3 (0.4) 43.3 (0.5) 37.2 (0.6) 34.3 (0.5) 32.3 (0.5) 37 45.1 (0.4) 47.0 (0.5) 40.2 (0.6) 35.7 (0.6) 33.1 (0.6) 42 45.9 (0.5) 46.5 (0.6) 38.3 (0.8) 35.7 (0.8) 31.1 (0.8) 47 43.1 (0.7) 43.8 (0.9) 35.6 (1.1) 27.0 (1.1) 28.7 (1.2) 52 37.9 (1.0) 36.8 (1.3) 27.7 (1.9) 27.4 (2.2) 20.3 (2.5) 58 36.4 (0.8) 32.4 (1.0) 20.9 (1.4) 17.1 (1.8) 15.0 (2.8) 62 29.0 (0.4) 27.1 (0.5) 19.4 (0.6) 15.5 (0.7) 12.6 (0.8) 67 23.5 (0.3) 21.9 (0.3) 16.6 (0.4) 12.0 (0.4) 10.3 (0.4) 72 18.4 (0.2) 16.7 (0.2) 12.0 (0.3) 9.9 (0.3) 7.4 (0.2) 77 13.3 (0.2) 11.8 (0.2) 8.5 (0.2) 6.7 (0.2) 5.4 (0.2) 82 9.1 (0.1) 7.9 (0.1) 5.9 (0.2) 4.7 (0.2) 3.7 (0.1) 87 6.6 (0.1) 5.9 (0.1) 4.1 (0.1) 3.0 (0.1) 2.6 (0.1) 92 4.6 (0.1) 4.0 (0.1) 2.7 (0.1) 2.2 (0.1) 1.5 (0.1) 97 3.0 (0.1) 2.8 (0.1) 1.4 (0.1) 1.0 (0.1) 0.8 (0.1) 102 2.1 (0.1) 1.9 (0.1) 0.4 (0.1) – – 107 0.2 (0.1) 0.8 (0.1) 0.3 (0.1) – – 112 1.3 (0.1) 1.0 (0.1) 0.8 (0.1) – – 117 1.1 (0.1) 1.2 (0.1) 0.4 (0.1) – – 122 1.1 (0.1) 1.0 (0.1) 0.7 (0.1) – – : (continued) $E_{\gamma}$ $\sigma_{tot}$ $E_{\gamma}$ $\sigma_{tot}$ -------------- ---------------- -------------- ---------------- 206 150.0 (0.8) 313 248.1 (0.9) 216 174.5 (0.8) 325 218.7 (0.6) 229 212.6 (0.9) 337 193.1 (0.6) 241 260.3 (1.0) 349 168.9 (0.6) 253 285.2 (1.0) 361 165.7 (0.7) 265 303.2 (1.0) 372 132.5 (0.8) 277 298.3 (0.9) 385 116.0 (0.8) 289 279.3 (0.8) 397 102.8 (0.8) 301 266.1 (0.8) : Unpolarized total cross section $\sigma_{tot}$ for coherent $\pi^{\circ}$ photoproduction on $^4$He in units of $\mu$b as a function of the photon laboratory energy $E_{\gamma}$ in MeV. The numbers in parentheses are 1$\sigma$ statistical errors. \[sigtot\] . \[fig5\]

--- abstract: 'This is a discussion of various concentration inequalities for U-statistics and the latest results. A special focus will be on providing proofs for bounds on the U-statistics using classical concentration inequalities, which, to our knowledge, are not in the literature – although the results are known.' address: | University of California, Berkeley\ Department of Statistics\ Berkeley CA 94709 author: - Yannik Pitcan title: 'A Note on Concentration Inequalities for U-Statistics' --- Introduction ============ U-statistics appear in statistics when producing minimum variance unbiased estimators. Naturally, they appear in clustering, ranking, and learning on graphs. In other words, any time we deal with exchangeable sequences, U-statistics are likely to come up. First, we begin with a definition of a U-statistic due to Joly and Lugosi (2015). Hoeffding (1948) was the first to introduce U-statistics in his paper “A class of statistics with asymptotically normal distributions.” \[U-Stat-defined\] Let $X$ be a random variable taking values in a measurable space $\mathcal{X}$ and let $h: \mathcal{X}^m\to \mathbb{R}$ be a measurable function of $m\geq 2$ variables. Let $P$ be the probability measure of $X$ and suppose we can access $n\geq m$ i.i.d. random variables $X_1,\ldots,X_n\sim X$. Then the U-statistics of order $m$ and kernel $h$ based on $\{X_i\}$ are defined as $$U_n(h)=\frac{(n-m)!}{n!} \sum_{(i_1,\ldots,i_m)\in I_n^m} h(X_{i_1},\ldots,X_{i_m}),$$ where $$I_n^m = \{(i_1,\ldots,i_m): 1\leq i_j\leq n,\, i_j\neq i_k \textrm{ if } j\neq k\}$$ is the set of all $m$-tuples of different integers between $1$ and $n$. Alternatively, if we restrict $I_n^m$ to be the set of increasing $m$-tuples of integers between $1$ and $n$, i.e. $i_j < i_k$ for $j<k$, then we have $$U_n(h)=\frac{1}{\binom{n}{m}} \sum_{(i_1,\ldots,i_m)\in I_n^m} h(X_{i_1},\ldots,X_{i_m}).$$ U-statistics are unbiased estimators of the mean $m_h = Eh(X_1,\ldots,X_m)$ and have minimal variance among all unbiased estimators. A few examples of U-statistics include the sample variance $s_n^2$, which is a U-statistic of order $2$ with kernel given by $h(x,y)=(1/2)(x-y)^2)$, and the third $k$-statistic, $ \frac{n}{(n-1)(n-2)} \sum_{i=1}^n (X_i - \bar{X}_n)^3$, which estimates the third cumulant. It is of interest to understand the concentration of a U-statistic around its expected value. We will discuss classical concentration bounds from decades ago and also put current results into context. Classical Bounds ================ First, we start by taking a look at some classical bounds that do not involve special conditions on the kernel. An inequality of Hoeffding (1963) established that $$P \left\{ \left| U_n(h)-m_h \right| > \left\Vert h\right\Vert _{\infty}\sqrt{\frac{\log(\frac{2}{\delta})}{2\left\lfloor n/m\right\rfloor }} \right\} \leq \delta$$ for $h$ bounded, for all $\delta>0$ and one can also derive $$P \left\{ \left| U_n(h)-m_h \right| > \sqrt{\frac{2\sigma^2 \log(\frac{2}{\delta})}{\left\lfloor n/m\right\rfloor }} \lor \frac{2\left\Vert h\right\Vert _{\infty} \log(\frac{2}{\delta})}{3\left\lfloor n/m\right\rfloor} \right\} \leq \delta,$$ where $\sigma^2$ is the variance of $h(\{X_i\})$. These bounds follow from applications of two classic inequalities. For our purposes, I will state the aforementioned inequalities in detail. The proofs can be seen in the book Concentration Inequalities by Lugosi, Massart, and Bousquet. **Hoeffding’s Lemma:** Let $X\in [a,b]$ with probability $1$ and $EX=\mu$. Then $${\mathrm{E}}[e^{s(X-\mu)}]\leq e^{s^2(b-a)^2/8}.$$ \[Hoeffding-Inequality\] **Hoeffding’s Inequality:** Let $X_1,\ldots,X_n$ be independent observations such that $E(X_i)=\mu$ and $X_i\in [a,b]$. Then, for any $\epsilon>0$, $$P(\left|\bar{X}_n - \mu\right|\geq \epsilon)\leq 2e^{-2n\epsilon^2/(b-a)^2}.$$ \[Hoeffding-Inequality-Corollary\] If $X_1,\ldots,X_n$ have the same properties as previously (independent, bounded in $[a,b]$, common mean $\mu$), then, with probability at least $1-\delta$, $$\left|\bar{X}_n - \mu\right|\leq \sqrt{\frac{(b-a)^2}{2n} \log\left(\frac{2}{\delta}\right)}$$ \[Bernstein-Inequality\] **Bernstein’s Inequality:** Let $X_1,\ldots,X_n$ be independent real-valued random variables with zero mean such that $\left|X_i\right|<c$ almost surely. Let $\sigma_i^2={\mathrm{Var}}(X_i)$ and $\sigma^2=\frac{1}{n}\sum_{i=1}^n \sigma_i^2.$ Then, $$P\{\bar{X}\geq \epsilon\}\leq \exp\left(-\frac{n \epsilon^2}{2(\sigma^2+c\epsilon/3)}\right).$$ **Bernstein’s Lemma:** Let $X$ be a random variable such that $EX=\mu$, ${\mathrm{Var}}(X)=\sigma^2$, and $\left|X-\mu\right|<c$ almost surely. Then $${\mathrm{E}}[e^{s(X-\mu)}]\leq \exp \left( \frac{\sigma^2}{c^2} (e^{sc}-1-sc)\right).$$ Proof of Hoeffding and Bernstein bounds for U-statistics ======================================================== We cannot immediately apply the above concentration inequalities to our U-statistics because there may be dependence between terms. Thus, in order to get concentration bounds on the U-statistics using these inequalities, we observe that, due to symmetry, we can rewrite U-statistics as sums of independent random variables. Let $k=\left\lfloor n/m\right\rfloor$ and $$V(X_1,\ldots,X_n)=\frac{1}{k} \left\{h(X_1,\ldots,X_m) + \cdots + h(X_{km-m+1},\ldots,X_{km}) \right\}.$$ One can see that $$U_n(h) = \frac{\sum_{\sigma\in S_n} V(X_{\sigma_1},\ldots,X_{\sigma_n})}{n!}$$ and each term on the right-hand side is a sum of $k$ independent random variables. Each of these can be bounded individually and then one may use the following lemma: Suppose $T$ is a random variable that can be written as $$T=\sum_{i=1}^{N} p_i T_i,\quad \textrm{where }\sum p_i=1, p_i\geq 0\quad \forall i$$ Then $P(T\geq t)\leq \sum_{i=1}^N p_i E[e^{s(T_i-t)}].$ Let $h>0$. The lemma follows from Markov’s inequality $$P(T\geq t)\leq e^{-st}E[e^{sT}]=E[e^{s(T-t)}]$$ and Jensen’s inequality. $$\exp(sT) = \exp\left(s\sum p_i T_i\right)\leq \sum p_i \exp(sT_i)$$ Combining these two, we get $P(T\geq t)\leq e^{-st}E[e^{sT}] \leq \sum p_i E[e^{s(T_i-t)}]$ Now we’ll finish the proof of the Hoeffding bound for U-statistics. In the following steps, $V$ is shorthand for $V(\{X_{\sigma(i)}\}_{i=1}^n)$. Denote $h_i$ to represent $h(X_{im-m+1},\ldots,X_{im})$ in the sum for $V(X_1,\ldots,X_n)$ for $i=1$ to $k$. Note that, in our case, $p_i=1/n!$ and $N=n!$, and using Hoeffding’s lemma, we have $$\begin{aligned} E[e^{s(V-m_h-t)}]&= {\mathrm{E}}\left[ \exp\left\{ s(\sum_{i=1}^{k}(\frac{h_{i}}{k}-\frac{m_{h}}{k}-\frac{t}{k}) \right\} \right]\\ &={\mathrm{E}}\left[ \exp\left\{s\left(\sum_{i=1}^{k}(\frac{h_{i}}{k}-\frac{m_{h}}{k})\right)\right\} \right] e^{-st}\leq [\exp\left\{s^{2}((2\|h\|_{\infty}/k)^{2}/8\right\}]^k e^{-st} \\ &=\exp\left\{s^2 \|h\|^2_{\infty}/2k - st\right\} \end{aligned}$$ and since this is independent of the permutation of $X_i$’s, we see that $$P(U_n-m_h\geq t)\leq (1/n!)\sum_{\sigma\in S_n} E[e^{s(V-m_h-t)}]=\exp\left\{\frac{s^2 \|h\|^2_{\infty}}{2k} - st\right\}$$ and minimizing the right-hand side over $s$, we see that it’s minimized when $s=kt/\left\|h\right\|_{\infty}$, and evaluating at this choice of $s$, we see that $$P(U_n-m_h\geq t)\leq \exp\left\{ - \frac{kt^2}{2\left\|h\right\|_{\infty}^2} \right\}$$ and thus $$P(\left|U_n-m_h\right|\geq t)\leq 2\exp\left\{ - \frac{t^2 \left\lfloor n/m\right\rfloor}{2\left\|h\right\|_{\infty}^2} \right\}$$ by union-bound. This completes the proof of the Hoeffding bound for U-statistics. We can use a similar approach to develop a bound involving the variance of $h(X_1,\ldots,X_m)$. For the Bernstein bound, we can repeat the same steps except now $$\begin{split} {\mathrm{E}}\left[ \exp\left\{s\left(\sum_{i=1}^{k}(\frac{h_{i}}{k}-\frac{m_{h}}{k})\right)\right\} \right]\leq \left[\exp\left\{\frac{\sigma^{2}}{\|h\|_{\infty}^{2}}\left(e^{\frac{s\|h\|_{\infty}}{k}}-1-\frac{s\|h\|_{\infty}}{k}\right)\right\}\right]^k \\ = \exp\left\{\frac{k\sigma^{2}}{\|h\|_{\infty}^{2}}\left(e^{\frac{s\|h\|_{\infty}}{k}}-1-\frac{s\|h\|_{\infty}}{k}\right)\right\} \end{split}$$ by letting $c=\left\|h\right\|_{\infty}/k$ and ${\mathrm{Var}}(\frac{h_i}{k}-\frac{m_h}{k})={\mathrm{Var}}(h_i/k)=\frac{\sigma^2}{k^2}$ in the Bernstein lemma. Then we once again find $s>0$ to minimize $$\frac{k\sigma^{2}}{\|h\|_{\infty}^{2}}\left(e^{\frac{s\|h\|_{\infty}}{k}}-1-\frac{s\|h\|_{\infty}}{k}\right) - st$$ . Setting $s=\frac{k}{\left\|h\right\|_{\infty}} \log(1+\frac{t\left\|h\right\|_{\infty}}{\sigma^2})$, we see that $$P(U_n-m_h\geq t)\leq \exp \left( \frac{-kt^2}{2(\sigma^2 + \frac{\left\|h\right\|_{\infty}t}{3})}\right)$$ and thus $$P(\left|U_n-m_h\right|\geq t)\leq 2 \exp \left( \frac{-kt^2}{2(\sigma^2 + \frac{\left\|h\right\|_{\infty}t}{3})}\right)$$ So far, all these results are general and don’t involve any assumptions about the kernel $h$ besides measurability. Discussion of Recent Work ========================= Under certain conditions, these concentration bounds can be tightened. Arcones and Giné required that the kernel was symmetric (invariant under permutation of the $X_i$’s) as well as bounded. In the presence of this and a degeneracy assumption, they were able to establish tighter bounds. A symmetric kernel $h$ is $P$-degenerate of order $q-1,\, 1<q\leq m$, if for all $x_1, \ldots, x_{q-1}\in \mathcal{X}$, $$\int h(x_1,\ldots,x_m)dP^{m-q+1}(x_q,\ldots,x_m) = \int h(x_1,\ldots,x_m)dP^{m}(x_1,\ldots,x_m)$$ and $$(x_1,\ldots,x_q)\mapsto \int f(x_1,\ldots,x_m)dP^{m-q}(x_{q+1},\ldots,x_m)$$ is not a constant function (Lugosi and Joly, 2015). When the kernel is $P$-degenerate with order $m-1$ and $m_h=E[h(X_1,\ldots,X_m)]=0$, then $h$ is considered to be $P$-canonical. The method used by Arcones and Giné to improve concentration guarantees in this scenario relied on decoupling arguments, where they replaced quadratic forms of random variables by bilinear forms. For example, a decoupling argument would be something along the lines of the following, taken from a monograph by Vershynin: Let $A$ be an $n\times n$ matrix with zero diagonal and $X=(X_1,\ldots,X_n)$ is a random vector with independent mean 0 coefficients. If $F$ is convex, then $${\mathrm{E}}F(\langle AX,X\rangle)\leq {\mathrm{E}}F(4\langle AX,X'\rangle)$$ where $X'$ is an independent copy of $X$. Arcones and Giné were able to show under the assumption that $h$ was canonical of $m$ variables, then there exist positive constants $c_1$ and $c_2$ depending only on $m$ such that for all $\delta\in (0,1)$, $$P\left\{ \left|U_{n}(h)-m_{h}\right|\geq c_{1}\left\Vert h\right\Vert _{\infty}\left(\frac{\log(\frac{c_{2}}{\delta})}{n}\right)^{m/2}\right\} \leq\delta,$$ and $$P\left\{ \left|U_{n}(h)-m_{h}\right|\geq\left(\frac{\sigma^{2}\log(\frac{c_{1}}{\delta})}{c_{2}n}\right)^{m/2}\lor\frac{\left\Vert h\right\Vert _{\infty}}{\sqrt{n}}\left(\frac{\log(\frac{c_{1}}{\delta})}{c_{2}}\right)^{(m+1)/2}\right\} \leq\delta.$$ If $m\geq 2$, then the first of the two proposed inequalities by Arcones and Giné provides a rate of convergence of $O_p(n^{-1})$, which is much faster than the $O_p(n^{-1/2})$ rate implied by the classical Hoeffding bound. In the case where the kernel had a unbounded but light-tailed distribution (subgaussian as an example), the above inequalities could be extended as done in a work by Giné et al. (2000). However, these results do not hold anymore if the kernel were heavy-tailed. To get around this roadblock, Nemirovsky and Yudin (1983) introduce a more robust median-of-means estimator that outperforms $U$-statistics for estimating the mean of the kernel function. What they did was take a confidence level $\delta\in (0,1)$ and divide the data into $V\approx \log \delta^{-1}$ blocks. Then they computed the empirical mean for each block and took the median of these empirical means, call this $\bar{\mu}$. In their paper, they show that $$\left\| \bar{\mu}-m_h \right\|\leq c\sqrt{{\mathrm{Var}}(X)}\sqrt{\frac{\log(1/\delta)}{n}}$$ with probability at least $1-\delta$ for a constant $c$. This is an improvement in that these guarantees would still hold even if $X$ were heavy-tailed. Joly and Lugosi use a median-of-means style estimator as well, except instead of taking the mean on each block, they calculated the empirical means over $m$-tuples of different blocks to compute (decoupled) U-statistics. Then they computed the medians over all these values. Notably, when the kernel was degenerate of order $m-1$, the rate of convergence of their estimator was $(\log (1/\delta)/n)^{m/2}$, which is similar to that of the U-statistics except without any assumption of boundedness. However, the tradeoff is degeneracy is required. Concluding Remarks ================== This is a relatively complete discussion and the first, to my knowledge, that explicitly provides derivations for standard concentration bounds on the U-statistics involving Hoeffding’s and Bernstein’s inequalities. These are well known to the statistics and machine learning communities, but it is difficult to find a proof that explains how to “create independence where there is not” before applying moment bounds – in our case we needed to use a representation of the U-statistics involving symmetric sums. Then we gave a summary of the tradeoffs in approaches of current work finding moment bounds of U-statistics, with emphasis on the techniques of decoupling and median-of-means estimators. [9]{} Arcones, M. A. and E. Giné (1993), *Limit theorems for U-processes.* The Annals of Probability 21, 1494-1542 St?phane Boucheron and G?bor Lugosi and Olivier Bousquet (2004), *Concentration inequalities.* Springer-Verlag. V. de la Pena, E. Gine (1999), *Decoupling. From dependence to independence. Randomly stopped processes. U-statistics and processes. Martingales and beyond.* Probability and its Applications (New York). Springer-Verlag, New York. W. Hoeffding (1963), *On Sequences of Sums of Independent Random Vectors* Journal of the American Statistical Association 58, 13-30 W. Hoeffding (1948), *A Class of statistics with asymptotically normal distribution* The Annals of Mathematical Statistics, 293-325 Joly, E. and Lugosi, G. (2015) *Robust estimation of U-statistics* Stochastic Processes and their Applications Volume 126, Issue 12, December 2016, Pages 3760-3773 Nemirovsky, A. and D. Yudin (1983), *Problem complexity and method efficiency in optimization* Vershynin, Roman (2011) A Simple Decoupling Inequality in Probability Theory

--- abstract: '> Artificial Intelligence (AI) has become an integral part of domains such as security, finance, healthcare, medicine, and criminal justice. Explaining the decisions of AI systems in human terms is a key challenge, as well as the potential implications on human interests, rights, and lives . While Explainable AI is an emerging field of research, there is no consensus on the definition, quantification, and formalization of explainability. In fact, the quantification of explainability is an open challenge. In our previous work, we incorporated domain knowledge for better explainability, however, we were unable to quantify the extent of explainability. In this work, we (1) briefly analyze the definitions of explainability from the perspective of different disciplines (e.g., psychology, social science), properties of explanation, explanation methods, and human-friendly explanations; and (2) propose and formulate an approach to quantify the extent of explainability. Our experimental result suggests a reasonable and model-agnostic way to quantify explainability.' author: - | Sheikh Rabiul Islam, William Eberle, and Sheikh K. Ghafoor\ Tennessee Technological University\ 1 William L Jones Dr, Cookeville, TN 38505\ bibliography: - 'bib.bib' title: Towards Quantification of Explainability in Explainable Artificial Intelligence Methods --- Introduction ============ The use of Artificial Intelligence (AI) has spread into a spectrum of high-stakes applications from different domains such as security, finance, healthcare, medicine, and criminal justice, impacting human interests, rights, and lives. Following the pace of the demand, the successful AI-based models have also become so complex that their decisions have become too complicated to express in human terms. This further complicates their adoption in many sensitive disciplines, raising concerns from the ethical, privacy, fairness, and transparency perspectives. The root cause of the problem is the lack of explainability and/or interpretability of the decision. We have seen the very confusing use of the terms ’explainability’ and ’interpretability’ throughout the literature. Some treat these as the same and stick to one, while some differentiate among the two, and others use them ambiguously. Research in Explainable Artificial Intelligence (XAI) is seeing a resurgence after three decades of slowed progress since some preliminary work on expert systems in the early 1980’s. Currently, there is no consensus on the definition these terms. In addition, the quantification of explainability is another open challenge, which will be difficult until there is a consensus on the concrete definition of those terms (e.g., explainability, interpretability). In our previous work [@islam2019infusing], we proposed an approach to incorporate domain knowledge in the model to make the prediction more explainable. However, we were unable to quantify the quality of explanations (i.e., validation of explainability). There has been some previous work [@molnar2019quantifying] that mentions three notions for quantificiaiton of explainability. The first two notions involve experimental studies with humans that mainly investigate whether a human can predict the outcome of the model [@dhurandhar2017tip], [@friedler2019assessing], [@huysmans2011empirical], [@poursabzi2018manipulating], [@zhou2018measuring]. However, the third notion (proxy tasks) does not involve a human. Instead they use known truth as a metric (e.g., the less the depth of the decision tree, the more explainable the model). In this work, our contributions are as follows: 1. We define explainability and interpretability from a multi-domain perspective and clarify their resemblance (Section \[sec:explanation\]). 2. We analyze the properties of explanation (Section \[subsec:properties\_explanation\]), explanation methods (Section \[subsec:properties\_explanation\_methods\]), and human friendly explanations (Section \[subsec:properties\_explanation\_human\]). 3. We present a potential way to formalize and quantify explainability with experimental results (Section \[sec:methods\] and \[sec:results\]). Explainability, and Interpretabiliy {#sec:explanation} =================================== [@miller2018explanation] argue that most of the work on XAI focuses on the researcher’s intuition of what constitutes a good explanation. However, there exists a vast area of research in philosophy, psychology, and cognitive science on how people generate, select, evaluate, and represent explanations and associated cognitive biases and social expectations towards the explanation process. Therefore, the author emphasizes that the research on Explainable AI should incorporate studies from these different domains. From the social science perspective, according to [@miller2018explanation] and [@lombrozo2006structure], explanation is both a process and product: - Cognitive explanation process: identifies the causes for an event, perhaps concerning particular counterfactual cases, and a subset of these causes is selected as the explanation. - Product: the explanation that results from the cognitive explanation process. - Social process: the process of transferring knowledge between explainer and explainee, generally an interaction between groups of people, in which the goal is to provide the explainee with enough information to understand the causes of the event. This definition of explanation insists that it is both a process and product (i.e., outcome), and focuses on understanding the causes of the event. Furthermore, according to psychologist [@lombrozo2006structure], ”Explanations are … the currency in which we exchanged belief”. This definition stresses the need for high fidelity for the explanation, in other words, gaining trust from the explanation recipient. Moreover, according to physicist [@deutsch1998fabric] who is the pioneer of quantum computation: *explanations* consist of *interpretations* of how the world works and why. This definition suggests that explanations are roughly equal to or a superset of interpretations. Interpretations are the building blocks of explanations; without interpretation, explanation is incomplete. Furthermore, the terms ”explainability” and ”interpretability” are simply the extent of explanation and interpretation accordingly. There are other mentionable definitions too. According to Miller (2017) [@miller2018explanation], ”interpretability is the degree to which a human can understand the cause of a decision”. Besides, according to Kim [@kim2017interpretability] ”Interpretability is the degree to which a human can consistently predict the model’s result”. Furthermore, according to [@ribeiro2016should] ”By ’explaining a prediction’, we mean presenting textual or visual artifacts that provide qualitative understanding of the relationship between the instance’s components (e.g. words in text, patches in an image) and the model’s prediction”. In other words, [@ribeiro2016should] put an emphasis on the qualitative understanding of the relationship between input and output (i.e., a selective/suitable set of information pieces that together can refer to a cause to an event, in contrast to the complete causal attributions) . To this end, in the case of an intelligent system (i.e., AI/ML-based system), it is evident that explainability is more than interpretability in terms of importance, completeness, and fidelity of prediction. Based on that, we choose to keep our focus entirely on explainability instead of interpretability. Finally, analyzing different definitions from the literature, we come up with following:\ **Definition**: *Explainability* of an AI model’s prediction is the extent of transferable *qualitative understanding* of the relationship between model input and prediction (i.e., selective/suitable causes of the event) in a recipient friendly manner.\ We proceed further based on this informal definition of explainability. Background {#sec:background} ========== There are two primary directions of research towards evaluation of explainability of an AI/ML model: (1) model complexity-based and (2) human study based. Model Complexity based Explainability Evaluation ------------------------------------------------ In the literature, model complexity and (lack of) model interpretability are often treated as the same [@molnar2019quantifying]. For instance, in [@furnkranz2012rule], [@yang2017scalable], model size is often used as a measure of interpretability (e.g., number of decision rules, depth of tree, number of non-zero coefficients). [@yang2017scalable] propose a scalable Bayesian Rule List (i.e., probabilistic rule list) consisting of a sequence of IF-THEN rules, identical to a decision list or one-sided decision tree. Unlike the decision tree that uses greedy splitting and pruning, their approach produces a highly sparse and accurate rule list with a balance between interpretability, accuracy, and computation speed. Similarly, the work of [@furnkranz2012rule] is also rule-based. They attempt to evaluate the quality of the rules from a rule learning algorithm by: the observed coverage, which is the number of positive examples covered by the rule, which should be maximized to explain the training data well; and consistency, which is the number of negative examples covered by the rule, which should be minimized to generalize well to unseen data. Furthermore, according to [@ruping2006learning], while the number of features and the size of the decision tree are directly related to the interpretability, the optimization of the tree size or features (i.e., feature selection) is costly as it requires generation of a large set of models and their elimination in subsequent steps. Besides, reducing the tree size (i.e., reducing complexity) increases error; however, they could not find a way to formulate the relation in simple functional form. More recently, [@molnar2019quantifying] attempt to quantify the complexity of the arbitrary machine learning model with a model agnostic measure. In that work, the author demonstrates that when the feature interaction (i.e., the correlation among features) increases, the quality of representations of explainability tools degrades. For instance, the explainability tool ALE Plot starts to show harsh lines (i.e., zigzag lines) as feature interaction increases. In other words, with more interaction comes a more combined influence in the prediction, induced from different correlated subsets of features (at least two), which ultimately makes it hard to understand the causal relationship between input and output, compared to an individual feature influence in the prediction. In fact, from our study of different explainability tools (e.g., LIME, SHAP, PDP), we have found that the correlation among features is a key stumbling block to represent feature contribution in a model agnostic way. Keeping the issue of feature interactions in mind, [@molnar2019quantifying] proposes a technique that uses three measures: number of features, interaction strength among features, and the main effect (excluding the interaction part) of features to measure the complexity of a post-hoc model for interpretation. Although, their work mainly focuses on model complexity for post-hoc models, that acted as a precursor to formulate our approach of explainability quantification. Our approach to quantify explainability is model agnostic and is for a model of any notion (e.g., pre-modeling, post-hoc). Human Study based Explainability Evaluation ------------------------------------------- The following work deals with the application- and human-level evaluation of explainability involving human studies. [@huysmans2011empirical] investigate the suitability of different alternative representation formats (e.g., decision tables, (binary) decision trees, propositional rules, and oblique rules) for classification tasks primarily focusing on the explainability of results rather than accuracy or precision. They discover that decision tables are the best in terms of accuracy, response time, confidence of answer, and ease of use. Furthermore, [@dhurandhar2017tip] argue that interpretability is not an absolute concept; rather, it is relative to the target model, and may or may not be relative to the human. Their finding suggests that a model is readily interpretable to a human when it uses no more than seven pieces of information [@miller1956magical]. Although, this might vary in task to task and person to person. For instance, a domain expert might consume a lot more detail information depending on their experience. The work of [@poursabzi2018manipulating] is a human-centered approach, focusing on previous work on human trust in a model from psychology, social science, machine learning, and human-computer interaction communities. In their experiment with human subjects, they vary factors (e.g., number of features, whether the model internals is clear or a black box) that makes a model more or less interpretable and measures how the variation impacts the prediction of human subjects. Their result suggests that participants who were shown a clear model with a small number of features were more successful in simulating the model’s predictions and trusted the model’s predictions. Furthermore, [@friedler2019assessing] investigate interpretability of a model based on two of its definitions: simulatability, which is a user’s ability to predict the output of a model on a given input; and ”what if” local explainability, which is a user’s ability to predict changes in prediction in response to changes in input, given the user has the knowledge of a model’s original prediction for the original input. They introduce a simple metric called runtime operation count that measures the interpretability, that is, the number of operations (e.g., the arithmetic operation for regression, the boolean operation for trees) needed in a user’s mind to interpret something. Their findings suggest that the interpretability decreases with an increase in the number of operations. Properties of Explanations and Explanation Methods {#sec:properties} ================================================== Although our main interest lies in explainable prediction, it depends on the model that generates the explanations. So, it is crucial to analyze the properties of both the explanation and the explanation method that generates the explanations. [@robnik2018perturbation] and [@molnar2018interpretable] attempt to define the properties of explanation methods and individual explanations, which we present in the following sections. Properties of Explanation Methods {#subsec:properties_explanation_methods} --------------------------------- Some properties of *explanation methods* include: - Expressive Power: Refers to the structure (e.g., decision trees, IF-THEN rules, weighted sum, and natural language) of the explanation method. - Translucency: The level in which the explanation method relies on looking into (e.g., coefficient for a linear regression, node splitting point in the tree-based approach) the ML model. For instance, explanation methods that rely on intrinsically interpretable methods like linear regression are highly translucent; however, counterfactual model agnostic explanation methods that leverage the changes in output in response to input for explanation have zero translucency as it does not look at model at all. The high translucency comes with the possible use of more information to generate the explanation. - Portability: The number of ML models that the explanation method covers. Usually low translucency (i.e., ML model usually remains black box) comes with more portability. - Algorithmic Complexity: The required time to generate the explanation. Properties of Individual Explanations {#subsec:properties_explanation} ------------------------------------- Some properties of *individual explanations* include: - Accuracy: The explanation needs to be accurate enough when fidelity is essential. Although, low accuracy of explanation might be a cause of the low accuracy of the ML/AI model. - Fidelity: How well the explanation approximates the prediction of the black-box model. Usually a highly accurate model with high fidelity leads to highly accurate explanations. Thus, fidelity and accuracy are related. - Consistency: Refers to the extent of consistency among explanations for different ML models on the same task. - Stability: Refers to the extent of similarity of explanations for similar instances. While consistency compares explanations between different models, stability compares the explanations of similar instances for a particular model. - Comprehensibility: The extent to which the recipient of explanation understands the explanation, which is very hard to measure. A few measures could be the number of features with non-zero weights in a linear model, or the number of rules in a tree. Usually a human can comprehend 7+-2 pieces of information at a time [@miller1956magical] - Certainty: Many ML models provide the probability of the target class. Similarly, explanations with a certainty value are expected to be useful. - Degree of Importance: How well the explanation covers the important features or how well the explanation reflects parts of the explanations. For example, from the decision rules, we can understand which of the rules are more important (e.g., important features are at the beginning of a ruleset, or the top of the tree). Recent laws [@goodman2016eu] [@algorithmic_accountability] focus on the impact of an algorithmic decision on human rights, interests, and lives. In that sense, we need a human-friendly explanation, which has slightly different properties than the explanation that does not consider humans as a key factor. Properties of Human-friendly Explanation {#subsec:properties_explanation_human} ---------------------------------------- According to [@miller2018explanation], humans usually prefer short explanations that contrast the current situation with a situation in which that event would not have occurred (a.k.a., counterfactual explanations). Sometimes abnormal causes provide good explanations. Besides, social context is also important for a good explanation as explanations are a social interaction among the explainer and the recipient of the explanation. Furthermore, the human friendly explanation does not consider all factors (i.e., selective in nature) for a particular prediction or behavior. However, if one needs to legally specify all influencing factors or need to debug the machine learning model, there is a need for a complete causal attribution [@molnar2018interpretable], which is out of the scope of human-friendly explanation. That is the reason behind mentioning the term *qualitative* instead of *quantitative* in our definition of explanation (see Definition of Explainability). In summary, according to [@vstrumbelj2011general], [@miller2018explanation] [@lipton1990contrastive], and [@molnar2018interpretable], human friendly explanations are: *contrastive*, *selected*, *social*, *abnormal behavior focused* *truthful*, *consistent*, and *general and probable*. Explainability Quantification Method {#sec:methods} ==================================== Usually, humans can relate and process 7+-2 pieces of information (i.e., cognitive chunks) to understand something [@miller1956magical]. For instance, suppose that, in the most generalized form, the quality of an explanation is dependent upon the number of cognitive chunks that the recipient has to relate to in order to understand an explanation (i.e., the less, the better). Lets assume, E [=]{} explainability; N~c~ [=]{} number of cognitive chunks; I [=]{} interaction; N~i~ [=]{} number of input cognitive chunks; and N~o~ [=]{} number cognitive chunks involved in the explanation representation (i.e., output cognitive chunks). $$\label{eq:formula1} E = \frac{1}{N_{c}}$$ However, sometimes, these cognitive chunks are correlated and their influence/contribution/abilities are not mutually exclusive. This interaction among cognitive chunks complicates the explainability. So we penalize Formula \[eq:formula1\] for having an interaction among cognitive chunks, resulting in Formula \[eq:formula2\]. $$\label{eq:formula2} E = \frac{1}{N_{c}} + (1 - I)$$ Where, the interaction *I* ranges in between 0 and 1, and the less the interaction, the better the explainability, so we take the complement of that. Formula \[eq:formula2\] is in a form that can be applied to any of the application, domain, or proxy level explanation evaluations described before. However, from the perspective of an application and domain level evaluation of explainability, to progress further, we need human studies that are out of the scope of this work. Instead, in this work, we focus on the proxy level evaluation of explainability that considers different properties of output representation (e.g., depth of decision tree, length of rule list) as a metric for evaluation. Furthermore, we need to breakdown the number of cognitive chunks more to get a better evaluation. Both the number of input cognitive chunks in the model and the number of output cognitive chunks involved in the representation of output are important to understand the causal relationship, which is vital for explanation. While the ideal explainability case would be when there is only one input and one output cognitive chunk (no chance of interaction), that is unusual in real-world situations. Following the segregation of input and output cognitive chunks, Formula \[eq:formula2\] can be re-written as Formulas \[eq:formula3\] and \[eq:formula4\]: $$\label{eq:formula3} E = \frac{1}{N_{c}} + (1 - I)$$ $$\label{eq:formula4} E = \frac{1}{N_{i}} + \frac{1}{N_{o}} + (1 - I)$$ where N~i~ refers to the number of input cognitive chunks and N~o~ refers to the number cognitive chunks involved in the explanation representation (i.e., output cognitive chunks). Usually, the more these cognitive chunks, the more complicated the explanation becomes. So, the ratio of the best case (i.e., one cognitive chunk) and observed case is added towards total explainability. Also, Formula \[eq:formula4\] has three predicates, which might have different influences on the quantification of explainability, and different domains might have different implications (e.g., accuracy vs explainability trade off). So, we add a weight term with each of the predicates, considering the weights are constant (e.g., .3333) by default, and their summation is equal to 1: w~1~ + w~2~ + w~3~ = 1. However, these weights can be set to a different distribution, perhaps dependant upon a particular domain (e.g., healthcare, finance). After the addition of the weight terms, Formula \[eq:formula4\] becomes Formula \[eq:formula5\]: $$\label{eq:formula5} E = \frac{w_{1}}{N_{i}} + \frac{w_{2}}{N_{o}} + w_{3} (1 - I)$$ Formula \[eq:formula5\] can then be used to quantify the explainability of the explanation method (i.e., global explainability). We can use Formula \[eq:formula5\] to also quantify the explainability of an instance level prediction (i.e., local explainability). In that case, the first predicate of Formula \[eq:formula5\] (including the weight term) remains the same (i.e., the same number of input chunks). However, predicate 2 and predicate 3 will be different from instance to instance as a different set of cognitive chunks with different interaction strengths might be involved in the representation of explanation for a particular instance as explanations are selective. Experimental Results {#sec:results} ==================== We provide a brief overview of the experimental settings of our previous work [@islam2019infusing] that we will then use for our proposed explainability quantification method. In this example, we incorporate domain knowledge in the model for mortgage bankruptcy prediction, and apply different supervised ML algorithms in three different ways: 1. Using *original features*: we use all original features that have a non-zero effect on prediction (features with zero effect were removed in a data pre-processing stage by observing the Random Forest’s feature importance). 2. Using *domain-related features*: we select a subset of features that match with extracted domain knowledge (e.g., 5 C’s of credit) from the domain. Traditionally, credit risk is assessed using the 5C’s of credit (character, capital, capacity, collateral, and cycle), a popular domain principle to for determining credit risk. We extract necessary domain knowledge and map to the 5 C’s of credit to get the domain features[@islam2019infusing]. 3. Using *newly constructed features*: we construct a very concise set of new features (i.e., five features) from the domain-related features using the quantitative measure of gain/compromises (i.e., the cumulative sum of related feature values times the correlation coefficient) associated with each element of the domain principle. Furthermore, we represent the predicted output as a composition of individual elements of the domain principle 5C’s of credit(i.e., *newly constructed features*) (Figure \[fig:breakdown\]). ![Breakdown of feature contributions for a random sample.[]{data-label="fig:breakdown"}](media/breakdown.jpeg){width=".99\linewidth"} One will notice that the representation (Figure \[fig:breakdown\]) of the prediction in terms of the *newly constructed features* provides better explainability as the final prediction is segregated into the individual influences of a very concise set of intuitive features (i.e., five compared to 30). However, there is no way to quantify the level of perceived explainability. To use our proposed formula in this work (Formula \[eq:formula5\]) to quantify explainability, we need to calculate the *interaction strength* (*I*) too. We measure the interaction strength among features using R’s iml package that uses the partial dependence of an individual features as the basis for calculating *interaction strength* (*I*). Original Domain Constructed ---------------------- ---------- -------- ------------- Input chunks (N~i~) 30 7 7 Output chunks (N~o~) 30 7 5 Int. Strength (I) 0.556 0.5233 0.5251 Explainability (E) 0.1701 0.2539 0.2723 : Comparison of explainability[]{data-label="tab:explainability"} ![Dispersion in performance[]{data-label="fig:original_vs_domain"}](media/original-domain.JPG){width=".99\linewidth"} ![Dispersion in performance[]{data-label="fig:original_vs_constructed"}](media/original-constructed.JPG){width=".99\linewidth"} Applying Formula \[eq:formula5\], on metadata (Table \[tab:explainability\]) of three different feature settings, we see that *newly constructed features* (5’C of credit) provide the best explainability score of .2723, which is an improvement of 60.14% compared to the 0.17 that we get using the *original features* (Table \[tab:explainability\]). In fact, even if we apply the state-of-the-art methods of post-hoc interpretability/explainability like SHAP, the explainability will be still limited to 0.17 as it does not reduce the number of cognitive chunks to represent output. Besides, using *domain related features*, the explainability score is .2538, which is better than using the original features, although a little worse than using the *newly constructed features*. Furthermore, this explainability gain comes with a negligible cost of performance (Figure \[fig:original\_vs\_domain\], Figure \[fig:original\_vs\_constructed\]) which is expected and a known trade off for explainability methods. In fact, for a few algorithms (e.g., Random Forest (RF), Gradient Boosting (GB)), *the newly constructed features*, and *domain related features* lead to better *recall*, which is crucial for anomaly detection where the target class instances are very few compared to non-target class instances. In that sense, besides a better explainability, the domain knowledge benefits us in terms of different performance (e.g., better recall, less features leads to less computation time) measures too. Conclusion ========== Explainable decisions from commercial AI systems are going to be a standard imposed by regulators to eliminate bias and discrimination, and ensure trust. Our work attempts to establish a concrete definition of explainability, properties of explanations and explanation methods, and a way to quantify explainability which is transferable to a variety of explainability methods or tools. For the quantification of explainability, we only present an approach for the proxy method. In the future, we would like to address human studies as an extension of this work.

--- abstract: | This work is concerned with the theory of graphical representation for the Ising and Potts models over general lattices with non-translation invariant external field. We explicitly describe in terms of the random-cluster representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-state Potts models with general external fields. We also consider the Gibbs states for the Edwards-Sokal representation of the Potts model with non-translation invariant magnetic field and prove a version of the FKG inequality for the so called general random-cluster model (GRC model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the almost sure quasilocality of the Gibbs measures for the GRC model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs measures for the ferromagnetic Ising model with a positive power-law decay magnetic field with small enough power, as conjectured in [@BCCP15]. author: - Leandro Cioletti and Roberto Vila title: Graphical Representations for Ising and Potts Models in General External Fields --- ------------------------------------------------------------------------ ------------------------------------------------------------------------ Introduction ============ Graphical representations are extremely useful tools for the study of phase transition in Equilibrium Statistical Mechanics. Fortuin and Kasteleyn [@FK72], marked the beginning of four decades of intense activity that produced a rather complete theory for translation invariant systems. These representations were successfully employed to obtain non-perturbative and deep results for Ising and Potts models on the hypercubic lattice using percolation-type methods, namely the discontinuity of the magnetization at the phase transition point for the one-dimensional Ising and Potts models with $1/r^2$ interactions [@ACCN88], the knowledge of the asymptotic behavior of the eigenvalues of the covariance matrix of the Potts model [@BC96], the Aizenman-Higuchi Theorem on the Choquet decomposition of the two-dimensional Ising and Potts models [@Aiz80; @CDIV14; @CV12; @GeoHig00; @Hig81] and the proof that the self-dual point on the square lattice $p_{sd}(q)=\sqrt{q}/(1+\sqrt{q})$ is the critical point for percolation in the random-cluster model ($q\geqslant 1$) [@beffara-duminil], see also the review [@vila]. For a detailed introduction to the random-cluster model we refer the reader to [@Dum15; @GHM01; @Grimmett2; @Hag98]. The relationship between graphical representations and phase transitions in Ising/Potts-type models is typically considered with respect to the random-cluster model (RC model) and in view of the Edwards-Sokal coupling [@ES]. Most papers employing such representations use spin models with null or translation invariant magnetic field, whereas we shall analyze graphical representations of the Ising and Potts models under arbitrary and non-translation invariant external fields, which is a significantly more complicated task for several reasons: when general boundary conditions are considered, the FKG property is harder to prove - as previously noticed by [@BBCK00], this property does not even hold for certain boundary conditions. In the absence of the magnetic field, phase transitions in the spin system can be directly detected by the random-cluster representation, but now this relationship is subtle since in some cases the phase transitions (in the percolation sense) in the random-cluster model does not correspond to a transition in the corresponding spin model. Such difficulties also appear in the analyses of Dobrushin-like states [@GG02], large $q$ order-disorder at the transition temperature [@CK03] and the effect of “weak boundary conditions” in the $q$-state Potts model [@BKM02]. Here the absence of symmetry brings questions regarding the color(s) of the infinite connected component(s), which need not be addressed in the case of null magnetic field, for instance. Furthermore, non-translation invariance causes many technical issues when using basic results from the classical theory of spin models and Ergodic Theory. To avoid confusion, on this paper the terms phase transition and critical inverse temperature shall hereby be solely employed to express changing in the number of the Gibbs measures when the temperature varies. This paper is motivated by some recent works on ferromagnetic Ising model in non-uniform external fields [@AGB07; @BCCP15; @BC10; @JS99; @NOZ99; @Navarrete]. Here, we are interested in developing the theory of graphical representation for non-translation invariant models whilst aiming for the problem of classifying which are the positive magnetic fields such that the ferromagnetic Ising model on the square lattice passes through a first order phase transition, in terms of its power law decay exponent. The formal Hamiltonian of this model is given by $$\label{hamiltoniano-Ising-formal-introducao} H(\sigma) = -J\sum_{ \{i,j\} } \sigma_i\sigma_j -\sum_{i} h_{i}\sigma_{i},$$ where the first sum ranges over the pairs of nearest neighbors. In this model, if the magnetic field $\pmb{h}=(h_i: i\in\mathbb{Z}^d)$ satisfies $\liminf h_i>0$, it has been proved [@BC10] that for any positive temperature the set of the Gibbs measures is a singleton, therefore for essentially bounded-from-below positive magnetic fields the conclusion is similar to the one obtained by Lee and Yang [@LY]. In [@BCCP15], the authors considered a positive, decreasing magnetic field and employed the Isoperimetric inequality and a Peierls-type argument to show that if the magnetic field is given by $h_i=h^{*}/|i|^{\alpha}$, where $h^*$ is a positive constant, then the model presents first order phase transition in every dimension $d\geqslant 2$, for any fixed exponent $\alpha>1$. On the other hand, if $\alpha<1$, they proved by means of a contour expansion that the uniqueness of the Gibbs measures holds at very low and by other methods at very high temperatures, and conjectured that the set of Gibbs measures at any positive temperature should be a singleton. The authors in [@BCCP15] justified why the extension of their results to any positive temperature is not obvious by resorting to most of the known techniques, but we prove as an application of the theory to be developed that the conjecture holds true. This is done by extending some results of the seminal work [@BBCK00] to the non-translation invariant setting. The paper is organized in three parts: the first part presents the relevant background material, including notation and the basic definitions of the models to be treated in subsequent parts. The second part is comprised of the theory on general finite graphs with free boundary conditions, the main results of which are the extension of the Edwards-Sokal coupling for general external fields and the explicit computation (in terms of the RC model) of the distribution function of a single spin of the Ising model with general external field and its expected values. These results are also generalized to the $q$-state Potts model in general external fields. The third part is concerned with the Potts, Edwards-Sokal and General random-cluster models in the non-translation invariant external fields setting with general boundary conditions. It is inspired by the reference [@BBCK00], but extends their results to non-translation invariant magnetic fields - a task that was occasionally non-trivial. In some cases, their results were essentially proved for very general fields and our work was simply to point out the necessary technical modifications. Fundamental results such as the FKG inequality required non-trivial adaptations and for this reason we presented its detailed proof for both free and wired boundary conditions in the so called general random-cluster model (GRC model) with non-translation invariant external field. Even with null external field, the random cluster measures lacked the key property of the quasilocality of the Gibbs measures, although it is possible to have the said property almost surely by assuming the uniqueness of the infinite connected component. For a null magnetic field on the hypercubic lattice, this fact was first proved in [@Pfister], however the geometry of the graph in this type of question is very important because for some non-amenable graphs such as regular trees even almost sure quasilocality fails, see [@FF; @Hag96]. For the random-cluster measures with translation invariant magnetic field, defined over amenable graphs, almost quasilocality was shown in [@BBCK00] for those measures having almost surely at most one infinite connected component. These results were recovered here for GRC models with non-translation invariant magnetic fields. The proofs of both the uniqueness of the infinite connected component and of the quasilocality of the Gibbs measures are given and new ideas are introduced to circumvent the lack of translation invariance. The conjecture stating the uniqueness of the Gibbs measures for the Ising model with power-law-decay magnetic field ($\alpha<1$) is proved in the last section of the third part. As a corollary of one of the main results of this part (Theorem \[teo-unicidade-gibbs-percolacao\]), we have obtained a characterization of the critical [^1] inverse temperature $\beta_c(\pmb{J},\pmb{h})$ of the ferromagnetic Ising model given by where $h_i=h^{*}/|i|^{\alpha}$, with $\alpha>1$ on the hypercubic lattice. Few facts are known about this inverse critical temperature. For example, in the positive external field case of the two-dimensional model with the coupling constant $\pmb{J}\equiv 1$ and $\sum_{i\in\mathbb{Z}^2}h_i<\infty$, it follows from the Onsager exact solution and a general result [@Georgii88] about summable perturbations of the Gibbs measures that $\beta_c(\pmb{J},\pmb{h})=\log(1+\sqrt{2})$. From [@BC10] it follows that $\beta_c(\pmb{J},\pmb{h}) = +\infty$ as long as $\liminf h_i>0$ in any dimension. The last section contains the proof that $\beta_c(\pmb{J},\pmb{h})$ is also trivial, i.e., $\beta_c(\pmb{J},\pmb{h}) = +\infty$ when $h_i=h^{*}/|i|^{\alpha}$, with $\alpha<1$. The most interesting cases are those where we do have phase transition and the magnetic field is given by $h_i=h^{*}/|i|^{\alpha}$, with $1<\alpha<2$ (not summable on entire lattice). For such cases, to the best of our knowledge, the only known fact about this critical point is that $\log(1+\sqrt{2})\leqslant \beta_c(1,\pmb{h})$, which is derived from the correlation inequalities. It is not known whether the Lieb-Simon inequality [@Sim; @Lieb], the Aizenman-Barsky-Fernández Theorem [@ABF] and other characterizations of the critical point (for example, [@DumTas15] )can be extended for the case $h_i=h^{*}/|i|^{\alpha}$, with $1<\alpha<2$. Background in graph theory ========================== We say that a graph $G=(V,E)$ is a countable graph if its vertex set $V$ is countable. As usual, a [**path**]{} $\gamma$ on $G$ is an alternating sequence of vertices and edges $\gamma=(v_0,e_1,v_1,e_2,\ldots,e_n,v_n)$, such that $v_i\neq v_j$ for all $0\leqslant i,j\leqslant n-1$, $v_n\in V\setminus\{v_1,v_2,\ldots,v_{n-1}\}$ and $e_j=\{v_{j-1},v_j\}$ for $1\leqslant j\leqslant n$. In case $v_0=v_n$ we say that $\gamma$ is a closed path or a circuit. The vertices $v_0$ and $v_n$ of $\gamma$ are called initial and final vertices, respectively. We say that $x,y\in V$ are [**connected**]{} if $x=y$ or there is a path $\gamma$ on $G$ so that $x=v_0$ and $y=v_n$, denoted $ x \longleftrightarrow_{\hspace*{-0.49cm}\scriptscriptstyle G}\ y $, and whenever it is clear from context, we shall remove the subscript $G$ from the notation. The length of a path $\gamma=(v_0,e_1,v_1,e_2,\ldots,e_n,v_n)$ is defined as $\vert \gamma \vert=n$. A graph $G$ is said to be a [**connected graph**]{} if any two vertices $i,j\in V$ are connected, otherwise we say that $G$ is disconnected. The connected component of $x\in V$ is the vertex set $C_x\equiv \{y\in V: y\leftrightarrow x\}$. The [**distance**]{} $d_G(x,y)$ between $x,y\in \mathbb{V}$ is defined by $d_G(x,y)=0$ if $x=y$; $d_G(x,y)=+\infty$ if $x\notin C_y$ and $ d_G(x,y) = \inf\{\vert \gamma \vert: \gamma\ \text{is a path connecting}\ x\ \text{to}\ y \} $, if $x\in C_y$. A graph $H=(\tilde{V},\tilde{E})$ is a subgraph of $G=(V,E)$ if $\tilde{V}\subset V$ and $\tilde{E}\subset E$. A subgraph $H$ of $G$ is an [**induced subgraph**]{} if it has the same vertex set as $G$ and a random subgraph of $G$ is an induced subgraph such that the edges are chosen randomly. Any infinite countable connected graph $\mathbb{L}\equiv (\mathbb{V}, \mathbb{E})$ will be called a lattice and from now on a finite subgraph of $\mathbb{L}$ will be denoted by $G=(V,E)$. The vertex set $V$ will sometimes be called [*the volume*]{} $V$. There are several definitions for the boundary of a vertex set $V$ contained in $\mathbb{L}$. In this work, the boundary is defined as follows. \[fronteira para vertices\] The boundary of an arbitrary vertex set $V$ in $\mathbb{L}$ is defined by $ \partial V \equiv \{i\in \mathbb{V}\setminus V: \text{d}_{\mathbb{L}}(i,V)=1\}, $ where $d_{\mathbb{L}}$ is the distance on the lattice $\mathbb{L}$. See Figure \[cfl\]. \[cfl\] The Ising model on countable graphs =================================== Let $\mathbb{L}=(\mathbb{V}, \mathbb{E})$ be an arbitrary lattice and $\Sigma$ the standard configuration space of the Ising model, i.e., $$\Sigma\equiv \big\{ \sigma = (\sigma_i: i\in \mathbb{V}): \sigma_i\in \{-1,+1\}, \ \forall i\in \mathbb{V} \big\} = \{-1,+1\}^\mathbb{V}.$$ This configuration space has a standard metric, for which the distance between any pair of configurations $\sigma,\omega\in\Omega$ is given by $$d(\sigma,\omega) = \frac{1}{2^{R}}, \ \text{and}\ R = \inf \left\{ r>0:\ \begin{array}{c} \sigma_i = \omega_i,\ \forall i\in B(o,r) \ \text{and} \\ \ \exists\ j\in\partial B(o,r)\ \text{such that}\ \sigma_j\neq \omega_j \end{array} \right\},$$ where $B({o,r})$ is the open ball in $\mathbb{L}$ of center $o\in \mathbb{V}$ (fixed) and radius $r>0$. Since the metric $d$ induces the product topology on $\Sigma$, it follows from Tychonoff’s Theorem that $(\Sigma,d)$ is a compact metric space. As a measure space, we always consider $\Sigma$ endowed with the Borel $\sigma$-algebra $\mathscr{B}(\Sigma)$, which is generated by the open sets on $(\Sigma,d)$. #### The Hamiltonian of the Ising model on a finite volume $V\subset \mathbb{L}$ with boundary condition $\mu\in\Sigma$ is given by $$\label{hamiltoniano-uni} \mathscr{H}^{\mu,\mathrm{{Ising}}}_{\pmb{h},V}(\sigma) \equiv - \sum_{ \substack{ i,j\in V \\ \{i,j\}\in \mathbb{E} } } J_{ij}\, \sigma_i \sigma_j - \sum_{i\in V} h_i\, \sigma_i - \!\!\!\! \sum_{ \substack{ i\in V, \ j\in \partial V \\ \{i,j\}\in \mathbb{E} } } \!\!\!\!\!\! J_{ij}\, \sigma_i \mu_j,$$ where the coupling constant $\pmb{J}\equiv (J_{ij}:{\{i,j\}\in \mathbb{E}})\in [0,+\infty)^{\mathbb{E}}$ satisfies the regularity condition $ \sum_{j\in \mathbb{V}} J_{ij} <+\infty, \ \forall i\in \mathbb{V} $ and the magnetic field is $ \pmb{h}\equiv (h_i:{i\in \mathbb{V}})\in \mathbb{R}^{\mathbb{V}} $. #### Gibbs measures. For any fixed finite volume $V$ and boundary condition $\mu$, we define the (finite) set of configurations compatible with $\mu$ outside $V$ as being the set of configurations $ \Sigma_{V}^{\mu} \equiv \{ \sigma\in \Sigma: \sigma_i = \mu_i \ \text{for} \ i\in \mathbb{V}\setminus V \}. $ The Gibbs measures of the Ising model on the finite volume $V$ with boundary condition $\mu$ at the inverse temperature $\beta>0$ is the probability measure $ \lambda^{\mu}_{\beta, \pmb{h},V}: \mathscr{B}(\Sigma)\to \mathbb{R} $ given by $$\begin{aligned} \lambda^{\mu}_{\beta, \pmb{h},V}(\{\sigma\}) = \left\{ \begin{array}{lll} \label{i1} {1\over \mathscr{Z}^{\mu,\mathrm{{Ising}}}_{\beta, \pmb{h},V}} \exp \big(-\beta \mathscr{H}_{\pmb{h},V}^{\mu,\mathrm{{Ising}}} (\sigma) \big), & \text{if} \ \sigma\in \Sigma_ V^{\mu} \\\\ 0, \ &\text{otherwise} \end{array} \right. \end{aligned}$$ where $\mathscr{Z}_{\beta, \pmb{h},V}^{\mu,\mathrm{{Ising}}}$ is a normalizing constant called the [**partition function**]{} given by $$\mathscr{Z}_{\beta, \pmb{h},V}^{\mu,\mathrm{{Ising}}} = \sum_{\sigma\in\Sigma_ V^{\mu}} \exp \big( -\beta \mathscr{H}_{\pmb{h},V}^{ \mu,\mathrm{{Ising}}}(\sigma) \big).$$ We denote by $\mathscr{G}^{\mathrm{{Ising}}}_{\beta}(\pmb{J},\pmb{h})$ the set of infinite-volume Gibbs measures which is given by the closure of the convex hull of the set of all the weak limits $ \lim_{V_n \uparrow \mathbb{V}} \lambda^{\mu}_{\beta, \pmb{h},V}, $ where $V_n\subset V_{n+1}$ and $\mu$ runs over all possible sequences of boundary conditions. #### The Ising model with free boundary condition. The Gibbs measure of the Ising model on a finite subgraph $G\subset \mathbb{L}$ with free boundary condition is given by $$\begin{aligned} \lambda_{\beta, \pmb{h},V}(\{\sigma\}) = {1\over \mathscr{Z}^{\mathrm{{Ising}}}_{\beta, \pmb{h},V}} \exp \big( -\beta \mathscr{H}_{\pmb{h},V}^{\mathrm{{Ising}}} (\sigma) \big),\end{aligned}$$ where $\mathscr{Z}_{\beta,\pmb{h},V}^{\mathrm{ Ising}}$ is the partition function and the Hamiltonian is given by $$\mathscr{H}_{\pmb{h},V}^{\mathrm{ Ising}} = - \sum_{ \{i,j\}\in E } J_{ij}\, \sigma_i \sigma_j - \sum_{i\in V} h_i\, \sigma_i.$$ The expected value of a random variable $f:\Sigma\to\mathbb{R}$, with respect to $\lambda^{\mu}_{\beta, \pmb{h},V}$ is given by $$\lambda^{\mu}_{\beta, \pmb{h},V}(f) \equiv \sum_{\sigma\in\Sigma^{\mu}_{V}} f(\sigma) \lambda^{\mu}_{\beta, \pmb{h},V}(\{\sigma\}).$$ The Potts model with inhomogeneous magnetic field {#sec-Potts-com-campo-cc-livre} ================================================= Let $q\in \mathbb{Z}^{+}$ be a fixed positive integer. The state space of the $q$-state Potts model on the lattice $\mathbb{L}$ is defined as $$\Sigma_{q}\equiv \big\{ \hat{\sigma}=(\hat{\sigma}_i: i\in \mathbb{V}): \hat{\sigma}_i\in \{1,2, \ldots, q\}, \ \forall i\in \mathbb{V} \big\} = \{1,2, \ldots, q\}^\mathbb{V}.$$ To define a $q$-state Potts model with magnetic field, we fix a family of coupling constants $ \pmb{J}\equiv \left(J_{ij}:{\{i,j\}\in \mathbb{E}}\right)\in [0,\infty)^{\mathbb{E}} $ and magnetic fields $ \pmb{\hat{h}}\equiv (h_{i,p}:i\in \mathbb{V}; p=1,\ldots,q)\in \mathbb{R}^{\mathbb{V}} \times \cdots\times \mathbb{R}^{\mathbb{V}} $. The Hamiltonian on a finite volume $G$ with boundary condition $\hat{\mu}\in\Sigma_{q}$ is given by $$\begin{aligned} \label{Potts-Geral} \mathscr{H}^{\hat{\mu},\mathrm{{Potts}}}_{\pmb{\hat{h}},q,V} (\hat{\sigma}) \equiv - \sum_{ \substack{ i,j\in V \\ \{i,j\}\in \mathbb{E} } } J_{ij}\delta_{\hat{\sigma}_i, \hat{\sigma}_j} - \sum_{p=1}^q \sum_{i\in V} {h_{i,p}\over q}\delta_{\hat{\sigma}_{i,p}} - \sum_{ \substack{ i\in V, \ j\in \partial V \\ \{i,j\}\in \mathbb{E} } } J_{ij} \delta_{\hat{\sigma}_i, \hat{\mu}_j},\end{aligned}$$ where $\delta_{\hat{\sigma}_i,\hat{\sigma}_j}$ is the Kronecker delta function. The Gibbs measure of Potts model on a finite volume $G$ with boundary condition $\hat{\mu}$ is defined analogously to the Ising model. We consider the set of all compatible configurations with the boundary condition $\hat{\mu}$, i.e., $ \Sigma_{q,V}^{\hat{\mu}} \equiv \{ \hat{\sigma}\in \Sigma_q: \hat{\sigma}_i = \hat{\mu}_i \ \text{for} \ i\in \mathbb{V}\setminus V \} $ and define the Gibbs measure on the volume $G$ with boundary condition $\hat{\mu}$ as the probability measure $\pi^{\hat{\mu}}_{\beta, \pmb{\hat{h}},q, V}$ on $\big(\Sigma_q, \mathscr{B}(\Sigma_q)\big)$ such that $$\pi^{\hat{\mu}}_{\beta, \pmb{\hat{h}},q,V}(\hat{\sigma}) = \left\{ \begin{array}{lll} { 1 \over \mathscr{Z}^{\hat{\mu},\mathrm{{Potts}}}_{\beta,\pmb{\hat{h}},q,V} } \exp \big( -\beta \mathscr{H}_{\pmb{\hat{h}},q,V}^{\hat{\mu},\mathrm{{Potts}}} (\hat{\sigma}) \big), & \text{if} \ \hat{\sigma}\in \Sigma_{q, V}^{\hat{\mu}} \\ \\ 0, \ &\text{otherwise} \end{array} \right.$$ where $ \mathscr{Z}_{\beta,\pmb{\hat{h}},q,V}^{\hat{\mu},\mathrm{{Potts}}} $ is the partition function. The free boundary condition case can be treated similarly to the previous section. The expected value of a random variable $f:\Sigma_{q}\to\mathbb{R}$ in this model is denoted by $ \pi^{\hat{\mu}}_{\beta, \pmb{\hat{h}},q,V}(f) $. The set of infinite-volume Gibbs measures is defined similarly to the previous section and denoted by $\mathscr{G}^{\mathrm{{Potts}}}_{\beta}(\pmb{J},\pmb{\hat{h}})$. In general, we use $\pmb{\hat{h}}$ to denote the magnetic field. In the special case where $q=2$ and the magnetic field satisfies $h_{i,1}=-h_{i,2}\equiv h_i$ we drop the hat from notation $\pmb{\hat{h}}$ and write the Hamiltonian, for example in the free boundary condition case, as follows $$\begin{aligned} \label{rescrever-Potts} \mathscr{H}^{\mathrm{{Potts}}}_{\pmb{h},2,V}(\hat{\sigma}) \equiv - \sum_{ \{i,j\}\in E } J_{ij}\delta_{\hat{\sigma}_i, \hat{\sigma}_j} - \sum_{i\in V} {h_i\over 2} ( \delta_{\hat{\sigma}_{i,1}} - \delta_{\hat{\sigma}_{i,2}} ). \end{aligned}$$ \[Medeq\] Fix a finite graph $G=(V,E)$ and assume that the magnetic field of the $2$-state Potts model satisfies $h_{i,1}=-h_{i,2}\equiv h_i$ for all $i\in V$. If $\hat{\sigma}\in \{1,2\}^V$ denotes the configuration obtained from $\sigma\in\{-1,1\}^V$ using the spins identification $-1\leftrightarrow 2$ and $1\leftrightarrow 1$, then we have for any $\beta>0$ that $$\lambda_{{\beta},\pmb{h},V}^{\mu}(\{\sigma\}) = \pi_{{2\beta},\pmb{h},2,V}^{\hat{\mu}}(\{\hat{\sigma}\}).$$ The random-cluster model with external field {#secao-def-modelo-RC} ============================================ This section is devoted to the $q=2$ inhomogeneous random-cluster models on a finite graph $G=(V,E)$. The general random-cluster model in external field will be introduced in the Section \[secao-potts-geral\], more precisely by the expression . The state space over which these models are defined is the cartesian product $\{0,1\}^{E}$. A generic element of this space will be denoted by $\omega$ and called an edge configuration. We say that an edge $e$ is open in the configuration $\omega$ if $\omega_e=1$, and we otherwise say $e$ is closed. Given $\omega\in \{0,1\}^{E}$, its set of open edges is denoted by $\eta(\omega)=\{e\in E: \omega_e=1\}$. We say that a path $\gamma:=(v_0,e_1,v_1,e_2,\ldots,e_n,v_n)$ on the graph $G$ is an open path on $\omega$ if all of its edges belong to $\eta(\omega)$, i.e., $\omega_{e_i}=1, \ \forall i=1,\ldots,n$. Two distinct vertices $x,y\in V$ are said to be connected in $\omega$ if there exists an open path $\gamma:=(v_0,e_1,v_1,e_2,\ldots,e_n,v_n)$ on this edge configuration such that $v_0=x$ and $v_n=y$. If $x,y\in V$ are connected on $\omega$, we write $x \leftrightarrow y$. A subgraph $H$ of $G$ is connected on $\omega$ if any pair of vertices of $H$ can be connected through a open path entirely contained in $H$. The open connected component of a vertex $x\in V$ is defined by $ C_x(\omega) \equiv \{y\in V: x\leftrightarrow y \ \text{in}\ \omega \} \cup\{x\}. $ The set $C_x(\omega)$ is called the open connected component of $x$ in the configuration $\omega$. To define the probability measure of the random-cluster model with external field we fix two families $ \pmb{p} \equiv (p_{ij}\in[0,1]:\{i,j\}\in E)\in [0,1]^{E} $ and $\pmb{h}\equiv (h_i:{i\in V})\in \mathbb{R}^{V}$. For convenience we will assume that the family $\pmb{p}$ is given by a family of coupling constants $\pmb{J}=(J_{ij}\in [0,+\infty]: \{i,j\}\in E )$ and the inverse temperature $\beta>0$ so that $p_{ij}= 1-\exp(-2\beta J_{ij})$. Following [@CMR98], the probability measure of the random-cluster model with external field $\pmb{h}$ on the finite volume $G$ is defined for each $\omega\in \{0,1\}^{E}$ by $$\phi_{\pmb{p},\pmb{h},G}(\omega) = { 1 \over \mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{h},G} } B_{\pmb{J}}(\omega) \displaystyle \prod_{\alpha =1}^{k({\omega,G})} 2\cosh \big(\pmb{h}(K_\alpha(\omega))\big),$$ where $\pmb{h}(K_\alpha(\omega))\equiv \beta {\sum_{i\in K_\alpha(\omega)}} h_i$, with the sets $K_1(\omega),\ldots, K_{k({\omega,G})}(\omega)$ being composed by the connected components of $(V,\eta(\omega))$, $B_{\pmb{J}}(\omega)$ representing the Bernoulli factors $$\begin{aligned} \label{fator de bernoulli} \displaystyle{ B_{\pmb{J}} (\omega) \equiv \prod_{\{i,j\}: \omega_{ij}=1} p_{ij} \prod_{\{i,j\}: \omega_{ij}=0} (1-p_{ij}) }\end{aligned}$$ and $\mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{h},G}$ being the partition function $$\mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{h},G} = \sum_{\omega\in \{0,1\}^{E}} B_{\pmb{J}}(\omega) \prod_{\alpha =1}^{k({\omega,G})} 2\cosh \big(\pmb{h}(K_\alpha(\omega))\big).$$ From now on, in order to ease the notation, we shall omit the $\omega$-dependence from the components $K_1(\omega),\ldots, K_{k({\omega,G})}(\omega)$ and simply write $K_1,\ldots, K_{k({\omega,G})}$. The Edwards-Sokal Coupling ========================== In this section we present the Edwards-Sokal model on a finite graph $G=(V,E)$. The configuration space of this model is given by the cartesian product $\{-1,+1\}^V\times\{0,1\}^E$. A pair of configurations $\sigma\in \{-1,+1\}^V$ and $\omega\in\{0,1\}^{E}$ are deemed consistent if $ \omega_{ij}=1 \Rightarrow \sigma_i=\sigma_j, \ \forall \ \{i,j\}\in E. $ The indicator function of the consistency of a pair $(\sigma,\omega)\in \{-1,+1\}^V \times\{0,1\}^{E}$, is denoted by $$\Delta(\sigma, \omega)\equiv \mathds{1}_{\{(\xi, \eta)\in \{-1,+1\}^V \times\{0,1\}^{E} :\ \text{if}\ \eta_{ij}=1 \ \text{then}\ \xi_i=\xi_j \}}(\sigma, \omega) .$$ \[f2\] Similarly to the previous section, we fix $\beta>0$, coupling constants $\pmb{J}$ and magnetic field $\pmb{h}$, and put $ p_{ij}\equiv 1-\exp(-2\beta J_{ij}). $ In the Edwards-Sokal model, the probability of a configuration $(\sigma, \omega)$ on a finite volume $G=(V,E)\subset \mathbb{L}$ is defined by $$\begin{aligned} {\nu}_{\pmb{p},\pmb{h},G}(\sigma, \omega) \equiv {1 \over \mathscr{Z}^{\mathrm{ ES}}_{\pmb{p},\pmb{h},G}} \displaystyle{ B_{\pmb{J}}(\omega) \Delta(\sigma, \omega) \times \exp \big( \beta \sum_{i\in V} h_i \left(\delta_{\sigma_i,1} -\delta_{\sigma_i,-1} \right) \big), }\end{aligned}$$ where $B_{\pmb{J}}$ represents the Bernoulli factors introduced in , $\delta_{\sigma_i,\sigma_j}$ is Kronecker’s delta function and $\mathscr{Z}^{\mathrm{ ES}}_{\pmb{p},\pmb{h},G}$ is the partition function. \[lema-fev1\] Let $G=(V,E)$ be a finite graph and consider the $2$-state Potts model on $G$ with free boundary condition and Hamiltonian given by . Suppose that $p_{ij}\equiv 1-\exp(-2\beta J_{ij})$ and $\hat{\sigma}$ is obtained from $\sigma$ as in the Proposition \[Medeq\]. Then $$\begin{aligned} &\exp\big( -2\beta (\mathscr{H}^{\mathrm{ Potts}} _{\pmb{h},2,V}(\hat{\sigma}) +\sum_{\{i,j\}\in E}J_{ij} ) \big)& \\ &=& \\ & \displaystyle \sum_{\omega} \big( \prod_{\{i,j\}: \omega_{ij}=1} p_{ij} \delta_{\sigma_i, \sigma_j} \prod_{\{i,j\}: \omega_{ij}=0} (1-p_{ij}) \big) \times {\textstyle \exp \big( \beta\sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) }. \end{aligned}$$ Using the relation between $\hat{\sigma}$ and $\sigma$, we first obtain the following equality $$\begin{array}{c} \exp \big( -2\beta (\mathscr{H}^{\mathrm{ Potts}} _{\pmb{h},2,V}(\hat{\sigma}) +\sum_{\{i,j\}\in E}J_{ij} ) \big) \\ = \\ \textstyle \exp \big( 2\beta (\sum_{\{i,j\}\in E}J_{ij} (\delta_{\sigma_i,\sigma_j}-1) + \sum_{i\in V} h_i {1\over 2} ( \delta_{\sigma_i,1} -\delta_{\sigma_i,-1} ) ) \big). \end{array}$$ By using that $p_{ij}=1-\exp(-2\beta J_{ij})$ and the elementary properties of the exponential, a straightforward computation shows that the above expression equals $$\prod_{\{i,j\}\in E} \big( p_{ij} \delta_{\sigma_i, \sigma_j} + (1-p_{ij}) \big) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1} ) \big) }.$$ By expanding the product in the above expression we get $$\begin{aligned} \sum_{E'\subset E} \!\! \big( \prod_{\{i,j\}\in E'} \!\!p_{ij} \delta_{\sigma_i, \sigma_j} \!\! \prod_{\{i,j\}\in E\setminus E'} \!\!\!\!\!(1-p_{ij}) \big) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } . \end{aligned}$$ Since the collection of all the induced subgraphs of $G$ is in bijection with $\{0,1\}^E$ we can rewrite the last expression as follows $$\begin{aligned} \sum_{\omega} \big( \prod_{\{i,j\}: \omega_{ij}=1} \!\! p_{ij} \delta_{\sigma_i, \sigma_j} \prod_{\{i,j\}: \omega_{ij}=0} \!\!\!(1-p_{ij}) \big) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \end{aligned}$$ which completes the proof. \[Lema5dez\] Under the hypothesis of Lemma \[lema-fev1\] we can show that there exists a constant $C\equiv C(\beta,G)>0$ so that $$\begin{aligned} \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},2,V} = C \mathscr{Z}^{\mathrm{{ES}}}_{\pmb{p},\pmb{h},G}. \end{aligned}$$ Observe that $$\begin{array}{rcl} \mathscr{Z}^{\mathrm{{Potts}} }_{2\beta, \pmb{h },2,V} &=& \displaystyle \sum_{\hat{\sigma}\in\{1,2\}^V} \exp \big( -2\beta \mathscr{H}_{\pmb{h},2,V}^{\mathrm{ Potts}} (\hat{\sigma}) \big) \\\\[0,2cm] &=& {\displaystyle { 1 \over C } \sum_{\hat{\sigma}\in\{1,2\}^V} } \exp \big( -2\beta (\mathscr{H}_{\pmb{h},2,V}^{\mathrm{ Potts}} (\hat{\sigma}) +\sum_{\{i,j\}\in E}J_{ij} ) \big), \end{array}$$ where $ C\equiv \exp ( 2\beta\sum_{\{i,j\}\in E}J_{ij} ) >0. $ From Lemma \[lema-fev1\] it follows that the right-hand-side above is equal to $$\begin{array}{c} \displaystyle C \sum_{\sigma, \omega} \! \big( \! \prod_{\{i,j\}: \omega_{ij}=1} p_{ij} \delta_{\sigma_i, \sigma_j} \prod_{\{i,j\}: \omega_{ij}=0} \!\!\!\!(1-p_{ij}) \big) \!\! \times \! {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \\ = \\[0,2cm] \displaystyle C \sum_{\sigma, \omega} \! \big( \! \prod_{\{i,j\}: \omega_{ij}=1} \!\!\! p_{ij} \prod_{\{i,j\}: \omega_{ij}=0} \!\!\!\!(1-p_{ij}) \big) \Delta(\sigma, \omega) \!\! \times \! {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \\ = \\[0,2cm] \displaystyle C \sum_{\sigma,\omega} B_{\pmb{J}}(\omega) \Delta(\sigma, \omega) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } = C \mathscr{Z}_{\pmb{p},\pmb{h},G}^{\mathrm{ ES}}. \qedhere \end{array}$$ \[Lema-cont\] Let $G$ be a finite graph and fix an edge configuration $\omega\in\{0,1\}^E$. If $\pmb{h}(K_\alpha)\equiv \beta {\sum_{i\in K_\alpha}} h_i$, where $K_1,\ldots, K_{k({\omega,G})}$ denote the connected components of $(V,\eta(\omega))$ we have $$\begin{aligned} \sum_{\sigma\in\{-1,+1\}^V} \Delta(\sigma, \omega) \times \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) = \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\textbf{h}(K_\alpha) \right). \end{aligned}$$ For a fixed $\omega$, if $K_1,\ldots,K_{k(\omega,G)}$ denotes the decomposition of $V$ on its connected components, we get $$\sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) = \sum_{\alpha =1}^{k(\omega,G)} \sum_{i\in K_{\alpha }} h_i ( \delta_{\sigma_i,1}-\delta_{\sigma_i,-1} ).$$ For each spin configuration $\sigma\in\{-1,1\}^{V}$ so that the pair $ (\sigma,\omega)\in \{-1,1\}^{V}\times\{0,1\}^{E} $ satisfies $\Delta(\sigma,\omega)=1$, we have that the value of all the spins in the same component has the same sign, see Figure \[f2\]. From the elementary properties of the exponential function we obtain the following equality $$\begin{array}{c} \Delta(\sigma, \omega) \times \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) \\[0.3cm] = \\ {\displaystyle \Delta(\sigma, \omega) \times \prod_{\alpha=1}^{k(\omega,G)} } \exp \big(\beta \sum_{i\in K_\alpha} h_i ( \delta_{\sigma_i,1}-\delta_{\sigma_i,-1} ) \big) . \end{array}$$ Since $V=\sqcup_{\alpha=1}^{k(\omega,G)}K_{\alpha}$, we have a natural bijection between the following spaces: $$\begin{aligned} \{-1,+1\}^V \quad \text{and} \quad \prod_{\alpha=1}^{k(\omega,G)}\{-1,1\}^{K_\alpha}. \end{aligned}$$ For sake of simplicity, we denote a generic element of the cartesian product $\prod_{\alpha=1}^{k(\omega,G)}\{-1,1\}^{K_\alpha}$ by $(\sigma_{K_1},\ldots,\sigma_{K_{k(\omega,G)}})$, where $\sigma_{K_j}\equiv({\sigma_i:i\in K_j}),$ $\, \forall j=1,\ldots, k(\omega,G).$ In this way we can simply write $ \sigma= (\sigma_{K_1},\ldots,\sigma_{K_{k(\omega,G)}}). $ By abusing the notation, we write $$\Delta(\sigma,\omega)= \prod_{\alpha=1}^{k(\omega,G)} \Delta(\sigma_{K_\alpha},\omega).$$ Using the previous observations and $ h_i ( \delta_{\sigma_i,1}-\delta_{\sigma_i,-1} ) = h_i\sigma_i $ we obtain $$\begin{gathered} \sum_{\sigma} \Delta(\sigma, \omega) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \\ = \sum_{(\sigma_{K_1},\ldots,\sigma_{K_{k(\omega,G)}})} \prod_{\alpha=1}^{k(\omega,G)} \Delta(\sigma_{K\alpha}, \omega) {\textstyle \exp \big( \beta \sum_{i\in K_\alpha} h_i \sigma_i \big) } \\[0,2cm] = \prod_{\alpha=1}^{k(\omega,G)} \sum_{\sigma_{K_\alpha}} \Delta(\sigma_{K\alpha}, \omega) {\textstyle \exp \big( \beta \sum_{i\in K_\alpha} h_i \sigma_i \big). } \end{gathered}$$ Because of the consistency condition, for each fixed $\alpha$, the sum appearing on the right-hand-side above has exactly two non zero terms where the spins in $K_{\alpha}$ take the values $\pm 1$. Therefore the above expression reduces to $$\begin{aligned} \prod_{\alpha=1}^{k(\omega,G)} 2\cosh \big( \beta \sum_{i\in K_\alpha} h_i \big), \end{aligned}$$ thus the lemma is proved. \[lema-fev2\] For any finite graph $G=(V,E)$, $\pmb{p}=(p_{ij}:i,j\in \mathbb{V})$, $\pmb{h}=(h_i:i\in V)$ and $\beta>0$ we have $$\mathscr{Z}_{\pmb{p},\pmb{h},G}^{\mathrm{ ES}} = \mathscr{Z}_{\pmb{p},\pmb{h},G}^{\mathrm{ RC}}.$$ The proof of this lemma is trivial given the above results. It is enough to change the sum order of the partition function of the Edwards-Sokal model, apply Lemma \[Lema-cont\] and finally use the definition of the partition function of the random-cluster model as we show below $$\begin{aligned} \mathscr{Z}_{\pmb{p},\pmb{h},G}^{\mathrm{ ES}} &=& \sum_\omega B_{\pmb{J}}(\omega) \sum_\sigma \Delta(\sigma, \omega) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \\\\[0,2cm] &=& \sum_\omega B_{\pmb{J}}(\omega) \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\textbf{h}(K_\alpha) \right) = \mathscr{Z}_{\pmb{p},\pmb{h},G}^{\mathrm{ RC}}. \end{aligned}$$ In the sequel we prove the main result of this section. The technicalities of the proof were worked out in the previous lemmas and now the remaining task is to simply piece them together. \[lemin\] Let $G=(V,E)$ be a finite graph, $\beta>0$, $\pmb{p}=(p_{ij}:i,j\in \mathbb{V})$ as above and $\pmb{h}=(h_i:i\in V)$ an external field. Then - $ \begin{aligned} \displaystyle{\sum_{\omega\in \{0,1\}^E} {\nu}_{\pmb{p},\pmb{h},G}(\sigma, \omega) =\lambda_{\beta,\pmb{h},V}(\sigma)} \quad\qquad\quad \ (\text{spin-marginal of} \ {\nu}_{\pmb{p},\pmb{h},G} ) \end{aligned} $ - $ \begin{aligned} \displaystyle{\sum_{\sigma\in\{-1,+1\}^V} {\nu}_{\pmb{p},\pmb{h},G}(\sigma, \omega) =\phi_{\pmb{p},\pmb{h},G}(\omega)}. \quad\qquad (\text{edge-marginal of} \ {\nu}_{\pmb{p},\pmb{h},G}) \end{aligned} $ We first prove *(1)*. Using the definition of Bernoulli factors $B_{\pmb{J}}$, and Lemma \[Lema5dez\] with $C\equiv\exp (2\beta\sum_{\{i,j\}\in E} J_{ij})>0$, we obtain $$\begin{gathered} \sum_{\omega} {\nu}_{\pmb{p},\pmb{h},G}(\sigma, \omega) \\[0,1cm] = { 1 \over \mathscr{Z}^{\mathrm{{ES}}}_{\pmb{p},\pmb{h},G} } \sum_{\omega} B_{\pmb{J}}(\omega) \Delta(\sigma, \omega) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) } \\[0,2cm] = { C \over \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},2,V} } \sum_{\omega} {\textstyle \big( \prod_{\{i,j\}: \omega_{ij}=1} p_{ij} \delta_{\sigma_i, \sigma_j} \prod_{\{i,j\}: \omega_{ij}=0} (1-p_{ij}) \big) }\times \\[0,2cm] \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) }. \end{gathered}$$ By applying Lemma \[lema-fev1\], it follows that the rhs above is equal to $$\begin{aligned} {C \over \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},V}} \exp {\textstyle \big( -2\beta ( \mathscr{H}_{\pmb{h},2,V}^{\mathrm{ Potts}} (\hat{\sigma}) + \sum_{\{i,j\}\in E}J_{ij} ) \big) } = \pi_{{2\beta},\pmb{h},2,V}(\hat{\sigma}) = \lambda_{\beta, \pmb{h}, V}(\sigma). \end{aligned}$$ To prove *(2)* it is enough to use Lemmas \[Lema-cont\] and \[lema-fev2\] as follows $$\begin{aligned} \label{importante_1} \sum_{\sigma} {\nu}_{\pmb{p},\pmb{h},G}(\sigma, \omega) &=& { B_{\pmb{J}}(\omega) \over \mathscr{Z}^{\mathrm{{ES}}}_{\pmb{p},\pmb{h},G} } \sum_{\sigma} \Delta(\sigma, \omega) \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) }\nonumber \\[0,2cm] &=& { 1 \over \mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\textbf{h},G} } \displaystyle{ B_{\pmb{J}}(\omega) \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } = \phi_{\pmb{p},\pmb{h},G}(\omega). \end{aligned}$$ \[medida-conditional1\] Let $\omega\in\{0,1\}^E$ be a fixed edge configuration. For each $\sigma\in\{-1,+1\}^V$ we have that $$\nu_{\pmb{p},\pmb{h},G}(\sigma\vert \omega)= { \Delta(\sigma, \omega) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \times {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big). }$$ From Proposition \[Medeq\] and Lemma \[lema-fev1\] we have, for any random variable $g:\{-1,1\}^V\to \mathbb{R}$, that $$\begin{gathered} \lambda_{\beta,\pmb{h},V}(g) = \sum_{\hat{\sigma}\in\{1,2\}^V}g(\hat{\sigma}) \pi_{{2\beta},\pmb{h},2,V}(\hat{\sigma}) \\[0,2cm] = { C \over \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},2,V} } \sum_{\sigma\in\{-1,+1\}^V} g(\sigma) \sum_{\omega\in\{0,1\}^E} \big( \prod_{\{i,j\}: \omega_{ij}=1} p_{ij} \delta_{\sigma_i, \sigma_j} \prod_{\{i,j\}: \omega_{ij}=0} (1-p_{ij}) \big)\times \\[0,2cm] \hspace*{5cm} \times \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big), \end{gathered}$$ where $ C \equiv \exp ( 2\beta\sum_{\{i,j\}\in E}J_{ij} ) > 0 $. By changing the order of the sums in the last expression we get $$\begin{aligned} { C \over \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},2,V} } \sum_{\omega\in\{0,1\}^E} B_{\pmb{J}}(\omega) \!\! \sum_{\sigma\in\{-1,+1\}^E} \!\!\!\!\!\! g(\sigma)\, \Delta(\sigma,\omega)\, {\textstyle \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) }. \end{aligned}$$ According to Lemmas \[Lema5dez\] and \[lema-fev2\], we have $C^{-1} \mathscr{Z}^{\mathrm{{Potts}}}_{2\beta,\pmb{h},2,V} = \mathscr{Z}^{\mathrm{{ES}}}_{\pmb{p},\pmb{h},G} = \mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{h},G} $. By introducing the product appearing in the definition of the random-cluster model, we can see that the above expression is equal to $$\begin{gathered} \sum_{\omega} { B_{\pmb{J}}(\omega) \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) \over \mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{h},G} }\times \\ \times\sum_{\sigma} g(\sigma) { \Delta(\sigma,\omega) \times \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \\ \\[0,2cm] \!\!\!\! = \sum_{\omega,\sigma} g(\sigma) { \Delta(\sigma,\omega) \times \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) }\phi_{\pmb{p},\pmb{h},G}(\omega). \end{gathered}$$ Therefore $$\begin{aligned} \lambda_{\beta,\pmb{h},V}(g)= \sum_{\omega} \left[ \sum_\sigma g(\sigma) { \Delta(\sigma,\omega) \exp \big( \beta \sum_{i\in V} h_i (\delta_{\sigma_i,1}-\delta_{\sigma_i,-1}) \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega). \end{aligned}$$ On the other hand, we get from Theorem \[lemin\] that $$\begin{aligned} \label{mart2} \lambda_{\beta,\pmb{h},V}(g) = \sum_{\omega,\sigma} g(\sigma) \nu_{\pmb{p},\pmb{h},G}(\sigma,\omega) \nonumber = \sum_\omega \Big[ \sum_\sigma g(\sigma) \nu_{\pmb{p},\pmb{h},G}(\sigma\vert\omega) \Big] \phi_{\pmb{p},\pmb{h},G}(\omega). \end{aligned}$$ The proof is completed upon comparison of the two previous expressions. Two-point function ================== The [**two-point function**]{} of the $q$-state Potts model is defined by $$\tau_{\beta,\pmb{\hat{h}},q,V}(x,y) \equiv \pi_{\beta,\pmb{\hat{h}},q,V} (\hat{\sigma}_x=\hat{\sigma}_y) - {1\over q}.$$ The term $1/q$ represents the probability that two independent spins uniformly chosen have the same value. In the random-cluster model, the [**connectivity function**]{} plays the role of the two-point function of the Potts model. This function is precisely the probability, with respect to $\phi_{\pmb{p},\pmb{h},G}$, that $x$ and $y$ are in the same connected component, notation $ \phi_{\pmb{p},\pmb{h},G}(x\leftrightarrow y). $ \[lema28fev\] Let $G=(V,E)$ be a finite graph and $x,y\in V$ two distinct vertices. Fix an edge configuration $\omega\in\{0,1\}^{E}$. If $x\not\leftrightarrow y$ in $\omega$, then $$\begin{gathered} \sum_{\sigma\in\{-1,+1\}^V} \mathds{1}_{\{\sigma_x=\sigma_y\}}\Delta(\sigma,\omega) {\textstyle \exp \big( \beta\sum_{i\in V}h_i\sigma_i \big) } \\ = 2\cosh\left(\pmb{h}(K_t)+\pmb{h}(K_u)\right) \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} 2\cosh\big(\pmb{h}(K_\alpha)\big), \end{gathered}$$ where $K_t\equiv K_t(\omega)$ and $K_u\equiv K_u(\omega)$ are two disjoint connected components containing the vertices $x$ and $y$, respectively. The basic ideas used to prove this lemma are the same we employed to prove Lemma \[Lema-cont\], which we once more present for sake of completeness. Let $V = \sqcup_{\alpha=1}^{k(\omega,G)}K_{\alpha}$ be a decomposition in terms of the connected components of the graph $(V,\eta(\omega))$. We recall that $ \{-1,+1\}^V \cong \prod_{\alpha=1}^{k(\omega,G)}\{-1,1\}^{K_\alpha} $ and its elements are denoted by $(\sigma_{K_1},\ldots,\sigma_{K_{k(\omega,G)}})$, with $\sigma_{K_j}\equiv({\sigma_i:i\in K_j}),$ $\ \forall j=1,\ldots, k(\omega,G).$ We also use the natural identification $ \sigma = (\sigma_{K_1},\ldots,\sigma_{K_{k(\omega,G)}}). $ Suppose that $x\not\leftrightarrow y$ in $\omega$. Denote $K_t$ and $K_u$ the components containing the vertices $x$ and $y$, respectively. Taking into account the decomposition of $V$ mentioned above, we have $$\begin{gathered} \sum_{\sigma} \mathds{1}_{\{\sigma_x=\sigma_y\}} \Delta(\sigma,\omega) {\textstyle \exp \big( \beta\sum_{i\in V}h_i\sigma_i \big) } \\ = \sum_{\sigma} \mathds{1}_{\{\sigma_x=\sigma_y=+1\}} \Delta(\sigma,\omega) \prod_{\alpha=1}^{k(\omega,G)} {\textstyle \exp \big( \beta\sum_{i\in K_\alpha}h_i\sigma_i \big) } \\ \hspace*{4cm}+ \sum_{\sigma} \mathds{1}_{\{\sigma_x=\sigma_y=-1\}} \Delta(\sigma,\omega) \prod_{\alpha=1}^{k(\omega,G)} {\textstyle \exp \big( \beta\sum_{i\in K_\alpha}h_i\sigma_i \big). } \end{gathered}$$ As previously observed, $ \Delta(\sigma,\omega) = \prod_{\alpha=1}^{k(\omega,G)} \Delta(\sigma_{K_\alpha},\omega) $, so from a simple computation we get that the expression above is equal to $$\begin{aligned} \label{bld} 2\cosh\big(\pmb{h}(K_t)+\pmb{h}(K_u)\big) \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} \sum_{\sigma_{K_\alpha}} \Delta(\sigma_{K_\alpha},\omega) \exp \big( \beta\sum_{i\in K_\alpha}h_i\sigma_i \big). \end{aligned}$$ Because of the consistency condition, the sums above over $\sigma_{K_\alpha}$ have actually two non-null terms. In each of such term the value of the spins is constant and therefore the product simplifies to $$\begin{gathered} \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} \sum_{\sigma_{K_\alpha}} \Delta(\sigma_{K_\alpha},\omega) {\textstyle \exp \big( \beta\sum_{i\in K_\alpha}h_i\sigma_i \big) } \\ = \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} \big( \exp(\pmb{h}(K_\alpha)) + \exp(-\pmb{h}(K_\alpha)) \big) \nonumber = \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} 2\cosh\big( \pmb{h}(K_\alpha)\big). \end{gathered}$$ Finally, by replacing the last expression in , we end the proof. Lemma \[lema28fev\] is vital for the most important result of this section, which is the next theorem. We state below the theorem for the $2$-state Potts model, but in fact the theorem is valid for much more general Potts models. The general case is treated in the last section. \[Correlação/quase-conectividade\] Let $G=(V,E)$ be a finite graph and $x,y$ two distinct vertices in $V$. Then $$\begin{gathered} \hspace*{-0.32cm}\tau_{2\beta,\pmb{h},2,V}(x,y) \! = \! {1\over 2} \phi_{\pmb{p},\pmb{h},G}(x\leftrightarrow y) + {1\over 2} \phi_{\pmb{p},\pmb{h},G} \big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot \tanh\left(\pmb{h}(K_t)\right) \cdot \tanh\left(\pmb{h}(K_u)\right) \!\big), \end{gathered}$$ where $K_t\equiv K_t(\omega)$ e $K_u\equiv K_u(\omega)$ are two disjoint connected components containing the vertices $x$ and $y$, respectively. By using the definition of the two-point function and Theorem \[lemin\] we get $$\begin{aligned} \tau_{2\beta,\pmb{h},2,V}(x,y) &=& \pi_{2\beta,\pmb{h},2,V}(\hat{\sigma}_x=\hat{\sigma}_y) - {1\over 2} \nonumber \\[0,2cm] &=& \sum_{\hat{\sigma}\in\{1,2\}^V} \Big(\mathds{1}_{\{\hat{\sigma}_x=\hat{\sigma}_y\}} - {1\over 2}\Big) \pi_{2\beta,\pmb{h},2,V}(\hat{\sigma}) \\[0,2cm] &=& \sum_{(\sigma,\omega)\in\{-1,+1\}^V\times\{0,1\}^E} \Big(\mathds{1}_{\{\sigma_x=\sigma_y\}}-{1\over 2}\Big) \nu_{\beta,\pmb{h},G}(\sigma,\omega) \nonumber \\[0,2cm] &=& \sum_{\omega\in\{0,1\}^E} \Big[ \sum_{\sigma\in\{-1,+1\}^V} \Big(\mathds{1}_{\{\sigma_x=\sigma_y\}}-{1\over 2}\Big) \nu_{\beta,\pmb{h},G}(\sigma\vert \omega) \Big] \phi_{\pmb{p},\pmb{h},G}(\omega).\nonumber \end{aligned}$$ Since $ \delta_{\sigma_i,\sigma_j} = {1\over 2}(1+\sigma_i\sigma_j) $, it follows from Corollary \[medida-conditional1\] that the rhs above is $$\begin{gathered} \label{I-1+I-2} = \sum_{\omega} \left[ \sum_{\sigma} \Big(\mathds{1}_{\{\sigma_x=\sigma_y\}}-{1\over 2}\Big) { \Delta(\sigma, \omega) \exp \big( \beta \sum_{i\in V} h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega)\nonumber \\[0,3cm] = \sum_{\omega} \mathds{1}_{\{x\leftrightarrow y\}}(\omega) \left[ \sum_{\sigma} \Big(\mathds{1}_{\{\sigma_x=\sigma_y\}}-{1\over 2}\Big) { \Delta(\sigma, \omega) \exp \big( \beta \sum_{i\in V} h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega)\nonumber \\[0,3cm] + \sum_{\omega} \mathds{1}_{\{x\not\leftrightarrow y\}}(\omega) \left[ \sum_{\sigma} \Big(\mathds{1}_{\{\sigma_x=\sigma_y\}}-{1\over 2}\Big) { \Delta(\sigma, \omega) \exp \big( \beta \sum_{i\in V} h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega)\nonumber \\[0,3cm] \equiv I_1+I_2. \end{gathered}$$ Notice that, as long as $x\leftrightarrow y$ in $\omega$ and the pair $(\sigma,\omega)$ is consistent, then $\sigma_x=\sigma_y$. From this observation and Lemma \[Lema-cont\], it follows that $$\begin{aligned} I_1={1\over 2}\phi_{\pmb{p},\pmb{h},G}(x\leftrightarrow y). \end{aligned}$$ On the other hand, applying Lemma \[Lema-cont\] again yields $$\begin{aligned} \label{I-2} I_2&=& -{1\over 2} \phi_{\pmb{p},\pmb{h},G}(x\not\leftrightarrow y) \nonumber \\[0,2cm] &&\hspace*{0,2cm} + \sum_{\omega} \mathds{1}_{\{x\not\leftrightarrow y\}}(\omega) \left[ \sum_{\sigma} \mathds{1}_{\{\sigma_x=\sigma_y\}} { \Delta(\sigma, \omega) \exp \big( \beta \sum_{i\in V} h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega)\nonumber \\[0,2cm] &\equiv& -{1\over 2}\phi_{\pmb{p},\pmb{h},G}(x\not\leftrightarrow y) +\tilde{I}_2. \end{aligned}$$ Now we work on $\tilde{I}_2$. By using Lemma \[lema28fev\], we have $$\begin{gathered} \tilde{I}_2 = \sum_{\omega} \mathds{1}_{\{x\not\leftrightarrow y\}} \left[ \sum_{\sigma} \mathds{1}_{\{\sigma_x=\sigma_y\}} { \Delta(\sigma, \omega) \exp \big( \beta \sum_{i\in V} h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega) \nonumber \\[0,3cm] = \sum_{\omega} \mathds{1}_{\{x\not\leftrightarrow y\}}(\omega) \!\! \left[ { 2\cosh\left(\pmb{h}(K_t)+\pmb{h}(K_u)\right) \prod_{\substack{\alpha=1 \\ \alpha\not=t,u}}^{k(\omega,G)} 2\cosh\big(\pmb{h}(K_\alpha)\big) \over \prod_{\alpha=1}^{k(\omega,G)} 2 \cosh\left(\pmb{h}(K_\alpha) \right) } \right] \!\! \phi_{\pmb{p},\pmb{h},G}(\omega) \nonumber \\[0,3cm] = {1\over 2} \phi_{\pmb{p},\pmb{h},G} \Big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot { \cosh\left(\pmb{h}(K_t)+\pmb{h}(K_u)\right) \over \cosh\left(\pmb{h}(K_t)\right)\cdot\cosh \left(\pmb{h}(K_u)\right) } \Big) \nonumber \\[0,3cm] = {1\over 2} \phi_{\pmb{p},\pmb{h},G} \big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot \big\{ 1 + \tanh\left(\pmb{h}(K_t)\right) \cdot \tanh\left(\pmb{h}(K_u)\right) \big\} \big). \end{gathered}$$ Replacing the last expression in we get that $$\begin{aligned} \label{I-2-1} I_2 = {1\over 2} \phi_{\pmb{p},\pmb{h},G} \big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot \tanh\left(\pmb{h}(K_t)\right) \cdot \tanh\left(\pmb{h}(K_u)\right) \big). \end{aligned}$$ Since $\tau_{2\beta,\pmb{h},2,V}(x,y)=I_1+I_2$, the theorem follows. Notice that in the absence of the magnetic field, i.e. $\pmb{h}\equiv 0$, the conclusion of the Theorem \[Correlação/quase-conectividade\] reduces to $$\tau_{2\beta,0,2,V}(x,y)= {1\over 2}\phi_{\pmb{p},0,G}(x\leftrightarrow y), \quad \forall x,y\in V$$ which is a well known identity for the Ising/Potts model with $q=2$, see [@Grimmett2] Theorem 1.16, p. 11. \[col\] The spin-spin correlation of the Ising model on the finite volume $V$ satisfies the following identity for any magnetic field $\pmb{h}\in \mathbb{R}^{V}$ $$\lambda_{{\beta},\pmb{h},V}(\sigma_x\sigma_y ) = \phi_{\pmb{p},\pmb{h},G}(x\leftrightarrow y) + \phi_{\pmb{p},\pmb{h},G} \big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot \tanh\left(\pmb{h}(K_t)\right) \cdot \tanh\left(\pmb{h}(K_u)\right) \big).$$ This follows easily from the definition of the expected value and Theorem \[Correlação/quase-conectividade\] since $ \lambda_{{\beta},\pmb{h},V}(\sigma_x\sigma_y ) = \lambda_{{\beta},\pmb{h},V}(\sigma_i=\sigma_j) - \lambda_{{\beta},\pmb{h},V}(\sigma_i\neq \sigma_j) = 2\lambda_{{\beta},\pmb{h},V}(\sigma_i=\sigma_j)-1 = 2 \left[ \pi_{2\beta,\pmb{h},2,V}(\hat{\sigma}_x =\hat{\sigma}_y)-{1\over 2} \right] = 2\tau_{2\beta,\pmb{h},2,V}(x,y) $. If we consider the Ising model on $G$ without magnetic field, from Corollary \[col\] we get $ \lambda_{{{\beta},0,V}} (\sigma_x\sigma_y ) = \phi_{\pmb{p},0,G}(x\leftrightarrow y), \ \forall x,y\in V. $ Applications ============ #### Spin-spin correlations. Corollary \[col\] can be used to obtain some correlation inequalities. Keeping the notation of Theorem \[Correlação/quase-conectividade\] and supposing that $h_i\geqslant 0$ for all $i\in V$, it follows from the monotonicity of the hyperbolic tangent that $\tanh(\beta h_x)\leqslant \tanh(\pmb{h}(K_t))$ and $\tanh(\beta h_y)\leqslant \tanh(\pmb{h}(K_u))$. These estimates together with Corollary \[col\] give us the following lower bound $ \phi_{\pmb{p},\pmb{h},G}\ (x\not\leftrightarrow y)\allowbreak \tanh(\beta h_x)\tanh(\beta h_y)\allowbreak \leqslant \allowbreak \lambda_{{\beta},\pmb{h},V}(\sigma_x\sigma_y). $ A simple computation shows that $ P_{\pmb{p}}(x\not\leftrightarrow y) \leqslant \phi_{\pmb{p},\pmb{h},G}(x\not\leftrightarrow y) $, where $P_{\pmb{p}}$ is the probability measure of the independent bond percolation model with parameter $\pmb{p}$. Supposing that $\pmb{p}\equiv p$ (the homogeneous model) and $p<p_c(\mathbb{V})$, for any given $\varepsilon>0$, if the distance between $x$ and $y$ is large enough then $ (1-\varepsilon) \tanh(\beta h_x)\tanh(\beta h_y) \leqslant \lambda_{{\beta},\pmb{h},V}(\sigma_x\sigma_y) , $ which, of course, can also be (better) obtained by the GKS inequality. Under the above assumptions, Corollary \[col\] also gives us an upper bound in terms of the iid Bernoulli bond percolation model, which is $ \lambda_{{\beta},\pmb{h},V} (\sigma_x\sigma_y)\allowbreak \leqslant\allowbreak e^{-C(\beta) d_{G}(x,y)}\allowbreak + P_p \big( \tanh\left(\pmb{h}(K_t)\right) \cdot \tanh\left(\pmb{h}(K_u)\right) \big), $ where at this point we are assuming $J_{ij}\equiv J$ and $p=1-e^{-\beta J}$. To obtain the asymptotic behavior of the second term in the rhs above, one needs to impose extra conditions on the geometry of the graph and the decay ratio of the magnetic field. #### Expected value and distribution function of a single spin. \[lema anteprincipal\] Consider a finite graph $G=(V,E)$, $x\in V$ and $\omega\in\{0,1\}^E$ a fixed edge configuration. Then $$\begin{gathered} \sum_{\sigma} \! \mathds{1}_{\{\sigma_x=\pm 1\}} \Delta(\sigma,\omega) {\textstyle \exp \big(\beta\sum_{i\in V}h_i\sigma_i\big) } \\ {\textstyle \!=\! \exp\left( \pm \pmb{h}(K_t)\right) \prod_{\substack{\alpha=1 \\ \alpha\not=t}}^{k(\omega,G)} } 2\cosh\big(\pmb{h}(K_\alpha)\big), \end{gathered}$$ where $K_t\equiv K_t(\omega)$ is the connected component containing the vertex $x$. To prove this lemma we proceed, mutatis mutandis, as in the proof of Lemma \[lema28fev\]. \[teorema de medida\] Let $G=(V,E)$ be a finite graph. We have, for any fixed $x\in V$, that $$\lambda_{\beta,\pmb{h},V}(\sigma_x=\pm 1) = {1\over 2} \pm {1\over 2} \phi_{\pmb{p},\pmb{h},G} \big( \tanh\left(\pmb{h}(K_t)\right) \big),$$ where $K_t(\omega)\equiv K_t$ is the connected component containing $x$. From Theorem \[lemin\], it follows that $$\begin{gathered} \label{espera1} \lambda_{\beta,\pmb{h},V}(\sigma_x=\pm1) = \sum_{(\sigma,\omega) \in \{-1,+1\}^V\times\{0,1\}^E} \mathds{1}_{\{\sigma_x=\pm 1\}} \nu_{\beta,\pmb{h},G}(\sigma,\omega) \nonumber \\[0,2cm] = \sum_{\omega\in\{0,1\}^E} \Big[ \sum_{\sigma\in\{-1,+1\}^V} \mathds{1}_{\{\sigma_x=\pm1\}} \nu_{\beta,\pmb{h},G}(\sigma\vert \omega) \Big] \phi_{\pmb{p},\pmb{h},G}(\omega).\nonumber \end{gathered}$$ Using Corollary \[medida-conditional1\], the above expression can be rewritten as $$\sum_{\omega\in\{0,1\}^E}\left[ \sum_{\sigma\in\{-1,+1\}^V} \mathds{1}_{\{\sigma_x=\pm 1\}}{\Delta(\sigma,\omega) \exp \big( \beta\sum_{i\in V}h_i\sigma_i \big) \over \prod_{\alpha=1}^{k(\omega,G)}2 \cosh\left(\pmb{h}(K_\alpha)\right)}\right] \phi_{\pmb{p},\pmb{h},G}(\omega).$$ Using now Lema \[lema anteprincipal\], we can see that the above expression is equal to $$\begin{gathered} \sum_{\omega\in\{0,1\}^E}\left[ { \exp\left( \pm \pmb{h}(K_t)\right) \prod_{\substack{\alpha=1 \\ \alpha\not=t}}^{k(\omega,G)} 2\cosh\big(\pmb{h}(K_\alpha)\big) \over \prod_{\alpha=1}^{k(\omega,G)} 2\cosh\big(\pmb{h}(K_\alpha)\big) } \right] \phi_{\pmb{p},\pmb{h},G}(\omega) \\[0,2cm] = {1\over 2} \sum_{\omega\in\{0,1\}^E}\left[ 1\pm\tanh\left(\pmb{h}(K_t)\right) \right] \phi_{\pmb{p},\pmb{h},G}(\omega). \end{gathered}$$ Under the hypothesis of Theorem \[teorema de medida\], we have that $$\lambda_{\beta,\pmb{h},V}(\sigma_x) = \phi_{\pmb{p},\pmb{h},G} \big( \tanh\left(\pmb{h}(K_t)\right) \big).$$ The proof follows directly from Theorem \[teorema de medida\]. General Potts models in external fields {#secao-potts-geral} ======================================= In this last section we state two propositions establishing a graphical representation for the two-point function of the $q$-state Potts model with general external fields, defined in the Section \[sec-Potts-com-campo-cc-livre\], in terms of the connectivity of the random-cluster model introduced below. The techniques employed to prove these results are similar to the ones we used in the previous section and therefore the proofs are omitted. Given a finite graph $G=(V,E)$, coupling constants $\pmb{J}=(J_{ij}\geqslant 0:\{i,j\}\in E)$ and $\pmb{\hat{h}}$ a magnetic field as defined in the Section \[sec-Potts-com-campo-cc-livre\], for each $\omega\in \{0,1\}^{E}$ we define the finite-volume Gibbs measure of the (general) random-cluster model in external field by $$\label{def-rcm-q-geral} \phi_{\pmb{p},\pmb{\hat{h}},q,G}(\omega) = { 1 \over \mathscr{Z}^{\mathrm{{RC}}}_{\pmb{p},\pmb{\hat{h}},q,G} } B_{\pmb{J},q}(\omega) \prod_{\alpha =1}^{k({\omega,G})} \sum_{p=1}^q \exp \big( \beta\sum_{i\in K_\alpha}h_{i,p} \big),$$ where $K_\alpha$ is defined exactly as in the Section \[secao-def-modelo-RC\] and $B_{\pmb{J},q}(\omega)$ is similar to the Bernoulli factor of the Section \[secao-def-modelo-RC\] with exception that $p_{ij}=1-\exp(-q\beta J_{ij})$. The Edwards-Sokal measure is generalized to $$\begin{aligned} \label{ES1-q} {\nu}_{\pmb{p},\pmb{\hat{h}},q,G}(\hat{\sigma}, \omega) \equiv {1 \over \mathscr{Z}^{\mathrm{ ES}}_{\pmb{p},\pmb{\hat{h}},q,G}} \displaystyle{ B_{\pmb{J},q}(\omega) \Delta_q(\hat{\sigma}, \omega) \times \exp \big( \beta \sum_{i\in V}\sum_{p=1}^q h_{i,p}\delta_{\hat{\sigma}_i,p} \big). }\end{aligned}$$ \[Correlação/quase-conectividade-geral\] Consider the Potts model with Hamiltonian given by , densities $p_{ij}\equiv1-\exp(-q\beta J_{ij})$ and $q\in\{2,3, \ldots\}$ fixed. For any pair of vertices $x,y\in V$ we have that $$\begin{gathered} \tau_{q\beta,\pmb{\hat{h}},q,V}(x,y) = \Big(1-{1\over q}\Big) \phi_{\pmb{p},\pmb{\hat{h}},q,G}(x\leftrightarrow y) \\ + \phi_{\pmb{p},\pmb{\hat{h}},q,G} \Big( \mathds{1}_{\{x\not\leftrightarrow y\}} \cdot \big\{ H_{\pmb{\hat{h}}}(K_t,K_u) - {1\over q} \big\} \Big), \end{gathered}$$ where the random variable $H_{\pmb{\hat{h}}}(K_t,K_u)$ is given by $$H_{\pmb{\hat{h}}}(K_t,K_u) \equiv { \sum_{r=1}^q \exp \big( \beta\sum_{i\in K_t}h_{i,r}+ \beta\sum_{i\in K_u}h_{i,r} \big) \over \sum_{r=1}^q \exp \big( \beta\sum_{i\in K_t}h_{i,r} \big) \cdot \sum_{r=1}^q \exp \big( \beta\sum_{i\in K_u}h_{i,r} \big) } ,$$ with $K_t\equiv K_t(\omega)$ and $K_u\equiv K_u(\omega)$ being the disjoint connected components containing the vertices $x$ and $y$, respectively. We omit the proof of this proposition because it is similar to the one given for Theorem \[Correlação/quase-conectividade\]. Notice that in case $\pmb{\hat{h}}\equiv 0$, we have for any $\omega\in \{0,1\}^{E}$ that $$H_0(K_t,K_u)(\omega)= {q\over q^2}={1\over q},$$ so Proposition \[Correlação/quase-conectividade-geral\] gives us the following identity $$\tau_{q\beta,0,q,V}(x,y) = \Big(1-{1\over q}\Big) \phi_{\pmb{p},0,q,G}(x\leftrightarrow y).$$ This is also a very well know identity, as can be seen in [@Grimmett2] Theorem 1.16, p. 11. Furthermore in case $q=2$ and $h_{i,1}=-h_{i,2}={h_i}$ for all $i\in V$, we have for any pair $x,y\in V$ that $$H_{\pmb{h}}(K_t,K_u) = {1\over 2} \Big\{ 1+\tanh \big({\beta}\sum_{i\in K_t}h_i\big). \tanh\big({\beta}\sum_{i\in K_u}h_i\big) \Big\}.$$ In other words, Proposition \[Correlação/quase-conectividade-geral\] generalizes Theorem \[Correlação/quase-conectividade\]. \[theor2\] Let $G=(V,E)$ be a finite graph and $x\in V$. For each $m\in\{1,\ldots,q\}$ with $q\geqslant 1$, we have $$\begin{aligned} \pi_{q\beta,\pmb{\hat{h}},q,V}(\hat{\sigma}_x=m) = \phi_{\pmb{p},\pmb{\hat{h}},q,G} \Big( { \exp ( \beta\sum_{i\in K_t}h_{i,m} ) \over \sum_{p=1}^q \exp ( \beta\sum_{i\in K_t}h_{i,p} ) } \Big), \end{aligned}$$ where $K_t \equiv K_t(\omega)$ is the connected component of $x$. *Sketch of the Proof.* To prove this theorem one needs to compute the marginals of the Edwards-Sokal coupling given in . The computation is similar to the one presented in the previous sections. The next step is to prove the identity $$\pi_{q\beta,\pmb{\hat{h}},q,V}(\hat{\sigma}_x=m) = \sum_\omega \Big[\sum_{\hat{\sigma}} \mathds{1}_{\{\hat{\sigma}_x=m\}} \nu_{\pmb{p},\pmb{\hat{h}},q,G}(\hat{\sigma}\vert \omega) \Big] \phi_{\pmb{p},\pmb{\hat{h}},q,G}(\omega)$$ and then one proves that the rhs above is exactly $$\begin{aligned} \sum_\omega \left[ \sum_{\hat{\sigma}} \mathds{1}_{\{\hat{\sigma}_x=\hat{\sigma}_y\}} { \Delta_q(\hat{\sigma},\omega) \exp \big( \beta\sum_{i\in V} \sum_{p=1}^q h_{i,p}\delta_{\hat{\sigma}_i,p} \big) \over \prod_{\alpha=1}^{k(\omega,G)} \sum_{p=1}^q \exp \big( \beta\sum_{i\in K_\alpha}h_{i,p} \big) } \right] \phi_{\pmb{p},\pmb{\hat{h}},q,G}(\omega). \end{aligned}$$ From this point, the result follows from the combinatorial arguments presented before. The general random-cluster model ================================ In this section we define the so called general random-cluster model on the lattice $\mathbb{L}=(\mathbb{V},\mathbb{E})$ (this terminology, GRC model, comes from [@BBCK00]) with inhomogeneous magnetic field of the form $ \pmb{\hat{h}} \equiv (h_{i,p}:i\in \mathbb{V};\ p=1,\ldots,q) \in \mathbb{R}^{\mathbb{V}}\times \cdots\times \mathbb{R}^{\mathbb{V}} $ and boundary conditions. The Bernoulli factors introduced before will be replaced in this section by (abusing notation) $$\begin{aligned} \label{nova def. Bernoulli} B_{\pmb{J}}(\omega)\equiv \prod_{\{i,j\}:\omega_{ij}=1} r_{ij},\end{aligned}$$ where $\pmb{J}=(J_{ij}\geqslant 0:\{i,j\}\in E)$, $r_{ij}\equiv \exp(q\beta J_{ij})- 1$ and $q\in\mathbb{Z}^+$ fixed. Although $r_{ij}\geqslant 0$, in general, they are not bounded by one, but mind that the random-cluster measure obtained with such “Bernoulli factors” is the same one gets when considering the old Bernoulli factors, since the weights in both cases are related by an overall normalization factor that cancels out because of the partition function. Fix a random subgraph $G=(V,E)$ on the lattice $\mathbb{L}$, let $\partial E = \{ e\in\mathbb{E}:\ e\cap V\neq \emptyset\ \text{and}\ e\cap \partial V\neq \emptyset \}. $ We denote by $\mathbb{B}_0(V)$ the set of all edges $\{x,y\}\in\mathbb{E}$ so that $\{x,y\}\subset V$. With this definition we have $\mathbb{B}_0(V)=E$. We use the notation $\mathbb{B}(V)$ to denote the set of all edges with at least one vertex in $V$. Note that $\mathbb{B}(V)=E\cup\partial E$. For any $\widetilde{E}\subset\mathbb{B}_0(\mathbb{V})$, we define $\mathbb{V}(\widetilde{E})$ as the set of sites which belong to at least one edge in $\widetilde{E}$. #### GRC model with general boundary condition. Fix a finite subgraph $G=(V,E)$ of the lattice $\mathbb{L}$. For each $i\in\mathbb{V}$ we define $ h_{i,\mathrm{ max}} \equiv \max\{h_{i,p}: p=1,\ldots, q\}. $ If $\omega\in \{0,1\}^{\mathbb{E}}$ and $C(\omega)$ denotes a generic connected component on $(\mathbb{V},\eta(\omega))$, the GRC measure with general boundary condition is obtained by normalizing the followings weights $$\begin{aligned} \label{definição de general boundary condition} \mathcal{W}^{\mathrm{ GRC}}_{E}(\omega_E|\omega_{E^c}) \equiv B_{\pmb{J}}(\omega) \!\!\!\!\!\!\!\! \prod_{ \substack{C(\omega):\\ \mathbb{V}(C(\omega))\cap V\not=\emptyset}} \sum_{p=1}^q q_p \ \exp \big( {-\beta\!\!\!\underset{i\in C(\omega)}{\sum} (h_{i,\mathrm{ max}}-h_{i,p})} \big),\end{aligned}$$ where $\{q_p:p=1,\ldots, q\}$ are positive constants, $B_{\pmb{J}}(\omega)$ is given by and the product runs over all the connected components $C(\omega)$ of the graph $(\mathbb{V},\eta(\omega))$. In the above expression we are using the convention $e^{-\infty}=0$. This measure is denoted by $ \phi^{\mathrm{ GRC}}_{E}. $ #### GRC model with free boundary condition. Let $G=(V,E)$ be a finite graph and $\omega\in\{0,1\}^E$ a configuration. If $C(\omega)$ denotes a generic connected component on $(V,\eta(\omega))$, we define $$\begin{aligned} \Theta_{V,\mathrm{ free}}(C(\omega)) \equiv \sum_{p=1}^q q_p\ {\textstyle \exp \big( \beta\underset{i\in C(\omega)}{\sum}h_{i,p} \big). }\end{aligned}$$ The GRC measure with free boundary condition is obtained by normalizing the weights $$\begin{aligned} \label{free peso} \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\omega) \equiv B_{\pmb{J}}(\omega) \prod_{C(\omega)} \Theta_{V,\mathrm{ free}}(C(\omega)),\end{aligned}$$ where $B_{\pmb{J}}(\omega)$ is given by and the product runs over all the connected components $C(\omega)$ of the graph $(V,\eta(\omega))$. This measure is denoted by $\phi^{\mathrm{ GRC}}_{V,\mathrm{ free}}$ and for each $\omega\in\{0,1\}^E$ it satisfies $ \phi^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\omega) \propto \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\omega), $ where the proportionality constant is exactly the (inverse of the) partition function of the GRC model. #### GRC model with wired boundary condition. Fix $\mathrm{m}\in \{1,\ldots, q\}$ and a finite subgraph $G=(V,E)$ of the lattice $\mathbb{L}$. If for each $\omega\in \{0,1\}^{E\cup\partial E}$. $C(\omega)$ denotes a connected component on $(V\cup\partial V,\eta(\omega))$, then we define $$\begin{aligned} \Theta_{V,\mathrm{ m}}(C(\omega)) \equiv \begin{cases} \Theta_{V,\mathrm{ free}}(C(\omega)), & \text{if}\ C(\omega)\cap \partial V=\emptyset \\[0,2cm] \exp {\textstyle \big(\beta\underset{i\in C(\omega)}{\sum}h_{i,\mathrm{m}}\big), } & \text{otherwise}. \end{cases}\end{aligned}$$ Similarly, the GRC measure with m-wired boundary condition is obtained by normalizing the weights $$\begin{aligned} \label{def.rcm-wired condition} \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ m}}(\omega) \equiv B_{\pmb{J}}(\omega)\prod_{C(\omega)} \Theta_{V,\mathrm{ m}}(C(\omega)),\end{aligned}$$ where the product runs over all the connected components $C(\omega)$ of the graph $(V\cup\partial V,\eta(\omega))$. This measure is denoted by $ \phi^{\mathrm{ GRC}}_{V,\mathrm{ m}}. $ One can easily see that when $E\equiv\mathbb{B}(\Lambda)$, $\Lambda\subset \mathbb{V}$ finite, in $$\begin{aligned} \mathcal{W}^{\mathrm{ GRC}}_{\mathbb{B} (\Lambda)}(\omega_{\mathbb{B}(\Lambda)} | \omega_{\mathbb{B}(\Lambda)^c}^{(1)}) &= e^{-\beta\sum_{i\in \Lambda}h_{i,\mathrm{ max}}} q_{\mathrm{ m}} \cdot B_{\pmb{J}}(\omega)\prod_{C(\omega)} \Theta_{\Lambda,\mathrm{ m}}(C(\omega)) \\[0,3cm] &\stackrel{\eqref{def.rcm-wired condition}}{=} e^{-\beta\sum_{i\in \Lambda}h_{i,\mathrm{ max}}} q_{\mathrm{ m}} \cdot \mathcal{W}^{\mathrm{ GRC}}_{\Lambda,\mathrm{ m}}(\omega),\end{aligned}$$ where $\omega^{(i)}$ is the configuration with $\omega_{e}^{(i)}=i$ for all $e\in\mathbb{B}_0(\mathbb{V})$ $(i=0,1)$. Therefore $$\phi^{\mathrm{ GRC}}_{\mathbb{B}(\Lambda)} (\omega_{\mathbb{B}(\Lambda)} | \omega_{\mathbb{B}(\Lambda)^c}^{(1)}) = \phi^{\mathrm{ GRC}}_{\Lambda,\mathrm{ m}}(\omega).$$ Similarly we obtain in with $E\equiv\mathbb{B}_0(\Lambda)$ $$\begin{aligned} \mathcal{W}^{\mathrm{ GRC}}_{\mathbb{B}_0 (\Lambda)}(\omega_{\mathbb{B}_0(\Lambda)} | \omega_{\mathbb{B}_0(\Lambda)^c}^{(0)}) &= e^{-\beta\sum_{i\in \Lambda}h_{i,\mathrm{ max}}} \cdot B_{\pmb{J}}(\omega)\prod_{C(\omega)} \Theta_{\Lambda,\mathrm{ free}}(C(\omega)) \\[0,3cm] &\stackrel{\eqref{free peso}}{=} e^{-\beta\sum_{i\in \Lambda}h_{i,\mathrm{ max}}} \cdot \mathcal{W}^{\mathrm{ GRC}}_{\Lambda,\mathrm{ free}}(\omega),\end{aligned}$$ then $$\phi^{\mathrm{ GRC}}_{\mathbb{B}_0(\Lambda)} (\omega_{\mathbb{B}_0(\Lambda)}|\omega_{\mathbb{B}_0(\Lambda)^c}^{(0)}) = \phi^{\mathrm{ GRC}}_{\Lambda,\mathrm{ free}}(\omega).$$ The FKG inequality ------------------ Throughout this section we assume that $\{q_p:p=1,\ldots,q\}$ introduced in and the magnetic field $\pmb{\hat{h}}$ satisfy $$\begin{aligned} \label{qmmaiorque1} \sum_{p\in {\cap}_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}})}q_p \geqslant 1,\end{aligned}$$ where $ \mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) \equiv \big\{p\in\{1,\ldots,q\}:h_{i,p}=h_{i,\mathrm{ max}}\big\}. $ We consider as usual the partial order on $\{0,1\}^\mathbb{E}$ where $ \omega\preceq \tilde{\omega} \ \Longleftrightarrow \ \omega_e\leqslant \tilde{\omega}_e, \ \forall \ e\in \mathbb{E}. $ We also use the standard notations $\omega_1\vee\omega_2$ and $\omega_1\wedge\omega_2$ for $ (\omega_1\vee\omega_2)_{e}=\max\{\omega_1(e),\omega_2(e)\} \ \text{and}\ (\omega_1\wedge\omega_2)_{e}=\min\{\omega_1(e),\omega_2(e)\} $ with $ e\in \mathbb{E}$, respectively. Let $(\Omega,\preceq)$ be a partially ordered space. A measure $\mu$ over $\Omega$ said to have the $\mathrm{FKG}$ property if $$\mu(fg)\geqslant \mu(f)\mu(g)$$ for any increasing (with respect to $\preceq$) measurable functions $f,g:\Omega\to \mathbb{R}$. Furthermore, if $\Omega$ is a cartesian product $\Omega=\prod_{e\in B}\Omega_e$, with $|\Omega_{e}|<\infty$, then $\mu$ is said to have the [**strong FKG property**]{}, if $\mu(\cdot\vert A)$ has the $\mathrm{FKG}$ property for each cylinder event $ A = \{ \omega\in\Omega:\omega_e=\alpha_e, \ \forall e\in \tilde{B} \} $, where $\tilde{B}\subset{B}$ is finite and $\alpha_e\in\Omega_e$ for all $e\in\tilde{B}$. If $\mathrm{m},\mathrm{\widetilde{m}}\in$ ${\cap}_{i\in \mathbb{V}} \mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}})$, then $ \Theta_{V, \mathrm{\widetilde{m}}}(C) = \Theta_{V,\mathrm{m}}(C) $ and therefore $\phi^{\mathrm{ GRC}}_{V,\mathrm{\widetilde{m}}}= \phi^{\mathrm{ GRC}}_{V,\mathrm{m}}$. This measure is denoted by $\phi^{\mathrm{ GRC}}_{V,\mathrm{ max}}$. \[Strong FKG Property\] Let $q\in \mathbb{Z}^+$, $\beta\geqslant 0$, $ \pmb{J} = (J_{ij}:\{i,j\}\in \mathbb{E})$ $\in [0,\infty)^\mathbb{E} $, $ \pmb{\hat{h}} = ( h_{i,p}\in\mathbb{R}: i\in\mathbb{V}, 1\leqslant p\leqslant q ) $ and $\{q_p: p=1,\ldots,q\}$ satisfying . Then for any finite subgraph $G=(V,E)$ of $\mathbb{L}$, the measures $\phi^{\mathrm{ GRC}}_{V,\mathrm{ free}}$ and $\phi^{\mathrm{ GRC}}_{V,\mathrm{ max}}$ have the strong $\mathrm{FKG}$ property. For simplicity we assume that the magnetic field we are dealing with satisfies the following inequalities $$\begin{aligned} \label{crescente-campos} h_{i,1}\leqslant h_{i,2}\leqslant \ldots \leqslant h_{i,q}, \quad \forall \ i\in\mathbb{V}. \end{aligned}$$ The FKG lattice condition for the $\phi^{\mathrm{ GRC}}_{V,\mathrm{ free}}$ is equivalent to $$\mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}} (\omega^{(1)}\vee \omega^{(2)}) \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}} (\omega^{(1)}\wedge \omega^{(2)}) \geqslant \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\omega^{(1)}) \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\omega^{(2)}),$$ where $\omega^{(1)}$ and $\omega^{(2)}$ are arbitrary configurations. Similarly for $\phi^{\mathrm{ GRC}}_{V,\mathrm{ max}}$. It is well known that such condition implies the strong FKG property, see for example Theorem $2.19$, p. $25$ in [@Grimmett2]. By defining $$\mathcal{R}(\xi,\omega) \equiv { \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}} (\xi\vee \omega) \over \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}(\xi) },$$ one can see that the lattice condition holds if $$\begin{aligned} \label{min} \mathcal{R}(\omega^{(1)},\omega^{(2)}) \geqslant \mathcal{R}(\omega^{(1)}\wedge\omega^{(2)},\omega^{(2)}). \end{aligned}$$ For a fixed configuration $\omega$, we chose an arbitrary order for $\eta(\omega)$ and represent these open edges as $(e_1,\ldots,e_{\vert \eta(\omega)\vert})$. So for any configuration $\xi\in\{0,1\}^{E}$ we have that $$\mathcal{R}(\xi,\omega)=\prod_{k=1}^{\vert\eta(\omega)\vert} \mathcal{R}(\xi\vee\omega^{(e_1)}\vee\cdots \vee \omega^{(e_{k-1})},\omega^{(e_{k})}),$$ where $ (\omega^{(e)})_{e'}\equiv \delta_{e,e'}$. Therefore it is enough to prove for configurations $\xi$, $\omega^{(1)}$ and $\omega^{(2)}$ such that $\xi$ has at least two zero coordinates or at most one zero and $\omega^{(1)}\equiv \xi\vee\omega^{(b)}$ and $\omega^{(2)}\equiv \xi\vee\omega^{(b')}$. Let us begin assuming that $\xi$ has at least two zero coordinates and $$\xi\equiv (*,\ldots,*,\aunderbrace[D]{0}_{ b-\mathrm{th}}, *,\ldots,*,\aunderbrace[D]{0}_{b'\mathrm{-th}},*,\ldots,*),$$ where $b,b'\in E\cup \partial E $, $b\not=b'$ and the stars indicate generic elements in $\{0,1\}$ (not necessarily equal). If we define $$\xi^b\equiv (*,\ldots,*,\aunderbrace[D]{1}_{b\mathrm{-th}}, *,\ldots,*,\aunderbrace[D]{0}_{ b'\mathrm{-th}},*,\ldots,*)$$ and $$\xi^{b'}\equiv (*,\ldots,*,\aunderbrace[D]{0}_{b\mathrm{-th}}, *,\ldots,*,\aunderbrace[D]{1}_{b'\mathrm{-th}},*,\ldots,*),$$ then we have that $ \omega^{(1)}=\xi\vee\omega^{(b)}=\xi^b, \omega^{(2)}=\xi\vee\omega^{(b')}=\xi^{b'} \text{and}\allowbreak\ \omega^{(1)}\wedge\omega^{(2)}=\xi . $ So in order to prove it is enough to prove that $$\begin{aligned} \label{por provar} \mathcal{R}(\xi^b,\xi^{b'})\geqslant\mathcal{R}(\xi,\xi^{b'}), \quad \mbox{with} \ b\not = b'. \end{aligned}$$ Now we concentrate on proving . To do this we first observe that if $\prod_{\{i,j\}:\xi_{ij}=1}r_{ij}=k$, then $$\prod_{\{i,j\}:(\xi^b\vee\xi^{b'})_{ij}=1} \!\!\!\!\!\!\!\! r_{ij} = r_b r_{b'} k, \quad \prod_{\{i,j\}:\xi^b_{ij}=1} \!\!\!\! r_{ij} = r_b k \quad \text{and} \quad \prod_{\{i,j\}:(\xi\vee\xi^{b'})_{ij}=1} \!\!\!\!\!\!\!\! r_{ij} = r_{b'} k.$$ So it follows from the definition that $${B_{\pmb{J}}(\xi^b\vee\xi^{b'}) \over B_{\pmb{J}}(\xi^b)} = { \prod_{\{i,j\}:(\xi^b\vee\xi^{b'})_{ij}=1}r_{ij} \over \prod_{\{i,j\}:\xi^b_{ij}=1}r_{ij} } = r_{b'} = { \prod_{\{i,j\}:(\xi\vee\xi^{b'})_{ij}=1}r_{ij} \over \prod_{\{i,j\}:\xi_{ij}=1}r_{ij} } = {B_{\pmb{J}}(\xi\vee\xi^{b'}) \over B_{\pmb{J}}(\xi)}.$$ Because of the above observation and the definitions of $\mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ free}}$ and $\mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ m}}$, the proof of reduces to $$\begin{aligned} \label{Desigualdade mt importante} { \Theta_{V,\mathrm{ \#}}(C(\xi^b\vee\xi^{b'})) \over \Theta_{V,\mathrm{ \#}}(C(\xi^b)) } \geqslant { \Theta_{V,\mathrm{ \#}}(C(\xi\vee\xi^{b'})) \over \Theta_{V,\mathrm{ \#}}(C(\xi)) }, \end{aligned}$$ where $\mathrm{\#}$ stands for “free”or “m”.\ \ [Free boundary condition case.]{}\ We broke the proof of in several cases. Let $A_1,A_2,B_1$ and $B_2$ be connected components of $(V,\eta(\xi))$ and consider the cases showed in the picture below \[q15\] The case $(I)$ represents that the end vertices of $b$ belong to $A_1$ and $A_2$ and the end vertices of $b'$ belong to $B_1$ and $B_2$. In this case the left and right sides of are equal, since $$\begin{gathered} { \Theta_{V,\mathrm{ free}}(A_1\cup A_2) \Theta_{V,\mathrm{ free}}(B_1\cup B_2) \over \Theta_{V,\mathrm{ free}}(A_1\cup A_2) \Theta_{V,\mathrm{ free}}(B_1) \Theta_{V,\mathrm{ free}}(B_2) } \\[0.2cm] = { \Theta_{V,\mathrm{ free}}(B_1\cup B_2) \Theta_{V,\mathrm{ free}}(A_1) \Theta_{V,\mathrm{ free}}(A_2) \over \Theta_{V,\mathrm{ free}}(A_1) \Theta_{V,\mathrm{ free}}(A_2) \Theta_{V,\mathrm{ free}}(B_1) \Theta_{V,\mathrm{ free}}(B_2) }. \end{gathered}$$For the case $(II)$, we should prove that $${ \Theta_{V,\mathrm{ free}}(A\cup B\cup C) \over \Theta_{V,\mathrm{ free}}(C\cup A) \Theta_{V,\mathrm{ free}}(B) } \geqslant { \Theta_{V,\mathrm{ free}}(C\cup B) \Theta_{V,\mathrm{ free}}(A) \over \Theta_{V,\mathrm{ free}}(A) \Theta_{V,\mathrm{ free}}(B) \Theta_{V,\mathrm{ free}}(C) },$$ which is equivalent to $$\begin{aligned} \label{seq,nerv} \Theta_{V,\mathrm{ free}}(C) \Theta_{V,\mathrm{ free}}(A\cup B\cup C) \geqslant \Theta_{V,\mathrm{ free}}(C\cup A) \Theta_{V,\mathrm{ free}}(C\cup B). \end{aligned}$$ To help us prove inequality , we define for each $m\in \{1,\ldots,q\}$ the following numbers $$\begin{aligned} a_m \equiv \exp \big(\beta{\sum}_{i\in A} \!h_{i,m}\big), \, b_m \equiv \exp \big(\beta{\sum}_{i\in B} \!h_{i,m}\big) \ \text{and} \ c_m \equiv \exp \big(\beta{\sum}_{i\in C} \!h_{i,m}\big). \end{aligned}$$ The hypothesis implies immediately that $(a_m)$ and $(b_m)$ are non-decreasing in $m$. Using this notation, reads $$\begin{aligned} \label{serie} \sum_{m=1}^q q_m c_m \sum_{m'=1}^q q_{m'}a_{m'}b_{m'}c_{m'} \geqslant \sum_{m=1}^q q_m a_mc_m \sum_{m'=1}^q q_{m'}b_{m'}c_{m'}. \end{aligned}$$ Both sides of the above inequality can be written using a bilinear form $$\varphi(a,b)\equiv\sum_{m,m'=1}^q r_{m,m'}a_{m'}b_{m},$$ where $r_{m,m'}\equiv q_mc_mq_{m'}c_{m'}$ , $a\equiv(a_1,\ldots,a_q)$ and $b\equiv(b_1,\ldots,b_q)$. Note that $\varphi$ is a symmetric bilinear form and can be written as $$\begin{aligned} \label{var} \varphi(1,c)\geqslant \varphi(a,b),\quad \text{where}\ c\equiv (a_1b_1,\ldots,a_qb_q). \end{aligned}$$ Therefore it remains to prove , which clearly holds since $$r_{m,m'}(a_{m'}-a_m)(b_{m'}-b_m)\geqslant 0 \ \ \Longleftrightarrow \ \ \varphi(1,c)-\varphi(a,b)-\varphi(b,a)+\varphi(c,1)\geqslant 0.$$ We proceed with for the case $(III)$. Now we have to prove that $$1= { \Theta_{V,\mathrm{ free}}(A\cup B) \over \Theta_{V,\mathrm{ free}}(A\cup B) } \geqslant { \Theta_{V,\mathrm{ free}}(A\cup B) \over \Theta_{V,\mathrm{ free}}(A) \Theta_{V,\mathrm{ free}}(B) },$$ in other words $ \Theta_{V,\mathrm{ free}}(A) \Theta_{V,\mathrm{ free}}(B) \geqslant \Theta_{V,\mathrm{ free}}(A\cup B) $, or equivalently $$\textstyle \sum_{m=1}^qq_ma_m\sum_{m'=1}^qq_{m'}b_{m'} \geqslant \sum_{m=1}^qq_ma_mb_m.$$ This last inequality is actually true since $$\sum_{m=1}^qq_ma_m \sum_{m'=1}^qq_{m'}b_{m'} \geqslant \sum_{m=1}^qq_ma_m b_{\text{max}} \sum_{m'\in {\cap}_{i\in\mathbb{V}} \mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}})}^qq_{m'} \geqslant \sum_{m=1}^qq_ma_mb_m.$$ For the cases where the end vertices of $b$ or $b'$ are contained in the same connected component, the inequality is trivial.\ \ [[Max wired boundary condition case.]{}]{}\ Suppose that $ \mathrm{\widetilde{m}}\in {\cap}_{{i\in\mathbb{V}}}\mathcal{Q}_{i,\mathrm{max}} (\pmb{\hat{h}}) $. To prove the inequality we have to analyze again the three cases above. For case $(I)$, analogously to the free boundary condition case, we have $${ \Theta_{V,\mathrm{ \widetilde{m}}}(A_1\cup A_2) \Theta_{V,\mathrm{ \widetilde{m}}}(B_1\cup B_2) \over \Theta_{V,\mathrm{ \widetilde{m}}}(A_1\cup A_2) \Theta_{V,\mathrm{ \widetilde{m}}}(B_1) \Theta_{V,\mathrm{ \widetilde{m}}}(B_2) } = { \Theta_{V,\mathrm{ \widetilde{m}}}(B_1\cup B_2) \Theta_{V,\mathrm{ \widetilde{m}}}(A_1) \Theta_{\mathrm{ \widetilde{m}}}(A_2) \over \Theta_{V,\mathrm{ \widetilde{m}}}(A_1) \Theta_{V,\mathrm{ \widetilde{m}}}(A_2) \Theta_{V,\mathrm{ \widetilde{m}}}(B_1) \Theta_{V,\mathrm{ \widetilde{m}}}(B_2) },$$ independently on whether the components $A_1,A_2,B_1$ and $B_2$ and the possible combinations among them intersect $V^c$. For the case $(II)$ and all the configurations sketched on the figure below we have from the definition of $\Theta_{V,\mathrm{ \widetilde{m}}}$ that the following equality holds $$c_{\mathrm{\widetilde{m}}}a_{\mathrm{\widetilde{m}}} b_{\mathrm{\widetilde{m}}}c_{\mathrm{\widetilde{m}}} = a_{\mathrm{\widetilde{m}}}c_{\mathrm{\widetilde{m}}} b_{\mathrm{\widetilde{m}}}c_{\mathrm{\widetilde{m}}}.$$ For the following configurations that also appears in the case $(II)$ : For $(a)$, the inequality comes from $${\textstyle \left(\sum_{m=1}^q q_m c_m\right) a_{\mathrm{\widetilde{m}}}b_{\mathrm{\widetilde{m}}} c_{\mathrm{\widetilde{m}}} \geqslant a_{\mathrm{\widetilde{m}}}c_{\mathrm{\widetilde{m}}} \left(\sum_{m'=1}^q q_{m'}b_{m'}c_{m'}\right), }$$ which is always valid since we have that $b_{\mathrm{\widetilde{m}}}\geqslant b_{m'}, \ \forall m'=1,\ldots,q.$ In $(b)$ inequality , comes from $${\textstyle \left(\sum_{m=1}^q q_m c_m \right) a_\mathrm{\widetilde{m}}b_\mathrm{\widetilde{m}} c_\mathrm{\widetilde{m}} \geqslant \left(\sum_{m=1}^q q_m a_m c_m\right) b_\mathrm{\widetilde{m}} c_\mathrm{\widetilde{m}}, }$$ which is also true because $a_\mathrm{\widetilde{m}}\geqslant a_{m}, \ \forall m=1,\ldots,q.$ Finally, in $(c)$ inequality , is a consequence of $${\textstyle \left(\sum_{m=1}^q q_mc_m\right) a_\mathrm{\widetilde{m}} b_\mathrm{\widetilde{m}} c_\mathrm{\widetilde{m}} \geqslant a_\mathrm{\widetilde{m}} c_\mathrm{\widetilde{m}} b_\mathrm{\widetilde{m}} c_\mathrm{\widetilde{m}} }$$ and the validity of this inequality is ensured by $${\textstyle \sum_{m=1}^q q_mc_m \geqslant \underset{ m\in\cap_{i\in \mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) }{\sum} q_mc_{\mathrm{ max}} \geqslant c_{\mathrm{ max}}=c_\mathrm{\widetilde{m}}, }$$ which follows from . Now we consider the case $(III)$, by splitting its analysis in the following sub-cases For $(d)$ the inequality , is valid as long as $${\textstyle a_\mathrm{\widetilde{m}} \left( \sum_{m'=1}^q\ q_{m'}b_{m'} \right) \geqslant a_\mathrm{\widetilde{m}}b_\mathrm{\widetilde{m}}. }$$ This is in fact true because $${\textstyle \sum_{m'=1}^q\ q_{m'}b_{m'} \geqslant \underset{ m' \in \cap_{i\in \mathbb{V}} \mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) }{\sum} q_{m'}b_{\mathrm{ max}} \geqslant b_{\mathrm{ max}}=b_\mathrm{\widetilde{m}}. }$$ For $(e)$, the desired inequality follows from $${\textstyle \left(\sum_{m=1}^q q_m a_m\right)b_\mathrm{\widetilde{m}} \geqslant a_\mathrm{\widetilde{m}} b_\mathrm{\widetilde{m}}, }$$ but this inequality holds because $$\sum_{m=1}^q\ q_{m}a_{m} \geqslant \underset{ m\in\cap_{i\in \mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) }{\sum} \ q_{m}a_{\mathrm{ max}} \geqslant a_{\mathrm{ max}}=a_\mathrm{\widetilde{m}}.$$ For the last sub-case $(f)$, we have to prove that $ a_\mathrm{\widetilde{m}} b_\mathrm{\widetilde{m}} = a_\mathrm{\widetilde{m}} b_\mathrm{\widetilde{m}}, $ which is obviously true. In the max wired boundary conditions, if the end vertices of $b$ or $b'$ belong to the same component, the result follows.\ \ To finish the proof we need to address the case where $\xi$ has at most one zero and $\omega^{(1)}\equiv \xi\vee\omega^{(b)}$ and $\omega^{(2)}\equiv \xi\vee\omega^{(b')}$. Suppose that $$\xi\equiv (1,\ldots,1,\aunderbrace[D]{0}_{b\mathrm{-th}}, 1,\ldots,1,\aunderbrace[D]{1}_{b'\mathrm{-th}},1,\ldots,1)$$ where $b,b'\in E\cup \partial E$ with $b\not=b'$. By defining $$\xi^b\equiv (1,\ldots,1,\aunderbrace[D]{1}_{b\mathrm{-th}}, 1,\ldots,1,\aunderbrace[D]{1}_{b'\mathrm{-th}},1,\ldots,1)$$ and $$\xi^{b'}\equiv (1,\ldots,1,\aunderbrace[D]{0}_{b\mathrm{-th}}, 1,\ldots,1,\aunderbrace[D]{1}_{b'\mathrm{-th}},1,\ldots,1),$$ we can see that $ \omega^{(1)}=\xi\vee\omega^{(b)}=\xi^b \text{,} \quad \omega^{(2)}=\xi\vee\omega^{(b')}=\xi^{b'} \quad \text{and} \quad \omega^{(1)}\wedge\omega^{(2)}=\xi. $ In this case, to prove for both “free” and “max” wired boundary conditions, it is enough to prove that $ \mathcal{R}(\xi^b,\xi^{b'})=\mathcal{R}(\xi,\xi^{b'}) \ \mathrm{with} \ b\not = b', $ but this is trivial since $$\begin{gathered} \mathcal{R}(\xi^b,\xi^{b'})= { \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}} (\xi^b\vee\xi^{b'}) \over \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi^b) } = { \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi^b) \over \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi^b) } = { \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi) \over \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi) } \\ = { \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}} (\xi\vee\xi^{b'}) \over \mathcal{W}^{\mathrm{ GRC}}_{V,\mathrm{ \#}}(\xi) } = \mathcal{R}(\xi,\xi^{b'}). \end{gathered}$$ Edwards-Sokal model =================== #### Edwards-Sokal model with general boundary condition. Fix $q\in\mathbb{Z}^+$, for any finite set $V\subset\mathbb{V}$ and any fixed configurations $\sigma_{V^c},\omega_{\mathbb{B}(V)^c}$ prescribed outside of $V$, we define de Edwards-Sokal measure $\phi^{\mathrm{ ES}}_{V,\mathbb{B}(V)}$ as the normalization of the following weights $$\begin{aligned} \mathcal{W}(\sigma_V,\omega_{\mathbb{B}(V)}| \sigma_{V^c},\omega_{\mathbb{B}(V)^c}) = \prod_{\substack{\{i,j\}\in\mathbb{B}(V)\\ \omega_{ij}=1}}r_{ij} \delta_{\sigma_i,\sigma_j} \times \exp \big( \beta \underset{i\in V}{\sum} \sum_{p=1}^q h_{i,p}\delta_{\sigma_i,p} \big),\end{aligned}$$ where $r_{ij}$ has been defined in . #### Edwards-Sokal model with wired and free boundary conditions. From the previous definition we can observe that, for any finite volume $V\subset\mathbb{V}$, the state $ \phi^{\mathrm{ ES}}_{V,\mathbb{B}(V)} ( \cdot| \sigma_{V^c},\omega_{\mathbb{B}(V)^c} ) $ is independent of $\omega_{\mathbb{B}(V)^c}$, and we define $$\phi^{\mathrm{ ES}}_{V,\mathrm{ m}}(\cdot) \equiv \phi^{\mathrm{ ES}}_{V,\mathbb{B}(V)} ( \cdot| \sigma_{V^c}^\mathrm{m},\omega_{\mathbb{B}(V)^c} ),$$ where $\sigma^\mathrm{m}$ is the constant configuration, $\sigma^\mathrm{m}_i=\mathrm{m}$ for all $i\in \mathbb{V}$, with $\mathrm{m}\in\{1,\ldots,q\}$ fixed. This state is known as the $\mathrm{m}$-wired boundary condition state. By similar reasons we have that $ \phi^{\mathrm{ ES}}_{V,\mathbb{B}_0(V)} ( \cdot | \sigma_{V^c},\omega_{\mathbb{B}_0(V)^c} ) $ does not depend on $\sigma_{V^c}$, provided that the $\omega$-boundary condition is chosen as $\omega_{\mathbb{B}_0(V)^c}=\omega^0_{\mathbb{B}_0(V)^c}$, where $\omega^0$ denotes the configuration with $\omega^0_{ij}=0$ for all $\{i,j\}\in\mathbb{B}(\mathbb{V})$. In this case we introduce the notation $$\phi^{\mathrm{ ES}}_{V,\mathrm{ free}}(\cdot) \equiv \phi^{\mathrm{ ES}}_{V,\mathbb{B}_0(V)} ( \cdot| \sigma_{V^c},\omega^0_{\mathbb{B}_0(V)^c} ).$$ Gibbs states and limit states ============================= #### Gibbs states. Let $\mathscr{P}(\Omega)$ denote the set of probability measures defined on some probability space $\Omega$. Since the families $\{\phi_\mathbb{B}^{\mathrm{ GRC}}\}$ and $\{\phi_{V,\mathbb{B}(V)}^{\mathrm{ ES}}\}$ are specifications (see [@Georgii88]), we can define as usual the set of the Gibbs measures compatible with these specifications as follows $$\begin{aligned} \label{Gibbs-rcm} \mathscr{G}^{\mathrm{ GRC}} \equiv \left\{ \phi\in\mathscr{P}(\Omega): \begin{array}{l} \phi(f) \stackrel{\mathrm{ DLR}}{=} \displaystyle \int \phi_\mathbb{B}^{\mathrm{ GRC}}(f|\omega_{\mathbb{B}^c}) \phi(\textrm{d}\omega), \\[0.3cm] \qquad\qquad \mathrm{supp}(f)\subset \mathbb{B} \end{array} \right\}\end{aligned}$$ and $$\mathscr{G}^{\mathrm{ ES}} \equiv \left\{ \nu\in\mathscr{P}(\Omega\times\Sigma): \begin{array}{l} \nu(f) \stackrel{\mathrm{ DLR}}{=} \displaystyle\int \phi_{V,\mathbb{B}(V)}^{ \mathrm{ ES}}(f|\sigma_{V^c},\omega_{\mathbb{B}(V)^c})\, d\nu(\sigma,\omega), \\[0.3cm] \ \qquad\qquad \mathrm{supp}(f)\subset V\times\mathbb{B}(V) \end{array} \right\}.$$ That is, $\mathscr{G}^{\mathrm{ GRC}}$ and $\mathscr{G}^{\mathrm{ ES}}$ are the class of probability measures (Gibbs measures) that are preserved for their respective probability kernels. #### Limit states. On the other hand, we define the set of the thermodynamic limits of the specification $\{\phi_{\mathbb{B}_n}^{\mathrm{ GRC}}\}$, where $\{\mathbb{B}_n\}$ is a cofinal collection in $\mathbb{E}$ : $$\begin{aligned} \label{Gibbs-limit} \mathscr{G}^{\mathrm{ GRC}}_{\mathrm{ lim}} \equiv \big\{ \phi\in\mathscr{P}(\{0,1\}^\mathbb{\mathbb{E}}): \phi\stackrel{\mathrm{ weak}}{=}\lim_{n\to\infty} \phi_{\mathbb{B}_n}^{\mathrm{ GRC}}(\cdot|\omega_n) \big\}.\end{aligned}$$ In general, it is not easy to relate the sets $\mathscr{G}^{\mathrm{ GRC}}$ and $\mathscr{G}^{\mathrm{ GRC}}_{\mathrm{ lim}}$ due to the lack of quasilocality of the specifications $\phi_\mathbb{B}^{\mathrm{ GRC}}$. One case where these sets can be related is the case in which we assume the existence of at most one connected component with probability one. As a consequence of Lemma \[quasilocality\], one can prove the following relation: $ \mathscr{G}^{\mathrm{ GRC}}_{\mathrm{ lim}} \subset \mathscr{G}^{\mathrm{ GRC}} $, see Lemma \[continencia-Gibbs\] below. For more details see [@BBCK00]. By using the FKG property for the GRC model and the previous definitions, one can prove the following theorem which ensures the existence of thermodynamic limit. \[limite termodinamico\] Let $\beta\geqslant 0$, $\pmb{J}=(J_{ij}:\{i,j\}\in \mathbb{E})$ $\in [0,\infty)^\mathbb{E}$ and $ \pmb{\hat{h}} \equiv (h_{i,p}\in\mathbb{R}:i\in \mathbb{V};\ p=1,\ldots,q) $. For each increasing quasilocal function $f$ (see [@Georgii88]), - The following limits exist $$\phi_{\mathrm{{max}}}^{\mathrm{{GRC}}}(f)\equiv \lim_{V\uparrow \mathbb{V}} \phi_{V, \mathrm{{max}}}^{\mathrm{{GRC}}}(f) \quad \ \mathrm{and} \ \quad \phi_{\mathrm{{free}}}^{\mathrm{{GRC}}}(f)\equiv \lim_{V\uparrow \mathbb{V}} \phi_{V, \mathrm{{free}}}^{\mathrm{{GRC}}}(f).$$ - If in addition, $ \mathrm{m} \in {\cap}_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) $, then the following limits exist $$\phi_{\mathrm{{max}}}^{\mathrm{{ES}}}(f) \equiv \lim_{V\uparrow \mathbb{V}} \phi_{V, \mathrm{{max}}}^{\mathrm{{ES}}}(f) \quad \ \mathrm{and} \ \quad \phi_{\mathrm{{free}}}^{\mathrm{{ES}}}(f) \equiv \lim_{V\uparrow \mathbb{V}} \phi_{V, \mathrm{{free}}}^{\mathrm{{ES}}}(f).$$ - If $\phi\in\mathscr{G}_{\mathrm{ lim}}^{\mathrm{ GRC}}$ or $\phi\in\mathscr{G}^{\mathrm{ GRC}}$, then for each increasing quasilocal function $f$ we have $$\phi_{\mathrm{ free}}^{\mathrm{ GRC}}(f) \leqslant \phi(f) \leqslant \phi_{\mathrm{ max}}^{\mathrm{ GRC}}(f).$$ The proof is similar to the proof of Theorem III.1 in [@BBCK00]. Since we are also interested in monotonicity properties with respect to the magnetic field, it is needed to introduce a partial order between two fields [@BBCK00]. Given two arbitrary magnetic fields $\pmb{\hat{h}}$ and $\pmb{\hat{h}'}$, we say that $$\begin{aligned} \label{relação de ordem nos campos} \pmb{\hat{h}}\prec \pmb{\hat{h}'} \ \ \Longleftrightarrow \ \ \forall \ i\in\mathbb{V}: \ h_{i,k}-h_{i,l}\leqslant h'_{i,k}-h'_{i,l}, \ \ \ k,l=1,\ldots,q\end{aligned}$$ whenever $h_{i,k}-h_{i,l}>0.$ \[teo-monotonicidade-campo-externo-grc\] Let $\pmb{\hat{h}}$ and $\pmb{\hat{h}'}$ be two arbitrary magnetic fields such that $\pmb{\hat{h}}\prec\pmb{\hat{h}'}$. Denote by $\phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{\mathrm{ \#}}$ and $\phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{\mathrm{ \#}}$ their respective measures defined in Theorem \[limite termodinamico\], where $\mathrm{\#}$  stands for “$\mathrm{free}$” or “$\mathrm{max}$”. Then, for any quasilocal increasing function $f$ we have $$\begin{aligned} \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{\mathrm{ free}}(f) \leqslant \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{\mathrm{ free}}(f) \qquad\text{and}\qquad \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{\mathrm{ max}}(f) \leqslant \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{\mathrm{ max}}(f). \end{aligned}$$ By the Holley Theorem, the stochastic domination claimed in the statement of the theorem is proved as long as the following lattice condition is satisfied $$\begin{aligned} \label{cond. latice} \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{V,\mathrm{ \#}} (\omega^{(1)}\vee \omega^{(2)}) \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{V,\mathrm{ \#}} (\omega^{(1)}\wedge \omega^{(2)}) \geqslant \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{V,\mathrm{ \#}} (\omega^{(1)}) \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{V,\mathrm{ \#}} (\omega^{(2)}) \end{aligned}$$ for all $\omega^{(1)},\omega^{(2)}\in\{0,1\}^E$, where $\mathrm{\#}$ denotes the “free” and “max” wired boundary conditions. For details, see Theorem 2.3, item (c), p. 20 in [@Grimmett2]. It is also well known that is a consequence of $$\begin{aligned} \label{divisão de medidas} { \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{V,\mathrm{ \#}} (\zeta^e) \over \phi^{\mathrm{ GRC},\pmb{\hat{h}'}}_{V,\mathrm{ \#}} (\zeta_{(e)}) } \geqslant { \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{V,\mathrm{ \#}} (\xi^e) \over \phi^{\mathrm{ GRC},\pmb{\hat{h}}}_{V,\mathrm{ \#}} (\xi_{(e)}) }, \end{aligned}$$ for any $\xi\preceq \zeta$ and $e\in E$, where $\xi_{(e)}$ ($\xi^e$) is the configuration that agrees with $\xi$ in all edges, except in $e$ where its value is zero (one). We shall remark that the notations $\xi_e$ and $\xi_{(e)}$ have different meaning. Without loss of generality, we can assume that $\xi$ and $\zeta$ are of the form $$\xi\equiv(*,\ldots,*,\aunderbrace[D]{0}_{\mathrm{ e-th}},*, \ldots,*) \quad \text{and} \quad \zeta\equiv(*',\ldots,*',\aunderbrace[D]{0}_{\mathrm{ e-th}}, *',\ldots,*'),$$ with $\xi\preceq \zeta.$ Let $k'\equiv \underset{\{i,j\}:\zeta_{ij}=1}{\prod}r_{ij}$ and $k\equiv \underset{\{i,j\}:\xi_{ij}=1}{\prod}r_{ij}$. From the definitions we get that $$\zeta_{(e)}=\zeta, \quad \xi_{(e)}=\xi, \quad \prod_{\{i,j\}:\zeta^e_{ij}=1}r_{ij}=r_e k' \quad \text{and} \quad \prod_{\{i,j\}:\xi^e_{ij}=1}r_{ij}=r_e k.$$ Therefore $$\begin{aligned} { B_{\pmb{j}}(\zeta^e) \over B_{\pmb{j}}(\zeta_{(e)}) } = { \prod_{\{i,j\}:\zeta^e_{ij}=1}r_{ij} \over \prod_{\{i,j\}:\zeta_{ij}=1}r_{ij} } = r_e = { \prod_{\{i,j\}:\xi^e_{ij}=1}r_{ij} \over \prod_{\{i,j\}:\xi_{ij}=1}r_{ij} } = { B_{\pmb{j}}(\xi^e) \over B_{\pmb{j}}(\xi_{(e)}) }. \end{aligned}$$ So it follows from the equation above that is a consequence of $$\begin{aligned} \label{novotheta} { \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(\zeta^e) \over \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(\zeta_{(e)}) } \geqslant { \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(\xi^e) \over \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(\xi_{(e)}) }, \end{aligned}$$ for both “free” and “max” wired boundary conditions. If $e=\{x,y\}$ and $x\leftrightarrow y$ in $\xi$, then is an equality. On the other hand, if $x\not\leftrightarrow y$ in $\xi$, then there are two connected components $A\equiv C(x,\xi)$ and $B\equiv C(y,\xi)$ containing the vertices $x$ and $y$, respectively. If $e$ is an open edge in $\xi$, then the components $A$ and $B$ are connected and will be denoted by $C\equiv A\cup B$. So $\vert C\vert=\vert A\vert +\vert B\vert$, from where we deduce that $$\begin{aligned} { \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(\zeta^e) \over \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(\zeta_{(e)}) } \cdot { \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(\xi_{(e)}) \over \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(\xi^e) } = { \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(C) \over \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(C) } \cdot { \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(A) \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(B) \over \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(A) \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(B) }, \end{aligned}$$ for either free or max wired boundary conditions. In order to prove , it is enough to prove that $$\begin{aligned} \label{maior=1} { \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(C) \over \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(C) } . { \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(A) \Theta^{\pmb{\hat{h}}}_{V,\mathrm{ \#}}(B) \over \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(A) \Theta^{\pmb{\hat{h}'}}_{V,\mathrm{ \#}}(B) } \geqslant 1. \end{aligned}$$ To establish the above inequality, we analyze separately the “free” and “max” wired boundary condition cases.\ \ [Free boundary condition case.]{}\ Keeping the notation used in the proof of the FKG inequality, for each $m\in\{1,\ldots,q\}$ we define $${ a_m \equiv \exp \big( \beta\underset{i\in A}{\sum}h_{i,m} \big), \ b_m \equiv \exp \big(\beta\underset{i\in B}{\sum}h_{i,m} \big) \ \ \text{and} \ \ c_m \equiv \exp \big( \beta\underset{i\in C}{\sum}h_{i,m} \big). }$$ Similarly we define $a'_{m}, b'_{m}$ and $c'_{m}$ by replacing $(h_{i,m})$ for $(h'_{i,m})$. With this notation, reads $$\begin{gathered} \label{des1} { \textstyle (\sum_{j=1}^q q_j a'_j) (\sum_{k=1}^q q_k b'_k) (\sum_{l=1}^q q_l a_l b_l) } \\[0,2cm] \leqslant {\textstyle (\sum_{j=1}^q q_j a_j) (\sum_{k=1}^q q_k b_k) (\sum_{l=1}^q q_l a'_l b'_l). } \end{gathered}$$ The proof of is divided in two steps.\ \ **Step 1**:(move the primes from $a_j$’s) we claim that $$\begin{gathered} \label{des2} {\textstyle (\sum_{j=1}^q q_j a'_j) (\sum_{k=1}^q q_k b'_k) (\sum_{l=1}^q q_l a_l b_l) } \\[0,2cm] \leqslant {\textstyle (\sum_{j=1}^q q_j a_j) (\sum_{k=1}^q q_k b'_k) (\sum_{l=1}^q q_l a'_l b_l) }. \end{gathered}$$ In fact, we first remark that without loss of generality we can assume that $h_{i,l}-h_{i,j}>0$, $\forall i\in \mathbb{V}$. From the hypothesis we have $\pmb{\hat{h}}\prec\pmb{\hat{h}'}$, so we get $\forall \ l,j=1,\ldots,q$ and $\forall \ i\in \mathbb{V}$ that $h_{i,l}-h_{i,j}\leqslant h'_{i,l}-h'_{i,j}$. From the last inequality, it follows that $$\begin{aligned} \label{implicacion} {a_l\over a_j}\leqslant {a'_l\over a'_j}, \quad \text{which implies} \quad a'_ja_l-a_ja'_l \leqslant 0. \end{aligned}$$ On the other hand, since $h_{i,l}-h_{i,j}>0$, we have $b_l-b_j>0$. Putting together the last two inequalities yields $$\begin{aligned} (a'_ja_l-a_ja'_l)(b_l-b_j)\leqslant 0, \end{aligned}$$ and we conclude that $ a'_j a_l b_l \leqslant \left[a'_ja_l-a_ja'_l\right]b_j + a'_l a_j b_l \leqslant a_j a'_l b_l, $ where in the last inequality we have used . By multiplying the above inequality for $q_j q_k q_l b'_k$ and then summing over $j,k,l=1,\ldots, q$, we prove the claim.\ \ **Step 2**:(move the primes from $b_k$’s) we claim that $$\begin{gathered} \label{des3} {\textstyle (\sum_{j=1}^q q_j a_j) \left(\sum_{k=1}^q q_k b'_k\right) \left(\sum_{l=1}^q q_l a'_l b_l\right) } \\[0,2cm] \leqslant {\textstyle (\sum_{j=1}^q q_j a_j) (\sum_{k=1}^q q_k b_k) (\sum_{l=1}^q q_l a'_l b'_l). } \end{gathered}$$ The proof is similar to the one given for the Step 1. We assume that $h_{i,l}-h_{i,k}>0$, $\forall \ i\in \mathbb{V}$ and prove in place of that $b'_k b_l - b_k b'_l \leqslant 0$, proceeding similarly to reach the conclusion. Finally, by piecing together the inequalities and , we obtain .\ \ [Max wired boundary condition case.]{}\ We first observe that if $ \mathrm{m} \in {\cap}_{i\in\mathbb{V}}\mathcal{Q}_{i,m}(\pmb{\hat{h}}) $ and $ \mathrm{\widetilde{m}} \in {\cap}_{i\in\mathbb{V}} \mathcal{Q}_{i,\widetilde{m}}(\pmb{\hat{h}'}) $, then $\mathrm{m}=\mathrm{\widetilde{m}}$. Given two connected components $A$ and $B$, if $A\cap V^c=\emptyset$ and $B\cap V^c=\emptyset$, then the inequality follows from the free boundary condition case. The remaining cases will be analyzed by considering the following cases: In the case $(a)$, is equivalent to the inequality $$\label{fin1} \begin{array}{c} a'_\mathrm{m} \left(\sum_{k=1}^q q_k b'_k\right)a_\mathrm{m} b_\mathrm{m} \leqslant a_\mathrm{m} \left(\sum_{k=1}^q q_k b_k\right)a'_\mathrm{m} b'_\mathrm{m}. \end{array}$$ To see that this inequality holds, it is sufficient to observe that the ordering between the magnetic fields implies $ b_\mathrm{m} b'_k\leqslant b'_\mathrm{m} b_k. $ Multiplying this last inequality by $a'_m a_m$ and summing over $k=1,\ldots,q$, we obtain . In the case $(b)$ the inequality reduces to $$\label{fin2} \begin{array}{c} (\sum_{j=1}^q q_j a'_j)b'_\mathrm{m} a_\mathrm{m} b_\mathrm{m} \leqslant (\sum_{j=1}^q q_j a_j)b_\mathrm{m} a'_\mathrm{m} b'_\mathrm{m}. \end{array}$$ Now we use that the magnetic field ordering implies that $ a_\mathrm{m} a'_j\leqslant a'_\mathrm{m} a_j $ and then proceed similarly to the previous case. Finally, in the case $(c)$ the inequality is equivalent to $ a'_\mathrm{m} b'_\mathrm{m} a_\mathrm{m} b_\mathrm{m} = a'_\mathrm{m} b'_\mathrm{m} a'_\mathrm{m} b'_\mathrm{m}, $ which is trivial. Our next result is the monotonicity, in the FKG sense, with respect to the coupling constants $\pmb{J}$ in the special case where $J_{ij}\equiv J,$ $\forall$ $i,j\in\mathbb{V}$. \[teorema ge\] Suppose that $0\leqslant J_1<J_2$ are two coupling constants. For each finite $V\subset\mathbb{V}$ denote by $\phi^{\mathrm{ GRC},J_k}_{V,\mathrm{ \#}}$ , k=1,2; the measure defined by the weights or by the weights with $ \mathrm{m} \in {\cap}_{i\in\mathbb{V}} \in \mathcal{Q}_{i,\mathrm{ max}}$$(\pmb{\hat{h}}). $ Then $$\phi^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}}(f) \leqslant \phi^{\mathrm{ GRC},J_2}_{V, \mathrm{ free}}(f),$$ where $f$ is a cylindrical increasing function, and $\mathrm{\#}$ stands for “$\mathrm{free}$” or “$\mathrm{max}$”. By assuming $J_1<J_2$, we get that $e^{q\beta J_1}-1<e^{q\beta J_2}-1$. For any configuration $\omega\in\{0,1\}^{\mathbb{B}(V)}$, we define the function $g:\{0,1\}^{\mathbb{B}(V)}\to\mathbb{R}$ by $$g(\omega) \equiv \left[ \dfrac{e^{q\beta J_1}-1}{e^{q\beta J_2}-1} \right]^{o(\omega)} \times \displaystyle { \prod_{\substack{C(\omega): \\ \mathbb{V}(C(\omega))\cap \partial V\not= \emptyset}} } \exp \big( \beta\sum_{i\in C(\omega)}h_{i,\mathrm{ max}} \big),$$ where $o(\omega)$ denotes the numbers of open edges in $\omega$. One can easily see that the function $g$ is decreasing since $g$ is composed by the product of non-negative decreasing functions. Let $f:\{0,1\}^{\mathbb{B}(V)}\to\mathbb{R}$ be a cylindrical increasing arbitrary function. Since $J_{ij}\equiv J,$ we have the following expression for Bernoulli factor: $B_{J_k}(\omega)=\left(e^{q\beta J_k}-1\right)^{o(\omega)}$, $k=1,2$. From the definition of the expected values we obtain $$\begin{gathered} \label{r_1} \phi^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}}(f) = { 1 \over Z^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}} } \sum_{\omega\in\{0,1\}^{\mathbb{B}(V)}} f(\omega) { \left(e^{q\beta J_1}-1\right)^{o(\omega)} \displaystyle \prod_{C(\omega)} \Theta_{V,\mathrm{ max}}(C(\omega)) } \\[0,3cm] = { 1 \over Z^{\mathrm{ GRC},J_2}_{V,\mathrm{ max}} } \sum_{\omega\in\{0,1\}^{\mathbb{B}(V)}} f(\omega) g(\omega) \left(e^{q\beta J_2}-1\right)^{o(\omega)} \\[0,2cm] \hspace*{4,2cm} \times \prod_{{\substack{C(\omega): \\ \mathbb{V}(C(\omega))\cap \partial V= \emptyset}}} \sum_{p=1}^q e^{\beta\sum_{i\in C(\omega)}h_{i,p}} \times { Z^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}} \over Z^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}} } \\[0,3cm] = \phi^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}}(f\cdot g) \times { Z^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}} \over Z^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}} },\end{gathered}$$\ where $Z^{\mathrm{ GRC},J}_{V,\mathrm{ \#}}$ denotes the normalization constant of the measure $\phi^{\mathrm{ GRC},J}_{V,\mathrm{ \#}}$ and $\mathrm{\#}$ stands for or . By taking $f\equiv1$ in we get the following equality $$\phi^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}}(g) = { Z^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}} \over Z^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}} }.$$ Using the last equation, and the strong FKG property (Theorem \[Strong FKG Property\]) we finally conclude that $$\phi^{\mathrm{ GRC},J_1}_{V,\mathrm{ max}}(f) = { \phi^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}}(f\cdot g) \over \phi^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}}(g) } \stackrel{\mathrm{ FKG}}{\leqslant} \phi^{\mathrm{ GRC},J_2}_{V,\mathrm{ free}}(f).$$ Note that Theorem \[teorema ge\] can be extended using Item $(iii)$ of Theorem \[limite termodinamico\] for any pair of $\mathrm{GRC}$ Gibbs measures at $J=J_1,$ resp. $J=J_2$. As a particular case, we obtain the following corollary. \[coro ge0\] Suppose that $0\leqslant J_1<J_2$ are two coupling constants. For each finite $V\subset\mathbb{V}$ denote by $\phi^{\mathrm{ GRC},J_k}_{V,\mathrm{ \#}}$ , k=1,2; the measure defined by the weights or by the weights with $ \mathrm{m}\in {\cap}_{i\in\mathbb{V}} \in\mathcal{Q}_{i,\mathrm{ max}} $ $(\pmb{\hat{h}}).$ Then $$\phi^{\mathrm{ GRC},J_1}_{V,\mathrm{ \#}}(f) \leqslant \phi^{\mathrm{ GRC},J_2}_{V, \mathrm{ \#}}(f),$$ where $f$ is a cylindrical increasing function and $\mathrm{\#}$ stands for “$\mathrm{free}$” or “$\mathrm{max}$”. GRC model and quasilocality =========================== In what follows we study the quasilocality of the random-cluster model in non-homogeneous magnetic field. The next lemma tells us that the specifications $\{\phi_{\mathbb{B}}^{\mathrm{{GRC}}}\}$ are almost surely quasilocal (see [@Georgii88; @Pfister]). To give a precise statement of this lemma, we need to introduce some notation: $$\mathscr{M}(\Delta,\Lambda) \equiv \Big\{ \! \omega\in \{0,1\}^{\mathbb{E}} : \forall x,y\in\Lambda, \ x\leftrightarrow\Delta^c \ \text{and} \ y\leftrightarrow\Delta^c \Rightarrow x \xleftrightarrow[\vspace*{-1cm}\mathbb{B}_0(\Delta)]{}y \! \Big\}$$ where $\Lambda\subset\Delta$ are finite subsets in $\mathbb{V}$. The following lemma is an adaptation of Lemma VI.2 in [@BBCK00] for our model. \[quasilocality\] Let $\mathbb{B}\subset\mathbb{B}_0(\mathbb{E})$ be a finite set and $f$ a cylindrical function depending only on the edges in $\mathbb{B}$. Then, for each pair of finite subsets $(\Delta,\Lambda)$ with $\mathbb{V}(\mathbb{B})\subset \Lambda\subset \Delta$, the function $$\omega\mapsto \mathds{1}_{\mathscr{M}(\Delta,\Lambda)}(\omega) \phi_{\mathbb{B}}^{\mathrm{ GRC}}(f\vert \omega_{\mathbb{B}^c})$$ is quasilocal. If in addition $\phi\in \mathscr{G}^{\mathrm{{GRC}}}_{\mathrm{{lim}}}$ or $\phi\in \mathscr{G}^{\mathrm{{GRC}}}$ have at most one infinite connected component and $\Lambda\subset \mathbb{V}$, then $$\phi(\mathscr{M}(\Delta,\Lambda))\uparrow 1, \quad \text{whenever} \ \ \Delta\uparrow\mathbb{V}.$$ Recalling the definition of $ \phi_{\mathbb{B}}^{\mathrm{ GRC}} (\cdot |\omega_{\mathbb{B}^c}) $, we note that it is enough to prove that the function $$\begin{aligned} \label{quasilocalidade} \omega\mapsto \mathds{1}_{\mathscr{M}(\Delta,\Lambda)}(\omega) \phi_{\mathbb{B}}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c}), \quad \forall \ \overline{\omega}_\mathbb{B}\in \{0,1\}^{\mathbb{B}}\end{aligned}$$ is quasilocal. In the sequel we shall prove the quasilocality of the mapping defined in . Let $\tilde{\Delta}$ be a finite subset of $\mathbb{V}$ such that $ \Delta\subset\tilde{\Delta}$. Consider the following configurations: $$\begin{array}{c} \omega \equiv (*,\ldots,*,\aunderbrace[D]{0}_{b\mathrm{-th}},*,\ldots), \qquad \omega^b \equiv (*,\ldots,*,\aunderbrace[D]{1}_{b\mathrm{-th}},*,\ldots) \end{array}$$ where $*$ is an arbitrary element in $\{0,1\}$ and $b\in\mathbb{B}(\tilde{\Delta})^c$. Suppose that $\omega\in\mathscr{M}(\Delta,\Lambda)$ and that there exists a connected component $C^{*}$ connecting $\Lambda$ to $\mathbb{B}(\tilde{\Lambda})^c$ in $\omega$. By definition, we have that $\omega^b\in \mathscr{M}(\Delta,\Lambda)$ and the connected component $C^{*}$ is unique. Let us consider two cases: $$\mathrm{1)} \ \mathbb{V}(C^{*})\cap\mathbb{V}(\{b\})=\emptyset \quad \mathrm{and} \quad \mathrm{2)} \ \mathbb{V}(C^{*})\cap\mathbb{V}(\{b\})\neq \emptyset.$$ In the first case we trivially have $ |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c})|=0. $ The second case is more elaborate. We consider separately two cases. We first assume that there is some $\varepsilon>0$ such that $\varepsilon<|h_{i,\mathrm{ max}}-h_{i,m}|$ for all $i\in \mathbb{V}$ and $m\in \{1,\ldots,q\}$. For this case let us denote by $C^*_b$ the connected component $\mathbb{V}(C^{*})\cap\mathbb{V}(\{b\})$. Then $$\begin{gathered} |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c})| \leqslant \overline{B}_J(\omega) k(\omega)\times \\[0,3cm] \times \sum_{m=1}^qq_m \Big| \exp \big( -\beta\sum_{i\in C^*_b}(h_{i,\mathrm{ max}}-h_{i,m}) \big) - \exp \big( -\beta\sum_{i\in C^*}(h_{i,\mathrm{ max}}-h_{i,m}) \big) \Big|,\end{gathered}$$ where $$k(\omega)\equiv \prod_{\substack{C(\omega):\mathbb{V}(C) \cap\mathbb{V} (\mathbb{B})\not=\emptyset \\|C|<\infty}} \, \sum_{m=1}^q q_m \exp \big( -\beta\sum_{i\in C}(h_{i,\mathrm{ max}}-h_{i,m}) \big) < \infty$$ and $$\overline{B}_J(\omega) = \prod_{\{i,j\}\in\mathbb{B}:\omega_{ij}=1} r_{ij}.$$ Suppose that $ m \not\in \underset{i\in\mathbb{V}}{\cap}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) $, then we have $$d(b,\Lambda)\leqslant |\mathbb{V}(C^*_b)| \leqslant \sum_{i\in C^{*}_b} \frac{1}{\varepsilon} (h_{i,\mathrm{ max}}-h_{i,m}).$$ The last inequalities imply that if $d(b,\Lambda)\to \infty$, then $$\sum_{i\in C^{*}_b}(h_{i,\max}-h_{i,m})\to \infty \qquad \text{and} \qquad \sum_{i\in C^{*}}(h_{i,\mathrm{ max}}-h_{i,m})\to \infty,$$ whenever $ m \not\in \underset{i\in\mathbb{V}}{\cap} \mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) $. Therefore, whenever $d(b,\Lambda)\to\infty$, we have that $$\begin{aligned} \label{condicao fraca de quasilocalidade} |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c})| \to 0.\end{aligned}$$ In the case $\liminf_{i\in \mathbb{V}} |h_{i,\mathrm{ max}}-h_{i,m}|=0$ it is enough to analyze whether $$\sum_{i\in C^*}(h_{i,\mathrm{ max}}-h_{i,m})$$ is finite or not. If it is infinite, then the result is trivial. Otherwise we use the continuity of the exponential function and a suitable choice of $b$ so that $d(b,\Lambda)\to\infty$. Now we consider two different configurations: $$\hat{\omega} \equiv (*,\ldots,*,\aunderbrace[D]{0}_{b\mathrm{-th}},*,\ldots,*, \aunderbrace[D]{1}_{ b'\mathrm{-th}} ,*,\ldots)$$ and $$\tilde{\omega} \equiv (*,\ldots,*,\aunderbrace[D]{1}_{b\mathrm{-th}},*,\ldots,*, \aunderbrace[D]{0}_{ b'\mathrm{-th}} ,*,\ldots),$$ where $*$ is arbitrary in $\{0,1\}$ and $b, b'\in\mathbb{B}(\tilde{\Delta})^c$ . We also denote $$\omega \equiv (*,\ldots,*,\aunderbrace[D]{0}_{b\mathrm{-th}},*,\ldots,*, \aunderbrace[D]{0}_{ b'\mathrm{-th}} ,*,\ldots,*),$$ with $*$ arbitrary in $\{0,1\}$. See that $\hat{\omega}=\omega^{b'}$ and $\tilde{\omega}=\omega^b$. Then by and the triangle inequality we have $$\begin{gathered} \hspace*{3,7cm} |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \hat{\omega}^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \tilde{\omega}_{\mathbb{B}^c})| \\ \leqslant \\ |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega^{b'}_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c})| + |W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \omega_{\mathbb{B}^c})| \to 0,\end{gathered}$$ when $d(b',\Lambda), d(b,\Lambda)\to \infty.$ Following this reasoning, we prove that for any two distinct configurations $\hat{\omega}, \tilde{\omega}$ em $\mathbb{B}(\tilde{\Delta})^c$ we have $$\left|W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B}| \hat{\omega}^b_{\mathbb{B}^c}) - W_\mathbb{B}^{\mathrm{ GRC}}(\overline{\omega}_\mathbb{B} | \tilde{\omega}_{\mathbb{B}^c})\right| \to 0, \quad \text{whenever} \quad \min_{b\in A}d(b,\Lambda)\to\infty$$ and $A\equiv \{e\in\mathbb{B}(\tilde{\Delta})^c: \hat{\omega}_e\not=\tilde{\omega}_e\}$, thus proving the quasilocality of the application . Finally, in order to prove the second statement, it is enough to notice that $ \{ \mathscr{M}(\Delta,\Lambda):\Delta\subset \mathbb{V} \ \text{finite} \} $ is an increasing sequence of events. \[continencia-Gibbs\] Let $q\in \mathbb{Z}^+$, $\beta\geqslant 0$, $\pmb{J}=(J_{ij}:\{i,j\}\in \mathbb{E})$ $\in [0,\infty)^\mathbb{E}$, $ \pmb{\hat{h}} = (h_{i,p}\in\mathbb{R}: i\in\mathbb{V},\ p=1,\ldots, q) $ and $\{q_p: p=1,\ldots,q\}$ satisfying . If $\phi\in \mathscr{G}^{\mathrm{ GRC}}_{\mathrm{ lim}}$ and has at most one infinite connected component then $ \mathscr{G}^{\mathrm{ GRC}}_{\mathrm{ lim}} \subset \mathscr{G}^{\mathrm{ GRC}}. $ The proof of this lemma follows from Lemma \[quasilocality\] and the almost sure quasilocality. See [@BBCK00]. The next theorem gives sufficient conditions for quasilocality of the specifications under a geometric assumption of almost sure existence of an infinite connected component in the graph, thus facilitating many technical calculations. \[vec\] Let $\beta \geqslant 0, J_{ij}\geqslant 0,$ $h_{i,m}\in\mathbb{R}$, $\forall i,j\in \mathbb{V}$ and $q_m>0, m=1,\ldots,q$ satisfying . If $\phi\in\mathscr{G}^{\mathrm{{GRC}}}$ and has at most one infinite connected component almost surely, $\mathbb{B}\subset\mathbb{B}_0(\mathbb{V})$, and $f$ is a cylindrical function depending on the configuration $\omega_{\mathbb{B}}$, then $$\phi(f\vert \mathscr{F}_{\mathbb{B}^c}) = \phi^{\mathrm{ GRC}}_{\mathbb{B}}(f\vert \omega_{\mathbb{B}^c}), \quad \phi \mathrm{-a.s.}$$ *Sketch of the Proof.* The idea is the same as the one employed in the proof of the Theorem III.4 in [@BBCK00]. For the sake of completeness, we sketch a proof. Let $\mathbb{B}_1,$ $\mathbb{B}_2$ be finite sets of bonds with $\mathbb{B}_1\cap \mathbb{B}_2=\emptyset$ and $f$ and $g$ be bounded cylinder functions depending only on the bonds in $\mathbb{B}_1$ and $\mathbb{B}_2$, respectively. Using the DLR equation and the consistence of the specifications $\{\phi_\mathbb{B}^{\mathrm{ GRC}}\}$, for $\mathbb{B}\supset \mathbb{B}_1\cap\mathbb{B}_2$ we can easily obtain that $$\begin{aligned} \label{c31} \phi(gf)=\lim_{\mathbb{B}\uparrow\mathbb{E}} \int \phi_\mathbb{B}^{\mathrm{ GRC}} \big(g\phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f|\cdot) \big|\omega_{\mathbb{B}^c} \big) \phi(\mathrm{d}\omega).\end{aligned}$$ Let $\Delta\supset \mathbb{V}(\mathbb{B}_1)$, since both $g$ and $ \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))}(\cdot) \phi_{\mathbb{B}}^{\mathrm{ GRC}}(f\vert \cdot) $ are quasilocal, the function $ g\cdot \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))} \phi_{\mathbb{B}}^{\mathrm{ GRC}}(f\vert \cdot) $ can be approximated by local functions. Then by DLR equation , we have $$\begin{gathered} \label{c33} \phi \big( g\cdot \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))} \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f\vert \cdot) \big) \\[0,2cm] = \lim_{\mathbb{B}\uparrow\mathbb{E}} \int \phi_\mathbb{B}^{\mathrm{ GRC}} \big( g\cdot \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))} \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f|\cdot) \big|\omega_{\mathbb{B}^c} \big) \phi(\mathrm{d}\omega).\end{gathered}$$ From Lemma \[quasilocality\] we get $ \phi\big(\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))\big) \uparrow 1 $ whenever $\Delta \uparrow \mathbb{V}$. Since $f$ and $g$ are bounded, using the Dominated Convergence Theorem we have $$\lim_{\Delta\uparrow\mathbb{V}} \int \!\!\phi_\mathbb{B}^{\mathrm{ GRC}} \big( g\cdot \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))} \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f|\cdot) \big|\omega_{\mathbb{B}^c} \big) \phi(\mathrm{d}\omega) =\!\! \int \!\!\phi_\mathbb{B}^{\mathrm{ GRC}} \big(g \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f|\cdot) \big|\omega_{\mathbb{B}^c} \big) \phi(\mathrm{d}\omega)$$ and $$\lim_{\Delta\uparrow\mathbb{V}} \phi \big( g\cdot \mathds{1}_{\mathscr{M}(\Delta, \mathbb{V}(\mathbb{B}_1))} \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f\vert \cdot) \big) = \phi \big( g \phi_{\mathbb{B}_1}^{\mathrm{ GRC}}(f\vert \cdot) \big).$$ Combining the above limits, together with the items and , we have $$\phi(gf)=\phi \big( g\phi_{\mathbb{B}_1}^{\mathrm{ GRC}} (f\vert \cdot) \big)$$ for all bounded $g$ depending only on the configurations $\omega_{\mathbb{B}_1^c}$. From the almost sure uniqueness of conditional expectation with respect to $\phi$, the proof follows. Using the general theory of thermodynamic formalism, one can prove the following lemma. \[monotonicidade no volume\] Let $q\in \mathbb{Z}^+$, $\beta\geqslant 0$, $\pmb{J}=(J_{ij}:\{i,j\}\in \mathbb{E})$ $\in [0,\infty)^\mathbb{E}$, the magnetic field be $\pmb{\hat{h}}$ and the sequence $\{q_p: p=1,\ldots,q\}$ satisfy . If $\Lambda\subset V$ are finite subsets of $\mathbb{V}$, then for any cylindrical increasing function $f$ we have $$\phi_{\Lambda,\mathrm{ free}}^{\mathrm{ GRC}}(f) \leqslant \phi_{V,\mathrm{ free}}^{\mathrm{ GRC}}(f) \quad \text{and} \quad \phi_{\Lambda,\mathrm{ max}}^{\mathrm{ GRC}}(f) \geqslant \phi_{V,\mathrm{ max}}^{\mathrm{ GRC}}(f).$$ When $q_p=1$, for all $p=1,\ldots,q$ in , then we call the model simply the $\mathrm{RC}$ model. In this case, we define the set of Gibbs measures $\mathscr{G}^{\mathrm{ RC}}$ and $\mathscr{G}^{\mathrm{ RC}}_{\mathrm{ lim}}$ similarly to and . From now on, the study turns to some fundamental properties of the $\mathrm{RC}$ model. The following theorem is valid only for the random-cluster model. \[principal-marginal\] Let $q\in \mathbb{Z}^+$, $\beta\geqslant 0$, $\pmb{J}=(J_{ij}:\{i,j\}\in \mathbb{E})$ $\in [0,\infty)^\mathbb{E}$ and $\pmb{\hat{h}}$ be a magnetic field as previously defined. Given $ \nu\in\mathscr{G}^{\mathrm{{ES}}}$, let $\phi_\nu$ denote its edge-marginal. Then for any cylindrical increasing function $f$ we have $ \phi_{\nu}(f) \leqslant \phi_{\mathrm{ max}}^{\mathrm{ RC}}(f). $ For more details see the proof of Theorem III.2 reference [@BBCK00]. Uniqueness of the infinite connected component ============================================== We have so far developed the theory of the random-cluster model with non-uniform magnetic field for countably infinite graphs. We are interested in the situation in which the infinite connected component is (almost surely) unique, as is commonly the case for an “amenable graph”. The amenability hypothesis is important for the uniqueness of the infinite connected component in several models, [@Burton; @G06; @Hag06]. When the graph is non-amenable, the non-uniqueness of the infinite connected component is known for several models including the Bernoulli percolation and null magnetic field random-cluster model, see [@BS11; @G06; @Hag06] and references therein. Therefore, from now on we assume tacitly that the lattice $\mathbb{L}$ is amenable, that is, $ \inf \{|\partial_{\mathbb{E}} V|/|V|\} = 0, $ where the infimum ranges over all finite connected subsets $V$ of $\mathbb{V}$, and $\partial_{\mathbb{E}}V$ is the set of edges with one end-vertex in V and one in $\mathbb{V}\setminus V$. In what follows we denote by $N_\infty$ the random variable that counts the number of infinite connected components in both sample spaces $\Omega\equiv \{0,1\}^{\mathbb{E}}$ and $\Sigma_{q}\times\Omega$. \[teo-unicidade-aglo-infinito\] Let $\beta>0$ be the inverse temperature and $\pmb{\hat{h}}$ a magnetic field. Then $$\phi_{\mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (N_{\infty}\leqslant 1) = \phi_{\mathrm{ free}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (N_{\infty}\leqslant 1) =1.$$ We only present the argument for $ \phi_{\mathrm{ max}}^{\mathrm{GRC}, \pmb{\hat{h}}} \in \mathscr{G}^{\mathrm{ GRC}} $, since for the free boundary condition case the proof works similarly. Let $\Lambda\subset \Delta$ be finite subsets of $\mathbb{V}$ and $D_{\Lambda,\Delta}$ the set of all $\omega\in \Omega$ with the property: there exist two points $u,v\in \partial \Lambda$ such that both $u$ and $v$ are joined to $\partial\Delta$ by paths using $\omega$-open edges of $\mathbb{E}_{\Delta}\setminus \mathbb{E}_{\Lambda}$, but $u$ is not joined to $v$ by a path using $\omega$-open edges of $\mathbb{E}_{\Delta}$. For any fixed configuration $\eta\in\Omega$ the mapping $ \omega \mapsto \mathds{1}_{D_{\Lambda,\Delta}} (\omega_{\mathbb{E}_{\Lambda}} \eta_{\mathbb{E}_{ \mathbb{V}\setminus\Lambda}} ) $ is decreasing. Because of the definition of $D_{\Lambda,\Delta}$ we can abuse notation and simply write $ \mathds{1}_{D_{\Lambda,\Delta}} (\omega_{\mathbb{E}_{\Lambda}} \eta_{\mathbb{E}_{ \Delta\setminus\Lambda}} ). $ Given $\epsilon>0$ small enough, we consider the external magnetic field $ \epsilon\pmb{\hat{h}}\equiv( \epsilon h_{i,p}, \forall \ i\in\mathbb{V}, p=1,\ldots,q) $. A straightforward computation shows that $\epsilon\pmb{\hat{h}}\prec \pmb{\hat{h}}$, where the partial order is given by . If $V$ contains $\Delta$ then it follows from the Theorem \[teo-monotonicidade-campo-externo-grc\] that $$\phi_{V, \mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } ( \mathds{1}_{D_{\Lambda,\Delta}}( \cdot\, \eta_{\mathbb{E}_{ \Delta\setminus\Lambda}} ) ) \leqslant \phi_{V, \mathrm{ max}}^{\mathrm{ GRC}, \epsilon\pmb{\hat{h}} } ( \mathds{1}_{D_{\Lambda,\Delta}}( \cdot\, \eta_{\mathbb{E}_{ \Delta\setminus\Lambda}} ) ).$$ By summing over all $\eta_{\mathbb{E}_{ \Delta\setminus\Lambda}}$ the above inequality we get that $$\phi_{V, \mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } ( D_{\Lambda,\Delta} ) \leqslant \phi_{V, \mathrm{ max}}^{\mathrm{ GRC}, \epsilon\pmb{\hat{h}} } ( D_{\Lambda,\Delta} ).$$ Taking $\epsilon\to 0$, $V\uparrow\mathbb{V}$ and using the continuity of $\mathds{1}_{D_{\Lambda,\Delta}}$ and the Theorem \[limite termodinamico\] we get from the last inequality that $$\begin{aligned} \phi_{\mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (D_{\Lambda,\Delta}) \leqslant \phi_{\mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{0}} } (D_{\Lambda,\Delta}).\end{aligned}$$ Since $ \cap_{\Delta \supset \Lambda} D_{\Lambda,\Delta} \uparrow \{N_{\infty}>1\}, $ when $\Lambda\uparrow \mathbb{V}$ it follows from the continuity of the measure and Theorem III.3 of [@BBCK00] that $\phi_{\mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (N_{\infty} >1)=0$. To state our next theorem, which is the main theorem of the next section, we need to introduce the following parameters: $$\begin{aligned} P_\infty(\beta,\pmb{J},\pmb{\hat{h}}) \equiv \sup_{x\in\mathbb{V} }\, \sup_{\phi\in\mathscr{G}^{\mathrm{ GRC}}} \phi(|C_x|=\infty) \end{aligned}$$ and $$\begin{aligned} \widetilde{P}_\infty(\beta,\pmb{J},\pmb{\hat{h}}) \equiv \sup_{x\in\mathbb{V} }\, \inf_{\phi\in\mathscr{G}^{\mathrm{ GRC}}} \phi(|C_x|=\infty), \end{aligned}$$ where $C_{x}$ is the infinite connected component containing the vertex $x$. For the RC model, the parameters $P_\infty$ and $\widetilde{P}_\infty$ are defined similarly. We also define the critical parameter $$\begin{aligned} \beta_c(\pmb{J},\pmb{\hat{h}}) \equiv \inf\{\beta > 0: P_\infty(\beta,\pmb{J},\pmb{\hat{h}})>0\}. \end{aligned}$$ To lighten the notation we introduce for each $\mathrm{m}\in \{1,\ldots, q\}$ fixed, the event $$\mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty \! \equiv \! \left\{ \! (\sigma,\omega)\in\Sigma_{q}\times\Omega : \!\! \begin{array}{c} N_\infty(\sigma,\omega)\geqslant 1 \ \text{and all vertices in any infinite} \\ \text{connected} \ \text{ component satisfies} \ \sigma_x=\text{m} \! \end{array} \right\}.$$ Uniqueness and phase transition =============================== Now we are ready to state and prove one of the main theorems of this paper. We emphasize that this theorem was inspired by Theorem II.5 in [@BBCK00]. \[teo-unicidade-gibbs-percolacao\] Fix $q\in\mathbb{Z}^+$, $\beta\geqslant 0$, a magnetic field $ \pmb{\hat{h}} = (h_{i,p}\in\mathbb{R}: i\in\mathbb{V}, p=1,\ldots, q) $ and $\{q_p: p=1,\ldots,q\}$ satisfying . - For all $\pmb{J}\geqslant 0$ $ \mathrm{(}J_{ij} \geqslant 0, \ \forall\,\{i,j\}\in\mathbb{E}\mathrm{)} $, there is at most one probability measure $\mu_0$ in $\mathscr{G}^{\mathrm{ ES}}_0 \equiv \{\nu\in\mathscr{G}^{\mathrm{ ES}}:\nu(N_\infty=0)=1\}$. - If $P_\infty(\beta,\pmb{J},\pmb{\hat{h}})=0$, then $\vert \mathscr{G}^{\mathrm{ ES}} \vert = \vert \mathscr{G}^{\mathrm{ RC}}\vert =1.$ In particular, if $\beta<\beta_c(\pmb{J},\pmb{\hat{h}}),$ then $\vert \mathscr{G}^{\mathrm{ ES}} \vert = \vert \mathscr{G}^{\mathrm{ RC}}\vert =1.$ <!-- --> - If $\pmb{J}\geqslant 0$ is an uniform coupling constant $ \mathrm{(}J_{ij}\equiv J\geqslant 0, \ \forall$ $\{i,j\}\in\mathbb{E}\mathrm{)}$, then $ P_\infty(\beta,J,\pmb{\hat{h}}) = \sup_{x\in\mathbb{V}}\ \phi_{\mathrm{ max}}^{\mathrm{ GRC}} (|C_x|=\infty) $  and $ \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) = \sup_{x\in\mathbb{V}}\ \phi_{\mathrm{ free}}^{\mathrm{ GRC}} (|C_x|=\infty). $ - Let $\pmb{J}\geqslant 0$ $\mathrm{(}J_{ij}\equiv J\geqslant 0, \ \forall$ $\{i,j\}\in\mathbb{E}\mathrm{)}$. If $P_\infty(\beta,\pmb{J},\pmb{\hat{h}})>0$ then the states $\phi_{\mathrm{ m}}^{\mathrm{ ES}},$ $ \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) $ are extremal $\mathrm{ES}$ Gibbs states with $ \phi_{\mathrm{ m}}^{\mathrm{ ES}} (\mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty) = 1. $ Moreover under the strong assumption $\widetilde{P}_\infty(\beta,\pmb{J},\pmb{\hat{h}})>0$ we have that $\vert \mathscr{G}^{\mathrm{ ES}} \vert > 1$. - Let $\pmb{J}\geqslant 0$ $\mathrm{(}J_{ij}\equiv J\geqslant 0, \ \forall$ $\{i,j\}\in\mathbb{E}\mathrm{)}$. If $\beta<\beta_c$, then $ P_\infty(\beta,J,\pmb{\hat{h}}) = \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) = 0 $, while both $P_\infty(\beta,J,\pmb{\hat{h}}) > 0$ and $\widetilde{P}_\infty(\beta,J,\pmb{\hat{h}})>0$ whenever $\beta>\beta_c$. The whole proof follows closely reference [@BBCK00]. $(i)$ We prove that $\mathscr{G}^{\mathrm{ ES}}_{0}= \{\phi_{\mathrm{ free}}^{\mathrm{ ES}}\}$, In fact, let $\nu\in \mathscr{G}^{\mathrm{ ES}}_{0}$ and $\{\Delta_n:n\in\mathbb{N}\}$ be a cofinal sequence of subsets of $\mathbb{V}$. Then the sequence of random sets $\{\Lambda_n: n\in\mathbb{N}\}$ defined by $\Lambda_n(\omega)\equiv\{x\in\Delta_n: x\not\leftrightarrow\Delta_n^c \ \text{in} \ \omega\}$ is also increasing. Note that the set $\Lambda_n$ is well defined due to the absence of infinite connected components. By Theorem \[limite termodinamico\], given $\epsilon>0$, we can take $\Delta$ big enough so that for each function $f$ with support in $(\Delta,\mathbb{B}_0(\Delta))$ we have $$\begin{aligned} \label{2m1} |\phi_{V,\mathrm{ free}}^{\mathrm{ ES}}(f) - \phi_{\mathrm{ free}}^{\mathrm{ ES}}(f) |\leqslant \epsilon, \quad \forall \ V\supset\Delta.\end{aligned}$$ On the other hand, we have that $$\begin{gathered} \nu(f) = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\supset \Delta\}}) \nonumber \\ = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu(f \mathds{1}_{\{\Lambda_n(\cdot) = \overline{\Lambda}_n\}}). \nonumber\end{gathered}$$ By using the DLR equations and their equivalent version of conditional expectations for the specification $\{\phi^{\mathrm{ ES}}_{\Lambda,\mathbb{B}_0(\Lambda)}\}$, we can rewrite the above expression as $$\begin{gathered} = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big(\phi_{\overline{\Lambda}_n,\mathbb{B}_0 (\overline{\Lambda}_n)}^{\mathrm{ ES}} ( f \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}}| \cdot ) \big) \\ = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big( \nu ( f \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}}| \mathscr{F}_{\overline{\Lambda}_n^c,\mathbb{B}_0 (\overline{\Lambda}_n)^c} ) \big).\end{gathered}$$ For each fixed $n$, the random variable $\mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}}$ depends only on the states of the sites and edges in $(\overline{\Lambda}_n,\mathbb{B}_0(\overline{\Lambda}_n))$, so we have that this random variable is independent of the $\sigma$-algebra $\mathscr{F}_{\overline{\Lambda}_n^c, \mathbb{B}_0(\overline{\Lambda}_n)^c}$. Hence the latter expression can be rewritten as $$\begin{gathered} = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big( \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}} \nu (f| \mathscr{F}_{\overline{\Lambda}_n^c,\mathbb{B}_0 (\overline{\Lambda}_n)^c} ) \big) \\ = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big( \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}} \phi_{\overline{\Lambda}_n, \mathbb{B}_0(\overline{\Lambda}_n)}^{\mathrm{ ES}} ( f| \cdot ) \big) \\ = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big( \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}} \phi_{\overline{\Lambda}_n,\mathrm{ free}}^{\mathrm{ ES}} (f) \big),\end{gathered}$$ where in the second equality we have used again the equivalent version of conditional expectation for specification $\{\phi^{\mathrm{ ES}}_{\Lambda,\mathbb{B}_0(\Lambda)}\}$, and in the last one we use the definition of the measure $\phi_{\overline{\Lambda}_n,\mathrm{ free}}^{\mathrm{ ES}}$. So we have the identity $$\begin{aligned} \label{2m2} \nu(f) = \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + \sum_{\overline{\Lambda}_n(\cdot)\supset \Delta} \nu \big( \mathds{1}_{\{\Lambda_n(\cdot)= \overline{\Lambda}_n\}} \phi_{\overline{\Lambda}_n,\mathrm{ free}}^{\mathrm{ ES}} (f) \big).\end{aligned}$$ Combining the identities and we have $$\begin{gathered} \label{2m3} \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + [\phi_{\mathrm{ free}}^{\mathrm{ ES}}(f)-\epsilon] \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\supset \Delta\}}) \leqslant \nu(f) \\[0,2cm] \leqslant \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\not\supset \Delta\}}) + [\phi_{\mathrm{ free}}^{\mathrm{ ES}}(f)+\epsilon] \nu(f \mathds{1}_{\{\Lambda_n(\cdot)\supset \Delta\}}). \end{gathered}$$ Since the sequence $\{\Lambda_n(\omega):n\in\mathbb{N}\}$ is increasing, the sequence $\{A_n:n\in\mathbb{N}\}$, with $A_n\equiv\{\Lambda_n(\cdot)\supset \Delta\}$, is also increasing. Therefore $\mathds{1}_{A_n}\uparrow 1$. Since $f$ is bounded, taking $n\uparrow\infty$ in and using the Dominated Convergence Theorem yields $$|\nu(f)-\phi_{\mathrm{ free}}^{\mathrm{ ES}}(f)| \leqslant \epsilon.$$ Since $\epsilon>0$ was arbitrary, we conclude the proof of this item.\ \ $(ii)$ If $P_\infty(\beta,\pmb{J},\pmb{\hat{h}})=0$, then $$\begin{aligned} \label{sm4} \phi(|C_x|=\infty)=0, \quad \forall \ \phi\in \mathscr{G}^{\mathrm{ RC}} \ \text{and} \ x\in\mathbb{V}.\end{aligned}$$ By the uniqueness of the infinite connected component (Theorem \[teo-unicidade-aglo-infinito\]) and Lemma \[continencia-Gibbs\], it follows that property holds for $\phi_{\mathrm{ max}}^{\mathrm{ RC}}$. If $\phi_\nu$ denotes the edge-marginal of $\nu\in\mathscr{G}^{\mathrm{ ES}}$, by Theorem \[principal-marginal\] we have $$0 = \phi_{\mathrm{ max}}^{\mathrm{ RC}}(N_\infty>0) \geqslant \phi_\nu(N_\infty>0)= \nu(N_\infty>0),$$ which implies that $\nu\in \mathscr{G}^{\mathrm{ ES}}_0 $. Therefore it follows from the item $(i)$ that $\nu=\phi^{\mathrm{ ES}}_{\mathrm{ free}}$. That is, $\mathscr{G}^{\mathrm{ ES}}=\mathscr{G}^{\mathrm{ ES}}_0= \{\phi^{\mathrm{ ES}}_{\mathrm{ free}}\}.$ On the other hand, if we denote by $ \mathscr{G}^{\mathrm{ RC}}_{0} \equiv \{\phi \in\mathscr{G}^{\mathrm{ RC}}:\phi(N_\infty=0)=1\}, $ we have from and Theorem \[limite termodinamico\], $$0 = \phi_{\mathrm{ max}}^{\mathrm{ RC}}(N_\infty>0)\geqslant \phi(N_\infty>0), \quad \forall \ \phi \in \mathscr{G}^{\mathrm{ RC}}.$$ Thus $\phi\in \mathscr{G}^{\mathrm{ RC}}_0$. By repeating the proof of item $(i)$, using Theorem \[vec\] and the DLR equations , we have that $\mathscr{G}^{\mathrm{ RC}}= \{\phi^{\mathrm{ RC}}_{\mathrm{ free}}\}.$\ \ $(iii)$ Using Item $(iii)$ of Theorem \[limite termodinamico\] gives $$P_\infty(\beta,J,\pmb{\hat{h}}) \leqslant \sup_{x\in\mathbb{V}} \phi_{\mathrm{ max}}^{\mathrm{ GRC}} (|C_x|=\infty) \quad \!\! \text{and} \quad \!\! \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) \geqslant \sup_{x\in\mathbb{V}} \phi_{\mathrm{ free}}^{\mathrm{ GRC}} (|C_x|=\infty).$$ To prove that the equality is attained, it is enough to show that $ \phi_{\mathrm{ max}}^{\mathrm{ GRC}} \in \mathscr{G}^{\mathrm{ GRC}} $ and $ \phi_{\mathrm{ free}}^{\mathrm{ GRC}} \in \mathscr{G}^{\mathrm{ GRC}} $, respectively. By using Theorem \[teo-unicidade-aglo-infinito\], we have that $ \phi_{\mathrm{ max}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (N_{\infty}\leqslant 1) = \phi_{\mathrm{ free}}^{\mathrm{ GRC}, \pmb{\hat{h}} } (N_{\infty}\leqslant 1) = 1 $, thus we conclude from Lemma \[continencia-Gibbs\] that $ \phi_{\mathrm{ max}}^{\mathrm{ GRC}} \in \mathscr{G}^{\mathrm{ GRC}} $ and $ \phi_{\mathrm{ free}}^{\mathrm{ GRC}} \in \mathscr{G}^{\mathrm{ GRC}} $.\ \ $(iv)$ Using the same technique employed by [@BBCK00], one can prove that $$\begin{aligned} \phi_{\mathrm{ max}}^{\mathrm{ RC}} (x \leftrightarrow \infty) = \lim_{V\uparrow\mathbb{V}} \phi_{V,\mathrm{ max}}^{\mathrm{ RC}} (x \leftrightarrow V^c).\end{aligned}$$ As a consequence, we have that, for all $ \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) $, $$\begin{aligned} \phi_{\mathrm{ m}}^{\mathrm{ ES}} (x \leftrightarrow \infty) = \lim_{V\uparrow\mathbb{V}} \phi_{V,\mathrm{ m}}^{\mathrm{ ES}} (x \leftrightarrow V^c).\end{aligned}$$ Combining the two last identities with the trivial fact $$\begin{aligned} \phi_{V,\mathrm{ m}}^{\mathrm{ ES}} (x \leftrightarrow V^c,\sigma_x=\mathrm{\tilde{m}}) = \phi_{V,\mathrm{ m}}^{\mathrm{ ES}} (x \leftrightarrow V^c) \delta_{\mathrm{m},\mathrm{\tilde{m}}},\end{aligned}$$ and taking the thermodynamic limit, we have $$\begin{aligned} \label{sm7} \phi_{\mathrm{ m}}^{\mathrm{ ES}} (\sigma_x=\mathrm{\tilde{m}}|x \leftrightarrow \infty) = \delta_{\mathrm{m},\mathrm{\tilde{m}}}, \quad \forall \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}).\end{aligned}$$ Now we prove that the state $\phi_{\mathrm{ m}}^{\mathrm{ ES}}$ is extremal whenever $ \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) $. To this end, let us assume that $ \phi_{\mathrm{ m}}^{\mathrm{ ES}} ( \mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty ) = 1 $, this will be proved below. Suppose that $\phi_{\mathrm{ m}}^{\mathrm{ ES}}$ is not extremal - then there are two Gibbs measures in $\mathscr{G}^{\mathrm{ ES}}$ so that $$\begin{aligned} \label{sm8} \phi_{\mathrm{ m}}^{\mathrm{ ES}} = t \phi_1^{\mathrm{ ES}} + (1-t) \phi_2^{\mathrm{ ES}}, \quad \text{with} \ \phi_i^{\mathrm{ ES}} (\mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty)=1 \ \text{and} \ t\in(0,1).\end{aligned}$$ If $\phi_i^{\mathrm{ RC}}$ denotes the RC marginal of $\phi_i^{\mathrm{ ES}}$, it follows from Lemma VIII.1 in [@BBCK00] that $\phi_i^{\mathrm{ RC}}\in\mathscr{G}^{\mathrm{ RC}}$, $i=1,2$. This implies that $$\begin{aligned} \phi_{\mathrm{ max}}^{\mathrm{ RC}} = t \phi_1^{\mathrm{ RC}} + (1-t) \phi_2^{\mathrm{ RC}}, \quad t\in(0,1).\end{aligned}$$ By stochastic domination one can prove that $\phi_{\mathrm{ max}}^{\mathrm{ RC}}$ is an extremal probability measure, so $ \phi_1^{\mathrm{ RC}} = \phi_2^{\mathrm{ RC}} = \phi_{\mathrm{ max}}^{\mathrm{ RC}} $. Using Lemma VIII.3 in [@BBCK00], this fact implies $\phi_1^{\mathrm{ ES}}=\phi_2^{\mathrm{ ES}}$, hence the extremality of $\phi_{\mathrm{ m}}^{\mathrm{ ES}}$ is proved. Finally, we prove that $ \phi_{\mathrm{ m}}^{\mathrm{ ES}}( \mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty ) = 1 $. Since $P_\infty(\beta,\pmb{J},\pmb{\hat{h}})>0$, we get from Item $(iii)$ of Theorem \[limite termodinamico\] that $$\begin{aligned} \label{sm10} 0 < P_\infty(\beta,\pmb{J},\pmb{\hat{h}}) \leqslant \sup_{x\in\mathbb{V}} \phi_{\mathrm{ max}}^{\mathrm{ RC}}(x\leftrightarrow\infty) \leqslant \phi_{\mathrm{ max}}^{\mathrm{ RC}} ( N_{\infty}\geqslant 1 ).\end{aligned}$$ Since $\phi_{\mathrm{ max}}^{\mathrm{ RC}}$ is an extremal Gibbs state and $\{N_{\infty}\geqslant 1\}$ is a tail event it follows from the inequality and the uniqueness of the infinite connected component (Theorem \[teo-unicidade-aglo-infinito\]) that $$1= \phi_{\mathrm{ max}}^{\mathrm{ RC}} ( N_{\infty}= 1 ) = \phi_{\mathrm{ m}}^{\mathrm{ ES}} (N_{\infty}= 1), \quad \forall \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}).$$ The previous equation together with the identity implies, for each $ \mathrm{m} \in \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) $, that $ \phi_{\mathrm{ m}}^{\mathrm{ ES}} (\mathscr{A}_{\geqslant 1,\mathrm{ m}}^\infty) =1 $. We now prove the second statement of Item $(iv)$. As long as the set $ \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}} (\pmb{\hat{h}}) $ has more than one element, the result follows from the first statement of the Item $(iv)$. Otherwise, without loss of generality, we can assume that $ \cap_{i\in\mathbb{V}}\mathcal{Q}_{i,\mathrm{ max}}(\pmb{\hat{h}}) = \{1\} $. Let $\phi^{\tiny\mathrm{RC}} \in \mathscr{G}^{\mathrm{RC}}$ be a spin-marginal of $\phi_{\mathrm{ 2}}^{\mathrm{ES}}$, then $$\begin{gathered} 0 < \widetilde{P}_\infty(\beta,\pmb{J},\pmb{\hat{h}}) \le \sup_{x\in\mathbb{V}} \phi^{\mathrm{ RC}}(x\leftrightarrow\infty) = \sup_{x\in\mathbb{V}} \phi_{\mathrm{ 2}}^{\mathrm{ ES}}(x\leftrightarrow\infty) \\[0,3cm] = \sup_{x\in\mathbb{V}} \phi_{\mathrm{ 2}}^{\mathrm{ ES}} ( x \leftrightarrow \infty, \sigma_x=2 ) \le \phi_{\mathrm{ 2}}^{\mathrm{ ES}} (\mathscr{A}_{\geqslant 1,\mathrm{ 2}}^\infty).\end{gathered}$$ Since $\phi_{\mathrm{ 1}}^{\mathrm{ ES}} (\mathscr{A}_{\geqslant 1,\mathrm{ 1}}^\infty)=1$ and $ \mathscr{A}_{\geqslant 1,\mathrm{ 1}}^\infty \cap \mathscr{A}_{\geqslant 2,\mathrm{ 1}}^\infty = \emptyset $, it follows from the above inequality that $ \phi_{\mathrm{ 1}}^{\mathrm{ ES}} \neq \phi_{\mathrm{ 2}}^{\mathrm{ ES}}. $\ \ $(v)$ By Item $(iii)$ and Corollary \[coro ge0\], we get that the maps $J\mapsto P_\infty(\beta,J,\pmb{\hat{h}})$ and $J\mapsto\widetilde{P}_\infty(\beta,J,\pmb{\hat{h}})$ are increasing, and so are the maps $\pmb{\hat{h}}\mapsto P_\infty(\beta,J,\pmb{\hat{h}})$ and $\pmb{\hat{h}}\mapsto\widetilde{P}_\infty(\beta,J,\pmb{\hat{h}})$, with respect to the partial order . From the definition, one has $$\begin{aligned} \label{ar1} \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) \leqslant P_\infty(\beta,J,\pmb{\hat{h}}), \quad \forall \beta,J\ \text{and}\ \pmb{\hat{h}}.\end{aligned}$$ From Item $(iii)$ and Theorem \[teorema ge\], we get $ P_\infty(\beta,J_1,\pmb{\hat{h}}) \leqslant \widetilde{P}_\infty(\beta,J_2,\pmb{\hat{h}}) $ for all $J_1<J_2$. By Item $(iii)$, we have that $P_\infty$ and $\widetilde{P}_\infty$ are thermodynamical limits. Using the form of the Hamiltonian of this model and the monotonicity properties proved above, we get, for all $\beta_1<\beta_2$, that $$\begin{aligned} \label{ar2} P_\infty(\beta_1,J,\pmb{\hat{h}}) = P_\infty(1,\beta_1 J, \beta_1 \pmb{\hat{h}}) &\leqslant \widetilde{P}_\infty(1,\beta_2 J,\beta_1\pmb{\hat{h}}) \nonumber \\ &\leqslant \widetilde{P}_\infty(1,\beta_2 J,\beta_2\pmb{\hat{h}}) \nonumber \\ &= \widetilde{P}_\infty(\beta_2,J,\pmb{\hat{h}}).\end{aligned}$$ Combining inequalities and yields $(v)$. We now consider the $q$-state Potts model where each value of the spin is coupled to a distinct and site dependent external field. The formal Hamiltonian of the model is $$\label{hamiltoniano-potts-BBCK-generalizado} H(\hat{\sigma}) = -J\sum_{ \{i,j\} } \delta_{\hat{\sigma}_i}\delta_{\hat{\sigma}_j} -\sum_{p=1}^q \sum_{i} {h_{i,p}\over q} \delta_{\hat{\sigma}_{i,p}}.$$ Let $\mathscr{G}^{\mathrm{{Spin}}}$ denote the set of all spin Gibbs states, defined by means of the DLR condition and the above Hamiltonian (appropriately modified to incorporate boundary conditions). \[teo-isomorfismo-GES-GISING\] Let $\Pi_S:\mathscr{G}^{\mathrm{ ES}}\rightarrow \mathscr{G}^{\mathrm{{Spin}}}$ denote the mapping that assigns the spin-marginal to infinite volume $\mathrm{ES}$ measure. Then $\Pi_S$ is a linear isomorphism. A direct proof of this theorem can be found in [@BBCK00]. Application - Ising model with power law decay external field ============================================================= In this section we apply the results above obtained to prove the uniqueness of the Gibbs measures, at any positive temperature, for the Ising model in $\mathbb{L}\equiv (\mathbb{Z}^d,\mathbb{E}^d)$, where $\mathbb{E}^d$ is the set of the nearest neighbors in the $d$-dimensional hypercubic lattice, with the Halmiltonian given by $$\label{hamiltoniano-BCCP} \mathscr{H}^{\mu,\mathrm{{Ising}}}_{\pmb{h},V}(\sigma) \equiv - \sum_{ \substack{ i,j\in V \\ \{i,j\}\in \mathbb{E} } } J\, \sigma_i \sigma_j - \sum_{i\in V} h_i\, \sigma_i - \!\!\!\! \sum_{ \substack{ i\in V, \ j\in \partial V \\ \{i,j\}\in \mathbb{E} } } \!\!\!\!\!\! J\, \sigma_i \mu_j,$$ where $\alpha\geqslant 0$ and $h^*>0$ and $$h_i = \begin{cases} \frac{h^*}{\|i\|^{\alpha}},& \text{if}\ i\neq 0; \\[0.3cm] h^*,& \text{if}\ i= 0. \end{cases}$$ From now on, we write $\mathscr{G}^{\mathrm{{Spin}}}_{\beta}$ instead of $\mathscr{G}^{\mathrm{{Spin}}}$ to make clear its dependence on the inverse temperature. By Proposition \[Medeq\], it follows that the set $\mathscr{G}^{\mathrm{{Spin}}}_{\beta}$ (defined in the last section) is precisely $ \mathscr{G}^{\mathrm{{Potts}}}_{2\beta}(\pmb{J},\pmb{h}) = \mathscr{G}^{\mathrm{{Ising}}}_{\beta}(\pmb{J},\pmb{h}) $, the set of the Gibbs measures of the above Ising model, if we take in the Hamiltonian $q=2$ and the magnetic field given by $$\pmb{\hat{h}} = \big( (h_{i,1},h_{i,2}) \in \mathbb{R}^2 : \ h^{*}/\|i\|^{\alpha} = h_{i,1}=-h_{i,2},\ \forall i\in \mathbb{V}\ \big).$$ In order to apply the previous results to study the uniqueness of this Ising model with magnetic field decaying to zero with polynomial rate $0\leqslant \alpha <1$, we will consider in this section the GRC model defined in with $q=2$, the constants $q_p\equiv 1$ and the magnetic field $\pmb{\hat{h}}$ as above. In [@BCCP15] the authors proved that for any $\alpha \in [0,1)$ there is a positive inverse temperature $\beta_{\alpha}<+\infty$ so that, for any $\beta>0$ such that $\beta_{\alpha}<\beta$, the set of the Gibbs measures for the Ising model defined by is a singleton. By the Dobrushin Uniqueness Theorem, we know that for any $\beta< 1/(2d J)$ the set of Gibbs measures for this Ising model at these inverse temperatures is also singleton. In the reference [@BCCP15] it was conjectured that the set of the Gibbs measures for this model with $\alpha \in [0,1)$ is a singleton for any $\beta>0$. In this work we settle this conjecture. \[cfl\] Suppose that $$\begin{aligned} \beta_c(\pmb{J},\pmb{\hat{h}}) \equiv \inf\{\beta > 0: P_\infty(\beta,\pmb{J},\pmb{\hat{h}})>0\} = +\infty.\end{aligned}$$ In this case, it follows from Item $(ii)$ of Theorem \[teo-unicidade-gibbs-percolacao\] that for any $\beta>0$ we have $\vert \mathscr{G}^{\mathrm{ ES}}_{\beta} \vert =1$. By Theorem \[teo-isomorfismo-GES-GISING\] we get that $\vert \mathscr{G}^{\mathrm{{Spin}}}_{\beta}\vert =1$. Suppose that $\beta_c(\pmb{J},\pmb{\hat{h}})<+\infty$. By using once more Item $(ii)$ of Theorem \[teo-unicidade-gibbs-percolacao\], we obtain the uniqueness for $\beta < \beta_c(\pmb{J},\pmb{\hat{h}})$, that is, $\vert \mathscr{G}^{\mathrm{{Spin}}}_{\beta}\vert =1$ for such values of $\beta$. If $ \beta > \max\{ \beta_\alpha, \beta_c(\pmb{J},\pmb{\hat{h}}) \} $ it was proved in [@BCCP15] that $\vert \mathscr{G}^{\mathrm{{Spin}}}_{\beta}\vert =1$. We claim that $ \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) \equiv 0 $ for any $\beta >0$. Indeed, take $\beta>\beta_{\alpha}$ if $ \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}})>0 $, so by Item $(iv)$ of Theorem \[teo-unicidade-gibbs-percolacao\] we have at least two ES Gibbs measures and by Theorem \[teo-isomorfismo-GES-GISING\] two Gibbs measures for the Ising model which contradicts [@BCCP15]. Therefore $\widetilde{P}_\infty(\beta,J,\pmb{\hat{h}})=0$ whenever $\beta>\beta_{\alpha}$. Since the mapping $\beta \mapsto \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) $ is increasing, the claim follows. From Item $(v)$ we have for any $\beta>\beta_c(\pmb{J},\pmb{\hat{h}})$ that $ P_\infty(\beta,J,\pmb{\hat{h}}) >0 $ and $ \widetilde{P}_\infty(\beta,J,\pmb{\hat{h}}) > 0, $ but this contradicts the above claim. Therefore we have proved that for any $\alpha \in [0,1)$ that $ \beta_c(\pmb{J},\pmb{\hat{h}})=+\infty, $ which implies by Theorem \[teo-unicidade-gibbs-percolacao\] that $\vert \mathscr{G}^{\mathrm{{Spin}}}_{\beta}\vert =1$ for any $\beta>0$. #### Acknowledgements. The authors thank Aernout van Enter for fruitful discussions. We also acknowledge the anonymous referees for their helpful comments, suggestions and references provided in their reports. Leandro Cioletti is supported by FEMAT and Roberto Vila is supported by CNPq. 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--- abstract: 'Words (phrases or symbols) play a key role in human life. Word (phrase or symbol) representation is the fundamental problem for knowledge representation and understanding. A word (phrase or symbol) usually represents a name of a category. However, it is always a challenge that how to represent a category can make it easily understood. In this paper, a new representation for a category is discussed, which can be considered a generalization of classic set. In order to reduce representation complexity, the economy principle of category representation is proposed. The proposed category representation provides a powerful tool for analyzing conceptual systems, relations between words, communication, knowledge, situations. More specifically, the conceptual system, word relations and communication are mathematically defined and classified such as ideal conceptual system, perfect communication and so on; relation between words and sentences is also studied, which shows that knowledge are words. Furthermore, how conceptual systems and words depend on situations is presented, and how truth is defined is also discussed.' author: - | [**Jian YU** ]{}\ Beijing Key Lab of Traffic Data Analysis and Mining\ Beijing Jiaotong University,Beijing, China\ Email: [email protected] bibliography: - 'bibfile.bib' title: 'Communication: Words and Conceptual Systems' --- [**Keywords:**]{} Word, Category, Conceptual System, Relations between Words, Sentence, Communication, Knowledge, Situation, Truth Introduction ============ When studying objects in the real or virtual world, categorization is a common approach in order to communicate with each other. A word or phrase (sometimes, symbol) represents the corresponding category name. Every minute in daily life, we rely on words (phrases or symbols) to help us to deal with everything, such as talking, listening, watching, thinking, etc. Transparently, a word (phrase or symbol) usually represents a name of a category. How to represent a category becomes a pivotal question in word understanding. In the literature, a classical category has three representation: the corresponding name representation, the corresponding set representation, the corresponding proposition representation. In common sense, a word or phrase represent the name of a category, which is also called the concept name or the category name. Set representation represents a class of objects in the world belonging to the corresponding category, proposition representation represents the mental representation of the corresponding category. Up to now, all the above three representations seem in question. Firstly, a category usually has different names in different languages. What’s more, a category has different names in the same language. For example, when seeing a specific dog, we can say it “dog” in English, “¹·” in Chinese. Clearly, the word “dog” is different from that specific dog in the world, which only is the name of the category “dog”. Hence, the name of a category cannot fully reflect the meaning of this category. In extremity, the name of a category is only considered as a symbol and has no meaning. Furthermore, sometimes man cannot find the right name of a category for what he wants to say, and sometimes man does not like to say the right name of a category for what he should say. In other words, a category may have different name in the mind from that in the utterance. [@DePaulo1996] have stated that lying was an everyday event, and [@Heyman2009] have discovered that parents lie to their children even through they maintain that lying is unacceptable. Secondly, a category may be very fuzzy or vague such as beauty, badness and so on. Clearly, a classical set can not express this case. More surprised, [@watanabe1969knowing] has stated that it is impossible to distinguish a category from others without feature selection. Thirdly, [@wittgenstein1953philosophical] has claimed that important categories such as games may not be defined (represented) by a proposition. Such view has been widely accepted in cognitive science fields [@lako1987women]. Fourthly, it is taken for granted that all the men should have the same representation for a classical category. What a pity, such an assumption is also clearly not correct, otherwise, misunderstanding about the same category cannot occur among men in daily life. Cognition research results have proved that most concepts are not disembodied but embodied [@lako1987women]. In the final, it is usually supposed that set representation is equivalent to proposition representation with respect to categorization. However, such an assumption may be false in daily life. Although there exist the above drawbacks, man still can use words (phrases or symbols) to express their thoughts, feelings and observations. Why? In this paper, we devote to answering this question and make major contributions as follows. 1) Semantic set is proposed to represent a category in Section 2. 2\) Conceptual system is discussed in Section 3. 3\) Relations between words are studied in Section 4. 4\) How to communicate concepts between people is discussed in Section 5. 5\) Relation between words and sentences is discussed in Section 6. 6\) How conceptual systems and words depend on situations is presented in Section 7. 7\) The relation between semantic set and its referent is discussed in Section 8. 8\) The definitions of truth and falsity are investigated in Section 9. Category and Semantic Set ========================= It is well known in China that The Named is the mother of all things[@Laotzu1979] and the word is used to refer to its corresponding objects in the world [@Gongsunlongzi], which usually is limited in a specific domain. Therefore, let the discussed object domain be a set $O$ and $o$ represent an object in $O$, the discussed category domain be a conceptual system $L$ and $\mathfrak{A}$ denotes a category in $L$. For different categories in $L$, $o$ may have different representations. For example, when categorization by obesity, a man can be represented by his age, his weight, his height and so on; when categorization of different types of individuals, a man can be represented by his sensation, his intuition, his feeling,his thinking,his attitudes and so on. Transparently, the object representation depends on categorization. Ugly Duckling Theorem [@watanabe1969knowing] has shown that a category specializes its object representation, otherwise, objects cannot be distinguished by different categories. In this paper, when specific categorization is needed, the corresponding object representation is supposed to have been obtained. Therefore, let $\mathfrak{A_O}$ be the category name we will discuss, $O_A=\{o_A| o\in O\}$, where $o_A$ is the object representation of the object $o$ in the set $O$ with respect to the category $\mathfrak{A}$, $I_A$ is the corresponding membership function defined as follows. [**Membership function:**]{}\ $I_A$: $O_A\mapsto R_{+}$ is called the membership function of $\mathfrak{A}$ if $\forall o_A\in O_A$, the greater $I_A(o_A)$ means the more probability $o$ belongs to the category $\mathfrak{A}$. As $O_A$ and $I_A$ are observable, $\{O_A,I_A\}$ is called the outer set of $\mathfrak{A}$. If $\forall o\in O, I_A(o_A)\in \{0,1\}$, then it is easy to know that $\forall o$, $I_A(o_A)=1$ if and only if $o$ belongs to the category $\mathfrak{A}$, and $I_A(o_A)=0$ if and only if $o$ does not belong to the category $\mathfrak{A}$. In this case, $\{O_A,I_A\}$ represents a classic set. If $\forall o\in O, I_A(o_A)\in [0,1]$, then $\{O_A,I_A\}$ represents a fuzzy set[@Zadeh1965fuzzyset]. In order to understand a category, it is reasonable to suppose that any category should have its corresponding concept in mental space [@JianYu2015Categorization]. Therefore, let $\underline{A}$ represent the concept corresponding to the category $\mathfrak{A}$ and the name of $\underline{A}$ be $\mathfrak{A_I}$. When the concept representation for any category is defined, objects can be categorized based on the similarity between objects and the concept representation. The category similarity mapping can be defined by computing the similarity between objects and the concept representation as follows. [**Category Similarity Mapping:**]{}\ $Sim_A$: $O_A\times \underline{A} \mapsto R_{+}$ is called the category similarity mapping of $\mathfrak{A}$ if $\forall o\in O$, the increase of $Sim_A(o_A,\underline{A})$ means the greater similarity between the object $o$ and the category $\mathfrak{A}$, where $o_A$ is the object representation with respect to $\mathfrak{A}$. Similarly, the category dissimilarity mapping can be defined as follows: [**Category Dissimilarity Mapping:**]{}\ $Ds_A$: $O_A\times \underline{A} \mapsto R_{+}$ is called the category dissimilarity mapping of $\mathfrak{A}$ if $\forall o\in O$, the decrease of $Ds_A(o_A,\underline{A})$ means the greater similarity between the object $o$ and the category $\mathfrak{A}$, where $o_A$ is the object representation with respect to $\mathfrak{A}$. Generally speaking, $(\underline{A},Sim_A)$ or $(\underline{A},Ds_A)$ can be unobservable, therefore, $(\underline{A},Sim_A)$ or $(\underline{A},Ds_A)$ is called the inner set of the category $\mathfrak{A}$, its name is called $\mathfrak{A_I}$. For brevity, ($\mathfrak{A_O},O_A,I_A$) is called the outer representation of the category $\mathfrak{A}$,($\mathfrak{A_I},\underline{A},Sim_A$) is called the inner representation of the category $\mathfrak{A}$. In language, $\mathfrak{A_O}$ is the outer name of the category, $\underline{A}$ is the mental representation of $\mathfrak{A}$, $\mathfrak{A_I}$ is the inner name of of the category. Usually, $\mathfrak{A_O}$ corresponds to a word (phrase or symbol) in a specific language, $\mathfrak{A_I}$ also corresponds to a word (phrase or symbol) in a specific language. In summary, a category $\mathfrak{A}$ can be represented by a six tuple: $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Sim_A)$ or $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Ds_A)$. $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Sim_A)$ or $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Ds_A)$ is called a semantic set. Usually, a semantic set depends on its user. Hence, a semantic set is a generalization of a classic set, which overcomes the drawbacks of the classical category representation and accords with the view of point in [@LakofeJohnson1980]. In daily life, a category $\mathfrak{A}$ is often represented by its outer name $\mathfrak{A_O}$. However, only the outer name of the category is known, it will be a puzzle for man to understand this category as the same name sometimes refers to many categories. Many words (phrases or symbols) in ancient classics cannot be understood by modern man just because their corresponding outer sets and inner sets have been lost in the long river of history. Conceptual System ================= As [@lako1987women] has pointed out, our conceptual systems grow out of bodily experience; moreover, the core of our conceptual systems is directly grounded in perception, body movement, and experience of a physical and social character. Therefore, the meaning of any word depends on a man’s physical and social environments. Transparently, different men may have different category representations with respect to the same category $\mathfrak{A}$. Every man has at least one conceptual system, which includes many categories. According the above analysis, the man $\alpha$’s conceptual system can be defined as $L^{\alpha}=\{\mathfrak{A}^{\alpha}\}$, where the category $\mathfrak{A}^{\alpha}$ can be represented by $(\mathfrak{A_O^{\alpha}},O_A^{\alpha},I_A^{\alpha},\mathfrak{A_I^{\alpha}},\underline{A}^{\alpha},Sim_A^{\alpha})$. $\mathfrak{A_O^{\alpha}}$ is a word (phrase or symbol), represents the referring action with respect to $\mathfrak{A}^{\alpha}$, $(\mathfrak{A_I^{\alpha}},\underline{A}^{\alpha},Sim_A^{\alpha})$ represents the mental representation concerned on the category $\mathfrak{A}^{\alpha}$. If the above three parts of the category $\mathfrak{A}^{\alpha}$ are not contradictory, then $\mathfrak{A^{\alpha}}$ is self-consistent. How to say a category $\mathfrak{A^{\alpha}}$ is self-consistent? In order to answer this question, the inner referring operator $\sim$ can be defined as follows: $$\label{innerreferring} \widetilde{o_A^{\alpha}}= \arg\max_{\mathfrak{B_I^{\alpha}}}Sim_{B}^{\alpha}(o_B^{\alpha},\underline{B}^{\alpha})$$ where $o_A^{\alpha}=o_B^{\alpha}, \mathfrak{B^{\alpha}}\in L^{\alpha}$; The outer referring operator $\rightarrow$ can be defined as follows: $$\label{outerreferring} \overrightarrow{o_A^{\alpha}}=\arg\max_{\mathfrak{B_O^{\alpha}}}I_{B}^{\alpha}(o_B^{\alpha})$$ where $o_A^{\alpha}=o_B^{\alpha}$, $\mathfrak{B^{\alpha}}\in L^{\alpha}$. Sometimes, $\overrightarrow{o_A^{\alpha}}$ or $\widetilde{o_A^{\alpha}}$ is multiple value. When $\overrightarrow{o_A^{\alpha}}$ or $\widetilde{o_A^{\alpha}}$ is multiple value, it is very difficult for man $\alpha$ to recognize the object $o$. If $\forall o\in O (\overrightarrow{o_A^{\alpha}} $ and $\widetilde{o_A^{\alpha}}$ are single value), it is natural to define that $A_O^{\alpha}=\{o\in O|\overrightarrow{o_A^{\alpha}}=\mathfrak{A_O^{\alpha}}\}$ and $A_I^{\alpha}=\{o\in O|\widetilde{o_A^{\alpha}}=\mathfrak{A_I^{\alpha}}\}$, where $A_O^{\alpha}$ is called the outer referring set and $A_I^{\alpha}$ the inner referring set with respect to the category $\mathfrak{A^{\alpha}}$. If $o\in A_O^{\alpha}$, then $o$ can be externally called $\mathfrak{A_O^{\alpha}}$, in other words, $o$ has an outer name $\mathfrak{A_O^{\alpha}}$. If $o\in A_I^{\alpha}$, then $o$ can be internally called $\mathfrak{A_I^{\alpha}}$, in other words, $o$ has an inner name $\mathfrak{A_I^{\alpha}}$. In daily life, sometimes man can call black into white. Hence it is not always true that $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_O^{\alpha}}$ for the man $\alpha$ with respect to one object $o\in O$. If $A_O^{\alpha}=A_I^{\alpha}$ and $\mathfrak{A_O^{\alpha}}=\mathfrak{A_I^{\alpha}}$, then $\mathfrak{A}^{\alpha}$ is self-consistent. Self-consistent is very helpful to reduce the complexity of a semantic set. If a semantic set is self-consistent, it is the simplest because its inner representation can be ignored in some sense. Therefore, self-consistent is the economy assumption for category representation. However, self-consistent is not always true for a conceptual system. If $\forall \mathfrak{A^{\alpha}}\in L^{\alpha}$ such that $A_O^{\alpha}=A_I^{\alpha}$ , there exist the mappings between the inner name and the outer name as follows. [**Name Encoding Mapping:**]{}\ $N_e^{\alpha}$: $ \mathfrak{A_O^{\alpha}}\mapsto \mathfrak{A_I^{\alpha}}$ is called the name encoding mapping of $L^{\alpha}$, where $\forall \mathfrak{A^{\alpha}} \in L^{\alpha}$ such that $A_O^{\alpha}=A_I^{\alpha}$ . [**Name Decoding Mapping:**]{}\ $N_d^{\alpha}$: $ \mathfrak{A_I^{\alpha}}\mapsto \mathfrak{A_O^{\alpha}}$ is called the name decoding mapping of $L^{\alpha}$, where $\forall \mathfrak{A^{\alpha}}\in L^{\alpha}$ such that $A_O^{\alpha}=A_I^{\alpha}$. According to the above analysis, a conceptual system can be accurately expressed as $L^{\alpha}_{\{N_e^{\alpha},N_d^{\alpha}\}}=\{\mathfrak{A}^{\alpha}\}$ if $\forall \mathfrak{A^{\alpha}}\in L^{\alpha}$ such that $A_O^{\alpha}=A_I^{\alpha}$ . A conceptual system $L^{\alpha}_{\{N_e^{\alpha},N_d^{\alpha}\}}$ is called a plain conceptual system. When its name encoding mapping and its name decoding mapping are an identity function, it can be simply denoted by $L^{\alpha}_I$, such a conceptual system is called a self-consistent conceptual system. For a category in the conceptual system $L^{\alpha}_{\{N_e^{\alpha},N_d^{\alpha}\}}\neq L^{\alpha}_I$, it can not be guaranteed to be self-consistent. In this case, it is more difficult to understand the conceptual system $L^{\alpha}_{\{N_e^{\alpha},N_d^{\alpha}\}}$ than to understand the conceptual system $L^{\alpha}_I$. In daily life, translation among different languages can be considered to seek the optimal name encoding mapping and name decoding mapping. Certainly, when the mappings between the inner name and the outer name do not exist, the inner referring set and the outer referring set need to be calculated independently, hence, the corresponding conceptual system is the most difficult to be understood. When the name encoding mapping and the name decoding mapping do exist for a conceptual system, it is enough to compute either the inner referring set or the outer referring set as the inner referring set is equal to the outer referring set. In order to further reduce the difficulty of understanding, it is natural to require that the name encoding mapping and the name decoding mapping are an identity function, which means such a conceptual system is self-consistent. Moreover, if any concept can be defined by a proposition in a self-consistent conceptual system, such a conceptual system is called an ideal conceptual system. Certainly, an ideal conceptual system can be accurately expressed by natural language. If a conceptual system is not ideal, it may not be accurately expressed by a natural language. Therefore, an ideal conceptual system is the most easily understood and inherited. However, it is the most difficult for man to obtain an ideal conceptual system. In general, man only can continuously approximate an ideal conceptual system through making $\underline{A}$ be expressed by some proposition as accurately and precisely as possible for any category $\mathfrak{A}$ in his conceptual system. Relations Between Words ======================= A conceptual system $L$ has many words, its two words usually have complex relations. In order to simplify to study relations among words, two assumptions are made as follows: **1): $L$ is self-consistent.** **2): $\forall o\in O$, $\forall \mathfrak{A}\in L$, $\forall \mathfrak{B}\in L$, $o_A$ can be expressed by a vector with fixed length $l$ such as $[o_A^{1},o_A^{2},\cdots,o_A^{l}]$, $o_B$ can be expressed by a vector with fixed length $m$ such as $[o_B^{1},o_B^{2},\cdots,o_B^{m}]$,where $o_A^{i}$ has a feature name $A^f_i$ in $L$, and $A^f=\{A^f_1,A^f_2,\cdots,A^f_l\}$ is the feature set of the category $\mathfrak{A}$, $o_B^{i}$ has a feature name $B^f_i$ in $L$, and $B^f=\{B^f_1,B^f_2,\cdots,B^f_m\}$ is the feature set of the category $\mathfrak{B}$.** According to the above two assumptions, $\forall o\in O$, $o_A$ can be defined as $[A^f_1(o),A^f_2(o),\cdots,A^f_l(o)]$, $o_B$ can be defined as $[B^f_1(o),B^f_2(o),\cdots,B^f_m(o)]$, where $A^f_1(o),A^f_2(o),\cdots, A^f_l(o),B^f_1(o),B^f_2(o),\cdots,B^f_m(o)$ can be considered to be functions. If $\mathfrak{A_O}$=$\mathfrak{B_O}$ and $A_O\bigcap B_O=\emptyset$ and $A^f\bigcap B^f=\emptyset$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called homonymy, which have the same category name but with different meanings. Theoretically, every word is homonymy as every word can be self-referring or non self-referring. For example, the word “dog” can refer to not only the word “dog” itself but also a real dog in the the world. If $\mathfrak{A_O}$=$\mathfrak{B_O}$ and $(A_O \bigcap B_O\neq \emptyset)\bigvee (\exists i \exists j (A^f_i=B^f_j))$ and $A_O \neq B_O$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called polysemy. If $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $A_O=B_O$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called synonymy, which have the different category names but with the same meaning. If $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $A_O \subset B_O$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called hyponymy. More detailed, the word $\mathfrak{A_O}$ is called the hyponym of the word $\mathfrak{B_O}$, the word $\mathfrak{B_O}$ is called the hypernym of the word $\mathfrak{A_O}$. If $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $A_f = B_f$ and $A_O \cap B_O=\emptyset$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called antonymy. If $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $\forall o\in A_O \exists \dot{o}\in B_O (o\in \dot{o})$, then the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ are called meronymy, the word $\mathfrak{A_O}$ is called the meronym of the word $\mathfrak{B_O}$, the word $\mathfrak{B_O}$ is called the holonym of the word $\mathfrak{A_O}$. If $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $\mathfrak{A_O}\in B_O $, then the word $\mathfrak{B_O}$ is said to modify the word $\mathfrak{A_O}$. Usually, the word $\mathfrak{A_O}$ is called the modified word, the word $\mathfrak{B_O}$ is called the modifier. Sometimes, more complex relations of two words $\mathfrak{A_O}$ and $\mathfrak{B_O}$ are needed to be considered. For example, metaphor and metonymy also illustrate relation between two words that seem irrelevant. In theory, if $\mathfrak{A_O}\neq \mathfrak{B_O}$ and $ \exists i \exists j (A^f_i=B^f_j)$, then one can directly use the word $\mathfrak{A_O}$ to refer to the category $\mathfrak{B}$, or one can say that $\mathfrak{A_O}$ is $\mathfrak{B_O}$ in order to make description simple, vivid and accurate. Here, it should be pointed out that category feature mapping usually depends on the situation, which includes relevant people, places, times, objects and environments. In many applications, it is very important to compute the semantic similarity between two words. According to the proposed category representation, categories $\mathfrak{A}$ and $\mathfrak{B}$ are said to be dissimilar if and only if $A^f\bigcap B^f=\emptyset$. Otherwise, the semantic similarity between two words can be defined. For example, a trivial definition of semantic similarity between the word $\mathfrak{A_O}$ and the word $\mathfrak{B_O}$ can be defined as: $\frac{1}{2}(\frac{|A^f\bigcap B^f|}{|A^f\bigcup B^f|}+\frac{|A_O\bigcap B_O|}{|A_O\bigcup B_O|})$. Communication ============= According the above analysis, self-consistent is very important for category representation. However, even when $\mathfrak{A}^{\alpha}$ is self-consistent, $\mathfrak{A}^{\alpha}$ may be something wrong. For example, [@Putnam1963Brains] has stated that madmen sometimes have consistent delusional systems. Why? As the final goal of a category is to help communication, different personal conceptual systems concerned on the category $\mathfrak{A}$ maybe have different representations, which results in misunderstanding, even contradiction. For a communication between two men $\alpha$ and $\beta$ concerned on the category $\mathfrak{A}$, two simple cases about the category $\mathfrak{A}$ are discussed as follows. One case is that only one man knows the category $\mathfrak{A}$. Without loss of generality, it is assumed that the man $\alpha$ knows $\mathfrak{A}$, i.e. he knows $\mathfrak{A}^{\alpha}$ and the man $\beta$ has no idea of $\mathfrak{A}$, then the man $\beta$ must learn the knowledge about the category $\mathfrak{A}$ from $\mathfrak{A}^{\alpha}$. In order to simplify learning, it is supposed that the $\mathfrak{A}^{\beta}$ and $\mathfrak{A}^{\alpha}$ have the same outer name and all the relevant conceptual systems are plain. Under such assumptions, the man $\beta$ needs to learn how to represent $\mathfrak{A}$ in $L^{\beta}=\{\mathfrak{A}^{\beta}\}$. Hence, the above question can be described as follows: Let the input representation be $(O_A^{\alpha},I_A^{\alpha},\underline{A}^{\alpha},Sim_A^{\alpha})$ and the output representation be $(O_A^{\beta},I_A^{\beta},\underline{A}^{\beta},Sim_A^{\beta})$ with respect to a learning algorithm, if a subset of $(O_A^{\alpha},I_A^{\alpha})$ or $O_A^{\alpha}$ is known, try to output $(O_A^{\beta},I_A^{\beta},\underline{A}^{\beta},Sim_A^{\beta})$, which is a standard categorization problem and has been well studied in [@JianYu2014Categorization],[@JianYu2015Categorization]. The other case is that both sides in communication know the category $\mathfrak{A}$. Assume that the representation of the category $\mathfrak{A}$ for the man $\alpha$ is $(\mathfrak{A_O^{\alpha}},O_A^{\alpha},I_A^{\alpha},\mathfrak{A_I^{\alpha}},\underline{A}^{\alpha},Sim_A^{\alpha})$ and the representation of the category $\mathfrak{A}$ for the man $\beta$ is $(\mathfrak{A_O^{\beta}},O_A^{\beta},I_A^{\beta},\mathfrak{A_I^{\beta}},\underline{A}^{\beta},Sim_A^{\beta})$. In the following, we will discuss different cases under the above assumptions. If $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$ and $A_I^{\alpha}=A_I^{\beta}$ and $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_I}^{\beta}$ and $\underline{A}^{\alpha}=\underline{A}^{\beta}$, then it is a totally perfect communication between $\alpha$ and $\beta$ concerned on the category $\mathfrak{A}$. If $\mathfrak{A}^{\alpha}$ and $\mathfrak{A}^{\beta}$ are self-consistent, it is easy to prove that the number of constraints for totally perfect communication can be greatly reduced. Therefore, it takes for granted that categories in personal conceptual systems are self-consistent in communication. Unfortunately, the above self-consistent conditions are neither necessary nor sufficient for a totally perfect communication. If $A_O^{\alpha}=A_O^{\beta}$ and $A_I^{\alpha}=A_I^{\beta}$ and $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_I^{\beta}}$ and $\underline{A}^{\alpha}=\underline{A}^{\beta}$ but $\mathfrak{A_O^{\alpha}}\neq \mathfrak{A_O^{\beta}}$, then such a category at least has two outer names when $\alpha$ and $\beta$ make no mistake. If $\alpha$ or $\beta$ know that $\mathfrak{A_O^{\alpha}}$ and $\mathfrak{A_O^{\beta}}$ are two outer names of the category $\mathfrak{A}$, then $\alpha$ and $\beta$ still can understand each other about the category $\mathfrak{A}$, otherwise, $\alpha$ and $\beta$ can not understand each other about the category $\mathfrak{A}$. If $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$ and $A_I^{\alpha}=A_I^{\beta}$ and $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_I^{\beta}}$, then it is a perfect communication between $\alpha$ and $\beta$ concerned on the category $\mathfrak{A}$. If $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$, then it is a semi perfect communication between the man $\alpha$ and the man $\beta$ concerned on the category $\mathfrak{A}$. Compared with perfect communication, semi perfect communication can be easily judged. In daily life, there are few perfect communications but more semi perfect communications. Obviously, a semi perfect communication on the category $\mathfrak{A}$ can lead to misunderstanding in some cases. If $\mathfrak{A}^{\alpha}$ and $\mathfrak{A}^{\beta}$ are self-consistent, then a semi perfect communication becomes a perfect communication. In practical communication, if $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}$, then it is a proper communication between the man $\alpha$ and the man $\beta$ concerned on the category $\mathfrak{A}$ and the object $o$. In everyday functioning, a proper communication between the man $\alpha$ and the man $\beta$ concerned on the category $\mathfrak{A}$ and the object $o$ usually makes the man $\alpha$ and the man $\beta$ feel mutual understanding at least with respect to the category $\mathfrak{A}$ and the object $o$. Sometimes, only $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ or $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}$ is true in communications. In this case, misunderstanding often occurs if one side thinks communication is right. For instance, when $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$, $\alpha$ and $\beta$ will think they can understand each other although it is not true in many times. For a conceptual system, a category may have several outer names, several categories may share one outer name, which brings more challenges into communications. Furthermore,when both sides know the category $\mathfrak{A}$ but only partial information is known about $\mathfrak{A}^{\alpha}$ or $\mathfrak{A}^{\beta}$, communication can still go on between the man $\alpha$ and the man $\beta$. In this case, totally perfect communication is supposed to be true in order to make mutual understanding possible. For example, $\mathfrak{A_O^{\alpha}}$ or $\mathfrak{A_O^{\beta}}$ is illustrated, communications are still available by assuming that $\mathfrak{A^{\alpha}}=\mathfrak{A^{\beta}}$. When an object $o$ is referred to, communications still carry on by finding the category corresponding to the object $o$ by assuming that $o_A^{\alpha}=o_A^{\beta}$. Certainly, such the above cases can bring more misunderstandings and more challenges in communication as you may not see (or say) what I see (or say)[@Petrie1976Do]. Usually, perfect communication is supposed to be hold in daily life in order to simplify mutual understanding. However, communication is usually not perfect and misunderstanding can not be guaranteed to be avoided, which has resulted in so many errors, miracles, jokes, tragedies, comedies, dramas, quarrels, peace, wars and so on. In general, if $L^{\alpha}=L^{\beta}$, then man $\alpha$ and man $\beta$ can perfectly understand each other. If $L^{\alpha}\wedge L^{\beta}\neq \emptyset$, then man $\alpha$ and man $\beta$ are considered to have common words, otherwise, they have no common word. Usually, the larger $\frac{|L^{\alpha}\wedge L^{\beta}|}{|L^{\alpha} \vee L^{\beta}|}$, the easier man $\alpha$ and man $\beta$ communicate. For human beings, education can make the outer representation of a category to be as same as possible for all people. If the conceptual representation of a category is defined by a proposition, education can make the inner representation of a category to be the same for all people. Frankly speaking, eduction can make men’s conceptual systems share common words as many as possible. When men share many common words, it can greatly reduce the dialogue cost, which is the deep reason why man prefer to using propositions to represent the conceptual representation of a category. Throughout evolution, any culture has formed enough common words so that a man can share his feeling, thought, observation, instruction, plan and imagination by common words in his community, such common words can naturally form a language. A language reflects the common words among its users. Words, Sentences and Knowledge ============================== As [@wittgenstein1953philosophical] has stated, “the meaning of a word is its use in language". Transparently, any word is used in sentences. Usually, man uses not words (phrases or symbols) but sentences to communicate with each other. Sentences describe man’s feeling, thought, observation, instruction, plan and imagination through the relation between words and objects in the universe. What’s relation between words and sentences? When studying the relation between words and sentences, a category represents sentence pattern, word represents the sentence pattern name that a word is used in the corresponding sentence, an object in the discussed domain can be supposed to be a sentence. Based on the above assumption, the proposed category representation $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Sim_A)$ can be redefined as follows: $\forall o, o_A$ can be represented by a sentence (sometimes, an object $o$ is even a sentence. But an object $o$ is not guaranteed to be a sentence), $\mathfrak{A_O}$ is an abstract name of a sentence pattern $\mathfrak{A}$. Consequently, $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Sim_A)$ can represent any sentence pattern. In other words, a sentence is an concrete object representation of some sentence pattern. A sentence pattern is named by a word (phrase or symbol). In theory, when self-consistent holds for $\mathfrak{A}$, it is very important to express $\underline{A}$ in an explicit way. According to the above analysis, $o_A$ is an instantiations of $\underline{A}$ and $\underline{A}$ is the conceptualization of $O_A$. Hence, $\underline{A}$ can be considered as an operator as follows: $\underline{A}$: $o \mapsto o_A$ such that $\underline{A}(o)=o_A$. By this way, $\underline{A}$ can be called category feature mapping. Usually, $\underline{A}$ can be explicitly defined. For example, assuming that $\underline{A}$ is defined by a predicate $P_A()$, then $o_A=P_A(o)$ as $P_A(o)$ is a statement. Under such an assumption, it is easy to know that $I_A(o_A)=Sim_A(o_A,\underline{A})\in \{0,1\}$. Therefore, the proposed category representation bridges the gap between words and propositions (statements). In a broad sense, the proposed category representation establishes the relation between sentences and words. It is well known that knowledge can be expressed by sentences in natural language. Considered the relation between words and sentences, we can say that knowledge are words. When you know all words clearly, you know all knowledge as far as man can reach. Here, a word $\mathfrak{A_O}$ refers to $(\mathfrak{A_O},O_A,I_A,\mathfrak{A_I},\underline{A},Sim_A)$, which belongs to common words in a culture independent of any individual. By the above analysis, more words means more knowledge. A new word means some new knowledge. Eduction helps words propagate. Man continuously creates new words to make life more advanced. To our surprise, Stefan George stated that “where word breaks, nothing may be”, which also implies that no words, no knowledge. Situations ========== In the above analysis, a conceptual system is supposed to be independent of situations. However, a conceptual system may contain too many words and the corresponding instantiations (sentences). Furthermore, a man may have several conceptual systems. For example, a man may master several languages such as English, Chinese, Spanish and so on. When will he speak English, Chinese and Spanish? If he is in Australia, he had better speak English. When he met a native farmer in Australia, he had not better talk about Beijing opera or Beijing roast duck. Therefore, specific situation determines the optimal conceptual system and words. Of course, a key problem for a man is to determine which conceptual system should be used at his current situation. If situations are considered as objects and conceptual systems are considered as categories, the above problem is a standard categorization question. In the following, mathematical language will be used to describe relation between conceptual systems and situations. In mathematics, suppose that $S=\{s(t)| t\in R\}$ is the set of situations, where $s(t)$ is the situation at time $t$. $s(t)$ consists of all objects in the universe at time $t$. In this paper, objects can be referred to any thing, including locations, individuals, things, events,properties, relations and so on. For a man $\alpha$, he only can experience his personal situations. Man $\alpha$’s situations can be represented by a set $S^{\alpha}=\{s^{\alpha}(t)| t\in R\}$, where $s^{\alpha}(t)$ is the situation at time $t$ that $\alpha$ is actually in. In daily life, $s^{\alpha}(t)$ is very different from $s(t)$. In general, $s^{\alpha}(t)\subset s(t)$. Moreover, no matter what conceptual system a man will choose, he actually perceives the same situation at a fixed time $t$. Hence, it can be assumed that $s_a^{\alpha}(t)$ is the abstract situation with respect to situation $s^{\alpha}(t)$, which represents object representation that $\alpha$ actually perceives at the situation $s^{\alpha}(t)$. As [@Barwiseperry1999] pointed out, all the objects in $s^{\alpha}(t)$ are represented in $s_a^{\alpha}(t)$, but some objects represented in $s_a^{\alpha}(t)$ do not belong to $s^{\alpha}(t)$. In theory, objects represented in $s_a^{\alpha}(t)$ includes objects that $\alpha$ has remembered, is experiencing. Sometimes, objects represented in $s_a^{\alpha}(t)$ are imagined or remembered by the man $\alpha$, thus they may not be objects in $s^{\alpha}(t)$. Usually, $s^{\alpha}(t)\subseteq s_a^{\alpha}(t)$. In practice, a man $\alpha$ usually has a finite conceptual systems. Assume that $\mathfrak{L}^{\alpha}=\{\mathfrak{L}^{\alpha}(t)|t\in R\}$ and $\forall t, \mathfrak{L}^{\alpha}(t)\in \{L_1^{\alpha}(t),L_2^{\alpha}(t),\cdots, L_c^{\alpha}(t)\} $, where $L^{\alpha}(t)$ is the conceptual system that $\alpha$ adopts at time $t$. If a man adopts a unsuitable conceptual system for his current situation, embarrassments (sometimes, dangers) may occur. Therefore, when a man $\alpha$ is in situation $s^{\alpha}(t)$, he needs to find the optimal conceptual system $\mathfrak{L}^{\alpha}(t)$. In order to judge which conceptual system should be used, it needs to set a situation feature mapping $s_L: s_a^{\alpha}(t)\rightarrow s_L^{\alpha}(t)$, where $s_L^{\alpha}(t)$ is the situation representation corresponding to the situation $s(t)$ and the man $\alpha$. Clearly, a conceptual system corresponds to a category. Assuming that $\underline{L}_i^{\alpha}(t)$ is the concept corresponding to the conceptual system $L_i^{\alpha}(t)$, $Sim_L^{\alpha}$ is the category similarity mapping between situations and conceptual systems. According to the study in [@JianYu2015Categorization], the optimal conceptual system should be defined as $\mathfrak{L}^{\alpha}(t)$=$\widetilde{s_L^{\alpha}(t)}=\arg\max_{L_i^{\alpha}(t)}Sim_{L}^{\alpha}(s_L^{\alpha}(t),\underline{L}_i^{\alpha}(t))$. When the optimal conceptual system $\mathfrak{L}^{\alpha}(t)$ is selected, man $\alpha$ needs to select the optimal word for the objects in the current situation $s^{\alpha}(t)$, such issues have been well studied in Section 3. Here, the discussed domain is $O\bigcap s_a^{\alpha}(t)$ when selecting the optimal word for the current situation $s^{\alpha}(t)$. If a man has a unique conceptual system, he can skip the process of selecting conceptual systems when communicating with others. What a pity, a man usually has more than one conceptual system except in childhood, even though he only speaks one language. [@Freud1923] had discovered that there are id, ego and superego in a personal mental life. Such three parts generally have different (sometimes, contradictory) conceptual systems. Thus, it is doomed for man to select the optimal conceptual system to fit the specific situation. Moreover, a situation often consists of several sub situations and a conceptual system also has several sub conceptual systems. With respect to a special sub situation, an optimal sub conceptual system can be selected according to the same principle. The hierarchical situation structure corresponds to similar hierarchical conceptual system structure. More complex, $\widetilde{s_L^{\alpha}(t)}$ may not be one element, which shows that two and more conceptual systems are candidate. In this case, a candidate conceptual system usually is randomly selected. The similar thing can happen when selecting the optimal word by inner referring operator. Such cases inevitably brings mistakes or errors in daily life. In theory, avoiding such cases need more refined words, conceptual systems and situation representation. For the same situation $s(t)$, the man $\alpha$ and the man $\beta$ may have experience different situations $s^{\alpha}(t)$ and $s^{\beta}(t)$, they also uses different conceptual systems $\mathfrak{L}^{\alpha}(t)$ and $\mathfrak{L}^{\beta}(t)$. For example, the blind and the deaf can not have the same situation experience even though they are in the same situation. However, situation’s constraints on conceptual systems are the same for all men. Situations make it is more complex for man to understand each other. However, situations are powerful for word sense disambiguation. In daily life, the meaning of any pronoun is determined by situations. Without situations, it is impossible to understand what a pronoun is referred to. Clearly, so called context is often a part of a special situation. In some sense, man always hope to find the truth independent of situations in order to reducing the computational complexity. Certainly, it is an impossible mission for man. Usually, situation limitation can be weaken in some sense, but can never be totally ignored. Denotation, Connotation and Annotation ====================================== For communication, if the speaker is the man $\alpha$ and the listener is the man $\beta$, the most important thing for the listener $\beta$ is to know that ($A_I^{\alpha},\underline{A}^{\alpha}$) when man $\alpha$ said the outer name $\mathfrak{A_O^{\alpha}}$. However, ($A_I^{\alpha},\underline{A}^{\alpha}$) cannot be observed explicitly no matter whether self-consistent is true or not. Therefore, sometimes the speaker and the listener can play a game together to fool others. In theory, it is very important to know what an element in $\mathfrak{A_O^{\alpha}}$ is referred to. Based on the study of [@Popper1972] and [@Cassirer1944], an element in $\mathfrak{A_O^{\alpha}}$ can belong to the physical world, the mental world, and the symbolic world. When the elements in $A_I$ are objects in the physical world, $\mathfrak{A_I}$ is a denotation. When the elements in $A_I$ are psychological states in the mental world, $\mathfrak{A_I}$ is a connotation. Therefore, man can use words to refer to not only objects in the real world but also mental states such as pains, loves, dislikes, fears, hopes, beliefs and intentions. Generally speaking, denotation needs to rebuild its inner set in the corresponding conceptual system as its referred object is explicit, connotation must form its outer set in the corresponding conceptual system as the mental state is transparent for its owner but implicit for other men. When the denotations are investigated, it may be a little easy to make an agreement by referring to the objects in the physical world. When the connotations are discussed, it may be very challenge to reach a conclusion accepted by two sides as the inner referring cannot be objectively observed by two sides. In natural language, many words (phrases or symbols) can not be absolutely denotation or connotation. For example, the word “pig" can make people different feelings besides its denotation. Clearly speaking, many words have some affective character. What’s more, words (phrases or symbols) may be neither denotation nor connotation. In daily life, men can create new words (phrases or symbols) to express their plans, imaginations and dreams. Before plans, imagination and dreams come true, they are only words (phrases or symbols). In reality, most plans, imaginations and dreams have no chance of being fulfilled, hence they are not real things in the world independent of the corresponding words (phrases or symbols). According to [@Popper1972] and [@Cassirer1944], words (phrases or symbols) belong to the symbolic world. As [@Cassirer1944] has pointed out, language, myth, art, and religion are parts of the symbolic world. In modern times, television, film, advertisement, world wide web and media are also parts of the symbolic world. For a self-consistent category $\mathfrak{A}$, if elements in $A_I$ do not correspond to something real in the physical world or the mental world except for the corresponding word (phrase or symbol) in the symbolic world, such a category is called an annotation. Hence, an annotation belongs to the symbolic world. When the discussed domain is constrained in the physical world or the mental world except for the symbolic world, $A_I=\emptyset$ means that the corresponding category $\mathfrak{A}$ may be an literary invention but $\mathfrak{A_I}$ is an actual word (phrase or symbol) in the symbolic world, for example, unicorn, Escher’s pictures. If the discussed domain is in the symbolic world, especially in languages, $A_I\neq \emptyset$ for any category $\mathfrak{A}$ as $\mathfrak{A_O}$ can be self-referring. Generally speaking, most objects are literary invention in a fiction but indeed reflect the writer’s idea, feeling and view. In theory, the symbolic world not only reflects the physical world or the mental world, but also creates something not existed in the physical world or the mental world. For example, [@Chomsky1956] composed his famous sentence “colorless green ideas sleep furiously”. Here, “colorless green ideas” refers to nothing in the physical world, but can refer to “colorless green ideas” itself in language. In fact, “colorless green ideas” has its specific sound or its handwritten word image, “colorless green ideas” can refer to the corresponding sound or the corresponding handwritten word image. Any word can be used without referring a real object in the physical world or a real psychological state in the mental world. Frankly speaking, any language can be used in the symbolic world by itself, without directly referring to any thing in the physical world or the mental world. Furthermore, many languages have been used in the world, and one language can be represented by the other language, including itself. If the conceptual system $L^{\alpha}$ is self-consistent, elements in $A_I^{\alpha}$ are not objects in the physical world or the mental world, the category $\mathfrak{A^{\alpha}}$ belongs to the symbolic knowledge for the man $\alpha$. In fact, some people can use language loquaciously without referring to a real event in the physical world or the mental world such as Denyse[@Pinker1995]. When elements in $A_I^{\alpha}$ are objects in the physical world or the mental world, the man really knows the category $\mathfrak{A^{\alpha}}$, $\mathfrak{A^{\alpha}}$ becomes the real knowledge for the man $\alpha$. Transparently, that $A_I^{\alpha}=\emptyset$ holds in the physical world does not mean that $A_I^{\beta}=\emptyset$ holds in the physical world. The same word may not have the same meaning for the different men. Truth, Falsity and Uncertainty ============================== In any conceptual system, it is very important to define what truth or falsity is. According to [@LakofeJohnson1980], truth is based on understanding. Understanding includes “knowing yourself" and “knowing others". “Knowing yourself" occurs in the same conceptual system, and “knowing others" in two conceptual systems. In the following, we will discuss them respectively. Knowing Yourself and Inner Truth -------------------------------- Knowing yourself means that a man know his conceptual system well. In other words, the man $\alpha$ know which category he takes to be true in his conceptual system $L^{\alpha}$. Unfortunately, the category in the conceptual system $L^{\alpha}$ has more than the states of truth and falsity. More specifically, for a category $\mathfrak{A^{\alpha}}$ in the conceptual system $L^{\alpha}$, if $\exists o$ such that $\overrightarrow{o_A^{\alpha}} $ or $\widetilde{o_A^{\alpha}}$ is multiple value, then $\mathfrak{A^{\alpha}}$ is said to be uncertain, otherwise, $\mathfrak{A^{\alpha}}$ is said to be certain. When $\mathfrak{A^{\alpha}}$ is uncertain, it is impossible to decide whether or not $\mathfrak{A^{\alpha}}$ is absolutely true or false. If $\mathfrak{A^{\alpha}}$ can be said to be absolutely true or false, then $\forall o\in O$ ($\overrightarrow{o_A^{\alpha}} $ and $\widetilde{o_A^{\alpha}}$ are single value). If $\forall o\in A_O\vee A_I (\overrightarrow{o_A^{\alpha}} $ and $\widetilde{o_A^{\alpha}}$ are single value), $\mathfrak{A^{\alpha}}$ is said to be inner true if it is self-consistent, $\mathfrak{A^{\alpha}}$ is said to be inner false if $\mathfrak{A_I^{\alpha}}\neq \mathfrak{A_O^{\alpha}}$ or $A_I^{\alpha}\neq A_O^{\alpha}$. Clearly, inner true is equivalent to self-consistent. If $\mathfrak{A^{\alpha}}$ is uncertain, it is possible to compute whether or not $\mathfrak{A^{\alpha}}$ is true or false in probability. However, it still can be judged if relation between a specific object $o$ and the category $\mathfrak{A}$ is true or false. If $\overrightarrow{o_A^{\alpha}}=\widetilde{o_A^{\alpha}}$, then the category $\mathfrak{A^{\alpha}}$ is locally self-consistent with respect to the object $o$. In particular, if $\overrightarrow{o_A^{\alpha}}=\widetilde{o_A^{\alpha}}$ and $\overrightarrow{o_A^{\alpha}}=\mathfrak{A_O^{\alpha}}$, then it is inner true that the object $o$ belongs to the category $\mathfrak{A}^{\alpha}$ . Knowing Others and Outer Truth ------------------------------ In daily life, a conscious man $\alpha$ knows that his judgement about $\mathfrak{A}$ may not be objectively true even if $\mathfrak{A^{\alpha}}$ is inner true. Usually, a conscious man needs compare his judgement with others. Let us consider the simplest case. Man $\alpha$ only needs to consider the opinion of man $\beta$ with respect to $\mathfrak{A}$. If $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$, then man $\alpha$ thinks that $\mathfrak{A^{\alpha}}$ is outer true with respect to man $\beta$. If $\mathfrak{A_O^{\alpha}}\neq\mathfrak{A_O^{\beta}}$ or $A_O^{\alpha}\neq A_O^{\beta}$, then man $\alpha$ thinks that $\mathfrak{A^{\alpha}}$ is outer false with respect to man $\beta$. In practice, the above condition is still very demanding for man $\alpha$ to judge whether or not $\mathfrak{A^{\alpha}}$ is outer true with respect to man $\beta$. Therefore, the condition of the outer truth needs further simplified with respect to a category. It is very difficult to get $A_O^{\alpha}$ and $A_O^{\beta}$ in general cases, but it is easier to consider an object in $A_O^{\alpha}$ or $A_I^{\alpha}$. When outer referring operator is supposed to be single value, man $\alpha$ thinks that it is outer true with respect to man $\beta$ that the object $o$ belongs to $\mathfrak{A^{\alpha}}$ if $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}=\mathfrak{A_O^{\alpha}}$ , and $\alpha$ thinks that it is outer false with respect to man $\beta$ that the object $o$ belongs to $\mathfrak{A^{\alpha}}$ if $\overrightarrow{o_A^{\alpha}}=\mathfrak{A_O^{\alpha}}\neq \overrightarrow{o_A^{\beta}}$. Certainly, it is easier to judge whether or not $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}$ holds than $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$. In particular, if man $\beta$ is supposed to be an oracle, man $\alpha$ thinks that $\mathfrak{A^{\alpha}}$ is outer true if $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$. Man $\alpha$ thinks that it is outer true that the object $o$ belongs to $\mathfrak{A^{\alpha}}$ if $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}=\mathfrak{A_O^{\alpha}}$ and man $\alpha$ thinks that it is outer false that the object $o$ belongs to $\mathfrak{A^{\alpha}}$ if $\overrightarrow{o_A^{\alpha}}=\mathfrak{A_O^{\alpha}}\neq \overrightarrow{o_A^{\beta}}$. Transparently, the inner truth is not equivalent to the outer truth. Moreover, the inner truth is independent of the outer truth. Truth, Falsity, Uncertainty --------------------------- Transparently, inner truth and outer truth are not guaranteed to be true. In logic, it is always assumed that absolute truth is independent of different conceptual systems. In order to achieve this aim, a natural assumption is that self-consistent is self evident and man $\beta$ must be an oracle. Even when $\mathfrak{A^{\alpha}}$ is self-consistent, $\mathfrak{A^{\alpha}}$ still depends on man $\alpha$. In order to make category representation impersonal, a simple assumption is that $\alpha=\beta$. Under the above assumption, it is easy to judge whether or not $\mathfrak{A}$ is true. As a matter of fact, if $\mathfrak{A}$ is inner true, then $\mathfrak{A}$ is true. If $\mathfrak{A}$ is inner false, then $\mathfrak{A}$ is false. In common sense, if $\overrightarrow{o_A}=\widetilde{o_A}=\mathfrak{A_O}$, it is said to be true that the object $o$ belongs to $\mathfrak{A}$; if $\overrightarrow{o_A}\neq\widetilde{o_A}$ or $\overrightarrow{o_A} \neq\mathfrak{A_O}$, it is said to be false that the object $o$ belongs to $\mathfrak{A}$. Certainly, when making the above judgments, the presupposition is made as follows: $\forall o\in O$ (its object representation $o_A$ is independent of any observer) and $\forall o\in (A_O\vee A_I)$($\overrightarrow{o_A}$ and $\widetilde{o_A}$ are single value). Under the above assumptions and presuppositions, if $\mathfrak{A}$ is true, then it is not only inner true but also outer true for anyone. If $\exists o\in O$(($\overrightarrow{o_A}$ or $\widetilde{o_A}$ is multiple value) and ($\mathfrak{A_O}\in \overrightarrow{o_A}$ or $\mathfrak{A_I}\in \widetilde{o_A}$)), $\mathfrak{A}$ is uncertain. In particular, if the object $o\in O$(($\overrightarrow{o_A}$ or $\widetilde{o_A}$ is multiple value)and ($\mathfrak{A_O}\in \overrightarrow{o_A}$ or $\mathfrak{A_I}\in \widetilde{o_A}$)), then it is uncertain that the object $o$ belongs to the category $\mathfrak{A}$ without specification. Obviously, the above assumption about truth is too strong to be satisfied in daily life. Obviously, it is little chance for a man to be an oracle. Therefore, truth in daily life can not be defined independent of different conceptual systems. In common sense, $\mathfrak{A^{\alpha}}$ is empirically true with respect to man $\alpha$ and man $\beta$ if and only if $\mathfrak{A^{\alpha}}$ and $\mathfrak{A^{\beta}}$ are inner true and $\mathfrak{A^{\alpha}}$ is outer true with respect to man $\beta$. Man $\alpha$ thinks that $\mathfrak{A^{\alpha}}$ is empirically true with respect to man $\beta$ if and only if $\mathfrak{A^{\alpha}}$ is inner true and $\mathfrak{A^{\alpha}}$ is outer true with respect to man $\beta$. More specifically, it is empirically true with respect to man $\alpha$ and man $\beta$ that the object $o$ belongs to the category $\mathfrak{A}$ if and only if $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}=\mathfrak{A_O^{\alpha}}$ and $\overrightarrow{o_A^{\alpha}}=\widetilde{o_A^{\alpha}}$ and $\overrightarrow{o_A^{\beta}}=\widetilde{o_A^{\beta}}$. If and only if $\overrightarrow{o_A^{\alpha}}=\overrightarrow{o_A^{\beta}}=\mathfrak{A_O^{\alpha}}$ and $\overrightarrow{o_A^{\alpha}}=\widetilde{o_A^{\alpha}}$, man $\alpha$ thinks it is empirically true with respect to man $\beta$ that the object $o$ belongs to the category $\mathfrak{A}$. In daily life, inner truth, outer truth and empirical truth is more useful than truth. Sometimes, outer truth, inner truth and empirical truth are confused with truth in daily life. However, there are great differences among inner truth, outer truth, empirical truth and truth. Sometimes, inner truth holds but outer truth does not hold. A famous example in [@Betrandrusell1946] is that the lunatic who believes that he is a poached egg is to be condemned solely on the ground that he is in a minority, or rather on the ground that the government does not agree with him. Certainly, the category “the poached egg” is self-consistent with respect to such an lunatic, therefore, it is inner true with respect to such an lunatic, but it can not be accepted as outer truth by others that lunatic has the right knowledge about “the poached egg”. Therefore, the inner truth is not guaranteed to be the outer truth. Similarly, outer truth does not guarantee inner truth. The emperor’s new clothes clearly illustrated the difference between outer truth and inner truth. History of science has illustrated many times that empirical truth is equivalent to truth. More specifically, when talking about connotations, inner truth is more important than outer truth; when referring to denotations, outer truth plays more important role than inner truth; when discussing annotations, outer truth have the same influence as inner truth. In everyday life, man should carefully distinguish when inner truth, outer truth, empirical truth and truth holds. Related Work ============ In this paper, we will partially survey some papers in the literature related to communication and truth. Grice’s Cooperative Principle ----------------------------- For communication, [@Grice1975] presented cooperative principle as follows: make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged. [@Grice1975] thought that cooperative principle can yield four maxims, including quantity, quality, relation and manner as follows:\ **Quantity:** Make your contribution as informative as is required (for the current purposes of the exchange); do not make your contribution more informative than is required.\ **Quality:** Do not say what you believe to be false; do not say that for which you lack adequate evidence.\ **Relation:** Be relevant.\ **Manner:** Avoid obscurity of expression; avoid ambiguity; be brief (avoid unnecessary prolixity); be orderly. Transparently, the listener must understand the speaker’s meaning in order to follow the cooperative principle during conversation. When the accepted purpose or direction of the talk exchange between man $\alpha$ and man $\beta$ is about the category $\mathfrak{A}$, if man $\alpha$ says $\mathfrak{A_O^{\alpha}}$, man $\beta$ should assume that $\mathfrak{A_O^{\alpha}}$=$\mathfrak{A_O^{\beta}}$ and $A_O^{\alpha}=A_O^{\beta}$ and $A_I^{\alpha}=A_I^{\beta}$ and $\mathfrak{A_I^{\alpha}}$=$\mathfrak{A_I}^{\beta}$ and $\underline{A}^{\alpha}=\underline{A}^{\beta}$. Under such an assumption, maxim of quantity requires that the man $\alpha$ refers to the object $o$ with respect to the category $\mathfrak{A}$, the man $\beta$ should provide the information $o_A^{\beta}-o_A^{\alpha}$. For maxim of quality, its first condition in requires categories in personal conceptual systems are self-consistent, which means that $A_O^{\alpha}=A_I^{\alpha}$ and $\mathfrak{A_O^{\alpha}}=\mathfrak{A_I^{\alpha}}$; its second condition implies $A_O^{\alpha}\neq \emptyset$. The same requirements for the man $\beta$ is also true. Maxim of relation and maxim of manner do something with not only relation between situations and conceptual systems but also totally perfect communication condition, which require that people must choose the right conceptual system according to the specific situation, otherwise, maxim of relation and maxim of manner may be violated in great probability. As pointed out in Section 5, totally perfect communication condition does not ask for self-consistent, which implies that cooperative principle is not fully consistent with maxim of quality. In daily life, many stories demonstrate this point. If totally perfect communication condition is satisfied, cooperative principle may be obeyed. If totally perfect communication condition is not satisfied, cooperative principle can not be obeyed. [@Pinker2007] has cited an example as follows: In an episode of *Seinfeld*, George is asked by his date if he would like to come up for coffee. He declines, explaining that caffeine keeps him up at night. Later he slaps his forehead and realizes, “‘Coffee’ does not mean coffee! ‘Coffee’ means sex". In this example, George and his date clearly did not satisfy the totally perfect communication condition, and George’s date also violated the self-consistent condition. In summary, Grice’s cooperative principle is just a roughly description about self-consistent and the relation between word and the corresponding category feature representation, totally perfect communication condition and the proposed relation between situation and conceptual system offer more powerful explanation for communication than Grice’s cooperative principle. On Truth -------- In the literature, many papers have devoted to defining truth such as correspondence theory, coherence theory, experientialist theory and so on [@LakofeJohnson1980]. The most famous definition of truth is originally based on the well known words of [@Aristotle1908]: “To say of what is that it is not, or what is not that it is, is false, while to say of what is that it is , or what is not that it is not, is true". Such statement has been translated by [@Tarski1956] as “ a true sentence is one which says that the state of affairs is so and so, and the state of affairs indeed is so and so". Then it formulated the classic Tarski truth definition, which can be intuitively interpreted as “ Snow is white" is true if and only if snow is white. However, the well known words of [@Aristotle1908] about truth can be interpreted differently from classic Tarski truth definition. In some sense, the proposed definitions of inner truth, outer truth, empirical truth and truth also follow Aristotle truth definition. Therefore, the study of truth in this paper is also consistent with correspondence theory. If considering self-consistent as self understanding, and semi perfect communication as mutual understanding, then the definition of inner truth, outer truth, empirical truth and truth are based on understanding. So inner truth, outer truth, empirical truth and truth are consistent with [@LakofeJohnson1980], which says that “we understand a statement of being true in a given situation when our understanding of the statement fits our understanding of the situation closely enough for our purposes". Therefore, the proposed study of truth has all the characteristics of [@LakofeJohnson1980]’s experientialist theory of truth, in other words, the study of truth in this paper has some elements of a coherence theory and a pragmatic theory. Discussion and Conclusions ========================== When ignoring category name, [@JianYu2015Categorization] has presented a category representation. However, everything has a name, as [@Cassirer1944] pointed out. In this paper, a new approach to representing words (phrases or symbols) is given by considering the category name. This proposed category representation solves the drawbacks of the classical category representation, it can be taken as a generalization of a classic set. [@Betrandrusell1950] has stated that “some words have two non-verbal uses, (a) as indicating objects (b) as expressing states of mind”. In this paper, such observations are extended to all categories. The proposed category representation has three parts: one is about linguistic (word, phrase or symbol), one is about action (outer set), one is about idea (inner representation). According to [@Popper1972] , words (phrases or symbols) belong to the third world (symbolic world), outer set belongs to the first world (physical world) , and inner representation belongs to the second world (mental world). Hence, the category name can be categorized into denotation, connotation and annotation with respect to a conceptual system. Denotation refers to real object in the physical world. Connotation refers to psychological state in the mental world. Annotation corresponds to object in the symbolic world but without referring objects in the physical world or the mental world. Sometimes, man can turn some annotation into reality. Under such an case, an annotation becomes a denotation or connotation. According the above analysis, the proposed category representation establishes the relations among three worlds [@Popper1972]. In daily life, every part of the category representation can independently form one conceptual system, i.e. say one thing, do another, think differently from saying and doing. For instance, men can smartly construct temporary conceptual systems to fit the local environmental requirements, such as actors, pretenders, spies, translators, etc. Generally speaking, man’s conceptual systems vary with times and environments. Man continuously changes his own conceptual systems through learning from others or the world. When a man has several conceptual systems, he usually selects the optimal conceptual systems and words with respect to the specific situation. Situations determine the adoption and evolution of conceptual systems. However, situations are very complex. Usually, $s^{\alpha}(t)$ is different from $s_L^{\alpha}(t)$. And the objects in $s^{\alpha}(t)$ can be observed by other men in the time $t$, and the objects represented in $s_L^{\alpha}(t)$ may be not observed by other men in the time $t$ but can be imagined or remembered by the man $\alpha$. Based on the proposed category representation, self-consistent and perfect communication are defined. Naturally, self-consistent is another version of categorization equivalency axiom proposed by [@JianYu2015Categorization] and totally perfect communication reinterprets the uniqueness axiom of category representation. Self-consistent greatly reduces understanding complexity and cognitive effort because of representational simplicity. Therefore, self-consistent can be called the economy assumption of category representation. In this paper, every man has his own conceptual system, which is consistent with the experientialist theory [@LakofeJohnson1980]. When two conceptual systems communicate, self-consistent is the lowest cost requirement as the inner representation can not be observable. When $\mathfrak{A}^{\alpha}$ is self-consistent, the man $\alpha$ is said to tell his truth (be honest) with respect to the category $\mathfrak{A}$, otherwise, the man $\alpha$ is lying with respect to the category $\mathfrak{A}$. The above analysis offers a clear explanation that prohibitions again lying is basic moral rule is social life just because lying can greatly increase understanding complexity and social cost. However, when $\mathfrak{A}^{\alpha}$ is self-consistent, the man $\alpha$ cannot be guaranteed to be objectively true with respect to the category $\mathfrak{A}$. When self-consistent is false but the inner referring set equals the outer referring set, it is worth further investigation how to construct name decoding mapping and name encoding mapping in a conceptual system. Translation is such a typical task. However, an ordinary person usually has no mapping between the inner name and the outer name in his conceptual system. Obviously, when self-consistent is not true for a category, man is considered to tell a lie. For instance, [@Harari2011] stated that most Christians did not imitate Christ, most Buddhists failed to follow Buddha, and most Confucians would have caused Confucius a temper tantrum. Many researches have been devoted to studying lying, such as [@DePaulo1996; @Heyman2009] and so on. When self-consistent is supposed to be true for category representation, relations between words are mathematically defined such as homonymy, polysemy, synonymy, hyponymy, antonymy, metaphor and metonymy, similarity. An interesting conclusion is that every word has more than two meanings, which latently supports that every man may have more than one conceptual systems. In practice, different conceptual systems often have different influences, which can result in more complex practical communication [@YoungFitzgerald2011]. In this paper, all the analysis ignores that the influence of $L^{\alpha}$ is different from that of $L^{\beta}$, in other words, all conceptual systems are considered to have the same influence. Even under such an assumption, only two simple cases about communications are discussed in this paper. One case is about learning. The other case is about understanding about the same category when two sides of communication both think that they know such a category. The conditions for perfect communication and relevant cases are presented. More complex cases such as both communication sides only know something about $\mathfrak{A}$ need to further study. Theoretically, all the above research results can be generalized into any two agents with their own conceptual systems. Such two agents can be any two sides, maybe two men, two robots, one robot and one man, one man and one book, and so on. In particular, if one side in communication is supposed to be absolutely true, then the other side is considered to be true or false depending on whether the proper communication holds or not. Furthermore, when one conceptual system has superpower influence, other conceptual systems may have been assimilated continuously or suppressed totally. In addition, only word understanding in single turn dialogue is studied in this paper, it is natural to investigate how to understand multi-turn dialogue in the future. The proposed semantic set also establishes the relation between sentences and words, which clearly shows that sentences are instantiations of the corresponding words and words are conceptualization of the corresponding sentences. Therefore, a surprising conclusion can be made: words are all knowledge, i.e. there is no knowledge without words. Here, words include spoken words, body words, musical words, etc. When a new word is created, some new knowledge is obtained. When some words are obsolete, their corresponding knowledge is also out of date. When some words are updated, relevant knowledge is also renewed. Therefore, words are evolved with situations, knowledge are also evolved with situations. Words and knowledge also follow the evolutionary principle: survival of the fittest. Last but not least, the proposed semantic set also introduces new definition of truth and analyzes the difference between inner truth, outer truth, empirical truth and truth. In daily life, inner truth, outer truth, empirical truth may be called truth. However, there are a great gap between inner truth, outer truth, empirical truth and truth. Different men may have different inner truth, outer truth, empirical truth about the same category. Strictly speaking, inner truth, outer truth, empirical truth are constrained by specific situations, truth does not depend on specific situations in some sense. ### Acknowledgements {#acknowledgements .unnumbered} This work was partially supported by the NSFC grant (61370129).

--- abstract: 'The current flow in phosphorene pnp junctions is studied. At the interfaces of the junction, omni-directional total reflection takes place, named anti-super-Klein tunneling, as this effect is not due to an energetically forbidden region but due to pseudo-spin blocking. The anti-super-Klein tunneling confines electrons within the junction, which thus represents a perfect lossless electron waveguide. Calculating the current flow by applying the Green’s function method onto a tight-binding model of phosphorene, it is observed that narrow electron beams propagate in these waveguides like light beams in optical fibers. The perfect guiding is found for all steering angles of the electron beam as the total reflection does not rely on the existence of a critical angle. For low electron energies and narrow junctions, the guided modes of the waveguide are observed. The waveguide operates without any loss only for a specific orientation of the junction. For arbitrary orientations, minor leakage currents are found, which however decay for low electron energies and grazing incidence angles. It is shown that a crossroad shaped pnp junction can be used to split and direct the current flow in phosphorene. The proposed device, a phosphorene pnp junction as a lossless electron waveguide, may not only find applications in nanoelectronics but also in quantum information technology.' author: - 'Yonatan Betancur-Ocampo' - 'Emmanuel Paredes-Rocha' - Thomas Stegmann bibliography: - 'eofpj.bib' title: Phosphorene pnp junctions as perfect electron waveguides --- Introduction ============ Graphene is nowadays certainly the most studied two-dimensional (2D) material, due to its exceptional properties and possible technological applications. Nevertheless, the absence of an intrinsic band gap complicates the realization of certain nanoelectronic devices [@Neto2009]. Several other 2D materials have been predicted and synthesized in recent years [@Naumis; @Das2015; @Bhimanapati2015; @Geng2018; @Wehling2014; @Li2014; @Vogt2012; @Li2018b; @Li2014b]. One of them is phosphorene, a single layer of black phosphorous, which possesses an intrinsic band gap and high electron mobility [@Das2014; @Na2014; @Li2014; @Li2014a; @Koenig; @Ezawa; @Peng; @Rudenko; @Cakir; @Lv; @Chang2015; @Wu2015; @Kou; @Lew; @Sisakht; @Rudenko2; @Elahi; @Utt; @Ameen2016; @Li2; @Carvalho; @Mehboudi2016; @Liu2017; @Soleimanikahnoj2017; @Yang2017; @Lewenkopf; @Sarkar; @Quhe2018; @Zhu; @Partoens; @Quhe2018; @Wu2018; @Zhu; @Zhang; @Betancur-Ocampo2019; @Miao; @Watts; @Ray; @Jung2020; @Dana2020]. Moreover, unconventional properties like negative reflection and anti-super-Klein tunneling [@Betancur-Ocampo2019], make phosphorene a promising material for nanoelectronic applications. The ballistic electron propagation in 2D materials allows the observation of analogies from optics [@Betancur-Ocampo2019; @Betancur; @Betancur3; @Stegmann2016; @Stegmann2019; @Sajjad; @Fuchs; @Chen; @Pozzi; @Elahi2019; @Cheianov; @Cheianov2; @Lee; @Baeuerle2018; @Li2018a; @Sajjad2011; @Bai2018; @Graef2019; @Zhang2019; @Yang2019]. For example, it has been shown that an electron beam, which hits the interface of a pn junction, is refracted similar to a light beam at the interface of two different media. An important tool for electron optics in 2D materials are waveguides or fibers, which may find application as electron conveyors in quantum information processes [@Baeuerle2018]. One strategy to construct such waveguides is to use gates that generate regions of different electrostatic potential and thus, constitute a pnp junction or, in other words, a potential well. Electron optics in 2D materials is studied commonly in graphene. However, the Klein tunneling of electrons prevents the efficient confinement of electrons by means of pnp junctions [@Katsnelson; @Kim; @Rozhkov2011]. As a solution to this problem, it has been suggested to steer the electron beam with incidence angles larger than the critical angle or to use smooth bipolar junctions [@Wu2011; @Zhang2009; @Hartmann2010; @Yuan2011; @Myoung2011; @Huang2012; @Beenakker2009; @Zhao2010; @Williams2011; @Rickhaus2013; @Rickhaus; @Li2018a; @Wilmart2014; @Rickhaus2015; @Cupo; @Li2018; @Lee2018]. Nevertheless, these strategies do not prevent Klein tunneling at normal incidence and hence, reduce the efficiency of the device. In this paper, we show that these problems do not appear in phosphorene due to anti-super-Klein tunneling, the omni-directional total reflection at the interface of a pn junction. In the following, we analyze quantum transport in phosphorene pnp junctions. We calculate numerically the current flow by applying the nonequilibrium Green’s function method on a tight-binding model of phosphorene. These calculations demonstrate that phosphorene pnp junctions constitute perfect electron waveguides, without any leakage current through the sidewalls. This perfect guiding of electrons persists for all energies in the pnp regime and all incidence angles. The electron waveguide works perfectly only for a specific orientation of the junction with respect to the phosphorene lattice. For arbitrary orientations, minor leakage currents are observed, which however decay for low electron energies and grazing incidence angles. Using a continuum model of phosphorene, we solve the Schrödinger equation analytically, which will allow us to understand the appearance of guided modes for low electron energies and narrow waveguides. Finally, we demonstrate that the current can be divided and guided efficiently in a crossroad-shaped pnp junction. Tight-binding model of phosphorene ================================== Phosphorene, a monolayer of black phosphorous, is a puckered two-dimensional crystal, see [Figure \[fig:1\]]{} (a, top). Its tight-binding model is given by $$H = \sum_{\braket{i,j}} t_{ij}\ket{i}\bra{j} + \textrm{H.c.}, \label{Hc}$$ where $\ket{i}$ are the atomic states localized on the phosphorous atoms at position ${\ensuremath{\boldsymbol{r}}}_i$. The sum takes into account first and second nearest neighbors, which are coupled by the energies $t_1=-1.22{\ensuremath{\,\mathrm{eV}}}$ and $t_2=3.665 {\ensuremath{\,\mathrm{eV}}}$, respectively [@Betancur-Ocampo2019; @Rudenko; @Rudenko2; @Cakir; @Sisakht; @Ezawa; @Lewenkopf; @Sousa2017]. The tight-binding model of phosphorene can be understood more easily by projecting the atoms to the $xy$ plane, keeping their couplings constant, see [Figure \[fig:1\]]{} (a, bottom). This projection shows that the phosphorene lattice can be understood as a strongly deformed hexagonal lattice with lattice constants $a = 4.42$, $b = 3.27 $ and $\delta=0.8 {\ensuremath{\,\mathrm{\AA}}}$ which connects the two atoms of the unit cell [@Brown]. Using a plane-wave ansatz, the tight-binding Hamiltonian can be written as $$H({\ensuremath{\boldsymbol{k}}}) = \left( \begin{array}{@{}*{2}{c}@{}} 0 & g^*({\ensuremath{\boldsymbol{k}}})\\ g({\ensuremath{\boldsymbol{k}}}) & 0 \end{array} \right), \label{H}$$ where $$\label{gofk} g({\ensuremath{\boldsymbol{k}}}) = \textrm{e}^{-{\mathrm{i}}k_{\delta}}[t_2 + 2t_1\textrm{e}^{{\mathrm{i}}k_a/2}\cos(k_b/2)]$$ with $k_{\delta} = {\ensuremath{\boldsymbol{k}}}\cdot{\ensuremath{\boldsymbol{\delta}}}$, $k_{a} = {\ensuremath{\boldsymbol{k}}}\cdot{\ensuremath{\boldsymbol{a}}}$ $k_{b} = {\ensuremath{\boldsymbol{k}}}\cdot{\ensuremath{\boldsymbol{b}}}$. The energy bands are given by $$\label{eofk} \begin{array}{c} E({\ensuremath{\boldsymbol{k}}})= s {\left| g({\ensuremath{\boldsymbol{k}}}) \right|} \\[2mm] = s \sqrt{t_2^2 + 4 t_1 t_2 \cos(k_a/2)\cos(k_b/2) + 4 t_1^2 \cos^2(k_b/2)}, \end{array}$$ where $s=\text{sgn}(E)= \pm 1$. They show a band gap $2\Delta = 4t_1 + 2t_2$ and strong anisotropy around the $\Gamma$ point. In the armchair direction ($x$-axis in [Figure \[fig:1\]]{} (a), $k_b=0$), the electrons behave approximately as massive Dirac Fermions. Their dispersion relation is quadratic at low energies, followed by a long linear regime. In the zigzag direction ($y$-axis, $k_a=0$), the electrons have the parabolic dispersion of Schrödinger Fermions. Moreover, it has been shown that this anisotropy leads to negative reflection and anti-super-Klein tunneling in phosphorene pn junctions [@Betancur-Ocampo2019]. The eigenfunctions are given by $$\Psi_{{\ensuremath{\boldsymbol{k}}}}({\ensuremath{\boldsymbol{r}}}) = \frac{1}{\sqrt{2}}\left( \begin{array}{@{}*{1}{c}@{}} 1 \\ s\textrm{e}^{{\mathrm{i}}\phi({\ensuremath{\boldsymbol{k}}})} \end{array}\right) \textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}\cdot {\ensuremath{\boldsymbol{r}}}}, \label{wf}$$ where the phase (or pseudo-spin) reads $$\begin{aligned} \phi({\ensuremath{\boldsymbol{k}}}) & = & \arctan\left(\frac{\textrm{Im}[g({\ensuremath{\boldsymbol{k}}})]}{\textrm{Re}[g({\ensuremath{\boldsymbol{k}}})]}\right) \nonumber\\[2mm] & = & \arctan\left(\frac{\lambda_1({\ensuremath{\boldsymbol{k}}})\cos(k_{\delta}) -\lambda_2({\ensuremath{\boldsymbol{k}}})\sin(k_{\delta})} {\lambda_2({\ensuremath{\boldsymbol{k}}})\cos(k_{\delta}) +\lambda_1({\ensuremath{\boldsymbol{k}}})\sin(k_{\delta})}\right), \label{phis}\end{aligned}$$ with $\lambda_1({\ensuremath{\boldsymbol{k}}}) = 2t_1\sin(k_a/2)\cos(k_b/2)$ and $\lambda_2({\ensuremath{\boldsymbol{k}}}) = t_2 + 2t_1\cos (k_a/2)\cos (k_b/2)$. Phosphorene pn and pnp junctions ================================ A phosphorene pn junction, which will be the main building block to construct a phosphorene waveguide, is constituted by two regions of different electrostatic potential (or doping), which shifts the energy bands, see [Figure \[fig:1\]]{} (c). In order to understand the electron scattering at the interface of such pn junction, we make the following ansatz for the wavefunction in region I $$\Psi_{\text{I}}({\ensuremath{\boldsymbol{r}}}) = \frac{1}{\sqrt{2}}\left( \begin{array}{c} 1\\ s\textrm{e}^{{\mathrm{i}}\phi_i} \end{array}\right) \textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}_i\cdot{\ensuremath{\boldsymbol{r}}}} + \frac{r}{\sqrt{2}}\left( \begin{array}{c} 1\\ s\textrm{e}^{{\mathrm{i}}\phi_r} \end{array}\right) \textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}_r\cdot{\ensuremath{\boldsymbol{r}}}} \label{pwr1}$$ and in region II $$\Psi_{\text{II}}({\ensuremath{\boldsymbol{r}}}) = \frac{t}{\sqrt{2}}\left( \begin{array}{c} 1\\ s'\textrm{e}^{{\mathrm{i}}\phi_t} \end{array}\right)\textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}_t\cdot{\ensuremath{\boldsymbol{r}}}}. \label{pwr2}$$ The continuity of the wavefunction at the interface allows to calculate the transmission coefficient $$T = \frac{2\sin[\phi_t -(\phi_r+\phi_i)/2]\sin[(\phi_i-\phi_r)/2]}{ss'-\cos(\phi_t-\phi_r)} \label{T}$$ as a function of the pseudo-spin angles $\phi_i$, $\phi_r$, and $\phi_t$ of the incident, reflected and transmitted electrons [@Betancur-Ocampo2019]. Total reflection $R=1-T=1$ is obtained, if the pseudo-spins of the incident and reflected electrons are parallel and thus, their difference vanishes, $\phi_i - \phi_r=0$. This is the case when the interface of the pn junction is aligned parallel to the armchair edge ($x$-axis in [Figure \[fig:1\]]{} (a)), because in this case the sign change of the transverse momentum, $k_b^r= -k_b^i$, does not affect the pseudo-spins in [(\[phis\])]{}. These findings are confirmed by [Figure \[fig:2\]]{}, which shows the reflection coefficient $R$ as a function of the angle of incidence $\theta$. This omni-directional total reflection, named anti-super-Klein tunneling, appears for all pn junctions parallel to the armchair edge, independently of the precise value of the electron energy and electrostatic potential. It does not rely on the existence of an energetically forbidden region but on the pseudo-spin directions. Therefore, such phosphorene pn junctions can be used to confine and guide an electron beam within a wide range of parameters. If the pn junction is oriented parallel to the zigzag edge ($y$-axis in [Figure \[fig:1\]]{} (a)), the anti-super-Klein tunneling does not take place. Nevertheless, [Figure \[fig:2\]]{} shows that the transmission is finite and decreases for grazing angles of incidence and lower electron energies. Waveguides constructed by means of such junction are possible but less efficient. A perfect electron waveguide is formed by a phosphorene pnp junction that is aligned along the armchair edge, because the anti-super-Klein tunneling confines the electrons within the n channel, see [Figure \[fig:3\]]{} (a). Such junction is generated by the potential well $$\label{V} V({\ensuremath{\boldsymbol{r}}})= \begin{cases} V &\text{if $ y \leq 0$ \: (sidewall, region I),}\\ 0 &\text{if $0< y < w$ \: (channel, region II),}\\ V &\text{if $y \geq w$ \: (sidewall, region III),} \end{cases}$$ where $w$ is the width of the n channel. Experimentally, a pnp junction in phosphorene could be realized by metallic gates. Following [@Cheng2019], it could be formed also by means of a nanotube on top of a phosphorene layer and a back-gate. Another experimental strategy could be to deposit phosphorene (or few-layer black phosphorus) over an annealed Cu foil that produces the potential well, as realized in graphene [@Bai2018]. For the wavefunction inside the channel (region II), we make the ansatz $$\Psi_{\text{II}}({\ensuremath{\boldsymbol{r}}}) = \frac{r_1}{\sqrt{2}}\left( \begin{array}{c} 1\\ s\textrm{e}^{{\mathrm{i}}\phi_i} \end{array}\right) \textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}_i\cdot{\ensuremath{\boldsymbol{r}}}} + \frac{r_2}{\sqrt{2}}\left( \begin{array}{c} 1\\ s\textrm{e}^{{\mathrm{i}}\phi_r} \end{array}\right) \textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}_r\cdot{\ensuremath{\boldsymbol{r}}}}, \label{wfc}$$ while outside the channel (region I/III) we write $$\Psi_{\text{I/III}}({\ensuremath{\boldsymbol{r}}}) = \frac{t_\text{I/III}}{\sqrt{2}}\left( \begin{array}{c} 1\\ s'\textrm{e}^{{\mathrm{i}}\phi^\text{I/III}_t} \end{array}\right)\textrm{e}^{{\mathrm{i}}{\ensuremath{\boldsymbol{k}}}^\text{I/III}_t\cdot{\ensuremath{\boldsymbol{r}}}}. \label{wfs}$$ Using the continuity of the wavefunction at the interfaces, this ansatz yields the simple resonance condition ${\left| k_y \right|}w = n\pi$ with an integer number $n$. This condition can be understood easily by taking account that anti-super-Klein tunneling takes place at the interfaces of the channel. Thus, the sidewalls act as infinite potential walls, where the wavefunction vanishes. Note that if we consider a potential such that the sidewalls represent energetically forbidden zones, ${\left| V-E \right|} \leq \Delta $, the oscillatory waves in the region I and III have to be replaced by evanescent ones, but the resonance condition remains the same. Expanding the dispersion relation around the $\Gamma$ point, the quantized energy spectrum reads $$E_n(p_x) \approx \sqrt{\left(\Delta +\frac{p^2_x}{2m_x}+ \frac{\hbar^2n^2\pi^2}{2m_yw^2} \right)^2 + v^2p^2_x}, \label{qegm}$$ where $ m_x= 2/\left(2\delta(2at_1 - \delta\Delta)-t_1a^2\right)$ and $m_y= -2/\left(t_1b^2\right)$ are the anisotropic masses, and $v = at_1 - \delta\Delta$ is the velocity along the $x$ direction. The guided modes of the waveguide are given by $$\Psi_A(y) = \Psi_B(y) = \sqrt{\frac{2}{w}}\sin\left(\frac{n\pi y}{w}\right). \label{wfgm}$$ Green’s function method for quantum transport ============================================= In order to confirm the functionality of the proposed phosphorene waveguides, we calculate numerically the quantum coherent current flow by means of the nonequilibrium Green’s function method (NEGF) applied to the tight-binding model of phosphorene. In the following, we will summarize briefly the essential equations. A detailed introduction to the NEGF method can be found, for example, in Refs. [@Datta; @Datta2; @Lewenkopf2013; @Barraza-Lopez2012]. The Green’s function of the system is given by $$G(E) = (E - H - \Sigma_V - \Sigma_C)^{-1},$$ where $E$ is the energy of the electrons (times a unit matrix), $H$ is the tight-binding Hamiltonian and $\Sigma_V= \sum_n V({\ensuremath{\boldsymbol{r}}}_n) \ket{n}\bra{n}$ takes into account the electrostatic potential that generates the pnp junction. In order to suppress boundary effects and mimic an infinite system, we place a constant complex potential $\Sigma_C= -{\mathrm{i}}\sum_{n \in \text{edge}} \ket{n}\bra{n}$ at the edges of the system, which absorbs the electrons. We will use two different methods to inject the electrons in the n region at the left edge, see [Figure \[fig:4\]]{} (a). Within the first approach, the electrons are injected as plane waves with a Gaussian profile, $$A({\ensuremath{\boldsymbol{r}}}) = \exp\bigl( -{\left| {\ensuremath{\boldsymbol{r}}} - {\ensuremath{\boldsymbol{r}}}_{0} \right|}^2/d_0^2 \bigr),$$ where the parameter ${\ensuremath{\boldsymbol{r}}}_0$ and $d_0$ control the position and width of the electron beam. These parameters are chosen in such a way that a narrow electron beam is injected in the waveguide. The momentum ${\ensuremath{\boldsymbol{k}}}$ of the plane waves, which in general does not correspond to the direction of propagation due to the anisotropy of the electronic structure, is calculated from the input parameters $E$ and $\theta$, as shown in Ref. [@Betancur-Ocampo2019]. The injection is taken into account within the NEGF approach by the inscattering function $$\Sigma^{\text{in}}_S = \sum_{n,m \in \text{LE}} A({\ensuremath{\boldsymbol{r}}}_n)A({\ensuremath{\boldsymbol{r}}}_m) \psi^*_{{\ensuremath{\boldsymbol{k}}}}({\ensuremath{\boldsymbol{r}}}_n)\psi_{{\ensuremath{\boldsymbol{k}}}}({\ensuremath{\boldsymbol{r}}}_m) \ket{n}\bra{m},$$ where the sum is over all atoms at the left edge (LE). The $\psi_{{\ensuremath{\boldsymbol{k}}}}({\ensuremath{\boldsymbol{r}}}_m)$ are the eigenstates in [(\[wf\])]{} evaluated at the position ${\ensuremath{\boldsymbol{r}}}_n$ of the atoms at the edge. Second, we will use the wide-band model to inject the electrons. This model represents a metallic contact with a constant surface density of states (DOS), which injects electrons with energy $E$ from the surface of the Fermi sea without a specified direction. Within the NEGF approach, this injection is modelled by the inscattering function $$\Sigma^{\text{in}}_S = \sum_{n,m \in \text{nLE}} \eta \ket{n}\bra{m},$$ where $\eta \sim 1$ is a constant proportional to the DOS at the surface of the metallic contact. The sum runs only over the atoms at the left edge which belong to the n region (nLE). The first model has the advantage that the direction of the electron beam can be tuned precisely, which makes it possible to compare the current flow with semi-classical trajectories [@Stegmann2016; @Stegmann2019; @Betancur-Ocampo2019]. The wide-band model does not have this flexibility but it is one of the most general model that approximates various experimental situations [@Verzijl2013]. Finally, the current flowing between the atoms at positions ${\ensuremath{\boldsymbol{r}}}_n$ and ${\ensuremath{\boldsymbol{r}}}_m$ is calculated by means of $$I_{nm} = \textrm{Im}(t^*_{nm}G^{\text{in}}_{nm}),$$ where $$G^{\text{in}} = G \Sigma^{\text{in}}_SG^{\dagger}.$$ Current flow in phosphorene waveguides ====================================== In the following, we explore the current flow in phosphorene nanoribbons by means of the NEGF method and compare with our predictions from the previous sections. We consider phosphorene nanoribbons of size $(L_x,L_y)= 1000 \times 200 \,a \approx 442 \times 88 {\ensuremath{\,\mathrm{nm}}}$, which consist of approximately one million atoms. In [Figure \[fig:4\]]{}, we show the current flow for electrons at energy $E= \Delta +0.2{\left| t_1 \right|}$ that are injected as narrow plane-wave beams at the left ribbon edge. Note that from now on, we measure all energies in multiples of ${\left| t_1 \right|}$ and distances in multiples of $a$. In a pristine phosphorene nanoribbon without an external electrostatic potential, see [Figure \[fig:4\]]{} (a,b), the electrons propagate ballistically through the system, but the electron beams are broadened due to diffraction. Reflections at the edges of the nanoribbon are suppressed gradually due to the absorption of the complex potential. In presence of the electrostatic potential [(\[V\])]{}, which generates a pnp junction, the electrons do not reach the edges at the top or bottom of the nanoribbon but are reflected totally at the pn interfaces, see [Figure \[fig:4\]]{} (c-f). Note that the electrons are reflected also to some degree at the right edge of the system, which leads to a spurious counter-propagating current. The electrons are confined within the n region of the junction, which thus forms a perfect electron waveguide. There is no leakage current, although the electrons hit the interface under very different angles, which demonstrates the absence of critical angles. The omni-directional total reflection at the pn interfaces, called anti-super-Klein tunneling [@Betancur-Ocampo2019], is not due to an energetically forbidden region, see [Figure \[fig:3\]]{} (b), but due to pseudo-spin blocking. In the left column of [Figure \[fig:4\]]{}, we also observe clearly that the confinement of the electron beam in a narrow channel allows us to transfer it without diffraction. However, note that the shown waveguides are still so wide that the quantization of the transverse momentum $k_y$ is irrelevant and the propagation of the electron beams can be understood in terms of ray-optics [@Betancur-Ocampo2019]. In [Figure \[fig:5\]]{}, we demonstrate that the phosphorene waveguide also works perfectly for other electron energies and all potentials that establish a pnp junction, $2\Delta+E < V <\infty$. Moreover, total reflection also emerges for potentials $ E<V<2\Delta+E$ due to the intrinsic band gap of phosphorene. As expected, the electrons cannot be confined by a p’pp’ junction. In order to demonstrate the robustness of the proposed device, we show in [Figure \[fig:6\]]{} the current flow obtained by using the wide-band model for the contact that injects the electrons at the left edge. It can be observed clearly that an otherwise dispersing electron beam is confined and guided efficiently by the electrostatic potential of the pnp junction. The way how the electrons are injected does not alter the functionality of the waveguide. When the electron energy and the width of the junction is reduced significantly, the quantization of the transverse momentum $k_y$ becomes relevant. In this regime, only some few subbands are occupied, $n= w\sqrt{2m_y(E-\Delta)}/(\pi\hbar) \sim 1$, and the guided modes of the waveguide can be observed clearly in the current flow patterns, see [Figure \[fig:7\]]{}. We have shown that the electron waveguide works perfectly, if the pnp junction is aligned parallel to the armchair edge of the phosphorene lattice. If the channel is oriented parallel to the zigzag edge, see [Figure \[fig:8\]]{}, the confinement is not perfect because the anti-super-Klein tunneling is absent. Nevertheless, as shown in [Figure \[fig:2\]]{}, electrons that hit the interface at grazing incidence angles are largely reflected. This effect increases even if the electron energy is lowered. Hence, such pnp junctions constitute less efficient but applicable electron waveguides, provided that the electrons flow essentially parallel to the sidewalls. The difference between total reflection due to anti-super-Klein tunneling and typical high (but partial) reflectivity is shown in [Figure \[fig:9\]]{}, which gives the current density on a logarithmic color scale in order to visualize the leaking currents. In order to demonstrate the unique efficiency of the proposed device, we show in [Figure \[fig:10\]]{} the current flow in a graphene pnp junction. The graphene nanoribbon is modelled by the standard first-nearest neighbor tight-binding Hamiltonian [@CastroNeto2009; @Stegmann2015; @Stegmann2019]. Electrons are injected at energy $E=0.1t_1'$ by means of the wide-band model within the n region of the junction, generated by the electrostatic potential [(\[V\])]{} with $V=2E$. Note that in the case of graphene, energies and distances are measured in multiples of $t_1'= 2.8 {\ensuremath{\,\mathrm{eV}}}$ and $a'= 1.42 {\ensuremath{\,\mathrm{\AA}}}$, the coupling energy and distance of neighboring carbon atoms. It can be observed clearly that in graphene the electrons pass the interfaces of the pnp junction due to Klein tunneling and are absorbed finally at the top and bottom edges of the nanoribbon. In fact, no current density can be detected at the right edge of the system. It is therefore impossible to construct an electron waveguide based on a graphene pnp junction. Finally, we analyze the current flow in a crossroad-shaped pnp junction, see [Figure \[fig:11\]]{}. In order to benefit from the high reflection at grazing incidence angles along the zigzag direction, the current is injected at the bottom edge by means of the wide-band model. We observe that the current is split into the different branches at the crossroad. Moreover, if branches are blocked by raising the electrostatic potential in this region, we observe that the electron beam can be guided efficiently in different directions. Note that such electron highways have been proposed also in bilayers of graphene using the intrinsic band gap of the material [@Qiao2011], but they are impossible in monolayers of graphene. Conclusions =========== We have shown that phosphorene pnp junctions, generated by regions of different electrostatic potential, can be used to confine and guide efficiently electrons in this material. Electron beams propagate in such waveguides like light beams in optical fibers, see [Figure \[fig:4\]]{}. In narrow junctions at low electron energy, the quantization of the transverse momentum becomes relevant and the guided modes of the waveguide can be observed, see [Figure \[fig:7\]]{}. There is absolutely no leakage current in junctions that are aligned parallel to the armchair edge of the phosphorene lattice due to the anti-super-Klein tunneling, the omni-directional total resistance due to pseudo-spin blocking. The waveguides operate without any loss for all steering angles of the electron beam, because they do not rely on the existence of a critical angle. Moreover, they work perfectly for all electron energies and potentials that establish a pnp junction. Junctions that are not parallel to the armchair edge suffer from partial transmission through the sidewalls of the junction, see [Figure \[fig:8\]]{}. Nevertheless, the transmission decays strongly for grazing incidence angles and lower electron energies, see [Figure \[fig:2\]]{}, making such junctions not perfect but feasible waveguides, in particular if the steering angles of the electron beam are mostly grazing. In this way, we have shown that a crossroad-shaped pnp junction can be used to split and direct efficiently the current flow in phosphorene, see [Figure \[fig:11\]]{}. Our theoretical work paves the way to electron optics experiments in phosphorene. Moreover, the proposed nanoelectronic device, a perfect lossless electron waveguide, may find also application, for example, as electron conveyors in quantum information technology. Acknowledgments {#acknowledgments .unnumbered} =============== The authors gratefully acknowledge financial support from CONACYT Proyecto Fronteras 952, Proyecto A1-S-13469, and the UNAM-PAPIIT research grant IA-103020. The data that support the findings of this study are available from the corresponding author upon reasonable request.

--- abstract: 'The complete algebraic structure of semisimple finite group algebra of a generalized strongly monomial group is provided. This work extends the work of Broche and del R[í]{}o on strongly monomial groups. The theory is complimented by an algorithm and is illustrated with an example.' author: - | Gurmeet K. Bakshi and Gurleen Kaur[[^1] [^2]]{}\ [*Centre for Advanced Study in Mathematics,*]{}\ [*Panjab University, Chandigarh 160014, India*]{}\ [*email: [email protected] and [email protected]* ]{} bibliography: - 'Reference.bib' title: '** Semisimple finite group algebra of a generalized strongly monomial group**' --- [**Keywords**]{} : finite group algebra, generalized strongly monomial group, primitive central idempotents, simple components, Wedderburn decomposition, coding theory.\ [**MSC2000 :**]{} 16S34,16K20,16S35,20C05 Introduction ============ Let ${\mathbb{F}_{q}}$ be a finite field with $q$ elements and $G$ a finite group of order coprime to $q$. Let ${\mathbb{F}_{q}}G$ be the group algebra of $G$ over ${\mathbb{F}_{q}}$. During the last few years, lot of interest has been seen in understanding the primitive central idempotents and the Wedderburn decomposition of finite semisimple group algebra ${\mathbb{F}_{q}}G$, thus enabling coding theorist to determine the ideals of $\mathbb{F}_{q}G$ which are precisely all the group codes (e.g., see [@BGP2], [@BGP1], [@BGP3], [@BRm2], [@BRS], [@Bs], [@BR], [@BP], [@FLP], [@FM], [@GM], [@Mr], [@SBDR2], [@SBDR]). If $G$ is abelian, this problem has been dealt in series of papers by several authors. Moving further to non abelian groups, a major step was taken by Broche and del R[í]{}o in [@BR] where they provided the description of primitive central idempotents and the Wedderburn decomposition of ${\mathbb{F}_{q}}G$ when $G$ is a strongly monomial group. Recall that all abelian-by-supersolvable groups are strongly monomial and all strongly monomial groups are monomial. Recently, in [@BK2], we have defined generalized strongly monomial groups and proved that, beside strongly monomial groups, the class of generalized strongly monomial groups contain several important families of groups such as subnormally monomial groups (in particular Frobenius monomial groups) and solvable groups with all Sylow subgroups abelian extended by supersolvable groups. In this paper, we extend the work of Broche and del R[í]{}o [@BR] to generalized strongly monomial groups. In section 2, we show (Theorem \[t1\]) that if $(H,K)$ is a generalized strong Shoda pair of $G$ and $\mathcal{C}$ is a $q$-cyclotomic class of $\operatorname{Irr}(H/K)$ which contain generators of $\operatorname{Irr}(H/K)$, then the pair $((H, K), \mathcal{C})$ correspond to a primitive central idempotent of $\mathbb{F}_q G$. In this case, we also describe the structure of the corresponding simple component of ${\mathbb{F}_{q}}G$. This result allows us to describe the complete set of primitive central idempotents and the precise Wedderburn decomposition of semisimple finite group algebra of a generalized strongly monomial group (Corollary \[c2\]). In section 3, we write a precise algorithm using the theory developed in section 2. In section 4, we illustrate the theory by computing the Wedderburn decomposition of semisimple group algebra $\mathbb{F}_{q}G$ for a generalized strongly monomial group $G$ which is not strongly monomial. Wedderburn decomposition ======================== We begin by recalling the basic terminology related to Shoda pairs, strong Shoda pairs and generalized strong Shoda pairs. A pair $(H,K)$ of subgroups of $G$ is called a *Shoda pair* of $G$ (see [@JdR], Chapter 3) if the following conditions hold: (i) : $K\unlhd H$, $H/K$ is cyclic; (ii) : if $g \in G$ and $[H,g]\cap H\subseteq K$, then $g \in H$. For a Shoda pair $(H,K)$ of $G$, recall the following standard notations: $$\widehat{H}:=\frac{1}{|H|}\displaystyle\sum_{h \in H}h,$$ $$\varepsilon(H,K):=\left\{\begin{array}{ll}\widehat{K}, & \hbox{$H=K$;} \\\prod(\widehat{K}-\widehat{L}), & \hbox{otherwise,}\end{array}\right.$$ where $L$ runs over the normal subgroups of $H$ minimal with respect to the property of including $K$ properly, and $$e(G,H,K):= {\rm~the~sum~of~all~the~distinct~}G{\rm {\operatorname{-}} conjugates~of~}\varepsilon(H,K).$$ A Shoda pair $(H,K)$ of $G$ is called a *strong Shoda pair* of $G$ (see [@JdR], Chapter 3) if the following conditions hold: (i) : $H \unlhd \operatorname{Cen}_{G}(\varepsilon(H,K))$; (ii) : the distinct $G$-conjugates of $\varepsilon(H,K)$ are mutually orthogonal. More generally, a Shoda pair $(H,K)$ of $G$ is called a generalized strong Shoda pair of $G$ (see [@BK2]), if there is a chain $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ of subgroups of $G$ (called strong inductive chain from $H$ to $G$) such that the following conditions hold for all $0 \leq i < n$: (i) : $H_{i} \unlhd \operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$; (ii) : the distinct $H_{i+1}$-conjugates of $\varepsilon^{(i)}(H,K)$ are mutually orthogonal, where $\varepsilon^{(0)}(H,K)= \varepsilon(H,K)$ and $\varepsilon^{(i)}(H,K)$ is the sum of all the distinct $H_{i}$-conjugates of $\varepsilon^{(i-1)}(H,K)$ for $1 \leq i \leq n$. For notational convenience, we will denote $\varepsilon^{(n)}(H,K)$ by $\mathfrak{e}(G,H,K)$. Let $(H,K)$ be a Shoda pair of $G$ and let $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ be subgroups of $G$. By the repeated application of Lemma 3 of [@BK2], $\varepsilon^{(i)}(H,K)$ is a rational multiple of $e_{\mathbb{Q}}(\lambda^{H_{i}})$, where $\lambda$ is any complex linear character on $H$ with kernel $K$ and $e_{\mathbb{Q}}(\lambda^{H_{i}})$ is the primitive central idempotent of the rational group algebra $\mathbb{Q}H_{i}$ associated with the complex irreducible character $\lambda^{H_{i}}$. Therefore, the centralizers of $\varepsilon^{(i)}(H,K)$ and $e_{\mathbb{Q}}(\lambda^{H_{i}})$ in $H_{i+1}$ coincide. Hence the above definition of a generalized strong Shoda pair of $G$ is equivalent to that given in [@BK2]. Recall that every Shoda pair $(H,K)$ of $G$ realizes a primitive central idempotent of $\mathbb{Q}G$ namely $e_{\mathbb{Q}}(\lambda^{G})$, where $\lambda$ is any complex linear character on $H$ with kernel $K$. Two Shoda pairs of $G$ are called equivalent if they realize the same primitive central idempotent of $\mathbb{Q}G$. A group $G$ is called strongly monomial if every primitive central idempotent of $\mathbb{Q}G$ is realized by a strong Shoda pair of $G$ and a group $G$ is called generalized strongly monomial if every primitive central idempotent of $\mathbb{Q}G$ is realized by a generalized strong Shoda pair of $G$. Two generalized strong Shoda pairs of $G$ are called equivalent if they are equivalent as Shoda pairs of $G$. A set of representatives of distinct equivalence classes of generalized strong Shoda pairs of $G$ is called a complete and irredundant set of generalized strong Shoda pairs of $G$. For details on strongly monomial groups and generalized strongly monomial groups, we refer to [@BK2] and [@OdRS04]. Given a Shoda pair $(H,K)$ of $G$, $\operatorname{Irr}(H/K)$ denotes the set of irreducible characters on $H/K$ over $\overline{\mathbb{F}}_{q}$, the algebraic closure of $\mathbb{F}_{q}$. If $k=[H:K]$ and $\xi_{k}$ is a primitive $k$th root of unity in $\overline{\mathbb{F}}_{q}$, then there is a natural action of $\operatorname{Gal}(\mathbb{F}_{q}(\xi_{k})/\mathbb{F}_{q})$ on $\operatorname{Irr}(H/K)$ by composition. The orbits of $\operatorname{Irr}(H/K)$ under this action are called $q$-cyclotomic classes. Denote by $\mathcal{C}_{q}(H/K)$ those $q$-cyclotomic classes which contain generators of $\operatorname{Irr}(H/K)$. If $\mathcal{C} \in \mathcal{C}_{q}(H/K)$ and $\chi \in \mathcal{C}$, set: $$\varepsilon_{\mathcal{C}}(H,K):= |H|^{-1}\sum_{h \in H} \operatorname{tr}(\chi(hK))h^{-1}=[H:K]^{-1}\widehat{K}\sum_{X \in H/K}\operatorname{tr}(\chi(X))h_{X}^{-1},$$ where $\operatorname{tr}=\operatorname{tr}(\mathbb{F}_{q}(\xi_{[H:K]})/\mathbb{F}_{q}$) is the trace of the field extension $\mathbb{F}_{q}(\xi_{[H:K]})/\mathbb{F}_{q}$ and $h_{X}$ is a representative of $X \in H/K$.\ Define $$e_{\mathcal{C}}(G,H,K):={\rm~the~sum~of~all~the~distinct~}G{\rm {\operatorname{-}} conjugates~of~}\varepsilon_{\mathcal{C}}(H,K).$$ If $\psi$ is a character of a subgroup $H$ of $G$ and $x \in G$, then $\psi^{x}$ is a character of $H^{x}=x^{-1}Hx$ given by $\psi^{x}(g)=\psi(xgx^{-1})$, $g \in H^{x}$. Denote by $\psi^{G}$, the character $\psi$ induced to $G$. For $\alpha \in \mathbb{Q}G$ and $x \in G,$ $\alpha^{x}=x^{-1}\alpha x$.\ We recall the following theorem due to Broche and del R[í]{}o [@BR]: [([@BR], Theorem 7)]{} Let $G$ be a finite group and $\mathbb{F}_{q}$ be a finite field with $q$ elements such that $\mathbb{F}_{q}G$ is semisimple. 1 : Let $(H,K)$ be a strong Shoda pair of $G$ and $\mathcal{C} \in \mathcal{C}_{q}(H/K)$. Then $e_{\mathcal{C}}(G,H,K)$ is a primitive central idempotent of $\mathbb{F}_{q}G$ and $$\mathbb{F}_{q}Ge_{\mathcal{C}}(G,H,K)\cong M_{[G:H]}(\mathbb{F}_{q^{(o/[E:H])}}),$$ where $E=E_{G}(H/K)$ is the stabilizer of any element of $\mathcal{C}_{q}(H/K)$ under the natural action of $N_{G}(K)$ and $o$ is the multiplicative order of $q$ module $[H:K]$. 2 : Let $X$ be a set of strong Shoda pairs of $G$. If every primitive central idempotent of $\mathbb{Q}G$, the rational group algebra, is of the form $e(G,H,K)$ for $(H,K) \in X$ then every primitive central idempotent of $\mathbb{F}_{q}G$ is of the form $e_{\mathcal{C}}(G,H,K)$ for $(H,K) \in X$ and $\mathcal{C} \in \mathcal{C}_{q}(H/K)$. The above theorem raises the following natural question: > [*If $(H,K)$ is a Shoda pair of $G$ and $\mathcal{C} \in \mathcal{C}_{q}(H/K)$, then, does the pair $((H, K), \mathcal{C})$ correspond to a primitive central idempotent of $\mathbb{F}_{q}G$? If yes, can we describe the corresponding simple component?* ]{} The first part of the above question is not very difficult to see. Given such a pair $((H, K), \mathcal{C})$, we can see analogous to Theorem 2.1 of [@OdRS04] that there is a unique $\alpha \in \mathbb{F}_{q}$ such that $e_{\mathbb{F}_q}(\lambda^G)= \alpha e_{\mathcal{C}}(G,H,K)$ for an arbitrary $\lambda \in \mathcal{C}$. Thus there does exist a primitive central idempotent, namely $\alpha e_{\mathcal{C}}(G,H,K)$ of $\mathbb{F}_{q}G$ that is associated with the pair $((H, K), \mathcal{C})$. To answer the above question, the main problem lies in describing the simple component $\mathbb{F}_{q}Ge_{\mathcal{C}}(G,H,K)$. We’ll show, in Theorem \[t1\], that the structure of this simple component can be described by proceeding in steps, when $(H,K)$ is a generalized strong Shoda pair of $G$. Let us elaborate on how to proceed by steps. If $(H,K)$ is a generalized strong Shoda pair of $G$, $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ is a strong inductive chain from $H$ to $G$ and $\mathcal{C}\in \mathcal{C}_{q}(H/K)$, set $$\varepsilon_{\mathcal{C}}^{(0)}(H,K) = \varepsilon_{\mathcal{C}}(H,K),$$ $$\varepsilon_{\mathcal{C}}^{(1)}(H,K)={\rm~the~sum~of~all~the~distinct~}H_{1}{\rm {\operatorname{-}} conjugates~of~}\varepsilon_{\mathcal{C}}^{(0)}(H,K),$$ $$\varepsilon_{\mathcal{C}}^{(2)}(H,K) ={\rm~the~sum~of~all~the~distinct~}H_{2}{\rm {\operatorname{-}} conjugates~of~}\varepsilon_{\mathcal{C}}^{(1)}(H,K),$$ and finally $$\varepsilon_{\mathcal{C}}^{(n)}(H,K) ={\rm~the~sum~of~all~the~distinct~}H_{n}{\rm {\operatorname{-}} conjugates~of~}\varepsilon_{\mathcal{C}}^{(n-1)}(H,K).$$ Denote the final step $\varepsilon_{\mathcal{C}}^{(n)}(H,K) $ by $\mathfrak{e}_{\mathcal{C}}(G,H,K)$. Apparently, one thinks that the definition of $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ depends on strong inductive chain from $H$ to $G$. However, we will see that this is not the case, i.e., the final step $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ is invariant of strong inductive chain from $H$ to $G$ (see Remark \[r1\]). Observe that if $(H,K)$ is a strong Shoda pair of $G$, then $\mathfrak{e}_{\mathcal{C}}(G,H,K)= {e}_{\mathcal{C}}(G,H,K)$ on taking strong inductive chain: $H \leq G$. [\[t1\]]{} Let $\mathbb{F}_{q}$ be a finite field of order $q$ and $G$ a finite group of order coprime to $q$. Let $(H,K)$ be a generalized strong Shoda pair of $G$ and $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ a strong inductive chain from $H$ to $G$. If $\mathcal{C} \in \mathcal{C}_{q}(H/K)$, then the following statements hold: (i) : $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ is a primitive central idempotent of $\mathbb{F}_{q} G$. More generally, for any $\lambda \in \mathcal{C}$, $\mathfrak{e}_{\mathcal{C}}(G,H,K)= e_{\mathbb{F}_{q}}(\lambda^{G})$; (ii) : $\mathbb{F}_{q}G \mathfrak{e}_{\mathcal{C}}(G,H,K) \cong M_{[G:H]}(\mathbb{F}_{{q}^{l_{(H,K)}}}),$ where $l_{(H,K)}=\frac{o}{\prod_{0\leq i< n}[C_{i}:H_{i}]}$, $o$ is the multiplicative order of $q$ module $[H:K]$ and $C_{i} = \operatorname{Cen}_{H_{i+1}}(\varepsilon_{\mathcal{C}}^{(i)}(H,K))$. \[r1\] Since $ \mathfrak{e}_{\mathcal{C}}(G,H,K)= e_{\mathbb{F}_{q}}(\lambda^{G})$, it follows immediately that the construction of $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ is independent of strong inductive chain from $H$ to $G$. \[r2\] $\prod_{0\leq i < n}[C_{i}: H_{i}]$ always divides $o$. [\[l1\]]{}If $(H, K)$ is a Shoda pair of a finite group $G$ and $\lambda$ is an $\overline{\mathbb{F}}_{q}$-irreducible character on $H$ with kernel $K$, then $\lambda^G$ is irreducible. [**Proof.**]{} This follows from Corollary 45.4 of [@CR]. $\Box$ [\[l2\]]{} Let $S$ be a subgroup of a finite group $G$ and $\psi$ an $\overline{\mathbb{F}}_{q}$-irreducible character on $S$ such that $\psi^{G}$ is irreducible, then there exists $ \alpha \in \mathbb{F}_{q}$ such that $$e_{\mathbb{F}_{q}}(\psi^{G})=\alpha \sum_{g \in T}e_{\mathbb{F}_{q}}(\psi)^{g},$$ where $T$ is a right transversal of $\operatorname{Cen}_{G}(e_{\mathbb{F}_{q}}(\psi))$ in $G$. Furthermore $\alpha =1$, if the distinct $G$-conjugates of $e_{\mathbb{F}_{q}}(\psi)$ are mutually orthogonal and in this case, ${\mathbb{F}_{q}}G e_{\mathbb{F}_{q}}(\psi^{G})$ is isomorphic to $ M_{[G:C]}({\mathbb{F}_{q}}C e_{\mathbb{F}_{q}}(\psi) )$, where $C= {\operatorname{Cen}}_{G}(e_{\mathbb{F}_{q}}(\psi))$. [**Proof.**]{} The proof is analogous to that of ([@BK2], Lemma 3, Proposition 2(i)).\ Throughout the rest of the paper, $(H,K)$ is a generalized strong Shoda pair of $G$ and $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ is a strong inductive chain from $H$ to $G$. Let $p$ be the prime divisor of $q$ and let $\mathbb{Z}_{(p)}$ denote the localization of $\mathbb{Z}$ at $p$. Then $\mathbb{F}_{p}$ can be identified with the residue field of $\mathbb{Z}_{(p)}$. Consider the natural ring homomorphism from $\mathbb{Z}_{(p)}$ onto $\mathbb{F}_{p}$ and extend it linearly from $\mathbb{Z}_{(p)}G$ onto $\mathbb{F}_{p}G.$ Denote by $\overline{\alpha}$, the image of $\alpha \in \mathbb{Z}_{(p)}G$ under the above epimorphism. By ([@BK2], Lemma 3), it follows that $\varepsilon^{(i)}(H,K)$ is a primitive central idempotent of $\mathbb{Z}_{(p)}H_{i}$ and hence $\overline{\varepsilon^{(i)}(H,K)}$ is a central idempotent of $\mathbb{F}_{p}H_{i}$ for all $0\leq i\leq n$. [\[l3\]]{} If, for some $i$, $\overline{\varepsilon^{(i)}(H,K)}= e_{1} + \cdots + e_{l}$ is the decomposition of $\overline{\varepsilon^{(i)}(H,K)}$ into sum of primitive central idempotents of $\mathbb{F}_{q}H_{i}$, then the following statements hold for all $ 1 \leq k \leq l$: (i) : $\operatorname{Cen}_{H_{i+1}}(e_{k})$ is a subgroup of $\operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$; (ii) : $H_{i} \unlhd \operatorname{Cen}_{H_{i+1}}(e_{k})$; (iii) : the distinct $H_{i+1}$-conjugates of $e_{k}$ are mutually orthogonal. [**Proof.**]{} By renaming $e_{i}$’s, we may assume that $k=1$. On multiplying the equation $\overline{\varepsilon^{(i)}(H,K)}= e_{1} + \cdots + e_{l}$ with $e_{1}$, we obtain that $${\label{e1} } \overline{\varepsilon^{(i)}(H,K)}e_{1}= me_{1}= e_{1}\overline{\varepsilon^{(i)}(H,K)},$$ where $m$ is the number of $e_{j}$’s that are equal to $e_{1}$. Note that the characteristic of $\mathbb{F}_{q}$ does not divide $m$. Consider any $ g \in H_{i+1}\setminus {\operatorname{Cen}}_{H_{i+1}}(\varepsilon^{(i)}(H,K)) $. Conjugating eqn (\[e1\]) by $g$, we have $${\label{e2}} \overline{\varepsilon^{(i)}(H,K)^{g}} e_{1}^{g}= me_{1}^{g}= e_{1}^{g}\overline{\varepsilon^{(i)}(H,K)^{g}}.$$ Since $\varepsilon^{(i)}(H,K) \varepsilon^{(i)}(H,K)^{g} =0$, we get from eqns (\[e1\]) and (\[e2\]) that $e_{1}e_{1}^{g}=0$. This gives that ${\operatorname{Cen}}_{H_{i+1}}(e_{1})$ is a subgroup of ${\operatorname{Cen}}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$, which proves (i). The normality of $H_{i}$ in ${\operatorname{Cen}}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$ immediately yields (ii), in view of (i). Since $\operatorname{Cen}_{H_{i+1}}(e_{1})\leq \operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$ and $e_{1}e_{1}^{g}=0$ if $g \notin \operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$, to prove (iii), we only need to show that $e_{1} e_{1}^{g} =0$ if $ g \in {\operatorname{Cen}}_{H_{i+1}}(\varepsilon^{(i)}(H,K)) \setminus {\operatorname{Cen}}_{H_{i+1}}(e_{1}) $. But this clearly holds because if $ g \in {\operatorname{Cen}}_{H_{i+1}}(\varepsilon^{(i)}(H,K)) \setminus {\operatorname{Cen}}_{H_{i+1}}(e_1),$ then $e_{1}$ and $e_{1}^{g}$ are distinct primitive central idempotents of ${\mathbb{F}_{q}}H_{i}$, as $H_{i}\unlhd \operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}(H,K))$. Hence, the lemma is proved. [\[p1\]]{} There exist subsets $R_i$, $ 0 \leq i \leq n$, of $ \mathcal{C}_{q}(H/K)$ with $R_{0} = \mathcal{C}_{q}(H/K)$ and $R_{i}$ contained in $R_{i-1}$ such that $\operatorname{Cen}_{H_{i}}(\varepsilon^{(i-1)}(H,K))$ acts on $\{\varepsilon_{\mathcal{C}}^{(i-1)}(H,K)~|~\mathcal{C} \in R_{i-1}\}$ by conjugation and $\{\varepsilon_{\mathcal{C}}^{(i-1)}(H,K)~|~\mathcal{C} \in R_{i}\}$ is a set of representatives under this action for all $1 \leq i \leq n$. Furthermore, as a consequence, the following statements hold for all $0\leq i\leq n$: (i) : $\overline{\varepsilon^{(i)}(H,K)} = \sum_{\mathcal{C} \in R_{i}}\varepsilon_{\mathcal{C}}^{(i)}(H,K)$; (ii) : $\varepsilon_{\mathcal{C}}^{(i)}(H,K)$, $ \mathcal{C} \in R_{i}$, are distinct primitive central idempotents of $\mathbb{F}_{q}H_{i}$.Moreover, ${\varepsilon_{\mathcal{C}}^{(i)}(H,K)}= e_{{\mathbb{F}_{q}}}(\lambda^{H_{i}}) $ for an arbitrary $\lambda \in \mathcal{C}$. [**Proof.**]{} We will prove the result by induction on $i$. Suppose $i=0$. Then $\varepsilon_{\mathcal{C}}^{(0)}(H,K)= \varepsilon_{\mathcal{C}}(H,K)$ and the result follows from ([@BR], Corollary 3, Lemma 6). Assume that the result is true for $i=k$, where $0 \leq k < n$. We’ll prove the result for $i= k+1$. Let $ g \in \operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$. Then $$\overline{\varepsilon^{(k)}(H,K)} = \sum_{\mathcal{C} \in R_{k}}\varepsilon_{\mathcal{C}}^{(k)}(H,K)=\sum_{\mathcal{C} \in R_k}(\varepsilon_{\mathcal{C}}^{(k)}(H,K))^{g} .$$ By induction, $\varepsilon_{\mathcal{C}}^{(k)}(H,K)$, $ \mathcal{C} \in R_{k}$, are distinct primitive central idempotents of $\mathbb{F}_{q}H_{k}$. Since $ H_{k} \unlhd \operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$, we also have that $\varepsilon_{\mathcal{C}}^{(k)}(H,K)^{g}$, $ \mathcal{C} \in R_k$, are also distinct primitive central idempotents of $\mathbb{F}_{q}H_{k}$. But a central idempotent can be uniquely written as a sum of primitive central idempotents, therefore $$\{ (\varepsilon_{\mathcal{C}}^{(k)}(H,K))^{g} ~| ~ \mathcal{C} \in R_{k} \} = \{ \varepsilon_{\mathcal{C}}^{(k)}(H,K)~ | ~ \mathcal{C} \in R_{k} \} .$$ This gives an action of $\operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$ on $\{ \varepsilon_{\mathcal{C}}^{(k)}(H,K)~|~ \mathcal{C} \in R_k\}$ by conjugation. Under this action, let $R_{k+1}$ contained in $R_{k}$ be such that $\{ \varepsilon_{\mathcal{C}}^{(k)}(H,K)~ | ~ \mathcal{C} \in R_{k+1} \} $ is a set of representatives of distinct orbits. For any $ \mathcal{C} \in R_{k+1}$, let $E_{\mathcal{C}}$ be the stabilizer of $\varepsilon_{\mathcal{C}}^{(k)}(H,K)$ in $ \operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$. In view of Lemma \[l3\], we have $E_{\mathcal{C}} = \operatorname{Cen}_{H_{k+1}}(\varepsilon_{\mathcal{C}}^{(k)}(H,K))$. Let $T_{\mathcal{C}}$ be a right transversal of $E_{\mathcal{C}}$ in $\operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$. Let $T$ be a right transversal of $\operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$ in $H_{k+1}$ so that $\{xy~|~x \in T_{\mathcal{C}}, y \in T\}$ is a right transversal of $E_{\mathcal{C}}$ in $H_{k+1}$. Now, $$\begin{aligned} \overline{\varepsilon^{(k+1)}(H,K)} &= \sum_{y \in T}(\overline{\varepsilon^{(k)}(H,K)})^{y} \\ &= \sum_{y \in T} \sum_{\mathcal{C} \in R_{k}}(\varepsilon_{\mathcal{C}}^{(k)}(H,K))^{y}~{\rm (by~induction)}\\ &= \sum_{y \in T}( \sum_{\mathcal{C} \in R_{k+1}}\sum_{x \in T_{\mathcal{C}}}(\varepsilon_{\mathcal{C}}^{(k)}(H,K))^x)^y \\&= \sum_{\mathcal{C} \in R_{k+1}} \sum_{y \in T}\sum_{x \in T_{\mathcal{C}}}(\varepsilon_{\mathcal{C}}^{(k)}(H,K))^{xy} \\ &=\sum_{\mathcal{C} \in R_{k+1}}\varepsilon_{\mathcal{C}}^{(k+1)}(H,K).\end{aligned}$$\ This proves (i) for $i=k+1$. Next, we prove (ii) for $ i= k+1$. Consider any $ \mathcal{C} \in R_{k+1}$ and an arbitrary $\lambda \in \mathcal{C} $. By Lemma \[l1\], $\lambda^{H_{k+1}}$ is irreducible and by Lemma \[l2\], $$e_{{\mathbb{F}_{q}}}(\lambda^{H_{k+1}}) = \alpha ({\rm the~sum~of~all~the~distinct~}H_{k+1}{\rm {\operatorname{-}} conjugates~of~} e_{{\mathbb{F}_{q}}}(\lambda^{H_{k}})),$$ for some $\alpha \in {\mathbb{F}_{q}}$. By induction, $e_{{\mathbb{F}_{q}}}(\lambda^{H_{k}}) = \varepsilon_{\mathcal{C}}^{(k)}(H,K)$. Also, by induction, both (i) and (ii) hold for $i=k$, so we obtain using Lemma \[l3\] that the distinct $H_{k+1}$-conjugates of $\varepsilon_{\mathcal{C}}^{(k)}(H,K)$ are mutually orthogonal and thus, by Lemma \[l2\], $\alpha =1$. This gives that $e_{{\mathbb{F}_{q}}}(\lambda^{H_{k+1}})$ equals ${\rm the~sum~of~all~the~distinct~}H_{k+1}{\rm {\operatorname{-}} conjugates~of~}$ $ \varepsilon_{\mathcal{C}}^{(k)}(H,K)$, i.e., $e_{{\mathbb{F}_{q}}}(\lambda^{H_{k+1}})= \varepsilon_{\mathcal{C}}^{(k+1)}(H,K).$ Thus $\varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$ is a primitive central idempotent of ${\mathbb{F}_{q}}H_{k+1}$. Next, we show that if $\mathcal{C}' \in R_{k+1}$ is different from $\mathcal{C}$, then $\varepsilon_{\mathcal{C'}}^{(k+1)}(H,K)$ and $\varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$ are mutually orthogonal. Let $m$ be the number of $\mathcal{C}'$ in $R_{k+1}$ such that $\varepsilon_{\mathcal{C}'}^{(k+1)}(H,K) = \varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$. On multiplying $\overline{\varepsilon^{(k+1)}(H,K)}=\sum_{\mathcal{C}\in R_{k+1}}\varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$ with $\varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$, we get that $\overline{\varepsilon^{(k+1)}(H,K)}\varepsilon_{\mathcal{C}}^{(k+1)}(H,K) = m \varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$. Since both $\overline{\varepsilon^{(k+1)}(H,K)}\varepsilon_{\mathcal{C}}^{(k+1)}(H,K) $ and $\varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$ are idempotents, it follows that $m=1$. This proves (ii) for $i=k+1$ and hence completes the proof. [\[c1\]]{} For every $ \mathcal{C} \in \mathcal{C}_{q}(H/K)$ and for every $i$, $0\leq i\leq n$, there exists $\mathcal{C}_{i} \in R_{i} $ such that $ {\varepsilon_{\mathcal{C}}^{(i)}(H,K)}= \varepsilon_{\mathcal{C}_{i}}^{(i)}(H,K)$ and consequently, the following properties hold: (i) : $H_{i}\unlhd \operatorname{Cen}_{H_{i+1}}(\varepsilon_{\mathcal{C}}^{(i)}(H,K)),~i\neq n$; (ii) : the distinct $H_{i+1}$-conjugates of $\varepsilon_{\mathcal{C}}^{(i)}(H,K)$ are mutually orthogonal, $i\neq n.$ Moreover, $\varepsilon_{\mathcal{C}}^{(i)}(H,K)=e_{\mathbb{F}_{q}}(\lambda^{H_{i}})$ for an arbitrary $\lambda \in \mathcal{C}$. [**Proof.**]{} We assert by induction on $i$ that for every $ \mathcal{C} \in \mathcal{C}_{q}(H/K)$, there exists $\mathcal{C}_{i} \in R_{i} $ such that $ {\varepsilon_{\mathcal{C}}^{(i)}(H,K)}= \varepsilon_{\mathcal{C}_{i}}^{(i)}(H,K)$. The result trivially holds true for $i=0$, as $R_{0}= \mathcal{C}_{q}(H/K)$. Suppose that the result is true for $i=k$. Hence there is a $C_{k} \in R_{k}$ such that $ \varepsilon_{\mathcal{C}}^{(k)}(H,K) = \varepsilon_{\mathcal{C}_k}^{(k)}(H,K)$. Thus the sum of all $H_{k+1}$-conjugates of $\varepsilon_{\mathcal{C}}^{(k)}(H,K)$ and $\varepsilon_{\mathcal{C}_{k}}^{(k)}(H,K)$ are also same, i.e., $ \varepsilon_{\mathcal{C}}^{(k+1)}(H,K) = \varepsilon_{\mathcal{C}_{k}}^{(k+1)}(H,K)$. We now show that there is a $C_{k+1} \in R_{k+1}$ such that $ \varepsilon_{\mathcal{C}_{k}}^{(k+1)}(H,K) = \varepsilon_{\mathcal{C}_{k+1}}^{(k+1)}(H,K)$. Recall from Proposition \[p1\] that $\{ \varepsilon_{\mathcal{C}}^{(k)}(H,K)~ | ~ \mathcal{C} \in R_{k+1} \} $ is a set of representatives of distinct orbits under the action of $\operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$ on $\{ \varepsilon_{\mathcal{C}}^{(k)}(H,K)~|~ \mathcal{C} \in R_{k}\}$ by conjugation. So there is a $C_{k+1} \in R_{k+1}$ such that $\varepsilon_{\mathcal{C}_{k}}^{(k)}(H,K) = (\varepsilon_{\mathcal{C}_{k+1}}^{(k)}(H,K))^g$ for some $ g \in \operatorname{Cen}_{H_{k+1}}(\varepsilon^{(k)}(H,K))$. Thus the sum of all $H_{k+1}$-conjugates of $\varepsilon_{\mathcal{C}_{k}}^{(k)}(H,K)$ is equal to the sum of all $H_{k+1}$-conjugates of $(\varepsilon_{\mathcal{C}_{k+1}}^{(k)}(H,K))^g$, which is same as the sum of all $H_{k+1}$-conjugates of $\varepsilon_{\mathcal{C}_{k+1}}^{(k)}(H,K)$, as $ g \in H_{k+1}$. Hence $ \varepsilon_{\mathcal{C}_k}^{(k+1)}(H,K) = \varepsilon_{\mathcal{C}_{k+1}}^{(k+1)}(H,K)$, as desired. This proves the assertion. Furthermore, (i) and (ii) hold using Proposition 1 and Lemma 3. Moreover, in view of (ii), the repeated application of Lemma \[l2\] yields that $\varepsilon_{\mathcal{C}}^{(i)}(H,K)=e_{\mathbb{F}_{q}}(\lambda^{H_{i}})$ for an arbitrary $\lambda \in \mathcal{C}$ and $0\leq i\leq n$.\ Denote by $R*_{\tau}^{\sigma}G$, the crossed product of the group $G$ over the ring $R$ with action $\sigma$ and twisting $\tau$ ( for details on crossed product, see ([@JdR], Chapter 2)). [\[p2\]]{} The following statements hold for all $\mathcal{C}\in \mathcal{C}_{q}(H/K)$ and $0\leq i < n$: (i) : $\mathbb{F}_{q}H_{i+1}\varepsilon_{\mathcal{C}}^{(i+1)}(H,K) \cong M_{[H_{i+1}:C_{i}]} (\mathbb{F}_{q} C_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K))$, where $ C_{i} = \operatorname{Cen}_{H_{i+1}}(\varepsilon^{(i)}_{\mathcal{C}}(H,K)).$ (ii) : $\mathbb{F}_{q} C_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K)$ $\cong$ $\mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K) *^{\sigma_{i}}_{\tau_{i}} C_{i}/H_{i}$, where the action $\sigma_{i}:C_{i}/H_{i}\rightarrow \operatorname{Aut}(\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K))$ maps $x$ to the conjugation automorphism $(\sigma_{i})_{x}$ on $\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K)$ induced by the fixed inverse image $\overline{x}$ of $x$ under the natural map $C_{i}\rightarrow C_{i}/H_{i}$ and the twisting $\tau_{i}:$ $ C_{i}/H_{i}\times C_{i}/H_{i}\rightarrow \mathcal{U}(\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K))$ is given by $\tau_{i}(x,y)=h\varepsilon_{\mathcal{C}}^{(i)}(H,K)$, where $h \in H_{i}$ is such that $\overline{x}.\overline{y}=h\overline{xy}$ and $\mathcal{U}(\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K))$ is the unit group of $\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K)$. (iii) : $(\sigma_{i})_{x}$ is not an inner automorphism of $ \mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K)$ for every non identity $x \in C_{i}/H_{i}$. (iv) : $C_{i}/H_{i}$ acts faithfully on the center $\mathcal{Z}(\mathbb{F}_{q}H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K))$ of $\mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K)$ by conjugation. [**Proof.**]{} Let $\mathcal{C} \in \mathcal{C}_{q}(H/K)$ and $0 \leq i < n$. Consider an arbitrary $\lambda \in \mathcal{C}$.\ (i) In view of Corollary \[c1\], $e_{{\mathbb{F}_{q}}}(\lambda^{H_{i}})=\varepsilon_{\mathcal{C}}^{(i)}(H,K),$ $e_{{\mathbb{F}_{q}}}(\lambda^{H_{i+1}})=\varepsilon_{\mathcal{C}}^{(i+1)}(H,K)$ and the distinct $H_{i+1}$-conjugates of $e_{{\mathbb{F}_{q}}}(\lambda^{H_{i}})$ are mutually orthogonal. Hence, by taking $S=H_{i}$, $ G = H_{i+1}$ and $\psi = \lambda^{H_{i}}$ in Lemma \[l2\], we get $$\mathbb{F}_{q}H_{i+1}\varepsilon_{\mathcal{C}}^{(i+1)}(H,K) \cong M_{[H_{i+1}: C_{i}]}(\mathbb{F}_{q}C_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K)).$$ This proves (i).\ (ii) In view of Corollary \[c1\], we have $H_i \unlhd C_i$. Therefore, (ii) follows.\ (iii) Suppose $ x \in C_{i}/H_{i}$ is such that $(\sigma_{i})_{x}$ is an inner automorphism of $ \mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K)$. So there is a unit $u$ in $ \mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K)$ such that $$\label{e3} \overline{x}\alpha \overline{x}^{-1}= u\alpha u^{-1}~~ \forall ~\alpha \in \mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K).$$ In particular, $${\label{e4} } \overline{x}h{\varepsilon_{\mathcal{C}}^{(i)}(H,K)}\overline{x}^{-1}= u h{\varepsilon_{\mathcal{C}}^{(i)}(H,K)}u^{-1}~~ \forall~ h \in H_{i}.$$ Let $\rho_i$ be the representation of $H_{i}$ affording the character $\lambda^{H_{i}}$. Extending $\rho_{i}$ linearly on $\mathbb{F}_{q}H_{i}$ and applying on eqn (\[e4\]), we get $$\rho_{i}(\overline{x}h{\varepsilon_{\mathcal{C}}^{(i)}(H,K)}\overline{x}^{-1})= \rho_i(uh{\varepsilon_{\mathcal{C}}^{(i)}(H,K)}u^{-1})~~ \forall~ h \in H_{i}.$$ Since $\rho_{i}$ is a ring homomorphism and $\rho_i({\varepsilon_{\mathcal{C}}^{(i)}(H,K)})$ is the identity matrix, we obtain $$\rho_{i}(\overline{x}h \overline{x}^{-1})= \rho_i(u)\rho_{i}(h)\rho_{i}(u^{-1})~~ \forall~ h \in H_{i}.$$ On taking trace, $$(\lambda^{H_{i}})^{\overline{x}}(h) = \lambda^{H_{i}}(h)~~ \forall~ h \in H_{i}.$$ Thus $\overline{x}$ belongs to the inertia group of $\lambda^{H_{i}}$ in $C_{i}.$ Since $H_i \unlhd C_i$ and $\lambda^{C_{i}}$ is irreducible, it follows from Mackey’s irreducibility criterion that $\overline{x} \in H_{i}$, i.e., $x$ is identity.\ (iv) Clearly, $C_{i}/H_{i}$ acts on $\mathcal{Z}(\mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K))$ by conjugation. All we need to see is that the action is faithful. Let $ x \in C_{i}/H_{i}$ be such that $${\label{e5} } \overline{x}\alpha \overline{x}^{-1}= \alpha ~~ \forall ~\alpha \in \mathcal{Z}(\mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K)).$$ Consider an arbitrary $ h \in H_{i}$. Let $T$ be a right transversal of ${\operatorname{Cen}}_{H_{i}}(h)$ in $H_{i}$. Observe that $\sum_{ g \in T}h^{g}$ commutes with all the elements of $H_{i}$ and hence it belongs to $\mathcal{Z}(\mathbb{F}_{q} H_{i} \varepsilon_{\mathcal{C}}^{(i)}(H,K))$. Thus, from eqn (\[e5\]), $$\overline{x}(\sum_{ g \in T}h^g) \overline{x}^{-1}= \sum_{ g \in T}h^g .$$ Let $\rho_{i}$ be the representation afforded by $\lambda^{H_{i}}$. Extending $\rho_{i}$ linearly on $\mathbb{F}_{q}H_{i}$, we have $$\rho_{i}(\overline{x}(\sum_{ g \in T}h^g) \overline{x}^{-1})= \rho_{i}(\sum_{ g \in T}h^g).$$ On taking trace, it follows that $(\lambda^{H_{i}})^{\overline{x}}(h) = \lambda^{H_{i}}(h)$. This is true for all $h \in H_{i}$. Hence $\overline{x}$ belongs to the inertia group of $\lambda^{H_{i}}$ in $C_{i} $, which gives $\overline{x} \in H_{i}$ and hence $x$ is identity.\ [**Proof of Theorem \[t1\].**]{} From Corollary \[c1\], $e_{\mathbb{F}_{q}}(\lambda^{H_{i}})=\varepsilon_{\mathcal{C}}^{(i)}(H,K)$ for $0 \leq i \leq n$. In particular, $i=n$ gives that $e_{\mathbb{F}_{q}}(\lambda^{G})$ equals $\varepsilon_{\mathcal{C}}^{(n)}(H,K)$, i.e., $\mathfrak{e}_{\mathcal{C}}(G,H,K)$. This proves (i). We now proceed to prove (ii). We will show that the following holds for all $ 1 \leq i \leq n$: $${\label{e6}}\mathbb{F}_{q}H_{i}\varepsilon_{\mathcal{C}}^{(i)}(H,K) \cong M_{[H_{i}: H]}(\mathbb{F}_{q^{l_{i}}}),$$ where $l_{i}= \frac{o}{[C_{0}: H_{0}][C_{1}: H_{1}] \cdots [C_{i-1}: H_{i-1}] }.$ Then (ii) follows from eqn (\[e6\]) by taking $i=n$. To prove eqn (\[e6\]), we will induct on $i$. For $i=1$, the result follows from Theorem 7 of [@BR], as $(H,K)$ is a strong Shoda pair of $H_{1}$ and $\varepsilon_{\mathcal{C}}^{(1)}(H,K) = e_{\mathcal{C}}(H_1,H,K)$. Suppose that the result is true for $i=k$, where $1 \leq k < n$, i.e., $${\label{e7}} \mathbb{F}_{q}H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K) \cong M_{[H_k: H]}(\mathbb{F}_{q^{l_k}}).$$ Using Proposition \[p2\], $${\label{e8}} \mathbb{F}_{q}H_{k+1} \varepsilon_{\mathcal{C}}^{(k+1)}(H,K) \cong M_{[H_{k+1}: C_{k}]}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K) *^{\sigma_k}_{\tau_k} C_k/H_k).$$ Since $\mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K)$ is a simple ring and, by Proposition \[p2\], $(\sigma_{k})_{x}$ is not an inner automorphism of $ \mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K)$ for every non identity $x \in C_k/H_k$, it follows from Lemma 2.6.1 of [@JdR] that $\mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K)*^{\sigma_{k}}_{\tau_{k}}C_{k}/H_{k}$ is also a simple ring and its center $\mathcal{Z}(\mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K) *^{\sigma_{k}}_{\tau_{k}}C_{k}/H_{k})$ equals $\mathcal{Z}(\mathbb{F}_{q}H_{k} \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_{k}/H_{k}}$, the fixed subring of $\mathcal{Z}(\mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K))$. Consequently, $\mathbb{F}_{q}H_{k}\varepsilon_{\mathcal{C}}^{(k)}(H,K)*^{\sigma_{k}}_{\tau_{k}}C_{k}/H_{k}$ is a matrix ring over $\mathcal{Z}(\mathbb{F}_{q}H_{k} \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_{k}/H_{k}}$, say of size $m \times m$. This gives $$\mathbb{F}_{q}H_{k+1}\varepsilon_{\mathcal{C}}^{(k+1)}(H,K) \cong M_{m[H_{k+1}: C_{k}]}(\mathcal{Z}(\mathbb{F}_{q}H_{k} \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_k/H_k}).$$ The faithful action of $C_k/H_k$ on $\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K))$ (Proposition \[p2\]) gives that $ \operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_k/H_k}) = \frac{\operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K)))}{[C_k: H_k]}$. From eqn (\[e7\]), $\operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K)))= l_k$. Therefore, $ \operatorname{dim}_{\mathbb{F}_q}(\mathcal{Z}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K))^{C_k/H_k}) = \frac{l_k}{[C_k: H_k]} = l_{k+1}$. This gives $${\label{e9}} \mathbb{F}_{q}H_{k+1} \varepsilon_{\mathcal{C}}^{(k+1)}(H,K) \cong M_{m[H_{k+1}: C_{k}]}(\mathbb{F}_{q^{l_{k+1}}}).$$ On comparing the dimension of $\mathbb{F}_{q}H_{k+1} \varepsilon_{\mathcal{C}}^{(k+1)}(H,K)$ over $ \mathbb{F}_{q}$ in eqns (\[e8\]) and (\[e9\]), we get $ [H_{k+1} : C_k]^2 [C_k : H_k] \operatorname{dim}_{\mathbb{F}_q}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K) ) = m^2 [H_{k+1} : C_k]^2 l_{k+1}$. Since from eqn (\[e7\]), $ \operatorname{dim}_{\mathbb{F}_q}(\mathbb{F}_q H_k \varepsilon_{\mathcal{C}}^{(k)}(H,K)) = [H_k : H]^2 l_k$, we get $ m= [C_k : H]$. Consequently, $$\mathbb{F}_{q}H_{k+1} \varepsilon_{\mathcal{C}}^{(k+1)}(H,K) \cong M_{[H_{k+1}: H]}(\mathbb{F}_{q^{l_{k+1}}}),$$ as desired. This completes the proof of the theorem.\ Given a generalized strong Shoda pair $(H, K)$ of $G$, we denote by $R_{(H,K)}$, the subset $R_{n}$ of $\mathcal{C}_{q}(H/K)$ obtained in Proposition \[p1\]. \[c2\] If $G$ is a generalized strongly monomial group and $S$ is a complete and irredundant set of generalized strong Shoda pairs of $G$, then (i) : $\mathfrak{e}_{\mathcal{C}}(G, H, K)$, $(H, K) \in S$ and $\mathcal{C} \in R_{(H, K)}$ is the complete and irredundant set of primitive central idempotents of ${\mathbb{F}_{q}}G$; (ii) : ${\mathbb{F}_{q}}G \cong \bigoplus_{(H, K) \in \mathcal{S}} \bigoplus_{\mathcal{C} \in R_{(H, K)}} M_{[G:H]}(\mathbb{F}_{q^{l_{(H,K)}}})$, where $l_{(H, K)}$ is as defined in Theorem \[t1\]. [**Proof.**]{} (i) In view of ([@BK2], Lemma 3), $\mathfrak{e}(G,H,K)$ is a primitive central idempotent of $\mathbb{Q}G$ for every $(H,K) \in S$ and consequently, $\{\mathfrak{e}(G,H,K)~|~(H,K) \in S\}$ is the complete and irredundant set of primitive central idempotents of $\mathbb{Q}G$. Hence, by taking $i=n$ in Proposition \[p1\](i), we have $$\label{e10}1=\sum_{(H,K) \in S}\overline{\mathfrak{e}(G,H,K)}=\sum_{(H,K) \in S}\sum_{\mathcal{C}\in R_{(H,K)}}\mathfrak{e}_{\mathcal{C}}(G,H,K).$$ Thus, to prove (i) it is enough to show that the primitive central idempotents on the right hand side of eqn (\[e10\]) are distinct. Suppose $(H,K)$, $(H',K') \in S$, $C \in R_{(H,K)}$ and $C' \in R_{(H',K')}$. If $(H,K)=(H',K')$, then by Proposition \[p1\](ii), $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ and $\mathfrak{e}_{\mathcal{C'}}(G,H',K')$ are distinct. Assume that $(H,K)\neq (H',K')$. By Proposition \[p1\], $\overline{\mathfrak{e}(G,H,K)}\mathfrak{e}_{\mathcal{C}}(G,H,K)=\mathfrak{e}_{\mathcal{C}}(G,H,K)=\mathfrak{e}_{\mathcal{C}}(G,H,K)\overline{\mathfrak{e}(G,H,K)}$ and $ \overline{\mathfrak{e}(G,H',K')}\mathfrak{e}_{\mathcal{C'}}(G,H',K')=\mathfrak{e}_{\mathcal{C'}}(G,H',K')=\mathfrak{e}_{\mathcal{C'}}(G,H',K')\overline{\mathfrak{e}(G,H',K')}$. Since $\mathfrak{e}(G,H,K)\mathfrak{e}(G,H',K')= 0$, it follows immediately that $\mathfrak{e}_{\mathcal{C}}(G,H,K)\mathfrak{e}_{\mathcal{C'}}(G,H',K')= 0$, i.e., $\mathfrak{e}_{\mathcal{C}}(G,H,K)$ and $\mathfrak{e}_{\mathcal{C'}}(G,H',K')$ are distinct. This proves (i).\ (ii) From (i), it follows that $$\mathbb{F}_{q}G\cong \bigoplus_{(H,K) \in S}\bigoplus_{\mathcal{C}\in R_{(H,K)}}\mathbb{F}_{q}G\mathfrak{e}_{\mathcal{C}}(G, H, K)$$ and by Theorem \[t1\](ii), $\mathbb{F}_{q}G \mathfrak{e}_{\mathcal{C}}(G,H,K) \cong M_{[G:H]}(\mathbb{F}_{{q}^{l_{(H,K)}}})$. This proves (ii) and completes the proof. An algorithm ============ The theory developed in the previous section provides an algorithmic method to write a complete and irredundant set of primitive central idempotents and the precise Wedderburn decomposition of a generalized strongly monomial group. For this purpose, the following steps are to be followed: Step I : Write a complete and irredundant set $S$ of generalized strong Shoda pairs of $G$. Step II : For every $(H,K) \in S$, determine $R_{(H,K)}$ using the following steps: - Fix a strong inductive chain $H=H_{0}\leq H_{1}\leq \cdots \leq H_{n}=G$ from $H$ to $G$. - Determine $\mathcal{C}_{q}(H/K)$. - Find a subset $R_{1}$ of $\mathcal{C}_{q}(H/K)$ so that $\{\varepsilon_{\mathcal{C}}^{(0)}(H,K)~|~\mathcal{C} \in R_{1}\}$ is a set of representatives under the action of $\operatorname{Cen}_{H_{1}}(\varepsilon^{(0)}(H,K))$ on $\{\varepsilon_{\mathcal{C}}^{(0)}(H,K)~|~\mathcal{C} \in \mathcal{C}_{q}(H/K) \}$ by conjugation. Note that $R_{1}$ is a set of representatives under the action of $N_{H_{1}}(K)$ on $\mathcal{C}_{q}(H/K)$ by conjugation. - Find a subset $R_{2}$ of $R_{1}$ so that $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~\mathcal{C} \in R_{2}\}$ is a set of representatives under the action of $\operatorname{Cen}_{H_{2}}(\varepsilon^{(1)}(H,K))$ on $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~\mathcal{C} \in R_{1} \}$. - Continue the above process for $n$ steps. Finally, we obtain $R_{n}$ which is the desired $R_{(H,K)}$. Step III : After finding $R_{(H,K)}$ for every $(H,K) \in S$, one can write primitive central idempotents and Wedderburn decomposition of $\mathbb{F}_{q}G$ using Corollary \[c2\]. An Example ========== Let us now illustrate the above algorithm for $G = \operatorname{SmallGroup}(1000,86)$ in GAP library. Let $\mathbb{F}_{q}$ be a finite field with $q$ coprime to $2$ and $5$ so that $\mathbb{F}_{q}G$ is a semisimple group algebra. The group $G$ is generated by $x_{i},~1\leq i\leq 6$, with the following defining relations: > $x_{1}^{2}x_{2}^{-1}$=$x_{2}^{2}x_{3}^{-1}$=$x_{4}^{5}$=$x_{3}^{2}$=$x_{5}^{5}$=$x_{6}^{5}$=$1,$\ > $[x_{2},x_{1}]$=$[x_{3},x_{1}]$=$[x_{3},x_{2}]$=$[x_{6},x_{3}]$=$[x_{6},x_{4}]$=$[x_{6},x_{5}]$=$1,$\ > $[x_{5},x_{4}]$=$x_{6}$, $[x_{5},x_{1}]$=$x_{4}x_{5},$\ > $[x_{6},x_{1}]$=$x_{6}^{2}$, $[x_{4},x_{2}]$=$x_{4}x_{6}^{2}$, $[x_{6},x_{2}]$=$x_{6}^{3},$\ > $[x_{5},x_{2}]$=$x_{5}x_{6}^{2}$, $[x_{5},x_{3}]$=$x_{5}^{3}x_{6}^{2}$, $[x_{4},x_{3}]$=$x_{4}^{3}x_{6}^{2}$, $[x_{4},x_{1}]$=$x_{4}^{2}x_{5}^{3}x_{6}^{4}.$ In [@BK1], we have already pointed out that $G$ belongs to the class $\mathcal{C}$. The class $\mathcal{C}$ consists of all finite groups in which each subgroup and each quotient group of subgroup has a non central abelian normal subgroup. Also, in Theorem 1 of [@BK2], we have shown that all the groups in class $\mathcal{C}$ are generalized strongly monomial. Consequently, $G$ is a generalized strongly monomial group. It is also known that $G$ is not strongly monomial (see [@JdR], Example 3.4.6). Now we proceed to determine the Wedderburn decomposition of $\mathbb{F}_{q}G$. The first step is to write a complete and irredundant set of generalized strong Shoda pairs of $G$. In ([@BK1], Section 7), we have seen that $S$ $=\{ (G,G), \linebreak (G,\langle x_{2},x_{3},x_{4},x_{5},x_{6}\rangle), (G,\langle x_{3},x_{4},x_{5},x_{6}\rangle), (G,\langle x_{4},x_{5},x_{6}\rangle), (\langle x_{4},x_{5},x_{6}\rangle, \langle x_{4},x_{6}\rangle), \linebreak (\langle x_{4},x_{5},x_{6}\rangle,\langle x_{5},x_{6}\rangle), (\langle x_{4},x_{5},x_{6}\rangle,\langle x_{4}^{-1}x_{5},x_{6}\rangle), (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle, \langle x_{3}x_{4}^{2}x_{6}^{3},x_{5}\rangle), \linebreak (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle,\langle x_{5}\rangle)\}$ is a complete and irredundant set of Shoda pairs of $G$. Note that all Shoda pairs in $S$ are generalized strong Shoda pairs of $G$. The next step is to compute $R_{(H,K)}$ for every $(H,K) \in S$. Observe that all the Shoda pairs in $S$ except the last two are strong Shoda pairs. Also recall from the algorithm that if $(H, K)$ is a strong Shoda pair of $G$ then $R_{(H, K)}$ is $R_1$, i.e., a set of representatives under the action of $N_{G}(K)$ on $\mathcal{C}_{q}(H/K)$ by conjugation (by taking strong inductive chain: $H\leq G$). The first four Shoda pairs in $S$ are strong Shoda pairs of $G$ and satisfy that $N_{G}(K)=H$ which gives that the action of $N_{G}(K)$ on $\mathcal{C}_{q}(H/K)$ is trivial. Therefore, if $(H,K) = (G,G), (G,\langle x_{2},x_{3},x_{4},x_{5},x_{6}\rangle), (G,\langle x_{3},x_{4},x_{5},x_{6}\rangle)$ or $(G,\langle x_{4},x_{5},x_{6}\rangle)$, then $R_{(H,K)} $ coincides with $ \mathcal{C}_{q}(H/K)$. Now we compute $ \mathcal{C}_{q}(H/K)$ for these pairs. Clearly, $\mathcal{C}_{q}(G/G)$ contains only the $q$-cyclotomic class of the principal character on $G$ and so there is only one simple component of $\mathbb{F}_{q}G$ corresponding to $(G,G)$ which is isomorphic to $\mathbb{F}_{q}$. For $(H,K)=(G,\langle x_{2},x_{3},x_{4},x_{5},x_{6}\rangle)$, the factor group $H/K$ is cyclic of order $2$ generated by $Kx_{1}$ and hence there is only one $\overline{\mathbb{F}}_{q}$ irreducible character on $H/K$, namely $\sigma: H/K\rightarrow \overline{\mathbb{F}}_{q}$ given by $Kx_{1}\rightarrow \xi_{2}$. Hence $\mathcal{C}_{q}(H/K)$ has only one $q$-cyclotomic class. Consequently, corresponding to $(G,\langle x_{2},x_{3},x_{4},x_{5},x_{6}\rangle)$, there is precisely one simple component of $\mathbb{F}_{q}G$ isomorphic to $\mathbb{F}_{q}(\xi_{2})$, i.e., $\mathbb{F}_{q}$. Next, for $(H,K) = (G,\langle x_{3},x_{4},x_{5},x_{6}\rangle)$, $H/K$ is cyclic of order $4$ generated by $Kx_{1}$. There are two faithful $\overline{\mathbb{F}}_{q}$ irreducible characters namely $\sigma$ and $\sigma^{3}$ on $H/K$, where $\sigma$ sends $Kx_{1}$ to $\xi_{4}$. Observe that both $\sigma$ and $\sigma^{3}$ lie in different $q$-cyclotomic class if $q\equiv 1(\operatorname{mod}4)$, and lie in the same $q$-cyclotomic class if $q \not \equiv 1(\operatorname{mod}4)$. Thus, corresponding to $(G,\langle x_{3},x_{4},x_{5},x_{6}\rangle) $, there are two simple components of $\mathbb{F}_{q}G$ (both isomorphic to $\mathbb{F}_{q}(\xi_{4}),$ i.e., $\mathbb{F}_{q}$) if $q\equiv 1\operatorname{mod}4$, and only one simple component (isomorphic to $\mathbb{F}_{q}(\xi_{4})$, i.e., $\mathbb{F}_{q^2}$) if $q \not \equiv 1(\operatorname{mod}4)$. If $(H,K)=(G,\langle x_{4},x_{5},x_{6}\rangle)$, then $H/K$ is cyclic of order $8$ generated by $Kx_{1}$. Note that $\{ \sigma, \sigma^3, \sigma^5, \sigma^7\}$ is the set of faithful $\overline{\mathbb{F}}_{q}$ irreducible characters on $H/K$, where $\sigma:~H/K\rightarrow \overline{\mathbb{F}}_{q}$ maps $Kx_{1}$ to $\xi_{8}$. Clearly, if $q\equiv 1(\operatorname{mod}8)$, then $\sigma$,$\sigma^3$,$\sigma^5$,$\sigma^7$ lie in distinct $q$-cyclotomic classes. Also, if $q \not\equiv 1(\operatorname{mod}8)$, then $C_\sigma$, the $q$-cyclotomic class containing $\sigma$, equals $\{\sigma,\sigma^{3}\}$ and $C_{\sigma^5}= \{\sigma^{5},\sigma^{7}\}$. Hence $$\mathcal{C}_{q}(H/K)=\left\{\begin{array}{ll}\{C_\sigma, C_{\sigma^3}, C_{\sigma^5}, C_{\sigma^7}\}, & \hbox{ if~$q\equiv 1(\operatorname{mod}8)$;} \vspace{.2cm}\\\{C_{\sigma}, C_{\sigma^5}\}, & \hbox{if $q \not\equiv 1(\operatorname{mod}8)$.}\end{array}\right.$$ Therefore, corresponding to $(G,\langle x_{4},x_{5},x_{6}\rangle)$, there are $4$ simple components of $\mathbb{F}_{q}G$ (all isomorphic to $\mathbb{F}_{q}$) if $q\equiv 1(\operatorname{mod}8)$, and two simple components of $\mathbb{F}_{q}G$ (both isomorphic to $\mathbb{F}_{q^{2}}$) if $q \not \equiv 1(\operatorname{mod}8)$. The next three Shoda pairs, i.e., $(\langle x_{4},x_{5},x_{6}\rangle, \langle x_{4},x_{6}\rangle),~(\langle x_{4},x_{5},x_{6}\rangle,\langle x_{5},x_{6}\rangle)$ and $(\langle x_{4},x_{5},x_{6}\rangle,\langle x_{4}^{-1}x_{5},x_{6}\rangle)$ are also strong Shoda pairs of $G$. Consider $(H,K)= (\langle x_{4},x_{5},x_{6}\rangle, \langle x_{4},x_{6}\rangle)$. The factor group $H/K$ is cyclic of order $5$ generated by $Kx_{5}$ and $$\mathcal{C}_{q}(H/K)=\left\{\begin{array}{lll}\{C_{\sigma},C_{\sigma^{2}},C_{\sigma^{3}},C_{\sigma^{4}}\}, & \hbox{ if~$q\equiv 1(\operatorname{mod}5)$;} \vspace{.2cm}\\\{C_{\sigma}\}, & \hbox{ if~$q\equiv 2{\rm~or~}3(\operatorname{mod}5)$ ;}\vspace{.2cm}\\\{C_{\sigma},C_{\sigma^{2}}\}, & \hbox{if $q \equiv 4(\operatorname{mod}5)$.}\end{array}\right.$$ It can be seen that $N_{G}(K)=\langle x_{2},x_{3},x_{4},x_{5},x_{6}\rangle$. Also, one can check that all the $q$-cyclotomic classes lie in same orbit under the action of $N_{G}(K)$ for all the cases of $q$. Therefore, $R_{(H,K)}=\{C_{\sigma}\}.$ This gives, by Theorem \[t1\], that there is only one simple component of $\mathbb{F}_{q}G$ corresponding to the Shoda pair $(\langle x_{4},x_{5},x_{6}\rangle,\langle x_{4},x_{6}\rangle)$ of $G$ and is isomorphic to $M_{8}(\mathbb{F}_{q})$. The similar computations apply to the Shoda pairs $(\langle x_{4},x_{5},x_{6}\rangle,\langle x_{5},x_{6}\rangle)$ and $(\langle x_{4},x_{5},x_{6}\rangle,\langle x_{4}^{-1}x_{5},x_{6}\rangle)$ of $G$ and yield that for each of them there is precisely one simple component (isomorphic to $M_{8}(\mathbb{F}_{q})$) of $\mathbb{F}_{q}G$. The last two Shoda pairs $ (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle, \langle x_{3}x_{4}^{2}x_{6}^{3},x_{5}\rangle) $ and $ (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle,\langle x_{5}\rangle) $ are not strong Shoda pairs of $G$. Let us now compute $R_{(H,K)} $ for these two pairs. Consider $(H, K) = (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle, \langle x_{3}x_{4}^{2}x_{6}^{3},x_{5}\rangle)$. Observe that $H=H_{0}\leq H_{1}\leq H_{2}=G$, where $H_{1}=\langle x_{3},x_{4},x_{5},x_{6}\rangle$, is a strong inductive chain from $H$ to $G$. The factor group $H/K$ is cyclic of order $5$ generated by $Kx_{6}$. Consider $\sigma:~H/K\rightarrow \overline{\mathbb{F}}_{q}$ given by $Kx_{6}\mapsto \xi_{5}$. We have $$\mathcal{C}_{q}(H/K)=\left\{\begin{array}{lll}\{C_\sigma, C_{\sigma^2}, C_{\sigma^3}, C_{\sigma^4}\}, & \hbox{ if~$q\equiv 1(\operatorname{mod}5)$;} \vspace{.2cm}\\\{C_\sigma \}, & \hbox{ if~$q\equiv 2{\rm~or~}3(\operatorname{mod}5)$;}\vspace{.2cm}\\\{C_\sigma, C_{\sigma^2}\}, & \hbox{if $q \equiv 4(\operatorname{mod}5)$.}\end{array}\right.$$ Since $N_{H_{1}}(K)=H$, the action of $N_{H_{1}}(K)$ on $\mathcal{C}_{q}(H/K)$ is trivial and so $R_{1}=\mathcal{C}_{q}(H/K)$. The next step is to compute $R_2$, which is the desired $R_{(H,K)}$. Recall from the algorithm that $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~ \mathcal{C} \in R_{2}\}$ is a set of representatives under the action of $\operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))$ on $\{\varepsilon_{\mathcal{C}}^{(1)}(H,K)~|~ \mathcal{C} \in R_{1}\}$. Using Wedderga [@wedd], we can see that $\operatorname{Cen}_{G}(\varepsilon^{(1)}(H,K))=G$. Suppose $q\equiv 1 (\operatorname{mod}5)$. The relations between group elements yield that $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)^{x_{2}} = \varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Hence $x_2$ does not centralize $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$. Since $G/H_1$ is cyclic of order $4$ generated by $H_1 x_1$ and $x_1^2 = x_2$, it follows that $\operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K))=H_{1}$ and so the orbit of $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$ has $[G:H_1] =4$ elements, namely $\varepsilon_{\mathcal{C}_\sigma}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^2}}^{(1)}(H,K)$, $\varepsilon_{\mathcal{C}_{\sigma^3}}^{(1)}(H,K)$ and $\varepsilon_{\mathcal{C}_{\sigma^4}}^{(1)}(H,K)$. Thus $R_{(H, K)} = \{C_\sigma\} .$ Hence, if $q\equiv 1 (\operatorname{mod}5)$, there is only one simple component of $\mathbb{F}_{q}G$, which by Theorem \[t1\] is isomorphic to $M_{20}(\mathbb{F}_{q})$. Next, suppose $q\equiv 2 {\rm~or~}3 (\operatorname{mod}5)$. In this case, $\mathcal{C}_{q}(H/K)$ has only one $q$-cyclotomic class namely $C_\sigma = \{\sigma, \sigma^{2}, \sigma^{3}, \sigma^{4}\}$. So $R_{(H,K)}$ equals $ \mathcal{C}_{q}(H/K)$ and in this case also there is one simple component of $\mathbb{F}_{q}G$ isomorphic to $M_{20}(\mathbb{F}_{q})$. Next suppose $q\equiv 4 (\operatorname{mod}5)$. By Remark \[r2\], $\prod_{0\leq i<2}[\operatorname{Cen}_{H_{i+1}}(\varepsilon_{\mathcal{C_\sigma}}^{(i)}(H,K)):H_{i}]$ must divide $2$. In view of Lemma \[l3\], $\operatorname{Cen}_{H_1}(\varepsilon_{\mathcal{C_\sigma}}^{(0)}(H,K))$ is a subgroup of $\operatorname{Cen}_{H_1}(\varepsilon^{(0)}(H,K)) = N_{H_{1}}(K) = H$ and therefore, $[\operatorname{Cen}_{H_1}(\varepsilon_{\mathcal{C_\sigma}}^{(0)}(H,K)):H] =1$. Consequently $[\operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K)):H_{1}] =1$ or $2$. Observe that $x_2$ centralizes $\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K) $. This gives that $[\operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K)):H_{1}]=2$. Since $[G : H_1]=4$ we have $[G: \operatorname{Cen}_{G}(\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K))] =2$. Therefore, the orbit of $\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K)$ has two elements, namely $\varepsilon_{\mathcal{C_\sigma}}^{(1)}(H,K)$ and $\varepsilon_{\mathcal{C}_{\sigma^{2}}}^{(1)}(H,K)$. This gives $R_{(H,K)} = \{C_\sigma\} $ and corresponding to $ (\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle, \langle x_{3}x_{4}^{2}x_{6}^{3},x_{5}\rangle) $, there is precisely one simple component of $\mathbb{F}_{q}G$ isomorphic to $M_{20}(\mathbb{F}_{q})$. Now consider the last Shoda pair $(H,K)=(\langle x_{5},x_{6},x_{3}x_{4}^{2}\rangle, \langle x_{5}\rangle)$. Observe that $H=H_{0}\leq H_{1}\leq H_{2}=G$, where $H_{1}=\langle x_{3},x_{4},x_{5},x_{6}\rangle$, is a strong inductive chain. In this case, $H/K$ is cyclic of order $10$ generated by $Kx_{3}x_{4}^{2}$. Suppose $\sigma:~H/K\rightarrow \overline{\mathbb{F}}_{q}$ sends $Kx_{3}x_{4}^{2}$ to $\xi_{10}$. We have $$\mathcal{C}_{q}(H/K)=\left\{\begin{array}{lll}\{C_{\sigma},C_{\sigma^{3}},C_{\sigma^{7}},C_{\sigma^{9}}\}, & \hbox{ if~$q\equiv 1(\operatorname{mod}10)$;} \vspace{.2cm}\\\{C_{\sigma}\}, & \hbox{ if~$q\equiv 3{\rm~or~}7(\operatorname{mod}10)$;}\vspace{.2cm}\\\{C_{\sigma},C_{\sigma^{3}}\}, & \hbox{if $q \equiv 9(\operatorname{mod}10)$.}\end{array}\right.$$ Similar arguments as in the above case yield that, in all the cases of $q$, there corresponds only one simple component of $\mathbb{F}_{q}G$ isomorphic to $M_{20}(\mathbb{F}_{q})$. Finally, summing up the above computations, we derive the following Wedderburn decomposition of $\mathbb{F}_{q}G$: $$\mathbb{F}_{q}G\cong \left\{\begin{array}{lll} \mathbb{F}_{q}^{(8)}\oplus M_{8}(\mathbb{F}_{q})^{(3)}\oplus M_{20}(\mathbb{F}_{q})^{(2)}, & \hbox{ if~$q\equiv 1(\operatorname{mod}8)$;} \vspace{.3cm}\\ \mathbb{F}_{q}^{(2)}\oplus \mathbb{F}_{q^{2}}^{(3)}\oplus M_{8}(\mathbb{F}_{q})^{(3)}\oplus M_{20}(\mathbb{F}_{q})^{(2)}, & \hbox{ if~ $q\equiv 3 {\rm~or~}7 (\operatorname{mod}8)$;} \vspace{.3cm}\\ \mathbb{F}_{q}^{(4)}\oplus\mathbb{F}_{q^{2}}^{(2)}\oplus M_{8}(\mathbb{F}_{q})^{(3)}\oplus M_{20}(\mathbb{F}_{q})^{(2)}, & \hbox{ if~ $q\equiv 5 (\operatorname{mod}8).$}\end{array}\right.$$\ Here, $R^{(n)}$ denotes the sum of $n$ copies of $R$. [^1]: Research supported by NBHM(Ref No: 2/39(2)/2015/NBHM/R&D-II/7424), Department of Atomic Energy, Government of India, is gratefully acknowledged. [^2]: Corresponding author

--- abstract: '[We consider the photon field between an unusual configuration of infinite parallel plates: a perfectly conducting plate $(\epsilon\to\infty)$ and an infinitely permeable one $\mu\to\infty)$. After quantizing the vector potential in the Coulomb gauge, we obtain explicit expressions for the vacuum expectation values of field operators of the form $<{\hat E}_i{\hat E}_j>_0$ and $<{\hat B}_i{\hat B}_j>_0$. These field correlators allow us to reobtain the Casimir effect for this set up and to discuss the light velocity shift caused by the presence of plates (Scharnhorst effect [@Scharnhorst; @Barton; @BarScharn]) for both scalar and spinor QED.]{}' author: - | M. V. Cougo-Pinto[^1], C. Farina[^2], F. C. Santos[^3]\ and\ A. C. Tort[^4]\ \ [*Instituto de Física, Universidade Federal do Rio de Janeiro*]{}\ [*Caixa Postal 68528, 21945-970 - Rio de Janeiro, RJ, Brazil*]{} title: | **QED vacuum between an unusual\ pair of plates** --- : 11.10.-z, 11.10.Mn[*Key words: Casimir effect, Scharnhorst effect, non-linear electrodynamics*]{} **Introduction** ================ Ordinary QED deals with processes in unbounded spacetime, with no boundary conditions whatsoever or external fields imposed on and without compactification of any spatial dimension. Nonetheless, a number of physical interesting processes involving photons and electrons (bound or not) occur within the confines of physical boundaries, that is, within a cavity. As an example consider the spontaneous emission by an atom. This process is due to the coupling of electromagnetic vacuum oscillations to the bound electron in the atom and in free space is a position-independent observable. However, inside a cavity the vacuum electromagnetic field modes can change substantially and as a consequence the spontaneous emission rate is affected and can become position-dependent [@KlepHaroche; @Milonni73; @Philpott73] (see also the textbook by Milonni [@Milonnibook] and references therein). For a cavity” comprised by a single metallic wall, for instance, the spontaneous emission rate goes with the reciprocal of the fourth power of the distance of the atom to the wall. In a broader sense, we can say that inside the cavity we can think of the atom as probing the local fluctuations of the electromagnetic vacuum. The influence of the atom-cavity interaction on the atomic spontaneous emission rate is one among a large number of effects of the so-called cavity QED, a specific branch of QED that basically deals with the influences of the surroundings of a physical system on its radiative properties (see ref(s) [@Haroche; @Berman] for recent reviews). Although the first cavity QED effect is attributed to Purcell [@Purcell], who pointed out that the spontaneous emission process associated with nuclear magnetic moment transitions at radio frequencies could be enhanced if the system were coupled to a ressonant external electric circuit, we can say that the first detailed papers on this subject were those written by Casimir and Polder [@Polder] in which, among other things, forces between polarizable atoms and metallic walls were treated, and by Casimir in his seminal work [@CASIMIR]. In its electromagnetic version, the Casimir effect is the macroscopic attraction force between two parallel perfectly conducting infinite surfaces due to the redistribution of normal modes of the vacuum electromagnetic field between them. Experimentally, the Casimir effect between metallic surfaces was first observed by Sparnaay [@SPARNAAY] and recently with remarkable accuracy by Lamoreux [@LAMO] and Mohideen and Roy [@MOHIDEEN]. The various Casimir effects have been the subject of many studies, for a review see [@MosteTrunov; @PLUNIEN]. Still another spectacular instance of cavity QED is the Scharnhorst effect [@Scharnhorst; @Barton]. This effect is basically the velocity shift caused by the change in the zero-point energy density of the quantized electromagnetic field induced by the presence of Casimirlike plates. Recall that an external electromagnetic field such as that of a propagating light couples to the quantized radiation field through fermionic loops. The Scharnhorst effect is not the only example where non-trivial vacua affects the speed of light. In fact this subject has attracted the attention of many physicists in the last years [@Adler; @Drummond; @DanielsShore; @Shore; @Latorre; @Dittrich]. It is clear from what was stated above that an analysis of the QED vacuum inside cavities is crucial for an understanding of its observable properties. Here we shall consider the QED vacuum confined by an unsual pair of mirrors. Specifically, we shall place an infinite perfectly conducting ($\epsilon\to\infty$) surface parallel to a second infinite perfectly permeable ($\mu\to\infty$) surface held at fixed distance $L$ from the first. This setup was first considered by Boyer in order to compute the corresponding Casimir effect in the framework of random electrodynamics [@BOYER] and leads to a repulsive force. This result is somewhat intriguing, since it seems to contradict the explanation given for the usual attractive Casimir effect which suggests that there is a greater number of modes outside the plates than inside [@Milonnibook]. In fact, this is not true: there is only a rearrangement of modes, for a nice explanation of this problem see [@Hushwater]. For the generalized $\zeta$-function approach applied to the repulsive Casimir effect for parallel plates geometry see [@Andre; @SANTOS]. This paper is organized as follows: in section [**2**]{} we determine the photon field ${\mbox{\boldmath $A$}}({\mbox{\boldmath $r$}},t)$ in the region between Boyer’s plates making use of the Coulomb gauge. Next we also evaluate the field operator correlators $<{\hat E}_i{\hat E}_j>_0$ and $<{\hat B}_i{\hat B}_j>_0$ with the aid of a simple but efficient regularization prescription. In section [**3**]{} we apply our results to reobtain the repulsive Casimir pressure of this setup. In section [**4**]{} we discuss the Scharnhorst effect but for this different situation. In particular, we show that, contrary to the case with of the usual pair of Casimir plates considered by Scharnhorst [@Scharnhorst] and Barton [@Barton], Boyer’s plates lead to a decrease in the speed of a light for propagation perpendicular to the plates. In section [**5**]{} we discuss the Scharnhorst effect for the case of scalar QED trying to keep as much as possible a close analogy with the spinorial QED case. Section [**6**]{} is left for the final remarks and conclusions. We use natural units so that Planck’s constant $\hbar$ and the speed of light $c$ are set equal to one. For the electromagnetic fields we employ the unrationalized gaussian system. The fine structure constant reads $\alpha = e^2\approx 1/137$. **Vacuum electromagnetic field between Boyer’s\ plates** =============================================== The setup we will consider consists of two infinite parallel surfaces (the plates) one of which will be considered to be a perfect conductor ($\epsilon\to\infty$) while the other is supposed to be perfectly permeable ($\mu\to\infty$). Also, we will choose Cartesian axes in such a way that the axis ${\cal OZ}$ is perpendicular to both surafces. The perfectly conducting surface will be placed at $z=0$ and the permeable one at $z=L$. The electromagnetic fields must satisfy the following boundary conditions: [**(a)**]{} the tangential components $E_x$ and $E_y$ of the electric field as well as the normal component $B_z$ of the magnetic field must vanish on the metallic plate at $z=0$. ([**b)**]{} The tangential components $B_x$ and $B_y$ of the magnetic field must vanish on the permeable plate at $z=L$. It is convenient to work with the vector potential ${\mbox{\boldmath $A$}}({\mbox{\boldmath $r$}},t)$ in the Coulomb gauge in which ${\mbox{\boldmath $\nabla\cdot A$}}({\mbox{\boldmath $r$}},t)=0$, ${\mbox{\boldmath $E$}}({\mbox{\boldmath $r$}},t)=-\partial{\mbox{\boldmath $A$}}({\mbox{\boldmath $r$}},t)/\partial t$ and ${\mbox{\boldmath $B$}}({\mbox{\boldmath $r$}},t)={\mbox{\boldmath $\nabla\times$}}{\mbox{\boldmath ${A}({\mbox{\boldmath $r$}},t)$}}$. Then the physical boundary conditions combined with our choice of gauge permit us to translate the boundary conditions in terms of the vector potential components. At $z=0$ we have: $$A_x(x,y,0,t)=0\,;\;\;\;\; A_y(x,y,0, t)=0\,;\;\;\;\;{\partial \over\partial z}A_z(x,y,0,t)=0\,,$$ On the other hand, at $z=L$ we have: $${\partial\over\partial x} A_x(x,y,L, t)=0\,;\;\;\;\; {\partial \over\partial y}A_y(x,y,L,t)=0\,;\;\;\;\; A_z(x,y,L,t)=0\,.$$ The appropriate vector potential ${\mbox{\boldmath $A$}}({\mbox{\boldmath $r$}},t)$ that satisfies the wave equation, the Coulomb gauge condition and the previous boundary conditions can be written in the form: $$\begin{aligned} \label{VECPOT} \mbox{\bf A}(\mbox{\bf r},t) &=& {1\over \pi}\left({\pi\over L}\right)^{1\over 2}\sum_{n=0}^\infty \int\,{d^2\kappa\over\sqrt{\omega}} \left\{a^{(1)}({\mbox{\boldmath $\hat\kappa$}}, n) {\mbox{\boldmath $\hat\kappa$}}\times\hat{\mbox{\bf z}}\sin\left[\left(n+{1\over2}\right){\pi z\over L}\right] \right.\nonumber\\ &+& \left. a^{(2)}({\mbox{\boldmath $\kappa$}}, n)\left[{\mbox{\boldmath $\hat\kappa$}} {i(n+{1\over2})\over \omega L} \sin\left[\left(n+{1\over2}\right){\pi z\over L}\right] -{\mbox{\boldmath ${\hat z}$}} {\kappa\over\omega}\cos\left[\left(n+{1\over2}\right){\pi z\over L}\right]\right]\right\}e^{i( {\mbox{\boldmath $\kappa$}}\cdot {\mbox{\boldmath $\rho$}}-\omega t)}\;\nonumber\\ &+& \mbox{Hermitian conjugate}\; ,\end{aligned}$$ where ${\mbox{\boldmath $\kappa$}} = (k_x, k_y)$ and ${\mbox{\boldmath $\rho$}}$ is the position vector in the $xy$-plane. The normal frequencies are given by $$\omega =\omega({\mbox{\boldmath $\kappa$}},n)=\sqrt{{\mbox{\boldmath $\kappa$}}^2+\left(n+{1\over 2}\right)^2{\pi^2\over L^2}}\,.$$ The Fourier coefficients $a^{(\lambda)}({\mbox{\boldmath $\kappa$}}, n)$ where $\lambda=1,2$ is the polarization index, are operators acting on the photon state space and satisfy the commutation relation $$\left[a^{(\lambda)}({\mbox{\boldmath $\kappa$}}, n),a^{(\lambda^\prime)}({\mbox{\boldmath $\kappa$}}^\prime, n^\prime)\right]=\delta_{\lambda\lambda^\prime}\delta_{nn^\prime}\delta\left ({\mbox{\boldmath $\kappa$}}-{\mbox{\boldmath $\kappa$}}^\prime\right)\,.$$ It is convenient to write the vector potential in the general form: $${\mbox{\boldmath $A$}}({\mbox{\boldmath $r$}},t)=\sum_\alpha a^\alpha (0){\mbox{\boldmath $A$}}^\alpha ({\mbox{\boldmath $r$}})e^{-i\omega_\alpha t} +\mbox{H. c.}\,,$$ where ${\mbox{\boldmath $A$}}^\alpha ({\mbox{\boldmath $r$}})$ denotes the mode functions. The mode functions for each polarization state obey the Helmholtz equation and satisfy the boundary conditions stated above. In our case they are given by: $$\label{AUM} {\mbox{\boldmath $A$}}_{{\mbox{\boldmath $\kappa$}}n}^{(1)} ({\mbox{\boldmath $r$}})={1\over\pi}\left({\pi\over L}\right)^{1\over 2}{1\over\omega^{1\over 2}}\sin\left[\left(n+{1\over2}\right){\pi z\over L}\right]e^{-i{\mbox{\boldmath $\kappa\cdot{\mbox{\boldmath $\rho$}}$}}}\,{\mbox{\boldmath $\hat\kappa\times\hat z$}} \,,$$ and $$\label{ADOIS} {\mbox{\boldmath $A$}}_{{\mbox{\boldmath $\kappa$}}n}^{(2)} ({\mbox{\boldmath $r$}})={1\over\pi}\left({\pi\over L}\right)^{1\over 2}{1\over\omega^{1\over 2}}\left[{\mbox{\boldmath $\hat\kappa$}}{in\pi\over L\omega}\sin\left[\left(n+{1\over2}\right){\pi z\over L}\right]-{\mbox{\boldmath $\hat z$}}{\kappa\over\omega}\cos\left[\left(n+{1\over2}\right){\pi z\over L}\right]\right]e^{-i{\mbox{\boldmath $\kappa\cdot{\mbox{\boldmath $\rho$}}$}}}\,.$$ Next we evaluate the electric field operator ${\mbox{\boldmath $E$}}({\mbox{\boldmath $r$}},t)$. Recalling that $a^\alpha |0>=0$ and ${a^\alpha}^\dagger |0>=0$ we first write for the correlators $<E_i({\mbox{\boldmath $r$}},t)E_j({\mbox{\boldmath $r$}},t)>_0$ a general expression of the form: $$\label{E0} <E_i({\mbox{\boldmath $r$}},t)E_j({\mbox{\boldmath $r$}},t)>_0=\sum_\alpha E_{i\alpha}({\mbox{\boldmath $r$}})E_{j\alpha}^*({\mbox{\boldmath $r$}})\,.$$ In our case (\[AUM\]) and (\[ADOIS\]) yield $$\label{E1} {\mbox{\boldmath $E$}}_{i{\mbox{\boldmath $\kappa$}}n}^{(1)} ({\mbox{\boldmath $r$}})={i\over\pi}\left({\pi\over L}\right)^{1\over 2}{1\over\omega^{1\over 2}}\sin\left[\left(n+{1\over2}\right){\pi z\over L}\right]e^{-i{\mbox{\boldmath $\kappa\cdot{\mbox{\boldmath $\rho$}}$}}}\, ({\mbox{\boldmath $\hat\kappa\times\hat z$}})_i \,,$$ and $$\label{E2} {\mbox{\boldmath $E$}}_{i{\mbox{\boldmath $\kappa$}}n}^{(2)} ({\mbox{\boldmath $r$}})={i\over\pi}\left({\pi\over L}\right)^{1\over 2}{1\over\omega^{1\over 2}}\left[{\mbox{\boldmath $\hat\kappa$}}_i{in\pi\over L\omega}\sin\left[\left(n+{1\over2}\right){\pi z\over L}\right]-{\mbox{\boldmath $\hat z$}}_i{\kappa\over\omega}\cos\left[\left(n+{1\over2}\right){\pi z\over L}\right]\right]e^{-i{\mbox{\boldmath $\kappa\cdot{\mbox{\boldmath $\rho$}}$}}}\,.$$ Now we substitute (\[E1\]) and (\[E2\]) into (\[E0\]), write ${\mbox{\boldmath $\hat\kappa$}}_i=\cos\phi\,\delta_{ix}+\sin\phi\,\delta_{iy}$, ${\mbox{\boldmath $\hat z$}}_i=\delta_{iz}$ and $({\mbox{\boldmath $\hat\kappa\times\hat z$}})_i=\sin\phi\,\delta_{ix}-\cos\phi\,\delta_{iy}$, where $\phi$ is the azimuthal angle in the $xy$-plane and compute all angular integrals. In this way we wind up with $$\begin{aligned} \label{VEV} &&<{\hat E}_i({\mbox{\boldmath $r$}},t){\hat E}_j({\mbox{\boldmath $r$}},t)>_0 \;\;=\;\; \left({2\over \pi}\right)\left({\pi\over L}\right){\delta^{\|}_{ij}\over 2}\sum_{n=0}^\infty\sin^2{\left[\left(n+{1\over 2}\right){\pi z\over L}\right]}\int_0^\infty d\kappa\,\kappa\,\omega ({\mbox{\boldmath $\kappa$}}, n) \nonumber \\ &+& \left({2\over \pi}\right)\left({\pi\over L}\right)\left({\pi\over L}\right)^2{\delta^{\|}_{ij}\over 2}\sum_{n=0}^\infty\sin^2{\left[\left(n+{1\over2}\right){\pi z\over L}\right]}\left( n+{1\over2}\right)^2 \int_0^\infty d\kappa\,\kappa\,\omega^{-1} ({\mbox{\boldmath $\kappa$}}, n) \nonumber \\ &+& \left({2\over \pi}\right)\left({\pi\over L}\right)\delta^{\bot}_{ij}\sum_{n=0}^\infty\cos^2{\left[\left(n+{1\over2} \right){\pi z\over L}\right]}\int_0^\infty d\kappa\,\kappa^3\, \omega^{-1}({\mbox{\boldmath $\kappa$}}, n)\,,\end{aligned}$$ where $\delta^{\|}_{ij}:=\delta_{ix}\delta_{jx}+\delta_{iy}\delta_{jy} $ and $\delta^{\bot}_{ij}:=\delta_{iz}\delta_{jz}$. The previous equation is only a formal expression for the field correlator $\langle {\hat E}_i({\mbox{\boldmath $r$}},t){\hat E}_j({\mbox{\boldmath $r$}},t)\rangle _0$, since it is an ill-defined expression plagued by divergent terms. Therefore, it lacks of physical meaning unless we adopt a regularization prescription. We will first regularize the integrals in equation (\[VEV\]) by using a method based on analytical extension in the complex plane. The idea is the following: take for example the first integral that appears on the r.h.s. of (\[VEV\]), $${\cal I}_1(n,L):=\int_0^\infty d\kappa \kappa\left( \kappa^2+{(n+{1\over2})^2\pi^2\over L^2}\right)^{1/2}\; .$$ Since this integral diverges for large $\kappa$, it is natural to modify the integrand so that the integral becomes finite. Our choice will be simply $${\cal I}_1(n,L)\longrightarrow {\cal I}^{reg}_1(n,L;s):= \int_0^\infty d\kappa \kappa\left( \kappa^2+{(n+{1\over2})^2\pi^2\over L^2}\right)^{1/2-s} \;$$ and after the calculations we will take the limit $s\to 0$. For the moment, let us assume that $\Re\, s$ is large enough to give a precise mathematical meaning for the previous integral. Then, making use of the following integral representation of the Euler beta function, [*c.f.*]{} formula [**3.251**]{}.2 [@Grad]: $$\int_0^\infty dx\,x^{\mu-1}\left(x^2+a^2\right)^{\nu-1} = {B\over2}\left({\mu\over2},1-\nu-{\mu\over2}\right) a^{\mu+2\nu-2}\,,$$ where $B(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$, which holds for $\Re\,\left(\nu+{\mu\over 2}\right)< 1$ and $\Re\,\mu>0$, we get $${\cal I}^{reg}_1(n,L;s)={1\over2}\left[ (n+{1\over2}{\pi\over L} \right]^{3-2s}\; {\Gamma(s-3/2)\over \Gamma(s-1/2)} ={1\over (2s-3)}\left[(n+{1\over2}{\pi\over L} \right]^{3-2s}$$ Inserting this result into the first term of the r.h.s. of (\[VEV\]) (call it ${\cal T}_1)$, it takes the form: $${\cal T}_1=\left({1\over 2s-3}\right)\left({\pi\over L}\right)^{3-2s} {\delta^{\|}_{ij}\over 2L}\left\{ \zeta_H(2s-3,1/2)-\sum_{n=0}^\infty \left( n+{1\over 2}\right)^{3-2s}\cos\left[ {2(n+1/2)\pi z\over L}\right] \right\}\; ,$$ where $\zeta_H(z,a)$ is the well known Hurwitz zeta function. Making the analytical extension to the $s$-complex plane and taking the limit $s\rightarrow 0$, we get $${\cal T}_1=-{1\over 6\pi}\left({\pi\over L}\right)^4 \delta^{\|}_{ij}\left\{\left(-{7\over 8}\right)\times {1\over 120}-G(\pi z/L)\right\}\; ,$$ where we made use of $\zeta_H(-3,1/2)=(-7/8)\times(1/120)$ and defined $$\begin{aligned} \label{GFINAL} G\left(\xi\right)& = & -\frac{1}{8}\times {d^3\over d\xi^3}\left({1\over 2\sin{\xi}}\right) \nonumber \\ & = & \frac{1}{8}\left(3{\cos^3{\xi}\over \sin^4{\xi}}+\frac{5}{2}{\cos{\xi}\over \sin^2{\xi}}\right)\,.\end{aligned}$$ Analogous calculations can be performed with the other terms of the r.h.s. of (\[VEV\]). It is then straightforward to show that $$\label{COREIEJ} \langle{\hat E}_i({\mbox{\boldmath $r$}},t){\hat E}_j({\mbox{\boldmath $r$}},t)\rangle_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;{1\over 120}+\delta_{i,j}G(\pi z/L)\right]\,,$$ and proceeding in the same way we did in the evaluation of the electric field correlators we obtain $$\label{CORBIBJ} \langle{\hat B}_i({\mbox{\boldmath $r$}},t){\hat B}_j({\mbox{\boldmath $r$}},t)\rangle_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;{1\over 120}-\delta_{i,j}G(\pi z/L)\right]\,,$$ for the magnetic field correlators. A straightforward calculation along the lines given here or the use of time-reversal invariance shows that the correlators $<E_i({\mbox{\boldmath $r$}},t)B_j({\mbox{\boldmath $r$}},t)>_0=0$. In passing, observe that no substractions whatsoever were required in our regularization procedure. This is a common feature of regularization prescriptions based on the analytical extension. However, other methods where the subtraction of the field correlators involving no boundary conditions are present can be used yielding the same results. **The Casimir effect between Boyer’s plates** ============================================= In order to get confidence in the previous results for the field operator correlators between Boyer’s plates, let us reobtain Boyer’s result [@BOYER] concerning the Casimir effect for this unusual set up. First, recall that the zero-point energy density $\rho_o$ for the electromagnetic fields is defined by the following vacuum expectation value: $$\label{Casi} \rho_0=\frac{1}{8\pi}<{\mbox{\boldmath $E$}}^2+{\mbox{\boldmath $B$}}^2>_0\,.$$ Making use of (\[COREIEJ\]) and (\[CORBIBJ\]) we obtain the position-dependent correlators: $$<{\mbox{\boldmath $E$}}^2>_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[{7\over 8\times 120}+3G(\xi)\right]\,,$$ $$<{\mbox{\boldmath $B$}}^2>_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[{7\over 8\times 120}-3G(\xi)\right]\,.$$ If we add these two equations the position-dependent terms will cancel out and if we substitute the result into (\[Casi\]) we will obtain: $$\rho_0= {7\over 8}\times {\pi^2\over 720 L^4}\,,$$ which is the position-independent and positive Casimir energy density leading to a repulsive force per unit area between the plates [@BOYER; @Andre; @SANTOS]. It is also convenient to analyze the behavior of the correlators $<{\mbox{\boldmath $E$}}^2>_0$ and $<{\mbox{\boldmath $B$}}^2>_0$ in the situations where one of the plates is removed. Let us first consider the limit of a single metal plate located at $z=0$. This means that we are taking the limit $L\to\infty$ in our previous results. The results are: $$\label{SMLEO} <{\mbox{\boldmath $E$}}^2>_0\approx +{3\over 4\pi z^4}\,,$$ and $$\label{SMLBO} <{\mbox{\boldmath $B$}}^2>_0\approx - {3\over 4\pi z^4}\,,$$ in agreement with the literature [@LR] On the other hand, the limit of a single infinitely permeable plate is obtained by removing the metal plate. This can be accomplished if we consider the limits $L\to\infty$, $z\to\infty$ in the previous results, but with $L-z<<L$. For this case we obtain: $$\label{SMLEL} <{\mbox{\boldmath $E$}}^2>_0\approx -{3\over 4\pi (z-L)^4}\,,$$ and $$\label{SMLBL} <{\mbox{\boldmath $B$}}^2>_0\approx +{3\over 4\pi (z-L)^4}\,.$$ Equations (\[SMLEL\]) and (\[SMLBL\]) are new results. Let us turn our attention now to one of the most intriguing properties of the QED vacuum: its anisotropy and the concomitant consequences on the speed of light. **The Scharnhorst effect for the spinor QED** ============================================= The Scharnhorst effect [@Scharnhorst; @Barton] is basically the light velocity shift in the QED vacuum caused by the presence of two parallel plates for propagation inside the plates and perpendicular to them. This was shown to occur for small frequencies $\omega<<m$ (soft photon approximation) and in the weak field limit. For the case of metallic plates, Scharnhorst [@Scharnhorst] and later on Barton [@Barton] showed that the phase velocity, which for this case coincides with the phase velocity for small frequencies, is greater thatn its value in free space ($c$) for propagation perpendicular to the plates. However, this does not mean that the signal velocity can be greater than $c$ because to determine the wave front velocity it is necessary to investigate the dispersion relation in the infinite frequency limit (see reference [@Milonni2; @BarScharn; @MilonniSvozil; @Menahem] for some discussion on this issue). The Scharnhorst effect with a boundary condition other than the standard one for perfect metallic plates has also been considered [@Bartinho]. It can be understood as follows: the external field as that describing the propagation of a plane wave interacts with the quantized electromagnetic fields through the fermionic loops and hence, any change in the quantized field modes, as for example caused by imposition of boundary conditions, can in principle modify the wave propagation. In references [@Scharnhorst; @Barton] this change was induced by the presence of two perfect parallel conducting plates. Since these authors assumed that the plates do not impose any boundary condition on the fermionic field, the Scharnhorst effect appears only at the two-loop level. Also, because it is a perturbative effec, it can be obtained by direct computation of the relevant Feynman diagrams that contribute to the effective action, namely: the two possible diagrams for the photon polarization tensor at two-loop level. This was precisely Scharnhorst’s approach, who after using a previous representation for the photon propagator between two metallic plates obtained by Bordag, Robaschik and Wieczorek [@BORDAG] found for propagation perpendicular to the plates that $$v_\bot=1+{11\pi^2\over 2^2.3^4.5^2}{\alpha^2\over (mL)^4}\; .$$ Later on, the same result was rederived by Barton [@Barton] in a more economic way, where the connection to the Casimir energy density is more apparent. The starting point in Barton’s approach is the addition to the electromagnetic field lagrangian density of a correction term represented by the Euler-Heisenberg [@HL] effective lagrangian density, so that the full lagrangian density reads: $$\begin{aligned} \label{FULLL} {\cal L}& = & {\cal L}^{(0)}+{\cal L}^{(1)}\nonumber \\ & = & {1\over 8\pi}\left({\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2\right)+ g\left[\left({\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2\right)^2+7\left({\mbox{\boldmath $E\cdot B$}} \right)^2\right]\,,\end{aligned}$$ where $g:=\alpha^2/5\cdot 3^2\cdot 2^3\cdot\pi^2 m^4$. The lagrangian density represented by (\[FULLL\]) describes the first vacuum polarization effects on slowly varying fields for which the condition $\omega \ll m$ holds and is valid only in the weak field approximation. In other words, the first non-linear effects to Maxwell equations coming from QED are described by the quartic terms added to the usual Maxwell lagrangian density in the above formula. The corresponding vacuum polarization ${\mbox{\boldmath $P$}}$ and magnetization ${\mbox{\boldmath $M$}}$ are given by: $$\label{P1} {\mbox{\boldmath $P$}}={\partial{\cal L}^{(1)}\over\partial{\mbox{\boldmath $E$}}}=4g\left ({\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2\right){\mbox{\boldmath $E$}}+14g \left({\mbox{\boldmath $E\cdot B$}}\right) {\mbox{\boldmath $B$}}\,,$$ and $$\label{M1} {\mbox{\boldmath $M$}}={\partial{\cal L}^{(1)} \over\partial{\mbox{\boldmath $B$}}}=-4g\left({\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2\right){\mbox{\boldmath $B$}} +14g\left({\mbox{\boldmath $E\cdot B$}} \right){\mbox{\boldmath $E$}}\,.$$ In order to include a radiative correction into the formalism, we can follow reference [@Barton] and rewrite the fields in equations (\[P1\]) and (\[M1\]) as the sum of two parts, one describing the quantized fields and the other one describing the classical fields, that is, we write: ${\mbox{\boldmath $E$}}\to{\mbox{\boldmath $E$}}_q+{\mbox{\boldmath $E$}}_c$ and ${\mbox{\boldmath $B$}}\to{\mbox{\boldmath $B$}}_q+{\mbox{\boldmath $B$}}_c$ and substitute into (\[P1\]) and (\[M1\]). This procedure is tantamount to the coupling of the external fields to the quantized ones by means of the intermediary action of a fermionic loop. Keeping only the terms which are linear in the classical fields, we obtain the following expressions for the electric susceptibilty $\chi^{(e)}_{ij}$ and magnetic susceptibility $\chi^{(m)}_{ij}$ tensors of the vacuum: $$\label{DIE} \chi^{(e)}_{ij}= 4g\left[ <{\mbox{\boldmath $E_q$}}^2-{\mbox{\boldmath $B_q$}}^2>_0\delta_{ij}+2<E_{qi}E_{qj}>_0 \right]+14g<B_{qi}B_{qj}>_0\,,$$ $$\label{PER} \chi^{(m)}_{ij}= 4g\left[-<{\mbox{\boldmath $E_q$}}^2-{\mbox{\boldmath $B_q$}}^2>_0\delta_{ij}+2<B_{qi}B_{qj}>_0 \right]+14g<E_{qi}E_{qj}>_0\,.$$ The dieletric and permittivity tensors of the vacuum are: $$\epsilon_{ij}=\delta_{ij}+4\pi\chi^{(e)}_{ij}=\delta_{ij}+ \Delta\epsilon_{ij}\,,$$ $$\mu_{ij}=\delta_{ij}+4\pi\chi^{(m)}_{ij}=\delta_{ij}+\Delta\mu_{ij}\,,$$ The vacuum expectation values in (\[DIE\]) and (\[PER\]) can be easily calculated with the correlators given by (\[COREIEJ\]) and (\[CORBIBJ\]). If we do this, we obtain for $\Delta\epsilon_{ij}$ and $\Delta\mu_{ij}$ the results: $$\label{DIE2} \Delta\epsilon_{ij}=g\left({\pi\over L}\right)^4{16\over 3}\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({11\over 120}\right)+3\delta_{ij}G(\xi)\right]\,,$$ and $$\label{PER2} \Delta\mu_{ij}=g\left({\pi\over L}\right)^4{16\over 3}\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({11\over 120}\right)-3\delta_{ij}G(\xi)\right]\,.$$ We can also derive single plate limits for $\Delta\epsilon_{ij}$ and $\Delta\mu_{ij}$. Making use of the approximations to $G(\xi)$ in the limits $\xi\to 0$ and $\xi\to\pi$ we have near the conducting plate at $z=0$: $$\Delta\epsilon_{ij}=-\Delta\mu_{ij}=18g{\delta_{ij}\over z^4}\,,$$ and near the permeable plate at $z=L$: $$\Delta\epsilon_{ij}=-\Delta\mu_{ij}=-18g{\delta_{ij}\over \left(z-L\right)^4}\,.$$ Now, we are interested in the refraction index $n=\sqrt{\epsilon\mu}$ and its first order shift: $$\label{INDEX} \Delta n= \frac{1}{2}\left(\Delta\epsilon+\Delta\mu\right)\,,$$ for directions of propagation defined by the cartesian axis. Let us consider first a plane wave propagating in the $OX$-direction with the electric field vibrating in the $OZ$-direction. Then $\Delta\epsilon\to\Delta\epsilon_{33}$ and $\Delta\mu\to\Delta\mu_{22}$, and from (\[DIE2\]), (\[PER2\]) and (\[INDEX\]) we can easily verify that $\Delta n=\frac{1}{2}\left(\Delta\epsilon_{33}+\Delta\mu_{22}\right)=0$. We obtain the same result in all instances in which the propagtion is parallel to the plane of the plates. As a consequence the speed of light remains unchanged for propagtion parallel to the plates. Now consider a plane wave propagating along the $OZ$-axis, perpendicularly to the pair of plates. Consider the wave polarized in the $OX$-direction, for instance. Then $\Delta\epsilon\to\Delta\epsilon_{11}$ and $\Delta\mu\to\Delta\mu_{22}$, and from (\[DIE2\]), (\[PER2\]) and (\[INDEX\]) we now obtain: $$\begin{aligned} \Delta n_\bot & \approx & \frac{1}{2}\left(\Delta\epsilon_{11}+\Delta\mu_{22}\right)\nonumber \\ & = & +\frac{7}{8}\times{\alpha^2\over\left(mL\right)^4}{11\pi^2\over 2^2\cdot3^4\cdot 5^2}\end{aligned}$$ which is the result obtained by Scharnhorst [@Scharnhorst] and reobtained by Barton [@Barton] multiplied by the factor $-7/8$. The speed of light in that direction will be: $$v_\bot\approx 1-\frac{7}{8}\times{\alpha^2\over\left(mL\right)^4}{11\pi^2\over 2^2\cdot3^4\cdot 5^2}<1\; ,$$ as anticipated in the begining of this work. The direction-averaged light velocity between Boyer’s plates also satisfies the unifying formula written down by Latorre, Pascual and Tarrach [@Latorre] for spinor QED which reads $$\begin{aligned} \langle v\rangle=1-{44\alpha^2\over 135 m_e^4}\; \rho_0\; .\end{aligned}$$ It can be shown that this formula can be obtained in the weak field limit of Dittrich and Gies’ approach to the study of non-trivial vacua [@Dittrich]. We will return to this in the next section. **The Scharnhorst effect in scalar QED** ---------------------------------------- Although the interaction of charged fermions of spin $1/2$ with themselves and with the photon field is described in a very satisfactory way by spinor QED, we are not prohibited of thinking on other theories. It may be very instructive to study other theories that, though not realistic, respect all important physical principles as for instance, the gauge principle and relativistic invariance. This is the case of the so-called scalar QED, which describes charged bosons interacting with themselves and with the radiation field. Naively, we could think that the interaction between the pseudoscalars charged mesons $\pi^\pm$ and $K^\pm$ could be described by scalar QED, but this is not true, mainly because these mesons have an inner structure and their interaction is dominated by the strong interaction. In fact, since there are no fundamental charged bosons in Nature, scalar QED is of limited application. However, scalar QED can be viewed as a toy model in many situations and hence it may shed some light on interesting physical processes, as we shall see. Without further apologies, we shall consider in this section the Scharnhorst effect in the framework of scalar QED. In the case of scalar QED the analogue of the Euler-Heisenberg effective lagrangian reads [@Schwinger51]: $${\cal L}_0^{(1)}=g_0\left[{7\over 4}\left({\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2\right)^2+\left({\mbox{\boldmath $E\cdot B$}} \right)^2\right]\,,$$ with $g_0:=\alpha^2/5\cdot 3^2\cdot 2^5\cdot\pi^2\cdot m_o^4$, where $m_o$ is the mass of the hypothetical charged boson associated with $1$-loop scalar QED. As before, the polarization ${\mbox{\boldmath $P$}}$ and the magnetization ${\mbox{\boldmath $M$}}$ are defined by equations (\[P1\]) and (\[M1\]), and as before we make use of the substitutions ${\mbox{\boldmath $E$}}\to{\mbox{\boldmath $E$}}_q+{\mbox{\boldmath $E$}}_c$ and ${\mbox{\boldmath $B$}}\to{\mbox{\boldmath $B$}}_q+{\mbox{\boldmath $B$}}_c$ and keep only terms linear in the classical fields to obtain the corrections $\Delta\epsilon_{ij}$ and $\Delta\mu_{ij}$ to the dielectric and permittivity tensors of the scalar QED vacuum. The results are $$\label{DIECAS} \Delta\epsilon_{ij}= 28\pi g_0<{\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2>_0\delta_{ij}+56\pi g_0<E_iE_j>_0 + 8\pi g_0<B_iB_j>_0\,,$$ $$\label{PERCAS} \Delta\mu_{ij}= -28g_0<{\mbox{\boldmath $E$}}^2-{\mbox{\boldmath $B$}}^2>_0\delta_{ij}+ 56\pi g_0 <B_iB_j>_0 + 8\pi g_0<E_iE_j>_0\,.$$ Now we can make use of these results and analyze the speed of light in confined scalar QED vacuum. Since the Scahrnhorst effect for scalar QED has never been discussed before, we will evaluate the light velocity shifts for two cases, to wit, for Casimir’s plates and for Boyer’s plates. [*Casimir’s plates*]{}. We shall consider the two perfectly conducting plates at $z=0$ and $z=L$ respectively. Expressions for the electric and magnetic field correlators for the Casimir’s plates can be found in, for instance, [@Barton], here we merely state the results $$\label{COECAS} <E_i({\mbox{\boldmath $r$}},t)E_j({\mbox{\boldmath $r$}},t)>_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;{1\over 120}+\delta_{ij}F(\pi z/L)\right]\,,$$ and $$\label{COBCAS} <B_i({\mbox{\boldmath $r$}},t)B_j({\mbox{\boldmath $r$}},t)>_0=\left({\pi\over L}\right)^4{2\over 3\pi}\left[\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;{1\over 120}-\delta_{ij}F(\pi z/L)\right]\,,$$ where $F(\xi)$ is defined by: $$F(\xi):= -\frac{1}{8}\times {d^3\over d\xi^3}\left({1\over 2}\cot{\xi}\right)\,.$$ Now we take (\[COECAS\]) and (\[COBCAS\]) into (\[DIECAS\]) and (\[PERCAS\]) and after some simple manipulations we end up with $$\Delta\epsilon_{ij}={16\over 3}g_0\left({\pi\over L}\right)^4\left[\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({1\over 15}\right)+27\delta_{ij}F(\xi)\right]\,,$$ and, $$\Delta\mu_{ij}={16\over 3}g_0\left({\pi\over L}\right)^4\left[\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({1\over 15}\right)-27\delta_{ij}F(\xi)\right]\,.$$ With these results we can now calculate the first correction to the refraction index $\Delta n$ and, consequently, the correction to the speed of light between Casimir’s plates in scalar QED. As in the corresponding case of spinor QED, we find that the speed of light parallel to the plates remains unchanged, but the speed of light perpendicular to the plates is modified by an amount given by $$\label{CAS} \Delta v_\bot=-\Delta n=+{16\over 45} g_0\left({\pi\over L}\right)^4>0\,.$$ It is interesting to compare this result with the analogous effect that taked place in spinor QED. Assuming the same charge for the particles (bosons and fermions), we see that the ratio between the light velocity shifts for scalar and usual QED is given by $${\Delta v_\bot^b\over \Delta v_\bot}=8\times \left({m\over m_o}\right)^4\; .$$ [*Boyer’s plates*]{}. Now we repeat the procedure for the unusual pair of plates that we are discussing here. The electric and magnetic field correlators we need are given by equations (\[COREIEJ\]) and (\[CORBIBJ\]). Substituting into (\[DIECAS\]) and (\[PERCAS\]) we obtain $$\Delta\epsilon_{ij}={16\over 3}g_0\left({\pi\over L}\right)^4\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({1\over 15}\right)+27\delta_{ij}G(\xi)\right]\,,$$ and, $$\Delta\mu_{ij}={16\over 3}g_0\left({\pi\over L}\right)^4\left[\left(-{7\over 8}\right)\left(-\delta^{\|}+\delta^{\bot}\right)_{ij}\;\left({1\over 45}\right)-27\delta_{ij}G(\xi)\right]\,.$$ Hence, the speed of light between Boyer’s plates in the direction perpendicular to the plates is modified by the amount $$\label{BO} \Delta v_\bot=-\Delta n=-{7\over 8}\times {16\over 45} g_0\left({\pi\over L^4}\right)<0\,.$$ The results given by equations (\[CAS\]) and (\[BO\]) can be unified by considering the average taken over all directions of propagation of the speed of light between the plates. To acomplish this first we write, for instance, for Casimir’s plates: $$v(\theta)=1-{16\over 45}g_0\left({\pi\over L}\right)^4 \cos^2{\theta}\,,$$ where $\theta$ is the angle between the direction of propagation and the ${\cal OZ}$-axis. Next we take the average over all direction. The result is: $$\begin{aligned} \label{average} \langle v\rangle={1\over 4\pi}\oint v(\theta)d\Omega&=&1+{8\alpha^2\over 135 m_o^4}\left({\pi^2\over 720 L^4}\right)\nonumber\\ &=&1+{8\alpha^2\over 135 m_0^4}\; \rho_0\; .\end{aligned}$$ The same result can be obtained from equation (\[BO\]) with $\rho_0=(-7/8)\times (\pi^2/720 L^4)$. This is the scalar QED version of the unifying formula obtained by Pascual, Latorre and Tarrach for spinor QED [@Latorre], and also as in the spinor QED case, it corresponds to the weak field limit of a more general approach due to Dittrich and Gies [@Dittrich]. **Final remarks** {#final-remarks .unnumbered} ----------------- In this work we have endeavoured to give another example of the consequences of imposing boundary conditions on QED vacuum oscillations by discussing these oscillations confined by a somewhat unusual pair of plates. In particular, we have obtained through a simple regularization procedure the correlators for the vacuum oscillations of the electromagnetic field sandwiched between these plates, the associated Casimir energy density and the natural converse of the original Scharnhorst effect at zero temperature. 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--- abstract: 'Currently, thirty-nine astatine, thirty-nine radon, thirty-five francium, and thirty-four radium isotopes have so far been observed; the discovery of these isotopes is discussed. For each isotope a brief summary of the first refereed publication, including the production and identification method, is presented.' address: | National Superconducting Cyclotron Laboratory and\ Department of Physics and Astronomy, Michigan State University,\ East Lansing, MI 48824, USA author: - 'C. Fry' - 'M. Thoennessen' bibliography: - '../isotope-discovery-references.bib' title: 'Discovery of the astatine, radon, francium, and radium isotopes' --- Introduction {#s:intro} ============ The discovery of astatine, radon, francium, and radium isotopes is discussed as part of the series summarizing the discovery of isotopes, beginning with the cerium isotopes in 2009 [@2009Gin01]. Guidelines for assigning credit for discovery are (1) clear identification, either through decay-curves and relationships to other known isotopes, particle or $\gamma$-ray spectra, or unique mass and Z-identification, and (2) publication of the discovery in a refereed journal. The authors and year of the first publication, the laboratory where the isotopes were produced as well as the production and identification methods are discussed. When appropriate, references to conference proceedings, internal reports, and theses are included. When a discovery includes a half-life measurement the measured value is compared to the currently adopted value taken from the NUBASE evaluation [@2003Aud01] which is based on the ENSDF database [@2008ENS01]. In cases where the reported half-life differed significantly from the adopted half-life (up to approximately a factor of two), we searched the subsequent literature for indications that the measurement was erroneous. If that was not the case we credited the authors with the discovery in spite of the inaccurate half-life. All reported half-lives inconsistent with the presently adopted half-life for the ground state were compared to isomer half-lives and accepted as discoveries if appropriate following the criterium described above. The first criterium is not clear cut and in many instances debatable. Within the scope of the present project it is not possible to scrutinize each paper for the accuracy of the experimental data as is done for the discovery of elements [@1991IUP01]. In some cases an initial tentative assignment is not specifically confirmed in later papers and the first assignment is tacitly accepted by the community. The readers are encouraged to contact the authors if they disagree with an assignment because they are aware of an earlier paper or if they found evidence that the data of the chosen paper were incorrect. The discovery of several isotopes has only been reported in conference proceedings which are not accepted according to the second criterium. One example from fragmentation experiments why publications in conference proceedings should not be considered is $^{118}$Tc and $^{120}$Ru which had been reported as being discovered in a conference proceeding [@1996Cza01] but not in the subsequent refereed publication [@1997Ber01]. The initial literature search was performed using the databases ENSDF [@2008ENS01] and NSR [@2008NSR01] of the National Nuclear Data Center at Brookhaven National Laboratory. These databases are complete and reliable back to the early 1960’s. For earlier references, several editions of the Table of Isotopes were used [@1940Liv01; @1944Sea01; @1948Sea01; @1953Hol02; @1958Str01; @1967Led01]. A good reference for the discovery of the stable isotopes was the second edition of Aston’s book “Mass Spectra and Isotopes” [@1942Ast01]. For the isotopes of the radioactive decay chains several books and articles were consulted, for example, the 1908 edition of “Gmelin-Kraut’s Handbuch der anorganischen Chemie” [@1908Fri01], Soddy’s 1911 book “The chemistry of the radio-elements” [@1911Sod01], the 1913 edition of Rutherford’s book “Radioactive substances and their radiations” [@1913Rut01], and the 1933 article by Mary Elvira Weeks “The discovery of the elements. XIX. The radioactive elements” published in the Journal of Chemical Education [@1933Wee01]. In addition, the wikipedia page on the radioactive decay chains was a good starting point [@2011wik02]. The isotopes within the radioactive decay chains were treated differently. Their decay properties were largely measured before the concept of isotopes was established. Thus we gave credit to the first observation and identification of a specific activity, even when it was only later placed properly within in the decay chain. ![Original nomenclature of radon, radium, actinium, and thorium isotopes within the three natural occurring radioactive decay series. The grey squares connected by the grey arrows depict the activities labeled by Rutherford in his Bakerian lecture [@1905Rut01]. The black squares correspond to radioactive substances discovered later.[]{data-label="f:chain"}](chain2.pdf) Figure \[f:chain\] summarizes the isotopes of the three natural occurring radioactive decay series with their original nomenclature. This notation of the original substances introduced by Rutherford during his Bakerian lecture presented on May 19$^{th}$ 1904 and published a year later [@1905Rut01] are shown by grey squares and connected by the grey arrows representing $\alpha$ and $\beta$ decay. The decay from actinium to actinium X and from thorium to thorium X was later shown to be more complex. These isotopes are shown as black squares with the corresponding decays shown by black arrows. $^{191-229}$At ============== Astatine was discovered in 1940 by Corson et al. by bombarding a bismuth target with $\alpha$ particles [@1940Cor03]. A month later Minder reported the observation of element 85 naming it helvetium [@1940Min02] which was later shown to be incorrect [@1942Kar01]. Also a later claim by Leigh-Smith and Minder naming element 85 anglohelvetium [@1942Lei01] was not confirmed. An even earlier report of the discovery of element 85 by Allison et al. in 1931 [@1931All01] selecting the name alabamine was incorrect [@1935Mac01]. The name astatine was officially accepted at the 15$^{th}$ IUPAC conference in Amsterdam in 1949 [@1949IUC01]. Thirty-nine astatine isotopes from A = 191–229 have been discovered so far and according to the HFB-14 model [@2007Gor01] about 37 additional astatine isotopes could exist. Figure \[f:year-astatine\] summarizes the year of first discovery for all astatine isotopes identified by the method of discovery: radioactive decay (RD), fusion evaporation reactions (FE), light-particle reactions (LP), projectile fission or fragmentation (PF), and spallation (SP). In the following, the discovery of each astatine isotope is discussed in detail and a summary is presented in Table 1. The observation of $^{230}$At was reported in a preprint [@2010Ben01], however, the paper was never accepted for publication in a refereed journal. ![Astatine isotopes as a function of time when they were discovered. The different production methods are indicated.[]{data-label="f:year-astatine"}](astatine-year.pdf) $^{191}$At {#at-1 .unnumbered} ---------- Kettunen et al. reported the discovery of $^{191}$At in the 2003 paper “Alpha-decay studies of the new isotopes $^{191}$At and $^{193}$At” [@2003Ket01]. A $^{141}$Pr target was bombarded with 248$-$266 MeV $^{54}$Fe beams from the Jyväskylä K-130 cyclotron forming $^{191}$At in (4n) fusion-evaporation reactions. Recoil products were separated with the gas filled recoil separator RITU and implanted into a position sensitive Si detector which also measured subsequent $\alpha$ decay. “The corresponding mother activity with an alpha-decay energy E$_\alpha$ = 7552(11) keV and half-life T$_{1/2}$ = (1.7$^{+1.1}_{-0.5}$) ms was assigned to originate from the equivalent 1/2$^+$ state in $^{191}$At...” The quoted half-life is the currently accepted value. $^{192}$At {#at-2 .unnumbered} ---------- In the 2006 paper “$\alpha$-decay spectroscopy of the new isotope $^{192}$At”, Andreyev et al. announced the discovery of $^{192}$At [@2006And01]. A $^{144}$Sm target was bombarded with a 230 MeV $^{51}$V beam from the GSI UNILAC heavy ion accelerator producing $^{192}$At in the (3n) fusion-evaporation reaction. Recoil products were separated with the velocity filter SHIP and implanted in a 16-strip position-sensitive silicon detector which also measured subsequent $\alpha$ decay. “Two $\alpha$-decaying isomeric states with half-lives of 88(6) ms and 11.5(6) ms were identified in the new isotope $^{192}$At, both of them having complex decay paths to the excited states in the daughter nucleus $^{188}$Bi.” The quoted half-lives correspond to the currently accepted values for the ground state and an isomeric state, respectively. $^{193}$At {#at-3 .unnumbered} ---------- Kettunen et al. reported the discovery of $^{193}$At in the 2003 paper “Alpha-decay studies of the new isotopes $^{191}$At and $^{193}$At” [@2003Ket01]. A $^{141}$Pr target was bombarded with 264$-$272 MeV $^{56}$Fe beams from the Jyväskylä K-130 cyclotron forming $^{193}$At in (4n) fusion-evaporation reactions. Recoil products were separated with the gas filled recoil separator RITU and implanted into a position sensitive Si detector which also measured subsequent $\alpha$ decay. “The corresponding mother activity with the alpha-decay energy E$_\alpha$=7295(5) keV and half-life T$_{1/2}$=(28$^{+5}_{-4}$) ms was assigned to originate from the equivalent 1/2$^+$ state in $^{193}$At...” The quoted half-life is the currently accepted value. Previously, the observation of $^{193}$At was reported in a conference proceeding [@1995Lei01]. $^{194}$At {#at-4 .unnumbered} ---------- In 2009 Andreyev et al. reported the observation of $^{194}$At in the paper “$\alpha$ decay of $^{194}$At” [@2009And01]. $^{141}$Pr targets were bombarded with a 259 MeV $^{56}$Fe beam from the GSI UNILAC producing $^{194}$At in (3n) fusion-evaporation reactions. Residues were separated with the velocity filter SHIP and implanted in a 16-strip position-sensitive silicon detector which also recorded subsequent $\alpha$ decay. “Thus, two different half-life values for decays attributed to $^{194}$At identify two $\alpha$-decaying isomeric states in this nucleus. The 310(8) ms isomer decaying to $^{190}$Bi$^{m1}$ will further be denoted as $^{194}$At$^{m1}$ while the 253(10) ms isomer decaying to $^{190}$Bi$^{m2}$ will be denoted as $^{194}$At$^{m2}$.” These half-lives correspond to the currently accepted values for isomeric states. Previously, a half-life of 180(80) ms was reported in a conference proceeding [@1995Lei01]. $^{195}$At {#at-5 .unnumbered} ---------- Tagaya et al. reported the discovery of $^{195}$At in the 1999 paper “The $\alpha$-decay energies and halflives of $^{195g,m}$At and $^{199}$Fr” [@1999Tag01]. $^{169}$Tm targets were bombarded with a 215 MeV $^{36}$Ar beam from the RIKEN ring cyclotron to form $^{195}$At in ($\alpha$6n) fusion-evaporation reactions. Recoils were separated with the gas-filled recoil separator GARIS and implanted in a position sensitive detector which also recorded subsequent $\alpha$ decay. “We therefore assigned the corresponding $\alpha$1 events to the decay of $^{195g}$At, of which the E$_{\alpha}$ and T$_{1/2}$ values were determined to be 7105$\pm$30 keV and 146$^{+21}_{-17}$ ms.” Tagaya et al. also reported an 385$^{+69}_{-51}$ ms isomeric state which is currently assigned the ground state with a half-life of 328(20) ms. Previously, a half-life of 630$^{+320}_{-160}$ ms was reported in a conference proceeding [@1995Lei01]. $^{196-199}$At {#at-6 .unnumbered} -------------- Treytl and Valli identified $^{196}$At, $^{197}$At, $^{198}$At, and $^{199}$At in the 1967 article “Alpha decay of neutron deficient astatine isotopes” [@1967Tre01]. Enriched $^{185}$Re and $^{187}$Re targets were bombarded with 100–200 MeV $^{20}$Ne beams from the Berkeley HILAC. Reaction products were collected on a silver foil by a helium jet and rotated in front of a Si(Au) surface barrier detector. “ASTATINE-199: The peak at 6.638 MeV with a half-life of 7.2 sec clearly belongs to $^{199}$At... ASTATINE-198: The alpha peaks at 6.747 MeV with a half-life of 4.9 sec and at 6.847 MeV with 1.5 sec have excitation functions similar to $^{198}$Po. We have assigned the former one to the ground state and the second one to an isomeric state of $^{198}$At... ASTATINE-197: The peak at 6.957 MeV with a half-life of 0.4 sec belongs to $^{197}$At... ASTATINE-196: In subsequent runs, an alpha peak at 7.055 MeV with a half-life of 0.3 sec was observed. The excitation function shown in \[the figure\] clearly follows that of $^{196}$Po.” The measured half-lives of 0.3(1) s for $^{196}$At, 0.4(1) s for $^{197}$At, 4.9(5) s for $^{198}$At, and 7.2(5) s for $^{199}$At, are close to the currently accepted values of 0.388(7) s, 0.350(40) s, 3.8(4) s, and 7.03(15) s, respectively. $^{200-201}$At {#at-7 .unnumbered} -------------- In the 1963 paper “Alpha decay of neutron-deficient astatine isotopes”, Hoff et al. reported the first observation of $^{200}$At and $^{201}$At [@1963Hof01]. A gold foil was bombarded with a $^{12}$C beam with energies up to 125 MeV from the Berkeley Hilac. Alpha-particle spectra were measured with a 180$^\circ$ double-focusing spectrograph. “An $\alpha$-emitter with a half-life of 1.5$\pm$0.1 min and an $\alpha$-particle energy of 6.342$\pm$0.006 MeV has been assigned to $^{201}$At... Two $\alpha$-groups with a half-life 0.9$\pm$0.2 min and energies of 6.412$\pm$0.009 and 6.465$\pm$0.011 MeV have been tentatively assigned to $^{200}$At.” These half-lives of 0.9(2) min for $^{200}$At and 1.5(1) min for $^{201}$At agree with the currently accepted values of 43(1) s and 85.2(16) s, respectively. Earlier, Barton et al. reported half-lives of 43 s and 1.7 min, but were only able to assign them to astatine isotopes with A$<$202 and A$<$203, respectively [@1951Bar01]. $^{202}$At {#at-8 .unnumbered} ---------- The paper “$\alpha$-particle branching ratios for neutron-deficient astatine isotopes” by Latimer et al. reported the observation of $^{202}$At in 1961 [@1961Lat01]. Gold and platinum foils were irradiated with 50$-$125 MeV $^{12}$C and 65$-$130 MeV $^{14}$N beams, respectively, from the Berkeley HILAC. Alpha-particle spectra were measured with a gridded ionization chamber following chemical separation. “Using the reported $\alpha$-branching ratio of 0.02 for $^{202}$Po, we have calculated an alpha-branching ratio of 0.120$\pm$0.008 for $^{202}$At, corresponding to a partial $\alpha$-half-life of 25 min... The over-all half-lives observed are in agreement with those reported by Hoff et al. [@1959Hof01]...” The overall half-life quoted for $^{202}$At was 3.0(2) min which agrees with the currently accepted half-life of 184(1) s. The reference to Hoff et al. corresponds to a conference abstract. Hoff et al. published their results in a refereed journal two years later [@1963Hof01]. Also, about three months later Forsling et al. independently reported a 3(1) min half-life for $^{202}$At [@1961For01]. $^{203}$At {#at-9 .unnumbered} ---------- $^{203}$At was identified by Barton et al. and published in the 1951 paper “Radioactivity of astatine isotopes” [@1951Bar01]. $^{209}$Bi was irradiated with $^4$He beams of up to 380 MeV from the Berkeley 184-in. cyclotron. Alpha spectra were recorded with an alpha-pulse analyzer following chemical separation. “For the present we shall assume the 7-min 6.10-Mev group to be At$^{203}$ and designate the 6.35-Mev group with 1.7-min half-life as At$^{<203}$.” This value agrees with the currently adopted value of 7.4(2) min. About three months earlier Miller et al. [@1950Mil01] measured an 11 min half-life by bombarding a gold target with a $^{13}$C beam and suggested the possibility that they had formed the 7 min $^{203}$At activity based on a private communication with Barton et al. $^{204}$At {#at-10 .unnumbered} ---------- The paper “$\alpha$-particle branching ratios for neutron-deficient astatine isotopes” by Latimer et al. reported the observation of $^{204}$At in 1961 [@1961Lat01]. Gold and platinum foils were irradiated with 50$-$125 MeV $^{12}$C and 65$-$130 MeV $^{14}$N beams, respectively, from the Berkeley HILAC. Alpha-particle spectra were measured with a gridded ionization chamber following chemical separation. “In this study, an $\alpha$-group of 5.95 MeV energy and half-life of 9$\pm$1 min has been observed. Excitation functions support the assignment of this activity to $^{204}$At.” This value agrees with the currently accepted value of 9.12(11) min. An earlier report of a 22 min half-life [@1951Bar01] was evidently incorrect. Also, about three months later Forsling et al. independently reported a 9(3) min half-life [@1961For01] and in 1959 Hoff et al. had reported a half-life of 9.3(2) min in a conference abstract [@1959Hof01]. $^{205}$At {#at-11 .unnumbered} ---------- $^{205}$At was identified by Barton et al. and published in the 1951 paper “Radioactivity of astatine isotopes” [@1951Bar01]. $^{209}$Bi was irradiated with $^4$He beams of up to 380 MeV from the Berkeley 184-in. cyclotron. Alpha spectra were recorded with an alpha-pulse analyzer following chemical separation. “For the present, we shall attribute the alpha-particle, which was found to decay with a 25-min half-life to At$^{205}$.” This value is consistent with the currently adopted value of 26.9(8) min. About three months earlier Miller et al. [@1950Mil01] measured an 25 min half-life by bombarding a gold target with a $^{13}$C beam and suggested the possibility that they had formed the 24 min $^{205}$At activity based on a private communication with Barton et al. $^{206}$At {#at-12 .unnumbered} ---------- The paper “$\alpha$-particle branching ratios for neutron-deficient astatine isotopes” by Latimer et al. reported the observation of $^{206}$At in 1961 [@1961Lat01]. Gold and platinum foils were irradiated with 50$-$125 MeV $^{12}$C and 65$-$130 MeV $^{14}$N beams, respectively, from the Berkeley HILAC. Alpha-particle spectra were measured with a gridded ionization chamber following chemical separation. “A least-squares analysis of several of the curves for which the statistics were good yielded a value of 29.5$\pm$0.6 min for the half-life of $^{206}$At.” This value agrees with the currently accepted value of 30.6(8) min. An earlier report of a 2.6 h half-life [@1951Bar01] was evidently incorrect. Also, about three months later Forsling et al. independently reported a 20(10) min half-life [@1961For01] and in 1959 Hoff et al. had reported a half-life of 31.0(15) min in a conference abstract [@1959Hof01]. $^{207}$At {#at-13 .unnumbered} ---------- $^{207}$At was identified by Barton et al. and published in the 1951 paper “Radioactivity of astatine isotopes” [@1951Bar01]. $^{209}$Bi was irradiated with $^4$He beams of up to 380 MeV from the Berkeley 184-in. cyclotron. Alpha spectra were recorded with an alpha-pulse analyzer following chemical separation. “At 75 Mev a new activity appeared having a half-life of about 2 hr, and this has been assigned to At$^{207}$ formed by the ($\alpha$,6n) reaction.” This value agrees with the currently adopted value of 1.81(3) h. $^{208}$At {#at-14 .unnumbered} ---------- In 1950 Hyde et al. reported the first observation of $^{208}$At in the paper “Low mass francium and emanation isotopes of high alpha-stability” [@1950Hyd01]. Thorium foils were bombarded with up to 350 MeV protons from the Berkeley 184-inch cyclotron. $^{212}$Fr was chemically separated and $^{208}$At was populated by $\alpha$-decay. Alpha spectra were measured with an ionization chamber. “High volatility is characteristic of astatine, and this 5.65 Mev activity was judged to be the At$^{208}$ daughter of Fr$^{212}$.” The measured half-life of 1.7 h agrees with the currently adopted value of 1.63(3) h. $^{209}$At {#at-15 .unnumbered} ---------- $^{209}$At was identified by Barton et al. and published in the 1951 paper “Radioactivity of astatine isotopes” [@1951Bar01]. $^{209}$Bi was irradiated with $^4$He beams of up to 380 MeV from the Berkeley 184-in. cyclotron. Alpha spectra were recorded with an alpha-pulse analyzer following chemical separation. “An activity assigned to At$^{209}$ is characterized by a half-life of 5.5$\pm$0.3 hr and an alpha-particle of 5.65 Mev.” This half-life agrees with the currently adopted value of 5.41(5) h. $^{210}$At {#at-16 .unnumbered} ---------- Kelly and Segre first observed $^{210}$At and reported their results in the 1949 paper “Some excitation functions of bismuth” [@1949Kel01]. Bismuth targets were bombarded with 29 MeV $^4$He beams from the Berkeley 60-inch cyclotron. Resulting activities were measured with a parallel plate ionization chamber. “Careful investigation, which will be discussed in detail later, showed that the Po$^{210}$ came from the Bi($\alpha$,3n) reaction producing At$^{210}$ which in turn decays to Po$^{210}$ by orbital electron capture, with a half-life of 8.3 hr.” This value is included in the calculation of the currently accepted half-life of 8.1(4) h. $^{211}$At {#at-17 .unnumbered} ---------- The discovery of $^{211}$At was reported in “Artificially radioactive element 85” by Corson et al. in 1940 [@1940Cor01]. The Berkeley 60-inch cyclotron was used to bombard a bismuth target with 32 MeV alpha particles. Alpha particles, gamma-, and x-rays were measured following chemical separation. “All these radiations separate together chemically as element 85, and all show the same half-life of 7.5 hours. The probable explanation of these effects is the following: Bi$^{209}$, by an ($\alpha$,2n) reaction, goes to 85$^{211}$, which decays either by K-electron capture to actinium C’(Po$^{211}$) or by alpha-particle emission (range 4.5 cm) to Bi$^{207}$.” The measured half-life agrees with the currently accepted value of 7.214(7) h. The discovery of the element astatine in this experiment had been published earlier without a mass assignment [@1940Cor03]. $^{212}$At {#at-18 .unnumbered} ---------- Winn reported the observation of $^{212}$At in the 1954 paper “Short-lived alpha emitters produced by $^3$He and heavy ion bombardments” [@1954Win01]. 28 MeV $\alpha$ particles from the Birmingham cyclotron bombarded a bismuth target forming $^{212}$Bi in the reaction $^{209}$Bi($\alpha$,n). The alpha activity was measured with a zinc sulphide screen attached to a light guide and a magnetically shielded phototube. Results were summarized in a table, quoting an observed half-life of 0.22(3) s, which is close to the currently accepted value of 0.314(2) s. Winn did not consider this observation a new discovery referring to the 1948 Table of Isotopes which listed a half-life of 0.25 s based on a private communication [@1948Sea01]. $^{213}$At {#at-19 .unnumbered} ---------- In the 1968 article “New neptunium isotopes, $^{230}$Np and $^{229}$Np” Hahn et al. reported the observation of $^{213}$At [@1968Hah01]. Enriched $^{233}$U targets were bombarded with 32$-$41.6 MeV protons from the Oak Ridge Isochronous Cyclotron forming $^{229}$Np in (p,5n) reactions, respectively. Reaction products were implanted on a catcher foil which was periodically rotated in front of a surface barrier Si(Au) detector. Isotopes populated by subsequent $\alpha$ emission were measured. “The $\alpha$-particle energies found for the $^{225}$Pa series are more precise than the previously available values: $^{225}$Pa, 7.25$\pm$0.02 MeV (new value); $^{221}$Ac, 7.63$\pm$0.02 MeV; $^{217}$Fr, 8.31$\pm$0.02 MeV and $^{213}$At, 9.06$\pm$0.02 MeV.” The observation of $^{213}$At was not considered new, referring to an unpublished thesis [@1951Key01]. The currently accepted half-life is 125(6) ms. $^{214}$At {#at-20 .unnumbered} ---------- Meinke et al. reported the observation of $^{214}$At in the 1949 paper “Three additional collateral alpha-decay chains” [@1949Mei01]. Thorium was bombarded by 150 MeV deuterons from the Berkeley 184-inch cyclotron. Alpha-decay chain from $^{226}$Pa was measured following chemical separation. “Although the mass type has not yet been identified through known daughters as above, general considerations with regard to the method of formation and half-life of the parent substance, and the energies of all the members of the series suggest a collateral branch of the 4n+2 family: $_{91}$Pa$^{226}\stackrel{\alpha}{\longrightarrow}_{89}$Ac$^{222}\stackrel{\alpha}{\longrightarrow}_{87}$Fr$^{218}\stackrel{\alpha}{\longrightarrow}_{85}$Fr$^{214}\stackrel{\alpha}{\longrightarrow}_{83}$Bi$^{210}$(RaE).” In a table summarizing the energies and half-lives of the decay chain only the $\alpha$-decay energy was given for $^{214}$At stating a calculated half-life of 10$^{-6}$ s. The currently accepted half-life is 558(10) ns. $^{215}$At {#at-21 .unnumbered} ---------- In the 1944 paper “Das Element 85 in der Actiniumreihe”, Karlik and Bernert reported the first observation of $^{215}$At [@1944Kar01]. The range of $\alpha$ particles from a actinium emanation source was measured with an ionization chamber. “Wir fanden in einem Verhältnis von ungefähr 5$\cdot$10$^{-6}$ zur Actinium A-Strahlung eine $\alpha$-Strahlung mit einer extrapolierten Reichweite von 8,0 cm (15$^\circ$, 760 mm), was 8,4 MeV Zerfallsenergie entspricht. Dieser Betrag steht in sehr guter Übereinstimmung mit dem Wert, der sich ergibt, wenn man in dem Diagramm der Zerfallsenergie als Funktion der Massenzahl die Kurve von der Ordnungszahl 85 bis zur Massenzahl 215 extrapoliert, so daß uns die Zuordnung der neuen $\alpha$-Strahlung zu dem Kern 85$^{215}$ (entstanden aus Ac A durch $\beta$-Zerfall) berechtigt erscheint.” \[We found an $\alpha$ radiation with a ratio of approximately 5$\cdot$10$^{-6}$ relative to the actinium A radiation which has an extrapolated range of 8.0 cm (15$^\circ$, 760 mm), corresponding to a decay energy of 8.4 MeV. This value agrees very well with the extrapolated value for mass number 215 in a plot of the decay energy as a function of the mass number for atomic number 85. Thus it is reasonable to assign the new $\alpha$ radiation to the nuclide 85$^{215}$ (produced by $\beta$ decay from Ac A).\] The presently adopted half-life is 100(20) $\mu$s. $^{216}$At {#at-22 .unnumbered} ---------- In “Artificial collateral chains to the thorium and actinium families,” Ghiorso et al. discovered $^{216}$At in 1948 [@1948Ghi01]. Thorium targets were irradiated with 80 MeV deuterons from the Berkeley 184-inch cyclotron. The $\alpha$-decay chain beginning at $^{228}$Pa was measured following chemical separation. “After the decay of the above-described series, a second group of alpha-particle emitters can be resolved. This second series, which decays with the 22-hour half-life of its protactinium parent, is a collateral branch of the 4n radioactive family as follows: $_{91}$Pa$^{228}\stackrel{\alpha}{\longrightarrow}_{89}$Ac$^{224}\stackrel{\alpha}{\longrightarrow}_{87}$Fr$^{220}\stackrel{\alpha}{\longrightarrow}_{85}$At$^{216}\stackrel{\alpha}{\longrightarrow}$...” The measured half-life of about 10$^{-3}$ s is consistent with the presently adopted value of 0.3 ms. In 1940, Minder [@1940Min02] and later in 1942, Leigh-Smith and Minder [@1942Lei01] had reported the observation of $^{216}$At $\beta$-decay which was evidently incorrect [@1942Kar01]. Also the observation of $^{216}$At in the natural thorium radioactive decay chain [@1943Kar01] was not correct [@1950Per01]. $^{217}$At {#at-23 .unnumbered} ---------- Hagemann et al. discovered $^{217}$At in 1947 in “The (4n+1) radioactive series: the decay products of U$^{233}$” [@1947Hag01]. The half-lives and $\alpha$- and $\beta$-decay energies of the nuclides in the decay chain of $^{233}$U were measured. “These decay products, which constitute a substantial fraction of the entire missing, 4n+1, radioactive series are listed together with their radioactive properties, in \[the table\].” The measured half-life of 18 ms is within a factor of two of the presently accepted value of 32.3(4) ms. Hagemann et al. acknowledge the simultaneous observation of $^{217}$At by English et al. which was submitted only a day later and published in the same issue of Physical Review on the next page [@1947Eng01]. $^{218}$At {#at-24 .unnumbered} ---------- $^{218}$At was identified by Karlik and Bernert in the 1943 paper “Eine neue natürliche $\alpha$-Strahlung” [@1943Kar01]. The range of $\alpha$ particles from a radium A source was measured with an ionization chamber. “Eine $\beta$-Umwandlung von Radium A würde zu einem Isotop des Elementes 85 von der Massenzahl 218 führen... Die entsprechende Energie beträgt 6.6$_3$ MeV, bzw. die gesamte Zerfallsenergie 6,7$_5$ MeV. Aus der Geiger-Nutallschen Beziehung würde sich daraus eine Halbwertszeit in der Großenordnung von Sekunden ableiten, was mit unseren Beobachtungen im Einklang steht.” \[A potential $\beta$ decay of radium A would lead to an isotope of element 85 with a mass number of 218... The corresponding energy is 6.6$_3$ MeV, corresponding to a total decay energy of 6.7$_5$ MeV. From this energy a half-life on the order of seconds can be derived from the Geiger-Nutall relation which is consistent with our observations.\] The currently adopted half-life for $^{218}$At is 1.5(3) s. $^{219}$At {#at-25 .unnumbered} ---------- In 1953 $^{219}$At was first reported by Hyde and Ghiorso in “The alpha-branching of AcK and the presence of astatine in nature” [@1953Hyd01]. A 20-mC $^{227}$Ac source was used to study the nuclide of the 4n+3 decay series by chemical and physical separation and measuring the radioactivity with an alpha-ray differential pulse analyzer. “The observed branching rate is ca 4$\times$10$^{-5}$, and the At$^{219}$ daughter decays predominantly by the emission of 6.27 Mev alpha-particles with a half-life of 0.9 minute to the new isotopes Bi$^{215}$, which in turn emits $\beta^-$ particles with a half-life of 8 minutes.” The measured half-life of 0.9 min for $^{219}$At is included in the calculated average of the currently adopted value of 56(3) s. $^{220}$At {#at-26 .unnumbered} ---------- In 1989 Liang et al.  reported the first observation of $^{220}$At in “A new isotope $_{85}^{220}$At$_{135}$” [@1989Lia01]. Thorium oxide was bombarded with 200 MeV protons from the Orsay synchrocyclotron. $^{220}$At was separated with the ISOCELE II on-line mass separator and transported to a measuring station consisting of a 4$\pi$ $\beta$-detector and two Ge(Li) detectors. “A new isotope $^{220}$At has been identified among the mass-separated products of a spallation reaction of ThO$_2$. Its half-life has been found to be 3.71$\pm$0.04 min.” This half-life is the currently adopted value. Less than three months later, Burke et al. independently reported a half-life of 3.73(13) min [@1989Bur01]. $^{221-223}$At {#at-27 .unnumbered} -------------- In the 1989 paper “New neutron-rich isotopes of astatine and bismuth” Burke et al. described the observation of $^{221}$At, $^{222}$At and $^{223}$At [@1989Bur01]. A thorium/tantalum metal-foil target was bombarded with 600 MeV protons from the CERN synchro-cyclotron. Astatine isotopes were produced in spallation reactions and separated with the ISOLDE-II on-line separator. Beta-ray spectra were measured with a 4$\pi$ plastic scintillator. “Multiscaling of the 4$\pi$ plastic scintillator signal gave a half-life of 2.3(2) min. This can be assigned to $^{221}$At... At mass A=222, an activity with a half-life of 54(10) s has been observed and assigned to $^{222}$At... The most neutron-rich astatine isotope seen in the present experiment was $^{223}$At. Its half-life was measured to be 50(7) s.” The measured half-lives of 2.3(2) min, 54(10) s, and 50(7) s for $^{221}$At, $^{222}$At and $^{223}$At, respectively, are the currently accepted values. $^{224}$At {#at-28 .unnumbered} ---------- In the 2010 paper “Discovery and investigation of heavy neutron-rich isotopes with time-resolved Schottky spectrometry in the element range from thallium to actinium”, Chen et al. described the discovery of $^{224}$At [@2010Che01]. A beryllium target was bombarded with a 670 MeV/u $^{238}$U beam from the GSI heavy-ion synchrotron SIS and projectile fragments were separated with the fragment separator FRS. The mass and half-life of $^{224}$At was measured with time-resolved Schottky Mass Spectrometry in the storage-cooler ring ESR. “In \[the figure\] time traces and their projection into a frequency spectrum are shown for the new isotope $^{224}$At and close-lying ions.” The quoted half-life of 76$^{+138}_{-23}$ s is currently the only measured value for $^{224}$At. $^{225-229}$At {#at-29 .unnumbered} -------------- $^{225}$At, $^{226}$At, $^{227}$At, $^{228}$At, and $^{229}$At were discovered by Alvarez-Pol and the results were published in the 2010 paper “Production of new neutron-rich isotopes of heavy elements in fragmentation reactions of $^{238}$U projectiles at 1A GeV” [@2010Alv01]. A beryllium target was bombarded with a 1 A GeV $^{238}$U beam from the GSI SIS synchrotron. The isotopes were separated and identified with the high-resolving-power magnetic spectrometer FRS. “To search for new heavy neutron-rich nuclei, we tuned the FRS magnets for centering the nuclei $^{227}$At, $^{229}$At, $^{216}$Pb, $^{219}$Pb, and $^{210}$Au along its central trajectory. Combining the signals recorded in these settings of the FRS and using the analysis technique previously explained, we were able to identify 40 new neutron-rich nuclei with atomic numbers between Z=78 and Z=87; $^{205}$Pt, $^{207-210}$Au, $^{211-216}$Hg, $^{214-217}$Tl, $^{215-220}$Pb, $^{219-224}$Bi, $^{223-227}$Po, $^{225-229}$At, $^{230,231}$Rn, and $^{233}$Fr.” $^{193-231}$Rn ============== Although it is generally accepted that the element radon was discovered by Dorn in 1900 the references to the original papers is not straightforward [@1957Par01]. Also arguments have been made to credit Rutherford [@2003Mar01] or M. and P. Curie [@2000Bre01] with the discovery of radon. In 1923, the International Union of Pure and Applied Chemistry (IUPAC) named the three known emanations - radium, actinium, and thorium - radon (Rn), actinon (An), and thoron (Tn), respectively. Ramsay and Gray had suggested the name niton for radium emanation [@1910Ram01] in 1910. Later radon (Rn) was adopted for all radon isotopes, however, the name emanation with the symbol Em was commonly in use for a long time, see for example reference [@1967Val01]. Thirty-nine radon isotopes from A = 193–231 have been discovered so far and according to the HFB-14 model [@2007Gor01] about 48 additional radon isotopes could exist. Figure \[f:year-radon\] summarizes the year of first discovery for all radon isotopes identified by the method of discovery: radioactive decay (RD), fusion evaporation reactions (FE), light-particle reactions (LP), projectile fission or fragmentation (PF), and spallation (SP). In the following, the discovery of each radon isotope is discussed in detail and a summary is presented in Table 1. The observation of $^{232}$Rn was reported in a preprint [@2010Ben01], however, the paper was never accepted for publication in a refereed journal. ![Radon isotopes as a function of time when they were discovered. The different production methods are indicated.[]{data-label="f:year-radon"}](radon-year.pdf) $^{193,194}$Rn {#rn-1 .unnumbered} -------------- Andreyev et al. reported the first observation of $^{193}$Rn and $^{194}$Rn in the 2006 paper “$\alpha$ decay of the new isotopes $^{193,194}$Rn” [@2006And02]. A $^{144}$Sm target was bombarded with 231$-$252 MeV $^{52}$Cr beams from the GSI UNILAC forming $^{193}$Rn and $^{194}$Rn in the (3n) and (2n) fusion-evaporation reactions, respectively. Recoil products were separated with the velocity filter SHIP and implanted into a position-sensitive silicon detector which also recorded subsequent $\alpha$ decay. “By using all 26 full-energy correlated recoil-$\alpha_1$ decays a half-life of T$_{1/2}$=0.78(16) ms was deduced for $^{194}$Rn... The half-life of T$_{1/2}$($^{193}$Rn)=1.15(27) ms was deduced from 19 full-energy recoil-$\alpha_1$(7670 keV-7890 keV) decays, which includes 16 events with the full-energy deposition in the PSSD and 3 events in which the energy was shared between the PSSD and BOX detectors.” Both of these half-lives are the currently adopted values. $^{195}$Rn {#rn-2 .unnumbered} ---------- The discovery of $^{195}$Rn by Kettunen et al. was reported in the 2001 paper “$\alpha$ decay studies of the nuclides $^{195}$Rn and $^{196}$Rn” [@2001Ket01]. A $^{142}$Nd target was bombarded with 239$-$267 MeV $^{56}$Fe beams from the Jyväskylä K-130 cyclotron producing $^{195}$Rn in the (3n) fusion-evaporation reaction. Recoil products were separated with the gas-filled recoil separator RITU and implanted into a position sensitive silicon detector which also measured subsequent $\alpha$ decay. “Two $\alpha$ decaying isomeric states, with E$_\alpha$=7536(11) keV \[T$_{1/2}$=(6$^{+3}_{-2}$) ms\] for the ground state and E$_\alpha$=7555(11) keV \[T$_{1/2}$=(5$^{+3}_{-2}$) ms\] for an isomeric state were identified in $^{195}$Rn.” These half-lives are the currently accepted values. $^{196,197}$Rn {#rn-3 .unnumbered} -------------- In the 1995 article “New $\alpha$-decaying neutron deficient isotopes $^{197}$Rn and $^{200}$Fr,” Morita et al. announced the identification of $^{196}$Rn and $^{197}$Rn [@1995Mor01]. A 273.6 MeV $^{36}$Ar beam from the RIKEN ring cyclotron bombarded an enriched $^{166}$Er target forming $^{196}$Rn and $^{197}$Rn in (6n) and (5n) fusion-evaporation reactions, respectively. Reaction products were separated with the gas-filled recoil separator GARIS and implanted in a position-sensitive silicon detector which also measured subsequent $\alpha$ decay. “The $\alpha$-decay energies (half-lives) of $^{197}$Rn, $^{197m}$Rn and $^{200}$Fr have been determined to be 7261$\pm$30 keV (51$^{+35}_{-15}$ ms), 7370$\pm$30 keV (18$^{+9}_{-5}$ ms), and 7500$\pm$30 keV, (570$^{+270}_{-140}$ ms), respectively.” Only one $\alpha$-decay event was observed for $^{196}$Rn with 5 ms between the implant and the $\alpha$ particle. The same group reported the half-life of $^{196}$Rn as 3$^{+7}_{-2}$ ms which agrees with the presently accepted value of 4.4$^{+1.3}_{-0.9}$ ms a year later [@1997Pu01]. The measured half-life of 51$^{+35}_{-15}$ ms for $^{197}$Rn agrees with the present value of 65$^{+25}_{-14}$ ms. Three months later Enquist et al. [@1996Enq01] independently reported the observation of the isomeric state which agreed with the value of Morita et al. $^{198}$Rn {#rn-4 .unnumbered} ---------- The discovery of $^{198}$Rn was published in the 1984 paper “Alpha decay of $^{198}$Rn” by Calaprice et al. [@1984Cal01]. Thorium hydroxide targets were bombarded with 600 MeV protons from the CERN synchrocyclotron forming $^{198}$Rn in spallation reactions. Decay curves of $^{198}$Rn were measured following isotope separation with the online mass separator ISOLDE. “The new nuclide $^{198}$Rn was found to have an $\alpha$-decay energy of 7196$\pm$10 keV and a half-life of 50$\pm$9 ms.” This half-life agrees with the currently accepted value of 65(3) ms. $^{199}$Rn {#rn-5 .unnumbered} ---------- In 1980, DiRienzo et al. reported the observation of $^{199}$Rn in “New isotope $^{199}$Rn and evidence for an isomeric state $^{199}$Rn$^m$” [@1980DiR01]. A 200 MeV $^{35}$Cl beam from the BNL three-stage Tandem Accelerator bombarded a $^{169}$Tm target forming $^{199}$Rn in the (5n) fusion-evaporation reaction. Recoil products were separated with a zero-degree recoil separator and implanted in a surface barrier detector which also measured subsequent $\alpha$ decay. “The other two lines at 6.990$\pm$0.015 MeV and 7.060$\pm$0.012 MeV are assigned to a new isotope $^{199}$Rn.” The currently accepted half-life is 0.59(3) s. $^{200}$Rn {#rn-6 .unnumbered} ---------- Hornshoj et al. reported the identification of $^{200}$Rn in “Alpha decay of neutron-deficient radon and polonium isotopes” in 1971 [@1971Hor01]. Th(OH)$_4$ targets were bombarded with 600 MeV protons from the CERN synchrocyclotron forming $^{200}$Rn in spallation reactions. Alpha-decay spectra were measured following isotope separation with the online mass separator ISOLDE. “$^{200}$Rn decays by an $\alpha$-group of energy 6.909$\pm$0.008 MeV, see \[the figure\]. The half-life was found to be 1.0$\pm$0.2 s.” This value is included in the calculation of the currently accepted value. $^{201-205}$Rn {#rn-7 .unnumbered} -------------- Valli et al. reported the discovery of $^{201}$Rn, $^{202}$Rn, $^{203}$Rn, $^{204}$Rn, and $^{205}$Rn in the 1967 article “Alpha-decay properties of neutron-deficient isotopes of emanation” [@1967Val01]. Platinum, gold, mercury, and thallium targets were bombarded with $^{16}$O, $^{14}$N, and $^{12}$C beams from the Berkeley HILAC. Alpha-particle spectra were measured with a Si(Au) detector following chemical separation. “Emanation-201: ...The most prominent of the groups, at 6.768 MeV, had a half-life of 3$\pm$1 sec. We tentatively assign it to $^{201}$Em on the following incomplete evidence... Emanation-202: ...By examination of several spectra taken at 15-sec intervals, the half-life was determined to be 13$\pm$2 sec... The excitation function leads to a mass assignment of 202... Emanation-203 and Emanation-203m: ...We assign the 45-sec 6.497 MeV activity to the ground state of $^{203}$Em and the 28-sec 6.547 MeV activity to an isomeric state as this choice fits best in the energy-versus-mass-number curve... Emanation-204: An $\alpha$ activity at 6.416 MeV with a half-life of 75$\pm$2 sec was prominent in the emanation fraction from gold targets bombarded with $^{14}$N or $^{16}$O nuclei or from platinum targets bombarded with $^{16}$O nuclei... the agreement of the $\alpha$ energy with the approximate value to be expected from systematic trends in $\alpha$-decay energies confirm the assignment of the new activity to $^{204}$Em... Emanation-206 and Emanation-205: ...From an analysis of many decay curves of the 6.260-MeV $\alpha$ group we found a two-component mixture with half-life periods of 6.5$\pm$1 min and 1.8$\pm$0.5 min. The longer-lived component corresponds to the $^{206}$Em reported by Stoner and Hyde [@1957Sto01]. The 1.8-min period can be assigned to the previously unknown $^{205}$Em from arguments based on our excitation function results.” The measured half-lives of 3.0(15) s ($^{201}$Rn), 13(2) s ($^{202}$Rn), 45(5) s ($^{203}$Rn), 75(2) s ($^{204}$Rn), and 1.8(5) min ($^{205}$Rn) agree with the presently adopted values of 3.8(1) s, 9.7(1) s, 44(2) s, 74.5(14) s, and 170(4) s, respectively. The value for $^{201}$Rn corresponds to an isomeric state. Earlier, Stoner and Hyde had reported a 3 min half-life and assigned it to either $^{204}$Rn or $^{205}$Rn [@1957Sto01]. $^{206,207}$Rn {#rn-8 .unnumbered} -------------- In the 1954 paper “The $\alpha$-activity induced in gold by bombardment with nitrogen ions,” Burcham described the identification of $^{206}$Rn and $^{207}$Rn [@1954Bur01]. Gold foils were bombarded with a 75–120 MeV nitrogen beam from the Birmingham Nuffield 60-inch cyclotron forming $^{206}$Rn and $^{207}$Rn in the fusion-evaporation reactions $^{197}$Au($^{14}$N,5n) and $^{197}$Au($^{14}$N,4n), respectively. Alpha-decay curves of the irradiated samples were measured with an ionization chamber. “Assignment of the 6.25 MeV group of $\alpha$-particles to $^{206}$Em is based on predictions from $\alpha$-decay systematics... The 6.09 MeV group of $\alpha$-particles could come from $^{207}$Em according to the systematics.” The measured half-lives of 6.5(5) min for $^{206}$Rn and 11.0(10) min for $^{207}$Rn are close to the currently adopted values of 5.67(17) min and 9.25(17) min, respectively. $^{208}$Rn {#rn-9 .unnumbered} ---------- Momyer and Hyde reported the observation of $^{208}$Rn in the 1955 article “The influence of the 126-neutron shell on the alpha-decay properties of the isotopes of emanation, francium, and radium” [@1955Mom01]. Thorium foils were bombarded with 340 MeV protons from the Berkeley 184-inch cyclotron. Alpha-particle spectra and decay curves were measured with an ionization chamber following chemical separation. “In summary, Em$^{208}$ appears to be a 23$\pm$2-minute activity with alpha-particle energy 6.141 MeV.” This half-life agrees with the currently adopted value of 24.35(14) min. In a companion paper actually submitted a day earlier, Momyer et al. measured the $\alpha$-decay energies in a magnetic spectrograph [@1955Mom02]. $^{209-211}$Rn {#rn-10 .unnumbered} -------------- Momyer et al. identified $^{209}$Rn, $^{210}$Rn, and $^{211}$Rn in “Recent studies of the isotopes of emanation, francium and radium” in 1952 [@1952Mom01]. Thorium targets were bombarded with 340 MeV protons from the Berkeley 184-inch cyclotron. Alpha-decay spectra were measured following chemical separation. Results were summarized in a table, assigning half-lives of 31 min, 2.7 h, and 16 h to $^{209}$Rn, $^{210}$Rn, and $^{211}$Rn, which agree with the currently accepted half-lives of 28.5(10) min, 2.4(1) h, and 14.6(2) h, respectively. Half-lives of 23 min and 2.1 h had been previously reported without firm mass assignments [@1949Ghi01]. $^{212}$Rn {#rn-11 .unnumbered} ---------- In 1950 Hyde et al. reported the first observation of $^{212}$Rn in the paper “Low mass francium and emanation isotopes of high alpha-stability” [@1950Hyd01]. Thorium foils were bombarded with up to 350 MeV protons from the Berkeley 184-inch cyclotron. $^{212}$Fr was chemically separated and $^{212}$Rn was populated by electron capture. Alpha spectra were measured with an ionization chamber. “Em$^{212}$ is shown to be a 23-minute alpha-emitter.” This agrees with the currently adopted half-life of 23.9(12) min. The same group had reported this activity previously without a mass assignment [@1949Ghi01]. $^{213}$Rn {#rn-12 .unnumbered} ---------- Rotter et al. observed $^{213}$Rn in 1967 and reported their results in the paper “The new isotope Ac$^{216}$” [@1967Rot01]. A 78 MeV $^{12}$C beam from the Dubna 1.5 m cyclotron bombarded a lead target forming radium in (xn) reactions. $^{213}$Rn was populated by $\alpha$ decay of $^{217}$Ra. Recoil nuclei were collected on an aluminum foil and $\alpha$-particle spectra were measured with a silicon surface barrier detector. “We obtained the following $\alpha$-particle energies: Rn$^{213}$ - 8.14 MeV, Fr$^{214}$ - 8.53 MeV, and Ra$^{215}$ - 8.73 MeV.” Rotter et al. did not consider this observation a new discovery referring to a conference abstract [@1962Gri01]. $^{214}$Rn {#rn-13 .unnumbered} ---------- In 1970 Torgerson and MacFarlane reported the first observation of $^{214}$Rn in “Alpha decay of the $^{221}$Th and $^{222}$Th decay chains” [@1970Tor01]. A 10.6 MeV/nucleon $^{16}$O beam from the Yale heavy ion accelerator was used to bombard a $^{208}$Pb target forming $^{222}$Th in (2n) fusion-evaporation reactions. $^{214}$Rn was then populated by subsequent $\alpha$ decays. Recoil products were transported to a stainless steel surface with a helium jet and $\alpha$ spectra were measured with a Si(Au) surface barrier detector. “However, at $^{16}$O incident energies below 80 MeV, the 9.040 MeV group could be clearly resolved as shown in \[the figure\].” Only three days later Valli et al. submitted their measurement of a 9.035(10) MeV $\alpha$ energy assigned to $^{214}$Rn with a 0.27(2) $\mu$s half-life [@1970Val01]. Earlier, the assignment of a 11.7 MeV $\alpha$ energy to $^{214}$Rn [@1962Kar01] was evidently incorrect. $^{215}$Rn {#rn-14 .unnumbered} ---------- In 1952, $^{215}$Rn was discovered by Meinke et al. and the results were reported in the paper “Further work on heavy collateral radioactive chains” [@1952Mei01]. Thorium nitrate targets were irradiated with a $^4$He beam from the Berkeley 184-inch cyclotron. $^{227}$U was chemically separated and the decay and energy of $\alpha$-particles were measured with an alpha-particle pulse analyzer. “An additional short-lived chain collateral to the actinium (4n+3) natural radioactive family has also been partially identified. This chain decays as follows: U$^{227}\rightarrow$Th$^{223}\rightarrow$Ra$^{219}\rightarrow$Em$^{215}\rightarrow$Po$^{211}\rightarrow$Pb$^{207}$.” An $\alpha$ energy of 8.6(1) MeV was assigned to $^{215}$Rn. The presently adopted half-life os 2.3(1) $\mu$s. $^{216,217}$Rn {#rn-15 .unnumbered} -------------- Meinke et al. reported the observation of $^{216}$Rn and $^{217}$Rn in the 1949 paper “Three additional collateral alpha-decay chains” [@1949Mei01]. Thorium was bombarded with 100$-$120 MeV $^4$He beams from the Berkeley 184-inch cyclotron. Alpha-decay chains from $^{228}$U and $^{229}$U were measured following chemical separation. “The irradiation of thorium with 100-Mev helium ions resulted in the observation of the following collateral branch of the artificial 4n$+$1, neptunium, radioactive family shown with Po$^{213}$ and its decay products: $_{92}$U$^{229}\overset{\alpha}{\rightarrow}_{90}$Th$^{225}\overset{\alpha}{\rightarrow}_{88}$Ra$^{221}\overset{\alpha}{\rightarrow}_{86}$Em$^{217}\ldots$ Immediately after 120-Mev helium ion bombardment of thorium the uranium fraction contains another series of five alpha-emitters, which is apparently a collateral branch of the 4n family: $_{92}$U$^{228}\overset{\alpha}{\rightarrow}_{90}$Th$^{224}\overset{\alpha}{\rightarrow}_{88}$Ra$^{220}\overset{\alpha}{\rightarrow}_{86}$Em$^{216}\ldots$” In a table summarizing the energies and half-lives of the decay chain only the $\alpha$-decay energies were given for $^{216}$Rn and $^{217}$Rn stating calculated half-lives of $\sim$10$^{-5}$ s and $\sim$10$^{-3}$ s, respectively. The currently accepted half-lives of $^{216}$Rn and $^{217}$Rn are 45(5) $\mu$sand 540(50) $\mu$s, respectively. $^{218}$Rn {#rn-16 .unnumbered} ---------- Studier and Hyde announced the discovery of $^{218}$Rn in the 1948 paper “A new radioactive series - the protactinium series” [@1948Stu01]. Thorium metal targets were bombarded with 19 MeV deuterons and a 38 MeV $^4$He beam from the Berkeley 60-inch cyclotron forming $^{230}$Pa in (d,4n) and ($\alpha$,p5n) reactions. $^{218}$Rn was populated by subsequent $\alpha$ decay after the initial $\beta^-$ decay of $^{230}$Pa to $^{230}$U. Alpha-decay spectra were measured following chemical separation. “\[The figure\] shows the frequency distribution of the observed time intervals after correction for random events. The total number of observed coincidence periods equal to or less than a given time interval is plotted against the time interval. The integral curve so obtained is exponential within the errors of the experiment and represents the decay of Em$^{218}$. The mean interval is 0.027 sec. corresponding to a half-life of 0.019 sec.” This value is within a factor of two of the currently accepted half-life of 35(5) ms. $^{219}$Rn {#rn-17 .unnumbered} ---------- In the 1903 article “Ueber den Emanationskörper aus Pechblende und über Radium” Giesel identified a new emanation which was later identified as $^{219}$Rn [@1903Gie01]. The emanation was separated from a pitchblende sample. “In den erwähnten ca. 2 g müssten also mindestens 2 Milligramm des fraglichen Elementes enthalten sein. Dass dasselbe nicht Radium oder Polonium sein kann, ist nach dem Gegebenen wohl ausgeschlossen. Von einer sonst aus practischen Gründen üblichen Namengebung des hypothetischen Elementes sehe ich vorläufig ab...” \[At least 2 mg of the element in question should be in the mentioned 2 g. Based on the presented facts it is probably ruled out that this substance can be radium or polonium. For now I refrain from the customary naming of the hypothetical element.\] A month later Debierne independently observed the actinium emanation and reported that it disappear rapidly [@1903Deb01]. The half-life of $^{219}$Rn is 3.96(1) s. $^{220}$Rn {#rn-18 .unnumbered} ---------- Rutherford reported the observation of an activity from radium later identified as $^{220}$Rn in the 1900 article “A radio-active substance emitted from thorium compounds” [@1900Rut01]. Thorium oxide samples were used to study the activity of the “emanation”: “...I have found that thorium compounds continuously emit radio-active particles of some kind, which retain their radio-active powers for several minutes. This ‘emanation,’ as it will be termed for shortness, has the power of ionizing the gas in its neighbourhood and of passing through thin layers of metals, and, with great ease, through considerable thicknesses of paper... The emanation passes through a plug of cotton-wool without any loss of its radio-active powers. It is also unaffected by bubbling through hot or cold water, weak or strong sulphuric acid. In this respect it acts like an ordinary gas... The result shows that the intensity of the radiation has fallen to one-half its value after an interval of about one minute.” This half-life agrees with the currently accepted value of 55.6(1) s. $^{221}$Rn {#rn-19 .unnumbered} ---------- Momyer and Hyde reported the observation of $^{221}$Rn in the 1956 paper “Properties of Em$^{221}$” [@1956Mom01]. Thorium targets were bombarded with 110 MeV protons from the 184-inch Berkeley cyclotron. Alpha-decay spectra were measured following chemical separation. “These results lead directly to the conclusion that a beta-emitting Em$^{221}$ with a 25-minute half-life is present in the samples and is giving rise to the known Fr$^{221}$ chain.” The quoted value is the currently adopted half-life. $^{222}$Rn {#rn-20 .unnumbered} ---------- In 1899 P. Curie and M. Curie reported the observation of an activity in radium samples later identified as $^{222}$Rn in “Sur la radioactivité provoquée par les rayons de Becquerel” [@1899Cur01]. The radioactivity of polonium and radium samples was studied by measuring current due to the ionization of air. “Si l’on soustrait la plaque activée a l’influence de la substance radioactive, elle reste radioactive pendant plusieurs jours. Toutefois, cette radioactivité induite va en décroissant, d’abord très rapidement, ensuite de moins en moins vite et tend á disparaître suivant une loi asymptotique.” \[Subtracting the contribution of the activated plate due to the radioactive substance, it remains radioactive for several days. However, the induced radioactivity is decreasing, first very rapidly, then slower and slower and tends to disappear asymptotically.\] The currently accepted half-life of $^{222}$Ra is 3.8235(3) d. $^{223,224}$Rn {#rn-21 .unnumbered} -------------- Butement and Robinson announced the discovery of $^{223}$Rn and $^{224}$Rn in the 1964 paper “New isotopes of emanation” [@1964But01]. Thorium metal powder was irradiated with a 370 MeV proton beam from the Liverpool synchrocyclotron. The half-lives of $^{223}$Rn and $^{224}$Rn were determined by the milking technique, where the activities were measured with a ZnS-Ag alpha scintillation counter. “The half-life of $^{224}$Em was obtained by extrapolating the decay curves of 3.6 day $^{224}$Ra to the time of milking, and plotting these extrapolated values against time of milking. The value obtained for the half life of $^{224}$Em is 114$\pm$6 min., the error being the standard deviation of the mean of six experiments... These experiments were very similar to those on $^{224}$Em, except that the intervals between milkings were shorter \[because of the shorter half life of $^{223}$Em\], and it was necessary to count the radium samples for some 50-60 days in order to follow the decay of 11.6 day $^{223}$Ra after 3.6 day $^{224}$Ra had decayed out.. The value obtained for the half life of $^{223}$Em is 43$\pm$5 min, where the error is the standard deviation on the mean of six experiments.” The half-life of 43(5) min for $^{223}$Rn is within a factor of two of the accepted value of 24.3(4) min and the half-life of 114(6) min for $^{224}$Rn agrees with currently accepted value of 107(3) min. $^{225,226}$Rn {#rn-22 .unnumbered} -------------- Hansen et al. reported the first observation of $^{225}$Rn and $^{226}$Rn in the paper “Decay characteristics of short-lived radio-nuclides studied by on-line isotope separator techniques” in 1969 [@1969Han01]. Protons of 600 MeV from the CERN synchrocyclotron bombarded a molten tin target and $^{225}$Rn and $^{226}$Rn were separated using the ISOLDE facility. The paper summarized the ISOLDE program and did not contain details about the individual nuclei but the results were presented in a table. The measured half-lives of 4.5(3) min for $^{225}$Rn and 6.0(5) min for $^{226}$Rn agree with the currently adopted values of 4.66(4) min and 7.4(1) min, respectively. $^{227}$Rn {#rn-23 .unnumbered} ---------- In 1986 Borge et al. reported the observation of $^{227}$Rn in the article “New isotope $^{227}$Rn and revised halflives for $^{223}$Rn and $^{226}$Rn” [@1986Bor01]. ThC$_2$ was bombarded with 600 MeV protons from the CERN synchrocyclotron. Decay curves were measured with a 4$\pi$ plastic scintillation detector following mass separation with the ISOLDE on-line separator. “These results yielded halflives of 23$\pm$1 s for the previously unknown isotope $^{227}$Rn and 2.52$\pm$0.05 min for $^{227}$Fr.” This value is included in the calculation of the current half-life. $^{228}$Rn {#rn-24 .unnumbered} ---------- $^{228}$Rn was first discovered by Borge et al. and the results were published in the 1989 paper “The new neutron-rich isotope $^{228}$Rn” [@1989Bor01]. The CERN synchrocyclotron was used to bombard a $^{232}$Th target with 600 MeV protons. Decay curves were measured with a 4$\pi$ plastic scintillation detector following mass separation with the ISOLDE II on-line separator. “From the growth and decay pattern of the Ra K$_{\alpha1}$X-rays and the two strongest $\gamma$-lines from the decay of $^{228}$Fr at 141 and 474 keV a half-life of 36$\pm$2 s was obtained for $^{228}$Fr when the value of 65 s has been kept fixed for the precursor $^{228}$Rn, and a half-life of 62$\pm$3 s for $^{228}$Rn resulted when the value of 38 s has been kept fixed for the daughter nucleus $^{228}$Fr.” The quoted half-life is the currently accepted value. $^{229}$Rn {#rn-25 .unnumbered} ---------- Neidherr et al. announced the discovery of $^{229}$Rn in the 2009 article “Discovery of $^{229}$Rn and the structure of the heaviest Rn and Ra isotopes from Penning-trap mass measurements” [@2009Nei01]. A UC$_X$ target was bombarded with 1.4 GeV protons from the CERN proton synchroton booster accelerator. $^{229}$Rn was measured with the double Penning-trap mass spectrometer ISOLTRAP after mass separation with the on-line isotopes separator ISOLDE. “This measurement gives a half-life of 12$^{+1.2}_{-1.3}$ s for a nuclide with mass number 229 then delivered to ISOLTRAP.” The quoted value is the currently adopted half-life. $^{230,231}$Rn {#rn-26 .unnumbered} -------------- $^{230}$Rn and $^{231}$Rn were discovered by Alvarez-Pol and the results were published in the 2010 paper “Production of new neutron-rich isotopes of heavy elements in fragmentation reactions of $^{238}$U projectiles at 1A GeV” [@2010Alv01]. A beryllium target was bombarded with a 1 A GeV $^{238}$U beam from the GSI SIS synchrotron. The isotopes were separated and identified with the high-resolving-power magnetic spectrometer FRS. “To search for new heavy neutron-rich nuclei, we tuned the FRS magnets for centering the nuclei $^{227}$At, $^{229}$At, $^{216}$Pb, $^{219}$Pb, and $^{210}$Au along its central trajectory. Combining the signals recorded in these settings of the FRS and using the analysis technique previously explained, we were able to identify 40 new neutron-rich nuclei with atomic numbers between Z=78 and Z=87; $^{205}$Pt, $^{207-210}$Au, $^{211-216}$Hg, $^{214-217}$Tl, $^{215-220}$Pb, $^{219-224}$Bi, $^{223-227}$Po, $^{225-229}$At, $^{230,231}$Rn, and $^{233}$Fr.” $^{199-233}$Fr ============== The element francium was discovered by Perey in 1939 by observing the $\beta$-decay of $^{223}$Fr in the natural actinium radioactive decay chain [@1939Per01]. Earlier incorrect observations of francium are described and referenced in the 2005 article “Francium (atomic number 87), the last discovered natural element” on the occasion of the 30$^{th}$ anniversary of Marguerite Perey’s death in 1975 [@2005Adl01]. The name francium was officially accepted at the 15$^{th}$ IUPAC conference in Amsterdam in 1949 [@1949IUC01]. Perey had originally suggested the symbol Fa but agreed it to be changed to Fr [@2005Adl01]. Thirty-five francium isotopes from A = 193–231 have been discovered so far and according to the HFB-14 model [@2007Gor01] about 40 additional francium isotopes could exist. Figure \[f:year-francium\] summarizes the year of first discovery for all francium isotopes identified by the method of discovery: radioactive decay (RD), fusion evaporation reactions (FE), light-particle reactions (LP), projectile fission or fragmentation (PF), and spallation (SP). In the following, the discovery of each francium isotope is discussed in detail and a summary is presented in Table 1. ![Francium isotopes as a function of time when they were discovered. The different production methods are indicated.[]{data-label="f:year-francium"}](francium-year.pdf) $^{199}$Fr {#fr-1 .unnumbered} ---------- Tagaya et al. reported the discovery of $^{199}$Fr in the 1999 paper “The $\alpha$-decay energies and halflives of $^{195g,m}$At and $^{199}$Fr” [@1999Tag01]. $^{169}$Tm targets were bombarded with a 215 MeV $^{36}$Ar beam from the RIKEN ring cyclotron to form $^{199}$Fr in (6n) fusion-evaporation reactions. Recoils were separated with the gas-filled recoil separator GARIS and implanted in a position sensitive detector which also recorded subsequent $\alpha$ decay. “The E$_\alpha$ and T$_{1/2}$ of $^{199}$Fr are 7655$\pm$40 keV and 12$^{+10}_{-4}$ ms, respectively.” The quoted value is the currently accepted half-life. $^{200}$Fr {#fr-2 .unnumbered} ---------- In the 1995 article “New $\alpha$-decaying neutron deficient isotopes $^{197}$Rn and $^{200}$Fr,” Morita et al. announced the identification of $^{200}$Fr [@1995Mor01]. A 186 MeV $^{36}$Ar beam from the RIKEN ring cyclotron bombarded a $^{169}$Tm target forming $^{200}$Fr in (5n) fusion-evaporation reactions. Reaction products were separated with the gas-filled recoil separator GARIS and implanted in a position-sensitive silicon detector which also measured subsequent $\alpha$ decay. “The $\alpha$-decay energies (half-lives) of $^{197}$Rn, $^{197m}$Rn and $^{200}$Fr have been determined to be 7261$\pm$30 keV (51$^{+35}_{-15}$ ms), 7370$\pm$30 keV (18$^{+9}_{-5}$ ms), and 7500$\pm$30 keV, (570$^{+270}_{-140}$ ms), respectively.” This value does not agree with the currently accepted value of 49(4) ms. We credit Morita et al. with the discovery of $^{200}$Fr because they measured the correct decay energy and correlated the events with known properties of the daughter nucleus $^{196}$At. Three months later Enquist et al. [@1996Enq01] independently reported a half-life of 19$^{+13}_{-6}$ ms which also disagrees with the present value. $^{201,202}$Fr {#fr-3 .unnumbered} -------------- The first observation of $^{201}$Fr and $^{202}$Fr was reported in “Alpha decay studies of new neutron-deficient francium isotopes and their daughters” by Ewan et al. [@1980Ewa01]. A uranium target was bombarded with 600 MeV protons from the CERN synchrocyclotron producing $^{201}$Fr and $^{202}$Fr in spallation reactions. Alpha-particle spectra were measured with a silicon surface-barrier detector following mass separation with the isotope separator ISOLDE. “The only hitherto unreported line in the spectrum is the 7388$\pm$15 keV line, whose decay, as obtained from the measurement with the position-sensitive detector, is shown in the inset of \[the figure\]. This line is assigned to $^{201}$Fr, for which a half-life of 48$\pm$15 ms thus was derived... The singles alpha spectrum observed from the decay of a source collected at mass 202 is shown in the lower part of \[the figure\]. In addition to previously known lines, mainly coming from heavier francium isotopes in analogy with the A=201 spectrum, a strong alpha line with an energy of 7251$\pm$10 keV is present... The new line is assigned to $^{202}$Fr, and the half-life was deduced to be 0.34$\pm$0.04 s.” The measured half-lives of 48(15) ms for $^{201}$Fr and 0.34(4) s for $^{202}$Fr agree with the currently adopted values of 62(5) ms and 0.30(5) s, respectively. $^{203}$Fr {#fr-4 .unnumbered} ---------- Valli et al. announced the discovery of $^{203}$Fr in the 1967 article “Alpha decay of neutron-deficient francium isotopes” [@1967Val02]. The Berkeley heavy ion linear accelerator was used to bombard $^{197}$Au and $^{205}$Tl targets with $^{16}$O and $^{12}$C beams with energies up to 166 and 126 MeV, respectively. Recoil products were collected on a catcher foil which was quickly positioned in front of a Si(Au) surface-barrier detector which measured subsequent $\alpha$ decay. “Francium-203. The peak at 7.130 MeV with a half-life of 0.7$\pm$0.3 sec is visible only in the spectra taken at the highest beam energies. Comparison of excitation functions indicates that the peak belongs to a francium isotope lighter than $^{204}$Fr, most probably to $^{203}$Fr.” This value is consistent with the currently adopted value of 0.30(5) s. $^{204-211}$Fr {#fr-5 .unnumbered} -------------- In 1964, Griffioen and MacFarlane reported the identification of $^{204}$Fr, $^{205}$Fr, $^{206}$Fr, $^{207}$Fr, $^{208}$Fr, $^{209}$Fr, $^{210}$Fr, and $^{211}$Fr in the paper “Alpha-decay properties of some francium isotopes near the 126-neutron closed shell” [@1964Gri01]. $^{197}$Au, $^{203,205}$Tl, and $^{208}$Pb targets were bombarded with $^{16}$O, $^{12}$C, and $^{11}$B beams with energies up to 10.38 MeV/amu from the Berkeley HILAC. Recoil products were collected on a catcher foil which was positioned in front of a gold surface-barrier detector which measured subsequent $\alpha$ decay. “B. Fr$^{211}$ and Fr$^{210}$: ... These facts seem to indicate that this group is due to two different isotopes, Fr$^{211}$ and Fr$^{210}$, which were formed by (C$^{12}$,6n) and (C$^{12}$,7n) reactions, respectively... C. Fr$^{209}$ and Fr$^{208}$: ... Once again there were indications that this alpha group is a result of two different isotopes. The Tl$^{203}$+C$^{12}$ excitation function is somewhat broadened and distorted and no other alpha groups with an excitation function corresponding to a (C$^{12}$,7n) reaction were found. The excitation function from the Au$^{197}$+O$^{16}$ system is also consistent with the assignment to Fr$^{209}$ and Fr$^{208}$... D. Fr$^{207}$ and Fr$^{206}$: ... Since, as has been mentioned before and will be discussed later, the cross section for the Au$^{197}$(O$^{16}$,7n)Fr$^{206}$ reaction is probably about one fourth the value of the Au$^{197}$(O$^{16}$,6n)Fr$^{207}$ reaction cross section, we cannot see any great effect on the excitation function for this group. Several things have led us to the conclusion that this is both Fr$^{207}$ and Fr$^{206}$... E. Fr$^{205}$ and Fr$^{204}$: ... Since it follows the excitation function for the 6.91-MeV group, this would identify this group as the parent of At$^{201}$ namely, Fr$^{205}$. The assignment of the 7.02-MeV group to Fr$^{204}$ is based on the excitation-function data and on alpha decay systematics.” The measured half-lives of 2.0(5) s ($^{204}$Fr), 3.7(4) s ($^{205}$Fr), 15.8(4) s ($^{206}$Fr), 18.7(8) s ($^{207}$Fr), 37.5(20) s ($^{208}$Fr), 54.7(10) s ($^{209}$Fr), 159(5) s ($^{210}$Fr), and 186(4) s ($^{211}$Fr) are close to the currently adopted values of 1.8(3) s, 3.92(4) s, 15.9(1) s, 14.8(1) s, 59.1 (s), 50.0(3) s, 3.18(6) min, and 3.10(2) min, respectively. $^{212}$Fr {#fr-6 .unnumbered} ---------- Hyde et al. reported the first observation of $^{212}$Fr in the 1950 paper “Low mass francium and emanation isotopes of high alpha-stability” [@1950Hyd01]. Thorium foils were bombarded with up to 350 MeV protons from the Berkeley 184-inch cyclotron. $^{212}$Fr was chemically separated and alpha spectra were measured with an ionization chamber. “Fr$^{212}$, with an apparent half-life of 19.3 minutes for branching decay by alpha-emission (44 percent) to At$^{208}$ and by orbital electron-capture (56 percent) to Em$^{212}$, has been found.” This half-life is included in the calculation of the currently adopted value. $^{213}$Fr {#fr-7 .unnumbered} ---------- In 1964 Griffioen and MacFarlane reported the identification of $^{213}$Fr in the paper “Alpha-decay properties of some francium isotopes near the 126-neutron closed shell” [@1964Gri01]. $^{197}$Au, $^{203,205}$Tl, and $^{208}$Pb targets were bombarded with $^{16}$O, $^{12}$C, and $^{11}$B beams with energies up to 10.38 MeV/amu from the Berkeley HILAC. Recoil products were collected on a catcher foil which was positioned in front of a gold surface-barrier detector which measured subsequent $\alpha$ decay. “A. Fr$^{213}$: \[The figure\] shows an alpha-particle spectrum of the activity collected while bombarding Tl$^{205}$ with 86-MeV C$^{12}$ ions. A strong group is seen at 6.77$\pm$0.01 MeV alpha particle energy. This activity decays with a half-life of 33.7$\pm$1.5 sec.” This value is included in the calculation of the currently accepted value. $^{214}$Fr {#fr-8 .unnumbered} ---------- Rotter et al. observed $^{214}$Fr in 1967 and reported their results in the paper “The new isotope Ac$^{216}$” [@1967Rot01]. A 78 MeV $^{12}$C beam from the Dubna 1.5 m cyclotron bombarded a bismuth target forming actinium in (xn) reactions. $^{214}$Fr was populated by $\alpha$ decay of $^{218}$Ac. Recoil nuclei were collected on an aluminum foil and $\alpha$-particle spectra were measured with a silicon surface barrier detector. “We obtained the following $\alpha$-particle energies: Rn$^{213}$ - 8.14 MeV, Fr$^{214}$ - 8.53 MeV, and Ra$^{215}$ - 8.73 MeV.” Rotter et al. did not consider this observation a new discovery referring to a conference abstract [@1962Gri01]. The observation corresponds to an isomeric state with a currently accepted half-life of 3.35(5) ms. $^{215,216}$Fr {#fr-9 .unnumbered} -------------- In the 1970 article “Production and decay properties of protactinium isotopes of mass 222 to 225 formed in heavy-ion reactions,” Borggreen et al. identified $^{215}$Fr and $^{216}$Fr [@1970Bor01]. The Berkeley heavy-ion linear accelerator (HILAC) was used to bombard $^{208}$Pb and $^{205}$Tl targets with $^{19}$F and $^{22}$Ne beams forming $^{224}$Pa and $^{223}$Pa in (3n) and (4n) fusion-evaporation reactions, respectively. $^{216}$Fr and $^{215}$Fr were then populated by subsequent $\alpha$-decay. Recoil products were deposited by a helium gas stream on a metal surface located in front of a gold surface-barrier detector which recorded the subsequent $\alpha$ decay. “Francium-216 appears to emit a single $\alpha$ group of 9.005$\pm$0.010 MeV which fits smoothly on the francium curve in \[the figure\]... The time-parameter information associated with the data sorting displayed in \[the figure\] yielded a 0.70$\pm$0.02 $\mu$sec half-life for $^{216}$Fr... The assignment of the 9.365-MeV group to $^{215}$Fr seems particularly secure owing to the very restricted number of possible assignments of $\alpha$ groups above 9.3-MeV energy.” For $^{215}$Fr only an upper limit of $<$500 ns was given. The currently accepted value is 86(5) ns. The measured half-life for $^{216}$Fr is the presently adopted value. $^{217}$Fr {#fr-10 .unnumbered} ---------- In the 1968 article “New neptunium isotopes, $^{230}$Np and $^{229}$Np” Hahn et al. reported the observation of $^{217}$Fr [@1968Hah01]. Enriched $^{233}$U targets were bombarded with 32$-$41.6 MeV protons from the Oak Ridge Isochronous Cyclotron forming $^{229}$Np in (p,5n) reactions. Reaction products were implanted on a catcher foil which was periodically rotated in front of a surface barrier Si(Au) detector. Isotopes populated by subsequent $\alpha$ emission were measured. “The $\alpha$-particle energies found for the $^{225}$Pa series are more precise than the previously available values: $^{225}$Pa, 7.25$\pm$0.02 MeV (new value); $^{221}$Ac, 7.63$\pm$0.02 MeV; $^{217}$Fr, 8.31$\pm$0.02 MeV and $^{213}$At, 9.06$\pm$0.02 MeV.” The observation of $^{217}$Fr was not considered new, referring to an unpublished thesis [@1951Key01]. $^{218}$Fr {#fr-11 .unnumbered} ---------- Meinke et al. reported the observation of $^{218}$Fr in the 1949 paper “Three additional collateral alpha-decay chains” [@1949Mei01]. Thorium was bombarded with 150 MeV deuterons from the Berkeley 184-inch cyclotron. The $\alpha$-decay chain from $^{226}$Pa was measured following chemical separation. “General considerations with regard to the method of formation and half-life of the parent substance, and the energies of all the members of the series suggest a collateral branch of the 4n+2 family: $_{91}$Pa$^{226}\overset{\alpha}{\rightarrow}_{89}$Ac$^{222}\overset{\alpha}{\rightarrow}_{87}$Fr$^{218}\overset{\alpha}{\rightarrow}_{85}$At$^{214}\overset{\alpha}{\rightarrow}_{85}$Bi$^{210}$(RaE).” In a table summarizing the energies and half-lives of the decay chain only the $\alpha$-decay energy was given for $^{216}$Rn stating a calculated half-life of about 10$^{-2}$ s. The currently accepted half-life is 1.0(6) ms. $^{219,220}$Fr {#fr-12 .unnumbered} -------------- In “Artificial collateral chains to the thorium and actinium families,” Ghiorso et al. discovered $^{219}$Fr and $^{220}$Fr in 1948 [@1948Ghi01]. Thorium targets were irradiated with 80 MeV deuterons from the Berkeley 184-inch cyclotron. The $\alpha$-decay chains beginning at $^{227}$Pa and $^{228}$Pa were measured following chemical separation. “Prominent soon after bombardment are a number of alpha-particle groups, which decay with the 38-minute half-life of the protactinium parent. These are due to the following collateral branch of the 4n+3 radioactive family: $_{91}$Pa$^{227}\stackrel{\alpha}{\longrightarrow}_{89}$Ac$^{223}\stackrel{\alpha}{\longrightarrow}_{87}$Fr$^{219}\stackrel{\alpha}{\longrightarrow}_{85}$At$^{215}\stackrel{\alpha}{\longrightarrow}$... After the decay of the above-described series, a second group of alpha-particle emitters can be resolved. This second series, which decays with the 22-hour half-life of its protactinium parent, is a collateral branch of the 4n radioactive family as follows: $_{91}$Pa$^{228}\stackrel{\alpha}{\longrightarrow}_{89}$Ac$^{224}\stackrel{\alpha}{\longrightarrow}_{87}$Fr$^{220}\stackrel{\alpha}{\longrightarrow}_{85}$At$^{216}\stackrel{\alpha}{\longrightarrow}$...” The decay energies and half-lives of the decay chains were listed in a table, assigning half-lives of $\sim$10$^{-4}$ s and $\sim$30 s to $^{219}$Fr and $^{220}$Fr, respectively. The currently adopted half-lives for $^{219}$Fr and $^{220}$Fr are 20(2) ms and 27.4(3) s, respectively. $^{221}$Fr {#fr-13 .unnumbered} ---------- Hagemann et al. discovered $^{221}$Fr in 1947 in “The (4n+1) radioactive series: the decay products of U$^{233}$” [@1947Hag01]. The half-lives and $\alpha$- and $\beta$-decay energies of the nuclides in the decay chain of $^{233}$U were measured. “These decay products, which constitute a substantial fraction of the entire missing, 4n+1, radioactive series are listed together with their radioactive properties, in \[the table\].” The measured half-life of 4.8 min agrees with the presently accepted value of 4.9(2) min. Hagemann et al. acknowledge the simultaneous observation of $^{221}$Fr by English et al. which was submitted only a day later and published in the same issue of Physical Review on the next page [@1947Eng01]. $^{222}$Fr {#fr-14 .unnumbered} ---------- Westgaard et al. identified $^{222}$Fr in the 1975 paper “Beta-decay energies and masses of short-lived isotopes of rubidium, caesium, francium, and radium” [@1975Wes01]. Lanthanum, yttrium-lanthanum, and thorium-lanthanum targets were irradiated with 600 MeV protons from the CERN synchrocyclotron. Beta- and gamma-rays were measured following mass separation with the ISOLDE on-line separator at CERN. “The decay of 15 min $^{222}$Fr: ... The singles $\beta$ spectrum measured in our experiment showed a flat tail of low intensity, extending to much higher energies than the main portion of the data. After subtraction of this tail, presumably due to $\alpha$ particles from $^{222}$Ra, a FK analysis gave for the endpoint energy E$^{max}_\beta$=1.78$\pm$0.02 MeV.” They measured half-life of 14.8 min agrees with the currently adopted value of 14.2(3) min. $^{223}$Fr {#fr-15 .unnumbered} ---------- Perey discovered $^{223}$Fr in 1939 as reported in “Sur un élément 87, dérivé de l’actinium” [@1939Per01]. $^{223}$Fr was observed within the natural actinium radioactive decay chain and populated by $\alpha$ decay from $^{227}$Ac. Beta-decay curves were measured following chemical separation. “En ajoutant du chlorure de cæsium à l’eau mère et en précipitant par une solution de perchlorate de sodium, il se forme des cristaux qui entraînent l’activité: celle-ci d’ecroît exponentiellement avec la période de 21 minutes $\pm$ 1. \[By adding liquid cesium chloride and precipitating a solution of sodium perchlorate crystals are formed that cause an activity which decreases exponentially with the period of 21 minutes $\pm$ 1.\] This half-life agrees with the presently adopted value of 22.00(7) min. This observation of $^{223}$Fr also represented the discovery of the element francium. $^{224-226}$Fr {#fr-16 .unnumbered} -------------- Hansen et al. reported the first observation of $^{224}$Fr, $^{225}$Fr and $^{226}$Fr in the paper “Decay characteristics of short-lived radio-nuclides studied by on-line isotope separator techniques” in 1969 [@1969Han01]. Protons of 600 MeV from the CERN synchrocyclotron bombarded a molten tin target and $^{224}$Fr, $^{225}$Fr and $^{226}$Fr were separated using the ISOLDE facility. The paper summarized the ISOLDE program and did not contain details about the individual nuclei but the results were presented in a table. The measured half-life of 2.67(20) min for $^{224}$Fr agrees with the presently adopted value of 3.33(10) min and the 3.9(2) min for $^{225}$Fr is included in the calculation of the currently accepted half-life of 3.95(14) min. The half-life of 1.43(23) min for $^{226}$Fr is within a factor of two of the present value of 49(1) s. $^{227,228}$Fr {#fr-17 .unnumbered} -------------- In 1972 Klapisch et al. reported the first observation of $^{227}$Fr and $^{228}$Fr in “Half-life of the new isotope $^{32}$Na; Observation of $^{33}$Na and other new isotopes produced in the reaction of high-energy protons on U” [@1972Kla01]. Uranium targets were bombarded with 24 GeV protons from the CERN proton synchrotron. $^{227}$Fr and $^{228}$Fr were identified by on-line mass spectrometry and decay curves were measured. “Following the same procedure as for Na, the isotopes $^{48}$K, $^{49}$K, and $^{50}$K were found. However, their half-lives were not short compared with the diffusion time, and hence could not be determined. We also observed the new neutron-rich isotopes $^{227}$Fr and $^{228}$Fr produced in the spallation of the uranium target.” The presently accepted half-lives are 2.47(3) min and 38(1) s for $^{227}$Fr and $^{228}$Fr, respectively. $^{229}$Fr {#fr-18 .unnumbered} ---------- In 1975 the discovery of $^{229}$Fr by Ravn et al. was announced in the paper “Short-lived isotopes of alkali and alkaline-earth elements studied by on-line isotope separator techniques” [@1975Rav01]. A thorium plus lanthanum target was bombarded with protons from the CERN synchrocyclotron. Beta-ray decay curves were measured with a 4$\pi$ plastic detector following mass separation with the isotope separator ISOLDE. “The following half-lives of new nucleides have been determined: ... $^{229}$Fr (50$\pm$20) sec.” This half-life agrees with the presently adopted value of 50.2(20) s. $^{230}$Fr {#fr-19 .unnumbered} ---------- In the 1987 article “Collective states in $^{230}$Ra fed by $\beta^-$ decay of $^{230}$Fr,” Kurcewicz et al. identified $^{230}$Fr [@1987Kur01]. Francium was produced by spallation of $^{238}$U with 600 MeV protons from the CERN synchrocyclotron. Gamma-ray singles and $\gamma-\gamma$ coincidences were measured with Ge(Li) detectors after mass separation with the on-line separator ISOLDE II. “A half-life of 19.1$\pm$0.5 s for $^{230}$Fr has been obtained by means of multispectra analysis using cycles of 20 s collection time followed by 6$\times$7 s measuring time.” This value is the currently accepted half-life. $^{231}$Fr {#fr-20 .unnumbered} ---------- The discovery of $^{231}$Fr was reported in the 1985 paper “The new neutron-rich nuclei $^{231}$Fr and $^{231}$Ra” by Hill et al. [@1985Hil01]. Francium was produced by spallation of $^{238}$U with 600 MeV protons from the CERN synchrocyclotron. Beta-particles and $\gamma$-rays were measured with a plastic scintillator and two Ge(Li) detectors, respectively, following mass separation with the on-line separator ISOLDE II. “With three other $\gamma$-lines, which are assigned to the $^{231}$Fr decay due to their half-lives, a weighted average of 17.5(8) s is obtained for the half-life of $^{231}$Fr.” The quoted value is included in the calculation of the currently adopted half-life. $^{232}$Fr {#fr-21 .unnumbered} ---------- Mezlev et al. reported the discovery of $^{232}$Fr in the 1990 paper “Search for delayed fission in neutron-rich nuclei” [@1990Mez01]. A uranium target was bombarded with 1 GeV protons. Beta-, gamma-, and X-rays were measured with solid state detectors following mass separation with the on-line mass separator IRIS. “Due to this technique the new isotopes $^{232}$Fr (T$_{1/2}$=5$\pm$1 s), $^{233}$Ra(T$_{1/2}$=30$\pm$5 s) and $^{234}$Ra (T$_{1/2}$=30$\pm$10 s) have been identified using the solid state detectors for the registration of the beta-, gamma-, X-radiation.” The measured of 5(1) s half-life for $^{232}$Fr agrees with the currently adopted value of 5.5(6) s. $^{233}$Fr {#fr-22 .unnumbered} ---------- $^{233}$Fr was discovered by Alvarez-Pol and the results were published in the 2010 paper “Production of new neutron-rich isotopes of heavy elements in fragmentation reactions of $^{238}$U projectiles at 1A GeV” [@2010Alv01]. A beryllium target was bombarded with a 1 A GeV $^{238}$U beam from the GSI SIS synchrotron. The isotopes were separated and identified with the high-resolving-power magnetic spectrometer FRS. “To search for new heavy neutron-rich nuclei, we tuned the FRS magnets for centering the nuclei $^{227}$At, $^{229}$At, $^{216}$Pb, $^{219}$Pb, and $^{210}$Au along its central trajectory. Combining the signals recorded in these settings of the FRS and using the analysis technique previously explained, we were able to identify 40 new neutron-rich nuclei with atomic numbers between Z=78 and Z=87; $^{205}$Pt, $^{207-210}$Au, $^{211-216}$Hg, $^{214-217}$Tl, $^{215-220}$Pb, $^{219-224}$Bi, $^{223-227}$Po, $^{225-229}$At, $^{230,231}$Rn, and $^{233}$Fr.” $^{201-234}$Ra ============== The element radium was discovered in 1898 by P. Curie, M. Curie and G. Bémont [@1898Cur02]. Thirty-four radium isotopes from A = 201–234 have been discovered so far and according to the HFB-14 model [@2007Gor01] about 50 additional radon isotopes could exist. Figure \[f:year-radium\] summarizes the year of first discovery for all radon isotopes identified by the method of discovery: radioactive decay (RD), fusion evaporation reactions (FE), light-particle reactions (LP), neutron-capture reactions (NC), and spallation (SP). In the following, the discovery of each radon isotope is discussed in detail and a summary is presented in Table 1. ![Radium isotopes as a function of time when they were discovered. The different production methods are indicated.[]{data-label="f:year-radium"}](radium-year.pdf) $^{201,202}$Ra {#ra-1 .unnumbered} -------------- $^{201}$Ra and $^{202}$Ra were first observed by Uusitalo et al. and the results were published in the 2005 paper “$\alpha$ decay studies of very neutron-deficient francium and radium isotopes” [@2005Uus01]. A $^{141}$Pr target was bombarded with 278-288 MeV $^{63}$Cu beams from the Jyväskylä K-130 cyclotron forming $^{201}$Ra and $^{202}$Ra in (3n) and (2n) fusion-evaporation reactions, respectively. Reaction products were separated with the gas-filled recoil separator RITU and implanted in a position-sensitive silicon detector which measured subsequent $\alpha$ decay. “We associate these measured values with the known activities of $^{197}$Rn$^m$ (from the 13/2$^+$ isomeric state) with E$_\alpha$ = 7356(7) keV and T$_{1/2}$ = 19$^{+8}_{-4}$ ms and $^{193}$Po$^m$ (from the 13/2$^+$ isomeric state) with E$_\alpha$ = 7004(5) keV and T$_{1/2}$ = 240(10) ms \[13\], and thus the activity with E$_\alpha$ = 7905 keV and T$_{lt}$ = 2 ms can be identified to originate from a new even-odd radium isotope $^{201}$Ra... Thus the activity with E$_\alpha$ = 7740 keV and T$_{lt}$ = 46 ms can be identified to originate from the (0$^+$) ground state of a new even-even isotope $^{202}$Ra.” The measured half-lives of 1.6$^{+7.7}_{-0.7}$ ms and 16$^{+30}_{-7}$ ms are the current values for $^{201}$Ra and $^{202}$Ra, respectively. A previously measured half-life of 0.7$^{+3.3}_{-0.3}$ ms for $^{202}$Ra [@1996Lei01] was evidently incorrect. $^{203}$Ra {#ra-2 .unnumbered} ---------- In the 1996 paper “Alpha decay studies of neutron-deficient radium isotopes,” Leino et al. described the observation of $^{203}$Ra [@1996Lei01]. A $^{175}$Lu target was bombarded with 191$-$208 MeV $^{35}$Cl beams from the Jyväskylä K130 cyclotron to produce $^{203}$Ra in the (7n) fusion-evaporation reaction. Reaction products were separated with the gas-filled recoil separator RITU and implanted in a position-sensitive PIPS detector which measured subsequent $\alpha$ decay. “The alpha particle energy E$_\alpha$ and the half-life T$_{1/2}$ of an isomeric state of the new isotope, $^{203m}$Ra, have been determined to be (7615$\pm$20) keV and (33$^{+22}_{-10}$) ms, respectively. An assignment of another decay with E$_\alpha$ = (7577$\pm$20) keV and T$_{1/2}$ = (1.1$^{+5.0}_{-0.5}$) ms to $^{203g}$Ra is made on the basis of one observed three-decay chain.” While the half-life for the ground state is not consistent with the currently adopted value of 31$^{+17}_{-9}$ ms the half-life for the isomeric state agrees with the present value of 24$^{+6}_{-4}$ ms. $^{204}$Ra {#ra-3 .unnumbered} ---------- Leddy et al. reported the observation of $^{204}$Ra in the 1995 article “$\alpha$ decay of a new isotope, $^{204}$Ra” [@1995Led01]. $^{28}$Si beams with energies of 164 and 170 MeV from the Argonne Tandem-Linac Accelerator System bombarded a $^{182}$W target producing $^{204}$Ra in the (6n) fusion-evaporation reaction. Recoil products were separated with the Fragment Mass Analyzer (FMA) and implanted in a double-sided silicon strip detector which measured subsequent $\alpha$ decay. “This assignment, summarized in \[the table\], constitutes the first observation of $\alpha$ decay from the ground state of $^{204}$Ra with an $\alpha$ energy of 7.488(12) MeV and a half-life of 45$^{+55}_{-21}$ ms.” The quoted half-life agrees with the currently accepted value of 57($^{+11}_{-5}$) ms. $^{205}$Ra {#ra-4 .unnumbered} ---------- $^{205}$Ra was first observed by Heßberger et al. as reported in the 1987 paper “Observation of two new alpha emitters with Z = 88” [@1987Hes01]. A $^{159}$Tb target was irradiated with $^{51}$V beams from the GSI UNILAC. Recoils were separated by the velocity filter SHIP and implanted in an array of seven position-sensitive surface-barrier detectors. “For $^{205}$Ra an $\alpha$ energy of E$_\alpha$=(7360$\pm$20) keV and a half-life of T$_{1/2}$=(220$\pm$60) ms were obtained.” This value agrees with the currently adopted half-life of 210($^{+60}_{-40}$) ms. $^{206-212}$Ra {#ra-5 .unnumbered} -------------- In the 1967 paper “On-line alpha spectroscopy of neutron-deficient radium isotopes,” Valli et al. described the observation of $^{206}$Ra, $^{207}$Ra, $^{208}$Ra, $^{209}$Ra, $^{210}$Ra, $^{211}$Ra, and $^{212}$Ra [@1967Val03]. $^{197}$Au and $^{206}$Pb targets were bombarded with $^{19}$F and $^{12}$C beams from the Berkeley HILAC to produce $^{216}$Ra and $^{218}$Ra compound nuclei. Recoil products were deposited on a collector foil which was then placed in front of a Si(Au) surface barrier detector. “C. Radium-212 and Radium-211: ...In careful measurements made at beam energies where $^{212}$Ra was predominant over $^{211}$Ra we determined an $\alpha$ energy of 6.896 MeV and a half-life of 13$\pm$2 sec. At beam energies where $^{211}$Ra was the principal activity we determined values of 6.910 MeV and 15$\pm$2 sec... D. Radium-210 and Radium-209: ...From measurements made at a 105-MeV $^{19}$F beam energy where $^{210}$Ra predominates over $^{209}$Ra we measured an $\alpha$ energy of 7.018 MeV and a half-life of 3.8$\pm$0.2 sec. Properties of $^{209}$Ra were measured on samples prepared at a beam energy of 140 MeV. The $\alpha$ energy and half-life are 7.008 MeV and 4.7$\pm$0.2 sec, respectively. E. Radium-208 and Radium-207: ...The half-life of the group was also determined at several beam energies. Values between 1.1 and 1.4 sec were observed, the shorter ones coming systematically from measurements at lower beam energies. Therefore, 1.3$\pm$0.2 sec is reported for $^{207}$Ra and 1.2$\pm$0.2 sec for $^{208}$Ra... F. Radium-206: The weak $\alpha$ group at 7.270 MeV in \[the figure\] belongs to $^{206}$Ra. Its half-life was measured to be 0.4$\pm$0.2 sec.” These half-lives of 13(2) s, 15(2) s, 3.8(2) s, 4.7(2) s, 1.2(2) s, 1.3(2) s, and 0.4(2) s for $^{212}$Ra, $^{211}$Ra, $^{210}$Ra, $^{209}$Ra, $^{208}$Ra, $^{207}$Ra, and $^{206}$Ra, respectively agree with the currently adopted values of 13.0(2) s, 13(2) s, 3.7(2) s, 4.6(2) s, 1.3(2) s, 1.35$^{+0.22}_{-0.13}$ s, and 0.24(2) s. $^{213}$Ra {#ra-6 .unnumbered} ---------- Momyer and Hyde reported the observation of $^{213}$Rn in the 1955 article “The influence of the 126-neutron shell on the alpha-decay properties of the isotopes of emanation, francium, and radium” [@1955Mom01]. A lead target was bombarded with a $^{12}$C beam from the Berkeley 60-inch cyclotron to form $^{213}$Ra in the $^{206}$Pb($^{12}$C,5n) fusion-evaporation reaction. Alpha-particle spectra and decay curves were recorded with an $\alpha$ pulse analyzer following chemical separation. “The half-life of the activity was 2.7$\pm$0.3 minutes, and the energy of the alpha particle was 6.90$\pm$0.04 MeV. After decay of the short-lived activity, several counts per minute of Em$^{209}$ were observed on the plate.” This half-life agrees with the currently adopted value of 2.73(5) min. Earlier Momyer et al. presented indirect evidence for the production of a $\sim$2 min half-life for $^{213}$Ra [@1952Mom01]. $^{214,215}$Ra {#ra-7 .unnumbered} -------------- Rotter et al. observed $^{214}$Ra and $^{215}$Ra in 1967 and reported their results in the paper “The new isotope Ac$^{216}$” [@1967Rot01]. An 78 MeV $^{12}$C beam from the Dubna 1.5 m cyclotron bombarded a lead target forming radium in (xn) reactions. Recoil nuclei were collected on an aluminum foil and $\alpha$-particle spectra were measured with a silicon surface barrier detector. “...an accuracy of 0.02-0.03 MeV is possible when determining $\alpha$-line energies if the absolute energy scale is based on an ‘internal’ reference line. The latter was the 6.77 MeV line of Fr$^{213}$ in our experiments with bismuth bombardment; in the case of a lead target it was the 7.17 MeV line of Ra$^{214}$. These lines were clearly discriminated in all the measured spectra... We obtained the following $\alpha$-particle energies: Rn$^{213}$ - 8.14 MeV, Fr$^{214}$ - 8.53 MeV, and Ra$^{215}$ - 8.73 MeV.” Rotter et al. did not consider these observations new discoveries referring to an earlier conference abstract [@1962Gri01] and a book by Hyde et al. [@1964Hyd01] which most probably also referred to the same conference abstract. Over a year later Valli et al. independently reported a 7.136(5) MeV $\alpha$ energy for $^{214}$Ra [@1967Val03]. The presently accepted half-lives are 2.46(3) s and 1.55(7) ms for $^{214}$Ra and $^{215}$Ra, respectively. $^{216}$Ra {#ra-8 .unnumbered} ---------- In the article “In-beam alpha spectroscopy of N=128 isotones. Lifetimes of $^{216}$Ra and a new isotope $^{217}$Ac,” Nomura et al. reported the observation of $^{216}$Ra in 1972 [@1972Nom01]. A $^{206}$Pb target was bombarded with 65$-$85 MeV $^{12}$C beams from the RIKEN IPCR cyclotron forming $^{216}$Ra in (4n) fusion-evaporation reactions. Alpha-particle spectra and decay curves were measured with a surface barrier Si detector. “Time distributions of the ground-state decay of $^{216}$Ra and $^{217}$Ac are shown in \[the figure\], from which half-lives of $^{216}$Ra and $^{217}$Ac have been determined of 0.18$\pm$0.03$~\mu$s and 0.10$\pm$0.01 $\mu$s, respectively.” The quoted value for $^{216}$Ra agrees with the currently adopted half-life of 182(10) ns. $^{217,218}$Ra {#ra-9 .unnumbered} -------------- In 1970, Torgerson and MacFarlane reported the first observation of $^{217}$Ra and $^{218}$Ra in “Alpha decay of the $^{221}$Th and $^{222}$Th decay chains” [@1970Tor01]. A 10.6 MeV/nucleon $^{16}$O beam from the Yale heavy ion accelerator was used to bombard a $^{208}$Pb target forming $^{221}$Th and $^{222}$Th in (3n) and (2n) fusion-evaporation reactions, respectively. $^{217}$Ra and $^{218}$Ra were then populated by subsequent $\alpha$ decay. Recoil products were transported to a stainless steel surface with a helium jet and $\alpha$ spectra were measured with a Si(Au) surface barrier detector. “$^{217}$Ra: A direct measurement of the half-life was made by gating the two prominent $^{221}$Th groups (8.146 and 8.472 MeV) and the $^{217}$Ra group to produce start and stop pulses respectively on a time-to-amplitude converter. The output of the converter was then routed into a pulse-height analyser, and the decay curve was obtained. Using this procedure, we have measured the half-life of $^{217}$Ra to be 4$\pm$2 $\mu$sec... $^{218}$Ra: ...We observed a group at 8.392 MeV with an intensity relative to the $^{222}$Th group that was expected if it were an $\alpha$-decay daughter of that nucleide... No half-life measurement could be made because of the weak intensity of this group.” Only three days later Valli et al. submitted their measurements of 8.995(10) MeV and 8.385(1) MeV $\alpha$ energies for $^{217}$Ra and $^{218}$Ra, respectively [@1970Val01]. The measured half-life for $^{217}$Ra of 4(2) $\mu$s is close to the presently accepted value of 1.63(17) $\mu$s and the current half-life for $^{218}$Ra is 25.2(3) $mu$s. $^{219}$Ra {#ra-10 .unnumbered} ---------- In 1952, $^{219}$Ra was discovered by Meinke et al. and the results were reported in the paper “Further work on heavy collateral radioactive chains” [@1952Mei01]. Thorium nitrate targets were irradiated with a $^4$He beam from the Berkeley 184-inch cyclotron. $^{227}$U was chemically separated and the decay and energy of $\alpha$-particles were measured with an alpha-particle pulse analyzer. “An additional short-lived chain collateral to the actinium (4n+3) natural radioactive family has also been partially identified. This chain decays as follows: U$^{227}\rightarrow$Th$^{223}\rightarrow$Ra$^{219}\rightarrow$Em$^{215}\rightarrow$Po$^{211}\rightarrow$Pb$^{207}$.” An $\alpha$ energy of 8.0(1) MeV was assigned to $^{219}$Ra. The presently accepted half-life is 10(3) ms. $^{220,221}$Ra {#ra-11 .unnumbered} -------------- Meinke et al. reported the observation of $^{220}$Ra and $^{221}$Ra in the 1949 paper “Three additional collateral alpha-decay chains” [@1949Mei01]. Thorium was bombarded with 100$-$120 MeV $^4$He beams from the Berkeley 184-inch cyclotron. Alpha-decay chains from $^{228}$U and $^{229}$U were measured following chemical separation. “The irradiation of thorium with 100-Mev helium ions resulted in the observation of the following collateral branch of the artificial 4n$+$1, neptunium, radioactive family shown with Po$^{213}$ and its decay products: $_{92}$U$^{229}\overset{\alpha}{\rightarrow}_{90}$Th$^{225}\overset{\alpha}{\rightarrow}_{88}$Ra$^{221}\overset{\alpha}{\rightarrow}_{86}$Em$^{217}\ldots$ Immediately after 120-Mev helium ion bombardment of thorium the uranium fraction contains another series of five alpha-emitters, which is apparently a collateral branch of the 4n family: $_{92}$U$^{228}\overset{\alpha}{\rightarrow}_{90}$Th$^{224}\overset{\alpha}{\rightarrow}_{88}$Ra$^{220}\overset{\alpha}{\rightarrow}_{86}$Em$^{216}\ldots$” In a table summarizing the energies and half-lives of the decay chains only the $\alpha$-decay energy was given for $^{220}$Ra stating a calculated half-life of 10$^{-2}$ s. The currently accepted half-life is 18(2) ms. The measured half-life of 31.0(15) s for $^{221}$Ra agrees with the presently adopted value of 28(2) s. $^{222}$Ra {#ra-12 .unnumbered} ---------- Studier and Hyde announced the discovery of $^{222}$Ra in the 1948 paper “A new radioactive series - the protactinium series” [@1948Stu01]. Thorium metal targets were bombarded with 19 MeV deuterons and a 38 MeV $^4$He beam from the Berkeley 60-inch cyclotron forming $^{230}$Pa in (d,4n) and ($\alpha$,p5n) reactions. $^{222}$Ra was populated by subsequent $\alpha$ decay after the initial $\beta^-$ decay of $^{230}$Pa to $^{230}$U. Alpha-decay spectra were measured following chemical separation. “It was necessary to measure the activity at such short intervals of time because of the short half-life of Ra$^{222}$, and as a result the statistical fluctuations were rather severe. An integral decay curve was plotted to minimize the scatter of points... The half-life of Ra$^{222}$ based on four such determinations is 38.0 seconds.” The quoted half-life is included in the calculation of the currently adopted value of 38.0(5) s. $^{223}$Ra {#ra-13 .unnumbered} ---------- In the 1905 article “A new radio-active product from actinium,” Godlewski reported the discovery of a new activity in the natural actinium decay chain which was later identified as $^{223}$Ra [@1905God01]. The activity of actinium samples were measured following chemical separation. “Taking into consideration the similarity of actinium and thorium, both as regards their chemical and radioactive properties, I resolved to try if the method used by Rutherford and Soddy for the separation of ThX would not serve also to separate an analogous product from actinium. The experiments were at once successful... This substance, which is so similar in properties to ThX, will be called actinium X (AcX). The product AcX, immediately after its separation, weight for weight, was more than a hundred times more active than the original actinium. The activity increased in the first day after removal to about 15 per cent. of its original value, and then decayed with the time according to an exponential law, falling to half value in about ten days.” This half-life agrees with the currently accepted value of 11.43(5) d. A year earlier Giesel had reported a new substance separated from emanium (actinium) [@1904Gie01] without any measurements [@1905God02]. $^{224}$Ra {#ra-14 .unnumbered} ---------- In 1902 Rutherford and Soddy announced the discovery of a new activity extracted from thorium later identified as $^{224}$Ra in the paper “The cause and nature of radioactivity - part I” [@1902Rut01]. Activities from a thorium nitrate sample were observed with a photographic plate and an electrometer following chemical separation. “If for present purposes the initial periods of the curves are disregarded and the later portions only considered, it will be seen at once that the time taken for the hydroxide to recover one half of its lost activity is about equal to the time taken by the ThX to lose half its activity, viz., in each case about 4 days, and speaking generally the percentage proportion of the lost activity regained by the hydroxide over any given interval is approximately equal to the percentage proportion of the activity lost by the ThX during the same interval.” The quoted half-life is close to the currently adopted value of 3.66(4) d. $^{225}$Ra {#ra-15 .unnumbered} ---------- Hagemann et al. discovered $^{225}$Ra in 1947 in “The (4n+1) radioactive series: the decay products of U$^{233}$” [@1947Hag01]. The half-lives and $\alpha$- and $\beta$-decay energies of the nuclides in the decay chain of $^{233}$U were measured. “These decay products, which constitute a substantial fraction of the entire missing, 4n+1, radioactive series are listed together with their radioactive properties, in \[the table\].” The measured half-life of 14.8 d agrees with the presently accepted value of 14.9(2) d. Hagemann et al. acknowledge the simultaneous observation of $^{225}$Ra by English et al. which was submitted only a day later and published in the same issue of Physical Review on the next page [@1947Eng01]. $^{226}$Ra {#ra-16 .unnumbered} ---------- In 1898 Curie et al. announced the discovery of a new radioactive substance later identified as $^{226}$Ra in the paper “Sur une nouvelle substance fortement radio-active, contenue dans la pechblende” [@1898Cur02]. During the study of radioactivity of pitchblende in addition to polonium a second new radioactive element was chemically separated. “La nouvelle substance radio-active que nous venons de trouver a toutes les apparences chimiques du baryum presque pur: elle n’est précipitée ni par l’hydrogene sulfure, ni par le sulfure d’ammonium, ni par l’ammoniaque; le sulfate est insoluble dans l’eau et dans les acides; le carbonate est insoluble dans l’eau; le chlorure, très soluble dans l’eau, est insoluble dans l’acide chlorhydrique concentré et dans l’alcool. Enfin cette substance donne le spectre du baryum, facile à reconnaitre... Les diverses raisons que nous venons d’énumerér nous portent à croire que la nouvelle substance radio-active renferme un élément nouveau, auquel nous proposons de donner le nom de radium.” \[The new radio-active substance which we have found has all the chemical appearances of almost pure barium: it is neither precipitated by hydrogen sulfide nor by the ammonium sulfide, nor with ammonia; the sulfate is insoluble in water and in acids; the carbonate is insoluble in water, the chloride, very soluble in water, is insoluble in concentrated hydrochloric acid and alcohol. Finally, the spectrum of this substance is easily recognizable as barium... The various reasons we have enumerated lead us to believe that the new radioactive substance contains a new element, which we propose to name radium.\] The currently adopted half-life is 1600(7) y. $^{227}$Ra {#ra-17 .unnumbered} ---------- $^{227}$Ra was discovered by Butler and Adam and the results were published in the 1953 article “Radiations of Ra$^{227}$” [@1953But01]. $^{226}$Ra was irradiated with thermal neutrons from the Chalk River NRX reactor producing $^{227}$Ra in neutron capture reactions. Beta- and gamma-rays were measured with an end-window Geiger counter and a sodium iodide (thalliated) scintillation spectrometer, respectively, following chemical separation. “Ra$^{227}$ is a $\beta^-$ emitter with a half-life of 41.2$\pm$0.2 minutes and with a $\beta^-$ endpoint of 1.31$\pm$0.02 Mev.” This half-life agrees with the currently accepted value of 42.2(5) min. $^{228}$Ra {#ra-18 .unnumbered} ---------- Hahn first observed a new activity in the natural thorium decay series later identified as $^{228}$Ra in 1907 and published his results in the article “Ein neues Zwischenprodukt im Thorium” [@1907Hah01]. The activities from thorium nitrate samples of different ages were compared. “Es ist naheliegend, daß die allmähliche Abnahme der Aktivität der frisch bereiteten Thorpräparate vom Zerfall eines aus irgend einem Grunde vorliegenden Überschusses an Radiothor herrührt; und in der Tat ergibt sich als Größenordnung für die Abnahme eine Periode von etwa zwei Jahren... Von der definitiven Wahl eines Namens für den neuen Körper sehe ich ab, bis sich seine Natur genauer hat feststellen lassen. Im letzteren Fall würde mir der Name ‘Mesothorium’ als zweckmäßig erscheinen.” \[It seems evident that the slow reduction of the activity of the newly prepared thorium samples is due to the decay of an unexplained excess of radio thorium; and indeed the order of magnitude of the activity is about two years... I restrain from selecting a name for the new substance until its properties are better determined. In the latter case the name ‘mesothorium’ seems suitable.\] Hahn submitted the identical paper three days later to another journal [@1907Hah02]. In a subsequent paper Hahn estimated the half-life of mesothorium to be about 7 years [@1907Hah03]. A year later Hahn quoted a half-life of $\sim$5.5 y [@1908Hah01] which agrees with the presently accepted value of 5.75(3) y for $^{228}$Ra. In the same paper Hahn renamed mesothorium to mesothorium 1 because he identified a separated decay product (mesothorium 2 or $^{228}$Ac). $^{229}$Ra {#ra-19 .unnumbered} ---------- In 1975 the discovery of $^{229}$Fr by Ravn et al. was announced in the paper “Short-lived isotopes of alkali and alkaline-earth elements studied by on-line isotope separator techniques” [@1975Rav01]. A thorium plus lanthanum target was bombarded with protons from the CERN synchrocyclotron. Beta-ray decay curves were measured with a 4$\pi$ plastic detector following mass separation with the isotope separator ISOLDE. “The following half-lives of new nucleides have been determined: ... $^{229}$Ra (4.0$\pm$0.2) min.” This half-life is the current value. $^{230}$Ra {#ra-20 .unnumbered} ---------- In the 1978 article “Decay of $^{230}$Ra and $^{230}$Ac,” Gilat and Katcoff announced the discovery of $^{230}$Ra [@1978Gil01]. At the Brookhaven Medium Energy Intense Neutron (MEIN) facility thorium targets were irradiated with secondary neutrons produced by stopping 200 MeV protons from the Brookhaven AGS injector Linac in a copper beam stop. $^{230}$Ra was produced in $^{232}$Th(n,2pn) reactions. Gamma-ray spectra and decay curves were measured following chemical separation. “The half-life of $^{230}$Ra was determined by: (1) analysis of the decay curves of $\sim$20 of the most prominent lines in the $^{230}$Ra-$^{230}$Ac spectrum, some of which are shown in \[the figure\]; and (2) milking experiments, in which the $^{23°}$Ac daughter was milked successively at 20 min intervals over a period of 3 hr, and the parent half-life was extracted from the values of the daughter activity. Both methods gave the identical result of 93$\pm$2 min.” This half-life is the currently adopted value. The tentative assignment of a 45.5(15) min half-life to $^{230}$Ra was evidently incorrect [@1975Rav01]. An even earlier measurement of a $\sim$1 hr half-life assigned to $^{230}$Ra was only published in a conference abstract [@1952Jen01]. $^{231,232}$Ra {#ra-21 .unnumbered} -------------- The observation of $^{231}$Ra and $^{232}$Ra was reported by Ahmad et al. in the 1983 paper “Determination of nuclear spins and moments in a series of radium isotopes” [@1983Ahm01]. A UC$_2$ target was bombarded with 600 MeV protons from the CERN synchrocyclotron. $^{231}$Ra and $^{232}$Ra were identified by on-line collinear fast beam laser spectroscopy following mass separation with the isotope separator ISOLDE. “The optical high-resolution measurements on 18 Ra isotopes in the range A = 208–232 have been carried out in the 4826 Å line (7s$^2$ $^1$S$_0$-7s7p$^1$P$_1$) of the neutral Ra and the 4683 Å line (7s $^2$S$_{1/2}$-7p$^{2}$P$_{1/2}$) of the Ra$^+$ spectra.” The presently accepted half-lives for $^{213}$Ra and $^{232}$Ra are 103(3) s and 250(50) s. $^{233,234}$Ra {#ra-22 .unnumbered} -------------- Mezlev et al. reported the discovery of $^{233}$Ra and $^{234}$Ra in the 1990 paper “Search for delayed fission in neutron-rich nuclei” [@1990Mez01]. A uranium target was bombarded with 1 GeV protons. Beta-, gamma-, and X-rays were measured with solid state detectors following mass separation with the on-line mass separator IRIS. “Due to this technique the new isotopes $^{232}$Fr (T$_{1/2}$=5$\pm$1 s), $^{233}$Ra(T$_{1/2}$=30$\pm$5 s) and $^{234}$Ra (T$_{1/2}$=30$\pm$10 s) have been identified using the solid state detectors for the registration of the beta-, gamma-, X-radiation.” For $^{233}$Ra, the quoted half-life is the currently accepted value, and for $^{234}$Ra the quoted half-life agrees with the currently adopted value of 30(10) s. Summary ======= The discoveries of the known astatine, radon, francium, and radium isotopes have been compiled and the methods of their production discussed. Only the following six isotopes were initially identified incorrectly: $^{204,206,216}$At, $^{214}$Rn, and $^{202,230}$Ra. In addition, the half-lives of $^{200,201,211}$At and $^{205,212}$Ra were at first reported without definite mass assignments. This work was supported by the National Science Foundation under grants No. PHY06-06007 (NSCL) and PHY10-62410 (REU). Table 1.\[tbl1te\] Discovery of astatine, radon, francium, and radium isotopes =============================================================================== [0.95]{}[@@lp[5.5in]{}@]{}\ Isotope & Astatine, radon, francium, and radium isotope\ First author & First author of refereed publication\ Journal & Journal of publication\ Ref. & Reference\ Method & Production method used in the discovery:\ & FE: fusion evaporation\ & NC: Neutron capture reactions\ & LP: light-particle reactions (including neutrons)\ & RD: radioactive decay\ & SP: spallation reactions\ & PF: projectile fragmentation of fission\ Laboratory & Laboratory where the experiment was performed\ Country & Country of laboratory\ Year & Year of discovery\ \[tableI\] [@llllllll@]{} Isotope & First Author & Journal & Ref. & Method & Laboratory & Country & Year\ \ \ Isotope & First author & Journal & Ref. & Method & Laboratory & Country & Year\ \ $^{191}$At & H. Kettunen & Eur. Phys. J. A &[@2003Ket01]& FE & Jyväskylä & Finland &2003\ $^{192}$At & A.N. Andreyev & Phys. Rev. C &[@2006And01]& FE & Darmstadt & Germany &2006\ $^{193}$At & H. Kettunen & Eur. Phys. J. A &[@2003Ket01]& FE & Jyväskylä & Finland &2003\ $^{194}$At & A.N. Andreyev & Phys. Rev. C &[@2009And01]& FE & Darmstadt & Germany &2009\ $^{195}$At & Y. Tagaya & Eur. Phys. J. A &[@1999Tag01]& FE & RIKEN & Japan &1999\ $^{196}$At & W. Treytl & Nucl. Phys. A &[@1967Tre01]& FE & Berkeley & USA &1967\ $^{197}$At & W. Treytl & Nucl. Phys. A &[@1967Tre01]& FE & Berkeley & USA &1967\ $^{198}$At & W. Treytl & Nucl. Phys. A &[@1967Tre01]& FE & Berkeley & USA &1967\ $^{199}$At & W. Treytl & Nucl. Phys. A &[@1967Tre01]& FE & Berkeley & USA &1967\ $^{200}$At & R.W. Hoff & J. Inorg. Nucl. Chem. &[@1963Hof01]& FE & Berkeley & USA &1963\ $^{201}$At & R.W. Hoff & J. Inorg. Nucl. Chem. &[@1963Hof01]& FE & Berkeley & USA &1963\ $^{202}$At & R.M. Latimer & J. Inorg. Nucl. Chem. &[@1961Lat01]& FE & Berkeley & USA &1961\ $^{203}$At & G.W. Barton & Phys. Rev. &[@1951Bar01]& LP & Berkeley & USA &1951\ $^{204}$At & R.M. Latimer & J. Inorg. Nucl. Chem. &[@1961Lat01]& FE & Berkeley & USA &1961\ $^{205}$At & G.W. Barton & Phys. Rev. &[@1951Bar01]& LP & Berkeley & USA &1951\ $^{206}$At & R.M. Latimer & J. Inorg. Nucl. Chem. &[@1961Lat01]& FE & Berkeley & USA &1961\ $^{207}$At & G.W. Barton & Phys. Rev. &[@1951Bar01]& LP & Berkeley & USA &1951\ $^{208}$At & E.K. Hyde & Phys. Rev. &[@1950Hyd01]& LP & Berkeley & USA &1950\ $^{209}$At & G.W. Barton & Phys. Rev. &[@1951Bar01]& LP & Berkeley & USA &1951\ $^{210}$At & E.L. Kelly & Phys. Rev. &[@1949Kel01]& LP & Berkeley & USA &1949\ $^{211}$At & D.R. Corson & Phys. Rev. &[@1940Cor01]& LP & Berkeley & USA &1940\ $^{212}$At & M.M. Winn & Proc. Roy. Soc. A &[@1954Win01]& LP & Birmingham & UK &1954\ $^{213}$At & R.L. Hahn & Nucl. Phys. A &[@1968Hah01]& LP & Oak Ridge & USA &1968\ $^{214}$At & W.W. Meinke & Phys. Rev. &[@1949Mei01]& LP & Berkeley & USA &1949\ $^{215}$At & B. Karlik & Naturwiss. &[@1944Kar01]& RD & Wien & Austria &1944\ $^{216}$At & A. Ghiorso & Phys. Rev. &[@1948Ghi01]& LP & Berkeley & USA &1948\ $^{217}$At & F. Hagemann & Phys. Rev. &[@1947Hag01]& RD & Argonne & USA &1947\ $^{218}$At & B. Karlik & Naturwiss. &[@1943Kar01]& RD & Wien & Austria &1943\ $^{219}$At & E.K. Hyde & Phys. Rev. &[@1953Hyd01]& RD & Berkeley & USA &1953\ $^{220}$At & C.F. Liang & J. Phys. G &[@1989Lia01]& SP & Orsay & France &1989\ $^{221}$At & D.G. Burke & Z. Phys. A &[@1989Bur01]& SP & CERN & Switzerland &1989\ $^{222}$At & D.G. Burke & Z. Phys. A &[@1989Bur01]& SP & CERN & Switzerland &1989\ $^{223}$At & D.G. Burke & Z. Phys. A &[@1989Bur01]& SP & CERN & Switzerland &1989\ $^{224}$At & L. Chen & Phys. Lett. B &[@2010Che01]& PF & Darmstadt & Germany &2010\ $^{225}$At & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ $^{226}$At & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ $^{227}$At & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ $^{228}$At & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ $^{229}$At & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ & & & & & &\ & & & & & &\ $^{193}$Rn & A.N. Andreyev & Phys. Rev. C &[@2006And02]& FE & Darmstadt & Germany &2006\ $^{194}$Rn & A.N. Andreyev & Phys. Rev. C &[@2006And02]& FE & Darmstadt & Germany &2006\ $^{195}$Rn & H. Kettunen & Phys. Rev. C &[@2001Ket01]& FE & Jyväskylä & Finland &2001\ $^{196}$Rn & K. Morita & Z. Phys. A &[@1995Mor01]& FE & RIKEN & Japan &1995\ $^{197}$Rn & K. Morita & Z. Phys. A &[@1995Mor01]& FE & RIKEN & Japan &1995\ $^{198}$Rn & F. Calaprice & Phys. Rev. C &[@1984Cal01]& SP & CERN & Switzerland &1984\ $^{199}$Rn & A.C. DiRienzo & Phys. Rev. C &[@1980DiR01]& FE & Brookhaven & USA &1980\ $^{200}$Rn & P. Hornshoj & Nucl. Phys. A &[@1971Hor01]& SP & CERN & Switzerland &1971\ $^{201}$Rn & K. Valli & Phys. Rev. &[@1967Val01]& FE & Berkeley & USA &1967\ $^{202}$Rn & K. Valli & Phys. Rev. &[@1967Val01]& FE & Berkeley & USA &1967\ $^{203}$Rn & K. Valli & Phys. Rev. &[@1967Val01]& FE & Berkeley & USA &1967\ $^{204}$Rn & K. Valli & Phys. Rev. &[@1967Val01]& FE & Berkeley & USA &1967\ $^{205}$Rn & K. Valli & Phys. Rev. &[@1967Val01]& FE & Berkeley & USA &1967\ $^{206}$Rn & W.E. Burcham & Proc. Phys. Soc. A &[@1954Bur01]& FE & Birmingham & UK &1954\ $^{207}$Rn & W.E. Burcham & Proc. Phys. Soc. A &[@1954Bur01]& FE & Birmingham & UK &1954\ $^{208}$Rn & F.F. Momyer& J. Inorg. Nucl. Chem. &[@1955Mom01]& SP & Berkeley & USA &1955\ $^{209}$Rn & F.F. Momyer& Phys. Rev. &[@1952Mom01]& SP & Berkeley & USA &1952\ $^{210}$Rn & F.F. Momyer& Phys. Rev. &[@1952Mom01]& SP & Berkeley & USA &1952\ $^{211}$Rn & F.F. Momyer& Phys. Rev. &[@1952Mom01]& SP & Berkeley & USA &1952\ $^{212}$Rn & E.K. Hyde & Phys. Rev. &[@1950Hyd01]& SP & Berkeley & USA &1950\ $^{213}$Rn & H. Rotter & Sov. J. Nucl. Phys. &[@1967Rot01]& FE & Dubna & Russia &1967\ $^{214}$Rn & D.F. Torgerson & Nucl. Phys. A &[@1970Tor01]& FE & Yale & USA &1970\ $^{215}$Rn & W.W. Meinke & Phys. Rev. &[@1952Mei01]& LP & Berkeley & USA &1952\ $^{216}$Rn & W.W. Meinke & Phys. Rev. &[@1949Mei01]& LP & Berkeley & USA &1949\ $^{217}$Rn & W.W. Meinke & Phys. Rev. &[@1949Mei01]& LP & Berkeley & USA &1949\ $^{218}$Rn & M.H. Studier & Phys. Rev. &[@1948Stu01]& LP & Argonne & USA &1948\ $^{219}$Rn & F. Giesel & Ber. Deuts. Chem. Ges. &[@1903Gie01]& RD & Braunschweig & Germany &1903\ $^{220}$Rn & E. Rutherford & Phil. Mag. &[@1900Rut01]& RD & McGill & Canada &1900\ $^{221}$Rn & F.F. Momyer& Phys. Rev. &[@1956Mom01]& LP & Berkeley & USA &1956\ $^{222}$Rn & P. Curie & Compt. Rend. Acad. Sci. &[@1899Cur01]& RD & Paris & France &1899\ $^{223}$Rn & F.D.S. Butement & J. Inorg. Nucl. Chem. &[@1964But01]& LP & Liverpool & UK &1964\ $^{224}$Rn & F.D.S. Butement & J. Inorg. Nucl. Chem. &[@1964But01]& LP & Liverpool & UK &1964\ $^{225}$Rn & P.G. Hansen & Phys. Lett. B &[@1969Han01]& SP & CERN & Switzerland &1969\ $^{226}$Rn & P.G. Hansen & Phys. Lett. B &[@1969Han01]& SP & CERN & Switzerland &1969\ $^{227}$Rn & M.J.G. Borge & Z. Phys. A &[@1986Bor01]& SP & CERN & Switzerland &1986\ $^{228}$Rn & M.J.G. Borge & Z. Phys. A &[@1989Bor01]& SP & CERN & Switzerland &1989\ $^{229}$Rn & D. Neidherr & Phys. Rev. Lett. &[@2009Nei01]& SP & CERN & Switzerland &2009\ $^{230}$Rn & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ $^{231}$Rn & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ & & & & & &\ & & & & & &\ $^{199}$Fr & Y. Tagaya & Eur. Phys. J. A &[@1999Tag01]& FE & RIKEN & Japan &1999\ $^{200}$Fr & K. Morita & Z. Phys. A &[@1995Mor01]& FE & RIKEN & Japan &1995\ $^{201}$Fr & G.T. Ewan & Z. Phys. A &[@1980Ewa01]& SP & CERN & Switzerland &1980\ $^{202}$Fr & G.T. Ewan & Z. Phys. A &[@1980Ewa01]& SP & CERN & Switzerland &1980\ $^{203}$Fr & K. Valli & J. Inorg. Nucl. Chem. &[@1967Val02]& FE & Berkeley & USA &1967\ $^{204}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{205}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{206}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{207}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{208}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{209}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{210}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{211}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{212}$Fr & E.K. Hyde & Phys. Rev. &[@1950Hyd01]& SP & Berkeley & USA &1950\ $^{213}$Fr & R.D. Griffioen & Phys. Rev. &[@1964Gri01]& FE & Berkeley & USA &1964\ $^{214}$Fr & H. Rotter & Sov. J. Nucl. Phys. &[@1967Rot01]& FE & Dubna & Russia &1967\ $^{215}$Fr & J. Borggreen & Phys. Rev. C &[@1970Bor01]& FE & Berkeley & USA &1970\ $^{216}$Fr & J. Borggreen & Phys. Rev. C &[@1970Bor01]& FE & Berkeley & USA &1970\ $^{217}$Fr & R.L. Hahn & Nucl. Phys. A &[@1968Hah01]& LP & Oak Ridge & USA &1968\ $^{218}$Fr & W.W. Meinke & Phys. Rev. &[@1949Mei01]& LP & Berkeley & USA &1949\ $^{219}$Fr & A. Ghiorso & Phys. Rev. &[@1948Ghi01]& LP & Berkeley & USA &1948\ $^{220}$Fr & A. Ghiorso & Phys. Rev. &[@1948Ghi01]& LP & Berkeley & USA &1948\ $^{221}$Fr & F. Hagemann & Phys. Rev. &[@1947Hag01]& RD & Argonne & USA &1947\ $^{222}$Fr & L. Westgaard & Z. Phys. A &[@1975Wes01]& SP & CERN & Switzerland &1975\ $^{223}$Fr & M. Perey & Compt. Rend. Acad. Sci. &[@1939Per01]& RD & Paris & France &1939\ $^{224}$Fr & P.G. Hansen & Phys. Lett. B &[@1969Han01]& SP & CERN & Switzerland &1969\ $^{225}$Fr & P.G. Hansen & Phys. Lett. B &[@1969Han01]& SP & CERN & Switzerland &1969\ $^{226}$Fr & P.G. Hansen & Phys. Lett. B &[@1969Han01]& SP & CERN & Switzerland &1969\ $^{227}$Fr & R. Klapisch & Phys. Rev. Lett. &[@1972Kla01]& SP & CERN & Switzerland &1972\ $^{228}$Fr & R. Klapisch & Phys. Rev. Lett. &[@1972Kla01]& SP & CERN & Switzerland &1972\ $^{229}$Fr & H.L. Ravn & J. Inorg. Nucl. Chem. &[@1975Rav01]& SP & CERN & Switzerland &1975\ $^{230}$Fr & W. Kurcewicz & Nucl. Phys. A &[@1987Kur01]& SP & CERN & Switzerland &1987\ $^{231}$Fr & P. Hill & Z. Phys. A &[@1985Hil01]& SP & CERN & Switzerland &1985\ $^{232}$Fr & K.A. Mezlev & Z. Phys. A &[@1990Mez01]& SP & Leningrad & Russia &1990\ $^{233}$Fr & H. Alvarez-Pol & Phys. Rev. C &[@2010Alv01]& PF & Darmstadt & Germany &2010\ & & & & & &\ & & & & & &\ $^{201}$Ra & J. Uusitalo & Phys. Rev. C &[@2005Uus01]& FE & Jyväskylä & Finland &2005\ $^{202}$Ra & J. Uusitalo & Phys. Rev. C &[@2005Uus01]& FE & Jyväskylä & Finland &2005\ $^{203}$Ra & M. Leino & Z. Phys. 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--- abstract: | Reported here are theoretical calculations on the trifluoromethanesulfonic (triflic) acid and water molecules, establishing molecular scale information necessary to molecular modeling of the structure, thermodynamics, and ionic transport of Nafionmembranes. The optimized geometry determined for the isolated triflic acid molecule, obtained from [*ab initio*]{} molecular orbital calculations, agrees with previous studies. In order to characterize side chain flexibility and accessibility of the acid proton, potential energy and free energy surfaces for rotation of both carbon-sulfur and sulfur-oxygen(hydroxyl) bonds are presented. A continuum dielectric solvation model is used to obtain free energies of electrostatic interaction with the solvent. Electrostatic solvation is predicted to reduce the free energy barrier to rotation of the F$_3$C-SO$_3$ bond to about 2.7 kcal/mol. This electrostatic effect is associated with slight additional polarization of the CF bond in the eclipsed conformation. The energetic barrier to rotation of the acid hydroxyl group away from the sulfonic acid oxygen plane, out into the solvent is substantially flattened by electrostatic solvation effects. The maximum free energy change for those solvent accessible proton conformations is less than one kcal/mol. We carried out additional [*ab initio*]{} electronic structure calculations with a probe water molecule interacting with the triflic acid. The minimum energy structures found here for the triflic acid molecule with the probe water revise results reported previously. To investigate the reaction path for abstraction of a proton from triflic acid, we found minimum energy structures, energies, and free energies for: (a) a docked configuration of triflate anion and hydronium cation and (b) a transition state for proton interchange between triflic acid and a water molecule. Those configurations are structurally similar but energetically substantially different. The activation free energy for that proton interchange is predicted to be 4.7 kcal/mol above the reaction end-points. [*Keywords:*]{} chemical equilibrium, electric conductivity, hydration, Nafion, solvation theory, trifluoromethanesulfonic acid. address: 'Materials Science and Technology Division$^\ast$ and Theoretical Division$^\dagger$ Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA ' author: - 'Stephen J. Paddison$^\ast$, Lawrence R. Pratt$^\dagger$, Thomas Zawodzinski$^\ast$, and David W. Reagor$^\ast$' - 'LA-UR-97-1688' title: Molecular Modeling of Trifluoromethanesulfonic Acid for Solvation Theory --- Introduction ============ Nafion, a perfluorinated ionomer widely used in polymer electrolyte fuel cells [@Lemons], consists of a hydrophobic Teflon backbone with randomly attached side chains terminating with trifluoromethanesulfonic (triflic) acid groups. These sulfonic acid functional groups are hydrophilic and preferentially hydrated. The Nafionmembrane functions as both a separator and an electrolyte in the fuel cell and the overall performance of the fuel cell is strongly influenced by the membrane conductivity, itself a function of the hydration [@Zawod:91; @Springer; @Zawod:93a; @Zawod:93b]. The water content of the membrane is determined both by equilibrium absorption and by nonequilibrium currents of ions and water. Extensive experimental measurement of electroosmotic drag coupling proton and water fluxes for various sulfonated membranes over a wide range of water content [@Zawod:93c; @Zawod:95; @Xie] has suggested that sulfonic acid interactions dominate proton transport. A molecular level understanding of these processes in Nafionis not available. Here we work towards a molecular description of these systems by securing information on molecular structures, energies, and charge distribution upon which molecular scale solution theory might be based. Electrostatic interactions of triflic acid with the environment are expected to be of first importance in this solution chemistry. Continuum dielectric solvation modeling is an appropriate approach to studying these solution processes; it is physical, computationally feasible, and can provide a basis for more molecular theory . Computational Methods ===================== Electronic Structure Calculations --------------------------------- Standard [*ab initio*]{} molecular orbital calculations were performed using the GAUSSIAN 92/DFT and 94 [@G92-DFT; @G94] systems of programs. Both the 6-31G\*\* split valence basis set [@Hariharan] and the D95\*\* Dunning double zeta basis set [@Dunning] were employed at the Hartree-Fock plus second-order Møller-Plesset (MP2) level of theory. Full geometry optimizations were obtained for isolated triflic acid (CF$_3$SO$_3$H), the triflate anion (CF$_3$SO$_3^-$), and triflic acid with a probe water molecule (CF$_3$SO$_3$H+H$_2$O) using gradient methods [@Schlegel]. Atom centered partial charges were obtained by the CHelpG scheme [@Breneman]. Potential energy surfaces for triflic acid were constructed from calculations at the MP2/6-31G\*\* level by rotation of the carbon-sulfur and sulfur-oxygen(hydroxyl) bonds with optimization of all other degrees of freedom. Finally, the optimized structures and energies were determined for the triflate-hydronium ion pair in two ways: (a) by docking triflate and hydronium ions, each rigidly constrained at their separate equilibrium geometries; and (b) by finding a transition state for interchange of a proton in the triflic-acid-water complex as CF$_3$SO$_3$H$^*$ + HOH = CF$_3$SO$_3$H + HOH$^*$ using Synchronous Transit-guided Quasi-Newton (STQN) methods [@Peng]. Dielectric Continuum Model -------------------------- A continuum dielectric model of aqueous solution chemistry has become a standard theoretical tool in recent years and discussions of basic issues can be found elsewhere [@Pratt:97]. The model identifies a solute molecule in a solution environment, defines a realistic solute molecular volume based on the geometry of the molecule, positions partial charges describing the solute electric charge distribution in that molecular volume, and assumes that the solvent exterior to the molecular volume may be idealized as a dielectric continuum. The numerical task is then the solution of the Poisson equation for which a defined local dielectric constant jumps from one (1.0) inside the molecular volume to the experimental value for the solution in the region exterior to the solute molecular volume. The numerical methods used here were boundary integral techniques outlined in References [@Corcelli; @Pratt:97]. The molecular volume is modeled as the union of spherical volumes centered on solute atoms. The radii adopted for those spheres depend on atomic type and are essentially empirical. Here we used the values determined by Stefanovich and Truong [@Stefanovich]. The value 77.4 was used for the solvent dielectric constant. Results and Discussion ====================== The optimized geometries of both triflic acid and the triflate anion are essentially identical to the structures reported by Gejji [*et al.*]{} [@Gejji:93a; @Gejji:93b] at the same levels of theory. Both structures are in good agreement with experimentally determined structures [@Schultz; @Delaplane], the only difference being that the sulfur-oxygen and carbon-fluorine bond lengths are 0.03-0.08Ålonger in the theoretical structures. To characterize the flexibility of the side chains of the perfluorinated membrane and the accessibility of the acidic protons, potential energy surfaces for triflic acid were determined for rotation of the carbon-sulfur bond and the sulfur-oxygen(hydroxyl) bond. The relative energies for rotation of both of these bonds are reported in Tables 1 and 2. The conformational potential energy surface for rotation of the carbon-sulfur bond is given in Figure 1. The highest energy structure occurs when the oxygen and fluorine atoms are eclipsed and the minimum energy structure when those atoms are staggered. The activation energy is 3.5 kcal/mol but the solvation free energy lowers that barrier to about 2.7 kcal/mol. This is mostly due to a slight additional polarization of the C-F bonds in the eclipsed conformation. Figure 2 shows as function of rotation of the carbon-sulfur bond the atom centered partial charges obtained for the triflic acid molecule. The sulfur-oxygen(hydroxyl) rotational potential energy surface possesses two physically equivalent minima (Figure 3). The lowest barrier between these minima corresponds to the proton directed axially, out into the solvent and has an energy of 2.0 kcal/mol. Electrostatic solvation substantially flattens that barrier so that the maximum free energy between the two minima is less than one kcal/mol. This stabilization of the barrier configuration is due to the better access of the proton to the solvent. The position of a probe water molecule interacting with triflic acid was determined by fully optimizing the pair of molecules at the MP2 level using both the 6-31G\*\* and D95\*\* basis sets. Recently, Ricchiardi and Ugliengo [@Ricchiardi] reported a geometry optimization of triflic acid plus a water molecule at the SCF/DZP level. Their result is qualitatively similar but quantitatively substantially different from that presented below. Because our results differed from those previous ones, several different basis sets and starting conditions were considered. The differences we observed among those optimizations were slight and do not account for the differences with the previous work that are likely traceable to different level of effort in achieving the optimum structure. Our results are shown in Figure 4. That structure is of lower energy than the structure reported by Ricchiardi and Ugliengo. However, our experience with this optimization is that the triflic-acid-water configurations are floppy as might be expected from noncovalent hydrogen-bonding interactions. For example, we several times found a local minimum of the energy surface with the exterior proton of Figure 4 directed into the plane of that drawing. Potential energy minima of that sort are only slightly higher in energy, of the order of one kcal/mol, than the structure shown. Note that the dihedral angle positioning the triflic acid proton is nearly the same in the minimum energy complex and the minimum energy triflic acid molecule (Figure 3). The floppiness of these structures is consistent with the prediction of the dielectric solvation model that free energy varies only gently with disposition of the acid proton in the region between the two minima (Figure 3). Attempts were made to determine a minimum energy conformation for triflate-hydronium ion pair but those configurations always relaxed to the optimum geometry of the triflic-acid-water molecule pair. Separately optimized structures for the triflate anion and the hydronium cation were then docked in an optimization that adjusted only the relative positions of the two ions constrained separately to be rigid. That optimum ion pair geometry is presented in Figure 4 also. The energy of the docked ion pair is 27.0 kcal/mol higher than the fully optimized molecule pair. The electrostatic solvation free energy of the docked ion pair relative to that of the optimized molecular complex is computed to be -16.0 kcal/mol so that net free energy of the docked ion pair, as a complex, is approximately 11.0 kcal/mol above that of the minimum energy molecule complex. Further efforts to determine a reaction path for abstraction of the proton from the triflic acid molecule were based upon the observation (Figure 4) that another minimum energy structure for the triflic-acid-water complex is obtained by cycling the triflic acid proton with a distinct water proton to achieve a physically equivalent but different triflic-acid-water molecule pair. This can be alternatively described by considering that one of the two close protons of the docked hydronium ion will be donated to the triflate anion when it is neutralized. We defined those two possibilities as end-points for a reaction path and found a transition state (Figure 4) using STQN methods \[26\]. Structurally, the docked ion pair and the transition state are quite similar. The energy of this transition state is 9.3 kcal/mol higher than the end-points of the reaction path or about 18 kcal/mol lower in than the energy of the docked structure. The electrostatic solvation free energy relative to the reaction end-point is predicted to be -4.6 kcal/mol so the activation free energy is about 4.7 kcal/mol. Conclusions =========== The results reported here establish molecular scale information necessary to molecular modeling of the structure, thermodynamics, and ionic transport of Nafionmembranes. Because of a slight additional polarization of the CF bond in the eclipsed conformation, electrostatic solvation reduces the free energy barrier to rotation of the F$_3$C-SO$_3$ bond to about 2.7 kcal/mol. Solvation enhances the accessibility of the acid proton by substantially flattening the energetic barrier to rotation of the acid hydroxyl group away from the sulfonic acid oxygen plane, out into the solvent. The free energy barrier for those solvent accessible proton conformations is less than one kcal/mol relative to the minimum energy conformations. The minimum energy structures found for the triflic acid molecule with the probe water revise structures reported previously. A doubly hydrogen-bonded configuration is found in which the triflic acid proton forms a short hydrogen-bond. The second hydrogen-bond is of a more traditional type. Minimum energy structures, energies, and free energies were found for: (a) a docked configuration of triflate anion and hydronium cation and (b) a transition state for proton interchange between triflic acid and a water molecule. Those configurations are structurally similar but energetically substantially different. The activation free energy for that proton interchange is predicted to be 4.7 kcal/mol above the reaction end-points. Experimentally determined values of activation energies for protonic conductivity in Nafionand similar perfluorosulfonic acid membranes are roughly 3 kcal/mole for a fully hydrated membrane and increases with decreasing water content \[5\]. We thank P. J. Hay and R. L. Martin for helpful discussions. This work was supported by the Los Alamos National Laboratory LDRD program. R. Lemons, J. Power Sources, 29 (1990) 251. T. A. Zawodzinski, M. Neeman, L. D. Sillerud and S. Gottesfeld, J. Phys. Chem. 95 (1991) 6040. T. E. Springer, T. A. Zawodzinski and S. Gottesfeld, J. Electrochem. Soc. 138 (1991) 2334. T. A. Zawodzinski, T. E. Springer, J. Davey, R. Jestel, C. Lopez, J. Valerio and S. Gottesfeld, J. Electrochem. Soc. 140 (1993) 1981. T. A. Zawodzinski, T. E. Springer, F. A. Uribe and S. Gottesfeld, Solid State Ionics 60 (1993) 199. T. A. Zawodzinski, C. Derouin, S. 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Pratt, G. J. Tawa, G. Hummer, A. E. Garc’í’a, S. A. Corelli, Int. J. Quant. Chem. LA-UR-95-2659, (in press 1997). G. Hummer, L. R. Pratt, and A. E. Garcí’a, LA-UR-96-4479, J. Phys. Chem. (in press 1997). G. Hummer, L. R. Pratt, and A. E. Garcí’a, LA-UR-97-1232: “Multistate gaussian model for polar and ionic hydration.” M. J. Frisch, G. W. Truck, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. W. Wong, J. B. Foresman, M. A. Robb, M. Head-Gordon, E. S. Replogle, R. Gomperts, J. L. Andres, K.Raghavachari, J. S. Binkley, C. Gonzalez, R. L. Martin, D. J. Fox, D. J. Defrees, J. Baker, J. J. P. Stewart and J. A. Pople, Gaussian 92/DFT, Gaussian, Inc., Pittsburgh, PA, 1993. M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. A. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, Gaussian 94 (Revision D.1), Gaussian, Inc., Pittsburgh, PA, 1995. P. C. Hariharan and J. A. Pople, Theo. Chim. Acta., 28 (1973) 213. T. H. Dunning, Jr. and P. J. Hay, in H.F. Schaefer III (Ed.), Modern Theoretical Chemistry, Plenum, New York, 1976, pp. 1-28. H. B. Schlegel, J. Comp. Chem., 3 (1982) 214. C. M. Breneman and K. B. Wiberg, J. Comp. Chem. 11 (1990) 361. C. Peng and H. B. Schlegel, Israel J. Chem. 33 (1993) 449. E. V. Stefanovich and T. N. Truong, Chem. Phys. Letts. 244 (1995) 65. S. P. Gejji, K. Hermansson, and J. Lindgren, J. Phys. Chem. 97 (1993) 6986. S. P. Gejji, K. Hermansson, and J. Lindgren, J. Phys. Chem. 97 (1993) 3712. G. Schultz, I. Hargittai, and R. Seip, Naturforsch. 36A (1981) 917. R. G. Delaplane, J. O. Lundren, and I. Olovsson, Acta Crystallogr. B31 (1975) 2202. R. Ricchiardi and P. Ugliengo, in S.M. Bachrach (Ed.), Proc. First Electron. Comput. Chem. Conf. \[CD-ROM\], ARInternet Corp., Landover, MD, 1995. [dd]{} angle & MP2 relative energy\ (deg) & (kcal/mol)\ 60.0 & 0.07\ 50.0 & 0.03\ 40.0 & 0.38\ 30.0 & 1.07\ 20.0 & 1.96\ 10.0 & 2.84\ 0.0 & 3.44\ -10.0 & 3.50\ -20.0 & 2.99\ -30.0 & 2.12\ -40.0 & 1.19\ -50.0 & 0.45\ -60.0 & 0.05\ -65.42 & 0.00\ [dd]{} angle & MP2 rel ative energy\ (deg) & (kcal/mol)\ 180.0 & 0.74\ 170.0 & 0.31\ 160.0 & 0.07\ 150.0 & 0.00\ 140.0 & 0.09\ 130.0 & 0.31\ 120.0 & 0.62\ 110.0 & 0.98\ 100.0 & 1.34\ 90.0 & 1.66\ 80.0 & 1.89\ 70.0 & 1.98\ 60.0 & 1.92\ 50.0 & 1.73\ 40.0 & 1.43\ 30.0 & 1.07\ 20.0 & 0.71\ 10.0 & 0.38\ 0.0 & 0.14\ -10.0 & 0.01\ -20.0 & 0.03\ -30.0 & 0.23\ -40.0 & 0.61\ -50.0 & 1.18\ -60.0 & 1.94\ -70.0 & 2.87\ -80.0 & 3.90\ -90.0 & 4.93\ -100.0 & 5.79\ -110.0 & 6.28\ -120.0 & 6.02\ -130.0 & 5.23\ -140.0 & 4.23\ -150.0 & 3.17\ -160.0 & 2.19\ -170.0 & 1.37\ -13.52 & 0.00\

--- abstract: | We estimate the number $|\mathcal{A}_{{\boldsymbol}\lambda}|$ of elements on a nonlinear family $\mathcal{A}$ of monic polynomials of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree $r$ having factorization pattern ${\boldsymbol}\lambda:=1^{\lambda_1}2^{\lambda_2}\cdots r^{\lambda_r}$. We show that $|\mathcal{A}_{{\boldsymbol}\lambda}|= \mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}+\mathcal{O}(q^{r-m-{1}/{2}})$, where $\mathcal{T}({\boldsymbol}\lambda)$ is the proportion of elements of the symmetric group of $r$ elements with cycle pattern ${\boldsymbol}\lambda$ and $m$ is the codimension of $\mathcal{A}$. We provide explicit upper bounds for the constants underlying the $\mathcal{O}$–notation in terms of ${\boldsymbol}\lambda$ and $\mathcal{A}$ with “good” behavior. We also apply these results to analyze the average–case complexity of the classical factorization algorithm restricted to $\mathcal{A}$, showing that it behaves as good as in the general case. address: - '${}^{1}$Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina' - '${}^{2}$ National Council of Science and Technology (CONICET), Argentina' - '${}^{3}$Instituto de Ciencias, Universidad Nacional de Hurlingham, Av. Gdor. Vergara 2222 (B1688GEZ) Hurlingham, Buenos Aires, Argentina' - '${}^{4}$Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina' author: - 'Guillermo Matera${}^{1,2}$' - 'Mariana Pérez${}^{1,3}$' - 'Melina Privitelli${}^{2,4}$' title: Factorization patterns on nonlinear families of univariate polynomials over a finite field --- [^1] Introduction ============ The distribution of factorization patterns on univariate polynomials over a finite field ${{\mathbb F}_{\hskip-0.7mm q}}$ is a classical subject of combinatorics. Let ${\boldsymbol}\lambda:=1^{\lambda_1}2^{\lambda_2}\cdots r^{\lambda_r}$ be a factorization pattern for polynomials of degree $r$, namely $\lambda_1,\ldots,\lambda_r\in{\mathbb Z}_{\ge 0}$ satisfy $\lambda_1+2\lambda_2+\cdots+r\lambda_r=r$. A seminal article of S. Cohen ([@Cohen70]) shows that the proportion of elements of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree $r$ is roughly the proportion $\mathcal{T}({{\boldsymbol}{\lambda}})$ of permutations with cycle pattern ${\boldsymbol}\lambda$ in the $r$th symmetric group $\mathbb{S}_r$ (an element of $\mathbb{S}_r$ has cycle pattern ${\boldsymbol}\lambda$ if it has exactly $\lambda_i$ cycles of length $i$ for $1\le i\le r$). In particular, the number of irreducible polynomials, or more generally the distribution of factorization patterns, of polynomials of “given forms” has been considered in a number of recent articles (see, e.g., [@Pollack13], [@BaBaRo15], [@Ha16], [@CeMaPe17]). In [@Cohen72] a subset of the set of polynomials of degree $r$ is called [uniformly distributed]{} if the proportion of elements with factorization pattern ${\boldsymbol}\lambda$ is roughly $\mathcal{T}({\boldsymbol}\lambda)$ for every ${\boldsymbol}\lambda$. The main result of this paper ([@Cohen72 Theorem 3]) provides a criterion for a linear family of polynomials of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of given degree to be uniformly distributed in the sense above. [@BaBaRo15], [@Ha16] and [@CeMaPe17] provide explicit estimates on the number of elements with factorization pattern ${\boldsymbol}\lambda$ on certain linear families of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$, such as the set of polynomials with some prescribed coefficients. In [@GaHoPa99 Problem 2.2] the authors ask for estimates on the number of polynomials of a given degree with a given factorization pattern lying in [*nonlinear*]{} families of polynomials with coefficients parameterized by an affine variety defined over ${{\mathbb F}_{\hskip-0.7mm q}}$. Except for general results (see, e.g., [@ChDrMa92] and [@FrHaJa94]), very little is known on the asymptotic behavior of such a number. In this article we address this question, providing a general criterion for a nonlinear family $\mathcal{A}\subset{{\mathbb F}_{\hskip-0.7mm q}}[T]$ to be uniform distributed in the sense of Cohen and explicit estimates on the number of elements of $\mathcal{A}$ with a given factorization pattern. Then we apply our results on the distribution of factorization patterns to analyze the behavior of the classical factorization algorithm restricted to such families $\mathcal{A}$. The classical factorization algorithm (see, e.g., [@GaGe99]) is not the fastest one. Nevertheless, it is worth analyzing it, since it is implemented in several software packages for symbolic computation, and a number of scientific problems rely heavily on polynomial factorization over finite fields. A precise worst–case analysis is given in [@GaGe99]. On the other hand, an average–case analysis for the set of elements of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of a given degree is provided in [@FlGoPa01]. This analysis relies on methods of analytic combinatorics which cannot be extended to deal with the nonlinear families we are interested in this article. For this reason, we provide an analysis of its average–case complexity when restricted to any nonlinear family $\mathcal{A}$ satisfying our general criterion. Now we describe precisely our results. Let ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$ be the algebraic closure of ${{\mathbb F}_{\hskip-0.7mm q}}$. Let $m$ and $r$ be positive integers with $m<r$ and let $A_{r-1}{\ifm {,\ldots,} {$,\ldots,$}}A_{0}$ be indeterminates over ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$. For a fixed $k$ with $0\leq k\leq r-1$, we denote ${{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}{A_k}]:= {{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_{k+1},A_{k-1},\ldots,A_0]$. Let $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_m\in {{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}{A_k}]$ and let $W:=\{G_1=0,\ldots,G_m=0\}$ be the set of common zeros in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^r$ of $G_1,\ldots,G_m$. Denoting by ${{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ the set of monic polynomials of degree $r$ with coefficients in ${{\mathbb F}_{\hskip-0.7mm q}}$, we consider the following family of polynomials: $$\label{non-linear family A} \mathcal{A}:=\{T^r+a_{r-1}T^{r-1}+\cdots+a_0\in {{\mathbb F}_{\hskip-0.7mm q}}[T]_r: G_i(a_{r-1},\ldots,a_{k-1},a_{k+1},\ldots,a_{0})=0\,\,(1\leq i \leq m)\}.$$ Consider the weight ${\ifm {{\sf wt}} {{$\sf wt$}}}:{{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}{A_k}]\to{\mathbb N}_0$ defined by setting ${\ifm {{\sf wt}} {{$\sf wt$}}}(A_j):=r-j$ for $0\leq j \leq r-1$ and denote by $G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ the components of highest weight of $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_m$. Let $(\partial{\boldsymbol}{G}/\partial {\boldsymbol}{A})$ be the Jacobian matrix of $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_m$ with respect to ${\boldsymbol}{A_k}$. We shall assume that $G_1 {\ifm {,\ldots,} {$,\ldots,$}}G_m$ satisfy the following conditions: - $G_1,\ldots,G_m$ form a regular sequence[^2] of ${{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}{A_k}]$. - $(\partial{\boldsymbol}{G}/ \partial {\boldsymbol}{A_k})$ has full rank on every point of the $W$. - $G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ satisfy $({\sf H}_1)$ and $({\sf H}_2)$. In what follows we identify the set ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[T]_r$ of monic polynomials of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[T]$ of degree $r$ with ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^r$ by mapping each $f_{{\boldsymbol}a_0}:= T^ r + a_{r-1}T^ {r-1}+\cdots+a_0 \in {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[T]_r$ to ${\boldsymbol}a_0:=(a_{r-1},\ldots,a_0)\in{\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^r$. For $\mathcal{B}\subset {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[T]_r$, the set of elements of $\mathcal{B}$ which are not square–free is called the discriminant locus $\mathcal{D}(\mathcal{B})$ of $\mathcal{B}$ (see [@FrSm84] and [@MaPePr14] for the study of discriminant loci). For $f_{{\boldsymbol}a_0}\in \mathcal{B}$, let $\mathrm{Disc}(f_{{\boldsymbol}a_0}):=\mathrm{Res}(f_{{\boldsymbol}a_0},f'_{{\boldsymbol}a_0})$ denote the discriminant of $f_{{\boldsymbol}a_0}$, that is, the resultant of $f_{{\boldsymbol}a_0}$ and its derivative $f'_{{\boldsymbol}a_0}$. Since $f_{{\boldsymbol}a_0}$ has degree $r$, by basic properties of resultants we have $$\mathrm{Disc}(f_{{\boldsymbol}a_0})= \mathrm{Disc}(F({\boldsymbol}A_0, T))|_{{\boldsymbol}A_0={\boldsymbol}a_0} := \mathrm{Res}(F({\boldsymbol}A_0, T), F'({\boldsymbol}A_0, T), T)|_{{\boldsymbol}A_0={\boldsymbol}a_0},$$ where the expression $\mathrm{Res}$ in the right–hand side denotes resultant with respect to $T$. It follows that $\mathcal{D}(\mathcal{B}):=\{{\boldsymbol}a_0 \in \mathcal{B} : \mathrm{Disc}(F({\boldsymbol}A_0, T))|_{{\boldsymbol}A_0={\boldsymbol}a_0}= 0\}$. We shall need further to consider first subdiscriminant loci. The first subdiscriminant locus $\mathcal{S}_1(\mathcal{B})$ of $\mathcal{B}\subset{\overline{{\mathbb F}}_{\hskip-0.7mm q}}[T]_r$ is the set of ${\boldsymbol}a_0\in \mathcal{D}(\mathcal{B})$ for which the first subdiscriminant $\mathrm{Subdisc}(f_{{\boldsymbol}a_0}):=\mathrm{Subres}(f_{{\boldsymbol}a_0},f'_{{\boldsymbol}a_0})$ vanishes, where $\mathrm{Subres}(f_{{\boldsymbol}a_0},f'_{{\boldsymbol}a_0})$ denotes the first subresultant of $f_{{\boldsymbol}a_0}$ and $f'_{{\boldsymbol}a_0}$. Since $f_{{\boldsymbol}a_0}$ has degree $r$, basic properties of subresultants imply $$\mathrm{Subdisc}(f_{{\boldsymbol}a_0} )= \mathrm{Subdisc}(F({\boldsymbol}A_0, T))|_{{\boldsymbol}A_0={\boldsymbol}a_0} := \mathrm{Subres}(F({\boldsymbol}A_0, T),F'({\boldsymbol}A_0, T), T))|_{{\boldsymbol}A_0={\boldsymbol}a_0},$$ where $\mathrm{Subres}$ in the right–hand side denotes first subresultant with respect to $T$. We have $\mathcal{S}_1(\mathcal{B}):=\{{\boldsymbol}a_0\in\mathcal{D}(\mathcal{B}): \mathrm{Subdisc}(F({\boldsymbol}A_0, T))|_{{\boldsymbol}A_0={\boldsymbol}a_0}=0\}$. Our next conditions require that the discriminant and the first subdiscriminant locus intersect well $W$: 1. $\mathcal{D}(W)$ has codimension at least one in $W$. 2. $ (A_0\cdot \mathcal{S}_1)(W):=\{{\boldsymbol}a_0\in W: a_0=0\}\cup \mathcal{S}_1(\mathcal{B})$ has codimension at least one in $\mathcal{D}(W)$. 3. $\mathcal{D}(V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}))$ has codimension at least one in $V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}})\subset {\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^r$. We briefly discuss hypotheses $({\sf H}_1)$–$({\sf H}_6)$. Hypothesis $({\sf H}_1)$–$({\sf H}_2)$ merely state that $W$ has the expected dimension $r-m$ and it is smooth. These conditions are satisfied for any sequence $G_1,\ldots,G_m\in{{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}A_k]$ as above with general coefficients (see, e.g., [@Benoist12] or [@GaMa18]). Hypothesis $({\sf H}_3)$ requires that $G_1,\ldots,G_m$ behave properly “at infinity”, which is also the case for general $G_1,\ldots,G_m$. Hypotheses $({\sf H}_4)$–$({\sf H}_5)$ require that “most” of the polynomials of $\mathcal{A}$ are square–free, and among those which are not, only “few” of them have roots with high multiplicity or several multiple roots. As we are looking for criteria for uniform distribution, namely families which behave as the whole set ${{\mathbb F}_{\hskip-0.7mm q}}[T]_r$, it is clear that such a behavior is to be expected. Further, it is required that “few” polynomials in the family under consideration have $0$ as a multiple root, which is a common requirement for uniformly distributed families (see, e.g., [@Cohen72]). Finally, hypothesis $({\sf H}_6)$ requires that the discriminant locus at infinity is not too large. We provide significant examples of families of polynomials satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$, which include in particular the classical of polynomials with prescribed coefficients. Our main result shows that any family $\mathcal{A}$ satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$ is uniformly distributed in the sense of Cohen, and provides explicit estimates on the number $|\mathcal{A}_{{\boldsymbol}\lambda}|$ of elements of $\mathcal{A}$ with factorization pattern ${\boldsymbol}\lambda$. In fact, we have the following result (see Theorem \[theorem: estimate fact patterns\] for a more precise statement). \[theorem: estimate fact patterns into\] For $m<r$ and ${\boldsymbol}\lambda$ a factorization pattern, we have $$\big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|\le q^{r-m-1}\big(\mathcal{T}({\boldsymbol}\lambda) (D\delta\, q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta)+r^2\delta\big),$$ where $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ and $D:=\sum_{i=1}^m({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1)$. Our methodology differs significantly from that of [@Cohen70] and [@Cohen72], as we express $|\mathcal{A}_{{\boldsymbol}\lambda}|$ in terms of the set of common of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational zeros of certain symmetric multivariate polynomials defined over ${{\mathbb F}_{\hskip-0.7mm q}}$. This allows us to establish several facts concerning the geometry of the set of zeros of such polynomials over ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$. Combining these results with estimates on the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of such set of zeros (see, e.g., [@CaMa06] or [@CaMaPr15]), we obtain our main results. Then we consider the average–case complexity of the classical factorization algorithm restricted to $\mathcal{A}$. This algorithm works in four main steps. First it performs an “elimination of repeated factors”. Then it computes a (partial) factorization of the result of the first step by splitting its irreducible factors according to their degree (this is called the distinct–degree factorization). The third step factorizes each of the factors computed in the second step (the equal–degree factorization). Finally, the fourth step consists of the factorization of the repeated factors left aside in the first step (factorization of repeated factors). The following result summarizes our estimates on the average–case complexity of each of these steps (see Theorems \[costo paso 1\], \[average de DDF\], \[average de EDF\] and \[costo en promedio cuarto paso\] for more precise statements). \[th: average factorization\] Let $\delta_{{\boldsymbol}G}:=\deg G_1\cdots \deg G_m$. Denote by $E[\mathcal {X} _1] $, $E[\mathcal {X} _2]$, $E[\mathcal {X} _3]$ and $E[\mathcal {X} _4]$ the average cost on $\mathcal{A}$ of the steps of elimination of repeated factors, distinct–degree factorization, equal–degree factorization and factorization of repeated factors. For $q > 15\delta_{{\boldsymbol}G}^{13/3}$, assuming that fast multiplication is used, we have $$\begin{aligned} E [\mathcal {X} _1] \leq c \, \mathcal{U} (r) +o(1),\quad\! &E[\mathcal{X}_2] \le\xi\,(2\, \tau_1 \lambda(q)+\tau_1+\tau_2\log r) \,M(r)\,(r+1)\big(1+o(1)\big),\\[1ex] E[\mathcal{X}_3] \leq \tau\, M(r) \log q\,(1+o(1)),\quad\! &E[\mathcal{X}_4] \leq \tau_1 M(r)(1+o(1)),\end{aligned}$$ where $M(r):=r\log r\log\log r$ is the fast–multiplication time function, $\mathcal{U}(r):=M(r)\log r$ is the $\gcd$ time function, $\lambda (q)$ is the number of multiplications required to compute $q$–th powers using repeated squaring, $\xi\sim 0.62432945\dots$ is the Golomb constant, and $c$, $\tau_1$, $\tau_2$ and $\tau$ are constants independent of $q$ and $r$. Here, the $o(1)$ terms goes to zero as $q$ tends to infinity, for fixed $r$ and $\deg G_1,\ldots,\deg G_m$. See Theorems \[costo paso 1\], \[average de DDF\], \[average de EDF\] and \[costo en promedio cuarto paso\] for explicit expressions of these terms. This result significantly strengthens the conclusions of the average–case analysis of [@FlGoPa01], in that it shows that such conclusions are not only applicable to the whole set ${{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ of monic polynomials of degree $r$, but to any family $\mathcal{A}\subset{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$. Further, our estimates improve by roughly a factor $r$ those of [@FlGoPa01], up to logarithmic terms, due to the fact that we consider fast multiplication of polynomials, instead of the classical polynomial multiplication considered [@FlGoPa01]. The paper is organized as follows. In Section \[sec: notions of algebraic geometry\] we collect the notions of algebraic geometry we use. In Section \[sec: estimate of |A|\] we obtain a lower bound on the number of elements of the family $\mathcal{A}$ under consideration. Section \[the factorization patterns\] is devoted to describe our algebraic–geometry approach to the distribution of factorization patterns and to prove Theorem \[theorem: estimate fact patterns into\]. In Section \[sec: examples\] we exhibit examples of linear and nonlinear families of polynomials satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$. Finally, in Section \[sec: average-case complexity of factorization\] we perform the average–case analysis of the classical polynomial factorization restricted to $\mathcal{A}$, showing Theorem \[th: average factorization\]. Basic notions of algebraic geometry {#sec: notions of algebraic geometry} =================================== In this section we collect the basic definitions and facts of algebraic geometry that we need in the sequel. We use standard notions and notations which can be found in, e.g., [@Kunz85], [@Shafarevich94]. Let ${\mathbb K}$ be any of the fields ${{\mathbb F}_{\hskip-0.7mm q}}$ or ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$. We denote by ${\mathbb A}^r$ the affine $r$–dimensional space ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^{r}$ and by ${\mathbb P}^r$ the projective $r$–dimensional space over ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}{\!}^{r+1}$. Both spaces are endowed with their respective Zariski topologies over ${\mathbb K}$, for which a closed set is the zero locus of a set of polynomials of ${\mathbb K}[X_1,\ldots, X_r]$, or of a set of homogeneous polynomials of ${\mathbb K}[X_0,\ldots, X_r]$. A subset $V\subset {\mathbb P}^r$ is a [*projective variety defined over*]{} ${\mathbb K}$ (or a projective ${\mathbb K}$–variety for short) if it is the set of common zeros in ${\mathbb P}^r$ of homogeneous polynomials $F_1,\ldots, F_m \in{\mathbb K}[X_0 ,\ldots, X_r]$. Correspondingly, an [*affine variety of ${\mathbb A}^r$ defined over*]{} ${\mathbb K}$ (or an affine ${\mathbb K}$–variety) is the set of common zeros in ${\mathbb A}^r$ of polynomials $F_1,\ldots, F_{m} \in {\mathbb K}[X_1,\ldots, X_r]$. We think a projective or affine ${\mathbb K}$–variety to be equipped with the induced Zariski topology. We shall denote by $\{F_1=0,\ldots, F_m=0\}$ or $V(F_1,\ldots,F_m)$ the affine or projective ${\mathbb K}$–variety consisting of the common zeros of $F_1,\ldots, F_m$. In the remaining part of this section, unless otherwise stated, all results referring to varieties in general should be understood as valid for both projective and affine varieties. A ${\mathbb K}$–variety $V$ is [*irreducible*]{} if it cannot be expressed as a finite union of proper ${\mathbb K}$–subvarieties of $V$. Further, $V$ is [*absolutely irreducible*]{} if it is ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$–irreducible as a ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$–variety. Any ${\mathbb K}$–variety $V$ can be expressed as an irredundant union $V=\mathcal{C}_1\cup \cdots\cup\mathcal{C}_s$ of irreducible (absolutely irreducible) ${\mathbb K}$–varieties, unique up to reordering, called the [*irreducible*]{} ([*absolutely irreducible*]{}) ${\mathbb K}$–[*components*]{} of $V$. For a ${\mathbb K}$–variety $V$ contained in ${\mathbb P}^r$ or ${\mathbb A}^r$, its [*defining ideal*]{} $I(V)$ is the set of polynomials of ${\mathbb K}[X_0,\ldots, X_r]$, or of ${\mathbb K}[X_1,\ldots, X_r]$, vanishing on $V$. The [*coordinate ring*]{} ${\mathbb K}[V]$ of $V$ is the quotient ring ${\mathbb K}[X_0,\ldots,X_r]/I(V)$ or ${\mathbb K}[X_1,\ldots,X_r]/I(V)$. The [*dimension*]{} $\dim V$ of $V$ is the length $n$ of a longest chain $V_0\varsubsetneq V_1 \varsubsetneq\cdots \varsubsetneq V_n$ of nonempty irreducible ${\mathbb K}$–varieties contained in $V$. We say that $V$ has [*pure dimension*]{} $n$ if every irreducible ${\mathbb K}$–component of $V$ has dimension $n$. A ${\mathbb K}$–variety of ${\mathbb P}^r$ or ${\mathbb A}^r$ of pure dimension $r-1$ is called a ${\mathbb K}$–[*hypersurface*]{}. A ${\mathbb K}$–hypersurface of ${\mathbb P}^r$ (or ${\mathbb A}^r$) can also be described as the set of zeros of a single nonzero polynomial of ${\mathbb K}[X_0,\ldots, X_r]$ (or of ${\mathbb K}[X_1,\ldots, X_r]$). The [*degree*]{} $\deg V$ of an irreducible ${\mathbb K}$–variety $V$ is the maximum of $|V\cap L|$, considering all the linear spaces $L$ of codimension $\dim V$ such that $|V\cap L|<\infty$. More generally, following [@Heintz83] (see also [@Fulton84]), if $V=\mathcal{C}_1\cup\cdots\cup \mathcal{C}_s$ is the decomposition of $V$ into irreducible ${\mathbb K}$–components, we define the degree of $V$ as $$\deg V:=\sum_{i=1}^s\deg \mathcal{C}_i.$$ The degree of a ${\mathbb K}$–hypersurface $V$ is the degree of a polynomial of minimal degree defining $V$. We shall use the following [*Bézout inequality*]{} (see [@Heintz83], [@Fulton84], [@Vogel84]): if $V$ and $W$ are ${\mathbb K}$–varieties of the same ambient space, then $$\label{eq: Bezout} \deg (V\cap W)\le \deg V \cdot \deg W.$$ Let $V\subset{\mathbb A}^r$ be a ${\mathbb K}$–variety, $I(V)\subset {\mathbb K}[X_1,\ldots, X_r]$ its defining ideal and $x$ a point of $V$. The [*dimension*]{} $\dim_xV$ [*of*]{} $V$ [*at*]{} $x$ is the maximum of the dimensions of the irreducible ${\mathbb K}$–components of $V$ containing $x$. If $I(V)=(F_1,\ldots, F_m)$, the [*tangent space*]{} $\mathcal{T}_xV$ to $V$ at $x$ is the kernel of the Jacobian matrix $(\partial F_i/\partial X_j)_{1\le i\le m,1\le j\le r}(x)$ of $F_1,\ldots, F_m$ with respect to $X_1,\ldots, X_r$ at $x$. We have $\dim\mathcal{T}_xV\ge \dim_xV$ (see, e.g., [@Shafarevich94 page 94]). The point $x$ is [*regular*]{} if $\dim\mathcal{T}_xV=\dim_xV$; otherwise, $x$ is called [*singular*]{}. The set of singular points of $V$ is the [*singular locus*]{} $\mathrm{Sing}(V)$ of $V$; it is a closed ${\mathbb K}$–subvariety of $V$. A variety is called [*nonsingular*]{} if its singular locus is empty. For projective varieties, the concepts of tangent space, regular and singular point can be defined by considering an affine neighborhood of the point under consideration. Let $V$ and $W$ be irreducible affine ${\mathbb K}$–varieties of the same dimension and $f:V\to W$ a regular map with $\overline{f(V)}=W$, where $\overline{f(V)}$ denotes the closure of $f(V)$ with respect to the Zariski topology of $W$. Such a map is called [*dominant*]{}. Then $f$ induces a ring extension ${\mathbb K}[W]\hookrightarrow {\mathbb K}[V]$ by composition with $f$. We say that the dominant map $f$ is [*finite*]{} if this extension is integral, namely each element $\eta\in{\mathbb K}[V]$ satisfies a monic equation with coefficients in ${\mathbb K}[W]$. A dominant finite morphism is necessarily closed. Another fact we shall use is that the preimage $f^{-1}(S)$ of an irreducible closed subset $S\subset W$ under a dominant finite morphism $f$ is of pure dimension $\dim S$ (see, e.g., [@Danilov94 §4.2, Proposition]). Rational points --------------- Let ${\mathbb P}^r({{\mathbb F}_{\hskip-0.7mm q}})$ be the $r$–dimensional projective space over ${{\mathbb F}_{\hskip-0.7mm q}}$ and ${\mathbb A}^r({{\mathbb F}_{\hskip-0.7mm q}})$ the $r$–dimensional ${{\mathbb F}_{\hskip-0.7mm q}}$–vector space ${{\mathbb F}_{\hskip-0.7mm q}}^n$. For a projective variety $V\subset{\mathbb P}^r$ or an affine variety $V\subset{\mathbb A}^r$, we denote by $V({{\mathbb F}_{\hskip-0.7mm q}})$ the set of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $V$, namely $V({{\mathbb F}_{\hskip-0.7mm q}}):=V\cap {\mathbb P}^r({{\mathbb F}_{\hskip-0.7mm q}})$ in the projective case and $V({{\mathbb F}_{\hskip-0.7mm q}}):=V\cap {\mathbb A}^r({{\mathbb F}_{\hskip-0.7mm q}})$ in the affine case. For an affine variety $V$ of dimension $n$ and degree $\delta$, we have the following bound (see, e.g., [@CaMa06 Lemma 2.1]): $$\label{eq: upper bound -- affine gral} |V({{\mathbb F}_{\hskip-0.7mm q}})|\leq \delta\, q^n.$$ On the other hand, if $V$ is a projective variety of dimension $n$ and degree $\delta$, we have the following bound (see [@GhLa02a Proposition 12.1] or [@CaMa07 Proposition 3.1]; see [@LaRo15] for more precise upper bounds): $$\label{eq: upper bound -- projective gral} |V({{\mathbb F}_{\hskip-0.7mm q}})|\leq \delta\, p_n,$$ where $p_n:=q^n+q^{n-1}+\cdots+q+1=|{\mathbb P}^n({{\mathbb F}_{\hskip-0.7mm q}})|$. Complete intersections {#subsec: complete intersections} ---------------------- Elements $F_1,\ldots, F_m$ in $\mathbb{K}[X_1,\ldots,X_r]$ or $\mathbb{K}[X_0,\ldots,X_r]$ form a *regular sequence* if $F_1$ is nonzero and no $F_i$ is zero or a zero divisor in the quotient ring $\mathbb{K}[X_1,\ldots,X_r]/ (F_1,\ldots,F_{i-1})$ or $\mathbb{K}[X_0,\ldots,X_r]/ (F_1,\ldots,F_{i-1})$ for $2\leq i \leq m$. In that case, the (affine or projective) $\mathbb{K}$–variety $V:=V(F_1,\ldots,F_{r-n})$ is called a [*set–theoretic complete intersection*]{}. We remark that $V$ is necessarily of pure dimension $r-m$. Further, $V$ is called an [*(ideal–theoretic) complete intersection*]{} if its ideal $I(V)$ over ${\mathbb K}$ can be generated by $m$ polynomials. We shall frequently use the following criterion to prove that a variety is a complete intersection (see, e.g., [@Eisenbud95 Theorem 18.15]). \[theorem: eisenbud 18.15\] Let $F_1,\ldots,F_m\in\mathbb{K}[X_1,\ldots,X_r]$ be polynomials which form a regular sequence and let $V:=V(F_1,\ldots,F_m)\subset{\mathbb A}^r$. Denote by $(\partial{\boldsymbol}{F}/\partial {\boldsymbol}{X})$ the Jacobian matrix of $F_1,\ldots,F_m$ with respect to $X_1,\ldots,X_r$. If the subvariety of $V$ defined by the set of common zeros of the maximal minors of $(\partial{\boldsymbol}{F}/\partial {\boldsymbol}{X})$ has codimension at least one in $V$, then $F_1,\ldots,F_m$ define a radical ideal. In particular, $V$ is a complete intersection. If $V\subset{\mathbb P}^r$ is a complete intersection defined over ${\mathbb K}$ of dimension $r-m$, and $F_1 ,\ldots, F_m$ is a system of homogeneous generators of $I(V)$, the degrees $d_1,\ldots, d_m$ depend only on $V$ and not on the system of generators. Arranging the $d_i$ in such a way that $d_1\geq d_2 \geq \cdots \geq d_m$, we call $(d_1,\ldots, d_m)$ the [*multidegree*]{} of $V$. In this case, a stronger version of holds, called the [*Bézout theorem*]{} (see, e.g., [@Harris92 Theorem 18.3]): $$\label{eq: Bezout theorem} \deg V=d_1\cdots d_m.$$ A complete intersection $V$ is called [*normal*]{} if it is [*regular in codimension 1*]{}, that is, the singular locus $\mathrm{Sing}(V)$ of $V$ has codimension at least $2$ in $V$, namely $\dim V-\dim \mathrm{Sing}(V)\ge 2$ (actually, normality is a general notion that agrees on complete intersections with the one we define here). A fundamental result for projective complete intersections is the Hartshorne connectedness theorem (see, e.g., [@Kunz85 Theorem VI.4.2]): if $V\subset{\mathbb P}^r$ is a complete intersection defined over ${\mathbb K}$ and $W\subset V$ is any ${\mathbb K}$–subvariety of codimension at least 2, then $V\setminus W$ is connected in the Zariski topology of ${\mathbb P}^r$ over ${\mathbb K}$. Applying the Hartshorne connectedness theorem with $W:=\mathrm{Sing}(V)$, one deduces the following result. \[theorem: normal complete int implies irred\] If $V\subset{\mathbb P}^r$ is a normal complete intersection, then $V$ is absolutely irreducible. Estimates on the number of elements of $\mathcal{A}$ {#sec: estimate of |A|} ==================================================== Let $X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_r$ be indeterminates over ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$. Denote by $\Pi_1 {\ifm {,\ldots,} {$,\ldots,$}}\Pi_{r}$ the elementary symmetric polynomials of ${{\mathbb F}_{\hskip-0.7mm q}}[X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_r]$. Observe that $f:=T^r+ a_{r-1}T^{r-1}+\ldots +a_0 \in \mathcal{A}$ if and only if there exists ${\boldsymbol}x\in {\mathbb A}^r$ such that $a_j=(-1)^{r-j} \Pi_{r-j}({\boldsymbol}x)$ for $0 \leq j \leq r-1$ and $$R_i:=G_i(-\Pi_1({\boldsymbol}x) {\ifm {,\ldots,} {$,\ldots,$}}{(-1)^{r-k-1}\Pi_{r-k-1}({\boldsymbol}x)}, {(-1)^{r-k+1}\Pi_{r-k+1}({\boldsymbol}x)} {\ifm {,\ldots,} {$,\ldots,$}}(-1)^{r}\Pi_{r}({\boldsymbol}x))=0$$ for $1\leq i \leq m$. Thus, we associate to $\mathcal{A}$ the polynomials $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_m\in{{\mathbb F}_{\hskip-0.7mm q}}[X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_r]$ and the variety $V \subset {\mathbb A}^r$ defined by $R_1 {\ifm {,\ldots,} {$,\ldots,$}}R_m$. Our estimates on the distribution of factorization patterns in $\mathcal{A}$ require asymptotically–tight estimates on the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $V$, and for the average–case analysis of the classical factorization algorithm restricted to $\mathcal{A}$ we need asymptotically–tight lower bounds on the number of elements of $\mathcal{A}$. For this purpose, we shall prove several facts concerning the geometry of the affine varieties $V$ and $W$. Hypothesis $({\sf H}_1)$ implies that $W$ is a set–theoretic complete intersection of dimension $r-m$. Furthermore, by $({\sf H}_2)$ it follows that the subvariety of $W$ defined by the set of common zeros of the maximal minors of $(\partial{\boldsymbol}{G}/\partial {\boldsymbol}{A_k})$ has codimension at least one in $W$. Applying Theorem \[theorem: eisenbud 18.15\] we deduce the following result. $W\subset {\mathbb A}^r$ is a complete intersection of dimension $r-m$. Consider the following surjective morphism of affine ${{\mathbb F}_{\hskip-0.7mm q}}$–varieties: $$\begin{aligned} \label{definition of Pir} {{\boldsymbol}\Pi^r}: {\mathbb A}^r & \rightarrow {\mathbb A}^{r} \\ \notag {\boldsymbol}{x} & \mapsto (-\Pi_1({\boldsymbol}{x}),\ldots,(-1)^{r}\Pi_{r}({\boldsymbol}{x})).\end{aligned}$$ It is easy to see that ${\boldsymbol}\Pi^r$ is a finite, dominant morphism with ${\boldsymbol}\Pi^r(V)=W$. By hypothesis $({\sf H}_1)$ the variety $W^j:=V(G_1,\ldots,G_j)\subset {\mathbb A}^r$ has pure dimension $r-j$ for $1\leq j \leq m$. This implies that $V^j:=({\boldsymbol}\Pi^r)^{-1}(W^j)=V(R_1{\ifm {,\ldots,} {$,\ldots,$}}R_j)$ has pure dimension $r-j$ for $1\leq j \leq m$. We conclude that $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_{m}$ form a regular sequence of ${{\mathbb F}_{\hskip-0.7mm q}}[X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r]$, namely we have the following result. \[lemma: V is complete inters\] $V$ is a set–theoretic complete intersection of dimension $r-m$. Next we study the singular locus of $V$. For this purpose, we make some remarks concerning the Jacobian matrix of $(\partial{\boldsymbol}\Pi^r/\partial{{\boldsymbol}X})$ of ${\boldsymbol}\Pi^r$ with respect to $X_1,\dots,X_r$. Denote by $A_r$ the $(r\times r)$–Vandermonde matrix $$A_r:=(X_j^{i-1})_{1\leq i,j\leq r}.$$ Taking into account the following well–known identities (see, e.g., [@LaPr02]): $$\frac{\partial \Pi_i}{\partial X_{j}}= \Pi_{i-1}-X_{j} \Pi_{i-2} + X_{j}^2 \Pi_{i-3} +\cdots+ (-1)^{i-1} X_{j}^{i-1}\quad (1\leq i,j \leq r),$$ we conclude that $(\partial{\boldsymbol}\Pi^r/\partial{{\boldsymbol}X})$ can be factored as $$\label{eq: factorization Jacobian elem sim pols} \left(\frac{\partial {{\boldsymbol}\Pi^r}}{\partial {\boldsymbol}X}\right):=B_r\cdot A_r := \left( \begin{array}{ccccc} -1 & \ 0 & 0 & \dots & 0 \\ \ \ \Pi_1 & -1 & 0 & & \\ -\Pi_2 & \ \ \Pi_1 & -1 & \ddots & \vdots \\ \vdots &\vdots & \vdots & \ddots & 0 \\ (-1)^{r}\Pi_{r-1} & (-1)^{r-1}\Pi_{r-2} & (-1)^{r-2}\Pi_{r-3} & \cdots &\!\! -1 \end{array} \!\!\right) \cdot A_r.$$ Since $\det B_r=(-1)^r$, we see that $$\det \left(\frac{\partial {{\boldsymbol}\Pi^r}}{\partial {\boldsymbol}X}\right) =(-1)^{r} \prod_{1\le i < j\le r}(X_j-X_i).$$ A critical point in the study of the singular locus of $V$ is the analysis of the zero locus of the $(r-1)\times(r-1)$ minors of $\left({\partial {{\boldsymbol}\Pi^r}}/{\partial {\boldsymbol}X}\right)$. For this purpose, we have the following result. \[prop: determinant minors order r-1\] Fix $k$ with $ 0 \leq k \leq r-1$ as in the introduction and $l$ with $ 1 \leq l \leq r$. Denote by $M_{r-k,l}$ the $(r-1)\times (r-1)$–matrix obtained by deleting the row $r-k$ and the column $l$ of $(\partial{{\boldsymbol}\Pi^r}/\partial{{\boldsymbol}X})$. Then $$\label{det maximal menor} \det M_{r-k,l}=(-1)^{r-k-1}\Delta_l\cdot X_l^k,$$ where $\Delta_l:=\prod_{1\le i < j\le r, \, \, i,j \neq l}(X_j-X_i).$ According to the factorization (\[eq: factorization Jacobian elem sim pols\]), we have $$M_{r-k,l}=B_{r}^{r-k} \cdot A_{r}^l,$$ where $B_{r}^{r-k}$ is the $(r-1)\times r$–submatrix of $B_r$ obtained by deleting its $(r-k)$th row and $A_{r}^l$ is the $r \times (r-1)$–submatrix of $A_{r}$ obtained by deleting its $l$th column. By the Cauchy–Binet formula, it follows that $$\det M_{r-k,l}=\sum_{j=1}^r \det B_r^{r-k,j}\cdot \det A_r^{j,l},$$ where $B_{r}^{r-k,j}$ is the $(r-1)\times (r-1)$–matrix obtained by removing the $j$th column of $B_r^{r-k}$ and $A_r^{j,l}$ is the $(r-1)\times (r-1)$–matrix obtained by removing the $j$th row of $A_r^l$. From [@Ernst00 Lemma 2.1] we deduce that $$\label{det A} \det A_r^{j,l}= \Delta_l \cdot \Pi_{r-j}^*,$$ where $\Pi_{r-j}^*=\Pi_{r-j}(X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_{l-1},X_{l+1} {\ifm {,\ldots,} {$,\ldots,$}}X_r).$ Next, we obtain an explicit expression of $\det B_{r}^{r-k,j}$ for $1\le j \le r$. Observe that $B_r^{r-k}$ has a block structure: $$\label{formula Br-1,l} B_{r}^{r-k}:=\left( \begin{array}{cc} B_{r-k-1} & \mathbf{0}\\ * & \mathcal{T}_k^*\\ \end{array} \right),$$ where $B_{r-k-1}$ is the $(r-k-1)\times(r-k-1)$ principal submatrix of $B_r$ consisting on its first $r-k-1$ rows and columns and $\mathcal{T}_k^*$ is the $k\times(k+1)$–matrix $$\mathcal{T}_k^*:=\left( \begin{array}{ccccccc} \Pi_1 &\!\! -1 & 0 & \dots & 0 & 0 \\ -\Pi_2 & \!\! \ddots & \ddots& &\vdots & \vdots \\ \vdots &\!\!\ddots & \ddots & \ddots & 0 & 0 \\ \vdots & & \ddots & \ddots &\!\! -1 &0 \\ (-1)^{k+1}\Pi_k \!\!&\!\! \dots & \dots & -\Pi_2& \Pi_1 &\!\!-1 \end{array} \right).$$ From we readily deduce that $$\label{det B} \det B_r^{r-k,j}=\left\{\begin{array}{cl} 0 & \textrm{for }1\le j \le r-k-1, \\ (-1)^{r-1} & \textrm{for }j=r-k, \\ (-1)^{r-i-1} \det \mathcal{T}_i & \textrm{for }j=r-k+i,\,1\le i\le k, \end{array}\right.$$ where $\mathcal{T}_i$ is the following $i \times i$ Toeplitz–Hessenberg matrix: $$\mathcal{T}_i:=\left( \begin{array}{ccccc} \Pi_1 & -1 & 0 & \dots & 0 \\ -\Pi_2\ \ & \ddots & \ddots& & \vdots \\ \vdots &\ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots &\!\! -1\, \\ (-1)^{i+1}\Pi_{i} & \dots & \dots & -\Pi_2&\!\! \Pi_1 \end{array} \!\right).$$ By the Trudi formula (see [@Muir60 Ch. VII]; see also [@Merca13 Theorem 1]) we deduce the following identity (see [@Merca13 Section 4]): $$\det \mathcal{T}_i=H_i,$$ where $H_i:=H_i(X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_r)$ is the $i$th complete homogeneous symmetric function. Therefore, combining and we conclude that $$\begin{aligned} \det M_{r-k,l} = \Delta_l\sum_{j=r-k}^r \det B_r^{r-k,j} \cdot \Pi_{r-j}^* &=\Delta_l \sum_{i=0}^k \det B_r^{r-k,\,i+r-k} \cdot \Pi_{k-i}^* \\ \notag & =\Delta_l \sum_{i=0}^k (-1)^{r-i-1} H_i \cdot \Pi^*_{k-i}.\end{aligned}$$ We claim that $$\label{Sk} S(k):=\sum_{i=0}^k (-1)^{r-i-1} H_i \cdot \Pi^*_{k-i}=(-1)^{r-k-1}X_l^k, \quad k=0 {\ifm {,\ldots,} {$,\ldots,$}}r-1.$$ We prove the claim arguing by induction on $k$. Since $H_0= \Pi_0^*=1$, the case $k = 0$ follows immediately. Assume now that holds for $k-1$ with $k>0$, namely $$\label{HI} (-1)^{r-1}\sum_{i=0}^{k-1}(-1)^i H_i\cdot \Pi_{k-1-i}^*=(-1)^{r-k}X_l^{k-1}.$$ It is well known that (see, e.g., [@CoLiOs92 $7$.§$1$, Exercise $10$]) $$\sum_{i=0}^k (-1)^i H_i\cdot \Pi_{k-i}=0.$$ Since $\Pi_{k-i}^*=\Pi_{k-i}(X_1 {\ifm {,\ldots,} {$,\ldots,$}}X_{l-1},X_{l+1} {\ifm {,\ldots,} {$,\ldots,$}}X_r)$, we deduce that $\Pi_{k-i}= X_l \cdot \Pi_{k-i-1}^*+ \Pi_{k-i}^*$. As a consequence, it follows that $$\sum_{i=0}^k(-1)^i H_i \cdot \Pi_{k-i}^*=X_l \sum_{i=0}^{k-1} (-1)^{i-1} H_i \cdot \Pi_{k-i-1}^*.$$ Combining this identity and the inductive hypothesis , we conclude that $$S(k)=-X_l \sum_{i=0}^{k-1} (-1)^{r-i-1} H_i \cdot \Pi_{k-i-1}^* =- X_l\, (-1)^{r-k} X_l^{k-1} =(-1)^{r-k-1}X_l^k.$$ This concludes the proof of the proposition. Denote by $(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X}):=(\partial R_i/\partial X_j)_{1\le i\le m,1\leq j \le r}$ the Jacobian matrix of $R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m}$ with respect to $X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r$. \[theorem: dimension singular locus V\] The set of ${\boldsymbol}{x}\in V$ for which $(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ does not have full rank, has codimension at least $2$. In particular, the singular locus $\Sigma$ of $V$ has codimension at least $2$. By the chain rule, we have the equality $$\left(\frac{\partial {\boldsymbol}{R}}{\partial {\boldsymbol}{X}}\right)=\left(\frac{\partial {\boldsymbol}{G}}{\partial {\boldsymbol}{A}}\circ{\boldsymbol}\Pi\right)\cdot \left(\frac{\partial{\boldsymbol}\Pi}{\partial {\boldsymbol}{X}}\right),$$ where ${\boldsymbol}\Pi:=(-\Pi_1,\ldots, (-1)^{r-k-1}\Pi_{r-k-1},(-1)^{r-k+1}\Pi_{r-k+1},\ldots,(-1)^{r}\Pi_{r})$. Fix a point ${\boldsymbol}{x}:=(x_1,\ldots,x_r)\in V$ such that $(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ does not have full rank, and let ${\boldsymbol}{v}\in{\mathbb A}^{m}$ be a nonzero element in the left kernel of $(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$. We have $${\boldsymbol}{0}= {\boldsymbol}{v}\cdot \left(\frac{\partial {\boldsymbol}{R}}{\partial {\boldsymbol}{X}}\right)({\boldsymbol}{x})={\boldsymbol}{v}\cdot\left(\frac{\partial {\boldsymbol}{G}}{\partial {\boldsymbol}{A}}\right)\big({\boldsymbol}\Pi({\boldsymbol}{x})\big)\cdot \left(\frac{\partial{\boldsymbol}\Pi}{\partial {\boldsymbol}{X}}\right)({\boldsymbol}{x}).$$ Since by hypothesis ($ {\sf H}_2$) the Jacobian matrix $(\partial{\boldsymbol}{G}/\partial {\boldsymbol}{A})\big({\boldsymbol}\Pi({\boldsymbol}{x})\big)$ has full rank, we see that ${\boldsymbol}{w}:={\boldsymbol}{v}\cdot\left({\partial {\boldsymbol}{G}}/{\partial {\boldsymbol}{A}}\right)\big({\boldsymbol}\Pi({\boldsymbol}{x})\big)\in{\mathbb A}^{r-1}$ is nonzero. As ${\boldsymbol}{w}\cdot \left({\partial{\boldsymbol}\Pi}/{\partial {\boldsymbol}{X}}\right)({\boldsymbol}{x})= {\boldsymbol}{0}$, all the maximal minors of $\left({\partial{\boldsymbol}\Pi}/{\partial {\boldsymbol}{X}}\right)({\boldsymbol}{x})$ must be zero. These minors are the determinants $\det M_{r-k,l}({\boldsymbol}{x})$, where $M_{r-k,l}$ are the matrices of Proposition \[prop: determinant minors order r-1\]. Since $\det M_{r-k,l}({\boldsymbol}x)=0$ for $1\leq l\leq r$, Proposition \[prop: determinant minors order r-1\] implies $$x_i^k\Delta_i(x)=x_j^{k}\Delta_{j}(x)=0\quad (1\leq i <j\leq r).$$ It follows that ${\boldsymbol}x$ cannot have its $r$ coordinates pairwise distinct. As a consequence, either ${\boldsymbol}x$ has $r-1$ pairwise–distinct coordinates, one of them being equal to zero, or ${\boldsymbol}x$ has at most $r-2$ pairwise–distinct coordinates. Let $$g:=(T-x_1) \cdots (T-x_r)=T^r -\Pi_1({\boldsymbol}x)T^{r-1}+ \cdots +(-1)^{r} \Pi_{r}({\boldsymbol}x).$$ Observe that ${\boldsymbol}\Pi^r({\boldsymbol}x) \in W$. If there is a coordinate $x_i=0$, then the constant coefficient of $g$ is zero. On the other hand, if ${\boldsymbol}x$ has at most $r-2$ pairwise–distinct coordinates, then there exist $i,j,l,h \in \{1 {\ifm {,\ldots,} {$,\ldots,$}}r\}$ with $i<j, l<h$ and $\{i,j\} \cap \{k,l\}= \emptyset$ such that $x_i=x_j$ and $x_h=x_l$. If $x_i\not= x_h$, then $g$ has two distinct multiple roots, while in the case $x_i=x_h$, $g$ has a root of multiplicity at least $4$. In both cases $g$ and $g'$ have a common factor of degree at least 2, which implies that $$\mathrm{Disc}(g)=0, \, \, \mathrm{Subdisc}(g)=0,$$ namely $g\in\mathcal{S}_1(W)$. In either case, ${\boldsymbol}\Pi^r({\boldsymbol}x) \in (A_0\cdot\mathcal{S}_1)(W)$. According to (${\sf H}_4$) and (${\sf H}_5$), $(A_0\cdot\mathcal{S}_1)(W)$ has codimension at least $2$ in $W$. Since ${\boldsymbol}\Pi^r$ is a finite morphism, we have that $({\boldsymbol}\Pi ^r)^{-1}\big((A_0\cdot\mathcal{S}_1)(W)\big)$ has codimension at least $2$ in $V$. In particular, the set of points ${\boldsymbol}{x}\in V$ with $\mathrm{rank}(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})< m$ is contained in a subvariety of codimension $2$ of $V$. Now let ${\boldsymbol}x$ be an arbitrary point of $\Sigma$. By Lemma \[lemma: V is complete inters\] we have $\dim T_{{\boldsymbol}x}V >r-m$. It follows that $\mathrm{rank}(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})<m$, for otherwise we would have $\dim T_{{\boldsymbol}x}V \leq r-m$, contradicting the hypothesis that ${\boldsymbol}x$ is a singular point of $V$. Therefore, from the first assertion the theorem follows. From Lemma \[lemma: V is complete inters\] and Theorem \[theorem: dimension singular locus V\] we obtain further consequences concerning the polynomials $R_i$ and the variety $V$. Theorem \[theorem: dimension singular locus V\] shows in particular that the set of points ${\boldsymbol}{x}\in V$ for which $(\partial {\boldsymbol}{R}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ does not have full rank has codimension at least one in $V$. Since $R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m}$ form a regular sequence, by Theorem \[theorem: eisenbud 18.15\] we conclude that $R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m}$ define a radical ideal of ${{\mathbb F}_{\hskip-0.7mm q}}[X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r]$, and thus $V$ is a complete intersection. In other words, we have the following result. \[coro: radicality and degree Vr\] $R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m}$ define a radical ideal and $V$ is a complete intersection. The geometry of the projective closure -------------------------------------- Consider the embedding of ${\mathbb A}^r$ into the projective space ${\mathbb P}^r$ defined by the mapping $(x_1,\ldots, x_r)\mapsto(1:x_1:\dots:x_r)$. The closure $\mathrm{pcl}(V)\subset{\mathbb P}^r$ of the image of $V$ under this embedding in the Zariski topology of ${\mathbb P}^r$ is called the projective closure of $V$. The points of $\mathrm{pcl}(V)$ lying in the hyperplane $\{X_0=0\}$ are called the points of $\mathrm{pcl}(V)$ at infinity. Denote by $F^h\in{{\mathbb F}_{\hskip-0.7mm q}}[X_0,\ldots,X_r]$ the homogenization of each $F\in{{\mathbb F}_{\hskip-0.7mm q}}[X_1,\ldots,X_r]$, and let $(R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m})^h$ be the ideal generated by all the polynomials $F^h$ with $F\in (R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m})$. We have that $(R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m})^h$ is radical because $(R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m})$ is a radical ideal (see, e.g., [@Kunz85 §I.5, Exercise 6]). It is well known that $\mathrm{pcl}(V)$ is the ${{\mathbb F}_{\hskip-0.7mm q}}$–variety of $\mathbb{P}^r$ defined by $(R_{1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m})^h$ (see, e.g., [@Kunz85 §I.5, Exercise 6]). Furthermore, $\mathrm{pcl} (V)$ has pure dimension $r-m$ (see, e.g., [@Kunz85 Propositions I.5.17 and II.4.1]) and degree equal to $\deg V$ (see, e.g., [@CaGaHe91 Proposition 1.11]). Next we discuss the behavior of $\mathrm{pcl} (V)$ at infinity. Consider the decomposition of each $R_i$ into its homogeneous components, namely $$R_i=R_i^{d_i}+R_i^{d_i-1}+\cdots+R_i^{0},$$ where each $R_i^j\in{{\mathbb F}_{\hskip-0.7mm q}}[X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r]$ is homogeneous of degree $j$ or zero, $R_i^{d_i}$ being nonzero for $1\le i\le m$. The homogenization of each $R_i$ is the polynomial $$\label{eq: Ri homogenization} R_i^h=R_i^{d_i}+R_i^{d_i-1}X_0+\cdots+R_i^{0}X_0^{d_i}.$$ It follows that $R_i^h(0,X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r)=R_i^{d_i}$ for $1\leq i \leq m$. To express each $R_i^{d_i}$ in terms of the component $G_i^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ of highest weight of $G_i$, let $A_0^{i_0}\cdots A_{k-1}^{i_{k-1}}A_{k+1}^{i_{k+1}}\cdots A_{r-1}^{i_{r-1}}$ be a monomial arising with nonzero coefficients in the dense representation of $G_i$. Then its weight $${\ifm {{\sf wt}} {{$\sf wt$}}}(A_0^{i_0}\cdots A_{k-1}^{i_{k-1}}A_{k+1}^{i_{k+1}}A_{r-1}^{i_{r-1}})=\mathop{\sum_{j=0}^{r-1}}_ {j\not=k} (r-j) i_j$$ equals the degree of the corresponding monomial $\Pi_r^{i_0} \cdots \Pi_{r-k+1}^{i_{k-1}}\Pi_{r-k-1}^{i_{k+1}}\cdots \Pi_{1}^{i_{r-1}}$ of $R_i$. We deduce the following result. \[lemma: rel between R\^di and S\^wt\] $R_i^{d_i}=G_i^{{\ifm {{\sf wt}} {{$\sf wt$}}}}(-\Pi_1 {\ifm {,\ldots,} {$,\ldots,$}}(-1)^{r-k-1}\Pi_{r-k-1},(-1)^{r-k+1}\Pi_{r-k+1} {\ifm {,\ldots,} {$,\ldots,$}}(-1)^{r}\Pi_{r})$ for $1\leq i \leq m$. In particular, $\deg R_i={\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ for $1\leq i \leq m$. Denote by $(\partial {\boldsymbol}{R}^{{\boldsymbol}{d}}/\partial {\boldsymbol}{X}):=(\partial R_i^{d_i}/\partial X_j)_{1\le i\le m,1\leq j \le r}$ the Jacobian matrix of $R_{1}^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_{m}^{d_m}$ with respect to $X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r$. Let $\Sigma^{\infty}\subset\mathbb{P}^r$ be the singular locus of $\mathrm{pcl}(V)$ at infinity, namely the set of singular points of $\mathrm{pcl}(V)$ lying in the hyperplane $\{X_0=0\}$. We have the following result. \[lemma: singular locus Vr at infinity\] The set of points ${\boldsymbol}{x} \in V(R_1^{d_1},\ldots,R_m^{d_m})\subset{\mathbb P}^{r-1}$ for which $(\partial {\boldsymbol}{R}^{{\boldsymbol}{d}}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ has not full rank, has codimension at least 1 in $V(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$. In particular, the singular locus $\Sigma^{\infty}\subset\mathbb{P}^r$ at infinity has dimension at most $r-m-2$. Consider the affine variety $V_{\mathrm{aff}}(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})\subset{\mathbb A}^r$ defined by $R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m}$. Hypothesis (${\sf H}_3$) asserts that $G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ satisfy hypotheses (${\sf H}_1$) and (${\sf H}_2$). Therefore, Lemma \[lemma: V is complete inters\] proves that $V_{\mathrm{aff}}(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$ is a set–theoretic complete intersection of dimension $r-m$. Denote by $\Sigma_{\mathrm{aff}}^\infty$ the set of points ${\boldsymbol}x\in V_{\mathrm{aff}}(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$ as in the statement of the lemma. Arguing as in the proof of Theorem \[theorem: dimension singular locus V\] we conclude that any ${\boldsymbol}x\in\Sigma_{\mathrm{aff}}^\infty$ cannot have its $r$ coordinates pairwise distinct. This implies that ${\boldsymbol}\Pi^r(\Sigma_{\mathrm{aff}}^\infty)$ is contained in the discriminant locus $\mathcal{D}(V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}))$. By hypothesis $({\sf H}_6)$ we have that $\mathcal{D}(V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}))$ has codimension at least 1 in $V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}})={\boldsymbol}\Pi^r(V_{\mathrm{aff}}(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m}))$. Since ${\boldsymbol}\Pi^r$ is a finite morphism, we deduce that $\Sigma_{\mathrm{aff}}^\infty$ has codimension at least 1 in $V_{\mathrm{aff}}(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$. The first assertion of the lemma follows. Now let ${\boldsymbol}{x}:=(0:x_1:\dots: x_r)$ be an arbitrary point of $\Sigma^{\infty}$. Since each $R_i^{h}$ vanishes identically in $\mathrm{pcl}(V)$, we have $R_i^{h}({\boldsymbol}{x})=R_i^{d_i}(x_1{\ifm {,\ldots,} {$,\ldots,$}}x_r)=0$ for $1\le i\le m$. Further, $(\partial {\boldsymbol}{R}^{{\boldsymbol}{d}}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ does not have full rank, since otherwise we would have $\dim \mathcal{T}_{{\boldsymbol}{x}}(\mathrm{pcl}(V))\le r-m$, which would imply that ${\boldsymbol}{x}$ is a nonsingular point of $\mathrm{pcl}(V)$, contradicting thus the hypothesis on ${\boldsymbol}{x}$. It follows that $\Sigma^{\infty}$ has codimension at least 1 in $V(R_1^{d_1},\ldots,R_m^{d_m})$, and thus dimension at most $r-m-2$. Our next result concerns the projective variety $V(R_1^{d_1},\ldots,R_m^{d_m})\subset {\mathbb P}^{r-1}$. \[Lemma: property variety Ri\^di\] $V(R_1^{d_1},\ldots,R_m^{d_m})\subset {\mathbb P}^{r-1}$ is a complete intersection of dimension $r-m-1$, degree $\prod_{i=1}^{m}d_i$ and singular locus of dimension at most $r-m-2$. Since $G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}{\ifm {,\ldots,} {$,\ldots,$}}G_m^{\ifm {{\sf wt}} {{$\sf wt$}}}$ satisfy hypothesis (${\sf H}_1$), Lemma \[lemma: V is complete inters\] shows that $V(R_1^{d_1},\ldots,R_m^{d_m})$ is of pure dimension $r-m-1$. Furthermore, Lemma \[lemma: singular locus Vr at infinity\] shows that the set of ${\boldsymbol}{x} \in V(R_1^{d_1},\ldots,R_m^{d_m})$ for which $(\partial {\boldsymbol}{R}^{{\boldsymbol}{d}}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ has not full rank, has codimension at least 1 in $V(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$. Then Theorem \[theorem: eisenbud 18.15\] proves that $R_1^{d_1},\ldots,R_m^{d_m}$ define a radical ideal, and therefore $V(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$ is a complete intersection. In particular, the singular locus of $V(R_1^{d_1} {\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$ is the set of ${\boldsymbol}{x} \in V(R_1^{d_1},\ldots,R_m^{d_m})$ for which $(\partial {\boldsymbol}{R}^{{\boldsymbol}{d}}/\partial {\boldsymbol}{X})({\boldsymbol}{x})$ has not full rank, and hence it has dimension at most $r-m-2$. Finally, the Bézout theorem proves the assertion concerning the degree. Now we prove our main result concerning $\mathrm{pcl}(V)$. \[theorem: proj closure of Vr is abs irred\] The identity $\mathrm{pcl}(V)=V(R_1^{h},\ldots,R_m^{h})$ holds and $\mathrm{pcl}(V)$ is a normal complete intersection of dimension $r-m$ and degree $\prod_{i=1}^r d_i$. Observe that the following inclusions hold: $$\begin{aligned} V(R_1^{h},\ldots,R_m^{h})\cap\{X_0\not=0\}& \subset V(R_1,\ldots,R_m),\\ V(R_1^{h},\ldots,R_m^{h})\cap\{X_0=0\}&\subset V(R_{1}^{d_1},\ldots,R_{m}^{d_m}). \end{aligned}$$ Lemma \[Lemma: property variety Ri\^di\] proves that $V(R_1^{d_1},\ldots,R_m^{d_m})\subset {\mathbb P}^{r-1} $ is a complete intersection of dimension $r-m-1$ and singular locus of codimension at least $1$. On the other hand, Lemma \[lemma: V is complete inters\] and Theorem \[theorem: dimension singular locus V\] show that $V(R_1,\ldots,R_m)\subset{\mathbb A}^r$ is of pure dimension $r-m$ and its singular locus has codimension at least $2$. We conclude that the same holds with $V(R_1^{h},\ldots,R_m^{h})\subset{\mathbb P}^r$. Since it is defined by $m$ polynomials, it is a set–theoretic complete intersection. Further, by Theorem \[theorem: dimension singular locus V\] and Lemma \[lemma: singular locus Vr at infinity\] the set of points ${\boldsymbol}x\in V(R_1^h,\ldots,R_m^h)$ for which $(\partial {\boldsymbol}{R}^{{\boldsymbol}{h}}/\partial {\boldsymbol}X)({\boldsymbol}x)$ has not full rank, has codimension at least 2 in $V(R_1^h,\ldots,R_m^h)$. Then Theorem \[theorem: eisenbud 18.15\] proves that $R_1^{h},\ldots,R_m^{h}$ define a radical ideal and therefore $V(R_1^{h},\ldots,R_m^{h})$ is a normal complete intersection. By Theorem \[theorem: normal complete int implies irred\] it follows that $V(R_1^{h},\ldots,R_m^{h})$ is absolutely irreducible. It is clear that $\mathrm{pcl}(V)\subset V(R_1^{h},\ldots,R_m^{h})$. Being both of pure dimension $r-m$ and $V(R_1^{h},\ldots,R_m^{h})$ absolutely irreducible, the identity of the statement of the theorem follows. Finally, since $R_1^h,\ldots,R_m^h$ define a radical ideal, the Bézout theorem proves the assertion on the degree. We end the section with the following result, which allows us to control the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $\mathrm{pcl}(V)$ at infinity. \[remark: dimension of pcl(V) at infinity\] $V_{ \infty}:=\mathrm{pcl}(V)\cap \{X_0=0\}\subset {\mathbb P}^{r-1}$ has dimension $r-m-1$. Indeed, recall that $\mathrm{pcl}(V)$ has pure dimension $r-m$. Hence, each irreducible component of $\mathrm{pcl}(V)\cap \{X_0=0\}$ has dimension at least $r-m-1$. From (\[eq: Ri homogenization\]) we deduce that $\mathrm{pcl}(V)\cap \{X_0=0\}\subset V(R_1^{d_1}{\ifm {,\ldots,} {$,\ldots,$}}R_m^{d_m})$. By Lemma \[Lemma: property variety Ri\^di\] we have that $V(R_1^{d_1},\ldots,R_m^{d_m})$ has dimension $r-m-1$. It follows that $\mathrm{pcl}(V)\cap \{X_0=0\}$ also has dimension $r-m-1$. Estimates on the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $W$ ---------------------------------------------------------------------------------- The results on $V$ allows us to estimate the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $W$. We start with the following result. \[theo: W absolutely irreducible\] $W\subset {\mathbb A}^r$ is absolutely irreducible. By Theorems \[theorem: proj closure of Vr is abs irred\] and \[theorem: normal complete int implies irred\] we have that $\mathrm{pcl}(V)$ is absolutely irreducible. As a consequence, $V$ is absolutely irreducible. Since ${\boldsymbol}\Pi^r(V)=W$, the assertion follows. As $|\mathcal{A}|=|W({{\mathbb F}_{\hskip-0.7mm q}})|$, we obtain estimates on the number of elements of $\mathcal{A}$. Combining Corollary \[theo: W absolutely irreducible\] with [@CaMa06 Theorem 7.1], for $q > \delta_{{\boldsymbol}G}:=\deg(G_1)\cdots\deg(G_m)$ we have the following estimate: $$\big||\mathcal{A}|-q^{r-m}\big| \le (\delta_{{\boldsymbol}G}-1)(\delta_{{\boldsymbol}G}-2)q^{r-m-{1}/{2}}+5 \delta_{{\boldsymbol}G}^{13/3}q^{r-m-1}.$$ On the other hand, according to [@CaMa06 Corollary 7.2], if $q >15\delta_{{\boldsymbol}G}^{13/3}$, then $$\big||\mathcal{A}|-q^{r-m}\big| \le (\delta_{{\boldsymbol}G}-1)(\delta_{{\boldsymbol}G}-2)q^{r-m-{1}/{2}}+7\delta_{{\boldsymbol}G}^2q^{r-m-1}.$$ We easily deduce the following result. \[estimation A\] For $q >15\delta_{{\boldsymbol}G}^{13/3}$, we have $$|\mathcal{A}|\geq q^{r-m}\bigg(1-\frac{3\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\textrm{ and }|\mathcal{A}|^{-1}\leq q^{m-r}\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg).$$ Further, $$|\mathcal{A}|\geq \frac{1}{2}q^{r-m}.$$ The distribution of factorization patterns in $\mathcal{A}$ {#the factorization patterns} =========================================================== Let $\lambda_1,\dots,\lambda_r$ be nonnegative integers such that $\lambda_1+2\lambda_2+\cdots+r\lambda_r=r$. Denote by ${\mathcal P}_{{\boldsymbol}\lambda}$ the set of $f\in {{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ with factorization pattern ${\boldsymbol}\lambda:=1^{\lambda_1}2^{\lambda_2}\cdots r^{\lambda_r}$, namely having exactly $\lambda_i$ monic irreducible factors over ${{\mathbb F}_{\hskip-0.7mm q}}$ of degree $i$ (counted with multiplicity) for $1\le i\le r$. Further, for $\mathcal{S}\subset{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ we denote $\mathcal{S}_{{\boldsymbol}\lambda}:=\mathcal{S}\cap\mathcal{P}_{{\boldsymbol}\lambda}$. In this section we estimate the number $|\mathcal{A}_{{\boldsymbol}\lambda}|$ of elements of $\mathcal{A}$ with factorization pattern ${\boldsymbol}\lambda$, where $\mathcal{A}\subset{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ is the family of . Factorization patterns and roots -------------------------------- Following the approach of [@CeMaPe17], we show that the set $\mathcal{A}_{{\boldsymbol}\lambda}$ can be expressed in terms of certain symmetric polynomials. Let $f\in{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ and $m\in {{\mathbb F}_{\hskip-0.7mm q}}[T]$ a monic irreducible factor of $f$ of degree $i$. Then $m$ is the minimal polynomial of a root $\alpha$ of $f$ with ${{\mathbb F}_{\hskip-0.7mm q}}(\alpha)={{\mathbb F}_{\hskip-0.7mm q^i}}$. Denote by $\mathbb G_i$ the Galois group $\mbox{Gal}({{\mathbb F}_{\hskip-0.7mm q^i}},{{\mathbb F}_{\hskip-0.7mm q}})$ of ${{\mathbb F}_{\hskip-0.7mm q^i}}$ over ${{\mathbb F}_{\hskip-0.7mm q}}$. We may express $m$ in the following way: $$m=\prod_{\sigma\in\mathbb G_i}(T-\sigma(\alpha)).$$ Hence, each irreducible factor $m$ of $f$ is uniquely determined by a root $\alpha$ of $f$ (and its orbit under the action of the Galois group of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$ over ${{\mathbb F}_{\hskip-0.7mm q}}$), and this root belongs to a field extension of ${{\mathbb F}_{\hskip-0.7mm q}}$ of degree $\deg m$. Now, for $f\in\mathcal{P}_{{\boldsymbol}\lambda}$, there are $\lambda_1$ roots of $f$ in ${{\mathbb F}_{\hskip-0.7mm q}}$, say $\alpha_1,\dots,\alpha_{\lambda_1}$ (counted with multiplicity), which are associated with the irreducible factors of $f$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree 1; we may choose $\lambda_2$ roots of $f$ in ${{\mathbb F}_{\hskip-0.7mm q^{2}}}\setminus{{\mathbb F}_{\hskip-0.7mm q}}$ (counted with multiplicity), say $\alpha_{\lambda_1+1},\dots, \alpha_{\lambda_1+\lambda_2}$, which are associated with the $\lambda_2$ irreducible factors of $f$ of degree 2, and so on. From now on we assume that a choice of $\lambda_1{\ifm {+\cdots+} {$+\ldots+$}}\lambda_r$ roots $\alpha_1{\ifm {,\ldots,} {$,\ldots,$}}\alpha_{\lambda_1 {\ifm {+\cdots+} {$+\ldots+$}}\lambda_r}$ of $f$ in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$ is made in such a way that each monic irreducible factor of $f$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ is associated with one and only one of these roots. Our aim is to express the factorization of $f$ into irreducible factors in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ in terms of the coordinates of the chosen $\lambda_1{\ifm {+\cdots+} {$+\ldots+$}}\lambda_r$ roots of $f$ with respect to certain bases of the corresponding extensions ${{\mathbb F}_{\hskip-0.7mm q}}\hookrightarrow{{\mathbb F}_{\hskip-0.7mm q^i}}$ as ${{\mathbb F}_{\hskip-0.7mm q}}$–vector spaces. To this end, we express the root associated with each irreducible factor of $f$ of degree $i$ in a normal basis $\Theta_i$ of the field extension ${{\mathbb F}_{\hskip-0.7mm q}}\hookrightarrow {{\mathbb F}_{\hskip-0.7mm q^i}}$. Let $\theta_i\in {{\mathbb F}_{\hskip-0.7mm q^i}}$ be a normal element and $\Theta_i$ the normal basis of the extension ${{\mathbb F}_{\hskip-0.7mm q}}\hookrightarrow{{\mathbb F}_{\hskip-0.7mm q^i}}$ generated by $\theta_i$, i.e., $$\Theta_i=\left \{\theta_i,\cdots, \theta_i^{q^{i-1}}\right\}.$$ The Galois group $\mathbb G_i$ is cyclic and the Frobenius map $\sigma_i:{{\mathbb F}_{\hskip-0.7mm q^i}}\to{{\mathbb F}_{\hskip-0.7mm q^i}}$, $\sigma_i(x):=x^q$ is a generator of $\mathbb{G}_i$. Thus, the coordinates in the basis $\Theta_i$ of all the elements in the orbit of a root $\alpha_k\in{{\mathbb F}_{\hskip-0.7mm q^i}}$ of an irreducible factor of $f$ of degree $i$ are the cyclic permutations of the coordinates of $\alpha_k$ in the basis $\Theta_i$. The vector that gathers the coordinates of all the roots $\alpha_1{\ifm {,\ldots,} {$,\ldots,$}}\alpha_{\lambda_1+\dots+\lambda_r}$ we choose to represent the irreducible factors of $f$ in the normal bases $\Theta_1{\ifm {,\ldots,} {$,\ldots,$}}\Theta_r$ is an element of ${{\mathbb F}_{\hskip-0.7mm q}}^r$, which is denoted by ${{\boldsymbol}x}:=(x_1,\dots,x_r)$. Set $$\label{eq: fact patterns: ell_ij} \ell_{i,j}:=\sum_{k=1}^{i-1}k\lambda_k+(j-1)\,i$$ for $1\le j \le \lambda_i$ and $1\le i \le r$. Observe that the vector of coordinates of a root $\alpha_{\lambda_1{\ifm {+\cdots+} {$+\ldots+$}}\lambda_{i-1}+j}\in{{\mathbb F}_{\hskip-0.7mm q^i}}$ is the sub-array $(x_{\ell_{i,j}+1},\dots,x_{\ell_{i,j}+i})$ of ${\boldsymbol}x$. With these notations, the $\lambda_i$ irreducible factors of $f$ of degree $i$ are the polynomials $$\label{eq: fact patterns: gij}m_{i,j}=\prod_{\sigma\in\mathbb G_i} \Big(T-\big(x_{\ell_{i,j}+1}\sigma(\theta_i)+\dots+ x_{\ell_{i,j}+i}\sigma(\theta_i^{q^{i-1}})\big)\Big)$$ for $1\le j \le \lambda_i$. In particular, $$\label{eq: fact patterns: f factored with g_ij} f=\prod_{i=1}^r\prod_{j=1}^{\lambda_i}m_{i,j}.$$ Let $X_1{\ifm {,\ldots,} {$,\ldots,$}}X_r$ be indeterminates over ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}$, set ${\boldsymbol}X:=(X_1,\dots,X_r)$ and consider the polynomial $M\in {{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}{X},T]$ defined as $$\label{eq: fact patterns: pol G} M:=\prod_{i=1}^r\prod_{j=1}^{\lambda_i}M_{i,j},\quad M_{i,j}:=\prod_{\sigma\in\mathbb G_i} \Big(T-\big(X_{\ell_{i,j}+1}\sigma(\theta_i)+ \dots+X_{\ell_{i,j}+i}\sigma(\theta_i^{q^{i-1}})\big)\Big),$$ where the $\ell_{i,j}$ are defined as in . Our previous arguments show that $f\in{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ has factorization pattern ${{\boldsymbol}\lambda}$ if and only if there exists ${\boldsymbol}x\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ with $f=M({{\boldsymbol}x},T)$. To discuss how many elements ${\boldsymbol}x\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ yield an arbitrary polynomial $f=M({\boldsymbol}x,T)\in\mathcal{P}_{{\boldsymbol}\lambda}$, we introduce the notion of an array of type ${\boldsymbol}\lambda$. Let $\ell_{i,j}$ $(1\le i\le r,\ 1\le j\le\lambda_i)$ be defined as in . We say that ${{\boldsymbol}x}=(x_1,\dots, x_r)\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ is of [*type ${\boldsymbol}\lambda$*]{} if and only if each sub-array ${\boldsymbol}x_{i,j}:=(x_{\ell_{i,j}+1},\dots,x_{\ell_{i,j}+i})$ is a cycle of length $i$. The following result relates the set $\mathcal{P}_{\boldsymbol{\lambda}}$ with the set of elements of ${{\mathbb F}_{\hskip-0.7mm q}}^r$ of type ${\boldsymbol}\lambda$ (see [@CeMaPe17 Lemma 2.2]). \[lemma: fact patterns: G(x,T) with fact pat lambda\] For any ${{\boldsymbol}x}=(x_1,\dots, x_r)\in {{\mathbb F}_{\hskip-0.7mm q}}^r$, the polynomial $f:=M({{\boldsymbol}x},T)$ has factorization pattern ${\boldsymbol}\lambda$ if and only if ${{\boldsymbol}x}$ is of type ${\boldsymbol}\lambda$. Furthermore, for each square–free polynomial $f\in \mathcal{P}_{{\boldsymbol}\lambda}$ there are $w({{\boldsymbol}\lambda}):=\prod_{i=1}^r i^{\lambda_i}\lambda_i!$ different ${{\boldsymbol}x}\in {{\mathbb F}_{\hskip-0.7mm q}}^r $ with $f=M({{\boldsymbol}x},T)$. Consider the polynomial $M$ of as an element of ${{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}X][T]$. We shall express the coefficients of $M$ by means of the vector of linear forms ${\boldsymbol}Y:=(Y_1{\ifm {,\ldots,} {$,\ldots,$}}Y_r)$, with $Y_i\in{\overline{{\mathbb F}}_{\hskip-0.7mm q}}[{\boldsymbol}X]$ defined in the following way for $1\le i\le r$: $$\label{eq: fact patterns: def linear forms Y} (Y_{\ell_{i,j}+1},\dots,Y_{\ell_{i,j}+i})^{t}:=A_{i}\cdot (X_{\ell_{i,j}+1},\dots, X_{\ell_{i,j}+i})^{t} \quad(1\le j\le \lambda_i,\ 1\le i\le r),$$ where $A_i\in{{\mathbb F}_{\hskip-0.7mm q^i}}^{i\times i}$ is the matrix $$A_i:=\left(\sigma(\theta_i^{q^{h}})\right)_{\sigma\in {\mathbb G}_i,\, 0\le h\le i-1}.$$ According to (\[eq: fact patterns: pol G\]), we may express the polynomial $M$ as $$M=\prod_{i=1}^r\prod_{j=1}^{\lambda_i}\prod_{s=1}^i(T-Y_{\ell_{i,j}+s})= \prod_{i=1}^r(T-Y_i)=T^r+\sum_{i=1}^r(-1)^i\,(\Pi_i({\boldsymbol}Y))\, T^{r-i},$$ where $\Pi_1({\boldsymbol}Y){\ifm {,\ldots,} {$,\ldots,$}}\Pi_r({\boldsymbol}Y)$ are the elementary symmetric polynomials of ${{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}Y]$. By we see that $M$ belongs to ${{\mathbb F}_{\hskip-0.7mm q}}[{{\boldsymbol}X},T]$, which in particular implies that $\Pi_i({\boldsymbol}Y)$ belongs to ${{\mathbb F}_{\hskip-0.7mm q}}[{{\boldsymbol}X}]$ for $1\le i\le r$. Combining these arguments with Lemma \[lemma: fact patterns: G(x,T) with fact pat lambda\] we obtain the following result. \[lemma: fact patterns: sym pols and pattern lambda\] A polynomial $f:=T^r+a_{r-1}T^{r-1}{\ifm {+\cdots+} {$+\ldots+$}}a_0\in{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ has factorization pattern ${\boldsymbol}\lambda$ if and only if there exists ${\boldsymbol}{x}\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ of type ${\boldsymbol}\lambda$ such that $$\label{eq: fact patterns: sym pols and pattern lambda} a_i= (-1)^{{r-i}}\,\Pi_{r-i}({\boldsymbol}Y({\boldsymbol}x)) \quad(0\le i\le r-1).$$ In particular, for $f$ square–free, there are $w({\boldsymbol}\lambda)$ elements ${\boldsymbol}x$ for which (\[eq: fact patterns: sym pols and pattern lambda\]) holds. Recall that the family $\mathcal{A}$ of is defined by polynomial $G_1 {\ifm {,\ldots,} {$,\ldots,$}}G_m \in {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[{\boldsymbol}{A_k}]$, for a fixed $k$ with $0 \leq k \leq r-1$. As a consequence, we may express the condition that an element of $\mathcal{A}$ has factorization pattern ${\boldsymbol}\lambda$ in terms of the elementary symmetric polynomials $\Pi_1 {\ifm {,\ldots,} {$,\ldots,$}}\Pi_{r-k-1},\Pi_{r-k+1} {\ifm {,\ldots,} {$,\ldots,$}}\Pi_{r}$ of ${{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}Y]$. \[coro: fact patterns: systems pattern lambda\] A polynomial $f:=T^r+a_{r-1}T^{r-1}{\ifm {+\cdots+} {$+\ldots+$}}a_0\in{{\mathbb F}_{\hskip-0.7mm q}}[T]_r$ belongs to $\mathcal{A}_{{\boldsymbol}\lambda}$ if and only if there exists ${\boldsymbol}{x}\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ of type ${\boldsymbol}\lambda$ such that and $$\label{eq: fact patterns: systems pattern lambda} G_j\big(-\Pi_{1}{\ifm {,\ldots,} {$,\ldots,$}}(-1)^{r-k-1}\Pi_{r-k-1},(-1)^{r-k+1}\Pi_{r-k+1}{\ifm {,\ldots,} {$,\ldots,$}}(-1)^r \Pi_{r} \big)({\boldsymbol}Y({\boldsymbol}x))=0 \quad(1\le j\le m)$$ hold, where $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_m$ are the polynomials defining the family $\mathcal{A}$. In particular, if $f:=M({\boldsymbol}x,T)\in\mathcal{A}_{{\boldsymbol}\lambda}$ is square–free, then there are $w({\boldsymbol}\lambda)$ elements ${\boldsymbol}x$ for which (\[eq: fact patterns: systems pattern lambda\]) holds. The number of polynomials in $\mathcal{A}_{{\boldsymbol}\lambda}$ ----------------------------------------------------------------- Given a factorization pattern ${\boldsymbol}{\lambda}$, in this section we estimate the number of elements of $\mathcal{A}_{{\boldsymbol}\lambda}$. For this purpose, in Corollary \[coro: fact patterns: systems pattern lambda\] we associate to $\mathcal{A}_{{\boldsymbol}\lambda}$ the polynomials $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_m\in{{\mathbb F}_{\hskip-0.7mm q}}[{\boldsymbol}X]$ defined as follows: $$\label{eq: geometry: def R_j} R_j:= G_j\big(-\Pi_{1}{\ifm {,\ldots,} {$,\ldots,$}}(-1)^{r-k-1}\Pi_{r-k-1},(-1)^{r-k+1}\Pi_{r-k+1}{\ifm {,\ldots,} {$,\ldots,$}}(-1)^r \Pi_{r} \big)({\boldsymbol}Y({\boldsymbol}x)).$$ Let $V:=V(R_1 {\ifm {,\ldots,} {$,\ldots,$}}R_m) \subset {\mathbb A}^r$ be the variety defined by $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_m$. Since $G_1 {\ifm {,\ldots,} {$,\ldots,$}}G_m$ satisfy hypotheses $({\sf H}_1)$–$({\sf H}_6)$, by Lemma \[lemma: V is complete inters\], Corollary \[coro: radicality and degree Vr\], Theorem \[theorem: proj closure of Vr is abs irred\] and Remark \[remark: dimension of pcl(V) at infinity\] we obtain the following result. \[Theo:geometry projective clousure\] Let $ m, r$ be positive integers with $m<r$. 1. $V\subset {\mathbb A}^r$ is a complete intersection of dimension $r-m$. 2. The projective variety $\mathrm{pcl}(V)\subset {\mathbb P}^r$ is a normal complete intersection of dimension $r-m$ and degree $\prod_{i=1}^m d_i$, where $d_i:=\deg(R_i)={\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ for $1\le i\le m$. 3. $V_{\infty} :=\mathrm{pcl}(V)\cap \{Y_0=0\}\subset {\mathbb P}^{r-1}$ has dimension $r-m-1$. Now we estimate the number of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of $V$. According to Theorem \[Theo:geometry projective clousure\], $\mathrm{pcl}(V)\subset{\mathbb P}^r$ is a normal complete intersection defined over ${{\mathbb F}_{\hskip-0.7mm q}}$, of dimension $r-m$ and multidegree ${\boldsymbol}d:=(d_1{\ifm {,\ldots,} {$,\ldots,$}}d_m)$. Therefore, [@CaMaPr15 Corollary 8.4] implies that the following estimate holds (see [@CaMa07], [@GhLa02a], [@GhLa02] and [@MaPePr16] for further explicit estimates of this type): $$\big||\mathrm{pcl}(V)({{\mathbb F}_{\hskip-0.7mm q}})|-p_{r-m}\big|\le (\delta (D-2)+2)q^{r-m-\frac{1}{2}}+14 D^2 \delta^2q^{r-m-1}.$$ where $p_{r-m}:=q^{r-m}+\cdots+ q+1=|{\mathbb P}^{r-m}({{\mathbb F}_{\hskip-0.7mm q}})|$, $\delta:=d_1\cdots d_m$ and $D:=\sum_{i=1}^m(d_i-1)$. On the other hand, the Bézout inequality implies $\deg V_{\infty}\le\delta$. Then by Theorem \[Theo:geometry projective clousure\] and we have $$\big|V_{\infty}({{\mathbb F}_{\hskip-0.7mm q}})\big|\le \delta p_{r-m-1}.$$ It follows that $$\begin{aligned} \label{ineq: estimation Vr} \big||V({{\mathbb F}_{\hskip-0.7mm q}})|-q^{r-m}\big|& = \big||\mathrm{pcl}(V)({{\mathbb F}_{\hskip-0.7mm q}})|-|V_{\infty} ({{\mathbb F}_{\hskip-0.7mm q}})|- p_{r-m}+p_{r-m-1}\big|\nonumber\\[1ex] & \le \big||\mathrm{pcl}(V)({{\mathbb F}_{\hskip-0.7mm q}})|-p_{r-m}\big|+ \big|V_{\infty}({{\mathbb F}_{\hskip-0.7mm q}})\big|+ 2{q^{r-m-1}} \nonumber\\[1ex] & \le \big((\delta(D-2)+2)q^{\frac{1}{2}}+14D^2 \delta^2+2 \delta+2\big)q^{r-m-1}.\end{aligned}$$ Let $V^{=}$ be the subvariety of $V$ defined as $$V^{ =}:=\mathop{\bigcup_{1\le i\le r}}_{ 1\leq j_1 <j_2\leq \lambda _i,\, \, 1\leq k_1 <k_2 \leq i} V\cap\{Y_{\ell_{i,j_1}+k_1}=Y_{\ell_{i,j_2}+k_2}\},$$ where $Y_{\ell_{i,j}+k}$ are the linear forms of . Let $V^{ \neq}({{\mathbb F}_{\hskip-0.7mm q}}):=V({{\mathbb F}_{\hskip-0.7mm q}})\backslash V^{ =}({{\mathbb F}_{\hskip-0.7mm q}})$. We claim that $V\cap\{Y_{\ell_{i,j_1}+k_1}=Y_{\ell_{i,j_2}+k_2}\}$ has dimension at most $r-m-1$ for every $1\le i\le r$, $1\leq j_1 <j_2\leq \lambda _i$ and $1\leq k_1 <k_2 \leq i$. Indeed, let ${\boldsymbol}x\in V\cap\{Y_{\ell_{i,j_1}+k_1}=Y_{\ell_{i,j_2}+k_2}\}$ for $i,j_1,j_2,k_1,k_2$ as above. By we conclude that $M({\boldsymbol}x,T)$ is not square–free, and therefore $\Pi^r({\boldsymbol}Y({\boldsymbol}x))\in\mathcal{D}(W)$. Since $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_m$ satisfy $({\sf H}_4)$, it follows that $\dim \mathcal{D}(W)\le r-m-1$, and the fact that $\Pi^r$ is a finite morphism implies that $\dim(\Pi^r)^{-1}(\mathcal{D}(W))\le r-m-1$. This proves our claim. The claim implies $\dim V^{ =}\le r-m-1$. By the Bézout inequality we have $$\deg V^{ =}\le \deg V\sum_{i=1}^r\frac{i^2\lambda_i^2}{4}\le \frac{r^2}{4}\delta.$$ As a consequence, by we see that $$\label{eq: estimates: upper bound V^=(fq)} |V^{ =}({{\mathbb F}_{\hskip-0.7mm q}})|\le \deg V^{ =}\,q^{r-m-1}\le \frac{r^2\delta}{4}\, q^{r-m-1}.$$ Finally, combining and we obtain the following result. \[ineq:estimation Vr distintos\] For $m<r$ we have $$\begin{aligned} \big||V^{ \neq}({{\mathbb F}_{\hskip-0.7mm q}})|-q^{r-m}\big|&\le q^{r-m-1} \Big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+2\delta+2+r^2\delta/4\Big), \end{aligned}$$ where $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ and $D:=\sum_{i=1}^m ({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1).$ By , $|V^{ =}({{\mathbb F}_{\hskip-0.7mm q}})|\le r^2\delta\, q^{r-m-1}/4$. Then, from we deduce that $$\begin{aligned} \big||V^{ \neq}({{\mathbb F}_{\hskip-0.7mm q}})|-q^{r-m}\big|& \leq \big||V({{\mathbb F}_{\hskip-0.7mm q}})|-q^{r-m}\big|+\big|V^{ =}({{\mathbb F}_{\hskip-0.7mm q}})\big|\nonumber\\[1ex] & \le \big((\delta(D-2)+2)q^{\frac{1}{2}}+14D^2 \delta^2+2 \delta+2\big)q^{r-m-1} + \frac{r^2\delta}{4} q^{r-m-1}. \end{aligned}$$ This shows the statement of the theorem. Next we use Corollary \[coro: fact patterns: systems pattern lambda\] to relate $|V({{\mathbb F}_{\hskip-0.7mm q}})|$ to the quantity $|\mathcal{A}_{{\boldsymbol}{\lambda}}|$. More precisely, let ${\boldsymbol}x:=({\boldsymbol}x_{i,j}:1\le i\le r,1\le j\le \lambda_i)\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ be an ${{\mathbb F}_{\hskip-0.7mm q}}$–rational zero of $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_m$ of type ${\boldsymbol}\lambda$. Then ${\boldsymbol}x$ is associated with $f\in\mathcal{A}_{{\boldsymbol}\lambda}$ having $Y_{\ell_{i,j}+k}({\boldsymbol}x_{i,j})$ as an ${{\mathbb F}_{\hskip-0.7mm q^i}}$–root for $1\le i\le r$, $1\le j\le\lambda_i$ and $1\le k\le i$, where $Y_{\ell_{i,j}+k}$ are the linear forms of . Let $\mathcal{A}_{{\boldsymbol}\lambda}^{sq}:=\{f\in \mathcal{A}_{{\boldsymbol}\lambda}: f \mbox{ is square--free}\}$ and $\mathcal{A}_{{\boldsymbol}\lambda}^{nsq}:=\mathcal{A}_{{\boldsymbol}\lambda}\setminus \mathcal{A}_{{\boldsymbol}\lambda}^{sq}$. Corollary \[coro: fact patterns: systems pattern lambda\] shows that any element $f\in \mathcal{A}_{{\boldsymbol}\lambda}^{sq}$ is associated with $w({\boldsymbol}\lambda):=\prod_{i=1}^r i^{\lambda_i}\lambda_i!$ common ${{\mathbb F}_{\hskip-0.7mm q}}$–rational zeros of $R_1{\ifm {,\ldots,} {$,\ldots,$}}R_m$ of type ${\boldsymbol}\lambda$. Observe that ${\boldsymbol}x\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ is of type ${\boldsymbol}\lambda$ if and only if $Y_{\ell_{i,j}+k_1}({\boldsymbol}x) \neq Y_{\ell_{i,j}+k_2}({\boldsymbol}x)$ for $1\leq i\leq r$, $1\leq j\leq \lambda_i$ and $1\leq k_1 <k_2 \leq i$. Furthermore, an ${\boldsymbol}x\in{{\mathbb F}_{\hskip-0.7mm q}}^r$ of type ${\boldsymbol}\lambda$ is associated with $f\in\mathcal{A}_{{\boldsymbol}\lambda}^{sq}$ if and only if $Y_{\ell_{i,j_1}+k_1}({\boldsymbol}x) \neq Y_{\ell_{i,j_2}+k_2}({\boldsymbol}x)$ for $1\leq i\leq r$, $1\leq j_1<j_2\leq \lambda_i$ and $1\leq k_1 <k_2 \leq i$. As a consequence, we see that $|\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| =\mathcal{T}({\boldsymbol}\lambda) \big|V^{\neq}({{\mathbb F}_{\hskip-0.7mm q}})\big|$, where $\mathcal{T}({\boldsymbol}\lambda):=1/w({\boldsymbol}\lambda)$. This implies $$\big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big| = \mathcal{T}({\boldsymbol}\lambda)\,\big||V^{ \neq}({{\mathbb F}_{\hskip-0.7mm q}})|-q^{r-m}\big|.$$ From Theorem \[ineq:estimation Vr distintos\] we deduce that $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big| &\le \,\mathcal{T}({\boldsymbol}\lambda)q^{r-m-1}\big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+2\delta+2+r^2\delta/4\big)\\ &\le \,\mathcal{T}({\boldsymbol}\lambda)q^{r-m-1}\big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta\big).\end{aligned}$$ Now we are able to estimate $|\mathcal{A}_{{\boldsymbol}\lambda}|$. We have $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|&= \big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}|+ |\mathcal{A}_{{\boldsymbol}\lambda}^{nsq}|-\mathcal{T}({\boldsymbol}\lambda)q^{r-m}\big|\nonumber\\ &\le\mathcal{T}({\boldsymbol}\lambda)q^{r-m-1}\big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta\big)+ |\mathcal{A}_{{\boldsymbol}\lambda}^{nsq}|. \label{eq: estimates: estimate A_lambda aux}\end{aligned}$$ It remains to bound $|\mathcal{A}_{{\boldsymbol}\lambda}^{nsq}|$. To this end, we observe that $f\in \mathcal{A}$ is not square–free if and only if its discriminant is equal to zero, namely it belongs to the discriminant locus $\mathcal{D}(W)$. By hypothesis $({\sf H}_4)$ the discriminant locus $\mathcal{D}(W)$ has dimension at most $r-m-1$. Further, by the Bézout inequality we have $$\deg\mathcal{D}(W)\le \deg W\cdot \deg \{{\boldsymbol}a_0 \in {\mathbb A}^r : \mathrm{Disc}(F({\boldsymbol}A_0, T))|_{{\boldsymbol}A_0={\boldsymbol}a_0}= 0\}\le \delta_{{\boldsymbol}G}\, r(r-1)\le \delta\, r^2.$$ Then implies $$\label{eq: estimates: upper bound discr locus} |\mathcal{A}_{{\boldsymbol}\lambda}^{nsq}|\le |\mathcal{A}^{nsq}|\le \delta_{{\boldsymbol}G}\, r(r-1) \,q^{r-m-1}\le \delta\, r^2q^{r-m-1}.$$ Hence, combining and we conclude that $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|&\le q^{r-m-1}\Big(\mathcal{T}({\boldsymbol}\lambda)\big((\delta (D-2)+2)q^{\frac{1}{2}}\!+\!14 D^2 \delta^2+r^2\delta\big)+r^2\delta\Big).\end{aligned}$$ In other words, we have the following result. \[theorem: estimate fact patterns\] For $m<r$, we have that $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|& \leq \mathcal{T}({\boldsymbol}\lambda)q^{r-m-1} \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+ r^2\delta\big),\\ \big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|&\le q^{r-m-1}\Big(\mathcal{T}({\boldsymbol}\lambda) \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta\big)+r^2\delta\Big), \end{aligned}$$ where $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ and $D:=\sum_{i=1}^m({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1)$. As we show in Section \[subsec: examples CeMaPe17\], Theorem \[theorem: estimate fact patterns\] extends [@CeMaPe17 Theorem 4.2]. More precisely, Theorem \[theorem: estimate fact patterns\] holds for families defined by linearly–independent linear polynomials $G_1,\ldots,G_m\in {{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots, A_2]$ with $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ not dividing $r(r-1)$, and linearly–independent linear polynomials $G_1,\ldots,G_m\in {{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots, A_3]$ with $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})>2$. The latter is precisely [@CeMaPe17 Theorem 4.2]. Examples of linear and nonlinear families {#sec: examples} ========================================= In this section we exhibit examples of linear and nonlinear families of polynomials satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$. Therefore, the estimate of Theorem \[theorem: estimate fact patterns\] is valid for these families. The linear families of [@CeMaPe17] {#subsec: examples CeMaPe17} ---------------------------------- Suppose that $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})>3$. Let $r,m,n$ be positive integers with $2\leq n \leq r-m$ and $L_1,\ldots,L_m\in{{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_n]$ linear forms which are linearly independent. In [@CeMaPe17] the distribution of factorization patterns of the following linear family is considered: $$\label{eq: family paper CMP17} \mathcal{A}:=\left\{T^r+a_{r-1}T^{r-1}{\ifm {+\cdots+} {$+\ldots+$}}a_0\in{{\mathbb F}_{\hskip-0.7mm q}}[T]: L_j(a_{r-1}{\ifm {,\ldots,} {$,\ldots,$}}a_n)=0\quad (1\le j\le m)\right\}.$$ Assume without loss of generality that the Jacobian matrix $(\partial L_i/\partial A_j)_{1\le i\le m,\,n\le j\le r-1}$ is lower triangular in row echelon form and denote by $1\le i_1<\cdots<i_m\le r-n$ the positions corresponding to the pivots. We have the following result. \[lemma: example MPP satisfies H1-H6\] If either $n=2$ and $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)$ or $n \geq 3$, then $L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m$ satisfy hypotheses $({\sf H}_1)$–$({\sf H}_6)$. It is clear that hypotheses $({\sf H}_1)$–$({\sf H}_2)$ hold. Further, since the component of highest weight of $L_k$ is of the form $L_k^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=b_{k,r-i_k}A_{r-i_k}$ for $1\le k\le m$, we conclude that $({\sf H}_3)$ holds. Now we analyze the validity of $({\sf H}_4)$. Denote $W:=V(L_1,\ldots,L_m)\subset{\mathbb A}^r$. It is clear that $${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[W]:={\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]/(L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$$ is a domain, where $\mathcal{J}:=\{{r-1} {\ifm {,\ldots,} {$,\ldots,$}}0\}\setminus \{r-i_1 {\ifm {,\ldots,} {$,\ldots,$}}r-i_m\}$. Therefore, it suffices to prove that the coordinate class $\mathcal{R}$ defined by $\mathrm{Disc}(F({\boldsymbol}{A}_0,T))$ in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[W]$ is a nonzero polynomial in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$, where $F({\boldsymbol}A_0,T):=T^r+A_{r-1}T^{r-1} +\dots+ A_0$ and ${\boldsymbol}A_0:=(A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0)$. If $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)$, then the nonzero monomial $r^rA_0^{r-1}$ occurs in the dense representation of $\mathcal{R}$. On the other hand, if $ \mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r$, then the nonzero monomial $A_1^r$ occurs in the dense representation of $\mathcal{R}$. Finally, if $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r-1$, then we have the nonzero monomial $A_0^{r-1}$ in the dense representation of $\mathcal{R}$. Next we show that $({\sf H}_5)$ is fulfilled. For this purpose, we first prove that $A_0, L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m$, $\mathrm{Disc}(F({\boldsymbol}{A}_0,T))$ form a regular sequence of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. We observe that $${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1}{\ifm {,\ldots,} {$,\ldots,$}}A_0]/(A_0, L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k : k \in \mathcal{J}_1]$$ is domain, where $\mathcal{J}_1:=\mathcal{J}\setminus \{0\}$. Hence, considering the class $\mathcal{R}_1$ of $\mathrm{Disc}(F({\boldsymbol}{A}_0,T))$ as an element of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k : k \in \mathcal{J}_1]$, it is enough to prove that it is nonzero. Indeed, if $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)$, then the monomial $(-1)^{r-1}(r-1)^{r-1}A_1^r$ occurs in the dense representation $\mathcal{R}_1$, while for $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ dividing $r$, the monomial $A_1^r$ appears in $\mathcal{R}_1$. Finally, for $n \geq 3$ and $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ dividing $r-1$, we have the nonzero monomial $(-1)^{r+1}A_1^2A_2^{r-1}$ in the dense representation of $\mathcal{R}_1$. Finally we prove that $L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m,\mathrm{Disc}(F({\boldsymbol}{A}_0,T)),\mathrm{Subdisc}(F({\boldsymbol}{A}_0,T))$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. Recall that ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]/(L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$ is a domain. Therefore, we may consider the classes $\mathcal{R}$ and $\mathcal{S}_1$ of $\mathrm{Disc}(F({\boldsymbol}{A}_0,T))$ and $\mathrm{Subdisc}(F({\boldsymbol}{A}_0,T))$ modulo $(L_1 {\ifm {,\ldots,} {$,\ldots,$}}L_m)$ as elements of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$. We have already shown that $\mathcal{R}$ is nonzero. On the other hand, if $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)$, then the nonzero monomial $r(r-1)^{r-2}A_1^{r-2}$ occurs in the dense representation of $\mathcal{S}_1$, while for $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ dividing $r(r-1)$, we have the nonzero monomial $2(-1)^r(r-2)^{r-2}A_2^{r-1}$ in the dense representation of $\mathcal{S}_1$. We conclude that $\mathcal{S}_1$ is nonzero. Further, [@MaPePr14 Theorem A.3] or [@Perez16 Teorema 3.1.7] show that $\mathcal{R}$ is an irreducible element of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$ and hence $ \mathbb{B}:={\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]/(\mathcal{R})$ is a domain. Thus, it suffices to see that the class of $\mathcal{S}_1$ in $\mathbb{B}$ is nonzero. If not, then $\mathcal{S}_1$ would be a nonzero multiple of $\mathcal{R}$ in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_k: k \in \mathcal{J}]$, which is not possible because $\text{max}\{\deg_{A_1}\mathcal{R},\deg_{A_2}\mathcal{R}\}=r$ and $\text{max}\{\deg_{A_1}\mathcal{S}_1,\deg_{A_2}\mathcal{S}_1\}= r-1$. Finally, we prove that $({\sf H}_6)$ holds. The components of highest weight of $L_1,{\ifm {,\ldots,} {$,\ldots,$}},L_m$ being of the form $L_k^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=b_{k,r-i_k}A_{r-i_k}$ for $k=1 {\ifm {,\ldots,} {$,\ldots,$}}m$, arguing as before we readily see that $({\sf H}_6)$ holds. From Lemma \[lemma: example MPP satisfies H1-H6\] it follows that the family $\mathcal{A}$ of satisfies the hypotheses of Theorem \[theorem: estimate fact patterns\]. Therefore, applying Theorem \[theorem: estimate fact patterns\] we obtain the following result. Suppose that $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})>3$. Let $\mathcal{A}$ be the family of and ${\boldsymbol}{\lambda}$ a factorization pattern. If either $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)$ and $L_k\in {{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_2]$ for $1\leq k\leq m$, or $L_k\in {{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_n]$ for $1\leq k\leq m$ and $ 3 \leq n \leq r-m$, then $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|& \leq \mathcal{T}({\boldsymbol}\lambda)q^{r-m-1} \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+ r^2\delta\big),\\ \big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-m}\big|&\le q^{r-m-1}\Big(\mathcal{T}({\boldsymbol}\lambda) \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta\big)+r^2\delta\Big),\end{aligned}$$ where $\delta:=\prod_{j=1}^m i_j$ and $D:=\sum_{j=1}^m(i_j-1)$. A linear family from [@GaHoPa99] -------------------------------- In [@GaHoPa99] there are experimental results on the number of irreducible polynomials on certain families over ${{\mathbb F}_{\hskip-0.7mm q}}$. Further, the distribution of factorization patterns on general families of polynomials of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of a given degree is stated as an open problem. In particular, the family of polynomials we now discuss is considered. Suppose that $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})>3$. For positive integers $s$ and $r$ with $3\leq s \leq r-2$, let $$\label{eq: linear family GaoPanarioHowell} \mathcal{A}:=\{T^r+g(T)T+1: \,\, g\in {{\mathbb F}_{\hskip-0.7mm q}}[T] \,\, \text{and}\, \deg g\leq s-1\}.$$ Observe that $\mathcal{A}$ is isomorphic to the set of ${{\mathbb F}_{\hskip-0.7mm q}}$-rational points of the affine ${{\mathbb F}_{\hskip-0.7mm q}}$–subvariety of ${\mathbb A}^r$ defined by the polynomials $$G_1:=A_{0}-1,\ G_2:=A_{s+1},\ldots, G_{r-s}:=A_{r-1}.$$ We show that hypotheses $({\sf H}_1)$–$({\sf H}_6)$ are fulfilled. It is easy to see that $({\sf H}_1)$ and $({\sf H}_2)$ hold, since $G_1{\ifm {,\ldots,} {$,\ldots,$}}G_{r-s}$ are linearly–independent polynomials of degree 1. Furthermore, taking into account that $$G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=A_{0},\ G_2^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=A_{s+1},\ldots, G_{r-s}^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=A_{r-1},$$ we immediately conclude that hypothesis $({\sf H}_3)$ holds. Now we analyze the validity of hypotheses $({\sf H}_4)$ and $({\sf H}_5)$. Let $W\subset {\mathbb A}^r$ be the ${{\mathbb F}_{\hskip-0.7mm q}}$–variety defined by the polynomials $G_1,\ldots,G_{r-s}$, and denote by $\mathcal{D}(W)\subset{\mathbb A}^r$ and $\mathcal{S}_1(W)\subset {\mathbb A}^r$ the discriminant locus and the first subdiscriminant locus of $W$ respectively. We first prove that $\mathcal{D}(W)$ has codimension one in $W$. It is clear that $G_1,\ldots,G_{r-s}$ form a regular sequence of ${{\mathbb F}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_0]$. Observe that $${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[W]={\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1},\ldots,A_0]/(G_1,\ldots,G_{r-s})\simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_s,\ldots,A_1]$$ is a domain. As a consequence, we may consider the coordinate function $\mathcal{R}$ defined by $\mathrm{Disc}(F({\boldsymbol}{A}_0,T))$ as an element of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_s {\ifm {,\ldots,} {$,\ldots,$}}A_1]$, where ${\boldsymbol}{A}_0:=(A_{r-1},\ldots,A_0)$ and $F({\boldsymbol}{A}_0,T):=T^r+A_{r-1}T^{r-1}+\dots+ A_0$. We observe that $\mathcal{R}\not=0$ in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_s {\ifm {,\ldots,} {$,\ldots,$}}A_1]$, because $F({\boldsymbol}A_0,T)$ is not a separable polynomial, and therefore it is not a zero divisor of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[W]$. It follows that $\mathcal{D}(W)$ has codimension one in $W$, namely hypothesis $({\sf H}_4)$ holds. Next we show that $(A_0\cdot\mathcal{S}_1)(W)$ has codimension at least one in $\mathcal{D}(W)$. Since $G_1:=A_0-1$ vanishes on $W$, the coordinate function of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[W]$ defined by $A_0$ is a unit, which implies $(A_0\cdot\mathcal{S}_1)(W)=\mathcal{S}_1(W)$. In what follows, we shall use the following elementary property. \[lemma: regular sequences 1\] Let $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m \in {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{0} {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$. If $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_i)[A_{i+1} {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$, then $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$. We shall also use the following property of regular sequences. \[lemma: regular sequences 2\] Let $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m \in {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$. For an assignment of positive integer weights ${\ifm {{\sf wt}} {{$\sf wt$}}}$ to the variables $A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}$, denote by $F_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}} {\ifm {,\ldots,} {$,\ldots,$}}F_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ the components of highest weight of $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m$. If $F_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}} {\ifm {,\ldots,} {$,\ldots,$}}F_m^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$, then $F_1 {\ifm {,\ldots,} {$,\ldots,$}}F_m$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_0 {\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}]$. Let $V_j:=V(F_1{\ifm {,\ldots,} {$,\ldots,$}}F_j)\subset {\mathbb A}^r$ for $1\le j\le m$. It is enough to see that $V_j$ has codimension $j$ for $1 \leq j \leq m$. By hypothesis, $V_j^{{\ifm {{\sf wt}} {{$\sf wt$}}}}:=V(F_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}} {\ifm {,\ldots,} {$,\ldots,$}}F_j^{{\ifm {{\sf wt}} {{$\sf wt$}}}})\subset {\mathbb A}^r$ has pure dimension $r-j$. Therefore, there exist $1\le {k_1}<\cdots< k_{r-j}\le m$ such that the variety $V:=V(F_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}} {\ifm {,\ldots,} {$,\ldots,$}}F_j^{{\ifm {{\sf wt}} {{$\sf wt$}}}},A_{k_1} {\ifm {,\ldots,} {$,\ldots,$}}A_{k_{r-j}} )\subset {\mathbb A}^r$ has dimension zero. Consider the following morphism of affine ${{\mathbb F}_{\hskip-0.7mm q}}$–varieties: $$\begin{aligned} {{\boldsymbol}\phi}: {\mathbb A}^r & \rightarrow {\mathbb A}^{r}\\ (a_0 {\ifm {,\ldots,} {$,\ldots,$}}a_{r-1}) & \mapsto (a_0^{{\ifm {{\sf wt}} {{$\sf wt$}}}(0)}, a_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}(1)},\ldots,a_{r-1}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(r-1)}), \end{aligned}$$ where ${\ifm {{\sf wt}} {{$\sf wt$}}}(0){\ifm {,\ldots,} {$,\ldots,$}}{\ifm {{\sf wt}} {{$\sf wt$}}}(r-1)$ are the weights assigned to $A_0{\ifm {,\ldots,} {$,\ldots,$}}A_{r-1}$ respectively. It is clear that ${\boldsymbol}\phi$ is a finite, dominant morphism. Observe that, if $F\in{\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_0,\ldots,A_{r-1}]$ is weighted homogeneous, then ${\boldsymbol}\phi(F)$ is homogeneous. We have that ${\boldsymbol}\phi(V) \subset {\mathbb A}^r$ is a zero–dimensional affine cone. Since ${\boldsymbol}\phi(V)$ is defined by the homogeneous polynomials $F_i^{{\ifm {{\sf wt}} {{$\sf wt$}}}}(A_0^{{\ifm {{\sf wt}} {{$\sf wt$}}}(0)},\ldots,A_{r-1}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(r-1)})$, $1\leq i \leq j$, and $A_{k_i} ^{{\ifm {{\sf wt}} {{$\sf wt$}}}(k_{i})}$, $1\leq i \leq r-j$, it must be ${\boldsymbol}\phi(V)=\{0\}$. Therefore, by, e.g., [@PaSa04 Proposition 18], the affine variety defined by the polynomials $$F_1(A_0^{{\ifm {{\sf wt}} {{$\sf wt$}}}(0)},\ldots,A_{r-1}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(r-1)}),\ldots, F_j(A_0^{{\ifm {{\sf wt}} {{$\sf wt$}}}(0)},\ldots,A_{r-1}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(r-1)}),A_{k_1}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(k_1)} {\ifm {,\ldots,} {$,\ldots,$}}A_{k_{r-j}}^{{\ifm {{\sf wt}} {{$\sf wt$}}}(k_{r-j})}$$ has dimension zero. Taking into account that ${\boldsymbol}\phi$ is a finite morphism, we conclude that the variety $\hat{V}_j\subset{\mathbb A}^r$ defined by $F_1,\ldots,F_j, A_{k_1},\ldots, A_{k_{r-j}}$ has also dimension zero. Finally, observe that the dimension of $V_j$ is at least $r-j$. On the other hand, $0=\dim\hat{V}_j\ge\dim V_j-(r-j)$. This finishes the proof of the lemma. It easy to see that $G_2 {\ifm {,\ldots,} {$,\ldots,$}}G_{r-s}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. Observe that ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]/(G_2 {\ifm {,\ldots,} {$,\ldots,$}}G_{r-s}) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{s}{\ifm {,\ldots,} {$,\ldots,$}}A_0]$. Therefore, to conclude that $({\sf H}_5)$ holds it suffices to prove that $\mathcal{G}_1$, $\mathcal{S}_1$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{s} {\ifm {,\ldots,} {$,\ldots,$}}A_0]$, where $\mathcal{G}_1$, $\mathcal{R}$ and $\mathcal{S}_1$ are the coordinate functions of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{r-1} {\ifm {,\ldots,} {$,\ldots,$}}A_0]/(G_2 {\ifm {,\ldots,} {$,\ldots,$}}G_{r-s})$ defined by $G_1$, $\mathrm{Disc}(F({\boldsymbol}A_0,T))$ and $\mathrm{Subdisc}(F({\boldsymbol}A_0,T))$, respectively. \[lemma: H5 ej lineal: regular sequence\] $\mathcal{G}_1$, $\mathcal{S}_1$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_{s} {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. We consider $\mathcal{R}, \mathcal{S}_1, \mathcal{G}_1$ as elements of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_{i+1})[A_i,\ldots,A_0]$ for an appropriate $i\in\{2,3\}$ and define a weight ${\ifm {{\sf wt}} {{$\sf wt$}}}_i$ by setting $${\ifm {{\sf wt}} {{$\sf wt$}}}_i(A_0):=r,\ {\ifm {{\sf wt}} {{$\sf wt$}}}_i(A_1):=r-1,\ldots,{\ifm {{\sf wt}} {{$\sf wt$}}}_i(A_i):=r-i.$$ Denote by $\mathcal{G}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$, $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ and $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ the components of highest weight of $\mathcal{G}_1$, $\mathcal{R}$ and $\mathcal{S}_1$ respectively. We have the following claim. $\mathcal{G}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$, $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_{i+1})[A_i,\ldots,A_0]$. Observe that $${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_{i+1})[A_i,\ldots,A_0]/(\mathcal{G}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_{i+1})[A_i,\ldots,A_1]$$ is a domain. As a consequence, it suffices to prove that the coordinate functions defined by $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ in this quotient ring form a regular sequence. With a slight abuse of notation, we shall also denote them by $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_i}$. The proof will be split into four parts, according to whether $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r$, $r-1$, $r-2$ or does not divide $r(r-1)(r-2)$. [**First case: $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r$**]{}. For $i:=2$, it is easy to see that in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]$, $$\begin{aligned} \mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}=A_1^r+(-1)^{r+1}2^{r-2}A_2^{r-1}A_1 ^2&\quad \text{and} \quad\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}=(2A_{2})^{r-1}.\end{aligned}$$ Observe that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ is a nonzero polynomial of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]$, and $${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]/(\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_1].$$ It follows that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ is not a zero divisor in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]/(\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2})$, which completes the proof of the claim in this case. [**Second case: $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r-1$**]{}. For $i:=3$, we prove that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3, A_2,A_1]$. Let $F:=T^r+A_3T^3+A_2T^2+A_1T$. It is easy to see that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=\mathrm{Disc}(F)$ and $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=\mathrm{Subdisc}(F)$. Observe that $F'=T^{r-1}+3A_3T^3+2A_2T^2+A_1$. By [@GeCzLa92 Lemma 7.1] we deduce that $$\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=(-1)^{{r}({r-1})} \mathrm{Res}(F',G)\ \textrm{ and } \ \mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=(-1)^{(r-1)(r-2)}\mathrm{Subdisc}(F',G),$$ where $G:=-2A_3T^3-A_2T^2$ is the remainder of the division of $F$ by $F'$. Therefore, applying the Poisson formula, it is easy to see that $$\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=(-1)^{r+1}A_1^2A_2^{r-1}+ 2^{r-1}A_1^2A_2^2A_3^{r-2}-2^{r-3}A_1^3A_3^{r-1}.$$ On the other hand, by, e.g., Theorem [@DaKrSz13 Theorem 2.5], we conclude that $$\begin{aligned} \mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}&=2 A_2^{r-1}+(-1)^r2^{r-2}A_2^2A_3^{r-2}+2A_1A_2^{r-3}A_3+3 (-1)^{r+1}2^{r-2}A_1A_3^{r-1}\\ &=2\big( A_2^{r-1}+A_1A_2^{r-3}A_3\big)+(-2)^{r-2}\big(A_2^2A_3^{r-2}-3 A_1A_3^{r-1}\big).\end{aligned}$$ In the second line we express $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ as the sum of two homogeneous polynomials of degrees $r-1$ and $r$ without common factors. Then [@Gibson98 Lemma 3.15] proves that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is an irreducible polynomial in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3,A_2,A_1]$. Next suppose that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is a zero divisor in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3,A_2,A_1]/(\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3})$. Since $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is irreducible, we have that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3} \in (\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3})$, which is easily shown to be not possible by a direct calculation. [**Third case: $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ divides $r-2$**]{}. For $i:=3$, we show that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3, A_2,A_1]$. As in the previous case, if $F:=T^r+A_3T^3+A_2T^2+A_1T$, then it can be seen that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=\mathrm{Disc}(F)$ and $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}:=\mathrm{Subdisc}(F)$. Since $F'=2T^{r-1}+3A_3T^3+2A_2T^2+A_1$, from [@GeCzLa92 Lemma 7.1] it follows that $$\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=(-1)^{{r}({r-1})} 2^{r-3}\mathrm{Res}(F',G)\ \textrm{ and } \ \mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}= (-1)^{(r-1)(r-2)}2^{r-3}\mathrm{Subdisc}(F',G),$$ where $G:=-\frac{1}{2}A_3T^3+\frac{1}{2}A_1T$ is the remainder the division of $F$ by $F'$. By the Poisson formula we obtain $$\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=\left\{ \begin{array}{rl} 4 A_1^3A_3^{r-1}-A_1^r -2A_2A_1{\!\!}^{\frac{r+2}{2}}A_3{\!\!}^{\frac{r-2}{2}}-A_1^2A_2^2A_3^{r-2}&\textrm{ for }r\textrm{ even},\\[1ex] 4A_1^3A_3^{r-1}+A_1^r+4A_1^{\frac{r+3}{2}}A_3{\!\!}^{\frac{r-1}{2}}-A_1^2A_2^2A_3^{r-2}&\textrm{ for }r\textrm{ odd}. \end{array} \right.$$ In the same vein, by, e.g., [@DaKrSz13 Theorem 2.5], we see that $$\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}=\left\{ \begin{array}{rl} 4A_2(A_1A_3)^{\frac{r-2}{2}} +2A_2^2A_3^{r-2}+2A_1^{r-2}-6A_1A_3^{r-2}&\textrm{ for }r\textrm{ even},\\[1ex] 7(A_1A_3)^{\frac{r-1}{2}} -2A_2^2A_3^{r-2}+2A_1^{r-2}+6A_1A_3^{r-1}&\textrm{ for }r\textrm{ odd}. \end{array} \right.$$ We observe that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is an irreducible polynomial in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3,A_2,A_1]$. To see this it suffices to apply the Eisenstein criterion, considering $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ as an element of the polynomial ring ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}((A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3,A_1])[A_2]$ and the prime $(A_1)$. Next, suppose that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is a zero divisor in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_4)[A_3,A_2,A_1]/(\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3})$. Since $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3}$ is irreducible, we have that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3} \in (\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_3})$, which can be shown to be not possible by a direct calculation. [**Fourth case: $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$ does not divide $r(r-1)(r-2)$**]{}. For $i:=2$, we prove that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ form a regular sequences in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]$. Arguing as before, we obtain $$\begin{aligned} \mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}&=(1-r)^{r-1}A_1^r-(r-2)r^{-1}A_1^2A_2^{r-1},\\ \mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}&=r(r-1)^{r-2}A_1^{r-2}+2(2-r)^{r-2}A_2^{r-1}.\end{aligned}$$ By the Stepanov criterion (see, e.g., [@LiNi83 Lemma 6.54]) we deduce that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ is an irreducible polynomial in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]$. Suppose that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$ is a zero divisor in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1]/(\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2})$. Since $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ is irreducible, we have that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2} \in (\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2})$, which can be seen not to be the case by a direct calculation. Therefore, we deduce that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2, A_1]$. By the claim and Lemma \[lemma: regular sequences 2\] it follows that $\mathcal{G}_1$, $\mathcal{S}_1$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_s {\ifm {,\ldots,} {$,\ldots,$}}A_{i+1})[A_i,\ldots,A_0]$, and Lemma \[lemma: regular sequences 1\] implies that $\mathcal{G}_1$, $\mathcal{S}_1$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_s {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. By Lemma \[lemma: H5 ej lineal: regular sequence\] we conclude that hypothesis $({\sf H}_5)$ holds. Finally, we prove that hypothesis $({\sf H}_6)$ holds. The components of higher weight of the polynomials $G_1,\ldots,G_{r-s}$ are $G_i^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=A_{s+i-1}$ for $2\leq i \leq r-s$ and $G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}}= A_0$. With the same arguments as above, we see that $\mathcal{D}(W^{{\ifm {{\sf wt}} {{$\sf wt$}}}})$ has codimension at least one in $W^{{\ifm {{\sf wt}} {{$\sf wt$}}}}$, where $W^{{\ifm {{\sf wt}} {{$\sf wt$}}}}:=V(G_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}},\ldots,G_{r-s}^{{\ifm {{\sf wt}} {{$\sf wt$}}}})$. Since the family satisfies hypotheses $({\sf H}_1)$–$({\sf H}_6)$, from Theorem \[theorem: estimate fact patterns\] we deduce the following result. Let $\mathcal{A}$ be the family and ${\boldsymbol}\lambda$ a factorization pattern. We have $$\begin{aligned} \big||\mathcal{A}_{{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^s\big|& \leq \mathcal{T}({\boldsymbol}\lambda)q^{s-1} \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+ r^2\delta\big),\\ \big||\mathcal{A}_{{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^s\big|&\le q^{s-1}\Big(\mathcal{T}({\boldsymbol}\lambda) \big((\delta (D-2)+2)q^{\frac{1}{2}}+14 D^2 \delta^2+r^2\delta\big)+r^2\delta\Big), \end{aligned}$$ where $\mathcal{A}_{{\boldsymbol}\lambda}$ is the set of elements of $\mathcal{A}$ with factorization pattern ${\boldsymbol}\lambda$, $\mathcal{A}_{{\boldsymbol}\lambda}^{sq}$ is the set of square–free elements of $\mathcal{A}_{{\boldsymbol}\lambda}$, $\delta:=r\cdot(r-s-1)!$ and $D:=r-1+{(r-s-2)(r-s-1)}/{2}$. We apply Theorem \[theorem: estimate fact patterns\] with $m:=r-s$ to the polynomials $$R_1:=(-1)^r\Pi_r-1,\ R_2:=(-1)^{r-s-1}\Pi_{r-s-1},\ldots, R_{r-s}:=-\Pi_1.$$ Therefore, we have $$\delta:=\prod_{i=1}^{r-s} \deg R_i=r\cdot(r-s-1)!\textrm{ and } D:=\sum_{i=1}^{r-s}(\deg R_i-1)=r-1+\frac{(r-s-2)(r-s-1)}{2}.$$ This finishes the proof. A nonlinear family ------------------ Let $r, t_1 {\ifm {,\ldots,} {$,\ldots,$}}t_r$ be positive integers with $r$ even. Suppose that $\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})>3$ does not divide $(r-1)(r+1)\big((r-1)^{r-1}+r^r\big)$. Consider the polynomial $G\in {{\mathbb F}_{\hskip-0.7mm q}}[A_1,\ldots,A_r]$ defined in the following way: $$G:=\sum_{t_1+2t_2 +\ldots +r t_r=r} (-1)^{\Delta(t_1,\ldots,t_r)}\frac{(t_1 +\dots+ t_r)!}{t_1! \dots t_r!} A_r^{t_1} \cdots A_1^{t_r},$$ where $\Delta(t_1,t_2,\ldots,t_r):= r-\sum_{i=1}^r t_i$. The polynomial $G$ arises as the determinant of the $n \times n$ generic Toeplitz–Hessenberg matrix, namely $$G=\det \left( \begin{array}{ccccc} A_r & 1 & 0 & \dots & 0 \\ \vdots &\ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ A_1 & \ldots & \ldots & A_r&\!\! 1 \end{array} \!\!\right).$$ This is the well–known Trudi formula (see [@Muir60 Ch. VII]; see also [@Merca13 Theorem 1]). We also remark that the polynomial $H_r:=G(\Pi_r,\ldots,\Pi_1)$ is critical in the study of deep holes of the standard Reed–Solomon codes (see [@CaMaPr12 Proposition 2.2]). We consider the following family of polynomials: $$\label{eq: nonlinear family} \mathcal{A}_{\mathcal{N}}:=\{T^{r+1} +a_rT^r+ \dots+a_0: G(a_r,\ldots, a_1)=0\}.$$ Observe $\mathcal{A}_{\mathcal{N}}$ may be seen as the set of ${{\mathbb F}_{\hskip-0.7mm q}}$–rational points of the ${{\mathbb F}_{\hskip-0.7mm q}}$–variety $W:=V(G) \subset {\mathbb A}^{r+1}$. Let ${\ifm {{\sf wt}} {{$\sf wt$}}}$ be the weight defined by ${\ifm {{\sf wt}} {{$\sf wt$}}}(A_i):=r+1-i$ for $i=0 {\ifm {,\ldots,} {$,\ldots,$}}r$. We shall see that this family of polynomials satisfies hypotheses $({\sf H}_1)$–$({\sf H}_6)$. It is clear that $({\sf H}_1)$ holds, because $G$ is nonzero. Further, since $G$ is a monic element of ${{\mathbb F}_{\hskip-0.7mm q}}[A_r,\ldots,A_2][A_1]$ of degree $1$ in $A_1$, we have that $$\nabla G({\boldsymbol}a_0)= \bigg(\frac{\partial{G}}{\partial{A_r}}({\boldsymbol}a_0){\ifm {,\ldots,} {$,\ldots,$}}\frac{\partial{G}}{\partial A_2}({\boldsymbol}a_0),1\bigg) \neq 0$$ for any ${\boldsymbol}a_0 \in W$. We deduce that hypothesis $({\sf H}_2)$ holds. Next we consider hypothesis $({\sf H}_3)$. Given an arbitrary nonzero monomial $$m_{G}:=\frac{(t_1 +\dots+ t_r)!}{t_1! \dots t_r!} A_r^{t_1} \dots A_1^{t_r}$$ arising in the dense representation of $G$, it is easy to see that ${\ifm {{\sf wt}} {{$\sf wt$}}}(m_G)=r$. It follows that $G$ is weighted homogeneous of weighted degree $r$. Then $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=G$, which readily implies that hypothesis $({\sf H}_3)$ holds. Now we analyze the validity of hypothesis $({\sf H}_4)$, namely that the discriminant locus $\mathcal{D}(W) \subset {\mathbb A}^{n+1}$ of $W$ has codimension at least 1 in $W$. For this purpose, it suffices to show that $\{G, \mathcal{R}\}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0]$, where $\mathcal{R}:=\mathrm{Disc}(F({\boldsymbol}A_0, T))$, $F({\boldsymbol}A_0,T):=T^{r+1}+A_rT^r +\dots+ A_0$ and ${\boldsymbol}A_0:=(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0)$. We consider $G$ and $\mathcal{R}$ as elements of the polynomial ring ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r,\ldots, A_2)[A_1, A_0]$ and the weight ${\ifm {{\sf wt}} {{$\sf wt$}}}_1$ on ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r,\ldots, A_2)[A_1, A_0]$ defined by setting $${\ifm {{\sf wt}} {{$\sf wt$}}}_1(A_1):=r,\quad {\ifm {{\sf wt}} {{$\sf wt$}}}_1(A_0):=r+1.$$ We claim that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_1}, \mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_1}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r,{\ifm {,\ldots,} {$,\ldots,$}}A_2)[A_1,A_0]$. It is easy see that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_1}=A_1$. Further, since ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_2)[A_1, A_0]/(G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_1}) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_2)[A_0]$ is a domain, to prove the claim it suffices to show that ${\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_1}}$ is nonzero modulo $(A_1)$. A direct calculation shows that ${\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}}}=(r+1)^{r+1}A_0^{r+1}$ modulo $(A_1)$, which proves the claim. As a consequence of the claim and Lemma \[lemma: regular sequences 2\] we see that $G$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_2)[A_1,A_0]$, and Lemma \[lemma: regular sequences 1\] implies that $G$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. In other words, hypothesis $({\sf H}_4)$ is satisfied. Next we show that hypothesis $({\sf H}_5)$ holds. To this end, we make the following claim. $A_0$, $\mathcal{R}$ and $G$ form a regular sequence of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_0]$. Since ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_0]/(A_0)\simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_1]$ and $G\in {\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_1]$, we have to show that $\mathcal{R}$ modulo $(A_0)$, and $G$, form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_1]$. We consider $G$ and $\mathcal{R}$ modulo $(A_0)$ as elements of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_{r-1},\ldots,A_2)[A_r,A_1]$, with the weight ${\ifm {{\sf wt}} {{$\sf wt$}}}_r$ defined by ${\ifm {{\sf wt}} {{$\sf wt$}}}_r(A_r):=1$ and ${\ifm {{\sf wt}} {{$\sf wt$}}}_r(A_1):=r$. We claim that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_{r-1},\ldots,A_2)[A_r,A_1]$. First we observe that $$G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}=A_1+A_r^r,$$ and the Stepanov criterion (see, e.g., [@LiNi83 Lemma 6.54]) proves that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ is an irreducible polynomial of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_{r-1},\ldots,A_2)[A_r,A_1]$. Thus, it is enough to prove that $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ is a nonzero polynomial of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_{r-1},\ldots,A_2)[A_r,A_1]/(G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r})$. We have $$\begin{aligned} \mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}&= - (r-1)^{r-1} A_r^rA_1^r+ r^rA_1^{r+1}\\ &\equiv -\big((r-1)^{r-1}+r^r\big)A_r^{r+r^2} \textrm{ modulo }G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}.$$ We conclude that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_r}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_{r-1},\ldots,A_2)[A_r,A_1]$. Combining Lemmas \[lemma: regular sequences 2\] and \[lemma: regular sequences 1\] as before we deduce that $G$ and $\mathcal{R}$ modulo $(A_0)$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_1]$, which implies that $A_0$, $\mathcal{R}$ and $G$ form a regular sequence of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r,\ldots,A_0]$. Next we make a second claim. $G$, $\mathcal{R}$ and $\mathcal{S}_1$ form a regular sequence of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. We consider $G$, $\mathcal{R}$ and $\mathcal{S}_1$ as elements of ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1,A_0]$, and consider the weight ${\ifm {{\sf wt}} {{$\sf wt$}}}_2$ defined by ${\ifm {{\sf wt}} {{$\sf wt$}}}_2(A_2):=r-1$, ${\ifm {{\sf wt}} {{$\sf wt$}}}_2(A_1):=r$, ${\ifm {{\sf wt}} {{$\sf wt$}}}_2(A_0):=r+1$. We claim that $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$, $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1,A_0]$. Since $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}=A_1$, we have that ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_1,A_0]/(G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}) \simeq {\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_0]$ is a domain. Therefore, it suffices to see that $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ modulo $(A_1)$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ modulo $(A_1)$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2,A_0]$. It is easy to see that $$\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}\textrm{ modulo }(A_1)=-2(r-1)^{r-1}A_2^r.$$ Further, we have $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}\textrm{ modulo }(A_1,A_2)=(r+1)^{r+1}A_{0}^r$. As a consequence, $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$, $\mathcal{S}_1^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ and $\mathcal{R}^{{\ifm {{\sf wt}} {{$\sf wt$}}}_2}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}(A_r {\ifm {,\ldots,} {$,\ldots,$}}A_3)[A_2, A_1, A_0]$. From Lemmas \[lemma: regular sequences 2\] and \[lemma: regular sequences 1\] it follows that $G$, $\mathcal{S}_1$ and $\mathcal{R}$ form a regular sequence in ${\overline{{\mathbb F}}_{\hskip-0.7mm q}}[A_r {\ifm {,\ldots,} {$,\ldots,$}}A_0]$. From the first claim we conclude that $\mathcal{D}(W)\cap\{A_0=0\}$ has codimension two in $W$, while the second claim shows that $\mathcal{S}_1(W)$ has codimension two in $W$. As a consequence, $\mathcal{D}(W)\cap(A_0\cdot\mathcal{S}_1)(W)$ has codimension two in $W$, that is, hypothesis $({\sf H}_5)$ is satisfied. Finally, since $G^{{\ifm {{\sf wt}} {{$\sf wt$}}}}=G$, we readily deduce that hypothesis $({\sf H}_6)$ holds. As a consequence of the fact that the family satisfies hypotheses $({\sf H}_1)$–$({\sf H}_6)$, we obtain the following result. Let $\mathcal{A}_\mathcal{N}$ be the family and ${\boldsymbol}\lambda$ a factorization pattern. We have $$\begin{aligned} \big||\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-1}\big|& \leq \mathcal{T}({\boldsymbol}\lambda)q^{r-2} (r^2q^{\frac{1}{2}}+14r^4),\\ \big||\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-1}\big|&\le q^{r-2}\big(\mathcal{T}({\boldsymbol}\lambda)(r^2q^{\frac{1}{2}}+14 r^4)+r^3\big), \end{aligned}$$ where $\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}$ is the set of elements of $\mathcal{A}_\mathcal{N}$ with factorization pattern ${\boldsymbol}\lambda$ and $\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}^{sq}$ is the set of square–free elements of $\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}$. This is a simple consequence of Theorem \[theorem: estimate fact patterns\] with $m:=1$ and the polynomial $$R_1:=G(-\Pi_1,\Pi_2,\ldots,(-1)^r\Pi_r).$$ As previously remarked, the weighted degree of $G$ is $r$, which implies that $\deg R_1=r$. Therefore, we have $$\delta:=\deg R_1=r\textrm{ and } D:=\deg R_1-1=r-1.$$ As a consequence, Theorem \[theorem: estimate fact patterns\] implies $$\begin{aligned} \big||\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}^{sq}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-1}\big|& \leq \mathcal{T}({\boldsymbol}\lambda)q^{r-2} \big((r(r-3)+2)q^{\frac{1}{2}}+14 (r-1)^2 r^2+ r^3\big),\\ \big||\mathcal{A}_{\mathcal{N},{\boldsymbol}\lambda}| -\mathcal{T}({\boldsymbol}\lambda)\,q^{r-1}\big|&\le q^{r-2}\Big(\mathcal{T}({\boldsymbol}\lambda) \big((r(r-3)+2)q^{\frac{1}{2}}+14 (r-1)^2 r^2 +r^3\big)+r^3\Big). \end{aligned}$$ This immediately implies the statement of the theorem. Average–case analysis of polynomial factorization over $\mathcal{A}$ {#sec: average-case complexity of factorization} ==================================================================== In this section we analyze the average–case complexity of the classical factorization algorithm applied to any family $\mathcal{A}$ as in satisfying hypotheses $({\sf H}_1)$–$({\sf H}_6)$. Given $f \in {{\mathbb F}_{\hskip-0.7mm q}}[T]$, the classical factorization algorithm finds the complete factorization $f=f_1^{e_1} \dots f_n^{e_n}$, where $f_1, \dots, f_n$ are pairwise distinct monic irreducible polynomials in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ and $e_1, \dots,e_n$ are strictly positive integers. The algorithm contains three main routines: - **elimination of repeated factors (ERF)** replaces a polynomial by a square–free one that contains all the irreducible factors of the original one with exponent $1$; - **distinct–degree factorization (DDF)** splits a square–free polynomial into a product of polynomials whose irreducible factors have all the same degree; - **equal–degree factorization (EDF)** splits completely a polynomial whose irreducible factors have all the same degree. More precisely, the algorithm works as follows: ${}$ 1. Input: a monic polynomial $f \in {{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree $r>0$. 2. Output: the complete factorization of $f$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$. **factor procedure ($f\in {{\mathbb F}_{\hskip-0.7mm q}}[T]$)** 1. $a_f:=\mathrm{ERF}(f)$ 2. ${\boldsymbol}b_f:=\mathrm{DDF}(a_f)$ 3. $F:=1$ 4. For $k$ from $1$ to $s$ ($s \leq r$) do 1. $F:=F\cdot \mathrm{EDF}(b_f[k],k)$ 5. end do 6. $c:=\mathrm{factor}(f/a_f)$ 7. Return $F\cdot c$. In [@FlGoPa01], the authors analyze the average–case complexity of the classical factorization algorithm applied to all the monic polynomials of degree $r$ of $ {{\mathbb F}_{\hskip-0.7mm q}}[T] $. Unfortunately, the results of this analysis cannot be directly applied to the family $\mathcal{A}$, because there is a small probability that a random monic polynomial of degree $r$ of $ {{\mathbb F}_{\hskip-0.7mm q}}[T] $ belongs to $\mathcal{A}$. As a consequence, we shall perform an analysis of the behavior of this algorithm applied to elements of $\mathcal{A}$, using the results on the distribution of factorization patterns of Section \[the factorization patterns\]. Considering the uniform probability on $\mathcal{A}$, let $\mathcal{X}: \mathcal{A} \rightarrow \mathbb{N}$ be the random variable that counts the number $ \mathcal {X} (f) $ of arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$ performed by the classical factorization algorithm to obtain the complete factorization in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of any $f\in\mathcal{A}$. We may describe this algorithm as consisting of four stages, and thus the random variable $\mathcal{X}$ may be decomposed as the sum of the random variables that count the cost of each step of the algorithm. More precisely, we consider the random variable $\mathcal{X}_1: \mathcal{A} \rightarrow \mathbb{N}$ that counts the number of arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ performed in the ERF step, namely $$\label{ERF} \mathcal{X}_1(f):=\mathrm{Cost}(\mathrm{ERF}(f)).$$ Further, we introduce a random variable $\mathcal{X}_2: \mathcal{A} \rightarrow \mathbb{N}$ that counts the number of arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ performed during the DDF step, namely $$\label{DDF} \mathcal{X}_2(f):=\mathrm{Cost}(\mathrm{DDF}(a_f)),$$ where $a_f:=\mathrm{ERF}(f)$ denotes the square–free polynomial obtained after the ERF step on input $f$. Denote by $${\boldsymbol}b_f:=\mathrm{DDF}(a_f)=(b_f(1),\dots, b_f(s))$$ the vector of polynomials obtained by applying the DDF step to the monic square–free polynomial $a_f:=\mathrm{ERF}(f)$, where $s$ is the degree of the largest irreducible factor of $a_f$. Each $b_f(k)$ consists of the product of all the monic irreducible polynomials in $ {{\mathbb F}_{\hskip-0.7mm q}}[T] $ of degree $k$ that divide $f$. With this notation, let $\mathcal{X}_3: \mathcal{A} \rightarrow \mathbb{N}$ be the random variable that counts the number of arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ of the EDF step, namely $$\label{EDF} \mathcal{X}_3(f):=\sum_{k=1}^s \mathcal{X}_{3,k}(f),\quad \mathcal{X}_{3,k}(f)\!\!:=\!\!\mathrm{Cost}(\mathrm{EDF}(b_f(k)))\quad(1\le k\le s).$$ Finally, we introduce a random variable $\mathcal{X}_4:\mathcal{A} \rightarrow \mathbb{N}$ that counts the number of operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ performed by the classical factorization algorithm applied to ${f}/\mathrm{ERF}(f)$. Our aim is to study the expected value of the random variable $\mathcal{X}$, namely $$\label{esperanza de la variable aleatoria costo} E[\mathcal{X}]:=\frac{1}{|\mathcal{A}|}\sum_{f \in \mathcal{A}}\mathcal{X}(f)= \frac{1}{|\mathcal{A}|} \sum_{k=1}^4 \sum_{f \in \mathcal{A}}\mathcal{X}_k(f).$$ We denote by $M(r)$ a *multiplication time*, so that the product of two polynomials in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree at most $r$ can be computed with at most $\tau_1 M(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Using fast arithmetic we can take $M(r):= r\log r\log\log r$ (see, e.g., [@GaGe99]). For $\tau_1$ suitably chosen, a division with remainder of two polynomials of degree at most $r$ can also be computed with at most $\tau_1 M(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Further, the cost of computing the greatest common divisor of two polynomials in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree at most $r$ is at most $\tau_2\, \mathcal{U}(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$, where $\mathcal{U}(r):= M(r) \log r$ (see, e.g., [@GaGe99]). Here, $\tau_1$ and $\tau_2$ are system and implementation dependent constants. Elimination of repeated factors ------------------------------- We consider in detail the step of elimination of repeated factors (ERF). Let $$f=f_1^{e_1}\dots f_n^{e_n}=\prod_{p\mid e_i}f_i^{e_i}\prod_{p \nmid e_i}f_i^{e_i}$$ be the factorization of $f \in \mathcal{A}$ into monic irreducible polynomials in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$, where $f_1, \dots,f_n$ are pairwise distinct, $e_1, \dots,e_n \in \mathbb{N}$ and $p:=\mathrm{char}({{\mathbb F}_{\hskip-0.7mm q}})$. It is clear that $f$ is square–free if and only if $\gcd (f,f')=1$ (see, e.g., [@GaGe99 Corollary 14.25]). Assume that $f$ is not square–free. Hence, $u:=\gcd (f,f') \neq 1$. It follows that $v:=f/u=\prod_{p \nmid e_i} f_i$ is the square–free part of the product $\prod_{p \nmid e_i}f_i^{e_i}$ (see, e.g., [@Shoup05 Theorem 20.4]). Since each $e_i \leq r:=\deg(f)$, we deduce that $\gcd(u,v^r)=\prod_{ p \nmid e_i} f_i^{e_i-1}$. Therefore, $$w:=\frac{u}{\gcd(u,v^r)}=\prod_{p \mid e_i} f_i^{e_i}$$ is the part of $f$ which is a power of $ p $. These are the foundations of the following procedure. ${}$ 1. Input: $f \in {{\mathbb F}_{\hskip-0.7mm q}}[T]$ monic of degree $r>0$. 2. Output: the square–free part of $f$, that is, the product of all distinct irreducible factors of $f$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$. **procedure ERF (f: polynomial)** 1. Compute $u:=\gcd(f,f')$ 2. Compute $v:=\frac{f}{u}$ 3. Compute $w:=\frac{u}{\gcd(u,v^r)}$ 4. Return $v \cdot \mathrm{ERF}(w^{1/p})$. According to [@GaGe99 Exercise 14.27], for $f\in {{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree at most $r$, the number of arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ performed by the ERF algorithm to obtain the square–free part of $f$ is $\mathcal{O}(M(r)\log r+ r \log (q/p))$. In this section we analyze the average–case complexity of the ERF algorithm restricted to elements of the family $\mathcal{A}$. More precisely, we analyze the expected value $E[\mathcal{X}_1]$ of the random variable $\mathcal{X}_1$ defined in , namely $$\label{esperanza ERF} E[\mathcal{X}_1]:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}} \mathcal{X}_1(f).$$ Let $\mathcal{A}^{sq}$ be the set of $f \in \mathcal {A} $ that are square–free and $\mathcal{A}^{nsq}:=\mathcal{A}\setminus \mathcal{A}^{sq}$. The probability that a random polynomial of $\mathcal{A}$ is square–free is $$P[\mathcal{A}^{sq}]=\frac{|\mathcal{A}^{sq}|}{|\mathcal{A}|}=1-\frac{|\mathcal{A}^{nsq}|}{|\mathcal{A}|}.$$ According to , we have $|\mathcal{A}^{nsq}|\leq r(r-1)\delta_{{\boldsymbol}G} q^{r-m-1}$. On the other hand, from Theorem \[estimation A\] it follows that, if $q > 15\delta_{{\boldsymbol}G}^{13/3}$, then $|\mathcal{A}|\geq\frac{1}{2} q^{r-m}$, where $ {\boldsymbol}G:= (G_1, \dots, G_m) $ are the polynomials defining the family $ \mathcal{A} $ and $\delta_{{\boldsymbol}G}:=\deg(G_1)\cdots\deg (G_m)$. As a consequence, $$P[\mathcal{A}^{sq}] \geq 1-\frac{2\,r^2\delta_{{\boldsymbol}G} \,q^{r-m-1}}{q^{r-m}} =1-\frac{2\,r^2\delta_{{\boldsymbol}G}}{q}.$$ In other words, we have the following result. \[mas libres de cuadrados\] For $q > 15\delta_{{\boldsymbol}G}^{13/3}$, the probability that a random polynomial of $\mathcal{A}$ is square–free is $P[\mathcal{A}^{sq}]\ge 1 - 2\,r^2\delta_{{\boldsymbol}G}/q$. In particular, if $q >\max\{ 15\delta_{{\boldsymbol}G}^{13/3},4\,r^2\delta_{{\boldsymbol}G}\}$, then $P[\mathcal{A}^{sq}]>1/2$. To estimate $E[\mathcal{X}_1]$, we decompose the family $\mathcal{A}$ into the sets $\mathcal{A}^{sq}$ and $\mathcal{A}^{nsq}$. We have $$E[\mathcal{X}_1]=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{sq}} \mathcal{X}_1(f) +\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{nsq}} \mathcal{X}_1(f)=:S_1^{sq}+S_1^{nsq}.$$ First we obtain an upper bound for $S_1^{sq}$. On input $f\in\mathcal{A}^{sq}$, the ERF algorithm performs the first three steps. Since $u:=\gcd(f,f')=1$ and $\gcd(u,v^r)=1$, its cost is dominated by the cost of calculating $u$, which is at most $\tau_2\,\mathcal{U}(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$, and the cost of calculating $v^r$, which at most $ \tau_1 \, \mathcal{U}(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. We conclude that, if $f \in \mathcal{A}^{sq}$, then $\mathcal{X}_1(f) \leq (\tau_1 +\tau_2)\, \mathcal{U}(r)$. Therefore, $$\label{cota Suma 1 de X1} S_1^{sq}:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{sq}} \mathcal{X}_1(f)\leq (\tau_1 +\tau_2)\, \mathcal{U}(r) \frac{|\mathcal{A}^{sq}|}{|\mathcal{A}|}.$$ On the other hand, if $f \in \mathcal{A}^{nsq}$, then [@GaGe99 Exercise 14.27] shows that the number of arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$ which performs the ERF algorithm on input $ f $ is bounded by $ \mathcal{X}_1(f) \leq c_1 \big( \mathcal{U}(r)+ r \log \big(\frac{q}{p}\big)\big)$, where $c_1$ is a constant independent of $q$ and $p:=\mathrm{Char}({{\mathbb F}_{\hskip-0.7mm q}})$. Hence, we have $$\label{cota suma S2 de X1} S_1^{nsq}:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{nsq}} \mathcal{X}_1(f)\leq c_1 \bigg(\mathcal{U}(r)+ r \log\Big(\frac{q}{p}\Big)\bigg) \frac{|\mathcal{A}^{nsq}|}{|\mathcal{A}|}.$$ Combining and we conclude that $$\begin{aligned} E[\mathcal{X}_1] & \leq (\tau_1 +\tau_2) \, \mathcal{U}(r) \frac{|\mathcal{A}^{sq}|}{|\mathcal{A}|} + c_1 \, \mathcal{U}(r) \frac{|\mathcal{A}^{nsq}|}{|\mathcal{A}|}+ c_1 \,r \log\Big(\frac{q}{p}\Big)\frac{|\mathcal{A}^{nsq}|}{|\mathcal{A}|}\\ \notag & \leq c_2 \, \mathcal{U}(r) + c_1\, r \log\Big(\frac{q}{p}\Big) \frac{|\mathcal{A}^{nsq}|}{|\mathcal{A}|},\end{aligned}$$ where $c_2:=\max\{ \tau_1 +\tau_2 ,c_1\}$. Hence, if $q>15\delta_{{\boldsymbol}G}^{13/3}$, then Lemma \[mas libres de cuadrados\] implies $$E[\mathcal{X}_1] \leq c_2\, \mathcal{U}(r) + 2\,c_1\, r^3 \delta_{{\boldsymbol}G}\log\Big(\frac{q}{p}\Big)\frac{1}{q}.$$ We obtain the following result. \[costo paso 1\] Let $q > 15\delta_{{\boldsymbol}G}^{13/3}$. The average cost $ E [\mathcal {X} _1] $ of the $\mathrm{ERF}$ algorithm applied to elements of $ \mathcal {A} $ is upper bounded as $ E [\mathcal {X} _1] \leq c_2 \, \mathcal {U} (r) +c_3 \log \big (\frac {q} {p} \big) \delta_{{\boldsymbol}G}\frac {r ^ 3} {q} $, where $c_2$ and $ c_3$ are constants independent of $ r$ and $ q $. We may paraphrase this result as saying that the average cost of the ERF algorithm applied to elements of $\mathcal{A}$ is asymptotically of order $\mathcal{U}(r)$, which corresponds to the cost of calculating the greatest common divisor $u:=\gcd (f,f')$. This generalizes the results of [@FlGoPa01 Section 2]. Distinct–degree factorization ----------------------------- Now we analyze the distinct–degree factorization (DDF) step. Recall that, given a square–free polynomial $a_f:=\mathrm{ERF}(f)$, the DDF routine outputs a list $(b(1),\ldots,b(s))$, where $b(k)$ is the product of all the irreducible factors of degree $k$ of the complete factorization of $a_f$ over ${{\mathbb F}_{\hskip-0.7mm q}}$. The output $(b(1),\dots,b(s))$ is called the [*distinct–degree factorization*]{} of $a_f$. The DDF procedure is based on the following property (see, e.g., [@LiNi83 Theorem 3.20]): for $k \geq 1$, the polynomial $T^{q^k}-T \in{{\mathbb F}_{\hskip-0.7mm q}}[T]$ is the product of all monic irreducible polynomials in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ whose degree divides $k$. It follows that $g_1:=\gcd(T^q-T,f)$ is the product of all the irreducible factors of $f$ of degree $ 1 $. Then, for $ 1 \leq k \leq r $, the polynomial $g_k:=\gcd(T^{q^k }-T, f/g_{k-1}) $ is the product of all the irreducible factors of $f$ of degree $k$. This proves the correctness of the following procedure. \[algoritmo DDF\]${}$ 1. Input: a monic square–free polynomial $a\in{{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree $r>0$. 2. Output: the distinct–degree factorization $(b(1),\dots,b(s))$ of $a$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$. 3. Let $g:=a$, $h:=T$ 4. While $g\neq 1$ do 1. Compute $h:=h^q \mod g$ 2. Compute $b(k):=\gcd(h-T,g)$ 3. Compute $g:=\frac{g}{b(k)}$ 4. $k:=k+1$ 5. End while 6. Return ${\boldsymbol}b$. In [@GaGe99 Theorem 14.4] it is shown that this algorithm performs $\mathcal{O}(s M(r) \log(rq))$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$, where $s$ is the maximum degree of the irreducible factors of the input polynomial $a$. In this section we analyze the average–case complexity of the DDF routine restricted to polynomials of the family $\mathcal{A}$. More precisely, we consider the expected value $E[\mathcal{X}_2]$ of the random variable $\mathcal{X}_2$ of , namely $$E[\mathcal{X}_2]:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}} \mathcal{X}_2(f).$$ We decompose as before the set of inputs $\mathcal{A}$ into the disjoint subsets $\mathcal{A}^{sq}$ (elements of $\mathcal{A}$ which are square–free) and $\mathcal{A} ^{nsq}:=\mathcal{A} \setminus \mathcal{A}^{sq}$. Hence, we have $$\label{esperanza DDF 1} E[\mathcal{X}_2]=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{sq}} \mathcal{X}_2(f)+\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{nsq}} \mathcal{X}_2(f).$$ First we obtain an upper bound for the first sum $S_2^{sq}$ in the right–hand side of . We express $\mathcal{A}^{sq}$ as a disjoint union as follows: $$\mathcal{A}^{sq}=\bigcup_{i=1}^r \mathcal{A}_{i}^{sq},$$ where $\mathcal{A}_{i}^{sq}$ is the set of elements of $\mathcal{A}^{sq}$ for which the maximum degree of the irreducible factors is $i$. Moreover, for $1 \leq i \leq r$, we can express each $\mathcal{A}_{i}^{sq}$ as the disjoint union $$\mathcal{A}_{i}^{sq}=\bigcup_{{\boldsymbol}\lambda \in \mathcal{P}_i} \mathcal{A}_{{\boldsymbol}\lambda}^{sq},$$ where $\mathcal{P}_i$ is the set of ${\boldsymbol}\lambda:=(\lambda_1, \dots, \lambda_i, 0,\dots,0)\in \mathbb{Z}_{\ge 0}^r$ such that $\lambda_1+\cdots+ i \, \lambda_i=r$ and $ \lambda_i>0$, and $\mathcal{A}_{ {\boldsymbol}\lambda}^{sq}$ is the set of elements of $\mathcal{A}_{i}^{sq}$ with factorization pattern ${\boldsymbol}\lambda$. Therefore, $$\label{suma 11 bis} S_2^{sq}=\frac{1}{|\mathcal{A}|} \sum_{i=1}^r \sum_{{\boldsymbol}\lambda\in \mathcal{P}_i}\sum _{f \in \mathcal{A}_{{\boldsymbol}\lambda }^{sq}} \mathcal{X}_2(f).$$ Fix $i$ with $ 1 \leq i \leq r$, let ${\boldsymbol}\lambda\in \mathcal{P}_i$ and $f \in \mathcal{A}_{{\boldsymbol}\lambda}^{sq}$. To determine the cost $\mathcal{X}_2(f)$, we observe that the procedure performs $i$ iterations of the main loop. Fix $l$ with $ 1\leq l\leq i$ and we consider the $l$th iteration of the DDF algorithm. The number of products modulo $g$ needed to compute $h^q \mod g$ is denoted by $\lambda(q)$. Using repeated squaring, and denoting by $\nu(q)$ the number of ones in the binary representation of $q$, the number of products required to compute $h^q \mod g$ is $$\lambda (q):= \lfloor \log q \rfloor + \nu (q) -1.$$ Thus the first step in the $ l $th iteration of the DDF algorithm requires at most $2\,\tau_1\, \lambda (q) M(r_l)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$, where $r_l:=\deg g$ (note that $r_1=r$ and $r_l \leq r$ for any $l$). Then the computation $b(k):=\gcd(h-T,g)$ requires at most $\tau_2 M(r_l) \log r_l$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Finally, the division $g/ b(k) $ requires at most $\tau_1 M(r_l)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. As a consequence, we see that $$\mathcal{X}_2(f) \leq \sum_{l=1}^i (2\,\tau_1 \lambda (q) + \tau_2 \log r_l + \tau_1) \, M(r_l).$$ Observe that, if $a \leq b$, then $M(a) \leq M(b)$ (see, e.g., [@GaGe99 §14.8])). It follows that $$\label{Costo X2} \mathcal{X}_2(f) \leq i\,c_{r,q},\quad c_{r,q}:= M(r)\,\big(2\,\tau_1\lambda(q)+ \tau_1+\tau_2 \log r\big).$$ Thus, we obtain $$S_2^{sq} \leq \frac{c_{r,q}}{|\mathcal{A}|} \sum_{i=1}^r \sum_{{\boldsymbol}\lambda \in \mathcal{P}_i}\sum _{f \in \mathcal{A}_{{\boldsymbol}\lambda }^{sq}} i= \frac{c_{r,q}}{|\mathcal{A}|}\sum_{i=1}^r i \sum_{{\boldsymbol}\lambda \in \mathcal{P}_i} |\mathcal{A}_{{\boldsymbol}\lambda}^{sq}|.$$ We have the following result. \[S2sq cota\] For $q > 15\delta_{{\boldsymbol}G}^{13/3}$, the sum $S_2^{sq}$ is bounded in the following way: $$\label{suma 11bisbisbisbis} S_2^{sq} \leq c_{r,q}\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q} \bigg)\xi (r+1)= c_{r,q}\,\xi (r+1)\big(1+o(1)\big),$$ where $M_{r}:=D\delta q^{\frac{1}{2}} +14\,D^2 \delta^2+ r^2\delta$, $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$, $D:=\sum_{i=1}^m ({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1)$ and $\xi\sim 0.62432945\dots$ is the Golomb constant. According to Theorem \[theorem: estimate fact patterns\], we have $$|\mathcal{A}_{{\boldsymbol}\lambda }^{sq}| \leq q^{r-m}\,\mathcal{T}({\boldsymbol}\lambda)\bigg(1 +\frac{M_{ r}}{q}\bigg),$$ where $\mathcal{T}({\boldsymbol}\lambda)$ is the probability of the set of permutations with cycle pattern ${\boldsymbol}\lambda $ in the symmetric group $\mathbb{S}_r$ of $r$ elements. Hence, $$\begin{aligned} \label{suma 11bisbisbis} S_2^{sq} & \leq \frac{c_{r,q}}{|\mathcal{A}|}\,q^{r-m}\bigg(1+\frac{M_{r}}{q} \bigg)\sum_{i=1}^r i \sum_{{\boldsymbol}\lambda \in \mathcal{P}_i} \mathcal{T}({\boldsymbol}\lambda ).\end{aligned}$$ Now we analyze the sum $E_r:=\sum_{i=1}^r i\sum_{{\boldsymbol}\lambda \in \mathcal{P}_i} \mathcal{T}({\boldsymbol}\lambda )$. Observe that the sum $ \sum_ {{\boldsymbol}\lambda \in \mathcal {P}_i} \mathcal{T} ({\boldsymbol}\lambda) $ expresses the probability of the set of permutations whose longest cycle has length $ i $. It follows that $ E_r $ is the largest expected length between cycles of a random permutation in $ \mathbb {S} _r $. In [@GoGa98] it is shown that $$\frac{E_r}{r+1}\leq \xi,$$ where $\xi$ is the Golomb constant (see, e.g., [@Knuth98]). Combining this upper bound, Theorem \[estimation A\] and , we readily deduce the statement of the lemma. Next we obtain an upper bound for the second sum $S_2^{nsq}$ of , namely $$S_2^{nsq}:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{nsq}} \mathcal{X}_2(f).$$ Given $f \in \mathcal{A}^{nsq}$, we bound $\mathcal{X}_2(f):=\mathrm{Cost}(\mathrm{DDF}(a_f))$, where $a_f:=\mathrm{ERF}(f)$ is the output square–free polynomial of the ERF procedure applied to $f$. By we have $$\mathcal{X}_2(f) \leq c_{N,q}\cdot s_a,$$ where $c_{N,q}:=M(N)\,\big(2\,\tau_1\lambda(q)+ \tau_1+\tau_2 \log N\big)$, $N:=\deg(a_f)$ and $s_a$ is the highest degree of the irreducible factors of $ a_f $. Since $f \in \mathcal{A}^{nsq}$, we have $N\leq r-1$ and $s_a\le r -2$. Moreover, it is easy to see that these bounds are optimal. Therefore we obtain $$\mathcal{X}_2(f) \leq c_{r-1,q}\, (r-2).$$ Combining this bound, Theorem \[estimation A\] and we see that, if $q> 15\delta_{{\boldsymbol}G}^{13/3}$, then $$\begin{aligned} \nonumber S_{2}^{nsq} \leq c_{r-1,q}\,(r-2) \frac{|A^{nsq}|}{|\mathcal{A}|} &\leq c_{r-1,q}\,(r-2)\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \frac{r^2\delta_{{\boldsymbol}G}\, q^{r-m-1}}{q^{r-m}}\\&\leq \,c_{r-1,q}\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\,\frac{r^3\delta_{{\boldsymbol}G}}{q}.\label{suma 2 bis}\end{aligned}$$ From the upper bounds of Lemma \[S2sq cota\] and we conclude that $$\begin{aligned} E[\mathcal{X}_2]&=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{sq}} \mathcal{X}_2(f)+\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}^{nsq}} \mathcal{X}_2(f)\\ &\le c_{r,q}\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q} \bigg)\xi (r+1)+\,c_{r-1,q}\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\,\frac{r^3\delta_{{\boldsymbol}G}}{q}.\end{aligned}$$ Since $c_{j,q}:=M(j)\,\big(2\, \tau_1 \lambda(q)+\tau_1+\tau_2\log j\big)$, we have $c_{r-1,q} \leq c_{r,q}$. As a consequence, we obtain the following result. \[average de DDF\] For $q> 15\delta_{{\boldsymbol}G}^{13/3}$, the average cost $E[\mathcal{X}_2]$ of the $\mathrm{DDF}$ algorithm restricted to $\mathcal {A} $ is bounded by $$\begin{aligned} E[\mathcal{X}_2] &\leq \xi\,(2\, \tau_1 \lambda(q)+\tau_1+\tau_2\log r) M(r)\,(r+1) \bigg(1 +\frac{M_r+r^2\delta_{{\boldsymbol}G}}{q}\bigg)\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\\ &=\xi\,(2\, \tau_1 \lambda(q)+\tau_1+\tau_2\log r) M(r)\,(r+1)\big(1+o(1)\big),\end{aligned}$$ where $M_{r}:=D\delta q^{\frac{1}{2}} +14\,D^2 \delta^2+ r^2\delta$, $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$, $D:=\sum_{i=1}^m ({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1)$ and $\xi\sim 0.62432945\dots$ is the Golomb constant. In [@FlGoPa01 Theorem 5] the authors prove that the average cost of the DDF algorithm applied to a random polynomial $ f \in {{\mathbb F}_{\hskip-0.7mm q}}[T] $ of degree at most $ r $ is of order $0.26689\, (2\,\tau_1\,\lambda(q) +\tau_2)\,r^3$. We prove that, assuming that fast arithmetic is used, the average cost of this algorithm restricted to $\mathcal{A}$ is of order $\xi(2\,\tau_1\,\lambda(q)+\tau_1 +\tau_2\log r)\,(r+1)\,M(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$, thus improving the result of [@FlGoPa01] (which assumes that standard arithmetic is used). The DDF algorithm does not completely factor any polynomial $f \in \mathcal{A}$ having distinct irreducible factors of the same degree. More precisely, the classical factorization algorithm ends in this step if the input polynomial $f$ has a factorization pattern ${\boldsymbol}\lambda\in \{0,1\}^r$. We conclude this section with a result on the probability that the DDF algorithm outputs the complete factorization of the input polynomial of $\mathcal {A}$. In [@FlSe09] it is shown that most factorizations are completed after the application of the DDF procedure. More precisely, it is proved that, when $ r $ is fixed and $ q $ tends to infinity, the probability that the DDF algorithm produces a complete factorization of a random polynomial of degree at most $ r $ in $ {{\mathbb F}_{\hskip-0.7mm q}}[T] $ is of order of $e^{-\gamma} \sim 0.5614\dots$, where $\gamma \sim 0.57721\dots$ is the Euler constant (see [@FlGoPa01 Theorem 6]). We generalize this result to the family $\mathcal{A}$. The probability that the $\mathrm{DDF}$ algorithm completes the factorization of a random polynomial of $\mathcal {A}$ is bounded from above by $\big(e^{-\gamma}+ e^{-{\gamma}}/{r}+O({\log r}/{r^2})\big)\big(1+o(1)\big)$, where $ \gamma $ is Euler’s constant. Let $\mathcal{A}_1$ be set of elements of $ \mathcal {A} $ whose irreducible factors have all distinct degrees. The probability that the DDF algorithm outputs the complete factorization of a random polynomial $f\in \mathcal {A} $ coincides with the probability that random $f\in \mathcal {A}$ belongs to $\mathcal {A}_1$. We may express $ \mathcal {A}_1 $ as the following disjoint union: $$\mathcal{A}_1=\bigcup_{{\boldsymbol}\lambda \in \mathcal{P}_{r}} \mathcal{A}_{1,{\boldsymbol}\lambda},$$ where $ \mathcal {P}_r $ is the set of all vectors $ {\boldsymbol}\lambda: = (\lambda_1, \dots, \lambda_r) \in \{0,1 \}^r $ such that $ \lambda_1 + \dots + r \, \lambda_r = r $ and $ \mathcal{A}_{1, {\boldsymbol}\lambda} $ is the set of elements of $ \mathcal {A}_1 $ having factorization pattern $ {\boldsymbol}\lambda $. Hence, $$\label{completar factorizacion DDF} P[\mathcal{A}_1]=\sum_{{\boldsymbol}\lambda \in \mathcal{P}_r} P [\mathcal{A}_{1,{\boldsymbol}\lambda}]=\frac{1}{|\mathcal{A}|} \sum_{{\boldsymbol}\lambda \in \mathcal{P}_r} |\mathcal{A}_{1,{\boldsymbol}\lambda}|.$$ Observe that, if $f \in \mathcal{A}_1$, then $f$ is square–free. By Theorem \[theorem: estimate fact patterns\], for $m<r$ we have $$|\mathcal{A}_{1,{\boldsymbol}\lambda}| \leq q^{r-m}\,\mathcal{T}({\boldsymbol}\lambda)\,\bigg(1+\frac{M_{r}}{q}\bigg),$$ where $M_{r}:=D\delta q^{\frac{1}{2}} +14\,D^2 \delta^2+ r^2\delta$, $\delta:=\prod_{i=1}^m {\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)$ and $D:=\sum_{i=1}^m ({\ifm {{\sf wt}} {{$\sf wt$}}}(G_i)-1)$. Theorem \[estimation A\] shows that, if $q > 15\delta_{{\boldsymbol}G}^{13/3}$, then $$P[ \mathcal{A}_1] \leq \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg)\sum_{{\boldsymbol}\lambda \in \mathcal{P}_r}\mathcal{T}({\boldsymbol}\lambda).$$ We observe that $\sum_{{\boldsymbol}\lambda \in \mathcal{P}_r} \mathcal{T}({\boldsymbol}\lambda)$ expresses the probability that a random permutation of $\mathbb{S}_r$ has a decomposition into cycles of pairwise different lengths. By [@GrKn90 (4.57)] (see also [@FlFuGoPaPo06 Proposition 1]), it follows that $$\sum _{{\boldsymbol}\lambda \in \mathcal{P}_r}\mathcal{T}( {\boldsymbol}\lambda)= e^{-\gamma}+\frac{e^{-\gamma}}{r}+O\bigg(\frac{\log_2r}{r^2}\bigg).$$ We deduce that $$P[\mathcal{A}_1]\leq \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \bigg(1+\frac{M_{r}}{q}\bigg)\bigg(e^{-\gamma}+ \frac{e^{-\gamma}}{r}+O\bigg(\frac{\log_2r}{r^2}\bigg)\bigg).$$ This finishes the proof of theorem. Equal–degree factorization -------------------------- After the first two steps of the classical factorization algorithm, the general problem of factorization is reduced to factorizing a collection of square–free polynomials $ b (k) $, whose irreducible factors have all the same degree $ k $. The procedure for equal–degree factorization (EDF) receives as input a vector $ {\boldsymbol}b_f: = \mathrm{DDF} (a_f) = (b_f (1), \dots, b_f (s)) $, where each $ b_f (k) $ is the product of the irreducible factors of degree $ k $ of the square–free part $ a_f: = \mathrm{ERF} (f) $ of $ f $. Its output is the irreducible factorization $b_f (k)=b_f (k, 1) \cdots b_f ( k, l)$ in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of each $b_f (k)$. The probabilistic algorithm presented here is based on the Cantor–Zassenhaus algorithm [@Zassenhaus69]. ${}$ 1. Input: a monic square–free polynomial $c\in{{\mathbb F}_{\hskip-0.7mm q}}[T]$ whose irreducible factors in ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ have all degree $k$. 2. Output: the complete factorization of $c$. **procedure EDF(c: square–free polynomial , $k$: integer)** 1. If $\deg c=k$, then return $c$ 2. End if 3. Choose a random $h\in{{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree $\deg{c}-1$. 4. Compute $g:=h^{(q^k-1)/2}-1 \mod c$ 5. Compute $d:=\gcd(g,c)$ 6. Return $\mathrm{EDF}(d,k)\cdot \mathrm{EDF}(c/d,k)$. The EDF algorithm is based on the principle we now briefly explain. Assume that the irreducible factorization of the input polynomial $ c $ is $c= f_1 \cdots f_j $, with each $f_i$ of degree $ k $. The Chinese remainder Theorem implies that $${{\mathbb F}_{\hskip-0.7mm q}}[T] /(c) \cong {{\mathbb F}_{\hskip-0.7mm q}}[T] / (f_1) \times \dots \times {{\mathbb F}_{\hskip-0.7mm q}}[T] / (f_j).$$ Under this isomorphism, a random $h\in{{\mathbb F}_{\hskip-0.7mm q}}[T]/(c)$ is associated to a $j$–tuple $(h_1,\dots,h_j)$, where each $h_i$ is a random element of ${{\mathbb F}_{\hskip-0.7mm q}}[T]/(f_i)$. Since each $f_i$ is irreducible, the quotient ring ${{\mathbb F}_{\hskip-0.7mm q}}[T]/(f_i)$ is a finite field, isomorphic to ${{\mathbb F}_{\hskip-0.7mm q^k}}$. The multiplicative group ${{\mathbb F}_{\hskip-0.7mm q^k}}^*$ being cyclic, there are the same number $(q^k-1)/2$ of squares and non–squares (see, e.g., [@GaGe99 Lemma 14.7]). Recall that $m \in {{\mathbb F}_{\hskip-0.7mm q^k}}^*$ is square if only if $m^{(q^k-1)/2}=1$. Therefore, testing whether $h_i^{(q^k-1)/2}=1$ discriminates the squares in ${{\mathbb F}_{\hskip-0.7mm q^k}}^*$. Thus, if $g:=h^{(q^k-1)/2}-1 \mod c$, then $\gcd(g,c)$ is the product of all the $f_i$ with $h$ a square in ${{\mathbb F}_{\hskip-0.7mm q}}[T]/(f_i)$. From the probabilistic standpoint, a random element $h_i$ of ${{\mathbb F}_{\hskip-0.7mm q}}[T]/(f_i)$ has probability $\alpha:= 1/2-1/(2q^k)$ of being a square and the dual probability $\beta:=1/2+1/(2q^k)$ of being a non–square. Then the EDF algorithm is applied recursively to the polynomials $ d = \gcd (g, c) $ and $ c / d $. In this way, all the irreducible factors of $c:=b(k)$ are extracted successively. Following [@FlGoPa01 Section 5], in this section we analyze the average–case complexity of the EDF algorithm applied to the family $ \mathcal {A} $, namely we consider the expected value $E[\mathcal{X}_3]$ of the random variable $\mathcal{X}_3$ of : $$E[\mathcal{X}_3]:=\frac{1}{|\mathcal{A}|} \sum _{f \in \mathcal{A}} \mathcal{X}_3(f).$$ We decompose $\mathcal {X} _3 $ as in in the form $$\mathcal{X}_3(f):=\sum_{k=1}^{\lceil r/2 \rceil} \mathcal{X}_{3,k}(f),\quad \mathcal{X}_{3,k}(f)\!\!:=\!\!\mathrm{Cost}(\mathrm{EDF}(b_f(k)))\quad(1\le k\le {\lceil r/2 \rceil}),$$ where $ b_f (k) $ is the $k$th coordinate of ${\boldsymbol}b_f: = \mathrm{DDF} (a_f) = (b_f (1), \dots, b_f (s)) $. Hence, we have $$E[\mathcal{X}_3]=\frac{1}{|\mathcal{A}|}\sum_{k=1}^{\lceil r/2 \rceil}\sum_{f\in \mathcal{A}}\mathcal{X}_{3,k}(f)=\sum_{k=1}^{\lceil r/2 \rceil} E[\mathcal{X}_{3,k}].$$ Fix $k$ with $ 1\leq k \leq \lceil r/2 \rceil$ and write $E[\mathcal{X}_{3,k}]$ as follows: $$E[\mathcal{X}_{3,k}]=\frac{1}{|\mathcal{A}|} \sum_{f \in \mathcal{A}^{sq}} \mathcal{X}_{3,k}(f)+\frac{1}{|\mathcal{A}|} \sum_{f \in \mathcal{A}^{nsq}} \mathcal{X}_{3,k}(f)=:S_{3,k}^{sq}+S_{3,k}^{nsq}.$$ We first bound $S_{3,k}^{sq}$. For this purpose, we express $\mathcal{A}^{sq}$ as the disjoint union $$\mathcal{A}^{sq}=\bigcup_{j=0}^{\lfloor r/k \rfloor} \mathcal{A}_{j,k}^{sq},$$ where $\mathcal{A}_{j,k}^{sq}$ is the set of all elements $f\in \mathcal{A}^{sq}$ having $j$ irreducible factors of degree $k$. Hence, $$\label{esperanza X3k libre de cuadrados} S_{3,k}^{sq}= \frac{1}{|\mathcal{A}|}\sum_{j=0}^{\lfloor r/k \rfloor } \sum_{f \in \mathcal{A}_{j,k}^{sq}} \mathcal{X}_{3,k}(f).$$ We first bound the cost $\mathcal{X}_{3,k}(f)$ of the $\mathrm{EDF}$ algorithm applied to any $f \in \mathcal{A}_{j,k}^{sq}$. \[Costo\] For any $f\in \mathcal{A}_{j,k}^{sq}$, we have $$\mathcal{X}_{3,k}(f) \leq \frac{j(j-1)}{\alpha \beta}\big(\tau_1\,\mu_k {M(r)}+ \tau_3\,\mathcal{U}(r)\big)\, \frac{k}{r},$$ where $\mu_k:=\lambda \big(\frac{q^{k} -1} {2} \big): =\lfloor \log(\frac{q^k-1}{2})\rfloor +\nu(\frac{q^k-1}{2})-1$ and $\tau_3:=\max\{\tau_1,\tau_2\}$. If $j=0$ or $j=1$, then the EDF procedure does not perform any computation, and the result trivially follows. Therefore, we may assume that $j\ge 2$. The cost of a recursive call to the EDF procedure for $ f \in \mathcal {A}_{j, k} ^{sq}$ is determined by the cost of computing $h^ {(q ^k-1) / 2} \mod f $, where $ h $ is a random element of $ {{\mathbb F}_{\hskip-0.7mm q}}[T] / (f) $, a greatest common divisor of $f$ with a polynomial of degree at most $ jk $ and a division of two polynomials of degree at most $ jk $. Observe that $ \mu_k$ products modulo $f$ are required to compute $ h^{ (q ^ k-1) / 2} \mod f $ using binary exponentiation. We conclude that $ h ^ {(q ^ k-1) / 2} \mod f $ can be computed with at most $2\, \tau_1\, \mu_k M (jk) $ arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$, while the remaining greatest common divisor and division are computed with at most $ \tau_2\, \mathcal{U} (jk) $ and $ \tau_1\, M(jk) $ arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$. In other words, we have $$2\, \tau_1\, \mu_k M (jk)+\tau_2\, \mathcal{U} (jk)+\tau_1\, M(jk)\le \Big(\tau_1\, \mu_k \frac{M (r)}{k\,r}+\tau_2\, \frac{\mathcal{U} (r)}{2\,k\,r}+\tau_1\, \frac{M(r)}{2\,k\,r}\Big)(jk)^2$$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Applying [@FlGoPa01 Lemma 4] with $\widetilde{\tau}_1:= \frac{\tau_1 M(r)}{k\,r}$ and $\widetilde{\tau}_2:=\frac{\tau_3\, \mathcal{U}(r)}{k\,r}$, we see that $$\mathcal{X}_{3,k}(f) \leq \bigg(\frac{j(j-1)}{2 \alpha \beta} + j\sum_{m=0}^{\infty}\sum_{l=0}^m \binom {m}{l} \alpha^{m-l} \beta^{l}\big(1-(1-\alpha^{m-l} \beta^l)^{j-1}\big)\bigg)\, (\mu_k \widetilde{\tau}_1 + \widetilde{\tau}_2)\, k^2.$$ Using the inequality $ 1- (1-u)^{j-1} \leq (j-1) u $ for $ j \geq 2 $ and $ 0 \leq u \leq 1 $, we obtain $$\begin{aligned} \sum_{m=0}^{\infty}\sum_{l=0}^m \binom {m}{l} \alpha^{m-l} \beta^{l}\big(1-(1-\alpha^{m-l} \beta^l)^{j-1}\big)&\le (j-1)\sum_{m=0}^{\infty}\sum_{l=0}^m \binom {m}{l} \alpha^{2(m-l)} \beta^{2l}\\ &\le (j-1) \sum_{m=0}^{\infty}(\alpha^2+ \beta^2)^m=\frac{j-1}{2\alpha \beta}.\end{aligned}$$ This easily implies the lemma. As a consequence of Lemma \[Costo\], we have $$\label{esperanza X3k libre de cuadrados bis} S_{3,k}^{sq}:= \frac{1}{\vert\mathcal{A}\vert}\sum_{j=2}^{\lfloor r/k \rfloor } \sum_{f \in \mathcal{A}_{j,k}^{sq}} \mathcal{X}_{3,k}(f) \leq \sum_{j=2}^{\lfloor r/k \rfloor } \frac{j(j-1)}{\alpha \beta}\, \big(\tau_1\,\mu_k {M(r)}+ \tau_3\,\mathcal{U}(r)\big)\, \frac{k}{r} \, \frac{|\mathcal{A}_{j,k}^{sq}|}{|\mathcal{A}|}.$$ In the next result we obtain an explicit upper bound for $S_{3,k}^{sq}$. \[cota N1\] For $q>15\delta_{{\boldsymbol}G}^{13/3}$, we have $$S_{3,k}^{sq}\leq \frac{1}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg) \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \bigg(1+\frac{M_{r}}{q}\bigg),$$ where $\mu_k$ and $\tau_3$ are as in Lemma \[Costo\] and $M_{r}$ is defined as in Theorem \[average de DDF\]. According to , we estimate the probability $P[\mathcal {A}_{j, k}^ {sq} ]$ that a random $ f \in \mathcal { A}$ is square–free and has $ j $ irreducible factors of degree $ k $. In [@KnKn90b] it is shown that, if $ q $ is sufficiently large, then the probability that a random $ f \in {{\mathbb F}_{\hskip-0.7mm q}}[T] $ of degree at most $r$ has $j$ distinct irreducible factors of degree $ k $ tends to $e^{-1/k}\frac{k^{-j}}{j!}$. We decompose the set $\mathcal{A}_{j,k}^{sq}$ into the disjoint union $$\mathcal{A}_{j,k}^{sq}=\bigcup_{{\boldsymbol}\lambda \in \mathcal{P}_r^{j,k}} \mathcal{A}_{j,{\boldsymbol}\lambda}^{sq},$$ where $\mathcal{P}_{r}^{j,k}$ is the set of all $ r $–tuples $ {\boldsymbol}\lambda: = (\lambda_1, \dots, \lambda_r) \in \mathbb{Z}_{\geq0}^r $ with $\lambda_1 + \dots + r \, \lambda_r = r $ and $ \lambda_k = j $. Hence, we have $$P[\mathcal{A}_{j,k}^{sq}]=\frac{1}{|\mathcal{A}|} \sum_{{\boldsymbol}\lambda \in \mathcal{P}_{r}^{j,k}} |\mathcal{A}_{j,{\boldsymbol}\lambda}^{sq}|.$$ From Theorem \[theorem: estimate fact patterns\] we deduce that $$|\mathcal{A}_{j,{\boldsymbol}\lambda}^{sq}| \leq q^{r-m}\,\mathcal{T}({\boldsymbol}\lambda)\bigg(1+\frac{M_{r}}{q}\bigg).$$ From Theorem \[estimation A\] it follows that, for $q> 15\delta_{{\boldsymbol}G}^{13/3}$, $$P[\mathcal{A}_{j,k}^{sq}]=\frac{1}{|\mathcal{A}|} \sum_{{\boldsymbol}\lambda \in \mathcal{P}_{r}^{j,k}} |\mathcal{A}_{j,{\boldsymbol}\lambda}^{sq}| \leq \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg) \sum_{{\boldsymbol}\lambda \in \mathcal{P}_{r}^{j,k}} \mathcal{T}({\boldsymbol}\lambda).$$ The sum of the right–hand side expresses the probability that a random permutation in $ \mathbb {S}_r $ has exactly $ j $ cycles of length $ k $. In [@ShLl96] it is shown that $$\sum_{{\boldsymbol}\lambda \in \mathcal{P}_{r}^{j,k}} \mathcal{T}({\boldsymbol}\lambda)=\frac{1}{j!k^j}\sum_{i=0}^{\lfloor r/k-j \rfloor}(-1)^i \frac{1}{i! k^i}.$$ We observe that the sum of all probabilities is $ 1 $, that is, $$\sum _{j=0}^{\lfloor r/k \rfloor} \frac{1}{j!k^j}\sum_{i=0}^{\lfloor r/k-j \rfloor}(-1)^i \frac{1}{i! k^i}=1.$$ As a consequence, by we deduce that $$\begin{aligned} S_{3,k}^{sq}& \leq \sum_{j=2}^{\lfloor r/k \rfloor } \frac{j(j-1)}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg) k^2 \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg) \frac{1}{j!k^j}\!\!\sum_{i=0}^{\lfloor r/k-j \rfloor} \frac{(-1)^i}{i! k^i}\\ & \leq \frac{1}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg) \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \bigg(1+\frac{M_{r}}{q}\bigg)\sum_{j=2}^{\lfloor r/k \rfloor} \frac{1}{(j-2)!k^{j-2}}\!\!\sum_{i=0}^{\lfloor r/k-j \rfloor} \frac{(-1)^i}{i! k^i}\\ & \leq \frac{1}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg) \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg) \bigg(1+\frac{M_{r}}{q}\bigg).\end{aligned}$$ This shows the lemma. Next we obtain an upper bound for $$\begin{aligned} \label{cota N2} S_{3,k}^{nsq}:=\frac{1}{|\mathcal{A}|} \sum_{f \in \mathcal{A}^{nsq}} \mathcal{X}_{3,k}(f).\end{aligned}$$ Let $f \in \mathcal{A}^{nsq}$ and ${\boldsymbol}b_f:=\mathrm{DDF}(a_f)=(b_f(1),\dots, b_f(s))$. Assume that $\deg(b_f(k))=m_k$. We have the following bound (see, e.g., [@GaGe99 Theorem 14.11]): $$\mathcal{X}_{3,k}(f) \leq c\, (k \log q+ \log m_k)M(m_k) \log\Big(\frac{m_k}{k}\Big),$$ where $c$ is a constant independent of $k$ and $q$. Taking into account the estimate of $|\mathcal{A}^{nsq}|$ of and Theorem \[estimation A\], we conclude that, if $q > 15\delta_{{\boldsymbol}G}^{13/3}$, then $$\begin{aligned} \label{Cota N2 bis} S_{3,k}^{nsq}& \leq c\, (k \log q+ \log m_k)M(m_k) \log\bigg(\frac{m_k}{k}\bigg) \frac{2\,r^2\delta_{{\boldsymbol}G}}{q}.\end{aligned}$$ Now we are able to bound the cost of the EDF procedure. \[average de EDF\] For $q > 15\delta_{{\boldsymbol}G}^{13/3}$, the average cost $E[\mathcal{X}_3]$ of the $\mathrm{EDF}$ algorithm restricted to $\mathcal{A}$ is upper bounded as $$E[\mathcal{X}_3] \leq \tau\, M(r) \log q\bigg(\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg)+\frac{r^2\delta_{{\boldsymbol}G}}{q}\bigg)= \tau\, \mathcal{U}(r) \log q\, (1+o(1)),$$ where $\tau$ is a constant independent of $q$ and $r$ and $M_{r}$ is defined as in Theorem \[average de DDF\]. Recall that $E[\mathcal{X}_3]=S_{3,k}^{sq}+S_{3,k}^{nsq}$. From Lemma \[cota N1\] and , we have $$\begin{aligned} S_{3,k}^{sq}&\le \bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg) \sum_{k=1}^{\lceil r/2 \rceil} \frac{1}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg),\\ S_{3,k}^{nsq}&\le \frac{2\,c\,r^2\delta_{{\boldsymbol}G}}{q} \sum_{k=1}^{\lceil r/2 \rceil}(k \log q+ \log m_k)M(m_k) \log\Big(\frac{m_k}{k}\Big).\end{aligned}$$ We first estimate the sum $$S_1:=\sum_{k=1}^{\lceil r/2 \rceil} \frac{1}{\alpha \beta}\bigg(\!\tau_1\mu_k \frac{M(r)}{k\,r}+ \tau_3\frac{\mathcal{U}(r)}{k\,r}\bigg).$$ Recall that $\mu_k:=\lfloor \log(\frac{q^k-1}{2})\rfloor +\nu(\frac{q^k-1}{2})-1$, $\alpha:={1}/{2}-{1}/(2 q^k)$ and $\beta:={1}/{2}+{1}/(2 q^k)$. It is easy to see that $$\frac{1}{\alpha \beta}\leq \frac{4q^2}{q^2-1}\leq \frac{16}{3},\quad \mu_k \leq 2\, k \log q.$$ As a consequence, $$\begin{aligned} S_1\!\leq\! \frac{64 \tau_1}{3}\,\frac{M(r) {\lceil r/2 \rceil} \log q}{r} + \frac{32 \tau_3}{3} \frac{\mathcal{U}(r)}{r}\sum_{k=1}^{\lceil r/2 \rceil} \frac{1}{k}\leq M(r) \log q\bigg(\frac{64 \tau_1}{3}+\frac{32 \tau_3}{3} \frac{H(\lceil r/2 \rceil)\log r}{r} \bigg),\end{aligned}$$ where $H(\lceil r/2 \rceil)$ is the $\lceil r/2 \rceil$–th harmonic number. Since $H(N)\leq 1+\ln N$ (see, e.g., [@GrKnPa94 §6.3]), we deduce that, if $r \geq 2$, then ${H(\lceil r/2 \rceil)\log r}/{r} \leq 1$. We conclude that $$\begin{aligned} \label{primera suma de X3 bisbis} S_1\leq M(r) \log q\bigg(\frac{64 \tau_1}{3}+\frac{32 \tau_3}{3}\bigg).\end{aligned}$$ We now estimate the sum $$S_2:=\sum_{k=1}^{\lceil r/2 \rceil}(k \log q+ \log m_k)M(m_k) \log\Big(\frac{m_k}{k}\Big).$$ We have the following inequalities: $$\sum_{k=1}^{\lceil r/2 \rceil}k\, M(m_k)\log\Big(\frac{m_k}{k}\Big)\leq M(r)\sum_{k=1}^{\lceil r/2 \rceil}m_k \frac{\log\big(\frac{m_k}{k}\big)}{\frac{m_k}{k}}\leq M(r)\sum_{k=1}^{\lceil r/2 \rceil} m_k \leq r\, M(r),$$ $$\sum_{k=1}^{\lceil r/2 \rceil} M(m_k)\log(m_k) \log\Big(\frac{m_k}{k}\Big) \leq M(r)\sum_{k=1}^{\lceil r/2 \rceil} \log^2(m_k) \leq M(r)\sum_{k=1}^{\lceil r/2 \rceil} m_k \leq r M(r).$$ Hence, we deduce that $$\begin{aligned} \label{segunda suma de X3 bis} S_2& \leq 2\, r M(r) \log q.\\ \notag\end{aligned}$$ From and we obtain the following upper bound for $E[\mathcal{X}_3]$: $$E[\mathcal{X}_3] \leq M(r)\log q \bigg(\bigg(1+\frac{15\delta_{{\boldsymbol}G}^{{13}/{6}}}{q^{{1}/{2}}}\bigg)\bigg(1+\frac{M_{r}}{q}\bigg)\bigg(\frac{64 \tau_1}{3}+\frac{32 \tau_3}{3}\bigg) +\frac{4\,c\,r^3\delta_{{\boldsymbol}G}}{q}\bigg).$$ Defining $ \tau: =\max \{\frac{64 \tau_1} {3}+\frac {32 \tau_3} {3},4\,c\}$, the statement of the theorem follows. In [@FlGoPa01 Theorem 9], using the classical multiplication of polynomials, it is shown that the EDF algorithm requires on average $ \mathcal {O} (r ^ 2 \log q)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$ on the set of elements of ${{\mathbb F}_{\hskip-0.7mm q}}[T]$ of degree at most $r$. Theorem \[average de EDF\] proves that, using fast multiplication, the EDF algorithm performs on average $r\, \log q $ arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$ on $\mathcal{A}$, up to logarithmic terms and terms which tend to zero as $q$ tends to infinity (for fixed $\delta_{{\boldsymbol}G}$ and $r$). Our analysis improves the worst–case analysis of [@GaGe99 Theorem 14.11], where it is proved that the EDF algorithm applied to a polynomial of degree at most $r$ having $j$ irreducible factors of degree $ k $ requires $ \mathcal {O} ((k \log q+\log r) M(r) \log j)$ arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$, that is, $\mathcal{O}^\sim (k\,r \log q )$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Average-case analysis of the classical algorithm ------------------------------------------------ Now we are able to conclude the analysis of the average cost of the factorization algorithm applied to elements of $ \mathcal {A} $. For this purpose, it remains to analyze the behavior of the classical factorization algorithm when the first three steps fail to find the complete factorization of the input polynomial, namely the expected value $E[\mathcal{X}_4]$ of the random variable $\mathcal{X}_4$ which counts the number of arithmetic operations in $ {{\mathbb F}_{\hskip-0.7mm q}}$ that the algorithm performs to factorize $f/\mathrm{ERF}(f) $, when $ f $ runs over all elements of $ \mathcal {A} $. We can rewrite $ E [\mathcal {X} _4] $ as follows: $$E[\mathcal{X}_4]=\frac{1}{|\mathcal{A}|}\sum_{f \in \mathcal{A}^{sq}} \mathcal{X}_4(f) + \frac{1}{|\mathcal{A}|}\sum_{f \in \mathcal{A}^{nsq}} \mathcal{X}_4(f)=:S_4^{sq}+S_4^{nsq}.$$ We estimate the first sum $S_4^{sq}$. If $f \in \mathcal{A}^{sq}$, then $f/\mathrm{ERF}(f)=1$ and the algorithm does not perform any further operation. Hence, the cost of this step is that of dividing two polynomials of degree at most $r$ at most, namely $\tau_1 M(r)$ arithmetic operations in ${{\mathbb F}_{\hskip-0.7mm q}}$. Thus, $$\label{Suma 1 de esperanza} S_4^{sq} :=\frac{1}{|\mathcal{A}|}\sum_{f \in \mathcal{A}^{sq}} \mathcal{X}_4(f)\leq \tau_1 M(r).$$ Now we estimate the second sum $S_4^{nsq}$. For this purpose, we decompose the set $\mathcal{A}^{nsq}$ into the disjoint union of the set $\mathcal{A}_{=2}^{nsq}$ of elements having all the irreducible factors of multiplicity at most $2$, and $\mathcal{A}_{\geq 2}^{nsq}:=\mathcal{A}^{nsq}\setminus \mathcal{A}_{=2}^{nsq}$. If $f \in \mathcal{A}_{=2}^{nsq}$, then $f$ is of the form $f=\prod_i f_i \prod_j f_j^2$, and we have $f/\mathrm{ERF}(f)=\prod_j f_j$. Consequently, in this case only the first three steps of the algorithm are executed, and the worst–case analysis of the classical algorithm of [@GaGe99 Theorem 14.14] shows that $\mathcal{X}_4(f) \leq c_3\, r\, M(r)\log(rq)$, where $c_3$ is a constant independent of $q$ and $r$. On the other hand, if $f \in \mathcal{A}_{\geq 2}^{nsq}$, then the four steps of the algorithm are executed. Observe that the last step is executed as many times as the highest multiplicity arising in the irreducible factors of $f/\mathrm{ERF}(f)$. Thus the worst–case analysis of [@GaGe99 Theorem 14.14] implies that $\mathcal{X}_4(f) \leq c_4\, r^2 M(r) \log(rq)$, where $c_4$ is a constant independent of $q$ and $r$. It follows that $$\begin{aligned} \label{Suma 2 esperanza} S_4^{nsq}&\leq c_3\, r\, M(r) \log(rq) \frac{|\mathcal{A}_{=2}^{nsq}|}{|\mathcal{A}|}+ c_4\, r^2 M(r) \log(rq)\frac{|\mathcal{A}_{\geq 2}^{nsq}|}{|\mathcal{A}|}\end{aligned}$$ Since $\mathcal{A}_{=2}^{nsq}$ is a subset of $\mathcal{A}^{nsq}$, from we have that $$\label{cota de no libres de cuadrados con un factor de grado 2} |\mathcal{A}_{=2}^{nsq}|\leq r(r-1)\delta_{{\boldsymbol}G}q^{r-m-1} \leq r^2\delta_{{\boldsymbol}G} q^{r-m-1}.$$ On the other hand, if $f \in \mathcal{A}_{\geq 2}^{nsq}$, then $\deg(\gcd(f,f')) \geq 2$. We deduce that $\mathrm{Res}(f,f')=\mathrm{Subres}(f,f')=0$. Hence, $\mathcal{A}_{\geq 2}^{nsq}$ is a subset of $\mathcal{S}_1(W)$, where $W\subset {\mathbb A}^r$ is the affine variety defined by $G_1, \dots, G_m$, $\mathcal{D}(W)$ is the discriminant locus of $W$ and $\mathcal{S}_1(W)$ is the first subdiscriminant locus of $W$. We deduce that $$\begin{aligned} \label{cota de no libres de cuadrados con otros factores} |\mathcal{A}_{\geq 2}^{nsq}| \leq r(r-1)^2(r-2)\delta_{{\boldsymbol}G} q^{r-m-2}\leq r^4\delta_{{\boldsymbol}G} q^{r-m-2}.\end{aligned}$$ Further, if $q >15\delta_{{\boldsymbol}G}^{13/3}$, then Theorem \[estimation A\] implies $|\mathcal{A}|\geq\frac{1}{2} q^{r-m}.$ Replacing , in we obtain $$\begin{aligned} \label{Suma 2 esperanza bis} \mathcal{S}_4^{nsq} & \leq 2\,c_3 M(r) \log(rq) \frac{r^3\delta_{{\boldsymbol}G}}{q}+ 2\,c_4 M(r) \log(rq) \frac{r^6\delta_{{\boldsymbol}G}}{q^2}.\end{aligned}$$ Combining and we obtain the following result. \[costo en promedio cuarto paso\] Let $q > 15\delta_{{\boldsymbol}G}^{13/3}$. 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[^1]: The authors were partially supported by the grants PIP CONICET 11220130100598 and PIO CONICET-UNGS 14420140100027 [^2]: This means that $\{G_1=0,\ldots,G_i=0\}$ has dimension $r-i$ for $1\le i\le m$; see Section \[subsec: complete intersections\] for details.

**Feasibility study on the least square method for fitting non-Gaussian noise data** Wei Xu, Wen Chen, Yingjie Liang Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering **Corresponding authors**:\ Wen Chen, Email address:\ Yingjie Liang, Email address:\ ***Abstract:*** This study is to investigate the feasibility of least square method in fitting non-Gaussian noise data. We add different levels of the two typical non-Gaussian noises, Lévy and stretched Gaussian noises, to exact value of the selected functions including linear equations, polynomial and exponential equations, and the maximum absolute and the mean square errors are calculated for the different cases. Lévy and stretched Gaussian distributions have many applications in fractional and fractal calculus. It is observed that the non-Gaussian noises are less accurately fitted than the Gaussian noise, but the stretched Gaussian cases appear to perform better than the Lévy noise cases. It is stressed that the least-squares method is inapplicable to the non-Gaussian noise cases when the noise level is larger than 5[%]{}.\ ***Keywords:*** Least square method, non-Gaussian noise, Lévy distribution, stretched Gaussian distribution, least square fitting\ **1. Introduction**\ Non-Gaussian noise is universal in nature and engineering.$^{1-3}$ In recent decades, non-Gaussian noise has widely been studied, especially in signal detection and processing,$^{\, 4-5}$ theoretical model analysis,$^{\, 6}$ and error statistics.$^{\, 7}$ It is known that the Gaussian distribution is the mathematical precondition to use the least square method.$^{4\, }$However, it is often directly used to process such non-Gaussian noise data, which may give wrong estimation.$^{\, 8}$ Thus, this study is to quantitatively examine the applicability of the least square method to analyze non-Gaussian noise data. Generally, non-Gaussian noise has detrimental influence on the stability of power system, and also can stimulate systems to generate ordered patterns.$^{9-10}$ To our best knowledge, Lévy and stretched Gaussian noises are two kinds of typical non-Gaussian noise, which are frequently used in fractional and fractal calculus.$^{11-12}$ Lévy noise has been observed in many complex systems, such as turbulent fluid flows,$^{13\, }$signal processing,$^{14}$ financial times series,$^{15-16}$ neural networks.$^{17\, }$We also note that the parameters estimation for stochastic differential equations driven by small Lévy noise were investigated.$^{18}$ Compared to the Lévy noise the stretched Gaussian noise is less studied, but its corresponding stretched Gaussian distribution has been explored,$^{19}$ such as in the motion of flagellate protozoa,$^{20\, }$SoL interchange turbulence simulation,$^{21\, }$anomalous diffusion of particles with external force,$^{22}$ not mentioned too much. It also should be pointed that processing of non-sinusoidal signals or sound textures has become an important research topic,$^{23}$ and the derived algorithms significantly improve the perceptual quality of stretched noise signals.$^{19\, }$ It is well known that the least square method is a standard regression approach to approximate the solutions of over determined systems, which is most frequently used in data fitting and estimation.$^{24}$ The core concept of the least square method is to identify the best match for the system by minimizing the square error.$^{25}$ Supposed that the data points are ${(x_1,\,y_1)},\,{(x_2,\,y_2)}\,{(x_3,\,y_3)},...,{(x_{200},\,y_{200})},$ where $x$ represents the independent variable and $y$ is the dependent variable. The fitting error $d$ characterizes the distance between $y$ and the estimated curve$f\left( x \right)$, i.e., $d_{1} =y_{1} -f\left( {x_{1} } \right),\;...,d_{n} =y_{n} -f\left( {x_{\mbox{n}} } \right)$. The best fitting curve is to minimize the square error $\delta =\;d_{1}^{2}+d_{2}^{2}+...+d_{\mbox{n}} ^{2}=\sum\limits_{i=1}^n {\left[ {y_{i} -f\left( {x_{i} } \right)} \right]} ^{2}$, where the errors $d_{i} $ are usually modeled by Gaussian distribution.$^{26}$ Field data are often polluted by noise$^{27\, }$and the Gaussian noise is the classical one, whose probability density function obeys Gaussian distribution. We have mentioned above that several types of noise data obey non-Gaussian distribution.$^{28}$ To examine the feasibility of the least square method in fitting non-Gaussian noise data, we generate the non-Gaussian random numbers as the noise, and then add different levels of the noise to the exact values of the selected functions including linear equations, polynomial and exponential equations as the observed values. By using the least square method, the maximum absolute and mean square errors are calculated and compared in the Gaussian and non-Gaussian applications. The rest of the paper is organized as follows. In Section 2, we introduce the Gaussian distribution, Lévy distribution, stretched Gaussian distribution, and the methods we use to analyze the noise data. In Section 3, we give the results and discussion. Finally, a brief summary is provided.\ **2. Theory and methods**\ ***2.1 Gaussian distribution***\ Gaussian distribution is also called as normal distribution, which is often encountered in mathematics, physics and engineering. The probability density function of Gaussian distribution is: $$\label{eq1} f\left( {x;\mu ,\sigma } \right)=\frac{1}{\sqrt {2\pi \sigma^{2}} }\exp \left( {-\frac{\left( {x-\mu } \right)^{2}}{2\sigma^{2}}} \right),$$ where $\mu $ and $\sigma $ respectively represent mean and standard deviation. When $\mu =0$ and $\sigma =1$, it degenerates into the standard Gaussian distribution. Figure1 gives four different cases of Gaussian density function.\ {width="2.5in" height="2in"} \[fig1\] ***2.2 Lévy distribution***\ Lévy distribution, named after Paul Lévy, is a rich class of probability distributions. The Gaussian and Cauchy distributions are its special cases. It is usually defined by its characteristic function $\phi_{\alpha ,\beta } \left( {k;\mu ,\sigma } \right)$.$^{29}$ $$\label{eq2} \phi_{\alpha ,\beta } \left( {k;\mu ,\sigma } \right)=F\left\{ {f_{\alpha ,\beta } \left( {k;\mu ,\sigma } \right)} \right\}=\exp \left[ {i\mu k-\sigma^{\alpha }\left| k \right|^{\alpha }\left( {1-i\beta \frac{k}{\left| k \right|}\omega \left( {k,\alpha } \right)} \right)} \right],$$ where $$\label{eq3} \omega \left( {k,\alpha } \right)=\left\{ {\begin{array}{l} \tan \frac{\pi \alpha }{2}\;\;\;\;\;\;\;\;\;\;\;\left( {\alpha \ne 1,0<\alpha <2} \right) \\ -\frac{2}{\pi }\ln \left| k \right|\;\;\;\;\;\;\;\;\;\left( {\alpha =1} \right) \\ \end{array}} \right.\quad ,$$ stability index $0<\alpha \le 2$, skewness parameter $-1\le \beta \le 1$, scale parameter $\sigma >0$, and location parameter $\mu \in \Re $. $\alpha $ and $\beta $ respectively determine the properties of asymptotic decay and symmetry. The standard Lévy distribution can be obtained by the following transformation. $$\label{eq4} p_{\alpha ,\beta } (k;\mu ,\sigma )=\frac{1}{\sigma }p_{\alpha ,\beta } (\frac{x-\mu }{\sigma };0,1).$$ When$\beta =1,\;\mu =0$, the probability density function of the Lévy distribution is stated as: $$\label{eq5} f\left( {x;\mu ,c} \right)=\sqrt {\frac{c}{2\pi }} \left( {-\frac{e^{-\frac{c}{2x}}}{x^{1.5}}} \right),$$ where $x\ge 0$, $\mu $ is the location parameter and $c$ is the scale parameter. Different cases of Eq.(5) are illustrated in Figure 2.\ {width="2.5in" height="2in"} \[fig2\] ***2.3 Stretched Gaussian distribution***\ The stretched Gaussian distribution has widely been used to describe anomalous diffusion and turbulence, especially in the porous media with fractal structure.$^{30}$ The solution to the fractal derivative equation in characterizing the fractal media has the form of stretched Gaussian distribution,$^{31\, }$whose probability density function is defined as: $$\label{eq6} f_{\beta } \left( x \right)=\frac{\beta }{2^{1+1/\beta }\Gamma \left( {1/\beta } \right)\sigma }\exp \left( {-\frac{1}{2}\left| {\frac{x-a}{\sigma }} \right|^{\beta }} \right)\;,\;\;\;\;\;\left( {-\infty <x<\infty ,\;\beta >0} \right).$$ where $\beta $ is the stretched exponent. When$\beta =2$ and $a=0$, it becomes to the standard Gaussian distribution. Figure 3 shows three cases of stretched Gaussian density function. {width="2.5in" height="2in"} \[fig3\] ***2.4 Generation of noise data***\ In this study, the noise data are obtained based on the above mentioned Gaussian and non-Gaussian random variables, which can be generated by using the inverse function method$^{32\, }$and the selection method.$^{33}$ Specifically, Chambers, Mallows, Stuck proposed the CMS method in Lévy random variables simulation,$^{\, 34}$ which is the fastest and most accuracy method. By using the CMS method, some variables need to be defined first. $$V=\pi \left( {U_{1} -\frac{1}{2}} \right),$$ $$W=-\ln U_{2} ,$$ $$L=\left\{ {\left. {1+\beta^{2}\tan^{2}\left( {\frac{\pi \alpha }{2}} \right)} \right\} } \right.^{1/2\alpha },$$ $$\theta_{0} =\frac{1}{\alpha }\arctan \left( {\beta \tan \left( {\frac{\pi \alpha }{2}} \right)} \right),$$ where $U_{1} $ and $U_{2} $are two independent uniform distribution on interval $\left( {0,1} \right)$. When $\alpha \ne 1$, the [Lévy]{} random number is $$\label{eq7} X=L\frac{\sin \left\{ {\left. {\alpha \left( {V+\theta_{0} } \right)} \right\} } \right.}{\left\{ {\cos \left( V \right)} \right\}^{1/\alpha }}.\left[ {\frac{\cos \left( {V-\alpha \left( {V+\theta_{0} } \right)} \right)}{W}} \right]^{\left( {1-\alpha } \right)/\alpha }.$$ When $\alpha =1$, $$\label{eq8} X=\frac{2}{\pi }\left\{ {\left( {\frac{\pi }{2}+\beta V} \right)\tan V-\beta \ln \left( {\frac{\frac{\pi }{2W}\cos V}{\frac{\pi }{2}+\beta V}} \right)} \right\}.$$ The general Lévy random numbers can be obtained based on some known properties.$^{35}$ For the stretched Gaussian distribution, we use the acceptance rejection method to generate its random numbers.$^{\, 33}$\ ***2.5 Methods***\ Both linear and nonlinear functions estimation are considered by using the least square method, in which the model function $f:\;R^{m}\to R$ is estimated as $$\label{eq9} y_{i} =f\left( {x_{i} } \right)+\varepsilon_{i} \;\left( {i=1,2,...,n} \right),$$ where $n$ is the number of observations, $y_{i} \in R$ the response variable, $x_{i} \in R^{m}$ the explanatory variable, the noise $$\label{eq9} \varepsilon_{i} \;\mbox{=}\;rand\times a\mbox{\% \, }\left( {a=1,\;5,\;10,\;15,\;20} \right).$$ In this study, we consider the case$n=200$, then the observed values $y_{1} ,\;y_{2} ,...,\;y_{200} $ can be constructed by adding the values of the random numbers to the exact values of the selected functions including linear equations, polynomial and exponential equations, finally the maximum absolute error and the mean square error are calculated for the above different cases in conjunction with the least square method. We give the following abbreviations in noise date processing for convenience:\ [FA: Gaussian noise least square error fitting, $\mu =5,\;\sigma =0.5.$]{}\ [FB: Lévy noise least square error fitting, $\alpha =1.8,\;\beta =0,\;\mu =1,\;\sigma =0.$]{}\ [FC: Stretched Gaussian noise least square error fitting,$\beta =2.5,\;a=1,\;\sigma =3.$]{}\ [Rerr1: Maximum absolute error: $\max \left| {F\left( {x_{i} } \right)-f\left( {x_{i} } \right)} \right|$.]{}\ [Rerr2: Mean square error: $\sqrt {\frac{\sum\limits_{i=1}^n {\left. {\left\{ {F\left( {x_{i} } \right)-f\left( {x_{i} } \right)} \right.} \right\}^{2}} }{n}} $.]{}\ **3. Results and discussion**\ In this section, we apply the least square method to fit various noise-polluted data by adding different levels of Gaussian and non-Gaussian noise to exact values of the selected functions including linear equations, polynomial and exponential equations, and give a brief discussion.\ a) The simplest typical model is the linear function. $$\label{eq12} f\left( x\right)=ax+b,\;\;\left( {a\ne 0} \right).$$ Here we select the following case as example: $$\label{eq1} f\left( x \right)=5x.$$ Tables 1 to 5 give the estimated parameters and the errors for five different levels of noise in the linear case. We can observe that the Gaussian noise fitting data maximum absolute error is in the range of $\left( {0.0005,\;0.0177} \right)$, the mean square error is in the range of $\left( {6.8544e-15,\;1.5782e-8} \right)$. The maximum absolute and the mean square errors of Gaussian noise are the least and Lévy distribution noise are the largest, with the relationship expressed as: $$\label{eq12} Rerr1\left( {FA} \right)\mbox{\, }<\mbox{\, }Rerr1\left( {FC} \right)\mbox{\, }<\mbox{\, }Rerr1\left( {FB} \right),$$ $$\label{eq13} Rerr2\left( {FA} \right)\mbox{\, }<\mbox{\, }Rerr2\left( {FC} \right)\mbox{\, }<\mbox{\, }Rerr2\left( {FB} \right).$$ The corresponding fitting curves are depicted in Figures 4-8. We can find that the results of Gaussian noise fitting have the best accuracy, and the stretched Gaussian noise fitting curves are closer to those of the Gaussian noise compared with the results of Lévy noise data fitting.\ Table 1. The estimated results for 1[%]{} noise in the linear case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{5}$ $_{0}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{4.9901}$ $_{0.0005}$ $_{0.0005}$ $_{6.8544e-15}$ $_{\mathbf{FB}}$ $_{4.9973}$ $_{-0.0073}$ $_{0.0076}$ $_{3.0886e-9}$ $_{\mathbf{FC}}$ $_{5.0087}$ $_{-0.0011}$ $_{0.0011}$ $_{2.9304e-13}$ \[tab1\] {width="2in" height="1.6in"} \[fig4\] [Table 2. ]{}The estimated results for 5[%]{} noise in the linear case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{5}$ $_{0}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{5.0195}$ $_{-0.0009}$ $_{0.0010}$ $_{1.0324e-13}$ $_{\mathbf{FB}}$ $_{5.0138}$ $_{0.0102}$ $_{0.0116}$ $_{1.4090e-8}$ $_{\mathbf{FC}}$ $_{5.0635}$ $_{-0.0119}$ $_{0.0109}$ $_{6.4216e-9}$ \[tab2\] {width="2in" height="1.6in"} \[fig5\] [Table 3. ]{}The estimated results for 10[%]{} noise in the linear case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{5}$ $_{0}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{5.0504}$ $_{-0.0032}$ $_{0.0032}$ $_{6.5200e-12}$ $_{\mathbf{FB}}$ $_{4.9793}$ $_{0.0104}$ $_{0.0104}$ $_{7.7237e-9}$ $_{\mathbf{FC}}$ $_{5.1412}$ $_{-0.0181}$ $_{0.0101}$ $_{1.9163e-8}$ \[tab3\] \[fig6\] [Table 4. ]{}The estimated results for 15[%]{} noise in the linear case. [[function]{}]{}$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ -------------------------------- ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{5}$ $_{0}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{4.8624}$ $_{0.0064}$ $_{0.0074}$ $_{2.6201e-10}$ $_{\mathbf{FB}}$ $_{4.9193}$ $_{0.0482}$ $_{0.0482}$ $_{3.8108e-6}$ $_{\mathbf{FC}}$ $_{5.1177}$ $_{-0.0322}$ $_{0.0322}$ $_{4.9674e-7}$ \[tab4\] \[fig7\] [Table 5. ]{}The estimated results for 20[%]{} noise in the linear case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{5}$ $_{0}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{4.8533}$ $_{0.0177}$ $_{0.0177}$ $_{1.5782e-8}$ $_{\mathbf{FB}}$ $_{5.0199}$ $_{0.1645}$ $_{0.1665}$ $_{7.5012e-4}$ $_{\mathbf{FC}}$ $_{5.1050}$ $_{-0.0355}$ $_{0.0355}$ $_{8.5735e-7}$ \[tab5\] \[fig8\] b\) A polynomial can be constructed by means of addition, multiplication and exponentiation to a non-negative power, which is usually written as the following form with a single variable $x$, $$\label{eq15} f\left( x \right)=a_{n} x^{n}+a_{n-1} x^{n-1}+...+a_{2} x^{2}+a_{1} x+a_{0} ,$$ where$a_{0} ,\;a_{1} ,\;...,\;a_{n-1} ,\;a_{n} $are constants. We select three parameters polynomial function. $$\label{eq16} F\left( x \right)=ax^{2}+bx+c\;\left( {a\ne 0} \right).$$ Here the following case is used as example: $$\label{eq17} y=4x^{2}+3x+2\;\left( {a\ne 0} \right).$$ Tables 6 to10 give the estimated parameters and the errors for five different levels of noise in the polynomial case. The corresponding fitting curves are depicted in Figures 9-13. Gaussian noise fitting maximum absolute error is in the range of $\left( {0.0019,\;0.0170} \right)$, and the mean square error is in the range of $\left( {3.2095e-13,\;2.2722e-9} \right)$. We notice that Eq. (13) and Eq. (14) also satisfied here. From Figures 9-13, the results of Gaussian noise fitting are the best, and the stretched Gaussian noise fitting curves are better than the results of Lévy noise data fitting.\ [Table 6. ]{}The estimated results for 1[%]{} noise in the polynomial case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------- ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{4}$ $_{3}$ $_{2}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{3.8776}$ $_{2.0197}$ $_{2.9991}$ $_{0.0019}$ $_{3.2095e-13}$ $_{\mathbf{FB}}$ $_{3.9672\, \, \, \, \, \, \, }$ $_{2.0072}$ $_{3.0042}$ $_{0.0046}$ $_{4.1653e-10}$ $_{\mathbf{FC}}$ $_{3.8480\, \, \, \, \, \, \, }$ $_{2.0262}$ $_{2.9995}$ $_{0.0013}$ $_{7.4921e-14}$ \[tab6\] **Figure 9. Polynomial model fitting to 1[%]{}noise.** \[fig9\] [Table 7. ]{}The estimated results for 5[%]{} noise in the polynomial case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------------------- ------------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{4}$ $_{3}$ $_{2}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{3.7673\, \, \, \, \, \, \, \, }$ $_{2.0500}$ $_{2.9974}$ $_{0.0026}$ $_{1.1552e-12}$ $_{\mathbf{FB}}$ $_{4.6455\, \, \, \, \, \, \, \, \, \, \, \, }$ $_{1.8825\, }$ $_{3.0166}$ $_{0.0190}$ $_{3.4974e-8}$ $_{\mathbf{FC}}$ $_{4.7226\, \, \, \, \, \, \, \, \, \, \, \, \, }$ $_{1.8644\, \, }$ $_{2.9985}$ $_{0.0079}$ $_{1.1658e-9}$ \[tab7\] **Figure 10. Polynomial model fitting to 5[%]{}noise.** \[fig10\] [Table 8. ]{}The estimated results for 10[%]{} noise in the polynomial case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ----------------- ----------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{4}$ $_{3}$ $_{2}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{4.4244}$ $_{1.8693}$ $_{3.0071}$ $_{0.0071}$ $_{7.7917e-11}$ $_{\mathbf{FB}}$ $_{4.6928}$ $_{1.8345}$ $_{3.0367}$ $_{0.0367}$ $_{7.6032e-7}$ $_{\mathbf{FC}}$ $_{3.5281}$ $_{2.1336}$ $_{2.9754}$ $_{0.0246}$ $_{9.9390e-8}$ \[tab8\] **Figure 11. Polynomial model fitting to 10[%]{}noise.** \[fig11\] [Table 9. ]{}The estimated results for 15[%]{} noise in the polynomial case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------------------------------------------------- ---------------------- ----------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{4}$ $_{3}$ $_{2}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{4.9474}$ $_{1.7866}$ $_{3.0070}$ $_{0.0128}$ $_{5.1818e-10}$ $_{\mathbf{FB}}$ $_{4.3098}$ $_{1.9391}$ $_{3.0338}$ $_{0.0379}$ $_{1.1459e-6}$ $_{\mathbf{FC}}$ $_{4.4846\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }$ $_{1.9551\, \, \, }$ $_{2.9692\, }$ $_{0.0318\, }$ $_{5.2768e-7}$ \[tab9\] **Figure 12. Polynomial model fitting to 15[%]{}noise.** \[fig12\] [Table 10. ]{}The estimated results for 20[%]{} noise in the polynomial case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------------------------------------------------------------------- ------------------- ------------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{4}$ $_{3}$ $_{2}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{6.1102\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }$ $_{1.6197\, \, }$ $_{3.0086\, \, \, }$ $_{0.0170}$ $_{2.2722e-9}$ $_{\mathbf{FB}}$ $_{3.0272\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }$ $_{2.2146}$ $_{3.0432\, \, \, }$ $_{0.0550}$ $_{7.1629e-6}$ $_{\mathbf{FC}}$ $_{4.0612\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, }$ $_{2.0446\, \, }$ $_{2.9506\, \, \, \, }$ $_{0.0494}$ $_{3.8487e-6}$ \[tab10\] **Figure 13. Polynomial model fitting to 20[%]{}noise.** \[fig13\] c\) Non-linear equations can be divided into two categories, one is polynomial equation, and the other is non-polynomial equation. In this part, we select the four parameters exponential function. $$\label{eq18} F\left( x \right)=ae^{3x}+be^{-t}-ct+d\;\left( {a\ne 0} \right).$$ Here the following case is used as example: $$\label{eq19} f\left( x \right)=0.5e^{3x}+0.2e^{-t}-t+\frac{1}{3}.$$ Tables 11 to 15 give the estimated parameters and the errors for five different levels of noise in the exponential function case. The corresponding fitting curves are shown in Figures 14-18. The Gaussian noise fitting data maximum absolute error is in the range of $\left( {0.0013,\;0.0091} \right)$, the mean square error is in the range of $\left( {2.5659e-14,\;1.5559e-9} \right)$. And the results of exponential cases have similar patterns with those shown in the linear and polynomial cases.\ [Table 11. ]{}The estimated results for 1[%]{} noise in the exponential case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $d \quad_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ------------------------------------------------- ----------------- ------------------- ----------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{0.5}$ $_{0.2}$ $_{1}$ $_{1/3}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{0.4651\, \, \, \, \, \, \, \, \, }$ $_{0.8096}$ $_{0.3043}$ $_{-0.2412}$ $_{0.0013}$ $_{2.5659e-14}$ $_{\mathbf{FB}}$ $_{0.4887\, \, \, \, \, \, \, \, \, \, \, \, }$ $_{0.4365}$ $_{0.7425}$ $_{0.1168}$ $_{0.0087}$ $_{4.3599e-9}$ $_{\mathbf{FC}}$ $_{0.4889\, \, \, \, \, \, \, \, \, \, }$ $_{0.3664}$ $_{0.8068\, \, }$ $_{0.1795}$ $_{0.0015}$ $_{7.7919e-13}$ \[tab11\] **Figure 14. Exponential model fitting to 1[%]{} noise data.** \[fig14\] [Table 12. ]{}The estimated results for 5[%]{} noise in the exponential case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $d \quad_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ------------------------------------------- ----------------- ----------------- ----------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{0.5}$ $_{0.2}$ $_{1}$ $_{1/3}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{0.4282\, \, \, \, \, \, \, \, \, \, }$ $_{1.5436}$ $_{-0.5101}$ $_{-0.9386}$ $_{0.002}$ $_{2.1618e-12}$ $_{\mathbf{FB}}$ $_{0.5493\, \, \, \, \, \, \, \, \, }$ $_{-0.7494}$ $_{2.0662}$ $_{1.2535}$ $_{0.0211}$ $_{1.5321e-7}$ $_{\mathbf{FC}}$ $_{0.6557\, \, \, \, \, \, \, \, \, }$ $_{-3.1130}$ $_{4.5957}$ $_{3.4751}$ $_{0.0156}$ $_{3.6588e-9}$ \[tab12\] **Figure 15. Exponential model fitting to 5[%]{} noise data.** \[fig15\] [Table 13 ]{}The estimated results for 10[%]{} noise in the exponential case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $d \quad_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------- ----------------- ----------------- ----------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{0.5}$ $_{0.2}$ $_{1}$ $_{1/3}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{0.7448\, \, \, \, \, \, \, \, \, }$ $_{-4.4825}$ $_{6.2056}$ $_{4.7692}$ $_{0.0137}$ $_{3.2450e-9}$ $_{\mathbf{FB}}$ $_{0.3697\, \, \, \, \, \, \, \, \, }$ $_{2.9448}$ $_{-1.9627}$ $_{-2.3200}$ $_{0.0545}$ $_{6.0328e-6}$ $_{\mathbf{FC}}$ $_{0.8745\, \, \, \, \, \, \, \, \, }$ $_{-6.9997}$ $_{9.0002}$ $_{7.1302}$ $_{0.0284}$ $_{6.2251e-8}$ \[tab13\] **Figure 16. Exponential model fitting to 10[%]{} noise.** \[fig16\] [Table 14. ]{}The estimated results for 15[%]{} noise in the exponential case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $d \quad_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------- ----------------- ------------------------- ----------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{0.5}$ $_{0.2}$ $_{1}$ $_{1/3}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{0.1986\, \, \, \, \, \, \, \, \, }$ $_{5.1211}$ $_{-4.7226}$ $_{-4.2859}$ $_{0.0097}$ $_{2.0389e-10}$ $_{\mathbf{FB}}$ $_{0.5151\, \, \, \, \, \, }$ $_{-0.6695}$ $_{1.8449\, \, \, \, }$ $_{1.1120}$ $_{0.0867}$ $_{3.3841e-5}$ $_{\mathbf{FC}}$ $_{0.4250\, \, \, \, \, \, \, \, }$ $_{1.1082}$ $_{-0.1989}$ $_{-0.5301}$ $_{0.0303}$ $_{1.8103e-7}$ \[tab14\] **Figure17. Exponential model fitting to 15[%]{} noise.** \[fig17\] [Table 15 ]{}The estimated results for 20[%]{} noise in the exponential case. ***function***$_{\mathbf{}}$ $a_{\mathbf{}}$ $b_{\mathbf{}}$ $c_{\mathbf{}}$ $d \quad_{\mathbf{}}$ $_{\mathbf{Rerr1}}$ $_{\mathbf{Rerr2}}$ ------------------------------ ---------------------------------------- ----------------- ----------------- ----------------------- --------------------- --------------------- $_{\mathbf{f}}$ $_{0.5}$ $_{0.2}$ $_{1}$ $_{1/3}$ $_{0}$ $_{0}$ $_{\mathbf{FA}}$ $_{0.3164\, \, \, \, \, \, \, \, \, }$ $_{3.9831}$ $_{-3.1994}$ $_{-3.2706}$ $_{0.0091}$ 1.5559e-9 $_{\mathbf{FB}}$ $_{0.3339}$ $_{3.7390}$ $_{-2.8727}$ $_{-2.9547}$ $_{0.0873}$ $_{4.5904e-5}$ $_{\mathbf{FC}}$ $_{0.6920}$ $_{-3.3436}$ $_{4.9695}$ $_{3.6399}$ $_{0.0451}$ $_{1.8947e-6}$ \[tab15\] **Figure 18. Exponential model fitting to 20[%]{} noise.** \[fig18\] To summarize all the above results, we can find that the maximum absolute and the mean square errors for the Gaussian noise cases are the smallest, but the values for the Lévy noise cases are the biggest, i.e., $$\label{eq20} Rerr1\left( {FA} \right)\mbox{\, }<\mbox{\, }Rerr1\left( {FC} \right)\mbox{\, }<\mbox{\, }Rerr1\left( {FB} \right),$$ $$\label{eq21} Rerr2\left( {FA} \right)\mbox{\, }<\mbox{\, }Rerr2\left( {FC} \right)\mbox{\, }<\mbox{\, }Rerr2\left( {FB} \right).$$ It can be observed from Figures 4 to 18, that the results of Gaussian noise fitting have the best accuracy, and the stretched Gaussian noise fitting curves are closer to those of the Gaussian noise compared with the results of Lévy noise data fitting. Thus, the least square method is less accurate when it is applied to the non-Gaussian noise data fitting compared with the cases of Gaussian noise, especially when the noise level is larger than 5[%]{}. This study mainly verifies the least square method is inapplicable to non-Gaussian noise when the noise level is high. To extend the results in more complicated systems, a mathematical proof to the conclusion should be derived in future study. The second goal of our further work is to modify the least square method in fitting non-Gaussian noises. Actually the core concept of the least square method is to minimize the square error $\delta =\sum\limits_{i=1}^n {\left[ {y_{i} -f\left( {x_{i} } \right)} \right]^{2}} $, i.e., in the linear case, $$\label{eq22} \delta =\sum\limits_{i=1}^n {\left[ {yi-ax-b} \right]^{2}} ,$$ to compute the minimum value of Eq. (22), the main task is to set the first-order derivatives of the parameters to be zero. $$\label{eq23} \left\{ {\begin{array}{l} \frac{\partial \delta }{\partial a}=0 \\ \frac{\partial \delta }{\partial b}=0 \\ \end{array}} \right.\quad .$$ The solutions of Eq. (23) are the target values of the parameters $a$ and $b$. Combining our previous work on fractional and fractal derivatives,$^{\, 30,36}$ we can employ the fractional and fractal derivatives to generalize Eq. (23), and the corresponding fitting errors can be defined by using the following power law transform: $$\label{eq24} \hat{{x}}=x^{\beta }.$$\ **4. Conclusions**\ This study examines the feasibility of least square method in fitting various noise data polluted by adding different levels of Gaussian and non-Gaussian noise to exact values of the selected functions including linear equations, polynomial and exponential equations. The maximum absolute error and the mean square error are calculated and compared for the different cases. Based on the foregoing results and discussions, the following conclusions can be drawn: 1\. The fitting results for the non-Gaussian noise are less accurate than those of the Gaussian noise, but the stretched Gaussian cases appear to perform better than the Lévy noise cases. 2\. 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--- abstract: 'We have incorporated a single crystal silicon shunt capacitor into a Josephson phase qubit. The capacitor is derived from a commercial silicon-on-insulator wafer. Bosch reactive ion etching is used to create a suspended silicon membrane; subsequent metallization on both sides is used to form the capacitor. The superior dielectric loss of the crystalline silicon leads to a significant increase in qubit energy relaxation times. $T_1$ times up to 1.6 $\mu$s were measured, more than a factor of two greater than those seen in amorphous phase qubits. The design is readily scalable to larger integrated circuits incorporating multiple qubits and resonators.' author: - 'U. Patel' - 'Y. Gao' - 'D. Hover' - 'G. J. Ribeill' - 'S. Sendelbach' - 'R. McDermott' title: Coherent Josephson phase qubit with a single crystal silicon capacitor --- The Josephson phase qubit is an attractive candidate for scalable quantum information processing in the solid state [@Clarke08; @Martinis02; @Lucero12; @Neeley10; @Ansmann09]. This qubit has achieved several important milestones, including realization of high-fidelity entangling gates in two and three qubit circuits [@Neeley10], violation of a Bell’s inequality [@Ansmann09], and full characterization of highly non-classical states in linear microwave resonators [@Hofheinz09]. However, qubit performance is limited by relatively short coherence times, under 1 $\mu$s. It has been shown that the dominant energy relaxation mechanism is dielectric loss induced by a continuum of low-energy defect states in the amorphous thin films of the circuit [@Martinis05]. In the most common approach to qubit realization, a $\sim$ 1 $\mu$m$^2$ Josephson junction is shunted by an external thin film capacitor of order 1 pF [@Steffen06]; in this case, qubit $T_1$ is determined solely by the loss tangent of the bulk capacitor dielectric: $T_{1}={1}/{\omega_{10}\tan\delta}$, where ${\omega_{10}/2\pi}$ is the qubit frequency. There have been efforts to develop amorphous dielectrics with improved intrinsic (low-temperature, low power) loss for superconducting qubit applications [@OConnell08]. An alternative approach is to incorporate crystalline, defect-free dielectrics into the qubit circuit. Molecular beam epitaxy techniques have been used to grow single-crystal Al$_{2}$O$_{3}$ tunnel barriers on crystalline Re underlayers, and phase qubit circuits incorporating these barriers demonstrate a significant reduction in the density of spurious microwave resonances [@Oh06; @Kline09]. Moreover, there has been progress in the incorporation of epitaxial Josephson junctions into transmon qubits, although $T_1$ times were under 1 $\mu$s [@Nakamura11; @Weides11]. Other work involves the development of grown bulk crystalline Al$_{2}$O$_{3}$ for phase qubit shunt capacitors [@Cho]. However, the robust, repeatable growth of crystalline dielectrics for superconducting qubit circuits remains a daunting challenge. ![Superconducting Josephson phase qubit with crystalline silicon shunt capacitor. (a) Fabrication process flow. The cavity etch is followed by Al deposition on the front and back sides. The front side Al is patterned into a base wiring layer; subsequent ion milling, oxidation, and counterelectrode deposition and etch steps form the Al-AlO$_x$-Al Josephson junction. (b) 3D view of the layer stackup for the completed device. (c) Phase qubit circuit layout. The qubit loop inductance is 680 pH, the capacitance is 1 pF, and the junction critical current is 1.5 $\mu$A.[]{data-label="fig:figure1"}](Fig1.pdf){width=".48\textwidth"} In this work, we demonstrate the realization of a Josephson phase qubit incorporating a crystalline silicon shunt capacitor. Silicon growth has been perfected over decades by the integrated circuit industry, and commercial-grade intrinsic silicon displays an internal quality factor in excess of $10^6$ in the low-power, low-temperature regime relevant to qubit operation. Moreover, silicon-on-insulator (SOI) technology provides a path to the incorporation of a crystalline silicon membrane into a thin film parallel plate capacitor. There was a recent demonstration of high quality-factor lumped-element ${LC}$ microwave resonators incorporating such crystalline silicon membranes, with intrinsic $Q$ in excess of $2\times 10^5$[@Weber11]. Using a variant of this process, we are able to integrate the low-loss SOI capacitor with a 1 $\mu$m$^2$ scale Josephson junction and the associated control and read-out circuitary to realize a phase qubit with significantly improved energy relaxation times. The process flow and layer stackup are shown in Fig. \[fig:figure1\](a), while the phase qubit circuit layout is shown in Figs. \[fig:figure1\](b-c). The devices were fabricated from commercial SOI wafers comprising a 400 $\mu$m silicon handle, 500 nm of buried SiO$_{2}$ (BOX), and a 2 $\mu$m crystalline silicon device layer. The wafer was photolithographically patterned on the back side, and Bosch reactive ion etch (RIE) and buffered oxide etch (BOE) steps were used to create a suspended silicon membrane with area 500 $\times$ 600 $\mu$m$^2$. Following an HF acid dip, blanket Al thin films were sputter deposited on the front and back sides of the wafer. The front side of the wafer was patterned and chemically etched to form the base electrode of the Josephson junction, including the qubit loop and the shunt capacitor plates; flux biasing and readout resonator structures were also formed in this layer. The native aluminum oxide was removed from the base electrode layer $via$ ion milling, the junction tunnel barrier was formed by thermal oxidation at ambient temperature (with a typical exposure of 100 mT O$_2$ for 10 minutes), and the Al junction counterelectrode was deposited and chemically etched to complete the circuit. A micrograph of the completed device is shown in Fig. \[fig:figure2\](a). The 680 pH qubit loop inductance consists of the parallel combination of two 300 $\mu$m $\times$ 500 $\mu$m Al loops with a 1.5 $\mu$A, 1 $\mu$m$^2$ Josephson junction fabricated at the middle of the central branch; this gradiometric configuration provides reduced sensitivity to fluctuations of the ambient magnetic field. The low-loss junction shunt capacitance comprises the series connection of two capacitors formed from the metallized back surface of the silicon membrane and two 200 $\times$ 200 $\mu$m$^2$ plates on the top surface of the chip, with the crystalline silicon device layer acting as capacitor dielectric. The dc bias flux and fast measurement pulses are coupled to the qubit $via$ an on-chip bias coil with mutual inductance 2.4 pH to the qubit loop, and the qubit is inductively coupled to a 3 GHz $LC$ resonator used for both excitation and readout. We thus avoid the need for any explicit galvanic or capacitive connection to the phase qubit loop. The device layout is carefully symmetrized to ensure that the floating bottom electrode of the qubit capacitance acts as a virtual ground. This arrangement minimizes spurious dielectric loss in the buried oxide at the edges of the membrane cavity due to any microwave potential that might develop on the bottom electrode. We anticipate that in future work the need to symmetrize the circuit can be eliminated by explicitly grounding the bottom electrode using microfabricated vias to connect the top and bottom metallization layers through the silicon dielectric. ![(a) Optical micrograph of the phase qubit device. The qubit is inductively coupled to an on-chip readout resonator and flux bias coil. (b) Electronic circuit diagram of the phase qubit with biasing and readout/excitation circuitry. (c) Qubit potential energy diagram showing different plasma frequencies $\omega_{p,0}$, $\omega_{p,1}$ for the device following qubit measurement. (d) Schematic diagram of transmission across readout resonator for different qubit measurement outcomes. The probe tone is applied at a frequency that gives the largest change in transmission amplitude.[]{data-label="fig:figure2"}](Fig2.pdf){width=".48\textwidth"} The 3 GHz readout resonator is capacitively coupled to a microwave feedline as shown in Fig. \[fig:figure2\](b); coupling to the feedline results in a loaded $Q$ of order 1000. By driving through the readout resonator at the qubit frequency (in the range from 5-6 GHz), we are able to perform coherent qubit rotations. Qubit measurement is performed by applying a fast (few ns) bias pulse through the flux line that projects the qubit ${|1\rangle}$ state out of the 01 manifold, resulting in a change in the circulating current state and effective inductance of the qubit loop. Thus, there are two distinct plasma frequencies associated with the measured qubit, as shown in Fig. \[fig:figure2\](c). To read out the qubit we perform a standard homodyne measurement of the transmission across the $LC$ resonator, relying on the small shift in resonant frequency associated with the two qubit measurement outcomes as shown in Fig. \[fig:figure2\](d). The qubit chip is mounted in an aluminum box and anchored at the 35 mK mixing chamber plate of a dilution refrigerator; the experiment is surrounded by a cold copper can coated with infrared-absorbing epoxy to suppress quasiparticle generation due to blackbody radiation from higher temperature stages of the cryostat [@Barends11]. The dc and microwave control lines are heavily filtered at 4.2 K and at the cold stage to suppress thermal excitation of the qubit. We have performed standard microwave pulse sequences to evaluate qubit coherence over a range of bias points; results are shown in Fig. \[fig:figure3\]. Qubit Rabi oscillations decay with a characteristic time in excess of 500 ns. Qubit Ramsey fringes decay on a shorter timescale of order 100 ns; as our qubit has no flux sweet spot Ramsey decay is dominated by excess low frequency magnetic flux noise [@Ithier05; @Bialzak07; @Sank12]. Qubit energy relaxation times up to 1.6 $\mu$s have been measured, and energy relaxation times greater than 1 $\mu$s have been measured on multiple devices. These SOI-based phase qubits thus display energy relaxation times that are a factor of 2-3 better than the $T_1$ times of state-of-the-art phase qubits incorporating low-loss a-Si:H shunt capacitors [@Wang08; @Neeley08]. It is worthwhile to consider the coherence budget of our qubit device. First, dielectric loss in the crystalline silicon shunt capacitor presents an ultimate limit to qubit $T_1$. However, the measured loss tangent of $5\times 10^{-6}$ would correspond in our device to an energy relaxation time around 10 $\mu$s, far in excess of the measured $T_1$. Next, qubit $T_1$ is limited by coupling to the heavily damped readout resonator, a phenomenon known as the Purcell effect [@Houck08]. However for our device and operating parameters the Purcell limit to qubit $T_1$ is around 20 $\mu$s. Instead the current devices appear to be limited by overcoupling to the flux bias coil: experiments reveal an oscillatory dependence of $T_1$ on qubit bias frequency which we have determined to arise from a frequency-dependent impedance transformation of the dissipation looking into the flux line. We anticipate that it will be straightforward to suppress this decoherence channel by re-engineering the bias tee on the flux line and by reducing the mutual inductance from the flux bias coil to the qubit. ![Characterization of SOI phase qubit coherence. (a) Rabi oscillations measured at a qubit operating frequency of 5.093 GHz. The solid line is a fit to the measured curve, giving a decay time of 510 ns. (b) Spontaneous decay of the qubit excited state and corresponding fit, yielding an energy relaxation time $T_1$ = 1.6 $\mu$s. (c) Qubit Ramsey fringes. The fit yields a dephasing time $T_2^* $ = 110 ns.[]{data-label="fig:figure3"}](Fig3.pdf){width=".48\textwidth"} ![Swap spectroscopy for qubit-$LC$ resonator system. (a) Optical micrograph of circuit with one SOI phase qubit coupled to two SOI-based microwave $LC$ resonators. (b) Qubit excited state occupation $P_1$ *versus* operating frequency and qubit-resonator interaction time. Coherent interaction between the qubit and the resonator yields a characteristic chevron pattern with a swap frequency of 190 MHz on resonance (dashed line).[]{data-label="fig:figure4"}](Fig4.pdf){width=".48\textwidth"} The SOI-based phase qubit can be adapted to realize more complex quantum circuits without any additional fabrication steps. We have fabricated a circuit in which one SOI phase qubit is inductively coupled to two on-chip $LC$ resonators; a micrograph of this circuit is shown in Fig. \[fig:figure4\](a). With this circuit we we have performed swap spectroscopy of the $LC$ resonators [@Marian11a; @Marian11b]. In this experiment, the qubit is initially detuned from the resonator and a microwave $\pi$-pulse is used to promote the qubit to the excited state. Next, a fast bias pulse is used to bring the qubit into or close to resonance with the $LC$ mode, and the two systems are allowed to interact for a variable period of time. Finally, the qubit is measured in the conventional way. In Fig. \[fig:figure4\](b) we show swap spectroscopy of an $LC$ mode at 5.89 GHz. The false color plot of qubit excited state occupation $P_1$ *versus* frequency and interaction time displays a characteristic chevron pattern, with coherent swaps on resonance at the coupling frequency 190 MHz. In summary, we have characterized the coherence of phase qubit circuits incorporating single crystal silicon capacitors derived from SOI wafers. The qubit $T_1$ times are more than a factor of two greater than those observed in state-of-the-art phase qubits incorporating low-loss amorphous shunt capacitors. The SOI phase qubits are readily integrated into more complex circuits incorporating multiple qubits and resonators. Thus SOI technology could provide a path to scalable, coherent multi-qubit circuits for rigorous investigations of quantum algorithms.\ This work was supported in part by IARPA through grant W911NF-09-1-0368 and by the National Science Foundation through grant DMR-0805051. Some work was performed at the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS-0335765). [99]{} J. Clarke and F. K. Wilhelm, Nature (London) **453**, 1031 (2008). J. M. Martinis, S. Nam, J. 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--- abstract: | The no-scalar-hair conjecture rules out the existence of asymptotically flat black holes with a scalar dressing for a large class of theories. No-scalar-hair theorems have been proved for the cases of neutral black holes and for charged black holes in the Maxwell electrodynamics. These theorems, however, do not apply in the case of non-linear electrodynamics. In the present work numerical solutions describing charged black holes coupled to Euler-Heisenberg type non-linear electrodynamics in scalar-tensor theories of gravity with massless scalar field are found. In comparison to the corresponding solution in General Relativity the presented solution has a simpler causal structure the reason for which is the presence of the scalar field. The present class of black holes has a single, non-degenerate horizon, i.e., its causal structure resembles that of the Schwarzschild black hole. Pacs numbers: 4.25.Dm, 04.40.-b, 04.70.-s, 95.30.Tg, 97.60.Lf author: - | Ivan Zh. Stefanov[^1], Stoytcho S. Yazadjiev [^2]\ [Dept. of Theoretical Physics, Faculty of Physics]{}\ [St.Kliment Ohridski University of Sofia]{}\ [5, James Bourchier Blvd., 1164 Sofia, Bulgaria ]{}\ \ Michail D. Todorov[^3]\ \[-1.mm\]\ \[-1.mm\]\ \[-1.mm\] [8, Kliment Ohridski Blvd., 1000 Sofia, Bulgaria]{} title: 'Scalar-tensor black holes coupled to Euler-Heisenberg nonlinear electrodynamics' --- Introduction ============ Scalar-tensor theories of gravity are the most natural generalization of General Relativity (GR) and arise naturally in string theory and higher dimensional gravity theories [@DP]. Different modifications of scalar-tensor theories are attracting much interest also in cosmology and astrophysics. The search of black holes in the scalar-tensor theories leads to the origin of the no-scalar-hair conjecture. It is similar to the no-hair conjecture in GR which states that in the exterior of a black hole the only information available regarding the black hole may be that of its mass, charge, and angular momentum. For neutral static, spherically symmetric black holes the only available information is about its mass, and the exterior of the black hole is reduced to the Schwarzschild solution. According to the above mentioned no-scalar-hair conjecture the presence of a scalar field would not lead to the existence of other preserved quantities which would allow a distant observer to distinguish between the Schwarzschild black hole and a black hole with a scalar dressing. A no-scalar-hair theorem which rules out the existence of static, spherically symmetric, asymptotically flat, neutral black holes with regular, non-trivial scalar field was proved by Saa [@Saa]. This theorem holds for a large class of scalar tensor theories in which the scalar field is non-minimally coupled to gravity. In order to prove the theorem Saa applied an explicit, covariant method to generate the exterior solutions for these theories through conformal mapping of solutions from the minimally coupled case. The scalar field in these theories becomes a constant and hence trivial if one demands that the essential singularity at the center of symmetry is hidden by an event horizon. A similar theorem treating also the case of charged scalar field with self-interaction was proved by Bekenstein [@Be]. Banerjee and Sen [@BSen] generalized Saa’s theorem for the case of charged black holes in Maxwell electrodynamics. In the case of non-linear electrodynamics, however, the energy-momentum tensor of the electromagnetic field has a non-zero trace a sequence of which is that the electromagnetic field is non-trivially coupled to the scalar field. Hence, we can expect that the no-scalar-hair theorems might not hold in that case. In the present work we prove that our assumption is correct and find numerical solutions describing black holes with a non-trivial scalar field coupled to non-linear electrodynamics. We consider a particular example of nonlinear electrodynamics – the Euler-Heisenberg electrodynamics. The nonlinear electrodynamics was first introduced by Born and Infeld in 1934 to obtain finite energy density model for the electron [@BI]. In recent years nonlinear electrodynamics models are attracting much interest, too. The reason is that the nonlinear electrodynamics arises naturally in open strings and $D$-branes [@L]. Nonlinear electrodynamics models coupled to gravity and stringy non-linear electrodynamics have been discussed in different aspects (see, for example, [@Demianski]-[@Y] and references therein).Here we obtain and discuss black hole solutions coupled to the nonlinear Euler-Heisenberg electrodynamics within scalar-tensor theories. Asymptotically flat black holes in Euler-Heisenberg theory coupled to Einstein gravity were studied in [@YT]. The effective Lagrangian for electrodynamics due to one-loop quantum corrections was calculated by Heisenberg and Euler [@EH]: $$L_{\rm EH} = -{1\over 4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{4} \,b^2 \left(F_{\mu\nu}F^{\mu\nu}\right)^2 + \gamma \left[ F_{\mu\nu}(\star F)^{\mu\nu} \right]^2$$ where $b^2= {8he^4/ (2880 \pi^2 m^4)}$, $\gamma = {7he^4/(5760\pi^2m^4)}$ and $h$, $e$, and $m$ are the Planck constant, electron charge, and electron mass, respectively, and the star “$\star$” stands for the Hodge operator. The star sign denotes the Hodge dual. From experimental aspect, the Euler-Heisenberg theory is more accurate classical approximation of QED than the Maxwell theory when the field has high intensity[@SB]. Regarding the electric-magnetic duality, the Euler-Heisenberg action breaks it as it was pointed in [@GR1]. Formulation of the problem ========================== The general form of the extended gravitational action in scalar-tensor theories is $$\begin{aligned} \label{JFA} S = {1\over 16\pi G_{*}} \int d^4x \sqrt{-{\tilde g}}\left[{F(\Phi)\tilde {\cal R}} - Z(\Phi){\tilde g}^{\mu\nu}\partial_{\mu}\Phi \partial_{\nu}\Phi \right. \nonumber \\ \left. -2 U(\Phi) \right] + S_{m}\left[\Psi_{m};{\tilde g}_{\mu\nu}\right] .\end{aligned}$$ Here, $G_{*}$ is the bare gravitational constant, ${\tilde R}$ is the Ricci scalar curvature with respect to the space-time metric ${\tilde g}_{\mu\nu}$. The dynamics of the scalar field $\Phi$ depends on the functions $F(\Phi)$, $Z(\Phi)$ and $U(\Phi)$. In order that the gravitons carry positive energy the function $F(\Phi)$ must be positive. The nonnegativity of the energy of the scalar field requires that $2F(\Phi)Z(\Phi) + 3[dF(\Phi)/d\Phi]^2 \ge 0$. The action of matter should depend only on the material fields $\Psi_{m}$ and the space-time metric ${\tilde g}_{\mu\nu}$ but not on the scalar field $\Phi$ so that the weak equivalence principle is satisfied. For convenience, it is a standard mathematical technique to study the scalar-tensor theory in the conformally related Einstein frame given by the metric: $$\label {CONF1} g_{\mu\nu} = F(\Phi){\tilde g}_{\mu\nu} . \label{conf}$$ The transition to the Einstein conformal frame includes not only a conformal transformation of the metric but also a redefinition of the scalar field. The new scalar field in the Einstein frame is defined in the following way: $$\label {CONF2} \left(d\varphi \over d\Phi \right)^2 = {3\over 4}\left\{{d\ln[F(\Phi)]\over d\Phi } \right\}^2 + {Z(\Phi)\over 2 F(\Phi)}$$ and $$\label {CONF3} {\cal A}(\varphi) = F^{-1/2}(\Phi) \,\,\, ,\nonumber \\ 2V(\varphi) = U(\Phi)F^{-2}(\Phi) .$$ In the Einstein frame action (\[JFA\]) takes the form $$\begin{aligned} \label{EFA} S= {1\over 16\pi G_{*}}\int d^4x \sqrt{-g} \left[{\cal R} - 2g^{\mu\nu}\partial_{\mu}\varphi \partial_{\nu}\varphi - 4V(\varphi)\right] \nonumber \\ + S_{m}[\Psi_{m}; {\cal A}^{2}(\varphi)g_{\mu\nu}]\end{aligned}$$ where $R$ is the Ricci scalar curvature with respect to the Einstein metric $g_{\mu\nu}$. We take the following Jordan frame nonlinear electrodynamics action $$S_{m} = {1\over 4\pi G_{*}}\int d^4x \sqrt{{-\tilde g}} L(X, Y)$$ where $$X = {1\over 4} F_{\mu\nu}{\tilde g}^{\mu\alpha} {\tilde g}^{\nu\beta} F_{\alpha\beta}, \,\,\, Y = {1\over 4} F_{\mu\nu}\left({\tilde \star} F\right)^{\mu\nu}$$ and ${\tilde \star}$ is the Hodge dual with respect to the Jordan frame metric ${\tilde g}_{\mu\nu}$. In the Einstein frame we have $$\label{EFNEDA} S_{m} = {1\over 4\pi G_{*}}\int d^4x \sqrt{-g} {\cal A}^4(\varphi) L(X, Y)$$ where $$X = {{\cal A}^{-4}(\varphi)\over 4} F_{\mu\nu}{g}^{\mu\alpha} {g}^{\nu\beta} F_{\alpha\beta}, \label{X} \,\,\, Y = {{\cal A}^{-4}(\varphi)\over 4} F_{\mu\nu}\left({ \star} F\right)^{\mu\nu}$$ and “$\star$” is the Hodge dual with respect to the Einstein frame metric $g_{\mu\nu}$. Through varying the action (\[EFA\]) with (\[EFNEDA\]) we obtain the following field equations $$\begin{aligned} &&{\cal R}_{\mu\nu} = 2\partial_{\mu}\varphi \partial_{\nu}\varphi + 2V(\varphi)g_{\mu\nu} - 2\partial_{X} L(X, Y) \left(F_{\mu\beta}F_{\nu}^{\beta} - {1\over 2}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} \right) \nonumber \\ &&-2{\cal A}^{4}(\varphi)\left[L(X,Y) - Y\partial_{Y}L(X, Y) \right] g_{\mu\nu}, \nonumber \\ &&\nabla_{\mu} \left[\partial_{X}L(X, Y) F^{\mu\nu} + \partial_{Y}L(X, Y) (\star F)^{\mu\nu} \right] = 0 \label{F},\\ &&\nabla_{\mu}\nabla^{\mu} \varphi = {dV(\varphi)\over d\varphi } - 4\alpha(\varphi){\cal A}^{4}(\varphi) \left[L(X,Y) - X\partial_{X}L(X,Y) - Y\partial_{Y}L(X, Y) \right], \nonumber\end{aligned}$$ where $\alpha(\varphi) = {d\ln{\cal A}(\varphi)\over d\varphi}$. In what follows we consider the truncated[^4] Euler-Heisenberg electrodynamics described by the Lagrangian $$L_{\rm EH}(X) = - X + 4b^2 X^2 . \label{EHL}$$ Basic equations and qualitative investigation ============================================= In this paper, we will be searching for black hole solutions, namely solutions which have an event horizon on which the scalar field $\varphi$ is regular. In order to ensure the smooth transition between the Einstein and the Jordan conformal frames, we will impose some restricting conditions on the coupling function ${\cal A}(\varphi)$. We will require that $0<{\cal A}(\varphi)<\infty$ for $r_{H}\leq r \leq \infty$, where $r_{H}$ is the radius of the horizon. The coupling function ${\cal A}(\varphi)$ (respectively $\alpha(\varphi)$) determines the properties of the solutions and contains the diversity of scalar-tensor theories. In the present work we will consider only theories for which $\alpha(\varphi)$ has a fixed positive sign for all values of $\varphi$. The manner of investigation of solutions within theories with negative $\alpha(\varphi)$ is similar. Theories in which the coupling function changes its sign are much more complicated (also from numerical point of view) since in them some interesting effects like bifurcation of solutions can appear, especially when $\alpha(\varphi)\sim\varphi$. Such solutions are currently being studied by the authors and the results will be given elsewhere. Magnetically charged black holes -------------------------------- The metric of a static, spherically symmetric spacetime can be written in the form $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu} = - f(r)e^{-2\delta(r)}dt^2 + {dr^2\over f(r) } + r^2\left(d\theta^2 + \sin^2\theta d\phi^2 \right).$$ In the magnetically charged case the electromagnetic field is given by $$F = P \sin\theta d\theta \wedge d\phi$$ and the magnetic charge is denoted by $P$. The field equations reduce to the following coupled system of ordinary differential equations: $$\begin{aligned} &&\frac{d\delta}{dr}=-r\left(\frac{d\varphi}{dr} \right)^2\label{MagnDelta},\\ &&\frac{d m}{dr}=r^2\left[\frac{1}{2}f\left(\frac{d\varphi}{dr}\right)^2-{\cal A (\varphi)}^{4}L(X) \right],\\ \label{MagnM} &&\frac{d }{dr}\left( r^{2}f\frac{d\varphi }{dr} \right)= r^{2}\left\{-4\alpha(\varphi){\cal A}^{4}(\varphi) \left[L(X) - X\partial_{X}L(X)\right] - r f\left(\frac{d\varphi}{dr} \right)^3 \right\} \label{MagnPhi} ,\end{aligned}$$ where $ X $ reduces to: $$X = {{\cal A}^{-4}(\varphi)\over 2} \frac{P^2}{r^4}.$$ Electrically charged black holes -------------------------------- In the electrically charged case we make the following ansatz for the electromagnetic field $$F = F_{\rm tr} dt \wedge dr,$$ and through eq.(\[X\]) and the equation for the electromagnetic field in (\[F\]) we obtain the following equation for $X$, $$64b^4 X^{3}-16b^2X^2+X=-\frac{1}{2}\frac{Q^2}{r^4{\cal A}^{4}(\varphi)}.$$ It has one real root, namely $$X=\frac{1}{24 b^2}\left( h^{1/3}+h^{-1/3}+2 \right),\label{Xsol}$$ where $$h=-1-\frac{54 Q^2 b^2}{r^4{\cal A}^{4}(\varphi)} + 6b\sqrt{3}\sqrt{\frac{Q^2\left[1+\frac {27 Q^2 b^2} {r^4{\cal A}^{4}(\varphi)}\right]}{r^4{\cal A}^{4}(\varphi)}}.$$ In the pure electric case eqs.(\[F\]) reduce to the following system of non-linear, ordinary differential equations $$\begin{aligned} &&\frac{d\delta}{dr}=-r\left(\frac{d\varphi}{dr} \right)^2, \label{ElDelta} \\ &&\frac{d m}{dr}=r^2\left\{\frac{1}{2}f\left(\frac{d\varphi}{dr} \right)^2 - {\cal A }^{4}(\varphi) \left[L(X) - 2X\partial_{X}L(X)\right] \right\}, \label{ElM}\\ &&\frac{d }{dr}\left( r^{2}f\frac{d \varphi}{dr} \right)=r^{2} \left\{ - 4\alpha (\varphi){\cal A}^{4}(\varphi) \left[L(X) - X\partial_{X}L(X)\right] -r f\left(\frac{d\varphi}{dr} \right)^3\right\}.\label{ElPhi}\end{aligned}$$ Qualitative investigation ------------------------- Some general properties of the solutions in both the magnetically and the electrically charged cases can be derived through an analytical investigation of the equations. We will use the fact that for the nonlinear electrodynamics under consideration the following relation holds: $$X\partial_{X}L(X)- L(X)>0\label{EDHAM}.$$ This inequality holds for a large class of nonlinear electrodynamics including the Euler-Heisenberg electrodynamics given by Lagrangian (\[EHL\]). The non-existence of inner horizons for our solution can be proved through an analytical analysis of the following equation $$\frac{d }{dr}\left( e^{-\delta}r^{2}f\frac{d\varphi }{dr} \right)=4 r^2 e^{-\delta} \alpha(\varphi) {\cal A}^{4}(\varphi)\left[X\partial_{X}L(X)- L(X)\right]>0,\label{phianl} \\$$ which is another form of equations (\[MagnPhi\]) and (\[ElPhi\]). Let us admit that more than one horizon exists. Then, we integrate eq.(\[phianl\]) in the interval $r\in[r_{-},r_{+}]$ where we denote he first inner and the outer horizons by $r_{-}$ and $r_{+}$, respectively, i.e., $$\begin{aligned} &&\left. \left( e^{-\delta}r^{2}f\frac{d\varphi }{dr} \right) \right|_{r_{+}}- \left.\left( e^{-\delta}r^{2}f\frac{d\varphi }{dr} \right) \right|_{r_{-}}\nonumber \\ &&\hspace{1cm}=4 \int\limits_{r_{-}}^{r_{+}}r^2 e^{-\delta} \alpha(\varphi) {\cal A}^{4}(\varphi)\left[X\partial_{X}L(X)- L(X)\right] dr>0, \nonumber \\ \label{varhu}\end{aligned}$$ Since $f(r_{-})=0=f(r_{+})$ the left-hand side (LHS) of (\[varhu\]) is equal to zero. The integral on the RHS, however, is positive. The contradiction we reach means that our admission is incorrect. So if a black hole exists it will have a single horizon, i.e., its causal structure will be Schwarzschild-like. In both conformal frames, inside the event horizon a space-like singularity is hidden. The qualitative behavior of $\delta(r)$ can easily be seen from eqs.(\[MagnDelta\]) and (\[ElDelta\]). It decreases monotonously with $r$ for both the magnetically and the electrically charged cases. The scalar field plays a crucial role in changing the causal structure of the magnetically charged black hole. In GR, for $(M/P)^2\leq 24/25$ a single horizon exists, but for $(M/P)^2>24/25$ a second and a third horizons occur. Extremal solutions exist only for $b^2\leq b_{\rm crit}^2=8/27P^2$. In the presence of the scalar field, for $b^2>b^2_{\rm crit}$, the causal structure is qualitatively the same as in the General Theory. For $b^2 \leq b^2_{\rm crit}$, however, the scalar field changes the causal structure significantly and the number of horizons reduces to one. For the electrically charged case in GR extremal solutions exist for all values of $b^2$. An inner horizon emerges for $ M <M_{0}$, where $$M_{0}={\Gamma ({1\over 4}) \over 2\Gamma({3\over 2})} {Q ^{{3 \over 2}} \over (2b^2)^{{1\over4}} }. \\$$ Again, in the electrically charged case with the presence of the scalar field the black hole has a single horizon, namely the event horizon. The existence of extremal solutions is possible only for theories in which $\alpha(\varphi)$ turns to zero on the horizon. This can be easily proved through eq.(\[phianl\]). For the extreme solution $f(r_{H})=f'(r_{H})=0$. Thus, the LHS of (\[phianl\]) turns to zero on the horizon. Since relation (\[EDHAM\]) holds and $0<{\cal A}(\varphi)<\infty$, the only chance that the RHS of the solution turns to zero is $\alpha[\varphi(r_{H})]=0$. Numerical results ================= The nonlinear systems (\[MagnDelta\])-(\[MagnPhi\]) and (\[ElDelta\])-(\[ElPhi\]) are inextricably coupled and the event horizon $r_H$ is [*a priori*]{} unknown boundary. In order to be solved, they are recast as a equivalent first order systems of ordinary differential equations. Following the physical assumptions of the matter under consideration the asymptotic boundary conditions are set, i.e., $$\lim_{r \to \infty}m(r) =M \quad (M \>{\rm is\> the\> mass\> of\> the\> black\> hole\> in\> the\> Einstein\> frame}),$$ $$\lim_{r \to \infty}\delta(r)=\lim_{r \to \infty}\varphi(r)=0.$$ At the horizon both the relationship $$f(r_H)=0$$ and the regularization condition $$\left.\left(\frac{df}{dr}\!\cdot\! \frac{d \varphi}{d r}\right)\right|_{r=r_H} =\left. \left\{ 4 \alpha(\varphi) {\cal A}^4(\varphi) [X \partial_X L(X)-L(X) ]\right\}\right|_{r=r_H}$$ must be held. Supplying the governing equations (\[ElDelta\])-(\[ElPhi\]) with the above five conditions we compose well-posed boundary-value problem (BVP) for functions $\delta(r)$, $m(r)$, $\varphi(r)$ as well as the spectral parameter $r_H$. We treat it by the Continuous Analog of Newton Method (see, for example [@gavurin],[@jidkov],[@YFBT]). After an appropriate linearization we render the original BVP to solving a vector two-point BVP. On a discrete level sparse (almost diagonal) linear algebraic systems with regard to increments of sought functions $\delta(r)$, $m(r)$, and $\varphi(r)$ have to be inverted. For our numerical solution we have considered theories with constant coupling parameter, which correspond to the Brans-Dicke theory. So the coupling function has the following form: $${\cal A}(\varphi)=e^{\alpha\varphi}, \\$$ where $\alpha$ is a positive constant and in this theory $\alpha(\varphi)=\alpha={\rm const}$ . We have studied the parametric space for fixed value of the coupling parameter $\alpha = 0.01$ (this value is close to the one established on the bases of experimental data) and for several values of the magnetic charge. Magnetically charged case ------------------------- For regions where the ratio $(P/M)^2$ is considerably less than the critical value $ 24/25 $ the behavior of the solutions resembles that the GR case. When we decrease the mass for a fixed value of $P$, however, the solutions starts to deviate from it considerably. In GR the solutions pass through an extremal black hole [@YT]. In the case we consider, the absolute value of the scalar field rises considerably and prevents the emergence of a degenerate horizon (see Fig.(\[PhiM\])). The numerical investigations also reveal that as the mass approaches zero, the radius of the black hole approaches a constant value, which means that solutions with negative masses and finite radius of the event horizon exist Fig.(\[RhM\]). The temperature of the magnetically charged black hole is shown in Fig.(\[TMagn\]). As it can be seen in Fig.(\[UvelT\]), which is a magnification of the flat region of the curves in Fig.(\[TMagn\]), when the mass $M$ approaches the critical value, the temperature decreases but rises suddenly before it reaches the zero. So an extremal solution is not reached. Electrically charged case ------------------------- As in the magnetically charged black hole in the electrically charged one the causal structure is simpler than in the corresponding solution in GR. Instead of having an inner and outer horizon, which merge in a single degenerate horizon for some value of the mass, as in the GR case, the electrically charged black hole has a single horizon. The properties of the electrically charged black hole, however, differ significantly from those of the magnetically charged case. The radius of the electrically charged black hole decreases with the decrease of its mass, and solutions with negative masses do not exist (see Fig.(\[RhEl\])). The behavior of the temperature, however, is complex as this can be seen in Fig.(\[TEl\]). Due to numerical difficulties which arise in this region of the parametric space, some of the curves are incomplete. According to the curves we have obtained, for some values of the electrical charge $Q$ the temperature approaches zero. A possibility exists, that with the decrease of the mass the temperature passes through a local minimum and then increases to infinity similarly to the magnetically charged case. To answer this question we should carry out some analytical investigations. In cases of low mass the radius of the black hole is small which allows us to make an approximation in the equations by leaving only the leading terms on the RHS. From the approximate system the following relation can be derived $$r^2e^{\delta} \left[ \alpha \left(e^{-2\delta}f \right)'- \left(e^{-2\delta}f \right) \varphi'\right]=C,$$ where $C$ is an arbitrary positive constant. On the horizon this relation takes the following form $$\alpha r_{H}^2 e^{-\delta_{H}}f'_{H}=C.$$ Hence, for the temperature of the electrically charged black hole we obtain $$T=\frac{1}{ 4\pi C \alpha r_{H}^2 },\label{Temp}$$ from where we can say that it rises to infinity as $r_{H}$ and $M$ approach zero, for all values of the parameters. Thermodynamics ============== For the solution we study the First Law (FL) of thermodynamics holds. It is naturally connected with the Einstein frame as this can be seen below. The formal derivation of the FL of black hole thermodynamics for the case of non-linear electrodynamics including the presence of a scalar field, in the Einstein frame, can be seen in the work of Rasheed [@Rasheed]. The formulation of the FL in the Jordan frame requires proper definition of the thermodynamic variables. The temperature of the event horizon is invariant under conformal transformations of the metric that are unity at infinity [@Jacobson]. The properly defined entropy is also preserved under conformal transformations. In the Einstein frame, the entropy of the black hole is one forth of the horizon area. In the Jordan frame, however, this is not so [@MVisser; @FordRoman] and the definition of the entropy needs to be generalized as follows: $$S_{J}=\frac{1}{4G_{*}}\int d^2x \sqrt{-^{(2)}{\tilde g}}F(\Phi).$$ Using relation (\[conf\]) we find that $$S_{J}=\frac{1}{4G_{*}}\int d^2x \sqrt{-^{(2)} g}=S_{E}=S.$$ In the last two equations quantities $^{(2)}{\tilde g}$ and $^{(2)} g$ are the determinants of the induced metrics on the horizon in the Jordan and in the Einstein frame, respectively. The presence of the scalar field adds a term ${\cal{D}}\delta\varphi_{\infty}$ in the FL, where ${\cal{D}}$ is the scalar field charge and $\delta\varphi_{\infty}$ is the variation of the asymptotic value of the scalar field. We define the scalar field charge as $${\cal{D}} = - \left. r^2 \frac{d\varphi}{dr}\right|_{r\rightarrow \infty}. \label{dilatoncharge}$$ This charge, however, is not independent. Let us integrate eq.(\[phianl\]) from the radius of the horizon to infinity. We obtain $${\cal{D}}=4 \int\limits_{r_{H}}^{\infty} r^2 \alpha(\varphi) {\cal A}^{4}(\varphi)\left[X\partial_{X}L(X)- L(X)\right]dr.$$ Hence, we see that the scalar field charge can be determined unambiguously once the mass and the magnetic charge of the black hole and the asymptotic value of the scalar field at infinity are known. In our case, since the scalar field is fixed and vanishing at infinity the term coming from the scalar field disappears. Scalar-tensor theories of gravity violate the strong equivalence principle. This results in the appearance of three different possible masses as a measure of the total energy of the compact objects. In both conformal frames in the FL of thermodynamics the Arnowitt-Deser-Misner (ADM) mass from the Einstein frame should be used. Similarly, for boson and fermion stars the proper measure for the energy of the system is again the ADM mass in the Einstein frame $M$. For more details on the subject we would refer the reader to the works [@Lee; @Shapiro; @Whinnett; @Yazadji]. The ADM masses in both frames are not equivalent. The ADM mass in the Jordan frame $M_{J}$ is related to the ADM mass in the Einstein frame $M$, and the scalar field charge ${\cal D}$ in the following way $$M_{J}=M+\alpha\cal{D}.$$ It can be interpreted as the Keplerian mass of the black hole. The terms in FL connected with the magnetic and the electric charges, respectively also remain invariant with the transition between the two conformal frames. To sum up, the properly defined FL looks in the same way in both conformal frames. Its explicit form for the magnetically charged case and for the electrically charged case, respectively, is presented below. Magnetically charged case ------------------------- In the magnetically charged case the FL of thermodynamics is $$\delta M=T \delta S + \Psi_{H}\delta P,\label{FirstLawM}$$ where $T$, $S$ and $P$ are the temperature, the entropy, and the magnetic charge of the black hole. The quantity $\Psi$ conjugate to the magnetic charge is the potential of the magnetic field which is given by the following definition $$H_{\mu}=\partial_{\mu}\Psi.\label{defH}$$ On the other hand the magnetic field is defined as $$H_{\mu}=-\star G_{\mu\nu}\xi^{\nu},$$ where $$G_{\mu\nu}=-\frac{1}{2} \frac{\partial\left( {\cal A}^{4}(\varphi)L\right)}{\partial F_{\mu\nu} },$$ $\xi=\frac{\partial}{\partial t}$ is the Killing vector generating time translations and “$\star$” is the Hodge star operator. Electrically charged case ------------------------- In the electrically charged case the FL of black hole thermodynamics takes the following form $$\delta M=T \delta S + \Phi_{H}\delta P,\label{FirstLawEl}$$ where $T$, $S$ and $Q$ are the temperature, the entropy, and the electric charge of the black hole. The quantity $\Phi$ conjugate to the electric charge is the potential of the magnetic field which is given by the following definition $$E_{\mu}=\partial_{\mu}\Phi.\label{defH}$$ On the other hand the electric field is defined as $$E_{\mu}=F_{\mu\nu}\xi^{\nu}.$$ Conclusion ========== In the present work numerical solutions describing charged black holes coupled to non-linear electrodynamics in the scalar-tensor theories with massless scalar field were found. Purely magnetically and purely electrically charged cases were studied. For the Lagrangian of the non-linear electrodynamics the truncated Euler-Heisenberg Lagrangian was chosen and scalar-tensor theories with positive coupling parameter were considered. As a result of the numerical and analytical investigation, some general properties of the solutions were found. Both the magnetically charged case and electrically charged case have a single, non-degenerate event horizon, i.e., their causal structure resembles that of the Schwarzschild black hole and is simpler compared to the corresponding solution in the frame of GR. In both conformal frames, inside the event horizon a space-like singularity is hidden. Some properties of the purely magnetically charged black hole differ significantly from those of the electrically charged one. As the mass decreases, the radius of the horizon in the first case approaches a finite, constant value and the temperature of the horizon rises. Thus, solutions with negative masses and finite radius of the event horizon exist. In the electrically charged case, however, the radius of the horizon turns to zero for a finite value of the mass $ M $ of the black hole. In this case, in the $r_{H}\rightarrow 0$ limit a naked singularity is reached and the solution cannot be continued for negative masses. Acknowledgments {#acknowledgments .unnumbered} =============== This work was partially supported by the Bulgarian National Science Fund under Grant MUF04/05 (MU 408) and the Sofia University Research Fund N60. [tbds]{} E. Ayon-Beato, A. Garcia, [*Phys.Rev.Lett.*]{} [**80**]{}, 5056 (1998) N. Banerjee and S. Sen, Phys. Rev. [**D47**]{}, 104024 (1998). M. Born and L. Infeld, Proc. R. Soc. London [**A143**]{}, 410 (1934). R. Cai, D. Pang and A. Wang, Rev. [**D70**]{}, 124034 (2004). G. Clement and D. Gal’tsov, Phys. Rev. [**D62**]{}, 124013 (2000). T. Damour and A. Polyakov, Nucl. Phys. [**B423**]{}, 532 (1994) M. Demianski, [*Found.Phys.*]{} [**16**]{}, 187 (1986) T. Dey, Phys. Lett. [**B595**]{}, 484 (2004). 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--- abstract: 'In this paper, persistence properties of solutions are investigated for a 4-parameter family ($k-abc$ equation) of evolution equations having $(k+1)$-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, Novikov and Fokas-Olver-Rosenau-Qiao equations. These properties will imply that strong solutions of the $k-abc$ equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters $k$, $a$, $b$, and $c$.' address: | Department of Mathematics\ University of North Georgia\ Dahlonega, GA 30597 author: - 'Ryan C. Thompson' date: 'October 5, 2016' title: | Decay Properties of Solutions to a\ 4-parameter family of wave equations --- Introduction ============ For $k \in {\mathbb}{Z}^+$, $k \geq 2$ and $a, b, c \in {\mathbb}{R}$, we consider the Cauchy problem for the following $k-abc$ family of equations $$\begin{aligned} \label{kabc} u_t &+u^ku_x-au^{k-2}u_x^3+D^{-2}{\partial}_x\left[\frac{b}{k+1}u^{k+1}+cu^{k-1}u_x^2-a(k-2)u^{k-3}u_x^4\right] \nonumber \\ & \ +D^{-2}\left[[k(k+2)-8a-b-c(k+1)]u^{k-2}u_x^3-3a(k-2)u^{k-3}u_x^3u_{xx}\right]=0\end{aligned}$$• where $D^{-2} \doteq (1-{\partial}_x^2)^{-1}$, and study its persistence properties. We show that if the initial data is endowed with exponential decay at infinity, then the corresponding solution will carry this property. When $a=0$ and $k$ a positive odd integer with $k\geq 1$ and $b \in [0,k(k+2)]$ then, utilizing the aforementioned behavior of strong solutions, we show that this equation exhibits unique continuation in the sense that if the initial value $u(x,0)$ is given the property of decaying exponentially, then the solution $u(x,t)$ must be identically zero if assumed to be decaying exponentially at some later time $t>0$. These are natural extensions of the results proved in Himonas, Misiołek, Ponce and Zhou [@hmpz] for the Camassa-Holm equation. The $k-abc$ equation was first studied by Himonas and Mantzavinos [@hm1] where they showed well-posedness in Sobolev spaces $H^s$ for $s>5/2$. They also provided a sharpness result on the data-to-solution map and proved that it is not uniformly continuous from any bounded subset of $H^s$ into $C([0,T];H^s)$. It was shown, however, that the solution map is Hölder continuous if $H^s$ is equipped with a weaker $H^r$ norm where $r \in [0,s)$. The equation was also studied in Barostichi, Himonas and Petronilho in [@bhp] where they exhibited a power series method in abstract Banach spaces provided analytic initial data, thereby establishing a Cauchy-Kovalevsky type theorem for the $k-abc$ equation . It is important to note that when Himonas and Mantzavinos ushered in the $k-abc$ equation , they unified many Camass-Holm type equations. For instance, when $k=2$ and $c = (6-6a-b)/2$ you obtain the $ab$-equation $$\label{ab} u_t+u^2u_x-au_x^3+D^{-2}{\partial}_x\left[\frac b3u^3+\frac{6-6a-b}{2}uu_x^2\right]+D^{-2}\left[\frac{2a+b-2}{2}u_x^3\right]=0,$$ which was also introduced by Himonas and Mantzavinos in [@hm2] and was shown to possess both periodic and non-periodic traveling wave solutions. In fact, the non-periodic multipeakon traveling wave solutions to the $ab$-equation take the form $$u(x,t) = \sum_{j=1}^np_j(t)e^{-|x-q_j(t)|},$$ provided that the positions $q_j$ and momenta $p_j$ satisfy an appropriate system of nonlinear differential equations. When $a=0$ and $c=(3k-b)/2$, the $k-abc$ equation yields the generalized Camassa-Holm equation (g-$kb$CH) $$\label{gkbch} u_t+u^ku_x+D^{-2}{\partial}_x\left[\frac {b}{k+1}u^{k+1}+\frac{3k-b}{2}u^{k-1}u_x^2\right]+D^{-2}\left[\frac{(k-1)(b-k)}{2}u^{k-2}u_x^3\right]=0.$$ Well-posedness in $H^s$ with $s>3/2$ was shown in [@hh1], while its multipeakon traveling waves were derived in [@gh]. Furthermore, Himonas and Thompson [@ht] explored asymptotic behavior of strong solutions, unique continuation and conditions needed to admit global solutions. If we let $k=1$ in the aforementioned then we also obtain the $b$-family of equations with quadratic nonlinearities which was introduced by Holm and Staley in [@hs1] and [@hs2]. We may also see its local form as $$\label{bfam} \underbrace{m_t}_\text{evolution}+\underbrace{um_x}_\text{convection}+\underbrace{bu_xm}_\text{stretching}=0, \ \ \ m = u-u_{xx},$$ which expresses a fine balance between, evolution, convection and stretching. It is within this balance that one may find a starting point for not only deriving peakon solutions, but also establishing infinite speed of propagation and other asymptotic behavior of solutions. When we let $a=1/3$ and $b=2$ in the $ab$-equation we obtain the Fokas-Olver-Rosenau-Qiao (FORQ) equation $$\label{forq} u_t+u^2u_x-\frac13u_x^3+D^{-2}{\partial}_x\left[\frac23u^3+uu_x^2\right]+D^{-2}\left[\frac13u_x^3\right]=0,$$ which was derived in a variety of ways by Fokas [@fokas], Olver and Rosenau [@or] and Qiao [@qiao] and also appeared in a work by Fuchsteiner [@fuch]. Observe that for $k=1$ and $b=2$ the g-$kb$CH equation gives the celebrated Camassa-Holm (CH) equation $$\label{ch} (1-{\partial}_x^2)u_t=uu_{xxx}+2u_xu_{xx}-3uu_x$$ which appears in the context of hereditary symmetries studied by Fokas and Fuchssteiner [@ff]. This equation was first written explicitly and derived from the Euler equations by Camassa and Holm in [@cam], where they also found its “peakon" traveling wave solutions. These are solutions with a discontinuity in the first spacial derivative at its crest. The simplest one in the non-periodic case is of the form $u_c(x, t)=ce^{-|x-t|}$, where $c$ is a positive constant. The CH equation possesses many other remarkable properties such as infinitely many conserved quantities, a bi-Hamiltonian structure and a Lax pair. For more information about how CH arises in the context of hereditary symmetries we refer to [@ff]. Concerning it’s physical relevance, we refer the reader to the works by Johnson [@j1], [@j2] and Constantine and Lannes [@const1]. For $k=1$ and $b=3$ in , we obtain the Degasperi-Procesi (DP) equation $$\label{dp} (1-{\partial}_x^2)u_t=uu_{xxx}+3u_xu_{xx}-4uu_x.$$ Degasperi and Procesi [@dp] discovered this equation in 1998 as one of the three equations to satisfy asymptotic integrability conditions in the following family of equations $$\label{fam} u_t+c_0u_x+{\gamma}u_{xxx}-\alpha^2u_{txx}=(c_{1,n}u^2+c_{2,n}u_x^2+c_3uu_{xx})_x,$$ where $\alpha, c_0, c_{1,n}, c_{2,n}, c_3 \in {\mathbb}{R}$ are constants. Other integrable members of the family are the CH and the KdV equations. Furthermore, the DP and CH are the only integrable members of the $b$-family equation which is obtained from the g-$kb$CH equation by letting $k=1$ and $b \in {\mathbb}{R}$ (see [@mn]). Also, like the CH equation , the DP equation possesses peakon solutions of the form $u_c(x,t)=ce^{-|x-ct|}$. Recently, Vladimir Novikov [@nov], in addition to several integrable equations with quadratic nonlinearities like the CH and DP, generated about 10 integrable equations with cubic nonlinearities while investigating the integrability of Camassa-Holm type equations of the form $$\label{gen} (1-{\partial}_x^2)u_t=P(u,u_x,u_{xx},...),$$ where $P$ is a polynomial of $u$ and its derivatives. One of these CH-type equations with cubic nonlinearities, which happened to be a new integrable equation, is the following $$\label{ne} (1-{\partial}_x^2)u_t=u^2u_{xxx}+3uu_xu_{xx}-4u^2u_x,$$ which can be found from the g-$kb$CH equation by letting $k=2$ and $b=3$. Equation is now called the Novikov equation (NE) and also possesses many properties exhibited by Camassa-Holm type equations. One of which is the existence of peakon solutions of the form $u_c(x,t)=\sqrt{c}e^{-|x-ct|}$. Unlike CH and DP, however, unique continuation has not been shown for NE due to difficulties in handling its cubic nonlinearities. In the direction of peakon traveling wave solutions, it has been shown in [@hm2] that the $k-abc$ equation possesses these solitary wave solutions on the circle and the real line. On the real line, these solutions take the form $$u(x,t)={\gamma}e^{-|x-(1-a){\gamma}^kt|}, \ \ {\gamma}\in {\mathbb}{R},$$ where $(1-a){\gamma}^k$ is the wave speed. On the circle, these solutions take the form $$u(x,t)={\gamma}\cosh\left(\left[x-\left[1+(1-a)\sinh^2\pi\right]\cosh^{k-2}(\pi){\gamma}^kt\right]_p-\pi\right), \ \ {\gamma}\in {\mathbb}{R},$$ where $$[x]_p \doteq x-2\pi\left\lfloor \frac{x}{2\pi} \right\rfloor$$ provided that $6a+b+2c=3k$. Furthermore, under appropriate conditions on the four parameters the $k-abc$ equation also admits multi-peakon solutions. In regards to conservation laws, we take note that the $H^1$ norm is conserved under particular parameter conditons. In fact, we see have $$\frac12\frac{d}{dt}\|u(t)\|_{H^1}=\int\left(\left[2k-2c-1-\frac2k(9a+b+2c-3k)\right]u^ku_xu_{xx}-\frac{3a}{4}(k-1)(k-2)u^{k-3}u_x^5\right)dx$$ and so we find the $H^1$ norm is conserved if $9a+b+4c=9$ when $k=2$ and if $a=0$ and $2c+\frac2k(b+2c-3k)+1=2k$ when $k \geq 3$. The well-posedness of the $k-abc$ equation in Sobolev spaces and the continuity properties of its solution map have been studied in [@hm1]. There, the following result was proved. \[t1\] Let $a,b,c \in {\mathbb}{R}$ with $a \neq 0$ and $k \in {\mathbb}{N}$ with $k \geq 2$. Then the Cauchy problem for the $k-abc$ equation with initial condition $u(x,0)=u_0(x) \in H^s$. $x \in {\mathbb}{R}$ or ${\mathbb}{T}$, $s > 5/2$, has a unique solution $u \in C([0,T];H^s)$ which admits the estimate $$\|u(t\|_{H^s} \leq 2^{1+\frac 1k}\|u_0\|_{H^s}, \ \ 0 \leq t \leq T \leq (2^{k+1}kc_s\|u_0\|_{H^s}^k)^{-1},$$• where the constant $c_s>0$ depends only on $s$. Furthermore, the data-to-solution map $u_0 \mapsto u(t)$ is continuous. It is also important to note that when $a=0$, the $k-abc$ equation closely resembles the g-$kb$CH equation and a similar well-posedness result was proven, albeit for $s>3/2$, in [@hh1]. From the above well-posedness result, we may now extract our persistence properties and unique continuation. Our basic assumption is that the initial data as well as its first spacial derivative decay exponentially. We also endow the solution at time $t>0$ with an analogous condition which guarantees that our solution will be identically zero. The following results are based on the work in [@hmpz] for the CH equation. \[t2\] Assume that $T>0$, $s>5/2$ and $u \in C([0,T];H^s)$ is a strong solution of . If the initial data satisfies certain decay conditions at infinity, more precisely, if there is some $\theta \in (0,1)$ such that $$|u_0|, \ |{\partial}_xu_0| \sim O(e^{-\theta |x|}) \ \ as \ \ |x| \to \infty,$$ then $$|u(x,t)|, \ |{\partial}_xu(x,t)| \sim O(e^{-\theta |x|}) \ \ as \ \ |x| \to \infty$$ uniformly with respect to $t \in [0,T]$. *Notation.* We shall say that $$|f(x)| \sim O(e^{\alpha x}) \ \ \text{as} \ \ x \to \infty \ \ \text{if} \ \ \lim_{x \to \infty}\frac{|f(x)|}{e^{\alpha x}}=L,$$ for some $L>0$, and $$|f(x)| \sim o(e^{\alpha x}) \ \ \text{as} \ \ x \to \infty \ \ \text{if} \ \ \lim_{x \to \infty}\frac{|f(x)|}{e^{\alpha x}}=0.$$ \[kabc-uniq-cont\] Assume that for some $T>0$ and $s>5/2$, $ u \in C([0,T];H^s({\mathbb}{R})) $ is a solution to the $k-abc$ initial value problem, $k$ is an odd integer with $k \geq 1$ and that $$a=0, \ \ c=\frac{k(k+2)-b}{k+1}, \ \ b \in [0, k(k+2)].$$ If $u_0(x)=u(x,0)$ satisfies that for some $\alpha \in \left(\frac{1}{k+1},1\right)$, $$\label{decay1} |u_0(x)| \sim o(e^{-x}), \ \ \ \ and \ \ \ \ |{\partial}_xu_0(x)| \sim O(e^{-\alpha x}) \ \ \text{as} \ x \rightarrow \infty,$$ and there exists $t_1 \in (0,T]$ such that $$\label{decay2} |u(x,t_1)| \sim o(e^{-x}) \ \ as \ \ x \rightarrow \infty,$$ then $u \equiv 0$. The paper is organized as follows. In Section 2, we prove the persistence properties of the $k-abc$ equation as listed in Theorem \[t2\]. Then in Section 3 we use Theorem \[t2\] to prove Theorem \[kabc-uniq-cont\]. Finally, we make a remark about the aymptotic profile of solutions when provided compactly supported initial data. Proof of Theorem \[t2\] ======================= Assume that $u \in C([0,T];H^s)$ is a strong solution to with $s > 5/2$. Let $$\begin{aligned} \label{fs} &F_1(u)=\frac{b}{k+1}u^{k+1}+cu^{k-1}u_x^2-a(k-2)u^{k-3}u_x^4, \\ &F_2(u)=[k(k+2)-8a-b-c(k+1)]u^{k-2}u_x^3-3a(k-2)u^{k-3}u_x^3u_{xx}.\end{aligned}$$• Then we have that the $k-abc$ equation takes the form $$\label{nonloc2} u_t+u^ku_x-au^{k-2}u_x^3+{\partial}_xG*F_1(u)+G*F_2(u)=0.$$• Also, let $$\label{m} M=\sup_{t \in [0,T]}\|u(t)\|_{H^s},$$• $$\label{phi} {\varphi}_N(x)= \begin{cases} e^{\theta|x|}, \ \ |x|<N \\ e^{\theta N}, \ \ \ |x| \geq N, \end{cases}•$$• where $N \in {\mathbb}{N}$ and $\theta \in (0,1)$. Observe that for all $N$ we have $$0 \leq |{\varphi}_N'(x)| \leq {\varphi}_N, \ \ \ a.e. \ \ x \in {\mathbb}{R}.$$ Multiplying by $(u{\varphi}_N)^{2p-1}{\varphi}_n$ for $p \in {\mathbb}{Z}^+$ and integrating over the real line we obtain $$\begin{aligned} \label{24} \frac{1}{2p}\frac{d}{dt}&\int_{\mathbb}{R}(u{\varphi}_N)^{2p}dx+\int_{\mathbb}{R}u^ku_x(u{\varphi}_N)^{2p-1}{\varphi}_Ndx \nonumber \\ &-\int_{\mathbb}{R} au^{k-2}u_x^3(u{\varphi}_N)^{2p-1}{\varphi}_N dx \nonumber \\ &+\int_{\mathbb}{R}({\partial}_xG*F_1(u))(u{\varphi}_N)^{2p-1}{\varphi}_Ndx \nonumber \\ &+\int_{\mathbb}{R}(G*F_2(u))(u{\varphi}_N)^{2p-1}{\varphi}_Ndx=0.\end{aligned}$$• By the Sobolev embedding theorem and we have $$\|u\|_{L^\infty}+\|u_x\|_{L^\infty}+\|u_{xx}\|_{L^\infty} \leq CM.$$ This leads us to achieve the following estimates $$\begin{aligned} \label{25} &\left|\int_{\mathbb}{R}u^ku_x(u{\varphi}_N)^{2p-1}{\varphi}_Ndx\right| \leq \|u\|_{\infty}^{k-1}\|u_x\|_{\infty}\|u{\varphi}_N\|_{2p}^{2p} \leq CM^k\|u{\varphi}_N\|_{2p}^{2p} \\ &\left|a\int_{\mathbb}{R} u^{k-2}u_x^3(u{\varphi}_N)^{2p-1}{\varphi}_N dx\right| \leq |a|\|u\|_{\infty}^{k-3}\|u_x\|_{\infty}^3\|u{\varphi}_N\|_{2p}^{2p} \leq CM^k\|u{\varphi}_N\|_{2p}^{2p}.\end{aligned}$$• By Hölder’s inequality, we have the following estimates $$\begin{aligned} \label{26} &\left|\int_{\mathbb}{R}({\partial}_xG*F_1(u))(u{\varphi}_N)^{2p-1}{\varphi}_Ndx\right| \leq \|u{\varphi}_N\|_{2p}^{2p-1}\|({\partial}_xG*F_1(u)){\varphi}_N\|_{2p}, \nonumber \\ &\left|\int_{\mathbb}{R}(G*F_2(u))(u{\varphi}_N)^{2p-1}{\varphi}_Ndx\right| \leq \|u{\varphi}_N\|_{2p}^{2p-1}\|(G*F_2(u)){\varphi}_N\|_{2p}.\end{aligned}$$• From and the above estimates, this implies $$\label{ode1} \frac{d}{dt}\|u{\varphi}_N\|_{2p} \leq CM^k\|u{\varphi}_N\|_{2p}+\|({\partial}_xG*F_1(u)){\varphi}_N\|_{2p}+\|(G*F_2(u)){\varphi}_N\|_{2p}.$$• By Gronwall’s inequality, implies the following estimate $$\label{est1} \|u{\varphi}_N\|_{2p} \leq \left(\|u_0{\varphi}_N\|_{2p}+\int_0^t\left[\|({\partial}_xG*F_1(u)){\varphi}_N\|_{2p}+\|(G*F_2(u)){\varphi}_N\|_{2p}\right]d\tau\right)e^{CM^kt}.$$• Now differentiating with respect to the spacial variable $x$, multiplying by $({\varphi}_Nu_x)^{2p-1}{\varphi}_N$ and integrating over the real line yields $$\begin{aligned} \label{29} \frac{1}{2p}\frac{d}{dt}&\int_{\mathbb}{R}(u_x{\varphi}_N)^{2p}dx+\int_{\mathbb}{R}ku^{k-1}(u_x)^2({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx \nonumber \\ &+\int_{\mathbb}{R}u^ku_{xx}({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx-\int_{\mathbb}{R}a(k-2)u^{k-3}u_x^4({\varphi}_Nu_x)^{2p-1}{\varphi}_N dx \nonumber \\ &-\int_{\mathbb}{R}3au^{k-2}u_x^2u_{xx}({\varphi}_Nu_x)^{2p-1}{\varphi}_N dx+\int_{\mathbb}{R}({\partial}_x^2G*F_1(u))({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx \nonumber \\ &+\int_{\mathbb}{R}({\partial}_xG*F_2(u))({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx=0.\end{aligned}$$• This leads us to obtain the following estimates $$\begin{aligned} \label{210} &\left|\int_{\mathbb}{R}ku^{k-1}(u_x)^2({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx\right| \leq CM^k\|u_x{\varphi}_N\|_{2p}^{2p}, \nonumber \\ &\left|\int_{\mathbb}{R}a(k-2)u^{k-3}u_x^4({\varphi}_Nu_x)^{2p-1}{\varphi}_N dx\right| \leq CM^k\|u_x{\varphi}_N\|_{2p}^{2p} \nonumber \\ &\left|\int_{\mathbb}{R}3au^{k-2}u_x^2u_{xx}({\varphi}_Nu_x)^{2p-1}{\varphi}_N dx\right| \leq CM^k\|u_x{\varphi}_N\|_{2p}^{2p} \nonumber \\ &\left|\int_{\mathbb}{R}({\partial}_x^2G*F_1(u))({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx\right| \leq \|u_x{\varphi}_N\|_{2p}^{2p-1}\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{2p}, \nonumber \\ &\left|\int_{\mathbb}{R}({\partial}_xG*F_2(u))({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx\right| \leq \|u_x{\varphi}_N\|_{2p}^{2p-1}\|({\partial}_xG*F_2(u)){\varphi}_N\|_{2p}.\end{aligned}$$• For the third integral in , we estimate as follows $$\begin{aligned} \label{211} \left|\int_{\mathbb}{R}u^ku_{xx}({\varphi}_Nu_x)^{2p-1}{\varphi}_Ndx\right|&=\left|\int_{\mathbb}{R}u^k({\varphi}_N u_x)^{2p-1}({\partial}_x({\varphi}_Nu_x)-{\varphi}_N' u_x)dx\right| \nonumber \\ &=\left|\int_{\mathbb}{R}u^k({\varphi}_Nu_x)^{2p-1}{\partial}_x({\varphi}_Nu_x)dx-\int_{\mathbb}{R}u^k({\varphi}_Nu_x)^{2p-1}{\varphi}_N'u_xdx\right| \nonumber \\ &=\left|\frac{1}{2p}\int_{\mathbb}{R}u^k{\partial}_x[({\varphi}_Nu_x)^{2p}]dx-\int_{\mathbb}{R}u^k({\varphi}_Nu_x)^{2p-1}{\varphi}_N'u_xdx\right| \nonumber \\ &=\left|\frac{-k}{2p}\int_{\mathbb}{R}u^{k-1}u_x({\varphi}_Nu_x)^{2p}dx-\int_{\mathbb}{R}u^k({\varphi}_Nu_x)^{2p-1}{\varphi}_N'u_xdx\right| \nonumber \\ & \leq CM^k\|u_x{\varphi}_N\|_{2p}^{2p}.\end{aligned}$$• From - , we achieve the following differential inequality $$\label{ode2} \frac{d}{dt}\|u_x{\varphi}_N\|_{2p} \leq CM^k\|u_x{\varphi}_N\|_{2p}+\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{2p}+\|({\partial}_xG*F_2(u)){\varphi}_N\|_{2p}.$$• From and Gronwall’s inequality, we obtain $$\label{est2} \|u_x{\varphi}_N\|_{2p} \leq \left(\|{\partial}_xu_0{\varphi}_N\|_{2p}+\int_0^t[\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{2p}+\|({\partial}_xG*F_2(u)){\varphi}_N\|_{2p}]d\tau\right)e^{CM^kt}$$• By adding and , we have the following $$\begin{aligned} \label{est3} \|u{\varphi}_N\|_{2p}+\|u_x{\varphi}_N\|_{2p} \ \leq & \ (\|u_0{\varphi}_N\|_{2p}+\|{\partial}_xu_0{\varphi}_N\|_{2p})e^{CM^kt} \nonumber \\ &+\left(\int_0^t[\|({\partial}_xG*F_1(u)){\varphi}_N\|_{2p}+\|(G*F_2(u)){\varphi}_N\|_{2p}]d\tau\right)e^{CM^kt} \nonumber \\ &+\left(\int_0^t[\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{2p}+\|({\partial}_xG*F_2(u)){\varphi}_N\|_{2p}]d\tau\right)e^{CM^kt}\end{aligned}$$• Now, for any function $f \in L^1 \cap L^\infty$, $\lim_{q \to \infty}\|f\|_q=\|f\|_\infty$. Since we have that $F_i(u) \in L^1 \cap L^\infty$ (for $i=1,2$) and $G \in W^{1,1}$, we know that ${\partial}_x^jG*F_i(u) \in L^1 \cap L^\infty$ (for $i=1,2$ and $j=0,1,2$). Thus, by taking the limit of as $p \to \infty$, we obtain $$\begin{aligned} \label{est4} \|u{\varphi}_N\|_{\infty}+\|u_x{\varphi}_N\|_{\infty} \ \leq & \ (\|u_0{\varphi}_N\|_{\infty}+\|{\partial}_xu_0{\varphi}_N\|_{\infty})e^{CM^kt} \nonumber \\ &+\left(\int_0^t[\|({\partial}_xG*F_1(u)){\varphi}_N\|_{\infty}+\|(G*F_2(u)){\varphi}_N\|_{\infty}]d\tau\right)e^{CM^kt} \nonumber \\ &+\left(\int_0^t[\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{\infty}+\|({\partial}_xG*F_2(u)){\varphi}_N\|_{\infty}]d\tau\right)e^{CM^kt}\end{aligned}$$• A simple calculation, which we give in the appendix, shows that for $\theta \in (0,1)$ $$\label{calc1} {\varphi}_N(x)\int_{\mathbb}{R}e^{-|x-y|}\frac{1}{{\varphi}_N(y)}dy \leq \frac{4}{1-\theta}=C_0.$$• Thus, for any functions $f,g \in L^\infty$ we have $$\begin{aligned} \label{calc2} |{\varphi}_NG*(f^2g)(x)|&=\left|\frac12{\varphi}_N\int_{\mathbb}{R}e^{-|x-y|}f^2(y)g(y)dy\right| \nonumber \\ &\leq \frac12\left({\varphi}_N\int_{\mathbb}{R}e^{-|x-y|}\frac{1}{{\varphi}_N(y)}dy\right)\|f\|_\infty\|g\|_\infty\|f{\varphi}_N\|_\infty \nonumber \\ &\leq C_0\|f\|_\infty\|g\|_\infty\|f{\varphi}_N\|_\infty.\end{aligned}$$• Similarly, we have $$\label{calc3} |{\varphi}_N{\partial}_xG*(f^2g)(x)| \leq C_0\|f\|_\infty\|g\|_\infty\|f{\varphi}_N\|_\infty$$• and $$\label{calc4} |{\varphi}_N{\partial}_x^2G*(f^2g)(x)|=|{\varphi}_N[G*(f^2g)-f^2g]| \leq C_0\|f\|_\infty\|g\|_\infty\|f{\varphi}_N\|_\infty.$$• Therefore, with $f=u_x$ we obtain the following estimates $$\begin{aligned} \label{est5} &\|(G*F_2(u)){\varphi}_N\|_{\infty} \leq CM^k\|u_x{\varphi}_N\|_\infty, \nonumber \\ &\|({\partial}_xG*F_2(u)){\varphi}_N\|_{\infty} \leq CM^k\|u_x{\varphi}_N\|_\infty.\end{aligned}$$• Then, by following the same procedure as in we obtain $$\begin{aligned} \label{est6} &\|({\partial}_xG*F_1(u)){\varphi}_N\|_{\infty} \leq CM^k\|u{\varphi}_N\|_\infty, \nonumber \\ &\|({\partial}_x^2G*F_1(u)){\varphi}_N\|_{\infty} \leq CM^k\|u{\varphi}_N\|_\infty.\end{aligned}$$• So, by estimates , and we achieve the following $$\begin{aligned} \|u{\varphi}_N\|_{\infty}+\|u_x{\varphi}_N\|_{\infty} &\leq C(\|u_0{\varphi}_N\|_{\infty}+\|{\partial}_xu_0{\varphi}_N\|_{\infty}) \nonumber \\ &+C\int_0^t(\|u{\varphi}_N\|_\infty+\|u_x{\varphi}_N\|_\infty)d\tau.\end{aligned}$$• Hence, for any $N \in {\mathbb}{Z}^+$ and any $t \in [0,T]$, we have by Gronwall’s inequality that $$\begin{aligned} \|u{\varphi}_N\|_{\infty}+\|u_x{\varphi}_N\|_{\infty} &\leq C(\|u_0{\varphi}_N\|_{\infty}+\|{\partial}_xu_0{\varphi}_N\|_{\infty}) \nonumber \\ &\leq C(\|u_0e^{\theta|x|}\|_{\infty}+\|{\partial}_xu_0e^{\theta|x|}\|_{\infty}).\end{aligned}$$ By taking the limit as $N \to \infty$ we have that for any $t \in [0,T]$ $$|u(x,t)|e^{\theta|x|}+|u_x(x,t)|e^{\theta|x|} \leq C(\|u_0e^{\theta|x|}\|_{\infty}+\|{\partial}_xu_0e^{\theta|x|}\|_{\infty}).$$ This concludes our proof of Theorem \[t2\]. **Remark 2.1:** In fact, let $\theta \in (0,1)$ and $j=0,1,2$, if the initial data $u(x,0) = u_0$ satisfy $${\partial}_x^ju_0 \sim O(e^{-\theta|x|}), \ \ as \ \ x \to \infty,$$ then the solution $u(x,t)$ carries the same exponential decay properties, i.e. $${\partial}_x^ju \sim O(e^{-\theta|x|}), \ \ as \ \ x \to \infty.$$ Theorem \[t2\] tells us that the solution $u(x,t)$ can decay as $e^{-\theta|x|}$ as $x \to \infty$ for $\theta \in (0,1)$. The next theorem provides us with further decay properties on the solution if the only assumptions we make on $k$, $a$, $b$, and $c$ are those stated in the introduction. Proof of Theorem \[kabc-uniq-cont\] =================================== First, we will assume the result from Theorem \[t2\]. Integrating over the time interval $[0,t_1]$, noting that $(1-{\partial}_x^2)^{-1}f=G*f$ for all $f \in L^2$ where $G=\frac12e^{-|x|}$ and assuming the stated restrictions on parameters $k$, $a$, $b$ and $c$ we get $$\label{9911} u(x,t_1)-u(x,0)+\int_0^{t_1}u^k{\partial}_xu(x,\tau)d\tau+\int_0^{t_1}{\partial}_xG*F_1(u)(x,\tau)d\tau=0,$$ where $$F_1(u)=\frac{b}{k+1}u^{k+1}+cu^{k-1}u_x^2.$$ By hypothesis and we have $$\label{9912} u(x,t_1)-u(x,0) \sim o(e^{-x}) \ \ \text{as} \ \ x \rightarrow \infty.$$ From and Theorem \[t2\] it follows that $$\label{9913} \int_0^{t_1}u^k{\partial}_xu(x,\tau)d\tau \sim O(e^{-\alpha(k+1) x}) \ \ \text{as} \ \ x \rightarrow \infty,$$ and so $$\label{9914} \int_0^{t_1}u^k{\partial}_xu(x,\tau)d\tau \sim o(e^{-x}) \ \ \text{as} \ \ x \rightarrow \infty.$$ We shall show that if $u \neq 0$, then the last two terms in is $O(e^{-x})$ but not $o(e^{-x})$. Thus, we have $$\begin{aligned} \label{9915} \int_0^{t_1}{\partial}_xG*F_1(u)(x,\tau)d\tau={\partial}_xG*\int_0^{t_1}F_1(u)(x,\tau)d\tau={\partial}_xG*f_1(x),\end{aligned}$$ where by and Theorem \[t2\], $$\label{9916} 0 \leq f_1(x) \sim O(e^{-\alpha(k+1) x}), \ \ \text{so that} \ \ f_1(x) \sim o(e^{-x}) \ \ \text{as} \ \ x \rightarrow \infty.$$ Therefore, $$\label{9917} {\partial}_xG*f_1(x)=-\frac12e^{-x}\int_{-\infty}^xe^yf_1(y)dy+\frac12e^x\int_x^\infty e^{-y}f_1(y)dy.$$ From it follows that $$\label{9918} e^x\int_x^\infty e^{-y}f_1(y)dy=o(1)e^x\int_x^\infty e^{-2y}dy \sim o(1)e^{-x} \sim o(e^{-x}),$$ and if $f_1 \neq 0$ one has that $$\label{9919} \int_{-\infty}^xe^yf_1(y)dy \geq c_0, \ \ \ \text{for} \ \ x \gg 1, \ c_0>0.$$ Hence, the first integral in satisfies $$\label{9920} \frac12e^{-x}\int_{-\infty}^xe^yf_1(y)dy \geq \frac{c_0}{2}e^{-x}, \ \ \text{for} \ \ x \gg 1,$$ which combined with - yields a contradiction. Thus, $f_1(x) \equiv 0$ and consequently $u \equiv 0$. $\Box$ **Remark on compactly supported initial data:** In this section, we reflect on the property of unique continuation which we have just shown the Cauchy problem for the $k-abc$ equation to exhibit. In the case of compactly supported initial data unique continuation is essentially infinite speed of propagation of its support. Therefore, it is natural to ask the question: How will strong solutions behave at infinity when given compactly supported initial data? Our intentions are to follow the methods established in [@hmpz] and Himonas and Thompson [@ht]. We typically need two ingredients in order to provide an asymptotic profile of our solution. First, we observe that the $k-abc$ equation may be written in the following local form $$\begin{aligned} \label{kabc_loc} u_t &-u_{xxt}+(b+1)u^ku_x+(2c-3k)u^{k-1}u_xu_{xx}-u^ku_{xxx} \nonumber \\ &+(3k-9a-b-2c)u^{k-2}u_x^3+6au^{k-2}u_xu_{xx}^2+3au^{k-2}u_x^2u_{xxx}=0.\end{aligned}$$ Our goal would now be to find the appropriate $m$-equation of the form $$\label{kabc_m} \underbrace{m_t}_\text{evolution}+\gamma_1\underbrace{u^km_x}_\text{convection}+\gamma_2\underbrace{(u^k)_xm}_\text{stretching}=0, \ \ \ m = u-u_{xx}, \ \ \gamma_1, {\gamma}_2 \in {\mathbb}{R},$$ which expresses a balance between evolution, convection and stretching. Furthermore, we consider the following particle trajectories on our solution $u(x,t)$ given by $$\label{parttraj} \begin{cases} \eta_t(x,t)=u^k(\eta(x,t),t) \\ \eta(x,0)=x, \end{cases}$$ and then seek to establish a following conservation law of the form $$\label{cons_1} m(\eta(x,t),t)\eta_x^{{\gamma}_3} = m_0(x), \ \ {\gamma}_3 \in {\mathbb}{R}.$$ With to , we can establish an asymptotic profile of our solution provided compactly supported initial data. To see this construction, we refer the reader to [@hmpz] and [@ht]. In our search for the aforementioned with appropriate ${\gamma}_j$’s, however, we found that due to the coefficients of terms $u_t-u_{xxt}$, $(b+1)u^ku_x$ and $u^ku_{xxx}$ in our local equation forces our ${\gamma}_j$’s to be $${\gamma}_1 = 1, \ \ \ {\gamma}_2 = \frac bk, \ \ \ {\gamma}_3 = \frac bk.$$ Unfortunately, this forces $a=0$ and $3k-9a-b-2c=0$ which reduces us to the g-$kb$CH equation as studied in [@ht]. 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--- abstract: | We consider a slotted wireless network in an infrastructure setup with a base station (or an access point) and $N$ users. The wireless channel gain between the base station and the users is assumed to be i.i.d., and the base station seeks to schedule the user with the highest channel gain in every slot (opportunistic scheduling). We assume that the identity of the user with the highest channel gain is resolved using a series of contention slots and with feedback from the base station. In this setup, we formulate the contention resolution problem for opportunistic scheduling as identifying a random threshold (channel gain) that separates the best channel from the other samples. We show that the average delay to resolve contention is related to the entropy of the random threshold. We illustrate our formulation by studying the opportunistic splitting algorithm (OSA) for i.i.d. wireless channel [@qin-berry-osa]. We note that the thresholds of OSA correspond to a maximal probability allocation scheme. We conjecture that maximal probability allocation is an entropy minimizing strategy and a delay minimizing strategy for i.i.d. wireless channel. Finally, we discuss the applicability of this framework for few other network scenarios. title: An Information Theoretic Point of View to Contention Resolution --- Introduction ============ The advancements in the physical layer technology has enabled cellular networks (e.g., 3G and 4G deployments like Mobile WiMAX, LTE Advanced) and WLANs (e.g., IEEE 802.11n) support hundreds of megabits per second. However, with more and more users now accessing the Internet using wireless as the last mile, there is a continuous necessity to judiciously use the available network resources. Cross-layer strategies have become extremely helpful in supporting the ever increasing demand for bandwidth and stringent QoS. Opportunistic scheduling and multiuser diversity (see [@liu-etal-os]) is one such popular cross-layer technique recommended in current cellular standards and in ad hoc deployments for increasing the available network capacity. Unlike the wired channel, the wireless channel will always be constrained by fading and interference. Multiuser diversity enhances the network performance by wisely scheduling the users when their relative channel conditions are better. Opportunistic scheduling is known to significantly improve the network performance, especially for elastic traffic with loose delay constraints. Opportunistic scheduling involves learning the channel state information of the contending users and scheduling the user with a relatively better channel. Centralized schemes like polling incur a lot of overhead and may not scale well with the number of users. For such schemes, the rate region of the channel and the set of feasible QoS are well known (see e.g., [@neely-etal-powerrouting]). The performance of the system with partial channel state information was studied in [@gopalan-etal-partialcsi]. There is a lot of interest in developing distributed and semi-distributed algorithms for opportunistic scheduling. One popular technique has been to adjust the backoff parameters of the nodes based on their instantaneous channel gain. A number of works have studied the optimal performance and the achievable throughput of such strategies (see e.g., [@qin-berry-multiuser]). In [@qin-berry-osa], the authors propose a splitting algorithm that resolves contention with feedback from the base station. The distributed strategies incur losses due to collisions but are known to very efficient especially for networks with a large number of users. In this work, we are interested in the contention resolution problem of resolving the identity of the user with the highest channel gain. We formulate the contention resolution problem for opportunistic scheduling as identifying a random threshold (channel gain) that separates the best channel from the other samples. We show that the average delay to resolve contention is related to the entropy of the threshold random variable. We illustrate our formulation by studying the opportunistic splitting algorithm [@qin-berry-osa]. We show that OSA is a maximal probability allocation scheme and we conjecture that MPA is an entropy minimizing strategy and a delay minimizing strategy as well. In this work, we have studied opportunistic scheduling for $N$ users with i.i.d. channel gains. We believe that our formulation of contention resolution as a source code can help develop optimal strategies for a variety of other network scenarios as well. Related Literature ------------------ The idea of splitting with ternary feedback was originally proposed for scheduling users in Aloha type networks (see [@gallager-conflictresolution]). In [@arrow-etal-binaryquestions], Arrow et al., study a problem of resolving the user with the highest sample value with binary type questions. The optimal strategy was studied when accurate feedback of the number of contending users involved in every slot was available. The near optimality of greedy strategies (like MPA studied in Section \[sec:osa\_source\_coding\]) was also discussed in [@arrow-etal-binaryquestions]. In [@anantharam-varaiya-conflictresolution], Anantharam and Varaiya prove the optimality of binary type questions to minimize the average delay in [@arrow-etal-binaryquestions]. The performance of binary type questions in the presence of ternary feedback was first reported in [@cohen-etal-ternaryfeedback]. The optimal thresholds were obtained and the relevance to opportunistic scheduling was discussed. In [@qin-berry-osa], Qin and Berry study splitting with ternary feedback for opportunistic scheduling for i.i.d. wireless channel. We have briefly described the algorithm in Section \[sec:osa\]; we motivate our formulation of contention resolution as a source coding problem by studying the opportunistic splitting algorithm presented in [@qin-berry-osa]. Splitting algorithms have been studied for other network and channel scenarios as well. In [@kessler-sidi-splittingnoisy], Kessler and Sidi study splitting algorithms for noisy channel feedback. In [@qin-berry-osafairness], Qin and Berry report the performance of splitting for different notions of fairness. In [@yu-giannakis-sic], Yu and Giannakis study the performance of splitting with successive interference cancellation in a tree algorithm. In this work, we restrict to i.i.d. wireless channel under ideal channel assumptions; our aim is to present an alternate formulation for contention resolution using a source coding framework. There are number of works concerning distributed opportunistic feedback schemes for wireless systems (see e.g., [@tang-etal-opportunisticfeedback]). In [@qin-berry-multiuser], Qin and Berry proposes a channel aware ALOHA and characterizes its performance. In [@patil-deveciana-reducingfeedback], Patil and de Veciana discuss about reducing feedback for opportunistic scheduling to support best effort and real time traffic. In this work, we consider a semi-distributed framework where the base station helps resolve contention with feedback. Outline ------- In Section \[sec:network\_model\], we describe the network model and the opportunistic resolution problem. In Section \[sec:osa\], we briefly describe the opportunistic splitting algorithm from [@qin-berry-osa] and motivate our formulation. In Section \[sec:source\_coding\], we present contention resolution problem for opportunistic scheduling as a source coding problem. In Section \[sec:osa\_source\_coding\], we characterize OSA using a maximal probability allocation scheme and study its performance. In Section \[sec:two\_examples\], we discuss the applicability of our framework for other network scenarios and in Section \[sec:conclusion\], we conclude the paper and discuss future work. Network Model {#sec:network_model} ============= We consider the downlink wireless channel of a single cell of a cellular data network (or of a single cell WLAN in an infrastructure setup). A fixed number of users, $N$, share the slotted wireless channel over time. We assume that the channel gain between the base station and the wireless users is independent and identically distributed with a common continuous distribution $F(\cdot)$. We also assume that the users have knowledge of the common channel distribution $F(\cdot)$ and the number of users in the network, $N$. Let $(H_{n,1}, H_{n,2}, \cdots, H_{n,N})$ represent the vector channel gain of the users in slot $n$. We assume that every user $i$ would know its instantaneous channel gain $H_{n,i}$ at the beginning of every slot, but that information is not available with other users in the network, including the base station. The channel state information $H_{n,i}$ can be made available to the user $i$ by the transmission of a pilot signal by the base station at the beginning of the slot. The base station seeks to identify and schedule the user with the highest channel gain in every slot (opportunistic scheduling), i.e., the base station seeks to schedule $$\arg \max_{\{i=1,\cdots,N\}} \{H_{n,1}, H_{n,2}, \cdots, H_{n,N}\}$$ in slot $n$. Define $X_{n,i} := F(H_{n,i})$, the cumulative distribution value in the slot $n$. Then, the vector $(X_{n,1}, X_{n,2}, \cdots, X_{n,N})$ is i.i.d. Uniform in $[0,1]$ for any channel distribution $F(\cdot)$. Further, the contention resolution problem can equivalently be described as $$\arg \max_{\{ i=1,\cdots,N\}} \{X_{n,1}, X_{n,2}, \cdots, X_{n,N}\}$$ Hence, without loss of generality, we will assume that $F(\cdot)$ is Uniform in $[0,1]$ and consider $(X_{n,1}, X_{n,2}, \cdots, X_{n,N})$ as the channel gain variables. The base station resolves the identity of the user with the highest channel gain by coordinating the contention resolution process and by providing necessary feedback to aid in the resolution. We assume that a time slot comprises of $K$ mini slots, where the mini slots are used to resolve the contention. For example, the users can transmit MAC packets (like RTS/CTS in IEEE 802.11 DCF), possibly with some channel information, to the base station in a minislot and the base station can feed back the state of the contention in that slot. We assume that the base station feeds back the result of the contention within the minislot and the feedback of the base station is received by all the nodes in the network without any error. At the end of the contention process, the user that succeeded in the contention is permitted to transmit data in the remainder of the slot. In this setup, an objective of the base station would be to minimize the average number of minislots required to identify the user with the highest channel gain. Opportunistic Splitting {#sec:osa} ======================= In this section, we briefly describe a contention resolution strategy, opportunistic splitting algorithm (OSA) from [@qin-berry-osa], for a fixed number of users $N$ and for i.i.d. block fading wireless channel. Polling for opportunistic scheduling requires $N$ minislots to identify the user with the highest channel gain. OSA is a distributed medium access control protocol that uses ternary feedback to identify the user with the best channel with a constant overhead. A time slot is assumed to comprise of a maximum of $K$ minislots which are used for contention resolution. In every minislot, OSA describes a continuous range in $[0,1]$ (the sample space of the Uniform random variable), $(y_{min}, y_{max}] \subset [0,1]$; only the user(s) whose channel gain values fall within the range will transmit contention resolution packets in the minislot. At the end of the minislot, every user receives a feedback from the base station of $0$ or $1$ or $e$, indicating if the minislot was idle (no transmission), contained a successful packet transmission or involved an error due to collision, respectively. If the feedback is $1$, the lone transmitter is declared the winner of the contention and is permitted to transmit data for the remaining duration of the slot. If the feedback is $0$ or $e$, then the range is suitably adjusted and the contention resolution process continues until either a success occurs or the time-slot ends. The following pseudo-code describes the OSA algorithm for a fixed number of users $N$ and for i.i.d. channel gain (see [@qin-berry-osa] for more details). In the pseudo-code, $f$ denotes the feedback in a minislot and $k$ is the count of the number of minislots used for contention resolution. Initialize: $y_{low} = 0, y_{min} = 1 - \frac{1}{N}, y_{max} = 1$ Initialize: $f = 0$ and $k = 1$ $f = (0,1,e)$ feedback from $(y_{min}, y_{max}]$ $y_{low} = y_{min}$ $y_{min} = \frac{ (y_{min} + y_{max})}{2}$ $y_{max} = y_{min}$ $y_{min} = \frac{(y_{low} + y_{max})}{2}$ $y_{min} = y_{max} (1 - \frac{1}{N})$ $k = k + 1$ \[rem:osa\] The key features of the opportunistic splitting algorithm are the following. 1. OSA aims to maximize the chances of success in every minislot. For example, with $N$ users independently and Uniformly distributed in $[0,1]$, the probability of success (identifying the user with the best channel) in a minislot with the range $(p,1]$, $N p (1 - p)^{N-1}$, is maximized at $p = 1 - \frac{1}{N}$. In fact, OSA begins contention resolution with the range $(1 - \frac{1}{N}, 1]$. 2. When a collision occurs, OSA assumes that the most likely scenario is that two users are involved in the collision, and hence, it updates the threshold from $(y_{min}, y_{max}]$ to $( \frac{y_{min} + y_{max}}{2}, y_{max}]$ (the optimal strategy if there are only two contending users). OSA is an effective contention resolution strategy with the average number of minislots required to resolve contention known to be less than 2.5070 slots, independent of the number of users and channel gain distribution. Two User Case ------------- In this section, we will discuss in detail the opportunistic splitting algorithm for the two user case. The example will help us motivate the source coding framework described in the Section \[sec:source\_coding\]. Let $N = 2$ and let $(X_1, X_2)$ correspond to the vector channel gain of the two users in a slot. Define $Y_1 := \min\left(X_1, X_2\right)$ and $Y_2 := \max \left( X_1, X_2 \right)$. Then, $(Y_1, Y_2)$ is the ordered pair of the channel gain values where $0 \leq Y_1 \leq Y_2 \leq 1$. OSA initializes with $y_{low} = 0, y_{min} = \frac{1}{2}$ and $y_{max} = 1$. In the first minislot, only the user(s) with $\frac{1}{2} < X_i$ transmit a control packet. A success (a single transmission) happens in the first minislot iff ($X_1 \leq \frac{1}{2} < X_2$) or ($X_2 \leq \frac{1}{2} < X_1$), i.e., a success happens iff $Y_1 \leq \frac{1}{2} < Y_2$. The probability of the event can easily be computed and is equal to $\frac{1}{2}$. **Thus, contention is resolved in the first minislot whenever $0 \leq Y_1 \leq \frac{1}{2} < Y_2 \leq 1$ and the probability of the event is $\frac{1}{2}$; the threshold that resolves the contention successfully for the set $\{ (Y_1,Y_2) : 0 \leq Y_1 \leq \frac{1}{2} < Y_2 \leq 1\}$ is $\frac{1}{2}$ and the base station feeds back a $1$ in this case**. In the first minislot, an error due to collision occurs iff $\frac{1}{2} < Y_1 \leq Y_2$ and the slot is left idle iff $Y_1 \leq Y_2 \leq \frac{1}{2}$. Suppose that the feedback in the first minislot is $e$. Then, OSA updates the variables as $y_{low} = \frac{1}{2}, y_{min} = \frac{3}{4}$ and $y_{max} = 1$. In the second minislot, only the user(s) with $\frac{3}{4} < X_i$ transmit a control packet. A success happens now iff $Y_1 \leq \frac{3}{4} < Y_2$ and the conditional probability of the event (conditioned upon a collision in the first minislot) is $\frac{1}{2}$. **Thus, contention is resolved in the second minislot whenever $\frac{1}{2} < Y_1 \leq \frac{3}{4} < Y_2 \leq 1$ and the probability of the event is $\frac{1}{8}$; the threshold that resolves the contention successfully for the set $\{ (Y_1,Y_2): \frac{1}{2} < Y_1 \leq \frac{3}{4} < Y_2 \leq 1\}$ is $\frac{3}{4}$ and the base station feeds back a $e1$ in the first two minislots**. Events Threshold Feedback Prob ------------------------------------------------------------- --------------- ---------- ---------------- -- $0 \leq Y_1 \leq \frac{1}{2} < Y_2 \leq 1$ $\frac{1}{2}$ 1 $\frac{1}{2}$ $\frac{1}{2} < Y_1 \leq \frac{3}{4} < Y_2 \leq 1$ $\frac{3}{4}$ e1 $\frac{1}{8}$ $0 \leq Y_1 \leq \frac{1}{4} < Y_2 \leq \frac{1}{2}$ $\frac{1}{4}$ 01 $\frac{1}{8}$ $\frac{3}{4} < Y_1 \leq \frac{7}{8} < Y_2 \leq 1$ $\frac{7}{8}$ ee1 $\frac{1}{32}$ $\frac{1}{2} < Y_1 \leq \frac{5}{8} < Y_2 \leq \frac{3}{4}$ $\frac{5}{8}$ e01 $\frac{1}{32}$ $\frac{1}{4} < Y_1 \leq \frac{3}{8} < Y_2 \leq \frac{1}{2}$ $\frac{3}{8}$ 0e1 $\frac{1}{32}$ $0 \leq Y_1 \leq \frac{1}{8} < Y_2 \leq \frac{1}{4}$ $\frac{1}{8}$ 001 $\frac{1}{32}$ : The probability distribution on the threshold/feedback corresponding to OSA for $N = 2$ users.[]{data-label="tab:osa_two_users"} In Table \[tab:osa\_two\_users\], we have listed sets of ordered two tuples along with the threshold ($y_{min}$) for OSA that resolves the set. The feedback from the base station corresponding to the threshold (equivalently, the set) and the probability of the threshold (equivalently, the feedback) is also listed in the table. \[rem:osa\_2\] We make the following observations from the Table \[tab:osa\_two\_users\]. 1. The threshold ($y_{min}$) that resolves $(Y_1,Y_2)$ is always such that $Y_1 \leq y_{min} < Y_2$, i.e., OSA resolves contention by identifying a threshold between the user channel gains. The threshold is fed back to the users in ternary alphabet $(0,1,e)$. The lone user with a channel gain strictly greater than the threshold value fed back by the base station would learn about its successful contention resolution and the other users would refrain from transmitting any further in the slot. 2. The feedback for a threshold $y_{min}$ is, in fact, the binary expansion of $y_{min}$ (when feedback $e$ and feedback $1$ is mapped to $1$ and feedback $0$ is mapped to $0$). The feedback $1$ is equivalent to feedback $e$ followed by an EoC (end of contention) in this case. 3. The thresholds that resolve contention for OSA form a countable set with a valid probability distribution (the probabilities sum up to $1$). 4. The average delay to resolve contention is equal to the average length of the feedback, which is a function of the probability distribution of the threshold random variable. The probability distribution is a function only of the contention resolution algorithm (for the i.i.d. case). An optimal choice of the thresholds can minimize the average description length of the feedback and the delay to resolve contention. In Section \[sec:source\_coding\], we will propose a general framework for contention resolution for opportunistic scheduling motivated by the above observations. A Source Coding Problem {#sec:source_coding} ======================= In this section, we will formulate contention resolution for opportunistic scheduling with ternary feedback as identifying a random threshold (channel gain) that separates the best channel from the other samples. Let $(X_1, X_2, \cdots, X_N)$ correspond to the vector of i.i.d. channel gain values in a slot and let $(Y_1, Y_2, \cdots, Y_N)$ be the ordered N-tuple of channel gain values of the $N$ users such that $0 \leq Y_1 \leq Y_2 \leq \cdots \leq Y_{N-1} \leq Y_N \leq 1$. The base station seeks to identify $\arg \max_{\{i=1,\cdots,N\}} \{ X_1, \cdots, X_N\}$, or, equivalently, $\arg \{Y_N\}$ in the slot. We aim to resolve the contention by identifying a threshold $Y$ such that $Y_{N-1} \leq Y < Y_N$; the base station will feedback the threshold $Y$ using ternary alphabet of $0,1$ and $e$ which aids in resolving the contention. $Y_{N-1}$ and $Y_N$ are random variables, and hence, the threshold $Y$ will also be a random variable. Obviously, the uncertainty in $Y$ would be a measure of the average description length of the threshold/feedback. Let ${\mathsf C}: [0,1] \times [0,1] \rightarrow [0,1]$ be a code (an allocation), which assigns for every 2-tuple $(Y_{N-1},Y_N)$ an element $Y := {\mathsf C}(Y_{N-1},Y_N) \in [0,1]$, such that $Y_{N-1} \leq Y < Y_N$. Let $Y$ have a discrete distribution, i.e., let there exist a set $\Omega_Y = \{y_1, y_2, \cdots \}$ and a set of probabilities $\{p_{y_1}, p_{y_2}, \cdots\}$ such that $\sum_{i=1} p_{y_i} = 1$, and $p_{y_i} := {\mathsf Pr}(Y = y_i)$. Then, the entropy of the random variable $Y$ (equivalently, the code ${\mathsf C}$) is defined as $$- \sum_{i=1}^{\infty} p_{y_i} \log_2(p_{y_i})$$ Clearly, the entropy would approximate the average length of the feedback required for a contention resolution algorithm that resolves a two tuple $(Y_{N-1},Y_N)$ with threshold ${\mathsf C}(Y_{N-1},Y_N)$. The code ${\mathsf C}(,)$ can, in general, take a continuous sample space, all of $[0,1]$ and a useful description of entropy may not be possible in such cases. For continuous distribution $F(\cdot)$, ${\mathsf Pr}(Y_{N-1} \neq Y_N) = 1$, and for every $(Y_{N-1}, Y_N)$ such that $Y_{N-1} \neq Y_N$, there is some rational $Q$ such that $Y_{N-1} \leq Q < Y_N$. Hence, we can always identify a code with a countable sample space for any continuous $F(\cdot)$ and define its entropy. Further, the feedback from the base station for any contention resolution algorithm is a finite sequence in ternary alphabet. Hence, we will always seek a code with a discrete distribution for $Y$. In such a framework, our objective could be to identify the code with the minimum entropy. 1. The maximal probability allocation scheme of OSA (see Section \[sec:osa\_source\_coding\]) provides us a discrete distribution for the random threshold $Y$ with a finite entropy. 2. The ternary description of the threshold does not use all the alphabets completely. For example, the alphabet $1$ appears only at the end of every code word (EoC). Further, the code is non-singular but need not be uniquely decodable as the codes are decoded one at a time. Hence, the entropy of the threshold need not exactly measure the average feedback length (and the average delay). 3. We have assumed that the code ${\mathsf C}$ is a function only of the two tuple $(Y_{N-1}, Y_N)$. For correlated wireless channels and for arbitrary feedback schemes, we may need to consider ${\mathsf C}$ as a function of the N-tuple $(Y_{1},\cdots,Y_N)$. OSA as a Source Code {#sec:osa_source_coding} ==================== The opportunistic splitting algorithm with ternary feedback identifies a threshold $Y$ for every $N$ tuple ($Y_1,\cdots,Y_N)$ such that $Y_{N-1} \leq Y < Y_N$. OSA chooses the thresholds in a minislot such that the probability of success is maximized in the minislot. The following pseudo-code describes the code ${\mathsf C}: (Y_{N-1}, Y_N) \rightarrow Y$ corresponding to OSA. $\Omega_{()} := \{ (Y_{N-1},Y_N) : 0 \leq Y_{N-1} \leq Y_N \leq 1\}$ Initialize $k = 1$ a) Define $y_k$ as [$$y_k := \arg \max_{y \in [0,1]} {\mathsf Pr}(Y_{N-1} \leq y < Y_N | (Y_{N-1}, Y_N) \in \Omega_{()})$$ ]{} b) Define $\Omega_{y_k}$ as [$$\Omega_{y_k} = \{(Y_{N-1},Y_N) : (Y_{N-1},Y_N) \in \Omega_{()}, Y_{N-1} \leq y_k < Y_N\}$$ ]{} c) Update $\Omega_{()}$ as [ $$\Omega_{()} = \Omega_{()} \backslash \Omega_{y_k}$$ ]{} $k = k + 1$ We define $\Omega_{()}$ as the set of all $2$ tuples $(Y_{N-1},Y_N)$. The code begins with identifying a threshold $y_1$ that maximizes the probability of success in $\Omega_{()}$. Every two tuple in $\Omega_{()}$ that contains the threshold $y_1$ is assigned to be resolved by the threshold; we define the above set as $\Omega_{y_1}$, the set resolved by the threshold $y_1$. The set $\Omega_{y_1}$ is now removed from $\Omega_{()}$ and the procedure continues. Define $\Omega_Y$ as the set of all thresholds defined by the above pseudo-code. From the construction of the above code, we see that, $${\mathsf C}(Y_{N-1},Y_N) = \arg \max_{y \in \Omega_Y} \{ Pr(\Omega_{y}) : Y_{N-1} \leq y < Y_N\}$$ For this reason, we call OSA as the maximal probability allocation code (MPA). The following theorem from [@cohen-etal-ternaryfeedback] identifies the exact threshold of OSA for a given range $(y_{min},y_{max}]$. Given $N$ users and thresholds $(y_{min}, y_{max}]$ (i.e., $y_{min} \leq Y_{N-1} \leq Y_N \leq y_{max}$), the $y$ that maximizes the probability of success in the interval $(y_{min}, y_{max}]$ is the unique stationary point of $(y_{max} - y) (y^{N-1} - y_{min}^{N-1})$. **Remarks** 1. For any $N$, and with $y_{min} = 0, y_{max} = 1$, the above expression becomes, $(1 - y)(y^{N-1}) = y^{N-1} - y^N$. The expression is maximized at $y = 1 - \frac{1}{N}$. Hence, for any $N$, $y_1 = 1 - \frac{1}{N}$. 2. As an example, for $N = 2$, repeating the above procedure will yield us $y_1 = 0.5, y_2 = 0.75, y_3 = 0.25, y_4 = 0.875, \cdots$. Note that the above values are in fact the thresholds reported in Table \[tab:osa\_two\_users\]. 3. In Remark \[rem:osa\_2\], for the $N = 2$ case, we noticed that the feedback from the base station corresponding to a threshold can be viewed as the binary representation of the threshold itself. For general $N$, the feedback from the base station can still be viewed as the binary representation of the threshold, however, with the weights corresponding to a position computed from the thresholds $\{ y_k \}$ obtained from the pseudo-code. For example, the weight of the first position will be equal to $y_1$. ![Plot of the average delay performance of OSA and MPA as a function of the number of users $N$. We have also plotted the entropy of the threshold random variable corresponding to MPA.[]{data-label="fig:osa_vs_mpa"}](osa_vs_mpa.pdf) In Figure \[fig:osa\_vs\_mpa\], we plot the average delay performance of OSA (as described in Section \[sec:osa\]) and the maximal probability allocation code. As expected, the performance of OSA and MPA are similar and in fact, MPA performs better than OSA as it identifies the optimal thresholds without any approximations (see Remark \[rem:osa\]). We have also plotted in the Figure \[fig:osa\_vs\_mpa\], the entropy of the maximal probability allocation code in bits. As expected, the entropy of the random variable reflects the average delay performance of the contention resolution algorithm as a function of $N$ very well. Entropy Minimization -------------------- Entropy is a concave function of the distribution. The maximal probability allocation code identifies a local minima in the space of probability distributions. From limited numerical work (not reported in this paper), we conjecture that the maximal probability allocation code is a globally entropy minimizing strategy as well. The following theorem proves the optimality of MPA for the $N = 2$ case. MPA is a delay minimizing strategy and an entropy minimizing strategy for $N = 2$ case. Let $(x, 1]$ be the contention range in the first minislot for the delay minimizing strategy. Conditioning on the first minislot, the optimal average delay $D$ can be written as $$D = 2 x (1 - x) + (1 + D) (1 - 2 x (1 - x))$$ The optimal solution of $D$ is $2$ and is obtained at $x = \frac{1}{2}$. We note from Figure \[fig:osa\_vs\_mpa\] that the average delay of MPA is $2$ and hence, MPA is a delay optimal strategy. Let $\{ p_i \}$ be the discrete distribution that minimizes the entropy of the code for $N = 2$ case. We will define $E(\{ p_i \}) := - \sum_i p_i \log_2(pi)$ as the minimum entropy. Then, conditioning on the first minislot (as before), we have, $$E(\{ p_i \}) = -2 x (1-x) \log(2 x (1-x)) + E(\{ x^2 p_i\}) + E(\{ (1-x)^2 p_i \})$$ where, $$E(\{a p_i\}) = - \sum_i a p_i \log_2(a p_i)$$ Substituting in the above expression and simplifying it, we have, [$$E(\{ p_i \}) = \frac{ - 2 x (1 - x) \log_2(2 x (1 - x)) - x^2 \log(x^2) - (1-x)^2 \log((1-x)^2)}{2 x (1 - x)}$$ ]{} The above expression is minimized at $x = \frac{1}{2}$ and the minimum entropy is $3$. From Figure \[fig:osa\_vs\_mpa\], we note that the entropy of the threshold random variable for MPA is $3$ and hence, MPA is an entropy minimizing strategy as well. Two Examples {#sec:two_examples} ============ In this section, we discuss contention resolution for two different channel scenarios, a constant channel and a correlated channel. We compare the performance of OSA/MPA with the source-coding framework and illustrate the generality of our proposed technique. Constant Channel ---------------- We consider a downlink wireless channel with $3$ users. We assume that the channel gain is a constant, say $1$ unit, for all users and for all time slots. The objective of the contention resolution algorithm is to identify a user from the set of $3$ users (akin to a distributed medium access problem). Suppose that the users pick a uniform random variable, $X_i$, in $[0,1]$ independent of the other users. Then, OSA can be used to resolve contention among the $3$ users by identifying the user with the largest value of $X_i$; this is a popular strategy to apply OSA for discrete channel distributions. The average number of slots required to resolve contention using OSA is then $2.12$ slots (obtained from simulations). The OSA, in every slot, attempts to identify a $y$ such that the probability of a unique user in the interval $(y,1]$ is maximized. Here, in this example, we note that it is more appropriate to identify a $y$ that maximizes the probability of success either in $(y,1]$ or in $[0,y)$. The following algorithm is a contention resolution strategy optimized for this problem. Initialize: $y_{low} = 0, y_{min} = 1 - \frac{1}{3}, y_{max} = 1$ Initialize: $f = 0, k = 1$ $f = (0,1,e)$ feedback for interval $(y_{min},y_{max}]$ $k = k + 1$ $f = (0,1,e)$ feedback from interval $[y_{low}, y_{min})$ $y_{low} = y_{min}$ $y_{min} = y_{min} + (y_{max} - y_{min}) \times (1 - \frac{1}{3})$ $y_{max} = y_{min}$ $y_{min} = y_{min} + (y_{max} - y_{low}) \times (1 - \frac{1}{3})$ $k = k + 1$ ($f \neq 1$) Using simulations, we observe that the average effort needed to resolve contention is $1.89$ slots much less than the $2.12$ slots required by OSA. The proposed algorithm makes use of the fact that, in the event of a collision, the probability that two users are involved is significantly higher than the probability that three users are involved in the collision. The contention resolution problem was formulated as identifying a random threshold $Y$ between $Y_1$ and $Y_2$ ($Y_1 < Y \leq Y_2$) or between $Y_2$ and $Y_3$ ($Y_2 \leq Y < Y_3$). The entropy of the proposed strategy was observed to be strictly smaller than the entropy of the maximal probability allocation scheme of OSA. Correlated Channel ------------------ We consider a wireless downlink channel with $N = 2$ users. We assume that the wireless channel of the two users is correlated with the sample space, $\Omega_H = \{ (4,2), (4,6), (8,6), (8,10), (12,10), (12,14), (16,14)\}$ and with the joint probabilities $p_{H} = \{ \frac{1}{7} - 6 \epsilon, \frac{1}{7} - 5 \epsilon, \frac{1}{7} - 4 \epsilon, \frac{1}{7} - 3 \epsilon, \frac{1}{7} - 2 \epsilon, \frac{1}{7} - \epsilon, \frac{1}{7} + 21 \epsilon \}$, where $0 < \epsilon << 1$. OSA would maximize the probability of success in every minislot and hence, would consider the thresholds in the following sequence (if we restrict to integer thresholds) $15, 13, 11, 9, 7, 5$ and $3$. The average number of minislots required to resolve contention with OSA/MPA is $\frac{1}{7} (1 + 2 + \cdots + 6 + 6) \approx \frac{27}{7} \approx 4$. In general, if there are $k$ channel states, then the average number of slots required to resolve contention is approximately $\frac{k}{2}$. Consider the following alternative strategy in resolving contention. In the first minislot, we consider the threshold value $9$ to resolve contention. If a collision occurs in the first minislot, then the next threshold would be $13$ for the second minislot and in the event of an idle first minislot, the next threshold would be set to $7$ for the second minislot. Similarly, if there is collision in the first two minislots, then, the threshold would be set to $15$ for the third minislot and so on. If there is a unique user attempting in a minislot, the contention resolution algorithm stops. The average number of minislots required to resolve contention with this strategy is approximately $3$; in general, if there are $k$ channel states, then the average number of minislots required would be $\log(k)$. We note that, for large $k$, the above strategy is strictly optimal than the OSA. The contention resolution problem can be formulated as identifying a random threshold $Y$ such that $Y_1 \leq Y < Y_2$. Clearly, the minimum entropy for the wireless channel is approximately $\log(k)$ and is equal to the average number of minislots required to resolve contention. The two examples clearly illustrate that a maximal probability strategy like the OSA is not optimal for all channel scenarios. Also, the source-coding technique could provide us a way to identify the optimal contention resolution strategy under general channel scenarios as well. Conclusion and Future Work {#sec:conclusion} ========================== In this paper, we have modeled contention resolution for opportunistic scheduling as a source-coding problem. The entropy of a certain random variable is seen to approximate the average number of slots required to resolve contention. We characterized OSA as a maximal probability allocation scheme and obtained the thresholds for contention resolution (in OSA) from its source code. We note that MPA provides us a local optima, and we conjecture that MPA is globally optimal as well (for i.i.d. channel conditions). We believe that the information theoretic view point can be used to develop contention resolution algorithms for a variety of other network scenarios as well (e.g., partial network information, limited channel feedback). 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--- abstract: 'To date, the Burst Alert Telescope (BAT) onboard [*Swift*]{} has detected $\sim$ 1000 gamma-ray bursts (GRBs), of which $\sim$ 360 GRBs have redshift measurements, ranging from $z=0.03$ to $z=9.38$. We present the analyses of the BAT-detected GRBs for the past $\sim$ 11 years up through GRB151027B. We report summaries of both the temporal and spectral analyses of the GRB characteristics using event data (i.e., data for each photon within approximately 250 s before and 950 s after the BAT trigger time), and discuss the instrumental sensitivity and selection effects of GRB detections. We also explore the GRB properties with redshift when possible. The result summaries and data products are available at http://swift.gsfc.nasa.gov/results/batgrbcat/index.html. In addition, we perform searches for GRB emissions before or after the event data using the BAT survey data. We estimate the false detection rate to be only one false detection in this sample. There are 15 ultra-long GRBs ($\sim 2\%$ of the BAT GRBs) in this search with confirmed emission beyond $\sim$ 1000 s of event data, and only two GRBs (GRB100316D and GRB101024A) with detections in the survey data prior to the starting of event data.' author: - 'Amy Lien$^{1,2,}$, Takanori Sakamoto$^{3,}$, Scott D. Barthelmy$^4$, Wayne H. Baumgartner$^{1,2}$, John K. Cannizzo$^{1,2}$, Kevin Chen$^5$, Nicholas R. Collins$^{1,2}$, Jay R. Cummings$^4$, Neil Gehrels$^4$, Hans A. Krimm$^{1,6}$, Craig. B. Markwardt$^4$, David M. Palmer$^7$, Michael Stamatikos$^8$, Eleonora Troja$^{1,9}$, T. N. Ukwatta$^{7}$' bibliography: - 'ref.bib' title: 'The Third *Swift* Burst Alert Telescope Gamma-Ray Burst Catalog' --- Introduction {#sect:intro} ============ Gamma-ray bursts (GRBs) are one of the most energetic explosions in the universe, and are important in many aspects of astrophysics and cosmology. The remarkable amount of energy released by GRBs in such a short time scale provides a unique opportunity to study physics in an extreme environment, and also challenges the physical models of the progenitors. Both the observational evidence and theoretical studies connect long GRBs (bursts with duration longer than $\sim 2$ s) with the death of massive stars [see e.g., @Woosley06; @Fryer07; @Gehrels12; @Kumar15 and references therein]. On the other hand, the origin of short bursts (durations $\lesssim 2$ s) remains mysterious. Current studies suggest that short GRBs are likely related to compact-object mergers and thus they are one of the candidate sources of gravitational waves [see e.g., @Eichler89; @Nakar07; @Berger14 and reference therein], one of which has recently been detected directly by LIGO [@Abbott16]. Moreover, GRBs are one of the few astrophysical objects that can be directly detected out to very high redshift ($z \gtrsim 8$) due to their extraordinary brightness, and thus they provide a valuable tool to study the early universe. [*Swift*]{}, a multi-wavelength telescope dedicated to GRB studies, was launched on Nov. 20, 2004 [@Gehrels04]. Over the past $\sim 11$ years, the Burst Alert Telescope (BAT) onboard [*Swift*]{} has detected $\sim 1000$ GRBs. The unique ability of [*Swift*]{} to observe a large portion of the sky, promptly localize the burst, and rapidly downlink and circulate the detection notice has enabled fast multi-wavelength follow-up observations, and vastly enhanced the scientific outcome. The BAT is one of the three telescopes onboard [*Swift*]{}, and is capable of detecting GRBs and localizing a burst to within a few arcmin. When the BAT detects a GRB, [*Swift*]{} will slew to the GRB position and observe the burst with the X-ray Telescope (XRT) and the UV-Optical Telescope (UVOT) onboard [*Swift*]{}, which can further refine the localization to $\lesssim$ arcsec. The BAT is composed of a detector plane that has 32,768 CdZnTe (CZT) detectors, and a coded-aperture mask that has $\sim 52,000$ lead tiles. The coded-mask technique is useful in X-ray and gamma-ray astronomy to obtain a large field of view while maintaining imaging capability. The basic idea is that each point source casts a unique shadow through the coded-aperture mask onto the detector plane, and thus one can re-construct the source image/position by deconvolving the illuminated pattern on the detector plane and the mask pattern. The BAT has a field of view of 2.2 sr when it is $> 10\%$ coded, and an energy range of 14-150 keV for imaging or up to 350 keV with no position information. Details of the BAT instruments can be found in @Barthelmy05 and the first BAT GRB catalog [@Sakamoto08]. The BAT adopts two main trigger methods for detecting GRBs: (1) the rate trigger criteria, which search for GRBs based on count rate increases in the light curves, and (2) the image trigger criteria, which discover bursts based on images created with different time intervals ($\gtrsim$ minute). In addition, sometimes a burst that was not triggered on-board can be recovered later by ground analysis. We refer to these events as ground-detected GRBs. These ground-detected bursts usually happen when the BAT is not capable of triggering bursts, such as during spacecraft slews or close to the South Atlantic Anomaly (SAA, which is an area that contains a high level of background high-energy particles). Moreover, a burst can occur at a location that is highly off-axis relative to the BAT detector plane and hence only generate weak signals. The search and discovery of ground-detected GRBs are usually motivated by detections from other instruments, such as [*Fermi*]{}, INTEGRAL, and MAXI. Several catalogs related to the BAT-detected GRBs have been published, including the first and second BAT GRB catalog from the [*Swift*]{}/BAT team [@Sakamoto08; @Sakamoto11 which will hereafter be referred as the BAT1 and BAT2 catalog, respectively]. Some catalogs with selected BAT GRBs for specific usages have also been presented, such as the catalog composed by @Salvaterra12 (also known as the BAT6 catalog) that selects bright bursts detected by BAT with optimal conditions for ground follow-up observations, in order to construct a GRB sample with redshift completeness. Furthermore, many online GRB tables are available. Those that are related to the BAT data include (1) the Swift GRB table[^1] compiled by J. D. Myers using information from GCN circulars, (2) an online GRB catalog[^2] maintained by Tilan Ukwatta that contains GRBs from [*Swift*]{}, (3) the [*Swift*]{} Burst Analyser[^3] maintained by Phil Evans, which includes plots of both the BAT and XRT light curves at selected energy bands [@Evans10; @Evans09; @Evans07], (4) the “Swiftgrb database”[^4] produced by Padgett et al., which is completed through December of 2012 and includes data product for BAT and XRT, and (5) an online repository[^5] generated by Nathaniel Butler that includes the XRT and BAT light curves, spectra, and GRB redshifts [@Butler07; @Butler10]. Moreover, the “GRB Online Index (GRBOX)[^6]” maintained by Daniel Perley compiles a list of GRB with redshift measurements and information of follow-up observations. The webpage maintained by Jochen Greiner[^7] also presents a comprehensive information of GRB localizations and redshifts. In this catalog, we update the results in the BAT2 catalog to include the GRBs detected by BAT after 2009. We include all the bursts through the 1000th [*Swift*]{} GRB, GRB151027B, which was officially announced by the [*Swift*]{} team[^8]. This 1000th burst is counted based on the list in the Swift GRB table compiled by J. D. Myers$^{\ref{Swift_table}}$, which is slightly different than the list we compile here in the third BAT GRB catalog. The Swift GRB table lists the GRBs that were first reported by ${\it Swift}$ in the GCNs. In the third BAT GRB catalog, we include all GRBs that were reported being seen by BAT (either triggered onboard or found by ground analyses, some of which may be motivated by detections from other instruments). For those GRBs without an XRT/UVOT afterglow detection, we will mark them as “questionable detections” if the signal-to-noise ratio is lower than 7 (see Section \[sect:event\_standard\] and \[sect:ground\_GRBs\] for details of how the signal-to-noise ratio is determined). To make sure the analyses for the new and old bursts are consistent, we reanalyze all the bursts in the BAT2 catalog as well, using the same up-to-date software. The main GRB characteristics (e.g., burst durations, spectral fits) are acquired from analyses using the event data (sometime also called the event-by-event data), which record information of individual photons and usually cover ranges between $\sim$ 250 s before and $\sim 950$ s after the BAT trigger time. For the event data analyses, we follow the general pipelines adopted in the BAT2 catalog. We report some extra information in this catalog regarding the GRB observation status, such as the partial coding fraction and the trigger method (rate or image trigger). We also include summaries and discussions of the BAT observational constraints (e.g., the Sun/Moon constraints, fractions of time when BAT is able to trigger a burst, changes in the number of active detectors). Furthermore, in addition to studies using event data, we perform further searches for possible extended emission beyond the event data range using the BAT survey data. The survey data are binned in $\sim$ 5-min intervals, and cover time periods that do not have event data. This paper is organized as follows: Section \[sect:BAT\_status\] reports the update of the BAT status related to GRB observations, which includes status of the in-orbit calibration using the Crab observations, and a summary of the BAT observing time. Section \[sect:event\_data\_analysis\] presents the method of the data analysis for the event data. Section \[sect:event\_results\] reports the results of the event data analyses and includes discussions of observed burst properties and rest-frame characteristics for those GRBs with redshift measurements. Section \[sect:survey\_data\_analysis\] describes the pipeline for analyzing the survey data and also discusses the false-detection rate of the survey data in order to search for weak emissions beyond the event data range. Section \[sect:survey\_results\] summarizes the results from the survey data search. The overall summary is presented in Section \[sect:summary\]. Updates of the BAT status {#sect:BAT_status} ========================= Status of the in-orbit calibrations ----------------------------------- Each year, the [*Swift*]{} team schedules special observations of the Crab Nebula to perform an on-orbit calibration of the BAT for both energy and position measurements. During the calibration observations, the BAT observes the Crab nebula at different incident angles to check that the measurements show consistent results. In 2015, five observations with different incident angles were performed: (1) on-axis, (2) off-axis with $\theta = 30$ deg, $\phi = -90$ deg. (3) off-axis with $\theta = 30$ deg, $\phi = 90$ deg, (4) off-axis with $\theta = 45$ deg, $\phi = 0$ deg, and (5) off-axis with $\theta = 45$ deg, $\phi = -180$ deg. $\theta$ is the polar angle measured from the BAT pointing direction, and $\phi$ is the azimuth angle. ![ The best fit Crab photon index vs. $\theta$ angle. The dashed line marks the assumed Crab photon index of -2.15 [@Rothschild98; @Jung89]. []{data-label="fig:Crab_alpha_theta"}](theta_phindex_2004_2015.pdf){width="100.00000%"} ![ The best fit Crab flux in 15-150 keV vs. $\theta$ angle. The dashed line marks the assumed Crab flux of $2.11 \times 10^{-8} \ \rm erg \ cm^{-2} \ s^{-1}$ [@Rothschild98; @Jung89]. []{data-label="fig:Crab_flux_theta"}](theta_flux_2004_2015.pdf){width="100.00000%"} Figure \[fig:Crab\_alpha\_theta\] and \[fig:Crab\_flux\_theta\] show the results of the spectral fits with different incident angles from year 2005 to 2015. The Crab spectrum is fitted with a simple power-law model (see definition in Eq. \[eq:PL\]). The photon indices of the simple power-law model $\alpha_{PL}$ are plotted in Fig. \[fig:Crab\_alpha\_theta\], while the fluxes are presented in Fig. \[fig:Crab\_flux\_theta\]. Results show that the photon index and the flux measured at different locations on the BAT detector plane can vary by up to $\sim \pm 5\%$ and $\sim \pm 10\%$, respectively, from the canonical values of Crab photon index of -2.15 and flux of $2.11 \times 10^{-8} \ \rm erg \ cm^{-2} \ s^{-1}$ [@Rothschild98; @Jung89]. @Wilson-Hodge11 suggests that the flux of the Crab Nebula can fluctuate on a time scale of months to years, with the value changes as much as $\sim10\%$ in the BAT energy range from 2008 to 2010. However, as seen in Fig. \[fig:Crab\_flux\_theta\], the systematic errors for the flux measurements at large incident angles can be as large as $\sim 10\%$. This is because of the unknown systematic uncertainties included in the BAT energy response function. Thus, it can be difficult to place tight constraints for flux variations less than $\sim 10\%$, if the spectral analysis is involved in the BAT data. Note that the BAT Crab light curve presented in @Wilson-Hodge11 is based on the survey data process which extracts the counts directly from the sky images. The BAT calibration data base is last updated in 2009 [@Sakamoto11]. Figure \[fig:Crab\_position\] shows the differences between the true Crab position (${\rm RA}=83.633$, ${\rm DEC} = 22.014$) and the Crab positions calculated using observations from the five different incident angles taken in 2015. Results show that the Crab position measurements can change up to $\sim 2$ arcmin (with respect to the assumed “true” location) when measuring with different incident locations. However, 90% of the measurements are within 0.93 arcmin from the “true” location. ![ Crab position difference from “true” Crab position at ${\rm RA}=83.633$ deg and ${\rm DEC}=22.014$ deg. The dashed circle with a radius of 0.93 arcmin encloses 90% of the measurements. []{data-label="fig:Crab_position"}](position_diff_2015.pdf){width="60.00000%"} Summary of the BAT observing time --------------------------------- Based on the BAT log files, BAT spends $\sim 78\%$ of the time performing observations and searching for GRBs. Figure \[fig:BAT\_goodtime\] shows the fraction of the time when BAT was capable of triggering bursts from year 2005 to 2015. BAT is unable to trigger a burst mainly due to spacecraft slewing and when the spacecraft passes through the SAA. Fig. \[fig:BAT\_slew\_SAA\_time\] shows the fraction of time during spacecraft slews and SAA. There are $\sim 12\%$ of spacecraft slew time. This fraction gradually increases with time as [*Swift*]{} observes more and more Target-of-Opportunity (ToO) targets. Record shows that the spacecraft spends $\sim 9\%$ of the time in SAA. Note that this is the SAA time as defined for BAT operations, which is determined based on instant count rate and backlog in the ring buffer. XRT and UVOT adopt a more strict criteria for SAA using the location, and thus have a larger SAA time fraction in general. As shown in the figure, the fraction of the BAT SAA time decreases slightly from 2005 to 2015. This is because the solar activity increases during these years, which results in a slightly lower particle density in the SAA region. Since BAT uses the count rate to define the SAA time, the fraction of SAA time decreases as fewer cosmic-ray particles appear in the SAA region. ![ Fraction of the BAT observing time (i.e., the time when BAT is able to trigger bursts) as a function of year. The fraction remains very stable at $\sim 0.78$. []{data-label="fig:BAT_goodtime"}](BAT_goodtime_without2004.pdf){width="100.00000%"} ![ Fraction of the BAT slew time (red bars), and the time in SAA (black bars), as a function of year. Note that this is the SAA time for BAT only, which determines the SAA time based on instant count rate and backlog in the ring buffer. XRT and UVOT adopts a more strict criteria for SAA using the sky location, and thus have a larger SAA time fraction in general. []{data-label="fig:BAT_slew_SAA_time"}](BAT_slew_SAA_time_without2004.pdf){width="100.00000%"} Data Analysis for BAT event data {#sect:event_data_analysis} ================================ Standard analysis {#sect:event_standard} ----------------- All the BAT event data used in this analysis are downloaded from HEASARC[^9]. We use the standard BAT software (HEASOFT 6.15[^10]) and the latest calibration database (CALDB[^11]) to perform analysis for event data. Specifically, we use the script [*bateconvert*]{} for energy calibration, and [*batgrbproduct*]{}[^12] to perform a series of standard analyses of the event data, which includes filtering out hot pixels of the detectors, occultation time periods, refined-position analysis, duration estimation, making light curves with different time segments and bin sizes, and generating spectra. We adopt the default options of [*batgrbproduct*]{}, except the minimum partial coding fraction (pcodethresh), which is set to 0.05 instead of 0.0. The burst durations estimated by [*batgrbproduct*]{} include $T_{100}$, $T_{90}$, and $T_{50}$, which correspond to the durations that contain 100%, 90%, and 50% of the burst emission, respectively. Specifically, the start and end times of $T_{90}$ in this standard pipeline are defined as the times when the fraction of photons in the accumulated light curve reaches 5% and 95%[^13]. Similarly, the start and end time of $T_{50}$ is defined as the times when the accumulated light curve reaches 25% and 75%. These definitions for $T_{90}$ and $T_{50}$ are commonly adopted for quantifying burst durations by other teams, including BATSE, [*BeppoSAX*]{}, and [*Fermi*]{} [@Koshut96; @Paciesas99; @Frontera09; @vonKienlin14]. In this paper, we follow the convention and use the $T_{90} = 2$ s as the separation between the long and short GRBs categories [@Kouveliotou93]. Specifically, the bursts that are referred to as short GRBs in this paper have $T_{90} \leq 2$ s, without taking into account of the uncertainty in $T_{90}$. The burst refined positions are generally also found by [*batgrbproduct*]{} by running [*batcelldetect*]{} on the image with the burst emission. This image will end before the spacecraft slews if $T_{100}$ lasts beyond the slew time. These refined positions are not used in any of the further analyses, such as calculating the mask-weighted light curve and spectrum. However, we do use the signal-to-noise ratio reported with these refined positions to determine whether the burst could be a questionable detection. For a few dozen bursts, the signal-to-noise ratio associated with the refined positions found by this auto pipeline is lower than 7 (the typical image-trigger threshold). However, most of these cases are due to a long quiescent period of the burst emission before the spacecraft slews. We thus rerun the search for detections for these bursts using images created with the time interval and energy range determined by the flight software. If the signal-to-noise ratio is still below 7 and there is not an accompanying an XRT afterglows, we will mark it as a “tentative detection” in Table \[tab:summary\_general\]. The readers are advised to treat these bursts with special caution. For spectral analysis, we use the commonly-adopted X-ray fitting package, XSPEC[^14]. When the spectrum covers time periods including spacecraft slew time, we generate multiple response files in this period in order to create an “average” response file for the whole period. [*Swift*]{} slews at a rate of $\sim$ 1 deg per second [@Markwardt07] and hence the telescope motion can be safely ignored within 5 seconds [@Sakamoto11_calibrate]. Therefore, we create a response file for each five seconds during slew time, and a response file for each time segment when the spacecraft is settled. We create an average response file for the whole time period using the HEASARC tool [*addrmf*]{}, with weighting factors equal to the fraction of photon counts in the specific time periods of each response file. Following the BAT2 catalog, we fit the GRB spectra with two different models: simple power law (PL) and cutoff power law (CPL). The simple power-law model is described by the following equation, \[eq:PL\] f(E) = K\^[PL]{}\_[50]{} ( )\^[\^[PL]{}]{}, where f(E) is the photon flux at energy E, $\alpha^{\rm PL}$ is the PL index, and ${K^{PL}_{50} }$ is the normalization factor at 50 keV, with units of photons $\rm cm^{-2} \ s^{-1} \ keV^{-1}$. The cutoff power-law model is expressed as, \[eq:CPL\] f(E) = K\^[CPL]{}\_[50]{} ( )\^[\^[CPL]{}]{} (), where $\alpha^{\rm CPL}$ is the CPL index, ${K^{CPL}_{50} }$ is the normalization factor at 50 keV, with units of photons $\rm cm^{-2} \ s^{-1} \ keV^{-1}$, and $E_{\rm peak}$ is the peak energy in the $\nu F_\nu$ (i.e., $E^2 f(E)$) spectrum, where $F_\nu = E f(E)$ is the energy flux density. The BAT spectra produced using the mask-weighting techniques have Gaussian statistics [@Markwardt07]. Therefore, we use the “fit” command in XSPEC with the default “statistic chi” option, which finds a fit with the maximum likelihood for Gaussian data (in other words, finds a fit with a minimum $\chi^2$; see detailed descriptions in the Appendix B of the XSPEC manual[^15]) We use the “error” command in XSPEC to estimate the 90% confidence region for each parameter. This command changes the assigned parameter values until it finds a value that gives a fit with statistics differing by a number $\Delta$ from the best-fit statistics. When the data follow a Gaussian distribution, $\Delta$ follows a $\chi^2$ distribution. In our case, we use the default $\Delta = 2.706$, which is equivalent to the 90% confidence region in the $\chi^2$ distribution. Based on the XSPEC manual, the “error” command is one of the more reliable and recommended methods for constraining uncertainties, and it is not as computationally expensive as the Monte Carlo techniques (see detailed descriptions in the Appendix B of the XSPEC manual^\[note1\]^). However, in order to use the “error” command to estimate the uncertainties of the implicit parameters, such as the fluxes, one would need to re-write the function to make flux one of the parameters (instead of the normalization factor K), and re-do the fit. Therefore, for the flux error estimation, we use the “cflux” and “cpflux” command in XSPEC, which performs this conversion for energy flux and photon flux, respectively. Ideally, if XSPEC does find the global minimum, all these fits should find the same solution. However, if XSPEC only finds a local minimum, fitting using the different forms of the same function can converge to different solutions. Thus, we cross check the fits that use different forms of the same functions and only accept the fits if they find solutions that are consistent with each other (i.e., the fitted values have overlapping uncertainty ranges). Note that in order to set up an automatic pipeline for all the bursts, we accept the solutions found by the “fit” command after about 100 iterations. We cannot rule out the possibility that this is a local minimum. Sometimes XSPEC finds better fits when going through the search with the “error” command. However, we always discard these new fits to make sure the uncertainties for all the parameters are estimate based on the same best fit, and to prevent XSPEC from going into an infinite loop if it keeps finding new fits when constraining parameter errors. As mentioned in @Sakamoto11, most of the GRB spectra in the BAT energy range can be well-fitted by the simple PL model, and show no significant improvement in their fit when changing to the CPL model. We adopt the same criteria as the one in @Sakamoto11 and determine when CPL is a better fit when $\Delta \chi^2 \equiv \chi^2_{\rm PL} - \chi^2_{\rm CPL} > 6$ (and when there is no problem in the CPL fit; see the criteria for acceptable fits below). In the following list, we summarize the criteria we use to decide whether a fit is acceptable: - All the parameters (normalization factor, photon index, photon and energy fluxes for different energy bands, and $E_{\rm peak}$ for the CPL fit) and their errors are constrained. - The parameters (photon index and $E_{\rm peak}$) found by fitting different forms of the same functions are consistent with each other (i.e., the 1-sigma uncertainty regions overlap). We compare the fits that are used to constrain the photon and energy fluxes in different energy bands with the original fit for constraining the normalization factor and photon index. - The normalization factors and fluxes for different energy bands are not consistent with zero (i.e., the lower limit found is greater than zero). - The lower limit is not larger than the upper limit (this should always be satisfied in principle, but we place this criteria regardless just in case). - The “probability of the null hypothesis” from the resulting XSPEC fit (estimated based on the $\chi^2$ distribution) needs to be larger than 0.1. That is, the null hypothesis needs to be consistent with the data within 90% of confidence range. - If $\Delta \chi^2 \equiv \chi^2_{\rm PL} - \chi^2_{\rm CPL} > 6$ and the fit satisfies all the criteria above, we adopt the CPL fit in the following discussions and mark these bursts as better-fitted by the CPL model in the summary tables (Appendix \[sect:appendix\_tables\]). Sect \[sect:unaccept\_spec\] contains further discussions regarding the bursts with unacceptable spectra under these criteria. Note that spectral fits from both the simple PL and CPL are presented for all GRBs in the summary tables (Appendix \[sect:appendix\_tables\]) regardless of whether or not they satisfy these criteria. The names of GRBs with their best-fit models are given in separate lists in Section \[sect:appendix\_tables\]. We perform spectral analyses for the following four types of spectra: (1) Time-averaged spectra, which are the spectra created using the $T_{100}$ duration[^16], (2) The 1-s peak spectra, which cover the 1-s peak time selected by [*battblocks*]{}, (3) The time-resolved spectra, which are a series of spectra for each burst created based on the sub-time-segments within $T_{100}$ that are selected by [*battblocks*]{}. Ideally, these sub-time-segments pick out the sub-structure in the light curve variations, but the selections are not always perfect. (4) The 20-ms peak spectra, which cover the 20-ms peak time selected by [*battblocks*]{} using the 4-ms binned light curve. This is not one of the standard products included by [*batgrbproduct*]{}, however, we create this additional peak time and spectral analyses for those extremely short GRBs in particular. Due to the extremely short time interval, we generate a spectrum with only 10 energy bins (equally spaced in the log-scale from 14.0 to 149.99 keV), following the pipeline in the BAT2 catalog. Ground-detected GRBs {#sect:ground_GRBs} -------------------- There are 81 GRBs found in ground analysis, including 25 discovered during spacecraft slews. That is, these GRBs did not pass the on-board trigger criteria, but were identified by miscellaneous ground processing, such as searching in the BAT data for those GRBs triggered by other spacecrafts, and/or searching for possible GRBs during spacecraft slews (since BAT cannot trigger during this time). Most of these bursts are found in either the “failed event data” or the “slew event data”. The “failed event data” are $\sim 10$-second-long event data that are downlinked when a burst passes the first-stage detection threshold (i.e., the rate-trigger criteria), but failed to pass the second-stage trigger criteria (i.e., the image-trigger criteria; see @Barthelmy05 for more information of the two-stage trigger criteria for BAT). The “slew event data” are event data collected during spacecraft slews. GRB150407A, GRB140909A, GRB110604A, GRB070125, and GRB060123 were only found manually in images created by the flight software, and no event data are available. When at least some event data exist for a ground-detected burst, we re-analyze the burst using standard BAT data analysis scripts, [*bateconvert*]{} and [*batgrbproduct*]{}, as mentioned in Section \[sect:event\_standard\]. To perform the analysis, [*batgrbproduct*]{} requires some information from the prompt data collected through the Tracking and Data Relay Satellite System (TDRSS), which includes the burst observation ID, trigger number, trigger time, the RA and DEC of the burst found in the onboard analysis, background duration used for the trigger, and information of whether this is a rate or image trigger. For bursts triggered onboard, the TDRSS data are downlinked to the ground within seconds to minutes of the trigger time. However, only limited data are transfered due to the downlink bandwidth. The complete data are downlinked to ground stations $\sim$ hours later. Since ground-detected GRBs are not triggered onboard, they do not have TDRSS data. We thus manually create fake TDRSS messages that contain the relevant information required by [*batgrbproduct*]{}. Bursts that have “failed event data” are assigned with unique trigger numbers because they have passed the rate trigger criterion. We use these trigger numbers in the fake TDRSS messages. Bursts found using the “slew event data” do not have a unique trigger number. Hence, we use the observation ID corresponding to the relevant slew event data as the trigger number in the fake TDRSS message. For the burst position, we use the best position reported in a GCN circular, which was found by previous manual analysis using the BAT data, or from the follow-up XRT/UVOT observations if the afterglow was detected (for simplicity, we do not use position from ground-based follow-up). The burst position is the only important information in the TDRSS data that is used for the actual analysis. Other information, such as the background duration, are only used in the summary report produced by [*batgrbproduct*]{}, and thus we use arbitrary numbers in the fake TDRSS messages. Because the “failed event data” are much shorter than regular event data, it is common that the burst duration lasts longer than the $\sim 3 - 10$ s event data range. In such cases, we use the un-maskweighted count rate data (the so-called “quad-rate data”[^17] to be specific, which is the raw count rate binned in 1.6 s). Many of the ground-detected bursts have very low partial coding fractions, we follow the same guidelines described in Section \[sect:low\_pcode\] for analysis of these bursts. There are 15 ground-detected bursts that were outside of the BAT calibrated field-of-view (i.e., the region where the FLUX mask[^18] is applicable) in the whole event data range. These bursts require using DETECTION mask$^{\ref{note_mask}}$ for finding burst durations and refined positions, and the spectral analyses are unavailable. There are 2 ground-detected bursts requiring DETECTION mask for finding the burst refined positions, the rest of the analyses are done using the FLUX mask. Similar to the onboard triggered bursts, any ground-detected GRBs without an XRT afterglow and with signal-to-noise ratio less than 7 is marked as a “tentative burst”. However, the limited event data and the lack of flight trigger information make it difficult to determine the time interval and energy range that enclose the maximum burst emission to estimate the signal-to-noise ratio. A trial-and-error approach might find a higher signal-to-noise ratio, but it also increases the expected number of false detections, which are hard to quantify in a manual process. We therefore estimate the signal-to-noise ratio from the image created in 15-350 keV in the time interval of $T_{100}$ range if possible, or the whole event-data range if $T_{100}$ extends beyond that. If the $T_{100}$ range includes some spacecraft-slewing periods, we create a mosaic image with small time steps (usually $\sim 0.5$ s). Moreover, we note that for many ground-detected bursts, the lack of XRT afterglows might be due to delayed observations, since the bursts was discovered on the ground after full data downlinks, and manually submitting a ToO observation request. It is hard to make a robust conclusion on this issue, however, since the information is lost forever. GRBs with low partial coding fraction {#sect:low_pcode} ------------------------------------- There are some bursts that are detected at the very edge of the BAT field of view, and thus have a very low partial coding fraction. We examine every burst with a partial coding fraction lower than 10%. If the standard analysis method fails to perform part or all of the analysis due to the small partial coding fraction, we redo the analysis using the “DETECTION” mask aperture setting. In comparison to the default setting of the “FLUX” mask aperture, the DETECTION mask aperture is the full aperture that includes the largest solid angle and most illuminated detectors. However, it also includes regions with shadows from the mounting brackets, and thus will reduce the accuracy of flux measurements and is only recommended to use for finding bursts [@Markwardt07]. We only use DETECTION mask for finding burst refined position and estimating burst duration if necessary. We only perform spectral analysis for the part of the light curve that is available with the FLUX mask setting. There are 62 bursts with partial-coding fractions lower than 10% (including the ground-detected GRBs). We examine and assign these bursts into three groups with different analysis approaches: 1. FLUX mask is okay: the [*batgrbproduct*]{} pipeline completes the analysis with the default FLUX mask setting, and there no need to perform further analysis. There are 28 bursts in this category. 2. DETECTION mask needed only for finding refined position: the pipeline successfully performs most of the analysis except finding the refined position. We redo the search for the refined position with the DETECTION mask setting. There are 13 bursts requiring such analysis. 3. DETECTION mask needed for finding burst duration and refined position: There are 21 bursts with either part or all of the burst durations outside of the BAT calibrated field of view (i.e., the region included in the FLUX mask). For these bursts, we use the DETECTION mask for estimating the burst durations and refined positions. The spectral analysis is only available for the time period when the burst is in the BAT calibrated field of view. Manual examinations of the analysis results: re-analyzing bursts with problems and/or adding comments for special bursts ------------------------------------------------------------------------------------------------------------------------ Occasionally, the standard analysis can fail or generate erroneous results for several reasons. For example, peculiar background behavior, such as rapid background rise when the telescope enters the South Atlantic Anomaly (SAA), can cause incomplete background subtraction and result in wrong estimations of burst durations. Additionally, if some bright X-ray sources appear in the same field of view as the burst, the background subtraction can be incorrect since the mask-weighting technique assumes the burst to be the brightest source in the BAT field of view. Thus, the light curve might be contaminated by these bright sources. Furthermore, sometimes there are gaps in the event data due to downlink problems. Therefore, we examine the result of each GRB by eye and add comments for those bursts that require special treatment. For problems that appear in more than one GRB, we make the comments in standard format, in order to enable automatic searches afterwards. We also mark the short GRBs with extended emission. However, we do not adopt any quantifiable criteria for determining short GRBs with extended emission. We simply follow the discussions from previous GCN circulars and eye inspections from the light curves. The adopted comments in standard format include: - The event data are not available. - The event data are only available for part of the burst duration. - battblocks failed because of the weak nature of the burst. - GRB found by the ground process (failed event data). - GRB found by the ground process (slew event data). - DETECTION mask with pcodethresh = 0.0 is used for finding the refined position. - DETECTION mask with pcodethresh = 0.0 is used for everything except spectral analysis. - Refined positions found by mosaic image (DETECTION mask with pcodethresh = 0.01, time bin = XX s, and energy band = XX-XX keV). - Spectral analysis is not available. - Spectral analysis is only available for part of the burst duration. - The detector plane histogram data are used for the spectral analysis. - T100, T90, and T50 are lower limits. - T100, T90, and T50 might be lower limits. - Only part of the event data are used in order to have a more reasonable estimation of burst durations. - Burst durations are found using quad-rate data from ${\rm T0}_{\rm GCN}-\rm XX$ s to ${\rm T0}_{\rm GCN}+{\rm XX}$ s. - Burst durations are found using FRED-model fitting. - Tentative detection. - Short GRB with extended emission. - Maybe short GRB with extended emission. - Refined position calculated with time interval and energy band determined by the flight software (T0 to T0+XX s; XX-XX keV). - Spectral analysis failed, likely because the burst is too weak. - Obvious data gap within the burst duration. In the following sub-subsections, we summarize further discussion of the common problems. There are only two bursts, GRB041219A and GRB071112C, which were triggered on-board and have no event data due to downlink problems. For GRB071112C, the burst duration is found by applying [*battblocks*]{} on the quad-rate data. The spectral analysis is not available, because the closest survey data bin lasts from $\sim T0-120$ s to $\sim T0+10$ s, which includes more background period than the burst duration. For GRB041219A, all the analyses are not available due to the lack of event data, rate data, and survey data, because this burst occurred at the very beginning of the mission. There are 77 bursts that have burst durations that last longer than the event data. For the 35 ground-detected bursts that last longer than the available $\sim 10$ s of event data, we apply [*battblocks*]{} on quad-rate data (un-maskweighted) to estimate burst durations. However, for the bursts triggered on-board, we add the comment “T100, T90, and T50 are lower limits”, instead of using the quad-rate data for quantifying the burst durations. This is because the on-board triggers have much longer event data. Thus, the bursts that extend beyond the event data ranges are usually those with fairly long durations, and quantifying the durations using un-maskweighted rate data becomes more inaccurate due to changes of the background levels. The spectral analyses for these bursts with incomplete event data are only available for the part of the burst emission that occurs within the event data range. Unusual background changes are most likely to occur when the spacecraft is entering or leaving the SAA. During these times, the background count rates can increase/decrease by $\sim \rm few \times 10^4 \rm \ count \ s^{-1}$ within a few hundreds of seconds, and cause problems in background subtraction and burst-duration estimations. We examine the burst light curves individually. When the burst duration seems to be incorrect, we re-do the analysis with a modified light curve that excludes the peculiar background period (when possible, i.e., when the problematic background period is far enough from the burst emission) to check if the burst duration changes significantly by excluding the problematic part of the data. It was necessary to recalculate the burst durations of 53 GRBs using only part of the event data. \[sect:bright\_src\] Other bright X-ray sources in the field of view that have similar or higher signal-to-noise ratios as the GRB can cause problems in background subtraction and give incorrect estimations of the GRB counts. We thus list the bursts with bright X-ray sources in their field of view in Table \[tab:brigh\_sources\] in Appendix \[sect:appendix\_tables\]. The analysis results for these bursts need to be treated with caution, particular the reliability of potential weak emissions in the light curves, and suspicious bumps or dips in the spectra. Extra manual analyses to remove the bright sources might be needed to obtain more reliable results. We adopt the following criterion for selecting bursts with bright X-ray sources in their field of view: (1) If the burst has signal-to-noise ratio ${\rm SNR}_{\rm GRB} \geq 10.0$, the X-ray sources in the field of view need to have signal-to-noise ratios ${\rm SNR}_{\rm source} \geq (0.9 \times {\rm SNR}_{\rm GRB})$, and (2) if the burst has signal-to-noise ratio ${\rm SNR}_{\rm GRB} < 10.0$, the X-ray sources in the field of view need to have signal-to-noise ratios ${\rm SNR}_{\rm source} \geq ({\rm SNR}_{\rm GRB} - 1.0)$. There are 315 bursts that satisfy this criterion, indicating that for a large fraction of the BAT-detected bursts ($\sim 31\%$), extra caution is needed when determining the reality of weak emissions in the light curves. For the bursts triggered onboard, there are only 12 GRBs that have data gaps in the T100 range. Most of the data gaps are around one or two seconds. The only one with a large data gap of $58$ s is GRB080319B because it happened shortly after the “A” burst and had some problem in data collection. The 12 GRBs are: GRB151027A, GRB131002A, GRB130907A, GRB111209A, GRB111022B, GRB110709B, GRB090516, GRB081017, GRB080928, GRB080319B, GRB060526, and GRB041224. There are some events with marginal BAT detections ($< 7 \sigma$; see Section \[sect:event\_standard\] and \[sect:ground\_GRBs\] for details of how the detection significance is estimated). We mark these as “tentative detections” in Table \[tab:summary\_general\] if there was no XRT afterglow detected. There are 16 bursts that are marked as tentative detections. These bursts should be treated with caution, as some of them might be due to noise fluctuation even though they have a given GRB name. However, we note that 10 out of 16 tentative detections are lacking XRT information due to observational constraints, and thus are difficult to determine the true nature of the event with the BAT detections alone. Information from other instruments (e.g., [*Fermi*]{}, Konus-Wind, etc) might provide further clues to identify the burst nature. We leave the judgement to the readers because it differs on a case-by-case basis. ### Comparison with the BAT2 Catalog We reprocess the main plots presented in the BAT2 catalog. All the figures are very similar to those in the BAT2 catalog. Therefore, we do not notice any significant differences between the data analyses in the BAT2 catalog and the current one. However, we do introduce some new criteria for selecting acceptable spectral fits in this catalog. Results for the BAT event data analyses {#sect:event_results} ======================================= The analysis results are available through the following interfaces: 1. Tables in Appendix \[sect:appendix\_tables\], which list all the numbers from the analyses, including GRB names, trigger IDs, trigger times, burst durations, spectral fits, energy and photon fluxes, redshift (if available)...etc (see detailed descriptions in Appendix \[sect:appendix\_tables\]). 2. Webpages that summarize the GRB light curves and spectral analyses in plots, and also include special comments for the burst if available. There is one webpage for each burst, with an index for all the pages at http://swift.gsfc.nasa.gov/results/batgrbcat/index.html 3. The data products from the analyses, including the light curve fits files, spectra created for different time ranges ($T_{100}$, 1-s peak, 20-ms peak, and the time-resolved spectra). The corresponding response files generated by averaging through the slew time interval is also included. The data products can be found via the GRB index page http://swift.gsfc.nasa.gov/results/batgrbcat/index.html. There are three bursts that triggered BAT twice (GRB111209A, GRB110709B, GRB140716A). We merged event data from triggers of GRB111209A and GRB110709B, and only listed one of the trigger numbers in the trigger ID column in the summary tables. However, GRB140716A is a ground detected burst, and there is a large data gap between the two triggers. Therefore, we listed both triggers independently in all the summary tables as GRB140716A-1 and GRB140716A-2, which correspond to different trigger IDs. BAT GRB demographics -------------------- The BAT has detected 1006 bursts to date (until GRB151027B), including 925 GRBs triggered on-board and 81 bursts found by ground analyses (within which 25 events are found during spacecraft slews). As mentioned in Section \[sect:intro\], although GRB151027B is the officially announced 1000th GRB detected by [*Swift*]{}, there are 1006 in our list due to a slightly different criteria of counting the BAT-detected GRBs. Table \[tab:GRBnum\_vs\_ndet\] lists the average number of GRB detected each year and the number of active detectors of BAT. Both the total number of GRB detections (i.e., including ground-detected GRBs) and the number of GRB triggered onboard are included in the table[^19]. The averaged number of active BAT detectors per year is gradually decreasing because some detectors get noisier and are thus turned off. Results show that the number of GRB detections per year remains similar throughout the ten years of [*Swift*]{} observations, despite the continuously decreasing number of active BAT detectors. Table \[tab:GRB\_category\] summarizes the number of bursts in the commonly adopted categories (long, short, short GRBs with extended emission, etc). In this paper, the term “ultra-long burst” refers to GRBs with the observed durations longer than 1000 s (the usual BAT event data range), and we only considered duration measured using the BAT emission. ------ ----------------------------- --------------------------- -------------------------- Year Number of detections Number of detections Average number of active (with ground-detected GRBs) (no ground-detected GRBs) BAT detectors 2005 88 86 29413 2006 102 100 26997 2007 87 80 27147 2008 105 96 26478 2009 91 81 24387 2010 85 72 24050 2011 82 75 22817 2012 92 89 23017 2013 96 85 22053 2014 94 84 20413 ------ ----------------------------- --------------------------- -------------------------- : \[tab:GRBnum\_vs\_ndet\] Summary of yearly GRB detection and the averaged number of BAT active detectors from 2004 to 2014. GRB category Number of bursts (percentage) -------------------------------------- ------------------------------- Long 850 (84.49%) Short 90 (8.95%) Short with Extended Emission 12 (1.19%) Ultra long ($T_{90} \gtrsim 1000$ s) 16 (1.59%) Bursts with un-constrained durations 66 (6.56%) : \[tab:GRB\_category\] Summary of number of GRBs in each category. ![ All-sky map in Galactic coordinates for BAT-detected GRBs. Short GRBs are marked as blue stars; short GRBs with extended emission (E. E.) are plotted as red square; GRBs with burst duration longer than 1000 s (i.e., longer than the event data range) are shown as green triangles. []{data-label="fig:all_sky_map"}](all_sky_map.pdf){width="100.00000%"} The sky distribution (in Galactic coordinate) of all the BAT-detected GRBs are plotted in Fig. \[fig:all\_sky\_map\], with blue stars representing short GRBs, red square showing the short GRBs with extended emission (E. E.), and green triangle marking the bursts with duration longer than 1000 s (i.e., longer than the event data range). Burst durations --------------- There are 990 GRBs that have available burst durations. However, there are 9 bursts that do not have available errors associated with the $T_{90}$, because the burst durations are determined by the FRED-model fit. For the 16 GRBs without $T_{90}$ measurements, 10 of them are missing burst durations because [*battblocks*]{} failed to find the burst durations due to the weak nature of the bursts, and the rest of the 6 GRBs are those without event data. In addition, there are 52 bursts that have incomplete GRB durations, i.e., the reported burst durations are lower limits, mostly because the burst durations last longer than the available event data time range. Two of these bursts, GRB101225A and GRB060218, have unusually long burst durations without obvious structure in the light curve, and thus are also in the list of GRBs for which [*battblocks*]{} failed to find the burst durations. The upper panel of Fig. \[fig:T90\] shows the $T_{90}$ distribution of 940 bursts that have burst durations successfully determined. That is, we exclude bursts with incomplete $T_{90}$, and/or bursts without $T_{90}$ that are found by [*battblocks*]{} or FRED-model fit. There are 850 GRBs with $T_{90} > 2$ s (long GRBs), and 90 GRBs with $T_{90} \leq 2$ s (short GRBs). When folding in the appropriate lower/upper limit, there are 17 long bursts with the $T_{90}$ lower limit shorter than 2 second, and 5 short bursts with $T_{90}$ upper limit longer than 2 second. For comparison, the histogram of $T_{90}$ upper limits and $T_{90}$ lower limits are also plotted in the figure. Results show that the uncertainties in $T_{90}$ measurements do not have significant effect on the overall $T_{90}$ distribution. The $T_{90}$ distribution remains very similar to the one shown in the BAT2 catalog, and thus is still significantly different from the distributions of GRBs detected by other instruments, such as [*Fermi*]{} and BATSE, as mentioned in the BAT2 catalog. The middle and bottom panels of Fig. \[fig:T90\] shows the $T_{90}$ distributions for GRBs detected by [*Fermi*]{} and BATSE for comparison. $T_{90}$ of the [*Fermi*]{} GRBs are obtained from the [*Fermi*]{} GBM burst catalog[^20] [@Gruber14; @vonKienlin14], and $T_{90}$ of the BATSE GRBs are from The Fourth Gamma-ray Bursts Catalog [@Paciesas99]. Compared to the short GRB fraction in the [*Swift*]{}/BAT GRB sample ($\sim 9\%$), the fractions of short bursts are larger in both the [*Fermi*]{} GRB sample ($\sim 17\%$) and the BATSE sample ($\sim 26\%$). ![ $T_{90}$ distribution for [*Swift*]{}/BAT (top panel), [*Fermi*]{}/GBM (middle panel), and [*CGRO*]{}/BATSE (bottom panel). For the BAT GRBs, only bursts with successfully determined are included in the plot. $T_{90}$ for [*Fermi*]{}/GBM bursts are obtained from the [*Fermi*]{} GBM burst catalog [@Gruber14; @vonKienlin14]. $T_{90}$ for [*CGRO*]{}/BATSE bursts are from The Fourth Gamma-ray Bursts Catalog [@Paciesas99]. Distributions using the upper and lower bounds of the $T_{90}$ uncertainty range are also plotted for comparison. The bin size of this plot is 0.2 in log scale. []{data-label="fig:T90"}](T90_all_with_limit_all_missions.pdf){width="80.00000%"} Spectral Analyses ----------------- ### Time-averaged spectra After applying the criteria described in Sect \[sect:event\_standard\], there are 877 bursts that have acceptable spectral fits in their time-averaged spectra (spectra made by photons in the $T_{100}$ range), in which 90 bursts are better fitted by the CPL model. In addition to the short and long categories using the burst durations, previous studies from GRBs detected by multiple instruments, including BATSE, [*Fermi*]{}, and [*Swift*]{}, have found that short bursts tend to be harder than long GRBs [e.g., @Kouveliotou93; @Qin00; @Ripa09; @Sakamoto11; @Qin13; @vonKienlin14]. Figure \[fig:hardness\_ratio\_T90\] shows an updated version of the hardness ratio (i.e., the fluence in 50-100 keV divided by fluence in 25-50 keV) versus $T_{90}$ to include all the new BAT-detected GRBs since the BAT2 catalog. The fluences are estimated from the better-fitted spectral model (see criterion described in Section \[sect:event\_standard\]). We only included bursts with acceptable spectral fits and with available values of $T_{90}$ and $T_{90}$ errors. In addition, we exclude bursts with incomplete $T_{90}$ (i.e., the burst durations are lower limits) and those bursts with $T_{90}$ consistent with zero (i.e., the lower limit of $T_{90}$ is equal or less than zero). There are 815 bursts included in this plot. There are 86 bursts that are better fitted by the CPL model in this plot (marked in red). This plot is very similar to the one presented in the BAT2 catalog. The conventional two GRB classes, short-hard bursts and long-soft bursts, can be roughly identified in this plot, though the separation of the two groups is not very obvious. The particularly soft short burst with the hardness ratio of 0.47 and $T_{90}$ of 0.132 s is GRB140622A. Despite the unusually soft spectrum, the fast fading XRT light curve of this burst is consistent with the normal behavior of a short burst [@GCN16438; @GCN16439]. Moreover, the redshift of $z \sim 0.96$ measured from the emission lines from the possible host galaxy [@GCN16437] suggests that this is not a Galactic source and is unlikely to be a soft gamma repeater [e.g., @Mereghetti08]. ![ Hardness ratio (i.e., fluence in 50-100 keV over fluence in 25-50 keV) versus $T_{90}$. The fluences are calculated using the best-fit models (either the simple PL or CPL). The bursts that are better fitted by the CPL model are marked in red. Note that GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:hardness_ratio_T90"}](Hardness_T90_witherror_best_model_onecolor.pdf){width="80.00000%"} To further explore the difference in spectral hardness of the short and long bursts, we plot the histogram of the photon index $\alpha$ for the bursts that are better fitted by the simple PL model in Fig \[fig:alpha\_PL\]. The upper panel shows the distribution for all 787 GRBs that have acceptable spectral fits and are better fitted by the simple PL model. The middle and bottom panel show the distributions for long and short GRBs, respectively. We need the $T_{90}$ information to distinguish short and long bursts. Thus, GRBs without available values of $T_{90}$ and/or $T_{90}$ errors are excluded from these two panels. Moreover, GRBs without complete burst durations (only lower limits reported) are also excluded. The figure shows that the short bursts are slightly harder (i.e. higher $\alpha_{\rm PL}$) than long bursts, but the difference is not significant. There are 671 long GRBs, and 58 short GRBs in these two panels. ![ Distributions of the GRB power-law indices $\alpha_{PL}$ for those bursts that are better fitted by simple PL model. Distributions using the upper and lower bounds of the $\alpha_{PL}$ value uncertainty range are also plotted for comparison. []{data-label="fig:alpha_PL"}](alpha_pow_bin_oneplot_only_good_pow.pdf){width="80.00000%"} \[sect:BAT\_sensitivity\] The BAT detector is a photon-counting instrument [@Barthelmy05], and thus the sensitivity roughly increases with $\sqrt{T}$ ($T$ is the exposure time), as the signal-to-noise ratio increases as $1/\sqrt{N}$ ($N$ is the number of photons) and the number of photons N increases as time $T$ (for BAT, the photon $N$ is dominated by background photons, and thus roughly increase linearly with $T$). Figure \[fig:energy\_photon\_flux\_T90\] shows this effect by plotting the correlation between the energy/photon fluxes of the BAT-detected GRBs versus the burst durations $T_{90}$[^21]. The fluxes are estimated by the best-fit model (either the simple PL or the CPL). The bursts that are better fitted by the CPL model are marked in red. The figure shows a clear correlation of the minimum fluxes of the detected GRBs and the burst durations. This correlation of the energy flux and $T_{90}$ is very similar to BAT sensitivity (as a function of exposure time) derived in @Baumgartner13 (Eq. 9), despite that in @Baumgartner13 the sensitivity is derived for non-GRB sources with Crab-like spectra, and for a signal-to-noise ratio of $5 \sigma$ (instead of the $\sim 6.5$ to $7 \sigma$ threshold used for GRB detections). However, note that this plot includes only the GRBs with “acceptable spectral fits” (as defined by criteria described in Section \[sect:event\_standard\]). Thus, this plots might exclude dim bursts that do not have data with low enough uncertainties to constraint the fits. In addition, due to the complexity of the BAT trigger algorithm, this correlation between the minimum detectable fluxes with the burst durations should only be treated as an approximation. For example, the burst durations are not usually identical to the actual exposure time used by the trigger algorithm for detecting the burst, because the trigger algorithm might not correctly bracket the burst period. Thus, this correlation does not necessary mean that GRBs with fluxes above this line will be certainly detected, since the foreground period of the trigger algorithm needs to first correctly select the optimal period that maximizes the signal-to-noise ratio [@Lien14]. Moreover, if the GRB flux decays significantly with time so that the average flux decreases faster than $T^{-1/2}$, there would be no gain in the signal-to-noise ratio by increasing the exposure time. The figure also shows that bursts that are better fitted by the CPL model tend to have higher fluxes. This is because a burst needs to be bright enough to obtain a decent spectrum (i.e., with smaller uncertainties in each energy bin) that is capable of distinguishing the two models. ![ [*Left panel:*]{} Time-average energy flux (15-150 keV) vs. $T_{90}$. [*Right panel:*]{} Time-average photon flux (15-150 keV) vs. $T_{90}$. For both plots, the fluxes are estimated by the best-fit model (either simple PL or CPL). The bursts that are better fitted by the CPL model are marked in red. Note that GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:energy_photon_flux_T90"}](energy_flux_vs_T90_best_model.pdf "fig:"){width="49.00000%"} ![ [*Left panel:*]{} Time-average energy flux (15-150 keV) vs. $T_{90}$. [*Right panel:*]{} Time-average photon flux (15-150 keV) vs. $T_{90}$. For both plots, the fluxes are estimated by the best-fit model (either simple PL or CPL). The bursts that are better fitted by the CPL model are marked in red. Note that GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:energy_photon_flux_T90"}](photon_flux_vs_T90_best_model.pdf "fig:"){width="49.00000%"} Figure \[fig:energy\_photon\_flux\_T100\] shows the distribution of energy fluxes (left panel) and photon fluxes (right panel) for all the 877 bursts with acceptable spectral fits. The fluxes are estimated from the best-fit model (either the simple PL for CPL). The distributions for both the energy and photon fluxes are roughly Gaussian, with an average of $5.90 \times 10^{-8}$ $\rm erg \ s^{-1} \ cm^{-2}$ and 0.75 $\rm ph \ s^{-1} \ cm^{-2}$ for the energy and photon fluxes, respectively. Again, because we only plots those bursts with acceptable spectral fits, the weak bursts with lower fluxes are likely to be removed from this sample. ![ [*Left Panel:*]{} Distributions of the GRB energy flux (15-150 keV) estimated from best-fit model (either the simple PL or CPL). [*Right Panel:*]{} Distributions of the GRB photon flux (15-150 keV) estimated from best-fit model (either the simple PL or CPL). Distributions using the upper and lower bounds of the flux uncertainty range are also plotted for comparison. []{data-label="fig:energy_photon_flux_T100"}](energy_flux_avg_pow_all_best_model.pdf "fig:"){width="49.00000%"} ![ [*Left Panel:*]{} Distributions of the GRB energy flux (15-150 keV) estimated from best-fit model (either the simple PL or CPL). [*Right Panel:*]{} Distributions of the GRB photon flux (15-150 keV) estimated from best-fit model (either the simple PL or CPL). Distributions using the upper and lower bounds of the flux uncertainty range are also plotted for comparison. []{data-label="fig:energy_photon_flux_T100"}](photon_flux_avg_pow_all_best_model.pdf "fig:"){width="49.00000%"} \[sect: BAT\_selection\_effect\_spectrum\] Both the theoretical predictions from the synchrotron shock model [@Rees92; @Preece98 and reference therein] and empirical fits from observations with instruments that have wide-energy coverages (e.g., BATSE and [*Fermi*]{}) suggest that the GRB spectrum (photon flux as a function of photon energy) can be roughly described by a power-law decay at lower energy, followed by some steepening after the energy $E_{\rm peak}$, the peak energy in the $\nu F_\nu$ spectrum, where $F_\nu$ is the energy flux density. The BAT has a relatively narrow energy range. Therefore, it can be difficult to constrain $E_{\rm peak}$ with BAT data alone. In fact, the $E_{\rm peak}$ distributions from the BATSE and [*Fermi*]{} GRB samples suggest that the $E_{\rm peak}$ distribution peaks at around few hundreds keV (see Fig. \[fig:Epeak\_other\_missions\]). Moreover, even for those bursts with $E_{\rm peak}$ occurring within the BAT energy range, the narrow energy coverage also requires a spectrum to have less uncertainty in order to be able to constrain $E_{\rm peak}$. ![ $E_{\rm peak}$ distributions for GRBs detected by [*Fermi*]{} [@Gruber14; @vonKienlin14] and BATSE [@Goldstein13]. Distributions using the upper and lower bounds of the $E_{\rm peak}$ uncertainty range are also plotted for comparison. []{data-label="fig:Epeak_other_missions"}](Epeak_all_with_limit_other_missions.pdf){width="80.00000%"} In the current BAT GRB sample, there are 90 bursts with acceptable spectra that are better fitted by the CPL model. The $E_{\rm peak}$ and photon index $\alpha_{\rm CPL}$ distributions for these events are plotted in the left and right panel of Fig. \[fig:Epeak\], respectively. The black and red lines show the distributions using the lower and upper limits. As expected, all of the values of $E_{\rm peak}$ lie in the range of $\sim 10$ keV to $\sim 300$ keV, which is within the BAT energy range. However, these might not be the only bursts in the BAT sample that have $E_{\rm peak}$ within the BAT energy range. It is possible that some other bursts also have $E_{\rm peak}$ in this range, but are not bright enough to present good spectra that can distinguish the two models [@Sakamoto08]. The $E_{\rm peak}$ distribution peaks at $\sim 80$ keV (the center of the bin with the largest number of bursts), which is similar to the one presented in the BAT2 catalog, but is significantly different than the distributions from GRBs detected by other instruments, as shown in Fig. \[fig:Epeak\_other\_missions\] [@Sakamoto11; @Goldstein13; @Gruber14; @vonKienlin14]. The difference is likely due to instrumental biases with each instrument sensitives to a different energy range. ![ [*Left panel*]{}: $E_{\rm peak}$ distributions for GRBs that are better fitted by the CPL model. [*Right panel*]{}: Photon index $\alpha_{\rm CPL}$ distributions for GRBs that are better fitted by the CPL model. Distributions using the upper and lower bounds of the $E_{\rm peak}$/$\alpha_{\rm CPL}$ uncertainty range are also plotted for comparison. []{data-label="fig:Epeak"}](Epeak_CPL.pdf "fig:"){width="49.00000%"} ![ [*Left panel*]{}: $E_{\rm peak}$ distributions for GRBs that are better fitted by the CPL model. [*Right panel*]{}: Photon index $\alpha_{\rm CPL}$ distributions for GRBs that are better fitted by the CPL model. Distributions using the upper and lower bounds of the $E_{\rm peak}$/$\alpha_{\rm CPL}$ uncertainty range are also plotted for comparison. []{data-label="fig:Epeak"}](alpha_cutpow_bin_oneplot_only_good_cutpow.pdf "fig:"){width="49.00000%"} Both the BAT1 and BAT2 catalog have shown that most of the BAT-detected GRBs have spectral hardnesses that are consistent with the spectral hardnesses calculated from a Band function fit with $E^{\rm obs}_{\rm peak}$ in the range of the BAT energy band. Here we updated the same plot that shows the spectral hardness as in the BAT1 and BAT2 catalog with the new GRB detections, as shown in the left panel of Fig. \[fig:fluence\_fluence\_flux\_flux\]. In addition, we also plot the spectral hardness in flux instead of fluence in the right panel of Fig. \[fig:fluence\_fluence\_flux\_flux\]. As usual, we only include in these plots bursts with acceptable spectral fits and complete burst durations with valid numbers of $T_{90}$ and $T_{90}$ errors. There are 756 long bursts (red dots) and 59 short bursts (blue dots) in these plots. Similar to the BAT1 and BAT2 catalog, we plot the lines using the Band function with $E_{\rm peak}$ of 15 keV (dashed line) and 150 keV (dash-dotted line), respectively. Both lines are calculated using canonical values of $\alpha = -1$, $\beta=-2.5$. Each line traces the fluence ratios from the same $\alpha$, $\beta$, and $E_{\rm peak}$, with a range of normalizations. Results show that $\sim 80\%$ of the bursts lie between the two lines (when including the errors of the burst fluences), indicating that most of the BAT-detected bursts have fluence ratios that are consistent with the ones from the Band function with $E_{\rm peak}$ lying within the BAT energy range. In other words, it is possible that these bursts have $E_{\rm peak}$ within the BAT energy range, though most of the spectra do not have small enough uncertainties to constrain the $E_{\rm peak}$. ![ [*Left Panel:*]{} Fluence in the 50-150 keV range versus fluence in the 15-50 keV range. Fluences are estimated by the best-fit model (either the simple PL or CPL). Red dot represents the GRBs with $T_{90} > 2$ s; blue points refers to GRBs with $T_{90} \leq 2$ s. [*Right Panel:*]{} Similar plot as the left panel, but with energy flux instead of energy fluence. The dash-dotted line and the dashed line traces the fluences calculated from the Band function with $E^{\rm obs}_{\rm peak} = 15$ and 150 keV, respectively. Both lines assume a canonical values of $\alpha = -1$ and $\beta = -2.5$. []{data-label="fig:fluence_fluence_flux_flux"}](fluence_fluence_witherror_best_model_twocolor.pdf "fig:"){width="49.00000%"} ![ [*Left Panel:*]{} Fluence in the 50-150 keV range versus fluence in the 15-50 keV range. Fluences are estimated by the best-fit model (either the simple PL or CPL). Red dot represents the GRBs with $T_{90} > 2$ s; blue points refers to GRBs with $T_{90} \leq 2$ s. [*Right Panel:*]{} Similar plot as the left panel, but with energy flux instead of energy fluence. The dash-dotted line and the dashed line traces the fluences calculated from the Band function with $E^{\rm obs}_{\rm peak} = 15$ and 150 keV, respectively. Both lines assume a canonical values of $\alpha = -1$ and $\beta = -2.5$. []{data-label="fig:fluence_fluence_flux_flux"}](flux_flux_witherror_best_model_twocolor.pdf "fig:"){width="49.00000%"} Furthermore, @Sakamoto08 studies the potential confusion between the simple PL, CPL, and Band function [@Band93] spectral fit in the BAT observations using simulated spectra. These authors found that most of the BAT-detected GRBs probably have $E_{\rm peak}$ within the BAT energy range, and derived an equation to estimate $E_{\rm peak}$ (for the Band function) using the photon index from the simple PL fit. Figure \[fig:Epeak\_estimator\] shows $E_{\rm peak}$ distribution from the $E_{\rm peak}$ estimator derived in @Sakamoto08. Results show that majority ($\sim 78\%$) of the GRBs detected by BAT might have $E_{\rm peak}$ within the BAT energy range. This fraction is very similar to the fraction ($\sim 80\%$) of the bursts that fall between the lines in Fig \[fig:fluence\_fluence\_flux\_flux\], which traces the fluence ratios with $E^{\rm obs}_{\rm peak} = 15$ keV and $150$ keV, respectively. Because the $E_{\rm peak}$ estimator only works for GRBs that have $E_{\rm peak}$ within $15-150$ keV, all the bursts estimated to have $E_{\rm peak}$ below or above this energy range are placed in single bins in light red. ![ $E_{\rm peak}$ distributions for GRBs that are better fitted by the CPL model. []{data-label="fig:Epeak_estimator"}](logEpeak_bin_plot_epeak_estimator.pdf){width="80.00000%"} For bursts with similar total energies, it is reasonable to expect that BAT is most sensitive to those events that have $E_{\rm peak}$ within the instrument’s energy range, because in such cases most of the energy of these bursts are distributed in the detectable energy range. However, it is also possible to have a burst with $E_{\rm peak}$ outside of the BAT energy being detectable because it has higher fluxes overall, and it is not completely trivial how much the energy budget arrangement is sensitive to different spectral shapes. To quantify the energy budget in the BAT energy range, we calculate the energy flux in $15-150$ keV for a range of $E_{\rm peak}$ and photon indices in the CPL model, but with a fixed total flux (i.e., we assume a flux of $7.21 \times 10^{-7} \ \rm erg \ s^{-1} \ cm^{-2}$ from $1-10000$ keV, which is a relatively typical flux for GRB and it can be easily scaled up or down for bursts with different total flux/normalization). Figure \[fig:flux\_contour\] shows the contour plot of how flux changes as a function of $E_{\rm peak}$ and photon indices. We make the contour plot with the CPL model rather than the Band function simply because the Band function has one more parameter, which makes it difficult to present in a 2-dimensional plot. Moreover, since we are focusing on the flux in the narrow BAT energy range, the CPL model should be a good-enough approximation. We show an extremely wide range of $\alpha_{\rm CPL}$ in this plot for demonstration. However, observations from BATSE, [*Fermi*]{}, and [*Swift*]{} suggest that the value never exceeds 1 [@Sakamoto11; @Goldstein13; @Gruber14; @vonKienlin14]. Moreover, theoretical predictions from the synchrotron shock model enforce that $\alpha_{\rm CPL} < -2/3$ [@Rees92; @Preece98 and references therein]. Therefore, one can see in the plot that in the reasonable range of $\alpha_{\rm CPL}$ from $\sim -2$ to $\sim 1$, GRBs with normal energy output are indeed most likely to be detected by BAT if they have $E_{\rm peak}$ in the BAT energy range. For GRBs with much higher $E_{\rm peak}$, they would need to be roughly one or two orders of magnitude brighter in order to be detected, and the consequences become more significant at larger $\alpha_{\rm CPL}$. ![ A contour plot that shows how the flux in $15-150$ keV changes as a function of $E_{\rm} peak$ and $\alpha_{\rm CPL}$ in the CPL model. []{data-label="fig:flux_contour"}](CPL_flux_contour.pdf){width="80.00000%"} When comparing Fig. \[fig:Epeak\] and Fig. \[fig:alpha\_PL\] for the spectral index distributions from the simple PL and CPL fits, one would notice that in general the spectral indices from the CPL fits are higher than those from the simple PL fits. This can be explained if most of the bursts actually have $E_{\rm peak}$ within the BAT energy range. In such cases, the photon index from a simple PL fit will be an average of the true power law index before and after $E_{\rm peak}$, and thus will be lower than the real first slope of the spectrum. For those bursts that are better fitted by the CPL model, all except two have their lower limit of $\alpha_{\rm CPL}$ greater than -2/3. Therefore, almost all the bursts are consistent with the synchrotron shock model. The only two GRBs with the lower limits higher than -2/3 are GRB050219A and GRB130420B, and the $\alpha_{\rm CPL}$ range for these two bursts are (-0.41, 0.18) and (-0.61, 0.44), respectively. All the bursts with acceptable fits that are better fitted by the simple PL model also have spectral indices that are consistent with the the synchrotron shock model. However, the comparison using the simple PL fit might not be physically meaningful, if the simple PL model does not represent the true underlying spectral shape due to the uncertainty in the data. ### Peak spectra For the 1-s peak spectra, there are only 728 bursts that have acceptable spectral fits due to the smaller number of photon counts in the 1-s duration. Within these bursts with acceptable spectral fits, there are 68 GRBs that are better fitted by the CPL model. There are 51 bursts with $T_{100}$ shorter than one second. For these extremely short bursts, the 1-s peak flux is likely to include some background intervals, and the 20-ms peak flux discussed below (and reported in Section \[sect:20ms\_tables\]) is probably better represent the true peak flux. Nonetheless, we still present the 1-s peak flux for GRBs with $T_{100} < 1$ s here for completeness, and also because of the uncertainties in the burst duration measurements. There are 542 bursts that are better fitted by the simple PL model for both the $T_{100}$ spectra and the 1-s peak spectra; 22 bursts that are better fitted by the CPL for both of the $T_{100}$ spectra and the 1-s peak spectra; 42 bursts that are better fitted by the PL model for the $T_{100}$ spectra but change to the CPL fit for the 1-s peak spectra; and 50 bursts that are better fitted by the CPL model for the $T_{100}$ but switch to the simple PL fit for the 1-s peak spectra. Those bursts that switch between models for the two different spectral fits usually either have $\Delta \chi^2 \sim 5$ (close to our threshold for adopting the CPL model at $\Delta \chi^2 = 6$) for one of the spectra, or the fits have fairly low null probability (close to our threshold of 0.1 for rejecting the fit). Note that only one burst with $T_{100} < 1$ s (GRB081101) is better fitted by different models for the time-averaged spectrum and the 1-s peak spectrum. To be specific, the time-averaged spectrum for this burst is better fitted by the CPL model, while the 1-s peak spectrum is better fitted by the simple PL model, likely because the 1-s peak spectrum includes a larger interval than the true $T_{100}$ range and thus is contaminated by some background photons. ![ Comparison between the photon indices $\alpha_{\rm PL}$ from the $T_{100}$ spectral fits and the 1-s peak spectral fits. []{data-label="fig:alpha_compare"}](pow_alpha_T100_vs_pow_alpha_1speak.pdf){width="60.00000%"} Figure \[fig:alpha\_compare\] compares the photon indices $\alpha_{\rm PL}$ from the $T_{100}$ and 1-s peak spectral fits. The results show that the fits from two different kinds of spectra gives similar $\alpha_{\rm PL}$. Several studies have shown that it is not uncommon that the spectral evolution follows the shape of the light curve [e.g., @Zhang07; @Zhang11]. Therefore, one might expect that the 1-s peak spectrum is harder (i.e., has larger $\alpha_{\rm PL}$) than the time-averaged spectrum. Due to the large uncertainties in the values of $\alpha_{\rm PL}$, it is difficult to determine whether such trend exists. However, we do notice this trend of spectral evolution for some brighter bursts with decent spectrum (see the “Spectral Evolution” plot in the individual burst webpage[^22]). Similar to Fig. \[fig:energy\_photon\_flux\_T90\], we plot in Fig \[fig:1speak\_energy\_photon\_flux\_T90\] the 1-s peak energy flux (left panel) and the photon flux (right panel) as a function of $T_{90}$ to show the BAT detection limit with the 1-s flux. As expected, there is no obvious correlation between the minimum 1-s peak energy/photon flux with respect to $T_{90}$. Also, results show that the BAT sensitivity to 1-s flux is $\sim 3 \times 10^{-8} \ \rm erg \ s^{-1} \ cm^{-2}$ (or $\sim 0.3 \ \rm s^{-1} \ cm^{-2}$ for photon flux), which are similar to the detection limit shown in Fig. \[fig:energy\_photon\_flux\_T90\]. Similar to Fig. \[fig:energy\_photon\_flux\_T90\], the bursts that are fitted-better by the CPL model have higher minimum fluxes. ![ [*Left panel:*]{} 1-s peak energy flux (15-150 keV) vs. $T_{90}$. [*Right panel:*]{} 1-s peak photon flux (15-150 keV) vs. $T_{90}$. For both plots, the fluxes are estimated by the best-fit model (either simple PL or CPL). The bursts that are better fitted by the simple PL model are marked in blue, and GRBs that are better fitted by the CPL model are marked in red. []{data-label="fig:1speak_energy_photon_flux_T90"}](1speak_energy_flux_vs_T90_best_model.pdf "fig:"){width="49.00000%"} ![ [*Left panel:*]{} 1-s peak energy flux (15-150 keV) vs. $T_{90}$. [*Right panel:*]{} 1-s peak photon flux (15-150 keV) vs. $T_{90}$. For both plots, the fluxes are estimated by the best-fit model (either simple PL or CPL). The bursts that are better fitted by the simple PL model are marked in blue, and GRBs that are better fitted by the CPL model are marked in red. []{data-label="fig:1speak_energy_photon_flux_T90"}](1speak_photon_flux_vs_T90_best_model.pdf "fig:"){width="49.00000%"} Because most of the short bursts are shorter than one second, we also generate peak spectra for the 20-ms duration to have values that better represent the peak fluxes for short bursts. As mentioned in Sect \[sect:event\_standard\], the 20-ms peak spectra are made with larger energy bins and only ten energy bands, to have reasonable number of counts in each energy bin. However, despite the larger energy bins used, there are only 226 bursts with acceptable spectral fits for the 20-ms spectral analyses, which is significant lower than those from the 1-s peak spectral analyses and the time-averaged $T_{100}$ spectral analyses. Only five GRBs (GRB140209A, GRB110715A, GRB101023A, GRB060117, and GRB050525A) have the 20-ms spectrum better fitted by the CPL model. Figure \[fig:T100\_alpha\_20ms\_alpha\] shows the correlation between the photon indices $\alpha_{\rm PL}$ from the 20-ms peak spectral analyses and the time-averaged $T_{100}$ spectral analyses. Comparing the the similar plot made for the 1-s peak spectral fits, the correlation between the $\alpha_{\rm PL}$ from the 20-ms peak and $T_{100}$ spectra are less significant. Results also show that the long and short bursts follow similar trend. ![ Comparison between the photon indices $\alpha_{\rm PL}$ from the $T_{100}$ spectral fits and the 20-ms peak spectral fits. Long GRBs are shown in blue, and short GRBs are marked in red. []{data-label="fig:T100_alpha_20ms_alpha"}](pow_alpha_T100_vs_pow_alpha_20mspeak.pdf){width="60.00000%"} ### GRB that do not have acceptable spectral fits {#sect:unaccept_spec} In Sect \[sect:event\_standard\], we listed the criteria used for determining whether the results from spectral analysis are acceptable. In order to obtain a more complete picture of the BAT-detected bursts, we include further discussions about those GRBs that have the spectral analysis results excluded from these criteria, and hence are not included in the previous section. When we choose GRBs with acceptable spectra, we first select the bursts with $\Delta \chi^2 < 6$, and go through those CPL spectral fits to select the acceptable ones based on the criteria listed in Sect \[sect:event\_standard\]. We then go through the PL fits for the rest of the bursts, and choose those bursts with acceptable PL fits. Therefore, for all the GRBs without acceptable spectra, there must be some problems in their PL fits. However, even for the GRBs with acceptable PL fits, there might be some problems in their CPL fits (which is not adopted). Hence, when sorting out the reasons for those unacceptable spectral fits, we only look through the PL fits. We put the bursts with unacceptable spectral fits into three different categories: 1. GRBs with problematic spectral fits: These are GRBs with some problems in their fits, which includes GRBs with at least one unconstrained parameter (parameters here includes the photon index, normalization, energy and photon flux, the photon index for the fits used for finding energy and photon fluxes, and lower and upper limits for all these parameters), GRBs with fitted values outside of the uncertainty ranges (although this should not happen, we place this criteria here just to be safe), and bursts with inconsistent results from the original fits and those fits used to constrain the photon and energy fluxes. These are bursts that would need manual spectral analysis to figure out the exact causes of the problems, and whether a better fit could be found. In this category, there are 37 GRBs for the time-averaged spectral fits, 109 bursts for the 1-s peak spectral fits, and 352 events for the 20-ms peak spectral fits. 2. GRBs with lower limits consistent with zero (but not in group one): These are bursts with at least one of the values of either the normalization, the photon flux, or the energy flux, consistent with zero, and thus only the upper limit of these parameters can be obtained. In this category, there are 6 GRBs for the time-averaged spectral fits, 100 bursts for the 1-s peak spectral fits, and 418 events for the 20-ms peak spectral fits. 3. GRBs with spectral fits that have the null hypothesis probability $<$ 0.1 (but not in group one): These are fits that are likely to reject the null hypothesis[^23]. However, we note that many of these fits that are inconsistent with the null hypothesis are likely due to systematic problems in data reduction rather than physical reasons. For example, other bright X-ray sources in the field-of-view can cause problems in background subtraction (as discussed in Sect \[sect:bright\_src\]) and give an incorrect estimation of the GRB source counts. Therefore, a careful examination of the potential data reduction problems must be carried out before seeking for alternative models. In this category, there are 87 GRBs for the time-averaged spectral fits, within which 43 bursts have X-ray sources with similar or higher signal-to-noise ratio in their field of view. For the 1-s peak spectral fits, there are 77 bursts in this group, and 24 of these have X-ray sources with similar or higher signal-to-noise ratio in their field of view. For the 20-ms peak spectral fits, there are 48 events in this group, and 28 of them have X-ray sources with similar or higher signal-to-noise ratio in their field of view. For the time-averaged spectral fits, there are one burst, GRB061218, that belongs to both group 2 and group 3. For the 1-s peak and 20-ms peak spectral fits, there are more GRBs belonging to both group 2 and group 3 (7 bursts for the 1-s peak spectral fits, and 38 events for the 20-ms peak spectral fits). Also, note that since GRB140716A are listed as GRB140716A-1 and GRB140716A-2 in all the summary tables for the two separate triggers of this same burst (as described in Section \[sect:event\_results\]), this burst sometime shows up in both lists of the acceptable spectral fits and unacceptable spectral fits. Specifically, GRB140716A-1 has acceptable spectral fits for the time-averaged spectrum and the 1-s peak spectrum, while GRB140716A-2 has acceptable spectral fits for the time-averaged spectrum but unacceptable fits for the 1-s peak spectrum. For the 20-ms peak spectrum, both GRB140716A-1 and GRB140716A-2 has unacceptable spectral fits, but they only count as one burst in the unacceptable spectral fit list of the 20-ms peak spectrum. Partial coding fraction ----------------------- ![ Distribution of the partial coding fractions for long, short, and ground-detected GRBs. The bursts found by ground-analyses during spacecraft slews are not included because the partial coding fraction changes constantly during slews. []{data-label="fig:pcode"}](pcode_bin_plot_long_short_all_with_grd_and_slew.pdf){width="80.00000%"} Figure \[fig:pcode\] shows the histogram of the partial coding fraction for long, short, and ground-detected GRBs. The bursts found by ground-analyses during spacecraft slews are not included, because the partial coding fraction changes constantly for these events. There are no short GRBs triggered on-board with partial coding fraction less than 0.2, which is equivalent to an incident angle of $\sim 50^{\rm o}$. Short GRBs with Extended Emission {#sect:Short_with_EE} --------------------------------- Short GRBs with extended emissions have raised special interests among the GRB community, partially due to their mixed characteristics between the short and long bursts [e.g., @Norris06; @Gehrels06; @Troja08; @Kaneko15]. We therefore include a special section of short GRBs with extended emission. We also present both a more secure list of these bursts (Table \[tab:sGRB\_with\_EE\]), plus a list of possible short GRBs with extended emission (Table \[tab:maybe\_sGRB\_with\_EE\]). All of the short GRBs with extended emissions listed in Table \[tab:sGRB\_with\_EE\] that occurred before 2009 are included in the BAT2 catalog. Note that we do not apply any quantitative criteria for selecting these short GRBs with extended emissions. Most of the bursts before 2009 are adopted from the BAT2 catalog, which was classified by @Norris10. However, the bursts after 2009 are chosen based on reports in GCN circulars. We also double check by eye inspection for all GRBs to (1) make sure no other obvious GRBs with similar features are missed in previous GCN circulars, and (2) those bursts reported as short GRBs with extended emissions do show such a feature. ------------ ------------------- -------------------- ---------- ------------------------- ------------------------- GRB name Short Pulse Start Short Pulse End E.E. End Short Pulse E.E. \[s\] (E.E. Start) \[s\] \[s\] $\alpha_{\rm PL}$ $\alpha_{\rm PL}$ GRB150424A -0.060 0.468 95.012 $-0.78^{+0.06}_{-0.06}$ $-2.10^{+0.46}_{-0.54}$ GRB111121A -0.336 1.000 138.264 $-0.99^{+0.08}_{-0.08}$ $-1.83^{+0.15}_{-0.15}$ GRB090916 -0.040 0.392 68.520 $-1.58^{+0.27}_{-0.27}$ $-1.37^{+0.38}_{-0.37}$ GRB090715A -0.200 0.800 48.936 $-1.02^{+0.21}_{-0.20}$ $-1.54^{+0.62}_{-0.62}$ GRB090531B 0.252 1.300 56.132 $-0.99^{+0.16}_{-0.16}$ $-1.69^{+0.27}_{-0.28}$ GRB080503 -2.192 0.600 221.808 $-1.32^{+0.45}_{-0.43}$ $-1.89^{+0.12}_{-0.12}$ GRB071227 -0.144 0.908 150.552 $-1.01^{+0.24}_{-0.23}$ $-2.23^{+0.41}_{-0.49}$ GRB070714B -0.792 1.976 74.640 $-0.96^{+0.08}_{-0.08}$ $-1.92^{+0.43}_{-0.49}$ GRB061210 -0.004 0.080 89.392 $-0.69^{+0.12}_{-0.12}$ $-1.80^{+0.34}_{-0.37}$ GRB061006 -23.288 -22.000 137.720 $-0.91^{+0.08}_{-0.07}$ $-2.06^{+0.22}_{-0.23}$ GRB051227 -0.848 0.828 122.732 $-0.94^{+0.25}_{-0.23}$ $-1.48^{+0.27}_{-0.27}$ GRB050724 -0.024 0.416 107.140 $-1.51^{+0.14}_{-0.14}$ $-2.05^{+0.25}_{-0.26}$ ------------ ------------------- -------------------- ---------- ------------------------- ------------------------- : \[tab:sGRB\_with\_EE\] A list of definite short GRBs with extended emission (E. E.). The times listed in the table are relative to the burst trigger time. ![ The distribution of the photon index from the simple PL fit $\alpha_{\rm PL}$ with spectra from the short pulses (left panel) and extended emissions (right panel) of the short GRBs with extended emission (listed in Table \[tab:sGRB\_with\_EE\]). The $\alpha_{\rm PL}$ distributions from short and long bursts are also plotted for comparison. Furthermore, distributions using the upper and lower bounds of the $\alpha_{\rm PL}$ value uncertainty range are also plotted. []{data-label="fig:shortGRBs_with_EE"}](alpha_pow_bin_sGRB_with_EE.pdf){width="100.00000%"} GRB name Trigger ID/Observation ID Comment ------------ --------------------------- --------- GRB140302A 589685 1 GRB140209A 586071 1, 3 GRB140102A 582760 1 GRB130716A 561974 3 GRB130612A 557976 1, 2, 3 GRB110402A 450545 1, 3 GRB100816A 431764 1, 2, 3 GRB090831C 361489 1, 3 GRB090530 353567 1 GRB090518 352420 1, 3 GRB090510 351588 3 GRB080303 304549 1 GRB080123 301578 3 GRB081211B 00090053089 4 GRB060614 214805 1 : \[tab:maybe\_sGRB\_with\_EE\] A list of possible short GRBs with extended emission. The comments note the reasons for these bursts being in the “possible” category, with each number corresponding to the following descriptions: (1) The short pulse is slightly longer than 2 s. (2) The extended emission is not picked out by the auto-pipeline ([*battblocks*]{}). (3) The extended emission is weak, and there are some significant fluctuations in the background and/or bright X-ray sources in the field of view, which may cause extra residuals when performing the mask-weighting. (4) The bursts was found by ground analyses and there is not sufficient event data. The possible short GRBs with extended emission listed in Table \[tab:maybe\_sGRB\_with\_EE\] are GRBs with similar structure as those listed in Table \[tab:sGRB\_with\_EE\]. They are selected because some literature (mostly the GCN circulars) mentioned potential extended emissions. However, these are not included in Table \[tab:sGRB\_with\_EE\] because of at least one of the following reasons: (1) The short pulse is slightly longer than 2 s. (2) The extended emission is not picked out by the auto-pipeline ([*battblocks*]{}). (3) The extended emission is weak, and there are some significant fluctuations in the background and/or bright X-ray sources in the field of view, which may cause extra residuals when performing the mask-weighting. (4) The bursts was found by ground analyses and there is not sufficient event data. The corresponding comments for each burst are presented in the table. Note that although GRB081211B is a ground-detected burst (during a spacecraft slew) with only $\sim 120$ s of event data, several GCN circulars suggest that this burst is possibly a short GRB with extended emission, with the short spike detected by Konus-Wind while the burst was outside of the BAT field of view (Golenetskii et al. 2008/GCN 8676; Perley et al. 2008/GCN 8914). In fact, there is also a visible short spike at $\sim$ T0-150 s in the BAT raw light curve. However, due to the lack of event data at that time, we cannot confirm whether this short pulse is related to the GRB. There are five bursts (GRB091117, GRB100724A, GRB100625A, GRB101219A, and GRB090621B) for which the GCN circular mentioned an indication of extended emission, but further analysis suggests that the extended emissions of GRB091117, GRB100724A, GRB100625A, and GRB101219A are below $\sim 3 \sigma$ even when choosing an optimal time periods and optimal energy bands. GRB090621B shows no extended emission until $\sim$ T0+150 s, when a low-significant bump occurred and last $\sim 50$ s. Thus, we conclude that these extended emissions are likely not real. Figure \[fig:shortGRBs\_with\_EE\] shows the results of spectral fits from the simple PL model ($\alpha_{\rm PL}$) for both the short pulses (left panel) and the extended emission (right panel), for the confirmed short GRBs with extended emissions (Table \[tab:sGRB\_with\_EE\]). Because of the small sample of the short GRBs with extended emission, and all of these bursts have constrained $\alpha_{\rm PL}$ but not necessarily have constrained parameters in the CPL model, we only show the simple PL model fits in this figure. Moreover, we include all the bursts, even if the do not satisfied some of the criteria listed in Section \[sect:event\_standard\] (in fact, only a few simple PL fits here do not satisfied all the strict criteria in \[sect:event\_standard\], such as the lower limit of one energy band is consistent with zero). The $\alpha_{\rm PL}$ distributions for short and long GRBs are also plotted in gray bars in the left and right panel, respectively, to be compared with the distributions of short pulses and extended emissions. As mentioned in previous studies [e.g., @Sakamoto11], the spectra of short pulses are harder than the extended emission in general, and resemble more of short GRBs, while the spectra for extended emission parts are softer and match better with the $\alpha_{\rm PL}$ distributions of long GRBs. GRBs with redshift measurements ------------------------------- We include a list of GRBs with redshift measurements in this catalog (Table \[tab:redshift\_list\]). The information in this list is collected from and cross-checked between other online lists (e.g., GRBOX by Daniel Perley[^24], online list by Jochen Greiner[^25], online table by Nathaniel Butler[^26]), the Gamma-ray Coordinates Network (GCN) circulars [@Barthelmy95], and papers. The redshift list with full references are included in Table \[tab:redshift\_list\]. To date (till GRB151027B), there are 378 BAT-detected GRBs with redshift measurements (within which 18 redshifts are marked as potential questionable measurements). In Table \[tab:redshift\_list\], we mark the four common methods for redshift measurements: (1) absorption lines measurement from the GRB afterglow spectra (noted by symbol “ba”); (2) emission lines from the GRB host galaxy spectra (noted by symbol “he”); (3) photometric redshift from the GRB afterglow (noted by symbol “bp”); (4) photometric redshift from the GRB host galaxy (noted by symbol “hp”). Other less-common methods, such as the Lyman-alpha break, are described in short sentences in Table \[tab:redshift\_list\]. If we noticed that some questions were raised about the GRB redshifts (e.g., the potential host galaxy might not be related to the burst), the redshift value in the table will be followed by a question mark, and these redshift values are not included in the following summarized numbers and plots. There are 229 GRB spectroscopic redshifts from GRB afterglows, 96 spectroscopic redshifts measured from host galaxies, 17 photometric redshifts from GRB afterglows, and 12 photometric redshifts from host galaxies. The redshift distribution of the BAT-detected GRBs is shown in Fig. \[fig:redshift\_with\_image\]. The distribution of bursts found by the image trigger is plotted in red. Compared to GRBs detected by rate triggers, the image-triggered GRBs are more uniformly distributed throughout all redshifts. The image-triggered bursts compose of 20.0% of the events with redshift measurements, which is very similar to the fraction of image-triggered GRBs out of all BAT-detected bursts (17.5%). ![ Redshift distribution for the BAT GRBs that have redshift measurements (gray bars). Distribution for bursts that are detected by image trigger are shown in red bars. []{data-label="fig:redshift_with_image"}](redshift_bin_plot_all_image.pdf){width="80.00000%"} ### Energy flux versus redshift: exploring potential selection effects in redshift measurements with the BAT trigger characteristic The energy fluxes in 15-150 keV as a function of redshift is plotted in Fig. \[fig:energy\_flux\_vs\_z\]. There are 35 bursts with redshift measurements excluded from this plot, because those bursts do not have acceptable spectral fits. The image-triggered bursts are marked in red, and the GRBs with photometric redshifts are marked in blue, with the uncertainties shown when available. Note that these uncertainties are adopted from different sources (see Table \[tab:redshift\_list\]) and might refer to different confidence ranges. Thus, the values here are only presented as rough references. As expected, image triggers find bursts with lower fluxes in general throughout the redshift range. The decline in the number of detections of high-flux bursts at higher redshift is likely due to the shrinking of the sample size of detected GRBs (it is less likely to detect bursts from the tail of the distribution). There is also a slight decrease in the detections of low-flux bursts at higher redshift, though the effect is not obvious until higher redshift ($z \gtrsim 5$), which might imply that the majority of the bursts (i.e., the center of the intrinsic flux distribution) does not lie far from the BAT sensitivity threshold. ![ Redshift versus energy flux (15-150 keV) for the BAT GRBs that have redshift measurements. Bursts that are detected by the image trigger are marked in red; GRBs with photometric redshifts are marked in blue; all other bursts are plotted in black. The uncertainties for redshifts are shown when available. However, note that these uncertainties are adopted from different references and might refer to different confidence ranges (see table \[tab:redshift\_list\] for the original sources). []{data-label="fig:energy_flux_vs_z"}](flux_z_plot_best_model_mark_image.pdf){width="65.00000%"} ![ Comparison of the distributions of the time-averaged ($T_{100}$) energy flux in 15-150 keV for the GRBs with redshifts and all GRBs (only bursts with acceptable spectral fits are included). []{data-label="fig:energy_flux_compare"}](energy_flux_avg_pow_all_best_model_redshift_compare.pdf){width="65.00000%"} Figure \[fig:energy\_flux\_compare\] compares the distributions of the time-averaged ($T_{100}$) energy flux in $15-150$ keV for the GRBs with redshifts and all GRBs (but only those bursts with acceptable spectral fits are included). The plot is shown in the normalized number of GRBs, since there are $\sim 3$ times more bursts in the all-GRB sample than in the GRB-with-redshift sample. Results show that the two distributions are very similar, which implies that the successful redshift measurements from the ground-based follow-up facilities have no correlation with how bright the burst is in the BAT energy range. ### Burst duration versus redshift {#sect:T90_vs_z} Many studies have shown that the observed burst durations do not present a clear effect of time dilation for GRBs at higher redshift [e.g., @Sakamoto11; @Kocevski13; @Littlejohns14]. One likely explanation is the “tip-of-the-iceberg” effect, where a larger fraction of the burst emission becomes hidden underneath the background noise, as the brightness of the GRB decreases at higher redshift. @Kocevski13 demonstrates this effect with a single-pulse structure and concludes that the observed duration can miss up to $\sim 80\%$ of the true burst duration at high redshift $z \sim 5$. ![ $T_{90}$ versus redshift $z$. [*Upper Panel:*]{} the $T_{90}$ in the observer frame as a function of $z$. The blue-dotted line shows the expected correlation between the observed burst duration and redshift for bursts with luminosity of $L_{\rm band, rest} = 10^{52} \ \rm erg \ s^{-1}$ and $10^{53} \ \rm erg \ s^{-1}$, respectively. [*Bottom Panel:*]{} the $T_{90}$ in the rest frame as a function of $z$. Bursts with photometric redshifts are marked in red specifically due to the large uncertainties in their redshift measurements. The uncertainties for redshifts are also shown when available. However, note that these uncertainties are adopted from different references and might refer to different confidence ranges (see Table \[tab:redshift\_list\] for the original sources). Moreover, GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:T90_obs_rest_vs_z"}](T90_obs_vs_z_with_fit.pdf "fig:"){width="55.00000%"} ![ $T_{90}$ versus redshift $z$. [*Upper Panel:*]{} the $T_{90}$ in the observer frame as a function of $z$. The blue-dotted line shows the expected correlation between the observed burst duration and redshift for bursts with luminosity of $L_{\rm band, rest} = 10^{52} \ \rm erg \ s^{-1}$ and $10^{53} \ \rm erg \ s^{-1}$, respectively. [*Bottom Panel:*]{} the $T_{90}$ in the rest frame as a function of $z$. Bursts with photometric redshifts are marked in red specifically due to the large uncertainties in their redshift measurements. The uncertainties for redshifts are also shown when available. However, note that these uncertainties are adopted from different references and might refer to different confidence ranges (see Table \[tab:redshift\_list\] for the original sources). Moreover, GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:T90_obs_rest_vs_z"}](T90_rest_vs_z.pdf "fig:"){width="55.00000%"} Figure \[fig:T90\_obs\_rest\_vs\_z\] shows the comparison of the rest-frame $T_{90}$ and the observer-frame $T_{90}$ as a function of redshift $z$. The rest-frame $T_{90}$ is calculated simply by correcting the (1+z) time dilation effect (i.e., rest-frame $T_{90}$ = (observer-frame $T_{90}$)/(1+z)). Indeed, there is no obvious trend from time dilation. However, the observed $T_{90}$ seems to show some correlation between the minimum detectable $T_{90}$ with redshift. As discussed in Section \[sect:BAT\_sensitivity\], BAT can in general detect lower fluxes for bursts with longer burst durations. Therefore, one would generally expect that BAT can only detect longer bursts at higher redshift for GRBs with a specific intrinsic luminosity. In other words, the correlation between the minimum detectable flux (at $5 \sigma$) in the observed energy band $F_{\rm band, obs}$ and the exposure time $T$ [Eq. 9 in @Baumgartner13]: F\_[band, obs]{} = 1.18 mCrab ( )\^[-1/2]{} = 2.86 10\^[-11]{}  \[erg  s\^[-1]{}  cm\^[-2]{}\] ( )\^[-1/2]{} requires a longer exposure time to detect lower flux, and in the GRB case the maximum possible exposure time is determined by the burst duration. The observed flux in the observed energy band is correlated with the luminosity in the corresponding redshifted bandpass as F\_[band, obs]{} = , where $D_{\rm L}$ is the luminosity distance and $L_{\rm band, rest}$ here refers to the luminosity in an energy band that corresponds to the observed energy band, not the bolometric luminosity (see detailed descriptions in Appendix \[sect:lum\_flux\]). Thus, for a specific luminosity, one can get the following correlation between the exposure time and the redshift, ( )\^[1/2]{} = 2.86 10\^[-11]{}  \[erg  s\^[-1]{}  cm\^[-2]{}\]  . For simplicity, we approximate the exposure time by the burst duration, i.e., assuming T = $T_{90}$, and plot this relation as the blue-dotted lines in Fig. \[fig:T90\_obs\_rest\_vs\_z\] (for $L_{\rm band, rest} = 10^{52} \ \rm erg \ s^{-1}$ and $10^{53} \ \rm erg \ s^{-1}$, respectively). As expected, a larger luminosity corresponds to smaller burst duration throughout the redshift. As one can see in the plot, both of the blue-dashed lines lie below all but one of the detected burst with redshift measurements, leaving a relatively large empty space between the blue dotted line with $L_{\rm band, rest} = 10^{52} \ \rm erg \ s^{-1}$ and the detected bursts with the shortest durations at each redshift. Naively, one might expect that this indicates that there are no bursts with $L_{\rm band, rest} > 10^{52} \ \rm erg \ s^{-1}$, otherwise BAT should have detected them even if they have durations that lies below the blue dotted line. However, when we calculate the GRB luminosities in the following subsection (see Fig. \[fig:lum\_vs\_z\]), more than one bursts have luminosities exceeding $10^{52} \ \rm erg \ s^{-1}$. Therefore, we are only missing bright “short” bursts that should have been detectable, and not all detectable bright bursts. Another possibility to explain these missing bright short bursts in the plot would be the missing redshift measurements. In other words, we might have detected these bright short bursts, but do not have redshift measurements and thus they are not included in this plot. Unfortunately, it remains difficult to distinguish these two possibilities due to the biases and incompleteness of redshift measurements. The GRB community has commonly used the observed burst duration to classify GRBs and to infer different physical origins, with the long GRBs related to deaths of massive starts, and short bursts linked to compact-object mergers. However, the burst duration in the observed frame is affected by several biases, such as the tip-of-the-iceberg effect mention above and the time-dilation effect, and thus might not represent the true duration of a burst. Although it is difficult to recover the intrinsic total burst duration from the tip-of-the-iceberg effect, we can easily correct the time-dilation effect and calculate the rest-frame $T{90}$ by dividing the observed $T_{90}$ by the (1+z) factor. Figure \[fig:compare\_T90\] compares the rest-frame $T_{90}$ distribution and the observer-frame $T_{90}$ distribution. For the observer-frame $T_{90}$ distribution, we only include GRBs with redshift measurements, in order to have a fair comparison with the rest-frame $T_{90}$ distribution because there are more long bursts that have redshift measurements than short bursts. Results show that the “tail” of short bursts in the distribution become slightly less significant in the rest-frame $T_{90}$ histogram. There are 21 bursts with the observer-frame $T_{90} > 2$ s, but have the rest-frame $T_{90} \leq 2$ s. However, this rest-frame $T_{90}$ distribution is still unlikely to represent the distribution of the intrinsic burst duration due to other observational biases. ![ Comparison between the rest-frame $T_{90}$ distributions and the observer-frame $T_{90}$ distributions for the BAT GRBs that have redshift measurements. Note that GRBs with unconstrained durations, such as the ultra-long GRBs, are not included in this plot. []{data-label="fig:compare_T90"}](T90_bin_plot_with_redshift.pdf){width="60.00000%"} ### Spectral characteristics versus redshift: searching for hints of the intrinsic spectral shapes and energy outputs ![ The time-averaged ($T_{100}$) photon index $\alpha_{\rm PL}$ (for the bursts that are better fitted by the simple PL model) as a function of redshift $z$. Bursts with photometric redshifts are marked in blue specifically, due to the large uncertainties in their redshift measurements. The uncertainties for redshifts are also shown when available. However, note that these uncertainties are adopted from different references and might refer to different confidence ranges (see table \[tab:redshift\_list\] for the original sources). []{data-label="fig:alpha_vs_z"}](alpha_PL_z_plot_witherr.pdf){width="80.00000%"} Figure \[fig:alpha\_vs\_z\] shows the time-averaged ($T_{100}$) photon index $\alpha_{\rm PL}$ (for those GRBs that are better fitted by the simple PL model) as a function of redshift $z$. Again, the bursts with photometric redshifts are marked in blue specifically, due to the larger uncertainties in their redshift measurements. Similar to the same figure shown in the BAT2 catalog, there is no clear correlation between $\alpha_{\rm PL}$ and redshift. However, there are probably some instrumental selection biases for the photon indices of the BAT-detected GRBs. As discussed in Section \[sect: BAT\_selection\_effect\_spectrum\], BAT is most likely to detect bursts with a certain range of $\alpha_{\rm PL}$ that gives higher fluxes in the BAT energy range. Moreover, the burst needs to be bright enough to have a spectrum with uncertainties small enough to distinguish the simple PL and CPL model. Therefore, the distribution of $\alpha_{\rm PL}$ might not represent the true intrinsic distribution. ![ GRB luminosity in the rest-frame energy band that corresponds to the observed 15-150 keV band versus redshift. The luminosity is calculated using the best-fit model (either PL or CPL) in the T100 range, with assumption that the emission is isotropic. Note that these luminosities are for different rest-frame energy bands for GRBs at different redshifts. []{data-label="fig:lum_vs_z"}](lum_z_plot_best_model_15_150keV_with_line.pdf){width="80.00000%"} Figure \[fig:lum\_vs\_z\] shows the GRB luminosity in the observed 15-150 keV band as a function of redshift. Due to the limited energy range of the BAT, we can only constrain the energy emission within the BAT energy range. Extrapolating the spectral fits beyond the BAT energy range is probably not a good approximation because the turn-over point in the spectrum (i.e., $E_{\rm peak}$) might happen somewhere above the BAT energy limit, and thus the total luminosity calculated by extrapolating the spectral fits from the BAT spectrum can be over-estimated. We therefore calculate the luminosity in the observed 15-150 keV (the BAT energy range). However, this luminosity will correspond to a different rest-frame energy range for a GRB at a different redshift. Specifically, the luminosity is calculated by the following equations: $$\begin{aligned} L_{\rm band, rest} = \int^{E^{\rm max}_{\rm rest}}_{E^{\rm min}_{\rm rest}} L_{E,{\rm rest}} \ dE_{\rm rest} \nonumber &= \int^{E^{\rm max}_{\rm rest}}_{E^{\rm min}_{\rm rest}} \frac{4 \pi D^2_L}{1+z} \ F_{E,{\rm obs}} \ dE_{\rm rest} \nonumber \\ & = \int^{E^{\rm max}_{\rm obs}}_{E^{\rm min}_{\rm obs}} \frac{4 \pi D^2_L}{1+z} \ F_{E, {\rm obs}} \ (1+z) dE_{\rm obs} \nonumber \\ & = \int^{E^{\rm max}_{\rm obs}}_{E^{\rm min}_{\rm obs}} 4 \pi D^2_L \ F_{E,{\rm obs}} \ dE_{\rm obs} \nonumber \\ &= 4 \pi D^2_L \ F_{\rm band, obs} \end{aligned}$$ where $F_{\rm band, obs} = \int^{E^{\rm max}_{\rm obs}}_{E^{\rm min}_{\rm obs}} F_{E,{\rm obs}} \ dE_{\rm obs}$ refers to the flux in the observed energy range, and $L_{\rm band, rest}$ corresponds to the luminosity in the rest-frame energy range $E^{\rm min}_{\rm rest} = E^{\rm min}_{\rm obs}(1+z)$ to $E^{\rm max}_{\rm rest} = E^{\rm max}_{\rm obs}(1+z)$. More detail descriptions can be found in Appendix \[sect:lum\_flux\]. As expected, there is a clear correlation between the minimum luminosity of detected burst with redshift, which is mainly from the Malmquist bias. The black lines in Fig. \[fig:lum\_vs\_z\] demonstrate the effect of Malmquist bias due to the sensitivity of BAT. The solid line shows the minimum detectable luminosity in the observed 15-150 keV band assuming a $T_{90}$ of $100$ s. The dashed black line shows the same detectable luminosity but assuming a redshift-dependent $T_{90}$ of $100/(1+z)$ s. Note that this $T_{90}$ becomes shorter at higher redshift, contrary to the expectation of time dilation effect. As discussed in Section \[sect:T90\_vs\_z\], there is no clear evidence of the time-dilation effect in the burst durations, which is likely because a larger fraction of the bursts are buried under noisy background as the burst become dimmer at higher redshift. Some statistics of observational constraints -------------------------------------------- Figure \[fig:GRB\_sun\_angle\] shows the normalized distributions of both the GRB Sun angle (red bars) and the BAT boresight Sun angle (blue bars) (i.e., the angle between BAT’s pointing direction and the Sun). For the GRB Sun angles, a $1 \sigma$ error estimated from the Poisson distribution (i.e. $\sqrt{N}$) for each bin is also plotted. The normalized numbers for the BAT boresight Sun angle are calculated from the Sun angle recorded every 5 sec (excluding the time during SAA and spacecraft slews) from 2005 to August 2015. Thus, the number for the BAT boresight Sun angle represents the fraction of time that BAT spends at each Sun angle. Both distributions generally follow the sin$\theta$ shape (where $\theta$ is the Sun angle), which is expected from the amount of solid angle covered by the same angular range $\delta \theta$ (for example, there is more area covered within 80 to 90 degrees than from 170 to 180 degrees). This relation of the solid angle versus Sun angle is shown as the black lines in the plots. The sharp drop off for the BAT boresight Sun angle at around 40 degree is due to the Sun constraints for the XRT, UVOT, and the star-tracker at $\sim 44$ degrees[^27]. There are some GRBs detected with Sun angle less then $\sim 44$ degrees because the BAT has a large field of view. These bursts will have Sun constraints for prompt XRT and UVOT observations. This figure shows that the number of GRB detections generally follows the fraction of time that BAT spends at the location. The GRB detection rate seem to be slightly lower than average when BAT is pointing close to the Sun. However, this effect is not very significant and is still consistent with the BAT pointing time (the blue bars) within $\sim 2-3 \sigma$ of the Poisson errors from the number of detections. ![ Distributions of the Sun angles of the GRBs (red bars) and the BAT boresight (blue bars). The $1-\sigma$ poisson errors are plotted for the bins of GRB detections. For comparison, the black lines mark the solid angle area for the Sun angle range in each bin. []{data-label="fig:GRB_sun_angle"}](BAT_GRB_Sun_angle.pdf){width="70.00000%"} Similar to Fig. \[fig:GRB\_sun\_angle\], the distributions of the Moon angles for both the GRBs (red bars) and the BAT boresights (blue bars) are plotted in Fig. \[fig:GRB\_moon\_angle\]. The number of GRB detections also follows the fraction of time that BAT spends at each Moon angle location, as expected. Again, the drop off at $\sim 20$ degrees in the distribution of the BAT boresight moon angle is due to the Moon constraints for XRT, UVOT, and the star tracker at $\sim 19$ degrees^\[constraint\_web\]^). ![ Distributions of the Moon angles of the GRBs (red bars) and the BAT boresight (blue bars). The $1-\sigma$ poisson errors are plotted for the bins of GRB detections. For comparison, the black lines mark the solid angle area for the Sun angle range in each bin. []{data-label="fig:GRB_moon_angle"}](BAT_GRB_moon_angle.pdf){width="70.00000%"} GRB analysis with the BAT survey data {#sect:survey_data_analysis} ===================================== To perform further search of pre-burst or extended GRB emissions beyond the period of the event data, we use the results of the 104-month survey analysis, which is an extension of the previously published 70-month survey data analyses [@Baumgartner13]. The survey process performs standard survey analysis using the script [*batsurvey*]{}[^28], and generates mask-weighted, cleaned images for each observations in eight energy bands (14-20, 20-24, 24-35, 35-50, 50-75, 75-100, 100-150, 150-195 keV). We select a sub-sample of these images that have times close to the GRB trigger time. In addition, we exclude the images with times overlaps roughly with the event data range when we perform the search. In other words, we adopt the following criteria for the search: 1. For GRBs occurred after GRB060319, we search in survey images with time that covers $T0-0.2$ day to $T0-50$ s, and $T0+500$ s to $T0+1$ day, where $T0$ is the BAT trigger time. 2. For GRBs occurred before (and include) GRB060319, we search in survey images with time that covers $T0-0.2$ day to $T0-50$ s, and $T0+300$ s to $T0+1$ day. The two different criteria is required because BAT downlinks a shorter range of event data ($\sim T0-250$ s to $\sim T0+300$ s) in early mission time, and extends the event data range after GRB060319. We then estimate the signal-to-noise ratio of the GRB locations in these images using the standard BAT analysis script [*batcelldetect*]{}[^29]. We use GRB locations estimated by the XRT, which has a resolution of $\sim$ few arcsec. We also include a list of 21 very bright X-ray sources (e.g., Crab, Vela X-1, Cyg X-1...etc) in the input source catalog of [*batcelldetect*]{}, in case there is any residue of these bright sources in the background area that might affect detections at the GRB locations. Furthermore, we examine the resulting detection times in survey data to make sure those are not already reported using the event data analysis. Note that because the survey process only includes data from Dec. 2004 to Aug. 2013, we only search through possible GRB emissions in survey data until Aug. 2013. False-detection rate: searching for weak emission {#sect:false_detection-rate} ------------------------------------------------- We perform a study of the false-detection rate in order to find a reliable criteria to search for weak emission. To quantify the false detection rate, we estimate the signal-to-noise ratio using background locations around GRBs. We choose the background locations to be $\sim 1$ deg from the GRBs (so most of the time the background detection is from the same images as the GRB detections), and also $\sim 1$ deg from other X-ray sources. We adopt the X-ray source list from @Krimm13. We quantify the false-detection rate $R_{\rm false}$ in a particular energy band with a specific signal-to-noise ratio threshold as follows, R\_[false]{} = , where $\rm N ({>SNR_{lim}})$ is the number of survey images with the background signal-to-noise ratio at the specific location higher than the assigned threshold. $N_{\rm tot}$ is the total number of survey images we included in the search, that is, the subset of all survey images that are close to GRB trigger times, as described above. Note that because of the different event data ranges for the earlier mission time and the later mission time, the false-detection rate study uses images that satisfy criterion 1 mentioned in previous section. The image exposure times can vary from $\sim 300 \ \rm s$ to $\sim 2500 \ \rm s$, with the majority of the exposure time around few hundred seconds. Ideally, one would require the observation time of each image to be identical to have a fair comparison of the signal-to-noise ratio in each image. However, because the survey process [@Baumgartner13] only produces one image for each observation (i.e., using the “SNAPSHOT” option in [*batsurvey*]{}$^{\ref{note:batsurvey}}$), our estimation can only based on these images with different exposure times. To produce survey images with finer time bins in each observation (i.e., using the “DPH” option in [*batsurvey*]{}$^{\ref{note:batsurvey}}$), one would need to re-process all survey data and the process will take $\sim$ months to years to finish, and hence would be beyond the scope of this paper. We quantify the false-detection rate for a range of different signal-to-noise ratio thresholds (from $\sim$ 2.0 to $\sim$ 5.0) in different energy bands. The energy ranges we tried include the eight energy bands used by the survey process (14-20 keV, 20-24 keV, 24-35 keV, 35-50 keV, 50-75 keV, 100-150 keV, 150-195 keV), the total energy band (14-195 keV), an energy band that combines the three soft bands (14-35 keV), and an energy band that covers the three energy bands with the largest effective area (35-100 keV). Figure \[fig:snr\_histogram\] shows an example of the resulting histogram of the signal-to-noise ratio at the background locations in images with energy 14-195 keV. Table \[tab:snr\_bgd\] lists some of our calculations of false-detection rate in the most interesting ranges of signal-to-noise ratio threshold. The numbers in parenthesis following each false-detection rate are the expected numbers of false detections out of the whole image samples in our study (i.e., a total number of 19182 images). ![ The histogram of signal-to-noise ratio at the background locations in images with energy 14-195 keV, from Dec. 2004 to Aug. 2013. []{data-label="fig:snr_histogram"}](GRB_bgd_snr9.pdf){width="80.00000%"} We investigate the expected detection rate for each criteria, and select some potentially useful criteria to perform further tests by calculating the signal-to-noise ratios at the GRB locations. This gives us a total number of real detections plus false detections at each GRB location. We search through each criterion until we find one that gives the largest ratio of the number of detections at the GRB location $N_{\rm GRB\_locations}$ (i.e., number of real plus false detections) over the number of detections at the background locations $N_{\rm bgd\_location}$ (i.e. false detections). In other words, we demand the ratio r\_[detect]{} = to be as large as possible. Table \[tab:snr\_criteria\] presents a list of criteria that we tried, and the results of false-detection rates, actual number of detections at the GRB locations, and the number of detections at the background locations. Sometimes each location can have multiple detections at different times. Thus, the numbers in detected locations (i.e., each location only counted once even when they are detected at different times) are shown in the parenthesis. The criterion using images with the energy band 14-195 keV and signal-to-noise ratio threshold above 4.3 sigma turns out to be the one that has the highest $r_{\rm detect}$. We thus adopt this criterion to search for possible emissions in survey data. Results are summarized and discussed in Section \[sect:survey\_results\] [@|\*[15]{}[c|]{}]{} Energy & SNR $\geq$ 5.0 & SNR $\geq$ 4.9 & SNR $\geq$ 4.8 & SNR $\geq$ 4.7 & SNR $\geq$ 4.6 & SNR $\geq$ 4.5 & SNR $\geq$ 4.4\ 14-20 &0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&$5.21 \times 10^{-5}$ (1)&$1.56 \times 10^{-4}$ (3)\ 20-24 &$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)\ 24-35 &0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)\ 35-50 &$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)&$1.04 \times 10^{-4}$ (2)&$1.56 \times 10^{-4}$ (3)\ 50-75 &0.0 (0)&0.0 (0)&0.0 (0)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$1.04 \times 10^{-4}$ (2)\ 75-100 &0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&$1.56 \times 10^{-4}$ (3)&$2.08 \times 10^{-4}$ (4)\ 100-150 &$1.04 \times 10^{-4}$ (2)&$1.56 \times 10^{-4}$ (3)&$1.56 \times 10^{-4}$ (3)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$2.60 \times 10^{-4}$ (5)\ 150-195 &0.0 (0)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)\ 14-195 &0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&0.0 (0)&$5.21 \times 10^{-5}$ (1)\ 14-35 &0.0 (0)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$2.08 \times 10^{-4}$ (4)\ 35-100 &$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$5.21 \times 10^{-5}$ (1)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)\ Energy & SNR $\geq$ 4.3 & SNR $\geq$4.2 & SNR $\geq$ 4.1 & SNR $\geq$4.0 & SNR $\geq$ 3.9 & SNR $\geq$ 3.8 & SNR $\geq$ 3.7\ 14-20 &$1.56 \times 10^{-4}$ (3)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$6.77 \times 10^{-4}$ (13)&$6.77 \times 10^{-4}$ (13)&$1.20 \times 10^{-3}$ (23)\ 20-24 &$5.21 \times 10^{-5}$ (1)&$2.60 \times 10^{-4}$ (5)&$2.60 \times 10^{-4}$ (5)&$3.12 \times 10^{-4}$ (6)&$6.25 \times 10^{-4}$ (12)&$6.25 \times 10^{-4}$ (12)&$1.04 \times 10^{-3}$ (20)\ 24-35 &$1.04 \times 10^{-4}$ (2)&$2.60 \times 10^{-4}$ (5)&$2.60 \times 10^{-4}$ (5)&$4.17 \times 10^{-4}$ (8)&$6.25 \times 10^{-4}$ (12)&$6.25 \times 10^{-4}$ (12)&$1.25 \times 10^{-3}$ (24)\ 35-50 &$1.56 \times 10^{-4}$ (3)&$1.56 \times 10^{-4}$ (3)&$1.56 \times 10^{-4}$ (3)&$3.12 \times 10^{-4}$ (6)&$4.69 \times 10^{-4}$ (9)&$4.69 \times 10^{-4}$ (9)&$7.29 \times 10^{-4}$ (14)\ 50-75 &$1.04 \times 10^{-4}$ (2)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$3.12 \times 10^{-4}$ (6)&$5.73 \times 10^{-4}$ (11)&$5.73 \times 10^{-4}$ (11)&$9.89 \times 10^{-4}$ (19)\ 75-100 &$2.08 \times 10^{-4}$ (4)&$3.12 \times 10^{-4}$ (6)&$3.12 \times 10^{-4}$ (6)&$3.64 \times 10^{-4}$ (7)&$5.73 \times 10^{-4}$ (11)&$5.73 \times 10^{-4}$ (11)&$8.33 \times 10^{-4}$ (16)\ 100-150 &$2.60 \times 10^{-4}$ (5)&$3.64 \times 10^{-4}$ (7)&$3.64 \times 10^{-4}$ (7)&$3.64 \times 10^{-4}$ (7)&$5.21 \times 10^{-4}$ (10)&$5.21 \times 10^{-4}$ (10)&$9.89 \times 10^{-4}$ (19)\ 150-195 &$5.21 \times 10^{-5}$ (1)&$1.56 \times 10^{-4}$ (3)&$1.56 \times 10^{-4}$ (3)&$2.60 \times 10^{-4}$ (5)&$5.73 \times 10^{-4}$ (11)&$5.73 \times 10^{-4}$ (11)&$9.37 \times 10^{-4}$ (18)\ 14-195 &$5.21 \times 10^{-5}$ (1)&$2.08 \times 10^{-4}$ (4)&$2.08 \times 10^{-4}$ (4)&$5.73 \times 10^{-4}$ (11)&$7.81 \times 10^{-4}$ (15)&$7.81 \times 10^{-4}$ (15)&$1.30 \times 10^{-3}$ (25)\ 14-35 &$2.08 \times 10^{-4}$ (4)&$4.69 \times 10^{-4}$ (9)&$4.69 \times 10^{-4}$ (9)&$5.21 \times 10^{-4}$ (10)&$1.04 \times 10^{-3}$ (20)&$1.04 \times 10^{-3}$ (20)&$1.77 \times 10^{-3}$ (34)\ 35-100 &$2.08 \times 10^{-4}$ (4)&$3.12 \times 10^{-4}$ (6)&$3.12 \times 10^{-4}$ (6)&$3.64 \times 10^{-4}$ (7)&$6.25 \times 10^{-4}$ (12)&$6.25 \times 10^{-4}$ (12)&$1.25 \times 10^{-3}$ (24)\ ---------------------- ----------- ---------------------- ------------------ ------------------------- ------------------ Criterion SNR False-detection At GRB locations At background locations $r_{\rm detect}$ Energy bands threshold rate \# of detections \# of detections any 1 of the 8 bands 3.0 $3.9 \times 10^{-2}$ 778 (479 GRBs) 719 (338 backgrounds) 1.0 (1.42) any 1 of the 8 bands 4.0 $2.6 \times 10^{-3}$ 68 (58 GRBs) 49 (47 backgrounds) 1.39 (1.23) any 1 of the 8 bands 4.5 $7.8 \times 10^{-4}$ 24 (18 GRBs) 14 (13 backgrounds) 1.71 (1.38) any 1 of the 8 bands 4.8 $3.6 \times 10^{-4}$ 18 (14 GRBs) 6 (6 backgrounds) 3.0 (2.33) any 2 of the 8 bands 3.0 $7.0 \times 10^{-3}$ 27 (20 GRBs) 25 (23 backgrounds) 1.08 (0.87) any 2 of the 8 bands 3.5 $2.8 \times 10^{-5}$ 12 (7 GRBs) 3 (2 backgrounds) 4.0 (3.5) any 3 of the 8 bands 3.0 $7.0 \times 10^{-6}$ 11 (7 GRBs) 0 (0 backgrounds) N/A (N/A) any 3 of the 8 bands 2.5 $1.9 \times 10^{-4}$ 17 (12 GRBs) 11 (10 backgrounds) 1.55 (1.2) any 3 of the 8 bands 2.8 $2.9 \times 10^{-5}$ 12 (8 GRBs) 2 (2 backgrounds) 6.0 (4.0) both 50-75 keV and 75-100 keV bands 3.0 $2.1 \times 10^{-5}$ 5 (3 GRBs) 0 (0 backgrounds) N/A (N/A) 14-35 keV band 4.0 $5.2 \times 10^{-4}$ 18 (13 GRBs) 10 (10 backgrounds) 1.8 (1.3) 14-35 keV band 4.3 $2.1 \times 10^{-4}$ 16 (11 GRBs) 4 (4 backgrounds) 4.0 (2.75) 14-35 keV band 4.5 $5.2 \times 10^{-5}$ 14 (9 GRBs) 1 (1 backgrounds) 14.0 (9.0) 14-195 keV band 4.3 $5.2 \times 10^{-5}$ 19 (14 GRBs) 1 (1 backgrounds) 19.0 (14.0) 14-195 keV band 4.0 $5.7 \times 10^{-4}$ 28 (22 GRBs) 11 (11 backgrounds) 2.55 (2.0) 35-100 keV band 4.8 $5.2 \times 10^{-5}$ 11 (8 GRBs) 1 (1 backgrounds) 11.0 (8.0) 35-100 keV band 4.3 $2.1 \times 10^{-4}$ 14 (11 GRBs) 4 (4 backgrounds) 3.5 (2.75) 35-100 keV band 4.0 $3.6 \times 10^{-4}$ 18 (15 GRBs) 7 (7 backgrounds) 2.57 (2.14) ---------------------- ----------- ---------------------- ------------------ ------------------------- ------------------ : A summary of all the criteria we examined$^{\ref{survey_table_footnote}}$. []{data-label="tab:snr_criteria"} [^30] Results of BAT survey data analysis: possible ultra-long GRBs {#sect:survey_results} ============================================================= We find 21 detections (16 GRBs) beyond the event data range, which are summarized in Table \[tab:survey\_GRB\]. Most of these detections happened after the BAT trigger times. However, there are two detections (GRB100316D and GRB101024A) that occurred before the BAT trigger times. Within these detections, 7 GRBs are previously classified as ultra-long GRBs, which are GRB121027A, GRB111215A, GRB111209A, GRB101225A, GRB100316D, GRB090417B, GRB060218 [e.g., @Virgili13; @Gendre13; @Levan14; @Boer15]. Studies usually refer “ultra-long GRBs” to bursts with durations $\gtrsim$ kiloseconds [e.g., @Gendre13; @Stratta13; @Levan14; @Evans14; @Boer15], however, no unified definition has been adopted. In this paper, an ultra-long GRB is referred to those events with observed durations longer than 1000 s in BAT energy band. Note that the ultra-long GRB130925A [@Evans14; @Piro14] detected by BAT is not in the list of GRBs with confirmed detection in survey data (Table \[tab:survey\_GRB\_duration\]), because the currently existing survey data product required for the analysis ends before this burst. Table \[tab:survey\_GRB\_duration\] compares the total burst duration in BAT, and the $T_{90}$ estimated from the event data. The burst duration from both the event and survey data is a rough estimation from the beginning of the $T_{100}$ (or the beginning of the event data if the burst emission starts beforehand) plus the time of middle point of the last survey detection time bin. For those two precursor detection, the duration is estimated from the middle point of the survey detection to the end of $T_{100}$. There are 15 GRBs in Table \[tab:survey\_GRB\_duration\] with estimated duration longer than 1000 s when including the emissions in survey data, and thus are considered as ultra-long GRBs with our definition. However, we note the potentially large uncertainty in the burst duration estimations due to the large time bin (i.e., long exposure time) of some survey data. Moreover, we expect one false detection in this sample, based on the false-detection rate study mentioned in previous section. Seven bursts in this sample have durations exceeding the event data range (and thus only a lower limit of $T_{90}$ can be determined). All of these seven bursts are previously recognized as ultra-long GRBs. The rest of the bursts, however, show a rather diverse $T_{90}$, ranging from a few seconds to a few hundred seconds. If using the lower limit for bursts without constraint $T_{90}$), the median of $T_{90}$ from event data for these ultra-long bursts is 193 s. Only $\sim 6\%$ of the total BAT GRBs with constraint burst durations have $T_{90}$ larger than 193 s. Note that there are two detections (GRB060218 and GRB050730) that happen within 500 sec after the BAT trigger times. This is because earlier in the mission, BAT downlinked a shorter range of event data that only covers until $\sim 300$ s after trigger time. Therefore, these two detections would have been covered by event data range if the GRBs had occurred more recently. Also, GRB080319B is the well-known naked-eye burst [@Racusin08]. It is possible that the extraordinary brightness of this burst is the main reason for the event being detectable for a long time in BAT for $\sim 1340$ sec. Thus, one needs to be cautious when exploring potential physical causes of these late time BAT detections, particularly for brighter GRBs in our sample. -------------------- ---------------------------------------- --------------------------- ------------------- GRB name Start time of detection exposure \[s\] Image exposure time \[s\] SNR in 14-195 keV (relative to the BAT trigger time) GRB121027A$^\star$ 1327.45 496.0 19.32 GRB121027A$^\star$ 5351.45 732.0 11.64 GRB111215A$^\star$ 703.0 840.0 12.27 GRB111209A$^\star$ 4814.0 2600.0 40.98 GRB111209A$^\star$ 10606.0 2584.0 14.08 GRB111209A$^\star$ 16427.0 2400.0 7.58 GRB111209A$^\star$ 565.0 630.0 92.73 GRB101225A$^\star$ 1372.0 300.0 10.28 GRB101225A$^\star$ 4936.0 2601.0 4.55 GRB101024A -5252.13 779.0 4.73 GRB100728A 981.73 792.0 4.83 GRB100316D$^\star$ -775.0 600.0 9.01 GRB091127 5192.90 409.0 4.36 GRB090417B$^\star$ 662.0 1140.0 23.51 GRB090404 44356.93 557.0 4.31 GRB090309A 4075.176 2400.0 4.40 GRB080319B$^\star$ 938.1 799.0 11.26 GRB070518 57158.83 1381.0 4.92 GRB070419B 3724.13 2400.0 5.22 GRB060218$^\star$ 404.0 2327.0 19.20 GRB050730 356.2 390.0 8.53 -------------------- ---------------------------------------- --------------------------- ------------------- : \[tab:survey\_GRB\] List of GRBs detected in survey data with signal-to-noise ratio $> 4.3 \sigma$ in 14-195 keV. We expect on average $\sim 1$ false detection in our search sample. The GRBs that are detected in all the criteria we investigated (see Section \[sect:false\_detection-rate\]) are noted by $\star$. -------------------- ---------------------- ---------------------------- ----------------- GRB name Number of detections Burst duration from $T_{90}$ in survey data both event and survey data from event data GRB121027A$^\star$ 2 $\sim 5727$ s 80 s GRB111215A$^\star$ 1 $\sim 1121$ s $> 374$ s GRB111209A$^\star$ 4 $\sim 18181$ s $>811$ s GRB101225A$^\star$ 2 $\sim 6416$ s $> 1377$ s GRB101024A 1 $\sim 4883$ s 18.7 s GRB100728A 1 $\sim 1457$ s 193 s GRB100316D$^\star$ 1 $\sim 1270$ s $> 522$ s GRB091127 1 $\sim 5398$ s 7.42 GRB090417B$^\star$ 1 $\sim 1471$ s $> 267$ s GRB090404 1 $\sim 44671$ s 82 s GRB090309A 1 $\sim 5276$ s 3 s GRB080319B$^\star$ 1 $\sim 1341$ s $>125$ s GRB070518 1 $\sim 57851$ s 5.5 s GRB070419B 1 $\sim 4934$ s 238 s GRB060218$^\star$ 1 $\sim 1624$ s $> 602$ s GRB050730 1 $\sim 608$ s 155 s -------------------- ---------------------- ---------------------------- ----------------- : \[tab:survey\_GRB\_duration\] Comparison of the GRB duration estimated using only event data and using both event and survey data. Again, GRBs that are detected in all the criteria we investigated (see Section \[sect:false\_detection-rate\]) are noted by $\star$. Although we found relatively small number of detections in the survey data, this does not necessarily imply that the GRB emissions usually finish before the end of the event data. The survey data are binned in $\gtrsim 5$ minutes, and thus it is not sensitive to late-time burst emissions if they occur on a much shorter time scale. In fact, throughout the process of inspecting the burst light curves created by event data by eye, we noted 52 GRBs with incomplete burst durations, which are likely to have burst emissions beyond the event data range. Interestingly, there are 9 GRBs (GRB121027A, GRB101024A, GRB100728A, GRB091127, GRB090404, GRB090309, GRB070518, GRB070419B, and GRB050730) that have detections in the survey data but are not included in the list of burst with incomplete burst durations, which is consistent with the results shown in Table \[tab:survey\_GRB\_duration\] that there is no clear relation between the prompt BAT emission in the event data range and the late-time detections in the survey data. We further compare these late-time emissions to the [*Swift*]{}/XRT light curves generated by the Burst Analyser[^31] [@Evans10; @Evans09; @Evans07]. The Burst Analyser can plot the GRB light curves from both the BAT event data and the XRT data in the 15-50 keV range. The equivalent XRT fluxes in the 15-50 keV range are estimated by extrapolating the XRT spectrum in 0.3-10 keV. We also calculate the BAT fluxes in the 15-50 keV range by fitting the simple PL model to the BAT spectra generated from the survey data (eight energy bands). Figures \[fig:lc\_1\] to \[fig:lc\_precursor\] overlay the BAT detections in the survey data on top of the observations from the BAT event data and the XRT data. Results show that from most of the late-time detections in the survey data, the BAT extended emissions generally follow the behavior seen in the XRT light curves, and the photon indices from the simple PL fit ($\alpha^{\rm PL}$) are similar to the ones derived from the XRT data. However, there are three bursts (GRB070518, GRB090309, and GRB090404) that show late time fluxes much larger than what are measured in the XRT. The light curves of these bursts are shown in Fig. \[fig:lc\_weird\]. Within these three peculiar bursts, the spectral fits from GRB090309 is problematic with un-constrained energy flux and photon index. Moreover, GRB090309 turns out to have $T_{90} = 3 \pm 1.4$ s, which makes this a possible short burst. We found two bursts, GRB100316D and GRB101024A, with detections in survey data before the starting of event data. The light curves for these two bursts with precursor detections are plotted in Fig. \[fig:lc\_precursor\]. The time in the x-axis is shifted to prevent the problem of plotting a negative number on a log scale. Also, the spectral fits from GRB101024A is problematic and the flux might not be accurate. Precursors of GRB emissions are suggested in some of the theoretical models [e.g., @Yamazaki09]. However, the fraction of precursor we found in this study is very low, suggesting that the precursor, if exists, would be either weaker than the BAT’s sensitivity in a regular survey image exposure time of $\sim$ few thousand seconds, or the emission is much shorter than the survey data exposure time and thus the signal is greatly reduced by the background noise. ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB111209A_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB121027A_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB050730_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB060218_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB070419B_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_1"}](GRB080319B_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_2"}](GRB090417B_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_2"}](GRB091127_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_2"}](GRB100728A_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_2"}](GRB101225A_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). []{data-label="fig:lc_2"}](GRB111215A_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. These GRBs have late time BAT emissions that do not match the XRT emissions extrapolated to the same energy range. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). For GRB090309A, we plot the BAT $T_{100}$ flux instead because there are no BAT fluxes produced by the Burst Analyser [@Evans10] for this burst. []{data-label="fig:lc_weird"}](GRB070518_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. These GRBs have late time BAT emissions that do not match the XRT emissions extrapolated to the same energy range. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). For GRB090309A, we plot the BAT $T_{100}$ flux instead because there are no BAT fluxes produced by the Burst Analyser [@Evans10] for this burst. []{data-label="fig:lc_weird"}](GRB090309_survey.pdf "fig:"){width="49.00000%"} ![ Comparisons between detections in the BAT survey data, BAT event data, and the XRT data. These GRBs have late time BAT emissions that do not match the XRT emissions extrapolated to the same energy range. The end of $T_{100}$ is marked as black-dashed line in each figure. If the burst emissions extend beyond the event data range, the black-dashed line marks the lower-limit of $T_{100}$ (i.e., the end of $T_{100}$ found by [*battblocks*]{}, or the end of the event data range if [*battblocks*]{} failed to estimate the burst durations). For GRB090309A, we plot the BAT $T_{100}$ flux instead because there are no BAT fluxes produced by the Burst Analyser [@Evans10] for this burst. []{data-label="fig:lc_weird"}](GRB090404_survey.pdf "fig:"){width="49.00000%"} ![ Precursor detections in the BAT survey data, overlayed on the BAT event data and XRT data for comparison. The x-axes are shifted to avoid negative regions in the plots. The BAT trigger times $T0$ are marked as green-dashed lines. Both the start and stop time of $T_{100}$ are marked as black-dashed line for GRB100316D. These times are labeled as upper and lower limits because the burst emission extends beyond the event data range, that is, the burst starts before $T^{\rm start}_{100}$ and ends after $T^{\rm stop}_{100}$. The $T_{100}$ range (from $T0-0.25$ to $T0+20.13$s) for GRB101024A is not marked because it appears very close to the $T0$ line on the plot. []{data-label="fig:lc_precursor"}](GRB100316D_survey.pdf "fig:"){width="49.00000%"} ![ Precursor detections in the BAT survey data, overlayed on the BAT event data and XRT data for comparison. The x-axes are shifted to avoid negative regions in the plots. The BAT trigger times $T0$ are marked as green-dashed lines. Both the start and stop time of $T_{100}$ are marked as black-dashed line for GRB100316D. These times are labeled as upper and lower limits because the burst emission extends beyond the event data range, that is, the burst starts before $T^{\rm start}_{100}$ and ends after $T^{\rm stop}_{100}$. The $T_{100}$ range (from $T0-0.25$ to $T0+20.13$s) for GRB101024A is not marked because it appears very close to the $T0$ line on the plot. []{data-label="fig:lc_precursor"}](GRB101024A_survey.pdf "fig:"){width="49.00000%"} Summary {#sect:summary} ======= We present the temporal and spectral analyses of the event data for GRBs detected by BAT for the past eleven years (up to GRB151027B). Particularly, we include analyses of the burst durations $T_{100}$, $T_{90}$, and $T_{50}$, the refined positions, and report the spectral fits from both the simple power-law model (Eq. \[eq:PL\]) and the cutoff power-law model (Eq. \[eq:CPL\]). We include spectral fits for the spectra made with the $T_{100}$ duration, the 1-s peak interval, the 20-ms peak duration. Moreover, we perform spectral fits of the time-resolved spectra, with the time-resolved durations selected by the automatic pipeline [*battblocks*]{}. All the detailed numbers are summarized in tables in Appendix \[sect:appendix\_tables\], except the time-resolved durations and spectral fits due to the large amount of data. The results of the time-resolved analyses can be found on the online directories at http://swift.gsfc.nasa.gov/results/batgrbcat/index.html. Following the BAT1 and BAT2 catalog, we adopt the same criterion of $\Delta \chi^2 >6$ for determining whether the spectrum is better fitted by the CPL model than the simple PL model. Furthermore, we introduce a few more criteria to select the acceptable spectral fits, in order to ensure the parameters estimated from the spectral fits (e.g., photon index and flux) are reasonable. These criteria are listed in Section \[sect:event\_standard\]. Lists of bursts with acceptable PL and CPL fit are presented in Appendix \[sect:appendix\_tables\]. The analysis results show that the general temporal and spectral distributions of the GRBs remain similar to those reported in the BAT2 catalog, though BAT has doubled the number of GRB detections since 2009. Roughly $9\%$ of the BAT GRBs are short ($T_{90} \lesssim 2$ s), which is less than the short GRB fraction in the [*Fermi*]{} sample ($\sim 17\%$) and the BATSE sample ($\sim 26\%$). The spectral analyses show that short GRBs in the BAT sample are only slightly harder than the long bursts. A study of the distribution of GRB partial coding fraction (which is related to the burst incident angle) suggests that short GRBs are harder to detect than long bursts when the GRB is highly off-axis. Some further studies of the BAT sensitivity show that the fluxes of the BAT-detected GRBs decrease as a function of $T^{-1/2}_{90}$. That is, BAT can detect GRBs with lower flux if the burst is longer. This result is consistent with the expectation that the BAT sensitivity should improves with longer exposure time. Specifically, the sensitivity is inversely proportional to the square of the exposure time, because the noise fluctuation scales as $1/\sqrt{N}$ in counting statistics, where N is the number of detected photons. We construct a special list of 12 short GRBs with extended emissions in Section \[sect:Short\_with\_EE\], and also a list of 15 possible short GRBs with extended emissions, which either have a short pulse that is slightly longer than 2 sec, or have some complicated factors that make it difficult to determine the reliability of the extended tail. The spectral analyses confirm that the photon indices of short pulses are more similar to the short GRB population, while the spectral properties of the extended emissions resemble more closely the long burst population. Moreover, we compile a list of GRBs with redshift measurements, and discuss how some of their properties vary with redshift. We found that while the redshift distribution of the rate-trigger bursts are peaked at low redshift (as expected), the redshift distribution for the image-trigger bursts are more evenly distributed throughout the redshift range, and is composed of a higher fraction of bursts with larger redshift. The redshift properties of BAT-detected GRBs show a hint of luminosity evolution. The number of short GRBs at high redshift seems to decrease faster than one would expect by simply taking into account the BAT sensitivity as a function of burst duration, with the assumption of a non-evolving upper limit of the GRB luminosity (Fig \[fig:T90\_obs\_rest\_vs\_z\]). However, due to the incompleteness of the redshift sample, we cannot rule out the possibility of we are simply missing more redshift measurements for the short bright bursts at large redshifts. Furthermore, there seems to be a lack of low-flux GRBs at high redshift ($z \gtrsim 5$; see Fig. \[fig:energy\_flux\_vs\_z\]), and a missing population of GRBs with luminosity $\gtrsim 10^{51} \rm erg \ s^{-1} \ cm^{-2}$ at low redshift ($z \lesssim$ 1; see Fig. \[fig:lum\_vs\_z\]). In addition to the event data analyses, we search for possible emissions beyond the event data range using the survey data. This search uses the existing survey data product, which are processed survey data and are available for events before August 2013. After performing some studies of the false detection rate in the sample of survey data that we use, we adopted the criterion of signal-to-noise ratio $\geq 4.3$ in the energy band $14-195$ keV for determining a detection. We expect one false detection in our sample using this criterion. We found 21 detections (16 GRBs) beyond the event data range, with 7 of these bursts have been previously classified as ultra-long GRBs. We do not find an obvious relation between the detections in survey data and the $T_{90}$ estimated using the event data. That is, these bursts with detections in survey data are not necessarily longer in the event data range. However, we do found almost all the survey data detections match well with the late-time XRT light curves. The detections usually happen when the burst is also bright in XRT, such as during the flares in the XRT light curves. The overall summaries for the GRB analysis results are presented at\ http://swift.gsfc.nasa.gov/results/batgrbcat/index.html, which includes summary tables of GRB properties, and a quick-look webpage and the original data product for each burst. This web-version summary is expected to continue to be updated with new GRBs beyond those included in this paper. [**[Acknowledgments]{}**]{}\ We are grateful for valuable discussions with Sylvain Guiriec, Brett Hayes, and Antonino Cucchiara. We thank James Bubeck, Phillip Newman, and J. D. Myers for setting up the web space to host the summary results of this catalog. Moreover, we appreciate the helpful suggestions and careful number checking from the anonymous referee. The plots in the paper are made with the Python Matplotlib [@Hunter07]. The interactive plots on the webpages are made with the Bokeh package [@Bokeh]. Appendix: Tables {#sect:appendix_tables} ================ This section includes a list of summary tables from the analyses: Summary tables that includes general information of the bursts -------------------------------------------------------------- Table \[tab:summary\_general\] contains some general information of the bursts. Information from these two tables are shown in one single list in the complete online version. They are divided into two tables in the abbreviated paper version due to the limited space. Column Format Unit Description ----------------------- -------- ------ --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. Trig\_time\_met F13.3 s The BAT trigger time shown in the Swift machine time (MET). Trig\_time\_UTC A26 – The BAT trigger time shown in UTC. RA\_ground A12 deg RA of the GRB from the BAT refined position. DEC\_ground A13 deg DEC of the GRB from the BAT refined position. Image\_position\_err A12 deg The uncertainty of the BAT refined position. Image\_SNR A12 – The signal-to-noise ratio of the detection from the BAT image. T90 A12 s Burst duration $T_{90}$. T90\_err A12 s The uncertainty of $T_{90}$. T50 A12 s Burst duration $T_{50}$. T50\_err A12 s The uncertainty of $T_{50}$. Evt\_start\_sincetrig I4 s The time when the event data start, relative to the BAT trigger time. Evt\_stop\_sincetrig I4 s The time when the event data end, relative to the BAT trigger time. pcode F6.4 – Partial coding fraction of the burst. Trigger\_method A5 – The triggering method for the burst (rate trigger, image trigger, ground detected, or ground detected during slew). XRT\_detection A3 – Whether or not there is an XRT detection of the burst. : \[tab:summary\_general\] The format of the summary table that includes some general information of the bursts. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Table  \[tab:burst\_duration\] lists the start and end time of $T_{100}$, $T_{90}$, $T_{50}$, and the 1-s peak duration, relative to the BAT trigger time. Column Format Unit Description ----------------- -------- ------ --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. Trig\_time\_met F13.3 s The BAT trigger time shown in the Swift machine time (MET). T100\_start A11 s $T_{100}$ start time, relative to the BAT trigger time. T100\_stop A11 s $T_{100}$ end time, relative to the BAT trigger time. T90\_start A11 s $T_{90}$ start time, relative to the BAT trigger time. T90\_stop A11 s $T_{90}$ end time, relative to the BAT trigger time. T50\_start A11 s $T_{50}$ start time, relative to the BAT trigger time. T50\_stop A11 s $T_{50}$ end time, relative to the BAT trigger time. 1s\_peak\_start A11 s 1-s peak start time, relative to the BAT trigger time. 1s\_peak\_stop A11 s 1-s peak end time, relative to the BAT trigger time. : \[tab:burst\_duration\] The format of the burst-duration table that lists the start and end time of $T_{100}$, $T_{90}$, $T_{50}$, and the 1-s peak duration, relative to the BAT trigger time. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Summary tables of spectral analyses for the time-average $T_{100}$ spectra {#sect:T100_spectra_tables} -------------------------------------------------------------------------- Table \[tab:spec\_T100\_best\_model\] summarizes the best-fit model for each burst. “PL” refers to simple power-law model; “CPL” refers to cutoff power-law model. If there is no acceptable spectral fit for the burst (see Section \[sect:event\_standard\] for the criteria), the best-fit-model column is listed as “N/A”. Column Format Unit Description ------------------ -------- ------ --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. Best\_fit\_model A3 – The best-fit model for the GRB spectrum, either the simple power law model (PL) or cutoff power-law model (CPL.) : \[tab:spec\_T100\_best\_model\] The format of the table that summarizes the best-fit spectral model of the time-average ($T_{100}$) spectrum for each burst. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Table \[tab:spec\_T100\_pow\_parameter\] shows a list of parameters from the spectral fit using the simple power-law model. Column Format Unit Description ---------------- -------- ---------------------------------------- -------------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm PL}$ as defined in Eq. \[eq:PL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm PL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm PL}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm PL}_{50}$, as defined in Eq. \[eq:PL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm PL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm PL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. T100\_start A11 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_pow\_parameter\] The format of the table that presents the parameters from the PL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Table \[tab:spec\_T100\_pow\_photon\_flux\] shows the photon fluxes in different energy ranges from the simple power-law fit. Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_pow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ cm^{-2} \ s^{-1}$) from the PL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Table \[tab:spec\_T100\_pow\_energy\_flux\] shows the energy fluxes in different energy ranges from the simple power-law fit. Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_pow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the PL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Table \[tab:spec\_T100\_pow\_energy\_fluence\] shows the energy fluences in different energy ranges from the simple power-law fit. The energy fluences are calculated by multiply the flux by $T_{100}$. Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_pow\_energy\_fluence\] The format of the table that presents the energy fluences (in unit of $\rm erg \ cm^{-2}$) from the PL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column titles are shown here for guidance regarding its form and content.) Similar to the tables for the simply PL fits, Table \[tab:spec\_T100\_cutpow\_parameter\] to \[tab:spec\_T100\_pow\_energy\_fluence\] present the parameters, fluxes, and fluences for the CPL fits. Column Format Unit Description ---------------- -------- ---------------------------------------- ---------------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm CPL}$ as defined in Eq. \[eq:CPL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm CPL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm CPL}$. Epeak A12 keV $E_{\rm peak}$ as defined in Eq. \[eq:CPL\]. Epeak\_low A12 keV The lowe limit of $E_{\rm peak}$. Epeak\_hi A12 keV The upper limit of $E_{\rm peak}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm CPL}_{50}$, as defined in Eq. \[eq:CPL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm CPL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm CPL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. T100\_start A11 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_cutpow\_parameter\] The format of the table that presents the parameters from the CPL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_cutpow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ s^{-1} \ cm^{-2}$) from the CPL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_cutpow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the CPL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. T100\_start A12 s The start time of the spectrum, relative to the BAT trigger time. T100\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_T100\_cutpow\_energy\_fluence\] The format of the table that presents the energy fluence in unit of $\rm erg \ s^{-1}$ from the CPL fit for the time-averaged ($T_{100}$) spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_T100\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Summary tables of spectral analyses for the 1-s peak spectra ------------------------------------------------------------ In this section, we present the tables for the 1-s peak spectra, which are the same set of the tables as those in Section \[sect:T100\_spectra\_tables\] but for the 1-s peak spectral fits. Column Format Unit Description ------------------ -------- ------ --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. Best\_fit\_model A3 – The best-fit model for the GRB spectrum, either the simple power law model (PL) or cutoff power-law model (CPL.) : \[tab:spec\_1speak\_best\_model\] The format of the table that summarizes the best-fit spectral model of the 1-s peak spectrum for each burst. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ---------------- -------- ---------------------------------------- -------------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm PL}$ as defined in Eq. \[eq:PL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm PL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm PL}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm PL}_{50}$, as defined in Eq. \[eq:PL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm PL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm PL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. Peak\_start A11 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_pow\_parameter\] The format of the table that presents the parameters from the PL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_pow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ cm^{-2} \ s^{-1}$) from the PL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_pow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the PL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_pow\_energy\_fluence\] The format of the table that presents the energy fluences (in unit of $\rm erg \ cm^{-2}$) from the PL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column titles are shown here for guidance regarding its form and content.) Column Format Unit Description ---------------- -------- ---------------------------------------- ---------------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm CPL}$ as defined in Eq. \[eq:CPL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm CPL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm CPL}$. Epeak A12 keV $E_{\rm peak}$ as defined in Eq. \[eq:CPL\]. Epeak\_low A12 keV The lowe limit of $E_{\rm peak}$. Epeak\_hi A12 keV The upper limit of $E_{\rm peak}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm CPL}_{50}$, as defined in Eq. \[eq:CPL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm CPL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm CPL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. Peak\_start A11 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_cutpow\_parameter\] The format of the table that presents the parameters from the CPL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_cutpow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ s^{-1} \ cm^{-2}$) from the CPL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_cutpow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the CPL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_1speak\_cutpow\_energy\_fluence\] The format of the table that presents the energy fluence in unit of $\rm erg \ s^{-1}$ from the CPL fit for the 1-s peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_1speak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Summary tables of spectral analyses for the 20-ms peak spectra {#sect:20ms_tables} -------------------------------------------------------------- In this section, we present the tables for the 20-ms peak spectra, which are the same set of the tables as those in Section \[sect:T100\_spectra\_tables\]. Column Format Unit Description ------------------ -------- ------ --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. Best\_fit\_model A3 – The best-fit model for the GRB spectrum, either the simple power law model (PL) or cutoff power-law model (CPL.) : \[tab:spec\_20mspeak\_best\_model\] The format of the table that summarizes the best-fit spectral model of the 20-ms peak spectrum for each burst. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ---------------- -------- ---------------------------------------- -------------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm PL}$ as defined in Eq. \[eq:PL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm PL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm PL}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm PL}_{50}$, as defined in Eq. \[eq:PL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm PL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm PL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. Peak\_start A11 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_pow\_parameter\] The format of the table that presents the parameters from the PL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_pow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ cm^{-2} \ s^{-1}$) from the PL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_pow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the PL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_pow\_energy\_fluence\] The format of the table that presents the energy fluences (in unit of $\rm erg \ cm^{-2}$) from the PL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column titles are shown here for guidance regarding its form and content.) Column Format Unit Description ---------------- -------- ---------------------------------------- ---------------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with each triggers. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. alpha A13 – $\alpha^{\rm CPL}$ as defined in Eq. \[eq:CPL\]. alpha\_low A13 – The lower limit of $\alpha^{\rm CPL}$. alpha\_hi A13 – The upper limit of $\alpha^{\rm CPL}$. Epeak A12 keV $E_{\rm peak}$ as defined in Eq. \[eq:CPL\]. Epeak\_low A12 keV The lowe limit of $E_{\rm peak}$. Epeak\_hi A12 keV The upper limit of $E_{\rm peak}$. norm A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The normalization factor $K^{\rm CPL}_{50}$, as defined in Eq. \[eq:CPL\]. norm\_low A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The lower limit of $K^{\rm CPL}_{50}$. norm\_hi A12 $\rm ph \ cm^{-2} \ s^{-1} \ keV^{-1}$ The upper limit of $K^{\rm CPL}_{50}$. chi2 F6.2 – $\chi^2$ from the fitting, reported by XSPEC. dof I2 – degree of freedom from the fitting, reported by XSPEC. reduced\_chi2 F6.4 – reduced $\chi^2$ (i.e., $\chi^2$ divided by the degree of freedom), reported by XSPEC. null\_prob A12 – The null probability of the model, reported by XSPEC. enorm F7.4 KeV The normalization energy, which is set to 50 keV in our fits, as shown in Eq. \[eq:PL\]. Exposure\_time A6 s The time interval of the spectrum. Peak\_start A11 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A11 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_cutpow\_parameter\] The format of the table that presents the parameters from the CPL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ----------------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm ph \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm ph \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm ph \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_cutpow\_photon\_flux\] The format of the table that presents the photon flux (in unit of $\rm ph \ s^{-1} \ cm^{-2}$) from the CPL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- ------------------------------ --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2} \ s^{-1}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2} \ s^{-1}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2} \ s^{-1}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_cutpow\_energy\_flux\] The format of the table that presents the energy flux (in unit of $\rm erg \ s^{-1} \ cm^{-2}$) from the CPL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) Column Format Unit Description ------------------ -------- --------------------- --------------------------------------------------------------------- -- GRBname A9 – GRB name. Trig\_ID I11 – Special ID associated with the trigger. For GRBs found in ground analysis, the ID are associated with the trigger ID of the failed event data, or the full observation ID of the event data for the analysis. 15\_25kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-25$ keV. 15\_25kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-25$ keV. 15\_25kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-25$ keV. 25\_50kev A12 $\rm erg \ cm^{-2}$ Photon flux in $25-50$ keV. 25\_50kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $25-50$ keV. 25\_50kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $25-50$ keV. 50\_100kev A12 $\rm erg \ cm^{-2}$ Photon flux in $50-100$ keV. 50\_100kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $50-100$ keV. 50\_100kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $50-100$ keV. 100\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-150$ keV. 100\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-150$ keV. 100\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-150$ keV. 100\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $100-350$ keV. 100\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $100-350$ keV. 100\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $100-350$ keV. 15\_150kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-150$ keV. 15\_150kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-150$ keV. 15\_150kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-150$ keV. 15\_350kev A12 $\rm erg \ cm^{-2}$ Photon flux in $15-350$ keV. 15\_350kev\_low A12 $\rm erg \ cm^{-2}$ The lower limit of photon flux in $15-350$ keV. 15\_350kev\_hi A12 $\rm erg \ cm^{-2}$ The upper limit of photon flux in $15-350$ keV. Exposure\_time A12 s The time interval of the spectrum. Peak\_start A12 s The start time of the spectrum, relative to the BAT trigger time. Peak\_stop A12 s The end time of the spectrum, relative to the BAT trigger time. : \[tab:spec\_20mspeak\_cutpow\_energy\_fluence\] The format of the table that presents the energy fluence in unit of $\rm erg \ s^{-1}$ from the CPL fit for the 20-ms peak spectra. Note that this table includes the fit for every GRB, regardless of whether the fit is acceptable from the criteria listed in Section \[sect:event\_standard\]. A list of GRBs with acceptable fits can be found in Table \[tab:spec\_20mspeak\_best\_model\]. (This table is available in its entirety in a machine-readable form in the online journal. Only the column formats are shown here for guidance regarding its form and content.) A list of GRBs with bright X-ray sources in the same field of view ------------------------------------------------------------------ Column Format Unit Description --------- -------- ------ ------------- GRBname A9 – GRB name. : \[tab:brigh\_sources\] The format of the table that presents a list of GRBs with bright X-ray sources in the same field of view (see Section \[sect:bright\_src\] for detailed discussions). (This table is available in its entirety in a machine-readable form in the online journal. Only the column format is shown here for guidance regarding its form and content.) Redshift list ------------- Table \[tab:redshift\_list\] presents a list of GRB redshifts with the references. The methods of redshift measurements are noted with the following symbols: “a”: spectroscopic measurement from absorption lines; “h”: spectroscopic measurement from host galaxy; “p”: photometric redshift; “hp”: photometric redshift from host galaxy. The redshift uncertainty for photometric redshift is presented when available. Note that the redshift uncertainties is adopted from the reference, and thus might represent different confidence level. Moreover, some of the redshift, especially the one from host galaxies and photometric redshifts, can be less robust due to the uncertainty in their measurements. For example, sometime the reference reported two solution for the photometric redshifts, with one of them stated as a more likely solution. We thus record the more likely one in this table. However, the reader is strongly recommended to refer to the original reference for more information. Column Format Unit Description ------------------------------ -------- ------ -------------------------------------------------- GRBname A9 – GRB name. z A6 – GRB redshift. Method A2 – The method of how the redshift is determined: “ba”: spectroscopic measurement from the burst afterglow absorption lines. “he”: spectroscopic measurement from host galaxy emission lines. “bp”: photometric redshift from burst afterglow. “hp”: photometric redshift from host galaxy. Uncertainty (when available) A11 – The redshift uncertainty (when available). Ref. A100 – References of the GRB redshift measurements. The reference marked with “$\star$” is the one related to the value presented in this table. : \[tab:redshift\_list\] A list of GRB redshifts. The measurement methods are also listed: “ba”: spectroscopic measurement from burst afterglow absorption lines; “he”: spectroscopic measurement from host galaxy emission lines; “bp”: photometric redshift from burst afterglow; “hp”: photometric redshift from host galaxy. (This table is available in its entirety in a machine-readable form in the online journal. Only the column titles are shown here for guidance regarding its form and content.) Appendix: The relation between the rest-frame luminosity and the observed flux {#sect:lum_flux} ============================================================================== Calculating the rest-frame luminosity from the observed flux involves dealing with the redshifting effects of both the time and energy, which can be confusing. We thus write out detail derivation of the equation used to calculate the rest-frame luminosity in the rest-frame energy band in Section \[sect:T90\_vs\_z\]. In addition, we also include cross-checks of our equations with equations used for luminosity/energy calculations in other papers. The luminosity distance $D_L$ is defined by the following equation, D\_L = ( )\^[1/2]{} such that the relation between the bolometric luminosity $L_{\rm bol, rest}$ and the bolometric flux $F_{\rm bol, obs}$ is L\_[bol, rest]{} = 4 D\^2\_L F\_[bol, obs]{} which is in the same form as the usual one in the local frame. All the cosmological redshifting effects between the rest-frame bolometric luminosity $L_{\rm bol, rest}$ and the observer-frame bolometric flux $F_{\rm bol, obs}$ are handled by $D_L$. To be more specific, the rest-frame bolometric luminosity is defined as L\_[bol, rest]{} = (i.e., the energy output in the rest frame divided by the time measured in the rest frame), while the observer-frame bolometric flux is defined as F\_[bol, obs]{} = where dA is the surface area. Energy and time in the rest frame and the observer frame are related by E\_[rest]{} = (1+z) E\_[obs]{} t\_[rest]{} = In the real astronomy cases, it is common that the bolometric flux and luminosity cannot be measured directly due to the limited bandpass of the instrument. The relationship between the luminosity and flux in certain bandpass can be derived from the luminosity density (i.e., the differential luminosity) and the flux density (i.e., the differential flux). The luminosity density and flux density can be defined as either (1) luminosity per photon frequency ($dL_{\rm bol, rest}/d\nu_{\rm rest}$) and flux per photon frequency ($dF_{\rm bol, obs}/d\nu_{\rm obs}$), or (2) luminosity per photon energy ($dL_{\rm bol, rest}/dE_{\rm rest}$) and flux per photon energy ($dF_{\rm bol, obs}/dE_{\rm obs}$). The photon energy E can be simply related to the photon frequency $\nu$ by $E = h \nu$, where $h$ is the Planck constant. The high energy astrophysics community commonly refers to photon energy rather than photon frequency. We thus adopt the “per-energy” definition for the luminosity density and flux density hereafter. In other words, the luminosity density $L_E$ and the flux density $F_E$ are defined as follows: L\_[E, rest]{} = = F\_[E, obs]{} = = The energy in the denominator does not cancel with the energy in the nominator, because the one in the nominator refers to the energy output, and the one in the denominator refers to the photon energy, despite they have the same unit. The relation between the luminosity density $L_E$ in the rest frame and the flux density $F_E$ in the observer frame can be derived as follows, $$\begin{aligned} L_{E, \rm rest} = \frac{dL_{\rm bol, rest}}{dE_{\rm rest}} \nonumber &= \frac{d}{dE_{\rm rest}} ( 4 \pi D^2_L F_{\rm bol, obs}) \nonumber \\ &= 4 \pi D^2_L \frac{dF_{\rm bol, obs}}{dE_{\rm rest}} \nonumber \\ &= 4 \pi D^2_L \frac{dF_{\rm bol, obs}}{dE_{\rm obs}} \frac{dE_{\rm obs}}{dE_{\rm rest}} \nonumber \\ &= 4 \pi D^2_L \frac{dF_{\rm bol, obs}}{dE_{\rm obs}} \ \frac{1}{(1+z)} \nonumber \\ &= \frac{4 \pi D^2_L \ F_{E, \rm obs}}{(1+z)}\end{aligned}$$ The luminosity at a certain energy band $L_{\rm band, rest}$ can be calculated by integrating $L_{E, \rm rest}$, $$\begin{aligned} \label{eq:lum_flux} L_{\rm band, rest} = \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} L_{E, \rm rest} \ dE_{\rm rest} \nonumber &= \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \frac{dL_{\rm bol, rest}}{dE_{\rm rest}} \ dE_{\rm rest} \nonumber \\ &= \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \frac{4 \pi D^2_L}{1+z} \ F_{E, \rm obs} \ dE_{\rm rest} \nonumber \\ & = \int^{E_{2, \rm obs}}_{E_{1, \rm obs}} \frac{4 \pi D^2_L}{1+z} \ F_{E, \rm obs} \ (1+z) dE_{\rm obs} \nonumber \\ & \ \ \ \rm (Change \ of \ variables \ with \ dE_{\rm rest} = (1+z) dE_{\rm obs}) \nonumber \\ & = \int^{E_{2, \rm obs}}_{E_{1, \rm obs}} 4 \pi D^2_L \ F_{E, \rm obs} \ dE_{\rm obs} \nonumber \\ &= 4 \pi D^2_L \ \int^{E_{2, \rm obs}}_{E_{1, \rm obs}} \ F_{E, \rm obs} \ dE_{\rm obs} \nonumber \\ &= 4 \pi D^2_L \ F_{\rm band, obs} \end{aligned}$$ Where $F_{\rm band, obs} = \int^{E_{2, \rm obs}}_{E_{1, \rm obs}} \ F_{E, \rm obs} \ dE_{\rm obs}$ is the observed flux in a the observed energy band, $E_{1, \rm obs}$ to $E_{2, \rm obs}$, which correspond to the rest-frame energy band, $E_{2, \rm rest} = (1+z) \ E_{1, \rm obs}$ to $E_{2, \rm rest} = (1+z) \ E_{2, \rm obs}$. This is the equation we use to calculate the “luminosity” in Section \[sect:T90\_vs\_z\]. Since we use the flux measured in the BAT energy band 15-150 keV, the “luminosity” we presented in this paper corresponds to different rest-frame energy for GRBs at different redshifts. We cross-check our derivation with the equations mentioned in previous papers. [@Bloom01] mentioned the relation between the energy $E_{\rm band, rest}$ in some comoving bandpass $[E_{1, \rm rest}, E_{2, \rm rest}]$ and the observed fluence $S_{\rm band, obs}$ in the corresponded-redshifted bandpass of $[E_{1, \rm rest}/(1+z), E_{2, \rm rest}/(1+z)]$ to be \[eq:energy\_fluence\] E\_[band, rest]{} =  S\_[band, obs]{} This relationship can be derived from Eq. \[eq:lum\_flux\] by integrating the luminosity from time $t_{1, \rm rest}$ to $t_{2, \rm rest}$ in the rest-frame, $$\begin{aligned} E_{\rm band, rest} &= \int^{t2, \rm rest}_{t1,\rm rest} L_{\rm band, rest} \ dt_{\rm rest} \nonumber \\ &= \int^{t2, \rm rest}_{t1,\rm rest} 4 \pi D^2_L \ F_{\rm band, obs} \ dt_{\rm rest} \nonumber \\ &= \int^{t2, \rm obs}_{t1,\rm obs} 4 \pi D^2_L \ F_{\rm band, obs} \ \frac{dt_{\rm obs}}{(1+z)} \nonumber \\ & \ \ \ \rm (Change \ of \ variables \ with \ dt_{\rm rest} = \frac{dt_{\rm obs}}{(1+z)} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)} \int^{t2, \rm obs}_{t1,\rm obs} F_{\rm band, obs} \ dt_{\rm obs} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)} \ S_{E, \rm band, obs}\end{aligned}$$ @Amati02 uses a slightly different approach by first blue-shifting the GRB spectrum back to the rest frame to get the total fluences in the rest frame, which can be expressed as follows, $$\begin{aligned} S_{\rm band, rest} &= \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \ \bigg( \int^{t2, \rm rest}_{t1,\rm rest} F_{\rm band, rest} \ dt_{\rm rest} \bigg) \ dE_{\rm rest} \nonumber \\ &= \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \ \bigg( \int^{t2, \rm rest}_{t1,\rm rest} \frac{dF}{dE_{\rm rest}} \ dt_{\rm rest} \bigg) \ dE_{\rm rest} \nonumber \\ &= \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \ \bigg( \int^{t2, \rm rest}_{t1,\rm rest} \ \frac{dE_{\rm rest}}{dt_{\rm rest} \ dA \ dE_{\rm rest}} \ dt_{\rm rest} \bigg) \ dE_{\rm rest}\end{aligned}$$ $S_{\rm band, rest}$ corresponds to the $\int^{10000}_1 \ EN(E,\alpha,E_0,\beta,A)dE$ in the original equation in @Amati02. Using Eq. \[eq:energy\_fluence\] and converting the observer-frame fluence to the rest-frame fluence, one can get the equation used in @Amati02, $$\begin{aligned} E_{\rm band, rest} &= \frac{4 \pi D^2_L}{(1+z)} \ S_{\rm band, obs} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)} \ \int^{E_{2, \rm obs}}_{E_{1, \rm obs}} \ \bigg( \int^{t2, \rm obs}_{t1,\rm obs} \ \frac{dE_{\rm obs}}{dt_{\rm obs} \ dA \ dE_{\rm obs}} \ dt_{\rm obs} \bigg) \ dE_{\rm obs} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)} \ \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \ \bigg( \int^{t2, \rm rest}_{t1,\rm rest} \ \frac{dE_{\rm rest}/(1+z)}{dt_{\rm rest}(1+z) \ dA \ dE_{\rm rest}/(1+z)} \ dt_{\rm rest}(1+z) \bigg) \ \frac{dE_{\rm rest}}{(1+z)} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)^2} \ \int^{E_{2, \rm rest}}_{E_{1, \rm rest}} \ \bigg( \int^{t2, \rm rest}_{t1,\rm rest} \ \frac{dE_{\rm rest}}{dt_{\rm rest} \ dA \ dE_{\rm rest}} \ dt_{\rm rest} \bigg) \ dE_{\rm rest} \nonumber \\ &= \frac{4 \pi D^2_L}{(1+z)^2} \ S_{\rm band, rest}\end{aligned}$$ Therefore, the equation we use for calculating the rest-frame luminosity from the observed flux is consistent with the equations used in @Bloom01 and @Amati02. In summary, as mentioned in @Bloom01, if one would like to calculate the luminosity or energy in the rest-frame bandpass that matches perfectly with the redshifted band in the observer frame, one can use Eq. \[eq:lum\_flux\] and Eq. \[eq:energy\_fluence\] without extra factor of k-correction. However, if one would like calculate the energy or luminosity in other energy bands, an extra factor of “k-correction” based on the source spectrum would be required to adjust the flux difference between the original energy range ($E_{1, \rm rest}$ to $E_{2, \rm rest}$) and the desired energy range [for more details of the k-correction factor, see @Bloom01]. [^1]: \[Swift\_table\] http://swift.gsfc.nasa.gov/archive/grb\_table/ [^2]: http://grb.pa.msu.edu/grbcatalog [^3]: http://www.swift.ac.uk/burst\_analyser/ [^4]: http://heasarc.gsfc.nasa.gov/docs/swift/archive/grbsummary/ [^5]: http://butler.lab.asu.edu/swift/ [^6]: http://www.astro.caltech.edu/grbox/grbox.php [^7]: http://www.mpe.mpg.de/ jcg/grbgen.html [^8]: http://www.nasa.gov/feature/goddard/nasas-swift-spots-its-thousandth-gamma-ray-burst/ [^9]: http://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/swift.pl [^10]: http://heasarc.nasa.gov/lheasoft/ [^11]: http://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/ [^12]: http://heasarc.gsfc.nasa.gov/ftools/caldb/help/batgrbproduct.html [^13]: http://heasarc.gsfc.nasa.gov/ftools/caldb/help/battblocks.html [^14]: http://heasarc.gsfc.nasa.gov/xanadu/xspec/ [^15]: \[note1\]http://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/XSappendixStatistics.html [^16]: There are ten GRBs with unconstrained $T_{100}$ due to the weakness of the burst. In such cases, we use the time interval determined by the flight software to create the time-averaged spectrum. [^17]: Descriptions for the quad-rate data can be found at http://swift.gsfc.nasa.gov/archive/archiveguide1/node1.html [^18]: \[note\_mask\] See Section \[sect:low\_pcode\] for explanation of the FLUX and DETECTION masks. [^19]: The numbers for 2015 is not listed in the table because not all GRBs in 2015 are included in this catalog [^20]: http://heasarc.gsfc.nasa.gov/W3Browse/fermi/fermigbrst.html [^21]: For historical reason, the time-averaged flux reported by the BAT team always refers to the one in the $T_{100}$ range instead of the $T_{90}$ range, while the burst durations are reported in $T_{90}$. For most of the bursts, we do not expect the average flux using the $T_{100}$ range to differ significantly from the one using the $T_{90}$ range, since the $T_{90}$ range includes the majority of the burst emission. [^22]: http://swift.gsfc.nasa.gov/results/batgrbcat/index.html [^23]: The null hypothesis here refers to the simple PL model, though for these events, the CPL fits either also have low probability for the null hypothesis, or have some other problems. [^24]: http://www.astro.caltech.edu/grbox/grbox.php [^25]: http://www.mpe.mpg.de/ jcg/grbgen.html [^26]: http://butler.lab.asu.edu/swift/bat\_spec\_table.html [^27]: \[constraint\_web\]http://swift.gsfc.nasa.gov/analysis/uvot\_digest/numbers.html [^28]: \[note:batsurvey\] https://heasarc.gsfc.nasa.gov/ftools/caldb/help/batsurvey.html [^29]: https://heasarc.gsfc.nasa.gov/ftools/caldb/help/batcelldetect.html [^30]: \[survey\_table\_footnote\] Because we are using criterion 1 mentioned in Section \[sect:survey\_data\_analysis\] for selecting survey images for the false-detection rate study, the numbers here only includes detections that occur beyond $T0+500 s$. Therefore, there are only 14 detected GRBs listed under the criterion “3.4 sigma in 14-195 keV band” (the one we used for the official search). The two GRBs listed in Table \[tab:survey\_GRB\], GRB060218 and GRB050730, with detections before $T0+500$ s is not included. [^31]: http://www.swift.ac.uk/burst\_analyser/

--- abstract: 'In this paper we study how high-gain anti-windup schemes can be used to implement projected dynamical systems in control loops that are subject to saturation on a (possibly unknown) set of admissible inputs. This insight is especially useful for the design of autonomous optimization schemes that realize a closed-loop behavior which approximates a particular optimization algorithm (e.g., projected gradient or Newton descent) while requiring only limited model information. In our analysis we show that a saturated integral controller, augmented with an anti-windup scheme, gives rise to a perturbed projected dynamical system. This insight allows us to show uniform convergence and robust practical stability as the anti-windup gain goes to infinity. Moreover, for a special case encountered in autonomous optimization we show robust convergence, i.e., convergence to an optimal steady-state for finite gains. Apart from being particularly suited for online optimization of large-scale systems, such as power grids, these results are potentially useful for other control and optimization applications as they shed a new light on both anti-windup control and projected gradient systems.' author: - 'Adrian Hauswirth[^1]' - Florian Dörfler - 'Andrew R. Teel[^2]' bibliography: - 'bibliography\_static.bib' nocite: '[@*]' title: 'Anti-Windup Approximations of Oblique Projected Dynamics for Feedback-based Optimization[^3]' --- Differential Inclusions, Autonomous Optimization, Anti-Windup Control 93C30, 34A60, 34H05 Introduction ============ In recent years, the design of feedback controllers based on optimization algorithms has garnered significant interest as a new approach to real-time optimization of large-scale systems such as power grids [@dallaneseOptimalPowerFlow2018; @tangRealTimeOptimalPower2017; @hauswirthOnlineOptimizationClosed2017; @molzahnSurveyDistributedOptimization2017] and communication networks [@lowInternetCongestionControl2002; @kellyRatecontrolcommunication1998]. The goal of *autonomous* (or *feedback-based) optimization* is to implement feedback systems that robustly solve nonlinear optimization problems in closed loop with a physical system, often without requiring explicit knowledge of the problem parameters, because the physical plant itself enforces certain constraints. In this paper, we investigate a new approach to enforce constraints by exploiting physical saturation. More precisely, we study how anti-windup control, which is ubiquitous in feedback control to mitigate integrator windup, can be used to implement projected dynamical systems (PDS) which are at the basis of continuous-time algorithms for constrained optimization. In particular, [PDS ]{}form a class of discontinuous dynamical systems that encompasses projected gradient flow [@hauswirthProjectedGradientDescent2016], projected Newton flow [@hauswirthProjectedDynamicalSystems2018], subgradient flow [@cortesDiscontinuousdynamicalsystems2008] and projected saddle-flows [@goebelStabilityRobustnessSaddlepoint2017; @cherukuriRoleConvexitySaddlePoint2017]. More generally, [PDS ]{}arise in many contexts that include unilateral constraints, such as variational inequalities [@nagurneyProjectedDynamicalSystems1996; @facchineiFinitedimensionalvariationalinequalities2003], evolutionary games [@lahkarProjectionDynamicGeometry2008], and complementarity systems [@brogliatoEquivalenceComplementaritySystems2006; @aubinDifferentialInclusionsSetValued1984]. The main contribution of this paper is to establish a rigorous connection between [PDS ]{}and anti-windup controllers and to generalize [@hauswirthImplementationProjectedDynamical2020]. From an abstract point of view, we consider a class of parametrized dynamical systems, termed *anti-windup approximations* (AWA), and we show uniform convergence of trajectories to the solution of a [PDS ]{}as the *anti-windup gain* tends to infinity. Moreover, we establish semiglobal practical robustness of [PDS ]{}with respect to anti-windup approximations. For the special case of strongly monotone vector fields we further show robust asymptotic stability for finite gains. Compared to [@hauswirthImplementationProjectedDynamical2020] we make the following generalizations: 1. We do not require the feasible domain to be convex. Instead, we work with (non-convex) prox-regular sets and show, by means of a counter-example, that prox-regularity cannot, in general, be relaxed further. 2. We consider *oblique* [PDS ]{}that provide an additional degree in the form of a (Riemannian) metric that allows us to capture wider variety of dynamics (such as projected Newton flows) and that is required for coordinate-free formulations. 3. We require solutions to be neither unique nor complete. In particular, our results allow for solutions with finite escape time. Although these results may not be of practical relevance, they illustrate the necessity of our assumptions. 4. We establish requirements for the convergence of anti-windup approximations of monotone dynamics on convex domains, thus providing a (partial) solution to a previously open problem formulated in [@hauswirthImplementationProjectedDynamical2020]. Finally, in a largely self-contained section, we illustrate the possibilities of the proposed anti-windup approximations of [PDS ]{}and the applicability of our theoretical results in the context of autonomous optimization [@tangRunningPrimalDualGradient2018; @colombinorobustnessguaranteesfeedbackbased2019; @hauswirthProjectedGradientDescent2016; @lawrenceOptimalSteadyStateControl2018; @colombinoOnlineOptimizationFeedback2019]. The rest of the paper is organized as follows: In \[sec:prelim\] we fix the notation and recall relevant notions from variational analysis and dynamical systems. In \[sec:tech\] we define our problem and establish some technical lemmas. In \[sec:unif,sec:rob\] we present our first two main results (\[thm:loc,thm:rob\_stab\]) on uniform convergence and semiglobal practical robust stability. In \[sec:rob\_conv\] we provide a stronger stability guarantee (\[thm:monot\]) for the special case of monotone vector fields. In \[sec:aw\_appl\], we illustrate the consequences of our results in the context of feedback-based optimization. For this, we consider four different optimization dynamics (three gradient-based and one saddle-point flow) and discuss their convergence behavior observed in simulations. In \[sec:conc\] we summarize our results and discuss open problems. Preliminaries {#sec:prelim} ============= Notation -------- We consider ${\mathbb{R}}^n$ with the usual inner product $\left\langle \cdot, \cdot \right\rangle$ and 2-norm $\| \cdot \|$. We use ${\mathbb{R}}^n_{\geq 0}$ for the non-negative orthant. The closed (open) unit ball of appropriate dimension is denoted by ${\mathbb{B}}$ ($\operatorname{int}{\mathbb{B}}$). For a sequence $\{K_n \}$, $K_n \rightarrow 0^+$ implies that that $K_n > 0$ for all $n$ and ${\lim}_{n \rightarrow \infty} K_n = 0$. For a map $F: {\mathbb{R}}^n \rightarrow {\mathbb{R}}^m$, differentiable at $x \in {\mathbb{R}}^n$, $\nabla F(x) \in {\mathbb{R}}^{m \times n}$ denotes the *Jacobian of $F$ at $x$*. Given a set ${\mathcal{C}}\subset {\mathbb{R}}^n$, its *closure*, *boundary*, and *(closed) convex hull* are denoted by $\operatorname{cl}{{\mathcal{C}}}$, $\partial {\mathcal{C}}$, and $\operatorname{co}{{\mathcal{C}}}$ ($\operatorname{\overline{co}}{\mathcal{C}}$), respectively. If ${\mathcal{C}}$ is non-empty, we write $\| {\mathcal{C}}\| := \sup_{v \in {\mathcal{C}}} \| v \|$. The *distance to ${\mathcal{C}}$* is defined as $d_{\mathcal{C}}(x) := \inf_{\tilde{x} \in {\mathcal{C}}} \| x - \tilde{x} \|$, and the *projection* $P_{\mathcal{C}}: {\mathbb{R}}^n \rightrightarrows {\mathcal{C}}$ is given by $P_{\mathcal{C}}(x) := \{ \tilde{x} \in {\mathcal{C}}\, | \, \| x - \tilde{x} \| = d_{{\mathcal{C}}}(x) \}$. The *domain* of a set-valued map $H: {\mathbb{R}}^n \rightrightarrows {\mathbb{R}}^m$ is defined as $\operatorname{dom}H := \{ x \, | \, H(x) \neq \emptyset \}$. We use the standard definitions of *outer semicontinuity (osc)*, *local boundedness*, *graphical convergence*, etc. from set-valued analysis. In particular, unless noted otherwise, we follow the definitions and notation of [@goebelHybridDynamicalSystems2012 Chap. 5]. The identity matrix (of appropriate size) is denoted by ${\mathbb{I}}$. Given a square symmetric matrix $G \in {\mathbb{R}}^{n \times n}$, $\lambda^{\max}_G$ and $\lambda^{\min}_G$ denote its maximum and minimum eigenvalue, respectively. The set of symmetric, positive definite matrices of size $n$ is denoted by ${\mathbb{S}}_+^n$. A *metric* on ${\mathcal{C}}\subset {\mathbb{R}}^n$ is a map $G: {\mathcal{C}}\rightarrow {\mathbb{S}}^n_+$. A metric $G$ induces an *inner product* $\left\langle u, v \right\rangle_{G(x)} := u^T G(x) v$ and an associated *2-norm* $\| u \|_{G(x)} := ( \left\langle u, u \right\rangle_{G(x)})^{1/2}$ for all $x \in {\mathcal{C}}$ and all $u, v \in {\mathbb{R}}^n$. A metric is (*Lipschitz*) *continuous* if it is (Lipschitz) continuous as a map $G: {\mathcal{C}}\rightarrow {\mathbb{S}}^n_+$ with respect to the $\lambda^{\max}$-norm on ${\mathbb{S}}^n_+$. Variational Geometry -------------------- We use the following, slightly simplified[^4], notions of variational geometry. For a comprehensive treatment the reader is referred to [@rockafellarVariationalAnalysis2009]. Given a closed set ${\mathcal{C}}\subset {\mathbb{R}}^n$ and $x \in {\mathcal{C}}$, a vector $v$ is a *tangent vector to ${\mathcal{C}}$ at $x$* if there exist sequences $x_k \rightarrow x$ with $x_k \in {\mathcal{C}}$ for all $k$ and $\delta_k \rightarrow 0^+$ such that $\frac{x_k - x}{\delta_k}\rightarrow v$. The set of all tangent vectors at $x$ is called the *tangent cone at $x$* and denoted by $T_x {\mathcal{C}}$. If the set-valued map $x \mapsto T_x {\mathcal{C}}$ is inner semicontinuous then ${\mathcal{C}}$ is *Clarke regular* (or *tangentially regular*) [@rockafellarVariationalAnalysis2009 Cor. 6.29]. If ${\mathcal{C}}$ is Clarke regular then $T_x{\mathcal{C}}$ is closed convex and the *(Euclidean) normal cone at $x$* is defined as the polar cone of $T_x{\mathcal{C}}$, i.e., $N_x {\mathcal{C}}:= \{ \eta \, | \, \forall v \in T_x {\mathcal{C}}: \, \left\langle \eta, v\right\rangle \leq 0 \}$ [@rockafellarVariationalAnalysis2009 Cor. 6.30]. Further, the map $x \mapsto N_x {\mathcal{C}}$ is osc [@rockafellarVariationalAnalysis2009 Prop. 6.6]. We follow the convention that $T_x {\mathcal{C}}= N_x {\mathcal{C}}= \emptyset$ for all $x \notin {\mathcal{C}}$. We will mostly work with the special class of *prox-regular sets*. Given a Clarke regular set ${\mathcal{C}}$ and $\alpha > 0$, a normal vector $\eta \in N_x {\mathcal{C}}$ is *$\alpha$-proximal* if $\left\langle \eta, y - x \right\rangle \leq \alpha \| \eta \| \| y - x \|^2$ for all $y \in {\mathcal{C}}$. The set ${\mathcal{C}}$ is *$\alpha$-prox-regular at $x$* if all normal vectors at $x$ are $\alpha$-proximal. In other words, the normal cone coincides with the cone of $\alpha$-proximal normals. A set is *$\alpha$-prox-regular* if it is $\alpha$-prox-regular at all $x \in {\mathcal{C}}$ and *prox-regular* if it is $\alpha$-prox-regular for some $\alpha > 0$. A key property of prox-regular sets is that the projection on ${\mathcal{C}}$ is locally well-defined [@adlyPreservationProxRegularitySets2016 Thm. 2.2 & Prop. 2.3]: \[prop:prox\] If ${\mathcal{C}}\subset {\mathbb{R}}^n$ is $\alpha$-prox-regular, then, for every $x \in {\mathcal{C}}+ \frac{1}{2\alpha} \operatorname{int}{\mathbb{B}}$, the set $P_{\mathcal{C}}(x)$ is a singleton and $d^2_{\mathcal{C}}$ is differentiable at $x$ with $\nabla (d^2_{\mathcal{C}}(x)) = 2(x - P_{\mathcal{C}}(x))$. Further, $P_{\mathcal{C}}({\overline{x}} + v) = {\overline{x}}$ holds for every ${\overline{x}} \in {\mathcal{C}}$ and all $v \in N_{{\overline{x}}} {\mathcal{C}}\cap \frac{1}{2\alpha} \operatorname{int}{\mathbb{B}}$. For example, every closed convex set is Clarke regular as well as $\alpha$-prox-regular for all $\alpha > 0$. Further, every set of the form ${\mathcal{C}}= \{ x \, | \, h(x) \leq 0 \}$, where $h: {\mathbb{R}}^n \rightarrow {\mathbb{R}}^m$ is differentiable, is Clarke regular if constraint qualifications hold [@rockafellarVariationalAnalysis2009 Thm. 6.14]. If, in addition, $h$ has a globally Lipschitz derivative, then ${\mathcal{C}}$ is prox-regular [@hauswirthProjectedDynamicalSystems2018 Ex. 7.7]. Dynamical Systems & Stability ----------------------------- Given a closed set ${\mathcal{C}}\subset {\mathbb{R}}^n$ and a set-valued map $H: {\mathbb{R}}^n \rightrightarrows {\mathbb{R}}^n$, we say that $x:[0, T ] \rightarrow {\mathcal{C}}$ for some $T>0$ is a *(Carathéodory) solution* of the (constrained) differential inclusion $$\begin{aligned} \label{eq:gen_inclusion} \dot x \in H(x) \, , \qquad x \in {\mathcal{C}}\,\end{aligned}$$ if $x$ is absolutely continuous, and $x(t) \in {\mathcal{C}}$ and $\dot x(t) \in H(x(t))$ hold for almost all $t \in [0,T]$. A map $x : [ 0, \infty) \rightarrow {\mathcal{C}}$ is a *complete* solution, if its restriction to any compact subinterval $[0, T]$ is a solution of . An inclusion  is *well-posed* if ${\mathcal{C}}$ is closed, $H$ is osc and locally bounded relative to ${\mathcal{C}}$, and $H(x)$ is non-empty and convex for all $x \in {\mathcal{C}}$. Standard results (e.g., [@goebelHybridDynamicalSystems2012 Lem. 5.26]) guarantee that  admits a solution for every initial condition $x(0) \in {\mathcal{C}}$ if it is well-posed and $H(x) \cap T_x {\mathcal{C}}\neq \emptyset$ for all $x \in {\mathcal{C}}$. For convenience, we introduce the following notion of truncated solution: Consider  with ${\mathcal{C}}= {\mathbb{R}}^n$. Given $T, \epsilon > 0$ and $x_0 \in {\mathbb{R}}^n$, a solution $x:[0, T'] \rightarrow {\mathbb{R}}^n$ of  with initial condition $x(0) = x_0$ is *$(T, \epsilon)$-truncated* if $x(t) \in x_0 + \epsilon {\mathbb{B}}$ for all $t \in [0, T']$ and either $T' = T$ or $\| x(T') - x_0 \| = \epsilon$ holds. Recall that on a compact domain, solutions of an (unconstrained) inclusion can always be extended up to the boundary of the domain [@filippovDifferentialEquationsDiscontinuous1988 §7, Thm. 2]: \[thm:filippov\] Let  be well-posed with ${\mathcal{C}}= {\mathbb{R}}^n$ and let ${\mathcal{A}}\subset {\mathbb{R}}^n$ be compact. Then, every solution $x: [0, T] \rightarrow {\mathbb{R}}^n$ with $x(0) \in {\mathcal{A}}$ can be extended up to the boundary of ${\mathcal{A}}$, i.e., there is a solution for every $T > 0$ or there exists $T$ such that $x(T) \in \partial {\mathcal{A}}$. Therefore, by considering an augmented inclusion with $\hat{H}(x) := (H(x), 1)$, initial condition $\hat{x}(0) := (x(0), 0)$, $\hat{{\mathcal{C}}} = {\mathbb{R}}^n \times {\mathbb{R}}$, and $\hat{{\mathcal{A}}} = {\mathcal{A}}\times [0, T]$, \[thm:filippov\] guarantees the existence of truncated solutions for every $T$ and every $\epsilon$: Let be well-posed with ${\mathcal{C}}= {\mathbb{R}}^n$. Then, for every $T, \epsilon > 0$ and every $x(0) \in {\mathbb{R}}^n$ there exists a $(T, \epsilon)$-truncated solution to . Hence, truncated solutions are convenient if finite escape times cannot be precluded, since their graph is always a compact subset of $[0, T] \times (x(0) + \epsilon {\mathbb{B}})$. We also require the notion of $\sigma$-perturbation of an inclusion [@goebelHybridDynamicalSystems2012 Def. 6.27]: Given $\sigma > 0$, the *$\sigma$-perturbation of * is given by $$\begin{aligned} \dot x \in H_\sigma (x) \qquad x \in {\mathcal{C}}_\sigma \end{aligned}$$ where ${\mathcal{C}}_\sigma := {\mathcal{C}}+ \sigma {\mathbb{B}}$ and $H_\sigma(x) := \operatorname{\overline{co}}H(( x + \sigma {\mathbb{B}}) \cap {\mathcal{C}}) + \sigma {\mathbb{B}}$ for all $x \in {\mathcal{C}}_\sigma$. Note in particular that, for $\sigma' \geq \sigma$,we have ${\mathcal{C}}_{\sigma} \subset {\mathcal{C}}_{\sigma'}$, $H_{\sigma}(x) \subset H_{\sigma'}(x)$ for all $x \in {\mathcal{C}}_\sigma$, and every solution of the $\sigma$-perturbation is a solution of the $\sigma'$-perturbation. Next, recall that $\omega: {\mathbb{R}}_{\geq 0} \rightarrow {\mathbb{R}}_{\geq 0}$ is a *${\mathcal{K}}_\infty$-function* (denoted by $\omega \in {\mathcal{K}}_\infty$) if $\omega$ is continuous, strictly increasing, unbounded, and it holds that $\omega(0) = 0$. We require the following lemma about ${\mathcal{K}}_\infty$-functions: [@sontagCommentsIntegralVariants1998 Cor. 10]\[lem:kinf\] For every $\omega \in {\mathcal{K}}_\infty$, there exist $\sigma_1, \sigma_2 \in {\mathcal{K}}_\infty$ such that $\omega(rs) \leq \sigma_1(r)\sigma_2(s)$ for all $r, s \geq 0$. A function $\beta:{\mathbb{R}}_{\geq 0} \times {\mathbb{R}}_{\geq 0} \rightarrow {\mathbb{R}}_{\geq 0}$ is a *${\mathcal{KL}}$-function* (denoted by $\beta \in {\mathcal{KL}}$) if it is non-decreasing in its first argument, non-increasing in its second argument, $\lim_{r \rightarrow 0^+} \beta(r,s) = 0$ for each $s \in {\mathbb{R}}_{\geq 0}$, and $\lim_{s \rightarrow \infty} \beta(r,s) = 0 $ for each $t \in {\mathbb{R}}_{\geq 0}$. A closed set ${\mathcal{A}}\subset {\mathbb{R}}^n$ is *uniformly globally (pre-)asymptotically stable* for  if there exists $\beta \in {\mathcal{KL}}$ such that for every solution $x: [0, T] \rightarrow {\mathcal{C}}$ of it holds that $$\begin{aligned} d_{{\mathcal{A}}}(x(t)) \leq \beta( d_{{\mathcal{A}}}(x(0)), t) \qquad \forall t \in [0, T] \, .\end{aligned}$$ \[rem:pre\_stab\] The term “pre-asymptotic” refers to the fact that solutions of  need not be complete for the above definition of stability to apply [@goebelHybridDynamicalSystems2012 Def 3.6 & Thm 3.40]. However, if  is well-posed and ${\mathcal{A}}$ is compact it follows that, for any initial condition $x(0) \in {\mathcal{C}}$, the (compact) set $\{x \, | \, \beta(d_{\mathcal{A}}(x), 0) \leq \beta(d_{\mathcal{A}}(x(0)), 0) \}$ is invariant, thus implicitly guaranteeing the existence of a complete solution. Oblique Projected Dynamical Systems ----------------------------------- PDS are continuous-time dynamical systems that are constrained to a set by projection of the vector field at the boundary of the domain. Compared to traditional definitions [@brogliatoEquivalenceComplementaritySystems2006; @cornetExistenceslowsolutions1983; @aubinDifferentialInclusionsSetValued1984; @nagurneyProjectedDynamicalSystems1996], we incorporate the possibility of *oblique* projection directions by means of a variable metric [@hauswirthProjectedDynamicalSystems2018]. Namely, we consider [PDS ]{}as defined by the differential equation of the form $$\begin{aligned} \label{eq:pds_ivp} \dot x = \Pi^G_{\mathcal{C}}[ f(x)] (x) \, \qquad x \in {\mathcal{C}}\, ,\end{aligned}$$ where ${\mathcal{C}}\subset {\mathbb{R}}^n$ is a Clarke regular set, $G:{\mathcal{C}}\rightarrow {\mathbb{S}}_+^n$ is a metric on ${\mathcal{C}}$, and $f: {\mathbb{R}}^n \rightarrow {\mathbb{R}}^n$ is a vector field. Given $x \in {\mathcal{C}}$ and $w \in {\mathbb{R}}^n$, the operator $\Pi^G_{\mathcal{C}}$ projects $w$ onto the tangent cone of ${\mathcal{C}}$ at $x$ with respect to the metric $G$, i.e., $$\begin{aligned} \Pi^G_{\mathcal{C}}[ w ] (x) := \underset{v \in T_x {\mathcal{C}}}{\arg \min} \| v -w \|_{G(x)} \, .\end{aligned}$$ Note that if $x \in {\mathcal{C}}$, then $\Pi^G_{\mathcal{C}}[w](x)$ is single-valued since ${\mathcal{C}}$ is assumed to be Clarke regular which implies that $T_x {\mathcal{C}}$ is closed convex. If $x \notin {\mathcal{C}}$, we have $\Pi^G_{\mathcal{C}}[ w ] (x) = \emptyset$ and therefore $\operatorname{dom}\Pi^G_{\mathcal{C}}[ w ] = {\mathcal{C}}$ for all $w \in {\mathbb{R}}^n$. If $f$ is a vector field, we abuse notation and write $\Pi^G_{\mathcal{C}}[ f ] (x) := \Pi^G_{\mathcal{C}}[ f(x) ] (x)$ for brevity. Given a metric $G$, we define the *normal cone of ${\mathcal{C}}$ at $x$ with respect to $G$* as $N_x^G {\mathcal{C}}:= \{ \eta \, | \, \forall v \in T_x {\mathcal{C}}: \, \left\langle \eta, v \right\rangle_{G(x)} \leq 0\}$. Note in particular that we have $$\begin{aligned} \label{eq:norm_cone_equiv} \eta \in N_x {\mathcal{C}}\quad \Longleftrightarrow \quad G^{-1}(x) \eta \in N^G_x {\mathcal{C}}\, .\end{aligned}$$ As a consequence of Moreau’s Theorem [@hiriart-urrutyFundamentalsConvexAnalysis2012 Thm. 3.2.5] the operator $\Pi_{\mathcal{C}}^G$ has the following crucial properties (see also [@brogliatoEquivalenceComplementaritySystems2006; @cornetExistenceslowsolutions1983; @aubinDifferentialInclusionsSetValued1984]): [@hauswirthProjectedDynamicalSystems2018 Lem. 4.5]\[lem:pds\_decomp\] If ${\mathcal{C}}$ is Clarke regular then, for every $x \in {\mathcal{C}}$, there exists a unique $\eta \in N^G_x {\mathcal{C}}$ such that $\Pi^G_{\mathcal{C}}[f](x) = f(x) - \eta$. Furthermore, $\Pi^G_{\mathcal{C}}[f](x) = f(x) - \eta$ holds if and only if $\eta \in N^G_x {\mathcal{C}}$ and $\left\langle f(x) - \eta, \eta \right\rangle_{G(x)} = 0$. Using Cauchy-Schwarz, it also holds that $\| \eta \|_{G(x)} \leq \| f(x) \|_{G(x)}$. Existence and uniqueness results for  without a variable metric can be found in [@aubinDifferentialInclusionsSetValued1984; @nagurneyProjectedDynamicalSystems1996; @cornetExistenceslowsolutions1983] and others. For the case with a variable metric with bounded condition number, the following statement is a condensation of results in [@hauswirthProjectedDynamicalSystems2018]: \[thm:pds\_exist\] Consider  and let ${\mathcal{C}}$ be Clarke regular, and $f$ and $G$ be continuous. Then,  admits a solution for every initial condition $x(0) \in {\mathcal{C}}$. If, in addition, there exists $\kappa > 0$ such that $\sup_{x \in {\mathcal{C}}} \lambda^{\max}_{G(x)}/ \lambda^{\min}_{G(x)} \leq \kappa$ and $f$ is globally Lipschitz, then  admits a complete solution for every $x(0) \in {\mathcal{C}}$. If ${\mathcal{C}}$ is prox-regular, and if $f$ and $G$ are (locally) Lipschitz, then  admits a unique solution for every initial condition $x(0) \in {\mathcal{C}}$.[^5] It is known that the solutions to  are equivalent to the solutions of $\dot x \in f(x) - N^G_x {\mathcal{C}}$ (for $G = {\mathbb{I}}$ see [@aubinViabilityTheory1991; @cornetExistenceslowsolutions1983]; for general $G$ see [@hauswirthProjectedDynamicalSystems2018 Cor. 6.3]). In light of \[lem:pds\_decomp\], we can show (next) that solutions of  are equivalent to solutions of $$\begin{aligned} \label{eq:ivp_alt} \dot x \in F(x) := f(x) - N^G_x {\mathcal{C}}\cap \gamma {\mathbb{B}}\qquad x \in {\mathcal{C}}\,,\end{aligned}$$ where $\gamma \geq \sup_{x \in {\mathcal{C}}} \| f(x) \|_{G(x)}$ (assuming $\sup_{x \in {\mathcal{C}}} \| f(x) \|_{G(x)} < \infty$). The advantage of this latter inclusion is that the mapping $F$ is bounded. \[prop:pds\_alt\_form\] If ${\mathcal{C}}$ is Clarke regular and $x \mapsto \| f(x) \|_{G(x)}$ is bounded, then, $x: [0, T] \rightarrow {\mathcal{C}}$ with $T>0$ is a solution of  if and only if it is a solution of . Let $x:[0, T] \rightarrow {\mathcal{C}}$ be a solution of . Then, $\Pi^G_{\mathcal{C}}[f](x(t)) = f(x(t)) - \eta(t)$ for some $\eta(t) \in N^G_{x(t)}{\mathcal{C}}$ satisfying $\| \eta(t) \|_{G(x(t))} \leq \| f(x(t)) \|_{G(x(t))} \leq \gamma$ by \[lem:pds\_decomp\] and therefore $\eta(t) \in N^G_{x(t)} {\mathcal{C}}\cap \gamma {\mathbb{B}}$. Conversely, assume that $x$ solves . Whenever $\dot{x}(t)$ exists, it holds that $\dot{x}(t) \in T_{x(t)} {\mathcal{C}}\cap - T_{x(t)} {\mathcal{C}}$ [@cornetExistenceslowsolutions1983 eq. 2.6] and $\dot{x}(t) = f(x(t))- \eta(t)$ for some $\eta(t) \in N^G_{x(t)} {\mathcal{C}}\cap \gamma {\mathbb{B}}$. Thus, we have $$\begin{aligned} \left\langle f(x(t)) - \eta(t), \eta(t) \right\rangle_{G(x(t))} \leq 0 \qquad \text{and} \qquad \left\langle -f(x(t)) + \eta(t), \eta(t) \right\rangle_{G(x(t))}\leq 0 \, , \end{aligned}$$ and therefore $\left\langle f(x(t)) - \eta(t), \eta(t) \right\rangle_{G(x(t))} = 0$ which, in turn, implies that $f(x(t)) - \eta(t) = \Pi_{\mathcal{C}}^G[f](x(t))$ by \[lem:pds\_decomp\]. If $f$ and $G$ are continuous and ${\mathcal{C}}$ Clarke regular,  is well-posed. Non-emptiness and convexity of $F(x)$ are immediate because $N^G_x {\mathcal{C}}\cap \gamma {\mathbb{B}}$ is non-empty (in particular, $0 \in N^G_x {\mathcal{C}}$) and convex for all $x \in {\mathcal{C}}$ (and $f$ is single-valued). For outer semicontinuity recall that for a Clarke regular ${\mathcal{C}}$ and continuous $G$ the mapping $x \mapsto N^G_x {\mathcal{C}}$ is osc [@hauswirthProjectedDynamicalSystems2018 Lem. A.6]. It then follows that the truncation $ N^G_x {\mathcal{C}}\cap \gamma {\mathbb{B}}$ is osc and locally bounded [@rockafellarVariationalAnalysis2009 p.161]. Finally, since $f$ is continuous and single-valued, $x \mapsto f(x) - N^G_x {\mathcal{C}}\cap \gamma {\mathbb{B}}$ is osc and locally bounded. Problem Formulation & Technical Results {#sec:tech} ======================================= Throughout the paper, we consider the system given by the (unconstrained) inclusion $$\begin{aligned} \label{eq:sys_aw_gen} \dot z \in F_K(z) := f(z, P_{\mathcal{Z}}(z)) - \tfrac{1}{K} G^{-1}(P_{\mathcal{Z}}(z)) \big ( z - P_{\mathcal{Z}}(z) \big) \, ,\end{aligned}$$ where ${\mathcal{Z}}\subset {\mathbb{R}}^n$ is a closed set, $f: {\mathbb{R}}^n \times {\mathcal{Z}}\rightarrow {\mathbb{R}}^n$ is a continuous vector field, $G: {\mathcal{Z}}\rightarrow {\mathbb{S}}^n_+$ is a continuous metric, and $K > 0$ is a constant parameter. Because $P_{\mathcal{Z}}$ is in general not single-valued (unless ${\mathcal{Z}}$ is convex),  has to be treated as a differential inclusion. Systems of the form  arise in the context of anti-windup control for feedback loops with integral controllers, as will be discussed in \[sec:aw\_appl\]. Hence, we will refer to  as an *anti-windup approximation* (AWA). We study the behavior of solutions of  as $K \rightarrow 0^+$ and show that, under appropriate assumption on ${\mathcal{Z}}, f$, $G$, and for an initial condition $z(0) \in {\mathcal{Z}}$, these solutions *converge uniformly* to solutions of the projected dynamical system $$\begin{aligned} \label{eq:aw_pds} \dot z = \Pi^G_{\mathcal{Z}}[\hat{f}](z) \, , \qquad z \in {\mathcal{Z}}\, ,\end{aligned}$$ where we use $\hat{f}(z) := f(z, P_{\mathcal{Z}}(z))$. Further, we show that a compact globally asymptotically set for  is *semiglobally practically asymptotically stable* for  in $K$. Namely, if ${\mathcal{A}}$ is compact and asymptotically stable compact for the PDS \[eq:aw\_pds\], then for any compact set of initial conditions ${\mathcal{B}}$ and any $\zeta > 0$, there exists $K > 0$ such that all trajectories of the AWA  starting in ${\mathcal{B}}$ converge to a subset of ${\mathcal{A}}+ \zeta {\mathbb{B}}$. The key idea for studying  is to exploit $\alpha$-prox-regularity of ${\mathcal{Z}}$ which, according to \[prop:prox\] guarantees, that $P_{\mathcal{Z}}(z)$ is single-valued for all $$\begin{aligned} z \in {\mathcal{Z}}^\circ_\alpha := {\mathcal{Z}}+ \tfrac{1}{2\alpha} \operatorname{int}{\mathbb{B}}\, .\end{aligned}$$ Hence, on ${\mathcal{Z}}^\circ_\alpha$,  reduces to an ODE. Further, under appropriate conditions on the problem parameters and for small enough $K$, trajectories starting in ${\mathcal{Z}}$ remain in ${\mathcal{Z}}^\circ_\alpha$. This insight will be rigorously established in \[subsec:exist\]. In \[subsec:perturb\] we then show that the AWA  corresponds to a $\sigma$-perturbation of the PDS  as a function of $K$. We then apply standard results from [@goebelHybridDynamicalSystems2012] to establish uniform convergence and semiglobal practical asymptotic stability in \[sec:unif,sec:rob\], respectively. Existence, Local Uniform Boundedness, and Equicontinuity {#subsec:exist} -------------------------------------------------------- As a first step in studying , we prove the following lemma for future reference: \[lem:vec\_field\] Let ${\mathcal{Z}}\subset {\mathbb{R}}^n$ be closed and $f: {\mathbb{R}}^n \times {\mathcal{Z}}$ be continuous. Then, $z \mapsto \hat{f}(z) := f(z, P_{\mathcal{Z}}(z))$ is locally bounded and osc. Furthermore, if ${\mathcal{Z}}$ is $\alpha$-prox-regular for $\alpha > 0$, then $\hat{f}$ is single-valued and continuous for all $z \in {\mathcal{Z}}^\circ_\alpha$. The projection $P_{\mathcal{Z}}: {\mathbb{R}}^n \rightrightarrows {\mathcal{Z}}$ is osc and locally bounded [@rockafellarVariationalAnalysis2009 Ex. 5.23], and $P_{\mathcal{Z}}(z)$ is non-empty and closed for all $z \in {\mathbb{R}}^n$ (since ${\mathcal{Z}}$ is closed). By continuity of $f$ it follows that $\hat{f}$ is osc and locally bounded, since both properties are preserved under addition and composition [@rockafellarVariationalAnalysis2009 Prop. 5.51 & 5.52]. Using \[prop:prox\] it follows that $\hat{f}$ is single-valued (hence continuous) for $z \in {\mathcal{Z}}_\alpha^\circ$. \[lem:vec\_field\] and \[prop:prox\] imply that, on ${\mathcal{Z}}^\circ_\alpha$, $F_K$ is single-valued and continuous. Consequently, standard results for ODEs guarantee that  admits a (local) solution for every initial condition $z(0) \in {\mathcal{Z}}^\circ_\alpha$. However, outside of ${\mathcal{Z}}^\circ_\alpha$,  is a differential inclusion for which the existence of solutions is not immediately guaranteed. Nevertheless, one can establish the existence of so-called *Krasovskii solutions* [@goebelHybridDynamicalSystems2012]. For the main result of this section we consider the following (local) setup: \[ass:local\] Consider  and $z_0 \in {\mathcal{Z}}$. Let $M, \nu, \mu, \alpha, \epsilon > 0$ be such that $$\begin{aligned} \label{eq:lin_growth} \| f(z, P_{\mathcal{Z}}(z)) \| \leq M \qquad \text{and} \qquad \mu {\mathbb{I}}\preceq G^{-1}(P_{\mathcal{Z}}(z)) \preceq \nu {\mathbb{I}}\end{aligned}$$ hold for all $z \in (z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$, and ${\mathcal{Z}}$ is $\alpha$-prox-regular at every $z \in (z_0 + \epsilon {\mathbb{B}}) \cap Z$. Parameters $M, \mu, \nu, \epsilon$ that satisfy  can always be found for any $z_0 \in {\mathcal{Z}}$ since $z \mapsto f(z, P_{\mathcal{Z}}(z))$ is locally bounded by \[lem:vec\_field\], $G$ is continuous, and $P_{\mathcal{Z}}$ is single-valued on $(z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$. \[ass:local\] allows us to formulate the following proposition which combines the existence of truncated solutions, the invariance of a neighborhood of ${\mathcal{Z}}$, and equicontinuity (i.e. uniform Lipschitz continuity): \[prop:exist\] Let \[ass:local\] be satisfied for $z_0 \in {\mathcal{Z}}$. Given any $T > 0$ and $K < \frac{\mu}{2 \alpha M}$, there exists a $(T, \epsilon)$-truncated solution $z$ for  with $z(0) = z_0$ (where $\epsilon$ stems from \[ass:local\]). Furthermore, $z$ satisfies, for almost all $t \in \operatorname{dom}z$, $$\begin{aligned} z(t) \in {\mathcal{Z}}+ \tfrac{K M}{\mu} {\mathbb{B}}\qquad \text{and} \qquad \| \dot{z}(t) \| \leq \left(1 + \tfrac{\nu}{\mu} \right)M \, . \end{aligned}$$ First, we consider the existence of solutions: As mentioned, \[lem:vec\_field\] and \[prop:prox\] imply that, on $(z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$,  reduces to a continuous ODE which is a well-posed inclusion (trivially). Hence, \[thm:filippov\] guarantees the existence of a *maximal* solution $z: [0, T'] \rightarrow (z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$ starting at $z_0$ and with $x(T')$ on the boundary of $(z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$. Next, by \[prop:prox\], we have $\nabla d_{\mathcal{Z}}^2(z) = 2 (z - P_{\mathcal{Z}}(z))$ for all $z \in {\mathcal{Z}}^\circ_\alpha$. Hence, the Lie derivative of $d^2_{\mathcal{Z}}$ along $F_K$ for all $z \in (z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$ is well-defined and satisfies $$\begin{aligned} {\mathcal{L}}_{F_K} \left( \tfrac{1}{2} d^2_{\mathcal{Z}}(z) \right) & = (z - P_{\mathcal{Z}}(z))^T \left( f(z, P_{\mathcal{Z}}(z)) - \tfrac{1}{K} G^{-1}(P_{\mathcal{Z}}(z))(z - P_{\mathcal{Z}}(z) \right) \\ & \leq d_{\mathcal{Z}}(z) \| f(z, P_{\mathcal{Z}}(z))\| - \tfrac{1}{K} (z - P_{\mathcal{Z}}(z))^T G^{-1}(P_{\mathcal{Z}}(z)) \left(z - P_{\mathcal{Z}}(z) \right) \\ & \leq d_{\mathcal{Z}}(z) \| f (z, P_{\mathcal{Z}}(z)) \| - \tfrac{\mu}{K} d^2_{\mathcal{Z}}(z) = d_{\mathcal{Z}}(z) ( M - \tfrac{\mu}{K} d_{\mathcal{Z}}(z)) \, . \end{aligned}$$ It follows that ${\mathcal{L}}_{F_K} \left( \tfrac{1}{2} d^2_{\mathcal{Z}}(z ) \right) < 0$ whenever $d_{\mathcal{Z}}(z) > \frac{K M}{\mu}$. Since $K < \frac{\mu}{2\alpha M}$ and using an invariance argument, it follows that $z(t) \in {\mathcal{Z}}+ \tfrac{K M}{\mu} {\mathbb{B}}\subset {\mathcal{Z}}^\circ_\alpha$ for all $t \in [0, T']$. In other words, for small enough $K$, any solution of  starting at $z_0$ remains within a neighborhood of ${\mathcal{Z}}$ on which the projection $P_{\mathcal{Z}}$ is single-valued. Since $z(T')$ lies on the boundary of $(z_0 + \epsilon {\mathbb{B}}) \cap {\mathcal{Z}}^\circ_\alpha$, but at the same time $z(T') \in {\mathcal{Z}}+ \frac{KM}{\mu} {\mathbb{B}}$, it follows that $\| z(T') - z_0 \| = \epsilon$. In other words, $z(T')$ lies on the boundary of $z_0 + \epsilon {\mathbb{B}}$ (rather than the boundary of ${\mathcal{Z}}^\circ_\alpha$). Hence, (after restricting $z$ to $[0, T]$ if $T'> T$) it can be concluded that $z$ is a $(T, \epsilon)$-truncated solution of . Finally, we have that for all $z \in ({\mathcal{Z}}+ \frac{K M}{\mu} {\mathbb{B}}) \cap (z_0 + \epsilon {\mathbb{B}})$ it holds that $$\begin{aligned} \left\| \tfrac{1}{K} G(z)^{-1} (z - P_{\mathcal{Z}}(z)) \right\| \leq \tfrac{1}{K} \nu \tfrac{K M}{\mu} \leq M\tfrac{\nu}{\mu } \, . \end{aligned}$$ It then follows from the definition of $M$ and the triangle inequality that $\| F_K(z) \| \leq M + M \tfrac{\nu}{\mu}$, thus establishing the bound on $\| \dot z(t) \|$. The proof of \[prop:exist\] suggests that the prox-regularity assumption on ${\mathcal{Z}}$ is primarily required for $d_{\mathcal{Z}}^2(z)$ to have a single-valued derivative in a neighborhood of ${\mathcal{Z}}$. The following example shows, however, that prox-regularity is a more fundamental requirement which, in general, cannot be avoided. \[ex:pds\_exist\] Consider the set ${\mathcal{Z}}:= \{ (z_1, z_2) \in {\mathbb{R}}^2 \, | \, |z_2| \geq \max\{ 0, z_1\}^\kappa \}$ for any $\frac{1}{2} < \kappa < 1$. Further assume that $G(z) = {\mathbb{I}}$ and $f(z) = (1, 0)$ for all $z \in {\mathbb{R}}^n$. Hence, we can choose $M = \nu = \mu = 1$ and any $\epsilon > 0$ to satisfy \[ass:local\]. Note, however, that ${\mathcal{Z}}$ is not prox-regular at $(0,0)$. Namely, every point on the positive $z_1$-axis has a non-unique projection onto ${\mathcal{Z}}$ as illustrated in \[fig:prx\_counter\]. [0.20]{} ![Non-prox-regular set for \[ex:pds\_exist\]: (a) non-uniqueness of projection for every $(z_1, 0)$ (b) construction of Krasovskii regularization, namely $v \in \operatorname{\overline{co}}F_K(z_0)$.](Figs/prox_regset.pdf "fig:"){width="\columnwidth"} [0.20]{} ![Non-prox-regular set for \[ex:pds\_exist\]: (a) non-uniqueness of projection for every $(z_1, 0)$ (b) construction of Krasovskii regularization, namely $v \in \operatorname{\overline{co}}F_K(z_0)$.](Figs/kras_construct.pdf "fig:"){width="\columnwidth"} We claim that for every $K > 0$ there exists a Krasovskii[^6] solution (i.e., a solution of the inclusion $z \in \operatorname{\overline{co}}F_K(z)$) starting on the $z_1$-axis that leaves the set ${\mathcal{Z}}+ \frac{K M}{\mu} {\mathbb{B}}$ established in \[prop:exist\]. This can be deduced graphically from \[fig:prx\_counter\_kras\]. Namely, let $z_0 = (z_{01},0)$ be such that $d_{{\mathcal{Z}}} (z_0) = K$. Then, there exists $v = (v_1, 0)$ with $v_1 > 0$ in the Krasovskii-regularization of $F_K(z_0)$, i.e., $v \in \operatorname{\overline{co}}F_K(z_0)$. In other words, on the boundary of ${\mathcal{Z}}+ K {\mathbb{B}}$, the vector $v$ points out of the (supposedly) invariant set. This, in turn, can be used to rigorously establish that the set ${\mathcal{Z}}+ K {\mathbb{B}}$ is not invariant, illustrating that the conclusion of \[prop:exist\] does not hold without prox-regularity of ${\mathcal{Z}}$, even when considering more general Krasovskii solutions. Anti-Windup Trajectories as Perturbed PDS {#subsec:perturb} ----------------------------------------- As a key technical result, we establish that solutions of the AWA are also solutions of a $\sigma$-perturbation of the [PDS ]{}in its alternate form . To prove this claim, consider $z_0 \in {\mathcal{Z}}$, and let $M, \mu, \nu, \alpha, \epsilon > 0$ be such that \[ass:local\] is satisfied. It follows from \[prop:pds\_alt\_form\] that, for some $T>0$, every $(T, \epsilon)$-truncated solution $z: [0, T'] \rightarrow (z_0+ \epsilon {\mathbb{B}})$ of the [PDS ]{} with $z(0) = z_0$ is also a $(T, \epsilon)$-truncated solution of the inclusion $$\begin{aligned} \label{eq:aw_sys_altform} \dot z \in \hat{F}(z) := f(z, P_{\mathcal{Z}}(z)) - N^G_z {\mathcal{Z}}\cap \gamma {\mathbb{B}}\quad \text{where} \quad \gamma := \max \left\{ \tfrac{M}{\sqrt{\mu}}, \tfrac{\nu}{\mu} M \right\}\end{aligned}$$ and vice versa. This choice of $\gamma$ will be convenient in the proof of \[prop:perturbation\] below. For now, note that using Cauchy-Schwarz, it holds that $$\begin{aligned} \sup_{z \in z_0 + \epsilon {\mathbb{B}}} \| f(z, P_{\mathcal{Z}}(z)) \|_{G(z)} \leq \sup_{z \in z_0 + \epsilon {\mathbb{B}}} \underbrace{\sqrt{\| G(z) \|}}_{\leq 1/\sqrt{\mu}} \underbrace{\| f(z, P_{\mathcal{Z}}(z)) \|}_{\leq M} \leq \gamma \, ,\end{aligned}$$ thus satisfying the condition on $\gamma$ in  and \[prop:pds\_alt\_form\]. Furthermore, given $z_0 \in {\mathcal{Z}}$, let \[ass:local\] hold with some $\epsilon > 0$. By \[lem:vec\_field\] we have that $z \mapsto f(z, P_{\mathcal{Z}}(z))$ is continuous on $z \in {\mathcal{Z}}^\circ_\alpha$ and hence uniformly continuous on the bounded set ${\mathcal{Z}}^\circ_\alpha \cap (z_0 + \epsilon {\mathbb{B}})$. As a consequence of uniform continuity there exists $\omega \in {\mathcal{K}}_\infty$ such that, for all $z, z' \in {\mathcal{Z}}^\circ_\alpha \cap (z_0 + \epsilon {\mathbb{B}})$, we have $$\begin{aligned} \label{eq:f_cont} \| f(z, P_{\mathcal{Z}}(z)) - f(z', P_{\mathcal{Z}}(z')) \| \leq \omega( \| z - z' \|) \, .\end{aligned}$$ \[prop:perturbation\] Consider $z_0 \in {\mathcal{Z}}$ and let \[ass:local\] hold with $M, \nu, \mu, \alpha$ and $\epsilon$. Further, let $K < \frac{\mu}{2 \alpha M}$. Then, for some $T >0$, every $(T, \epsilon)$-truncated solution $z:[0, T'] \rightarrow (z_0 + \epsilon {\mathbb{B}})$ of  is a solution of the $\sigma$-perturbation of  with $\sigma := \max \left\{ \frac{K M}{\mu}, \omega \left(\frac{K M}{\mu} \right) \right\}$, where $\omega \in {\mathcal{K}}_\infty$ satisfies . We need to show that the $(T, \epsilon)$-truncated solution $z$ satisfies $$\begin{aligned} \label{eq:infl_incl} \dot z(t) \in {\hat{F}}_\sigma(z(t))\, , \qquad z(t) \in {\mathcal{Z}}_\sigma \end{aligned}$$ for almost all $t \in [0, T']$, where $\mathcal{Z}_\sigma := {\mathcal{Z}}+ \sigma{\mathbb{B}}$ and $ {\hat{F}}_\sigma(z) := \operatorname{co}{\hat{F}}((z + \sigma {\mathbb{B}}) \cap {\mathcal{Z}}) + \sigma {\mathbb{B}}$ for all $z \in {\mathcal{Z}}_\sigma$ and with ${\hat{F}}$ defined in . Note that for $z \in {\mathcal{Z}}_\sigma$ we have that $$\begin{aligned} \label{eq:proj_prop} P_{\mathcal{Z}}(z) \subset (z + \sigma {\mathbb{B}}) \cap {\mathcal{Z}}\, . \end{aligned}$$ \[prop:exist\] guarantees that $z(t) \in {\mathcal{Z}}+ \frac{K M}{\mu} {\mathbb{B}}$, and since $\sigma \geq \frac{KM}{\mu}$ it follows that $z(t) \in {\mathcal{Z}}_\sigma$ for all $t \in [0, T']$. For the remainder of the proof we omit the argument of $z(t)$ to simplify notation. All statements hold for almost all $t \in [0, T']$. Since $z - P_{\mathcal{Z}}(z) \subset N_{P_{\mathcal{Z}}(z)} {\mathcal{Z}}$ for all $z \in {\mathbb{R}}^n$ [@rockafellarVariationalAnalysis2009 Ex. 6.16] and using  we have $$\begin{aligned} \label{eq:trun_norm_prop1} \tfrac{1}{K} G(P_{\mathcal{Z}}(z))^{-1} \left( z - P_{\mathcal{Z}}(z) \right) \in N^G_{P_{\mathcal{Z}}(z)} {\mathcal{Z}}\, . \end{aligned}$$ Furthermore, since $z \in {\mathcal{Z}}+ \frac{K M}{\mu}$ and using $\gamma$ as defined in  we have that $$\begin{aligned} \label{eq:trun_norm_prop2} \left \| \tfrac{1}{K} G(P_{\mathcal{Z}}(z))^{-1} \left( z - P_{\mathcal{Z}}(z \right) \right\| \leq \tfrac{1}{K} \nu \tfrac{K M}{\mu} = \tfrac{\nu M}{\mu} \leq \gamma. \end{aligned}$$ Combining  and  we have $$\begin{aligned} \label{eq:incl_pre} \dot z \in f (z, P_{\mathcal{Z}}(z)) - N^G_{P_{\mathcal{Z}}(z)}{\mathcal{Z}}\cap \gamma {\mathbb{B}}\, . \end{aligned}$$ Note that, in contrast to , the normal cone is evaluated at $P_{\mathcal{Z}}(z)$. Next, using the fact that $\omega$, as defined in , is strictly increasing, and exploiting the definition of $\sigma$, we have $$\begin{aligned} \label{eq:lip_prop} \| f (z, P_{\mathcal{Z}}(z)) - f (P_{\mathcal{Z}}(z), P_{\mathcal{Z}}(z)) \| \leq \omega\left(\| z - P_{\mathcal{Z}}(z) \|\right) \leq \omega\left(\tfrac{KM}{\mu}\right) \leq \sigma\, . \end{aligned}$$ Therefore, in summary, using  on , as well as , we have that $$\begin{aligned} \dot z & \in f(z, P_{\mathcal{Z}}(z)) - N^G_{P_{\mathcal{Z}}(z)}{\mathcal{Z}}\cap \gamma {\mathbb{B}}\\ & \subset f (P_{\mathcal{Z}}(z), P_{\mathcal{Z}}(z)) + \sigma {\mathbb{B}}- N^G_{P_{\mathcal{Z}}(z)}{\mathcal{Z}}\cap \gamma {\mathbb{B}}\\ & = {\hat{F}}(P_{\mathcal{Z}}(z)) + \sigma {\mathbb{B}}\subset {\hat{F}}((z + \sigma {\mathbb{B}})\cap {\mathcal{Z}}) + \sigma {\mathbb{B}}\subset {\hat{F}}_\sigma(z) \, . \end{aligned}$$ Hence, $z(\cdot)$ satisfies  which completes the proof. Uniform Convergence {#sec:unif} =================== We establish the graphical/uniform convergence of solutions of the anti-windup approximation  to solutions of the projected dynamics . This proof requires two arguments: On the one hand, we need to show that a graphically convergent sequence of solutions of  converges to a solution of . On the other hand, we need that such a graphically convergent sequence exists. Starting with the latter requirement, we first recall that from a bounded sequence of sets, we can always extract a graphically convergent subsequence [@goebelHybridDynamicalSystems2012 Thm. 5.7]. This applies in particular to a sequence of (uniformly) truncated solutions: \[lem:subseq\_extr\] Consider a sequence $K_n \rightarrow 0^+$ and $z_{0} \in {\mathcal{Z}}$. Given $T, \epsilon > 0$, any sequence $\{z_n\}$ of $(T, \epsilon)$-truncated solution of  with $K = K_n$ and $z_n(0) = z_{0}$ has a graphically convergent subsequence. \[lem:subseq\_extr\] is purely set-theoretic and does not imply that the limit $\operatorname{gph}\lim_{n \rightarrow \infty} z_n$ is a single-valued map. Hence, we need the following simplification[^7] of [@goebelHybridDynamicalSystems2012 Thm. 5.29]: \[lem:unif\_cvg\] Let the inclusion  be well-posed and $z_0 \in {\mathcal{Z}}$. Further, given any $T, \epsilon, \rho > 0$ and $\delta_i \rightarrow 0^+$, let $z_i: [0, T_i] \rightarrow {\mathcal{X}}_i$ denote a $(T, \epsilon)$-truncated solution of the $\delta_i \sigma$-perturbation of . If the sequence $\{z_i\}$ converges graphically, then convergence is to a solution $z: [0, T] \rightarrow {\mathcal{X}}$ of , where $T = \lim_{i \rightarrow \infty} T_i$. \[rem:graph\_unif\_cvg\] In the context of \[lem:unif\_cvg\], graphical convergence implies uniform convergence to $z$ on every subinterval of $[0, T)$ [@goebelHybridDynamicalSystems2012 Lem. 5.28]. Furthermore, if $T_i \geq T$ for all $i$, then convergence is uniform on $[0, T]$. Since, by \[prop:perturbation\], solutions of  are solutions of a $\sigma$-perturbation of an alternate form PDS  we can use \[lem:unif\_cvg\] to establish the following result: \[prop:incl\] Given $z_0 \in {\mathcal{Z}}$, let \[ass:local\] be satisfied. Consider $T>0$ and a sequence $K_i \rightarrow 0^+$, and assume that a sequence of $(T, \epsilon)$-truncated solutions $z_i$ of the AWA  with $K = K_i$ and $z_i(0) = z_0$ for all $i$ converges graphically. Then, the limit is a $(T, \epsilon)$-truncated solution of the PDS . Let $M, \mu, \nu >0$ and $\omega \in {\mathcal{K}}_\infty$ be defined as in \[ass:local\] and , respectively. Using \[lem:kinf\], there exist $\sigma_1, \sigma_2 \in {\mathcal{K}}_\infty$ such that $\omega(rs) \leq \sigma_1(r) \sigma_2(r)$ for all $r, s \geq 0$. Hence, we define $\delta_i := \max\{K_i, \sigma_1(K_i)\}$ and $\rho := \max\left\{ \tfrac{M}{\mu}, \sigma_2 \left(\tfrac{M}{\mu} \right) \right\}$. \[prop:perturbation\] states that for every $K_i$, the solution $z_i$ of  is also a solution of the $\sigma$-perturbation of  with $\sigma := \max \left \{K_i \tfrac{M}{\mu}, \omega \left(K_i \tfrac{M}{\mu} \right) \right\}$. It follows that $z_i$ is also a solution of every $\sigma'$-perturbation of  with $\sigma' \geq \sigma$. In particular, we can set $$\begin{aligned} \sigma' := \delta_i \rho = \max\{K_i, \sigma_1(K_i)\} \max\left\{ \tfrac{M}{\mu}, \sigma_2(\tfrac{M}{\mu}) \right\} \geq \sigma \, , \end{aligned}$$ and thus we have that $z_i$ is a solution of the $\delta_i \rho$-perturbation of . Since, by assumption, $\{z_i\}$ converges graphically to $z$ it follows from \[lem:unif\_cvg\] that $z$ is a solution of , and, by \[prop:pds\_alt\_form\], $z$ is a solution of . Finally, we need to show that $z: [0, T'] \rightarrow (z_0 + \epsilon {\mathbb{B}})$ is a $(T, \epsilon)$-truncated solution. Namely, we need to show that either $T = T'$ or $\| z(T) - z_0 \| = \epsilon$. This requirement is equivalent to $(T', z(T'))$ lying on the boundary of the cylinder ${\mathcal{X}}:= [0, T] \times (z_0 + \epsilon {\mathbb{B}})$. Since, by definition, for every $i$, $z_i$ is a $(T, \epsilon)$-truncated solution of  we have that $(T_i, z_i(T_i)) \in \partial {\mathcal{X}}$ for all $i$. Since $\partial {\mathcal{X}}$ is closed, it follows that the limit also lies in $\partial {\mathcal{X}}$. Now, we can immediately combine \[lem:subseq\_extr\] and \[prop:incl\] to arrive at our first main result about the graphical convergence of truncated solutions (i.e., local) solutions of anti-windup approximations to a projected dynamical system: \[thm:loc\] Let \[ass:local\] be satisfied for some $z_0 \in {\mathcal{Z}}$. Given any $T>0$ (and $\epsilon > 0$ from \[ass:local\]), consider a sequence $K_n \rightarrow 0^+$ and let $\{z_n \}$ denote a sequence of $(T, \epsilon)$-truncated solutions of the AWA  with $K = K_n$ and $z_n(0)~=~z_0$. Then, there exists a subsequence of $\{ z_{n}\}$ that converges graphically to a $(T,\epsilon)$-truncated solution of the PDS . Under certain circumstances, it can be useful to know that, rather than a subsequence of gains $\{K_n\}$, any sequence $K_n \rightarrow 0^+$ will lead to a converging sequence of solutions. This is guaranteed if it is known that the [PDS ]{}  has a unique solution: \[cor:uniq\] Let \[ass:local\] be satisfied for some $z_0 \in {\mathcal{Z}}$. Given any $T>0$ (and $\epsilon > 0$ from \[ass:local\]), assume that the PDS  admits a unique $(T, \epsilon)$-truncated solution $z$ with $z(0) = z_0$. Then, any sequence $\{z_n \}$ of $(T, \epsilon)$-truncated solutions of the AWA  with $z_n(0)~=~z_0$ and $K = K_n$ with $K_n \rightarrow 0^+$ converges graphically to the (unique) $(T,\epsilon)$-truncated solution of the PDS . Assume, for the sake of contradiction, that $\{z_n\}$ does not converge to the unique solution $z$ of . This implies that there exists $\nu > 0$ and a subsequence $\{z_m\}$ of $\{z_n\}$ such that $d_\infty(\operatorname{gph}z_m, \operatorname{gph}z) \geq \nu$ for all $m$ where $d_\infty$ denotes the Hausdorff distance between two sets. (In particular, since $z$ is a truncated solution $\operatorname{gph}z$ is compact and thus graphical convergence is equivalent to convergence with respect to $d_\infty$ [@rockafellarVariationalAnalysis2009 Ex. 4.13].) However, by \[lem:subseq\_extr\], the sequence $\{z_m\}$ has a convergent subsequence that converges to some limit $\tilde{z}$. By \[prop:incl\], $\tilde{z}$ is a solution of , but we also have $\| \tilde{z} - z \|_\infty \geq \varepsilon$ which contradicts the uniqueness of $z$. Finally, we can state the following *ready-to-use* result about uniform convergence in the case when the existence of unique complete solutions is guaranteed: Consider the AWA , let ${\mathcal{Z}}$ be prox-regular, $f$ globally Lipschitz, and there exist $\mu, \nu > 0$ such that $\mu {\mathbb{I}}\preceq G^{-1}(z) \preceq \nu {\mathbb{I}}$ for all $z \in {\mathbb{R}}^n$. Given $z_0 \in {\mathcal{Z}}$ and a sequence $K_n \rightarrow 0^+$, every sequence of complete solutions $z_n$ of the AWA  with initial condition $z_0$ and $K = K_n$ converges uniformly to the unique complete solution of the PDS  on every compact interval $[0, T]$. Note that the assumptions on ${\mathcal{Z}}$, $G$, and $f$ guarantee that for every initial condition  admits a unique complete (Carathéodory) solution (\[thm:pds\_exist\]). Hence, given any $T > 0$, let $z:[0,T] \rightarrow {\mathcal{Z}}$ denote the unique solution of the [PDS ]{}  and define $\epsilon > \sup_{t \in [0, T]} \| x(t) - x_0 \|$. Since $f$ is continuous and hence bounded over a compact set, \[ass:local\] is satisfied with $\nu, \mu, \alpha$ and by choosing $M:= \max_{z \in z_0 + \epsilon {\mathbb{B}}} \|f(z, P_{\mathcal{Z}}(z)) \|$. \[thm:loc\] guarantees convergence of a subsequence to the $(T, \epsilon)$-truncated solution $z: [0, T'] \rightarrow {\mathcal{X}}$ of . Moreover, for the same reason as in \[cor:uniq\] the sequence itself converges. Finally, by definition of $\epsilon$, we have that $z$ is defined on $[0, T']$ with $T' = T$ and $\|z(T) - z_0\| < \epsilon$ and, in this case, graphical convergence of $(T, \epsilon)$-truncated solutions implies their uniform convergence on $[0, T]$ (see \[rem:graph\_unif\_cvg\]). \[thm:loc\] and its corollaries can be slightly generalized, albeit at the expense of additional technicalities. For instance, instead of considering a single initial condition $z_0 \in {\mathcal{Z}}$, it is in general possible to consider a sequence of initial conditions (under some additional restrictions) that converges to $z_0$. Semiglobal Practical Robust Stability {#sec:rob} ===================================== Since anti-windup approximations can be seen as perturbations of projected dynamical systems, we can establish semiglobal practical asymptotic stability in $K$ with the following simplified[^8] lemma: [@goebelHybridDynamicalSystems2012 Lem. 7.20]\[lem:pract\_robust\] Let the inclusion  be well-posed and let ${\mathcal{A}}\subset {\mathcal{X}}$ be a compact and asymptotically stable set for , i.e., $d_{\mathcal{A}}( x (t)) \leq \beta( d_{\mathcal{A}}(x(0)), t)$ for all $t \geq 0$ holds for some $\beta \in {\mathcal{KL}}$ and any (complete) solution $x$ of . Then, for every $\rho > 0$, every compact $\mathcal{B} \subset {\mathbb{R}}^n$, and every $\zeta > 0$ there exists $\delta \in (0,1)$ such that every solution $x_\delta$ of the $\delta\rho$-perturbation of  starting in $\mathcal{B} \cap C_{\delta \rho}$ satisfies $d_{\mathcal{A}}(x_\delta (t)) \leq \beta( d_{\mathcal{A}}( x_\delta(0)), t) + \zeta$ for all $t \geq 0$. Hence, using \[prop:perturbation\], we arrive at the following second main result: \[thm:rob\_stab\] Consider a [PDS ]{}  where ${\mathcal{C}}$ is Clarke regular, $f$ and $G$ are continuous, and for which the compact set ${\mathcal{A}}\subset {\mathcal{Z}}$ is globally asymptotically stable, i.e., there is $\beta \in {\mathcal{KL}}$ such that for every solution $z$ it holds that $$\begin{aligned} d_{\mathcal{A}}( z (t)) \leq \beta( d_{\mathcal{A}}(z(0)), t) \quad \forall t \geq 0 \, . \end{aligned}$$ Then, for every $\zeta > 0$ and every compact ${\mathcal{B}}\subset {\mathcal{Z}}$ there exists $K^\star > 0$ such that for all $K \in (0, K^\star)$ every solution $z_K$ of the AWA  with $z_K(0) \in {\mathcal{B}}$ satisfies $$\begin{aligned} d_{\mathcal{A}}( z_K(t)) \leq \beta( d_{\mathcal{A}}(z_K(0)), t) + \zeta \qquad \forall t \geq 0\, . \end{aligned}$$ First, we establish that \[ass:local\] holds for every $z_0 \in {\mathcal{B}}$. Since ${\mathcal{B}}$ is compact, let $\overline{\beta} := \max_{z \in {\mathcal{B}}} \beta(d_{\mathcal{A}}(z), 0)$. Since $\beta$ is strictly increasing and unbounded, and, since ${\mathcal{A}}$ is compact, the set ${\mathcal{V}}:= \{ z \, | \, \beta(d_{\mathcal{A}}(z), 0) \leq \overline{\beta}\}$ is compact. Hence, we can choose $\epsilon > 0$ such that ${\mathcal{V}}\subset {\mathcal{B}}+ \epsilon {\mathbb{B}}$. It follows that any solution of   starting in ${\mathcal{B}}$ remains in ${\mathcal{B}}+ \epsilon {\mathbb{B}}$. By continuity over the compact set ${\mathcal{B}}+ \epsilon {\mathbb{B}}$, we can further choose and $M, \mu, \nu > 0$ such that $\| f(z, P_{\mathcal{Z}}(z)) \| \leq M$ and $\mu {\mathbb{I}}\preceq G^{-1}(z) \preceq \nu {\mathbb{I}}$ holds for all $z \in {\mathcal{B}}+ \epsilon {\mathbb{B}}$. Thus, \[ass:local\] is satisfied for all $z_0 \in {\mathcal{B}}$. Further, every (complete) solution of the PDS  starting in ${\mathcal{B}}$ remains in ${\mathcal{B}}+ \epsilon {\mathbb{B}}$ and hence can be written in its alternate form . Next, fix any $\rho > 0$. \[lem:pract\_robust\] implies that for every $\zeta > 0$ and every compact ${\mathcal{B}}\subset {\mathcal{Z}}$ there exists $\delta \in (0,1)$ such that the $\delta \rho$-perturbation is $\zeta$-practically pre-asymptotically stable. Given such a $\delta$, we conclude that there exists $K^\star > 0$ that, for all $K' < K^\star$, $\max \{ K' \tfrac{M}{\mu}, \omega (K' \tfrac{M}{\mu} ) \} \leq \delta \rho$ since $\omega$ is strictly increasing and $\omega(0) = 0$. Thus, \[prop:perturbation\] states that the solution of  with $K = K'$ is a solution of the $\sigma$-perturbation of  with $\sigma = \max \{ K' \tfrac{M}{\mu}, \omega(K' \tfrac{M}{\mu}) \}$. Moreover, it is also solution to any $\sigma'$-perturbation with $\sigma' \geq \sigma$ and, in particular, for $\sigma' = \delta \rho$. Since the asymptotic stability of ${\mathcal{A}}$ can often be established with a smooth Lyapunov function (see [@goebelHybridDynamicalSystems2012 Thm. 3.18]), we can also state the following corollary: Consider the PDS  where ${\mathcal{C}}$ is Clarke regular, $f$ and $G$ are continuous. Further, consider a compact set ${\mathcal{A}}\subset {\mathcal{Z}}$ for which there exists a *Lyapunov function*[^9]. Then, for every $\zeta > 0$ and every compact set ${\mathcal{B}}\subset {\mathbb{R}}^n$, there exists $K^\star$ such that for all $K \in (0, K^\star)$ every solution of  converges to a subset of ${\mathcal{A}}+ \zeta {\mathbb{B}}$. Preservation of Equilibria & Robust Convergence {#sec:rob_conv} =============================================== Finally, we consider the special case of  when $f$ depends only on $P_{\mathcal{Z}}(z)$, i.e., we study the system $$\begin{aligned} \label{eq:sys_aw_gen_spec} \dot z \in F_K(z) := f(P_{\mathcal{Z}}(z)) - \tfrac{1}{K} G^{-1}(P_{\mathcal{Z}}(z)) ( z - P_{\mathcal{Z}}(z))\end{aligned}$$ where, as before, ${\mathcal{Z}}$ is an $\alpha$-prox-regular set, $G: {\mathcal{Z}}\rightarrow {\mathbb{S}}_+^n$ is a continuous metric, $K > 0$ is a scalar, and $f: {\mathcal{Z}}\rightarrow {\mathbb{R}}^n$ is a continuous vector field. All of the previous results for  also apply to . In particular, as $K \rightarrow 0^+$, trajectories of  converge uniformly to solutions of the [PDS ]{} . Also, the practical stability results of \[sec:rob\] apply, but we show next that a stronger result can be derived for . In the following, $z^\star$ is a *weak equilibrium* of  if the constant trajectory $z \equiv z^\star$ is a solution of . Since we consider only Carathéodory solutions, $z^\star$ is a weak equilibrium of  if and only if $0 \in F_K(z^\star)$. An important advantage of  over the more general system  is that equilibria of  are preserved in the following sense (which generalizes [@hauswirthImplementationProjectedDynamical2020 Prop. 4]): \[prop:equil\] If ${\overline{z}}^\star \in {\mathcal{Z}}$ is a weak equilibrium point of the PDS , then there exists $K^\star > 0$ such that for all $K \in (0, K^\star)$ there exists a weak equilibrium point ${z}_K^\star \in {\overline{z}}^\star + N_{{\overline{z}}^\star} {\mathcal{Z}}\cap \tfrac{1}{2\alpha} \operatorname{int}{\mathbb{B}}$ for the AWA . Conversely, if ${z}^\star_K \in {\mathcal{Z}}^\circ_\alpha$ is a weak equilibrium of  for some $K$, then $P_{\mathcal{Z}}(z^\star)$ is a weak equilibrium of . Given a weak equilibrium ${\overline{z}}^\star \in {\mathcal{Z}}$ of , let $z^\star_K := {\overline{z}}^\star - K G({\overline{z}}^\star)f({\overline{z}}^\star)$. For $K \in (0, K^\star) := 1/ (2\alpha \| G({\overline{z}}^\star) f({\overline{z}}^\star)\|)$, we have $z^\star_K \in {\mathcal{Z}}^\circ_\alpha$. Since ${\overline{z}}^\star$ is an equilibrium of  (by assumption) and using \[lem:pds\_decomp\], we have $f({\overline{z}}^\star) \in - N^G_{{\overline{z}}^\star} {\mathcal{Z}}$. It follows from  that $-K G({\overline{z}}^\star) f({\overline{z}}^\star) \in N_{{\overline{z}}^\star} {\mathcal{Z}}$ and consequently $z^\star_K \in {\overline{z}}^\star + N_{{\overline{z}}^\star} {\mathcal{Z}}$. By \[prop:prox\], it follows that $P_{\mathcal{Z}}(z_K^\star) = {\overline{z}}^\star$ and therefore $$\begin{aligned} F_K(z^\star_K) = f ({\overline{z}}^\star) - \tfrac{1}{K}G^{-1}({\overline{z}}^\star) \left({\overline{z}}^\star - K G({\overline{z}}^\star)f({\overline{z}}^\star) - {\overline{z}}^\star \right) = 0 \, . \end{aligned}$$ Thus, $z^\star_K$ is a weak equilibrium of . The converse case follows the same ideas. Although equilibria of the PDS  are preserved by the AWA  (after projection), it is not clear whether convergence properties are preserved, especially since we are primarily interested in the convergence of $t \mapsto P_{\mathcal{Z}}(z(t))$ rather than the convergence of the solution $z$ of . \[thm:rob\_stab\] suggests that, in general, convergence is only within a neighborhood of asymptotically stable equilibria of the PDS . However, as we shown below, under additional conditions on $f, G$ and ${\mathcal{Z}}$, the projected solutions $t \mapsto P_{\mathcal{Z}}(z(t))$ do indeed converge to an equilibrium of . Anti-Windup Approximations of Monotone Dynamics ----------------------------------------------- Next, we show that if $-f$ is monotone and $G \equiv {\mathbb{I}}$, then $F_K$, as defined in , is monotone for small enough $K$. This, in turn, allows us conclude asymptotic stability of . Since we require only monotonicity of $f$, the following results can be used not only when $f$ is chosen as the gradient of a convex cost function, but also for saddle-point flows (see \[subsec:saddle\]), and pseudo-gradients for Nash-equilibrium seeking [@nagurneyProjectedDynamicalSystems1996; @depersisDistributedaveragingintegral2019]. Given a set ${\mathcal{C}}\subset {\mathbb{R}}^n$, recall that a map $F: {\mathcal{C}}\rightrightarrows {\mathbb{R}}^n$ is (strictly; $\beta$-strongly) *monotone* if for all $x, x' \in {\mathcal{C}}$ and all $v \in F(x)$ and $v' \in F(x')$ it holds that $$\begin{aligned} \left\langle v - v', x - x' \right\rangle \geq 0 \, (> 0; \geq \beta \| x- x' \|^2) \ .\end{aligned}$$ Further, if ${\mathcal{C}}$ is $\alpha$-prox-regular, the map $x \mapsto N_x {\mathcal{C}}$ has a *hypomonotone localization* [@rockafellarVariationalAnalysis2009 Ex. 13.38], i.e., for all $x, x' \in {\mathcal{C}}$, all $\eta \in N_x {\mathcal{C}}\cap {\mathbb{B}}$, and all $\eta' \in N_{x'} {\mathcal{C}}\cap {\mathbb{B}}$ we have $$\begin{aligned} \left\langle \eta - \eta', x- x' \right\rangle \geq - 2 \alpha \| x - x' \|^2 \, .\end{aligned}$$ In particular, if ${\mathcal{C}}$ is convex, we have $\left\langle \eta' - \eta, x'- x \right\rangle \geq 0$ and $x \mapsto N_x {\mathcal{C}}$ is monotone. \[prop:monot\] Consider $F_K$ as defined in  with $G \equiv {\mathbb{I}}$ and ${\mathcal{C}}$ is assumed to be $\alpha$-prox-regular. Let $-f$ be $\beta$-strongly monotone and globally $L$-Lipschitz. Then $-F_K$ is strictly monotone on ${\mathcal{Z}}^\circ_\alpha$ for all $0< K < 4 (\beta - 2 \alpha)/ L^2$. Given any $z, z' \in {\mathcal{Z}}^\circ_\alpha$, let ${\overline{z}} := P_{\mathcal{Z}}(z)$ and ${\overline{z}}' := P_{\mathcal{Z}}(z')$. Further, let $\eta := z - {\overline{z}} \in N_{{\overline{z}}} {\mathcal{Z}}$ and $\eta' := z' - {\overline{z}}' \in N_{{\overline{z}}'} {\mathcal{Z}}$. We can work directly with the monotonicity of $f$, the hypomonotocity of $z \mapsto N_z {\mathcal{Z}}$, and Cauchy-Schwarz to derive $$\begin{aligned} \left\langle z - z', F_K(z) - F_K(z') \right\rangle & = \left\langle z - z', f({\overline{z}}) - f({\overline{z}}') - \tfrac{1}{K}(z - {\overline{z}}) + \tfrac{1}{K}(z' - {\overline{z}}')) \right\rangle \\ & = \left\langle {\overline{z}} - {\overline{z}}' + \eta - \eta', f({\overline{z}}) - f({\overline{z}}') - \tfrac{1}{K} (\eta - \eta') \right\rangle \\ & = \left\langle {\overline{z}} - {\overline{z}}', f({\overline{z}}) - f({\overline{z}}') \right\rangle - \tfrac{1}{K} \left\langle \eta - \eta', \eta - \eta' \right\rangle \\ & \qquad \qquad + \underbrace{\left\langle \eta - \eta', f({\overline{z}}) - f({\overline{z}}') \right\rangle}_{L \| \eta - \eta'\| \| {\overline{z}} - {\overline{z}}' \|} - \tfrac{1}{K} \underbrace{\left\langle {\overline{z}} - {\overline{z}}', \eta - \eta' \right\rangle}_{\geq - 2 \alpha \| {\overline{z}} - {\overline{z}}' \|^2} \\ & \leq - (\beta - 2 \alpha) \| {\overline{z}} - {\overline{z}}' \|^2 + L \| {\overline{z}} - {\overline{z}}' \| \| \eta - \eta' \| - \tfrac{1}{K} \| \eta - \eta' \|^2 \, . \end{aligned}$$ A sufficient condition for the righthand side to be negative for all ${\overline{z}} \neq {\overline{z}}'$ is that $\beta - 2\alpha > 0$ and that the determinant $\tfrac{1}{K}(\beta - 2\alpha) - \tfrac{1}{4}L^2$ is positive, i.e., if $0< K < 4(\beta - 2 \alpha)/L^2$. This leads us to our third theoretical result which establishes convergence of anti-windup approximations for strongly monotone dynamics on convex sets: \[thm:monot\] Consider the AWA  with $G \equiv {\mathbb{I}}$ and let ${\mathcal{C}}$ be closed convex. Assume that $-f$ is $\beta$-strongly monotone and globally $L$-Lipschitz. Then, for all $K < 4 \beta / L^2$, every trajectory of  converges to an equilibrium point $z^\star$ (which is unique) such that $P_{\mathcal{Z}}(z^\star)$ is the unique equilibrium of the [PDS ]{} . Because of convexity of ${\mathcal{Z}}$, $P_{\mathcal{Z}}(z)$ is single-valued and continuous for all $z \in {\mathbb{R}}^n$ and globally 1-Lipschitz (i.e., non-expansive). As a consequence, $F_K$ is globally Lipschitz continuous and there exists a unique complete solution of  for every initial condition $z(0) \in {\mathbb{R}}^n$. Furthermore, since $K < 4 \beta / L^2$ and ${\mathcal{Z}}$ is convex (which lets us take $\alpha \rightarrow 0^+$), \[prop:monot\] guarantees that $F_K$ is strictly monotone on ${\mathbb{R}}^n$. Next, recall that the strong monotonicity of $-f$ and convexity of ${\mathcal{Z}}$ imply that has a unique equilibrium ${\overline{z}}^\star$ [@nagurneyProjectedDynamicalSystems1996 Thm. 2.3]. Consequently, \[prop:equil\] guarantees the existence of an equilibrium point $z^\star$ of  such that $P_{\mathcal{Z}}(z^\star) = {\overline{z}}^\star$. Furthermore, $z^\star$ is unique by [@nagurneyProjectedDynamicalSystems1996 Thm. 2.2]. In particular, strict monotonicity of $F_K$ implies that $V(z):= \tfrac{1}{2} \| z - z^\star \|^2$ is a Lyapunov function for  which can be used to establish global asymptotic stability of $z^\star$. \[thm:monot\] can, presumably, be generalized to prox-regular sets as well as general metrics $G$. However, in that case, additional restriction on $z(0)$ are required, the threshold value for $K$ is less easily quantifiable, and convergence is likely only local. Application: Anti-Windup for Autonomous Optimization {#sec:aw_appl} ==================================================== Next, we show how the AWA  models physical systems and how anti-windup implementations can be used in the context of autonomous optimization to approximate closed-loop optimization dynamics that are formulated as projected dynamical systems. First, consider the feedback control loop illustrated in \[fig:aw1\]. Namely, we study a plant controlled by an integral feedback controller that is subject to input saturation modelled as an Euclidean projection. An anti-windup scheme is in place to avoid integrator windup. More precisely, we consider a dynamical system of the form \[eq:fb\_aw\_sys\] $$\begin{aligned} \dot x & \in \tilde{f}(x, P_{\mathcal{U}}(u)) & & x \in {\mathbb{R}}^m \\ \dot u & \in k(x, u, P_{\mathcal{U}}(u)) - \tfrac{1}{K} \tilde{G}^{-1}(P_{\mathcal{U}}(u)) ( u - P_{\mathcal{U}}(u) ) \qquad & & u \in {\mathbb{R}}^p \end{aligned}$$ where ${\mathcal{U}}\subset {\mathbb{R}}^p$ is prox-regular, $\tilde{f}:{\mathbb{R}}^m \times {\mathcal{U}}\rightarrow {\mathbb{R}}^m $ and $k: {\mathbb{R}}^m \times {\mathbb{R}}^p \times {\mathcal{U}}\rightarrow {\mathbb{R}}^p$ are continuous vector fields, $\tilde{G}: {\mathcal{U}}\rightarrow {\mathbb{S}}_p^+$ is a continuous metric, and $K>0$. ; (ctrl.north)|-(aw\_inf.west) node\[at end, below\][$+$]{}; (awgain.west)-|(aw\_inf.north) node\[near end, left\][$-$]{}; (aw\_inf.east)–(int.west); (int.east)–(br1.west) node\[at end, below\] [$u$]{}; (br1.east)–(sat.west); (sat.east)–(br2.west); (br2.east)-|(plant.north) node\[near start, above\][$\overline{u} := P_{\mathcal{U}}(u)$]{}; (plant.south)|-(edge.center); (edge.center)–(ctrl.south); (br2.north)|-(awsum.east) node\[at end, above\] [$-$]{}; (br1.north)–(awsum.south) node\[near end, right\] [$+$]{}; (awsum.west)–(awgain.east); The system  can be brought into the form of an AWA  with $n = m + p$ by defining $z := \left[\begin{smallmatrix} x \\ u \end{smallmatrix}\right]$, ${\mathcal{Z}}:= {\mathbb{R}}^m \times {\mathcal{U}}$, and $G(z) := \left[\begin{smallmatrix} {\mathbb{I}}& 0 \\ 0 & \tilde{G}(u) \end{smallmatrix}\right]$. Thus, we further have $$\begin{aligned} P_{\mathcal{Z}}(z) = \begin{bmatrix} x \\ P_{\mathcal{U}}(u) \end{bmatrix} \qquad \text{and} \qquad f(z, P_{\mathcal{Z}}(z)) := \begin{bmatrix} \tilde{f}(x, P_{\mathcal{U}}(u)) \\ k(x, u, P_{\mathcal{U}}(u)) \end{bmatrix} \,.\end{aligned}$$ With these definitions, the [PDS ]{} takes the form $$\begin{aligned} \dot x & = \tilde{f}(x, u) & & x \in {\mathbb{R}}^m \\ \dot u & = \Pi_{\mathcal{U}}^{\tilde{G}} [k(x, u, u)](u) \qquad & & u \in {\mathcal{U}}\, ,\end{aligned}$$ where we can ignore the projection onto ${\mathcal{U}}$ in the third argument of $k$, because any solution of the PDS  is viable (i.e., remains in ${\mathcal{U}}$) by definition. \[fig:aw1\] shows one limitation of our problem setup: Compared to existing work on anti-windup control [@zaccarianModernAntiwindupSynthesis2011; @tarbouriechAntiwindupDesignOverview2009], we do not model any proportional controller subject to input saturation. This is motivated, on one hand, by theoretical necessity. On the other hand, for our application scenario of autonomous optimization discussed below, stability of the physical plant is usually a prerequisite. Feedback-based Gradient Schemes for Quadratic Programs ------------------------------------------------------ To illustrate the design opportunities for autonomous optimization, we present three anti-windup schemes that approximate projected gradient flows for a quadratic program (QP). We consider the relatively simple problem of solving a QP as it allows for a concise presentation, easy implementation, and comparability. However, needless to say, our theoretical results in the previous sections cover much more general setups. Our goal is to design a feedback controller that steers a plant to a steady state that solves the optimization problem $$\begin{aligned} \label{eq:app_opt_prob} \begin{split} \begin{aligned} \text{minimize} \quad & \Phi(x) :=\tfrac{1}{2} x^T Q x + c^T x + d \\ \text{subject to} \quad & x = h(u) := H u + w \\ & u \in {\mathcal{U}}:= \{ v \, | \, A_u v \leq b_u \} \end{aligned} \end{split}\end{aligned}$$ where $x \in {\mathbb{R}}^m$ and $u \in {\mathbb{R}}^p$ denote the system state and control input, respectively, and $Q \in {\mathbb{S}}_+^m$, $A_u \in {\mathbb{R}}^{r \times p}$ and the remaining parameters are of appropriate size. The map $h$ denotes the steady-state input-to-state map of the plant subject to the disturbance $w$.[^10] The set ${\mathcal{U}}$ defines constraints which are enforced by physical saturation. For solving  we aim at approximating the projected gradient flow $\dot u = \Pi_{\mathcal{U}}^G [- G^{-1}(u)\nabla \hat{\Phi} (u)](u)$, where we have defined $\hat{\Phi}(u) := \Phi (h(u))$ to eliminate the state variable $x$. In particular, we have $\nabla \hat{\Phi}(u) = H^T \nabla \Phi(h(u))$. In the following, the metric $G$ will be either $G \equiv {\mathbb{I}}$ or $G \equiv Q$ (the latter yielding a projected Newton flow). To approximate $\dot u = \Pi_{\mathcal{U}}^G [- G^{-1}(u) \nabla \hat{\Phi}(u) ](u)$, we consider three systems that fall into the class of anti-windup approximations defined by , two of which can be implemented in a feedback loop as in \[fig:aw1\]. Their convergence behavior for the same problem instance and varying $K$ is illustrated in \[fig:gradient\_sim\] and discussed below. 1. *Penalty Gradient Flow:* As a reference system we consider the gradient flow of the potential function $\Psi(u) := \hat{\Phi}(u) + \tfrac{1}{2K} d^2_{\mathcal{U}}(u))$ which is given by $$\begin{aligned} \label{eq:sim_pen_grad} \dot u = - \nabla \Psi(u) = - H^T \nabla \Phi(h(u)) - \tfrac{1}{K}(u - P_{\mathcal{U}}(u) ) \,. \end{aligned}$$ In this case, we have $G \equiv {\mathbb{I}}$ and $K > 0$ takes the role of a *penalty parameter* for the soft penalty term $d_{\mathcal{U}}^2$ that approximately enforces the input constraint $u \in {\mathcal{U}}$.[^11] The system  is a special case of the AWA  and, as a consequence, \[thm:loc,thm:rob\_stab\] (uniform convergence and robust practical stability) and their corollaries apply as $K \rightarrow 0^+$. However,  is not of the special form  and convergence of to the optimizer of the problem  is not guaranteed for positive $K > 0$. Neither does  lend itself to a feedback implementation, because $\nabla {\Phi}$ is evaluated at $h(u)$ rather than at $h(P_{\mathcal{U}}(u))$ (which is the actual system state for the saturated input). 2. *Anti-Windup Gradient Scheme:* As a second type of dynamics we consider $$\begin{aligned} \label{eq:sim_aw_grad} \underbrace{\dot u = - H^T \nabla {\Phi}( \overline{x}) - \tfrac{1}{K} (u -\overline{u})}_{\text{controller}} \qquad \underbrace{\overline{u} := P_{\mathcal{U}}(u) \qquad \overline{x} := h(\overline{u})}_{\text{physical system}} \end{aligned}$$ which can be implemented in closed loop because the quantities ${\overline{u}}$ and ${\overline{x}}$ are “evaluated” by the physical system at no computational cost (and are assumed to be measurable), which is one of the key features of autonomous optimization. Furthermore, because ${\mathcal{U}}$ is convex and $\Phi$ is strongly convex (which implies strong monotonicity), Theorem  is applicable and guarantees that $\overline{z} = (\overline{u}, \overline{x})$ converges to the optimizer of . This is confirmed in \[fig:gradient\_sim\]. 3. *Anti-Windup Newton Scheme:* As the final gradient-based anti-windup scheme we consider an anti-windup approximation with $G \equiv Q$ and which is given by $$\begin{aligned} \label{eq:sim_aw_newt} \underbrace{\dot u = - Q^{-1} \left(H^T \nabla {\Phi}( \overline{x}) - \tfrac{1}{K} (u -\overline{u} )\right)}_{\text{controller}} \qquad \underbrace{\overline{u} := P_{\mathcal{U}}(u) \qquad \overline{x} := h(\overline{u})}_{\text{physical system}} \, . \end{aligned}$$ The system  can be implemented in closed loop with a physical system and approximates a *projected Newton flow* [@hauswirthProjectedDynamicalSystems2018 Ex. 5.6]. This fact is noteworthy, because, in general, projected Newton flows do not lend themselves to an easy implementation (e.g., as an iterative algorithm). Even though, as seen in \[fig:gradient\_sim\], ${\overline{u}}$ converges to the optimizer of , strictly speaking, Theorem  is not directly applicable because $Q \neq {\mathbb{I}}$. ![Convergence behavior of , , and  for a problem instance of  with $p = 100$ (input dimension) and $r = 300$ (\# of input constraints).[]{data-label="fig:gradient_sim"}](Figs/subplots_grad.pdf){width=".9\columnwidth"} The anti-windup gradient and Newton schemes defined above illustrate some of the key features of autonomous optimization and anti-windup implementations: 1. Under the conditions of \[thm:monot\], the actual system state and saturated control input converge to the optimizer $u^\star$ of , even though the internal control variable $u$ does not in general converge to $u^\star$. 2. In a feedback implementation exploiting input saturation, neither the set ${\mathcal{U}}$ nor the steady-state disturbance $w$ needs to be known (or estimated). The only model information required is $H$. Furthemore, recent preliminary theoretical [@colombinorobustnessguaranteesfeedbackbased2019] and experimental results for power systems [@ortmannExperimentalValidationFeedback2019] suggest that these feedback schemes are robust against uncertainties in $H$. 3. The simulations in \[fig:gradient\_sim\] suggest that the convergence rate of the “projected trajectory” of  is not affected by the value of $K$ and is equivalent to the convergence rate of the nominal projected gradient flow. In contrast, the convergence rate of the anti-windup Newton scheme  does depend on $K$ and one can recover the rate of projected Newton flow only in the limit $K \rightarrow 0^+$. An analysis of this observation remains, however, outside the scope of this paper. Feedback-based Saddle-Flows with Anti-Windup {#subsec:saddle} -------------------------------------------- In autonomous optimization, constraints on the system state (or output) cannot be enforced directly because they are not directly controllable and often subject to disturbances affecting the physical plant (e.g. an unknown value of $w$). For the purpose of enforcing state or output constraints, projected saddle-point flows have been proven effective [@tangRunningPrimalDualGradient2018; @dallaneseOptimalPowerFlow2018; @ortmannExperimentalValidationFeedback2019]. In this section, we indicate how anti-windup approximations can be combined with this type of dynamical system, even though this leads us slightly outside the scope of our theoretical results. We consider quadratic program $$\begin{aligned} \label{eq:app_opt_prob_out} \begin{split} \text{minimize} \quad &\Phi(x) \\ \text{subject to} \quad & x = h(u), \, u \in {\mathcal{U}}\\ & x \in {\mathcal{X}}:= \{ x \, | \, A_x x \leq b_x \} \, , \end{split} \end{aligned}$$ where $\Phi, h$, and ${\mathcal{U}}$ are defined as in  and ${\mathcal{X}}$ denotes a set of state constraints with $A_x \in {\mathbb{R}}^{s \times m}$ and $b_x \in {\mathbb{R}}^s$. To solve , we consider the projected saddle-point flow $$\begin{aligned} \label{eq:saddle_flow_proj} \dot u = \Pi_{\mathcal{U}}\left[- H^T \nabla \Phi(h(u)) - H^T A_x^T \mu \right] \qquad \qquad \dot \mu = \Pi_{{\mathbb{R}}^s_{\geq 0}} [ A_x h(u) - b_x] \, , \end{aligned}$$ where $\mu \in {\mathbb{R}}^s$ denotes the dual multipliers associated with the output constraints. The system  (and special cases in which either primal or dual variables are not projected) has been extensively studied and convergence is guaranteed, for instance, under strict convexity of $\Phi$. We refer the reader to [@goebelStabilityRobustnessSaddlepoint2017; @cherukuriRoleConvexitySaddlePoint2017] and references therein. We approximate  with a (partial) anti-windup implementation as $$\begin{aligned} \label{eq:saddle_flow_aw} \underbrace{ \begin{aligned} \dot u & = - H^T \nabla {\Phi}( \overline{x}) - H^T A_x^T \mu - \tfrac{1}{K} (u -\overline{u}) \\ \dot \mu & = \Pi_{{\mathbb{R}}^s_\geq 0} [ A_x {\overline{x}} - b_x] \end{aligned} }_{\text{controller}} \qquad \underbrace{\overline{u} := P_{\mathcal{U}}(u) \qquad \overline{x} := h(\overline{u})}_{\text{physical system}} \, . \end{aligned}$$ We do not approximate the projected integration of the dual variables with an anti-windup term, since the dual variables are often internal variables of the controller and the projection on the non-negative orthant is easily implementable. ![Convergence behavior of  (and the [PDS ]{} ) for a problem instance of  with $p = 3$ (input dimension), $m = 5$ (state dimension), $r = 10$ (\# of input constraints), and $s = 5$ (\# of state constraints).[]{data-label="fig:saddle_sim"}](Figs/subplots_saddle.pdf){width=".9\columnwidth"} \[fig:saddle\_sim\] illustrates the behavior of  and . Similarly to the results for the gradient anti-windup approximations, we observe that $u$ does not, in general, converge to its optimal value. However, the saturated control input $P_{\mathcal{U}}(u)$ (and thereby the actual system state) and the dual variable $\mu$ converge to the solution of . \[thm:monot\] (robust convergence) does not apply to . First, while the projected saddle-flow  is monotone, strong monotonicity is usually not guaranteed [@goebelStabilityRobustnessSaddlepoint2017; @cherukuriRoleConvexitySaddlePoint2017]. Second, by applying only a partial anti-windup approximation, the vector field remains discontinuous because of the projection of $\mu$ on ${\mathbb{R}}^s_{\geq 0}$. Conclusion {#sec:conc} ========== In this paper we have studied a general class of dynamical systems which are inspired by classical anti-windup control schemes. We have rigourosly established that these systems approximate oblique projected dynamical systems in terms of uniform convergence and semiglobal practical robust stability. Furthermore, we have shown that for a special case, and under an additional monotonicity assumption, these anti-windup approximations exhibit robust convergence to the equilibria of the limiting projected dynamical system. We have further illustrated several ways in which our results apply in the context of autonomous optimization. In particular, we have shown how physical saturation can be exploited to drive a plant to an optimal steady state without explicit knowledge of the physically-enforced input domain. Several points remain open: First, it is unclear whether our analysis can be extended to consider control laws that incorporate a proportional control component. Second, the strong monotonicity requirement for robust convergence to equilibria of a projected dynamical systems can presumably be relaxed. Third, our simulations suggest that certain anti-windup gradient schemes retain the same convergence rate as the limiting projected gradient flow, independently of the anti-windup gain. Fully understanding this surprising phenomenon requires further work. [^1]: Department of Information Technology and Electrical Engineering, ETH Zürich, Zurich, Switzerland (,). [^2]: Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA (). [^3]: Submitted to the editors DATE. [^4]: To allow for a more concise presentation, we limit ourselves to closed, Clarke regular subsets of ${\mathbb{R}}^n$ which allow for an unambiguous definition of tangent and normal cones. [^5]: A solution $x:[0, T] \rightarrow {\mathcal{C}}$ of  is unique if for every other solution $x':[0, T'] \rightarrow {\mathcal{C}}$ with the same initial condition it holds that $x(t) = x'(t)$ for all $t \in [0, \min\{T, T'\}]$. [^6]: We cannot rely on the existence of Carathéodory solutions because ${\mathcal{Z}}$ is not prox-regular and \[prop:exist\] does not apply, but every Carathéodory solution (if it exits) is a Krasovskii solution. [^7]: We require only the first of the two statements of the original theorem. Further, we consider the case where $\rho$ is constant. Finally, we work with truncated solutions which have, by definition, a compact domain (and thus are trivially *locally eventually bounded* [@goebelHybridDynamicalSystems2012 Def. 5.24]). [^8]: We consider only global asymptotic stability, which allows us to use the distance function instead of more general indicator functions. Further, we limit ourselves to $\rho$ being a positive constant instead of a function. As noted in \[rem:pre\_stab\], compactness and stability of ${\mathcal{A}}$ guarantee the existence of complete solutions since finite-time escape is not possible. [^9]: Namely, $V : {\mathbb{R}}^n \rightarrow {\mathbb{R}}_{\geq 0}$ is a Lyapunov function for ${\mathcal{A}}$ if it is differentiable everywhere on ${\mathcal{Z}}$, there exist $\underline{\alpha}, \overline{\alpha} \in {\mathcal{K}}_\infty$ such that $\underline{\alpha}(d_{\mathcal{A}}(z)) \leq V(z) \leq\overline{\alpha}(d_{\mathcal{A}}(z))$ for all $z \in {\mathcal{Z}}$, and $\left\langle \nabla V(z), \Pi_{\mathcal{Z}}^G [f](z) \right\rangle \leq - \alpha(z)$ for all $z \in {\mathcal{Z}}$ where $\alpha: {\mathbb{R}}^n \rightarrow {\mathbb{R}}_{\geq 0}$ is continuous and positive definite with respect to ${\mathcal{A}}$, i.e., $\alpha(z) > 0$ for all $z \notin {\mathcal{A}}$ and $\alpha(z) = 0$ for all $z \in {\mathcal{A}}$. [^10]: In contrast to , we assume for  that the physical plant is described by an steady-state input-to-state map $x = h(u)$ that satisfies $\tilde{f}(h(u), u) = 0$ for all $u \in {\mathcal{U}}$. This approximation can be motivated by singular perturbation ideas [@mentaStabilityDynamicFeedback2018; @hauswirthTimescaleSeparationAutonomous2019] which stipulate that the interconnection of *fast decaying* plant dynamics and *slow* optimization dynamics is asymptotically stable. The results in this section can be generalized to a dynamic plant accordingly. [^11]: The penalty $d^2_{\mathcal{U}}$ is illustrative in the context of autonomous optimization, however, it is not generally practical for numerical optimization, because evaluating $\nabla d^2_{\mathcal{U}}$ requires computing $P_{\mathcal{U}}$. Instead, in numerical applications, it is more common to use a penalty $\| \max \{ A_u u - b_u, 0 \}\|^2$.

Since the discovery of high-temperature superconductivity, doped antiferromagnets (AF) have been studied intensively. The pseudo-gap behavior observed in the high-$T_c$ cuprates has stimulated great interest in systems with spin gaps. Several new one- and two-dimensional spin gap systems have been found experimentally. These materials are characterized by a disordered singlet ground state and a finite gap to all spin excitations. Some of the compounds which have two-dimensional (2d) character include the coupled spin ladder systems, SrCu$_2$O$_3$[@srcu2o3], CaV$_2$O$_5$[@cav2o5], (VO$_2$)P$_2$O$_7$[@vopo], Cu$_2$(C$_5$H$_{12}$N$_2$)$_2$Cl$_4$[@cuHpCl], and the plaquette RVB system, CaV$_4$O$_9$[@cav4o9]. Recently the two-dimensional spin gap system [[SrCu$_2$(BO$_3$)$_2$]{}]{} has been found by Kageyama [*et al.*]{}[@kag98]. It has a spin-singlet ground state with a spin gap $\sim 30$ K. The substance has additional interesting features, e.g., the high-field magnetization was observed to have two plateaus at 1/4 and 1/8 of the full moment. Recent work [@miy98] suggests that the underlying physics can be understood on the basis of a two-dimensional $S={1\over 2}$ Heisenberg model with antiferromagnetic nearest-neighbor ($J_1$, on links $A$) and next-nearest-neighbor ($J_2$, on links $B$) couplings on the lattice shown in Fig. \[FIG\_LATTICE\]. The nearest-neighbor bonds $A$ define a unique singlet covering of the lattice. The Heisenberg model corresponding to Fig. \[FIG\_LATTICE\] (with $J_1$, $J_2$) is in fact topologically equivalent to the model considered by Shastry and Sutherland [@shastry]. For this model the singlet product state forms an exact eigenstate of the Hamiltonian at all couplings, and is the ground-state in a region where the nearest-neighbor coupling $J_1$ dominates. On the other hand, for $J_1\rightarrow 0$ the system becomes equivalent to the 2d square lattice AF (with nearest-neighbor coupling $J_2$) which has a Néel-ordered ground state. The Shastry-Sutherland model can be complemented by a coupling $J_3$ on the links $C$; the singlet product state is an eigenstate of this generalized model, too. For $J_2=J_3$ the total spin on each $A$ bond is conserved, i.e., there is a macroscopic number of conserved quantities, and each eigenstate is characterized by the number and positions of triplets. =6.8 truecm The Heisenberg model of Fig. 1 has been studied in Refs. using exact diagonalization, Schwinger boson and series expansion methods. For $J_2=J_3$ it can be mapped onto a spin-1 model which shows a first order transition between the singlet state and a Néel-ordered state at $J_2/J_1 = 0.4296$. For $J_2\neq J_3$ the situation is less clear. For $J_3=0$ the numerical results provide evidence for a weak first order singlet-Néel transition at $J_2/J_1=0.70\pm 0.01$. The disordered [[SrCu$_2$(BO$_3$)$_2$]{}]{} compound lies probably close to the transition line to a Néel state, which explains the unusual temperature dependence of magnetic properties. A fit of the susceptibility obtained from the model (with $J_1$ and $J_2$ only) to the experimental data[@miy98] leads to the estimates of $J_1=100$ K and $J_2/J_1=0.68$. However, the actual ratio of the couplings $J_3/J_2$ is not known. To include the doping degree of freedom into the Shastry-Sutherland model we consider a standard $t-J$ model on the [[SrCu$_2$(BO$_3$)$_2$]{}]{} lattice: $$\begin{aligned} \mathcal{H} &=& \sum_{(i,j)\in A} \left ( - t_1 \: ( \hat{c}_{i\sigma}^\dagger \hat{c}_{j\sigma}^{} + h.c. ) + J_1 \: {\bf S}_i\cdot {\bf S}_j \right ) \nonumber\\ &+& \sum_{(i,k)\in B} \left ( - t_2 \: ( \hat{c}_{i\sigma}^\dagger \hat{c}_{k\sigma}^{} + h.c. ) + J_2 \: {\bf S}_i\cdot {\bf S}_k \right ) \nonumber\\ &+& \sum_{(i,l)\in C} \left ( - t_3 \: ( \hat{c}_{i\sigma}^\dagger \hat{c}_{l\sigma}^{} + h.c. ) + J_3 \: {\bf S}_i\cdot {\bf S}_l \right ) \:. \label{H}\end{aligned}$$ The electron operators $\hat c_{i\sigma}^{\dagger}$ exclude double occupancies. We have included hopping $t_3$ and interaction $J_3$ along the $C$ bonds. If the $t-J$ model is derived as strong-coupling limit of a Hubbard model on the [[SrCu$_2$(BO$_3$)$_2$]{}]{} lattice with on-site repulsion $U$, the ratio of the parameters is given by $${t_1^2 \over J_1} \:=\: {t_2^2 \over J_2} \:=\: {t_3^2 \over J_3} \:=\: {U \over 4} \,. \label{HUBBARD_REL}$$ Although the compound [[SrCu$_2$(BO$_3$)$_2$]{}]{} has to our knowledge not been doped so far, the study of the hole dynamics in this environment is an interesting and challenging question. Furthermore it may be possible that finite doping leads to the formation of hole pairs and eventually to superconductivity. In this paper we shall discuss the dynamics of a single hole in an otherwise half-filled system using a generalized spin-polaron picture. The one-hole spectral function for this case corresponds directly to the result of an angle-resolved photoemission experiment on the undoped compound which may give important information on the electronic correlations and the exchange constants in [SrCu$_2$(BO$_3$)$_2$]{}. We will show that the ratio $t_3/t_2$ tunes a cross-over between a narrow-band quasiparticle behavior and a regime where a free-fermion peak dominates the spectrum. The narrow-band behavior found here is similar to other 2d AF systems [@Dagotto94], it originates from the motion of a hole dressed with spin fluctuations [@Dagotto94; @Strings; @MaHo; @Riera97]. To investigate the hole motion we consider a one-particle Green’s function describing the creation of a single hole with momentum $\bf k$ at zero temperature: $$G({\bf k},\omega) \;=\; \sum_\sigma \langle\psi_0^N| {\hat c}_{{\bf k}\sigma}^\dagger {1 \over {z-H}} {\hat c}_{{\bf k}\sigma} |\psi_0^N\rangle \label{HOLE_GF}$$ where $z$ is the complex frequency variable, $z=\omega+i\eta$, $\eta\rightarrow 0$. $|\psi_0^N\rangle$ is the ground state of undoped system, i.e., with $N$ electrons on $N$ lattice sites. To describe the dimerized phase we employ a bond operator representation [@SaBha; @Gopalan] for the spins on the nearest-neighbor bonds. For each $A$ bond containing two $S=1/2$ spins we introduce bosonic operators for creation of a singlet and three triplet states out of the vacuum $|0\rangle$: $s^{\dagger}|0\rangle = \frac{1}{\sqrt{2}}(|\uparrow\downarrow\rangle - |\downarrow \uparrow\rangle)$, $t_{x}^{\dagger}|0\rangle = \frac{-1}{\sqrt{2}} (|\uparrow \uparrow\rangle - |\downarrow \downarrow\rangle)$, $t_{y}^{\dagger}|0\rangle = \frac{i}{\sqrt{2}}(|\uparrow \uparrow\rangle + |\downarrow \downarrow\rangle)$, $t_{z}^{\dagger}|0\rangle = \frac{1}{\sqrt{2}}(|\uparrow \downarrow\rangle + |\downarrow \uparrow\rangle)$, where the constraint $ s^{\dagger} s^{} + \sum_\alpha t_{\alpha}^{\dagger}t_{\alpha}^{} = 1$ has to be imposed on each bond to restrict the possible states to the physical Hilbert space. The original spins are related to the new boson basis operators by $S_{1,2}^{\alpha} = \frac{1}{2} ( \pm s^{\dagger} t_{\alpha}^{} \pm t_{\alpha}^{\dagger} s^{} - i \epsilon_{\alpha\beta\gamma} t_{\beta}^{\dagger} t_{\gamma})$. The Hamiltonian of the Shastry-Sutherland model written in terms of the bond operators contains no terms which create triplet excitations from a state containing only singlets which means that the singlet product state $|\phi_0\rangle = \prod_i s_i^\dagger |0\rangle$ is an exact eigenstate of the undoped system at all couplings. If we remove one electron from an $A$ bond, a single-hole state on this bond is created. We introduce fermionic operators for bonding (symmetric) and antibonding (antisymmetric) states of one electron (or hole) on an $A$ bond: $$\begin{aligned} a_{s,\sigma}^{\dag} |0\rangle = \frac{1}{\sqrt{2}} ( {\hat c}^{\dag}_{1,\sigma} + {\hat c}^{\dag}_{2,\sigma}) |0\rangle \:, \nonumber\\ a_{a,\sigma}^{\dag} |0\rangle = \frac{1}{\sqrt{2}} ( {\hat c}^{\dag}_{1,\sigma} - {\hat c}^{\dag}_{2,\sigma}) |0\rangle \: .\end{aligned}$$ The operators ${\hat c}^{\dag}_{1,2,\sigma}$ create an electron with spin $\sigma$ on one of the two sites of an $A$ bond. Hopping via $t_1$ leads to an ‘on-bond’ energy of $\pm t_1$ for $a_a$ and $a_s$, respectively. The other hopping terms in $\mathcal{H}$ permit the hole to hop between bonds; the exchange interaction gives rise to spin fluctuations on neighboring bonds. The full interaction Hamiltonian in terms of $s$, $t$, and $a$ operators can be easily found by calculating all possible one-hole matrix elements of the initial Hamiltonian $\mathcal{H}$ (\[H\]) (see e.g. Ref. ). The evaluation of the Green’s function (\[HOLE\_GF\]) is done using the Mori-Zwanzig projection technique. The set of dynamic variables is constructed from generalized path operators  [@Strings; @Vojta98] which here create strings of triplet excitations attached to the hole. Details of the calculational procedure can be found in Refs. . In the disordered phase of the Shastry-Sutherland model the evaluation of the matrix elements is simplified by the fact that the undoped ground state does not contain background spin fluctuations, i.e., no cumulant expectation values are involved (cf. Ref. ). In the present calculations we have employed up to 1800 dynamic variables with a maximum path length of 3. The neglect of the self-energy terms leads to a discrete set of poles for the Green’s functions, so the present approach cannot account for linewidths. In all figures we have introduced an artificial linewidth of $0.2 t_1$ to plot the spectra. Before turning to the hole dynamics it is worth mentioning that even a static vacancy (equivalent to the limit $t\rightarrow 0$) has non-trivial consequences: The pure singlet state with one spin removed, ${\hat c}_{i\sigma}|\phi_0\rangle$, is no longer an eigenstate of $\mathcal{H}$. The unpaired spin leads to triplet excitations in its neighborhood (‘screening cloud’), these are spatially confined as long as the spin gap is non-zero. Now we consider the case of non-zero hopping. We start with the ‘symmetric’ choice $t_2 = t_3$, $J_2 = J_3$, which allows to obtain several results analytically. $J_2=J_3$ implies that triplets are strictly localized in the absence of the hole, so any triplet created by hole hopping remains on its bond until the hole returns. $t_2=t_3$ reduces the number of possible inter-bond hopping processes. The only non-zero hopping matrix elements are $|\langle a_a s|{\mathcal H} | s a_a \rangle| = |\langle a_s t_\alpha |{\mathcal H} | t_\alpha a_s \rangle| = |\langle a_s s|{\mathcal H} | t_\alpha a_a \rangle| = t_2$ where $|XY\rangle = X_{i}^\dagger Y_{j}^\dagger |0\rangle$ is a short-hand notation for a state of two neighboring bonds $i$ and $j$. It follows that a hole in the antisymmetric state $a_a$ can freely propagate in a singlet background without emission of triplet fluctuations (direct hopping), i.e., $|\psi_{a,\bf k}\rangle = \sum_i \exp({\rm i} {\bf k R}_i) a_{a,i}^\dag s_i |\phi_0\rangle$ is an exact eigenstate of $\mathcal{H}$. In contrast, a hole being in the symmetric state $a_s$ always creates a triplet (and converts into $a_a$) when it hops to a neighboring singlet bond. This in turns means that any one-hole eigenstate of $\mathcal{H}$ which contains components with $a_s$ also involves triplet excitations. Since the triplets are localized and can only be created/removed by the hole, the state itself is localized, and contributions from such eigenstates to the spectrum are momentum-independent (non-dispersive). =8.7 truecm The calculated spectrum (Fig. \[FIG\_SPEC\_CROSS\]a) is therefore easily understood: It shows two dispersing bands which correspond to hole hopping in the antisymmetric state $a_a$ with effective hopping amplitude $t_2=t_3$ through the square lattice of rungs (two bands arise from the fact that the unit cell contains two rungs). All other contributions are localized and involve symmetric hole states. The corresponding wave functions can be modeled by a particle moving in an attractive potential centered at a single site $i$. A reasonable approximation for the states contributing to the spectrum is $(u a_{s,i}^\dag s_i + \sum_{{\bf R},\alpha} v_{\bf R} a_{a,i+{\bf R}}^\dag s_{i+{\bf R}} t_{i\alpha}^\dag s_i) |\phi_0\rangle$ where $u,v$ are some coefficients (typically $|u|\gg |v_{\bf R}|$ and $v_{\bf R}$ rapidly decaying with distance). Components with more than one triplet have coefficients much smaller than the one-triplet coefficients $v_{\bf R}$. Variation of the model parameters (keeping $t_2 = t_3$, $J_2 = J_3$) only results in small changes in the spectrum of Fig. \[FIG\_SPEC\_CROSS\]a since the dominating bands are determined by $t_2$ only. Note, however, that varying $t_1/t_2$ shifts the dispersing bands w.r.t. to the localized peaks, i.e., it can induce a level crossing at momentum $(0,0)$, so $t_1\gg t_2$ eventually leads to a localized one-hole ground state. Having understood the special situation at $t_2 = t_3$, $J_2 = J_3$, we turn to the general case. We assume $J_3 < J_2$ \[and $t_3 < t_2$ because of (\[HUBBARD\_REL\]) \] since the model behavior is symmetric with respect to the interchange of the $B$ and $C$ bonds. In the following we discuss the $t$ and $J$ parameters separately, but we keep in mind that they are usually connected by the relation (\[HUBBARD\_REL\]). For $J_2 \neq J_3$ the triplets are no longer completely localized, however, as is known from the undoped model, triplet hopping does not occur up to 6th order of perturbation theory in $J_2$, and so the triplet dispersion is weak. Consequently, $J_2 \neq J_3$ introduces a weak dispersion into the ‘background features’ of the spectrum. The effect of $t_2 \neq t_3$ on the spectrum is more pronounced: Hopping of the antisymmetric hole state now also emits triplets, i.e., $|\psi_{a,\bf k}\rangle$ is no longer an exact eigenstate of $\mathcal{H}$. Therefore the well-defined dispersing bands of Fig. \[FIG\_SPEC\_CROSS\]a mix with the background features. Spectral weight is transferred to the bottom of the spectrum which is easily understood from the fact that any triplet excitation increases the energy by at least the gap size. Figs. \[FIG\_SPEC\_CROSS\]a - \[FIG\_SPEC\_CROSS\]d show the evolution of the spectrum when going from $t_3/t_2=1$ to $t_3/t_2=0$; we have fixed $U/t_1=4$ and chosen the remaining parameters such that the size of the spin gap is (approximately) preserved. For $t_3=J_3=0$, Fig. \[FIG\_SPEC\_CROSS\]d, we arrive at a situation with a narrow band at the bottom of the spectrum with dispersion minimum at $(0,0)$; this band corresponds to the motion of a hole surrounded by triplet fluctuations (spin polaron). At higher energies a weak background is visible, it arises from excited polaron states. In addition, we note that smaller relative hopping strengths $t/J$ suppress triplet fluctuations and reduce the high energy background, whereas larger values of $t/J$ lead to an incoherent spectrum since the size of the spin polaron increases. To make contact with possible experiments we examine briefly the properties of the pronounced bands at the lower edge of the spectrum which would be visible in a photoemission spectrum. The experimentally measured bandwidth can be used to determine the ratio $t/J$, or equivalently, the on-site repulsion $U$. For the (experimentally unrealistic) case of $t_2=t_3$, $J_2=J_3$, two bands should be observed as in Fig. \[FIG\_SPEC\_CROSS\]a. The width of each of the bands is given by $4 t_2$ since the dispersion is the one of a free fermion. More likely, the material has $t_3 \ll t_2$, $J_3 \ll J_2$, which corresponds to Fig. \[FIG\_SPEC\_CROSS\]c or d. As a guide we plot in Fig. \[FIG\_BANDW1\] the bandwidth for $t_3=J_3=0$ as function of $J_1/t_1$. In contrary to the well-known square-lattice case (shown for comparison) the bandwidth is finite for $J\rightarrow 0$. The reason is the possibility of direct hopping (without emission of spin excitations); this is not the case in a square lattice antiferromagnetic background where hopping always creates spin defects which have to be removed by exchange processes. For $t/J\rightarrow 0$ the bandwidth saturates (in units of $t_2$) at a non-trivial value which arises from the finite cloud of triplet excitations around a static hole. (Of course, for $J_2=J_3=0$ this effect is absent, and the bandwidth becomes $2 t_2$ in the limit of $t/J\rightarrow 0$.) =7.5 truecm We have also calculated the spin correlations near the mobile hole. Except for the case corresponding to Fig. \[FIG\_SPEC\_CROSS\]a which shows no inter-dimer correlations in the ground state, the hole always introduces antiferromagnetic correlations between different dimers in its vicinity. However, for the possibly relevant values ($J_2/J_1=0.68$, $U/t_1\sim 2-10$) the dimerization is strong and the polaron size can be estimated to be smaller than 3 lattice spacings. We briefly mention that the present analysis can be extended to the Néel-ordered phase of the Shastry-Sutherland model as well. Therefore a condensate of one type of triplets is introduced by a proper transformation of the basis states on each bond [@Vojta99]; the condensation amplitude can be extracted from series expansion results [@weih98]. The ground state of the undoped system is obtained by an expansion around a modified product state (which is the Néel state in the case of dominating $J_2$); here of cause the ground state contains fluctuations around the product state. The spin polaron consists of spin deviations from the undoped background state in the vicinity of the hole. The results show that with increasing antiferromagnetic correlations the band minimum is shifted from $(0,0)$ to $(\pi,\pi)$ (this is equivalent to $(\pi/2,\pi/2)$ in the Brillouin zone of the square lattice defined by the $B$ bonds); the bands approach the shape known from the square-lattice antiferromagnet. Summarizing, in this paper we have studied the one-hole dynamics in a $t-J$ model for the 2d spin gap material [SrCu$_2$(BO$_3$)$_2$]{}. Using a generalized spin-polaron concept together with an expansion around a singlet product state we have calculated the one-hole spectral function. It shows an interesting cross-over from free-fermion behavior to correlated behavior under a variation of the ratio $t_2/t_3$ of 2nd and 3rd-nearest-neighbor coupling. At $t_3=t_2$ the main contributions to the spectrum can be described by the hopping of the bare hole between the singlet rungs with a tight-binding dispersion. In contrast, for $t_3\ll t_2$ one finds a narrow band at the bottom of the spectrum and a background at higher energies; this structure can be attributed to the motion of a hole dressed with spin fluctuations (similar to other strongly correlated systems). We add that very large $t/J$ leads to one-hole ground states with higher spin and ferromagnetic correlations in spirit of the Nagaoka effect, this has not been considered here. The hole dynamics at finite doping and the possibility of hole pairing are interesting subjects of future research. Financial support from the DFG (VO 794/1-1) is gratefully acknowledged. M. Azuma [*et al.*]{}, , 3643 (1994). H. Iwase [*et al.*]{}, J. Phys. Soc. Jpn [**65**]{}, 2397 (1996). A.W. Garret [*et al.*]{}, , 745 (1997). G. Chaboussant [*et al.*]{}, Phys. Rev. B [**55**]{}, 3046 (1997); Phys. Rev. Lett. [**79**]{}, 925 (1997). S. Taniguchi [*et al.*]{}, J. Phys. Soc. Jpn, [**64**]{}, 2758 (1995). H. Kageyama [*et al.*]{}, , 3168 (1999). S. 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--- abstract: 'Solid solution between nonmagnetic narrow gap semiconductor FeSi and diamagnetic semi-metal CoSi gives rise to interesting metallic alloys with long-range helical magnetic ordering, for a wide range of intermediate concentration. We report various interesting magnetic properties of these alloys, including low temperature re-entrant spin-glass like behaviour and a novel inverted magnetic hysteresis loop. Role of Dzyaloshinski-Moriya interaction in the magnetic response of these non-centrosymmetric alloys is discussed.' address: 'Low Temperature Physics Laboratory, Centre for Advanced Technology,Indore 452013, India' author: - 'M. K. Chattopadhyay, S. B. Roy and Sujeet Chaudhary' title: 'Interesting magnetic properties of Fe$_{1-x}$Co$_x$Si alloys' --- The narrow-gap semiconductor FeSi has drawn attention of condensed matter physicists repeatedly since late nineteen thirties [@1]. The revival of strong interest [@2] in FeSi during last decade is mainly due to its similarities with those of narrow gap rare-earth intermetallics popularly known as “Kondo insulators ” [@3]. This comparison gives rise to the possibility of the study of complex many-body phenomena associated with Kondo-lattice systems. Doping with Al in FeSi leads to a heavy fermion metal through a metal-insulator transition with strong similarities with that for Si:P (Ref.4) with the exception of a strongly renormalized effective carrier mass. The Fe$_{1-x}$Co$_x$Si alloys are also remarkable in that they are magnetic for almost all the intermediate concentration regime [@5; @6; @7], while the end compounds FeSi and CoSi are nonmagnetic, the latter being a diamagnetic semi-metal[@8]. The recent discovery of unusual positive magnetoresistance[@7] in these supposedly helimagnetic Fe$_{1-x}$Co$_x$Si alloys[@6] alongwith the suggestion of the interplay of quantum coherence effect at relatively high temperature are quite exciting. The unusual square-root field-temperature dependence of electrical conductivity and the positive nature of the magnetoresistance are correlated to square-root singularity in the density of states probably associated with “enhanced electron-electron interactions” in a disordered ferromagnet with low carrier concentration[@7]. These results suggest a possible new microscopic mechanism of magnetoresistance that could lead to the development of new type of magnetic materials of technological importance[@9]. In the light of these unusual findings, we became motivated for a closer scrutiny of the magnetic properties of Fe$_{1-x}$Co$_x$Si alloys, especially in the low field and low temperature regime. There exist already some hints of unusual low field magnetic properties of Fe$_{1-x}$Co$_x$Si alloys in the form of an almost singular behaviour in magnetization and cusp-like minimum in magnetoresistance near H = 0 (Ref. 7). In this communication we report results of high resolution magnetization measurements in Fe$_{1-x}$Co$_x$Si alloys highlighting (i) low temperature low field re-entrant spin-glass like behaviour (ii) interesting thermomagnetic history effects including a novel “inverted hysteresis loop” with negative remanence. The observation of this latter effect (which was so far considered to be limited to thin-film type of magnetic materials [@10; @11]) in relatively simple alloys like the present (Fe,Co)Si, is definitely interesting. We shall argue that the occurrence of Dzyaloshinski-Moriya interaction in the present non-centrosymmetric cubic B20 Fe$_{1-x}$Co$_x$Si alloys[@6] is playing an important role for the observed magnetic properties. The polycrystalline samples of Fe$_{1-x}$Co$_x$Si; x = 0.1, 0.15, 0.35, 0.45 and 0.65 were prepared by argon arc melting from high purity starting materials. The samples were annealed for 90 hours in vacuum at 900$^0$C for improving the homogeneity. Magnetiztion measurements were performed using a commercial SQUID-magnetometer (Quantum Design, MPMS-5). A scan-length of 4 cm with 32 data points in each scan was used for the measurements. However, all the important results were checked by varying the scan-length from 2 to 8 cm, to rule out any possible role of the small field inhomogeneity of the superconducting magnet (which is actually scan-length dependent) in the observed magnetic properties. Also before the start of each experimental cycle the sample chamber is heated to 200 K and flushed with helium; this is to get rid of any oxygen leaking into the sample chamber over a period of time. In Fig.1(a) we plot magnetization (M) and inverse dc-susceptibility ($\chi ^{-1}$) versus temperature (T) for Fe$_{1-x}$Co$_x$Si with x = 0.15 and 0.35. Estimated Curie temperatures (T$_C$) agree well with those reported in the literature [@7]. In Fig.1(b)-(c) we plot M vs field (H) plots for these alloys at various T both below and above T$_C$. Data also exist for x = 0.1 and 0.45 but not shown here for the sake of clarity and conciseness. The almost singular behaviour in M(H) near H = 0 for T $<$ T$_C$ as reported in Ref. 7 is quite evident in Fig. 1(b)-(c). We shall now concentrate in the low H magnetic response of these alloys. In fig.2 we present M vs T plots for x = 0.35 alloy obtained both in the zero field cooled (ZFC) and field cooled (FC) mode in various applied H. We observe two distinct features for H $\leq$ 500 Oe, namely (1) a peak in M$_{ZFC}$(T) and a sharp change in slope in M$_{FC}$(T) at a temperature T$_P$ ($<$ T$_C$). (2) a distinct thermomagnetic irreversibility (TMI) i.e. M$_{ZFC} \neq$ M$_{FC}$ for T$\leq$ T$_P$. Same qualitative features have also been observed for x = 0.1,0.15 and 0.45. Both these features, which disappear with H $>$ 500Oe, have not been reported so far (to our knowledge) for these (Fe, Co)Si alloys. The low-T low-H magnetic response described above has appreciable resemblance with the re-entrant spin-glasses [@12; @13]. To investigate more in this regard we have studied the H dependence of magnetization in details in two different T-regimes : (1) T $<$ T$_P$, and (2) T$_P < $ T $<$ T$_C$. In Fig.3 we plot M vs H for x = 0.35 alloy at 4.5K highlighting the following striking features: 1. There is a distinct bulge in the virgin M-H curve obtained after zero-field cooling the sample from T $>$ T$_C$. This feature takes the virgin M-H curve in a limited H-regime outside the field descending (ascending) M-H curve obtained after field cycling to 50 kOe (-50 kOe). 2. In the field cycling process if the maximum field of excursion H$_{max}$ goes beyond the technical saturation point H$_{sat} (\approx$ 1 kOe at T = 4.5K), the M-H curve takes the shape of an inverted hysteresis loop, i.e., the descending field leg of the M-H curve lies below that of the ascending field leg with positive coercivity and negative remanence (see the lower inset of Fig.3)[@14]. 3. If H$_{max}$ is limited to H $<<$ H $_{sat}$, M remains perfectly reversible. However, as H$_{max}$ enters the H-regime where the virgin M-H curve starts showing the non-linear behaviour in the form of a bulge, a small but distinct positive hysteresis is observed (see the upper inset of Fig.3). This hysteresis disappears as H approaches H = 0 in the descending field cycle and M merges with the virgin M-H curve. With H$_{max}>$ H$_{sat}$ this positive hysteresis changes sign giving rise to an “inverted hysteresis loop” in the low field regime (H $<$ H$_{sat}$) while the M-H curve remains perfectly reversible (within our experimental resolution) in the high field regime (H $>$ H$_{sat}$). In the T-regime T$_P <$ T $<$ T$_C$ the bulge in the virgin M-H curve and the associated positive hysteresis are not observed. However, inverted hysteresis loop behaviour continues to exist at H $<$ H$_{sat}$) even for T $>$ T$_P$. And as before, the M-H curves remain reversible for H $>$ H$_{sat}$. All these features of the M-H curve are also observed in x = 0.1, 0.15 and 0.45 alloys in the same qualitative manner. The observed peak in M$_{ZFC}$(T) and TMI in M-T plots in Fig.2 with H $\leq$ 500 Oe can naively be interpreted in terms of the hindrance of domains’ motion in a ferromagnetic system[@15]. However, even if the various anomalous aspects of the M-H curves decsribed above are ignored, the estimated coercivity field $|$H$_C|$ of the order of 15 Oe in our x = 0.35 alloy at T = 4.5K rules out such a simple explanation in our measurements with applied H of 500 Oe which is much larger than $|$H$_C|$ . Moreover the distinct change in slope in M$_{FC}$(T) cannot be associated with any domain-related phenomena. These results suggest that there exist probably a re-entarnt spin-glass like magnetic phase[@13] for T $<$ T$_P$ in these alloys. This low T phase appears to be quite fragile and can easily be erased with moderate applied magnetic field. It is interesting to note here that the anomalous bulge in the virgin M-H curve is observed below T$_P$ only, and it is quite clear from the above arguments that it is not associated with any domain related phenomenon either. We suggest that this non-linear beahviour in the virgin M-H curve probably represents a field-induced transition from a low-H magnetic state to a high-H one. The bulge in the virgin M-H curve has been reported earlier for (Fe,Co)Si in passing [@5; @7], and in the absence of a detailed magetization study it was attributed to domain related effects in a ferromagnet [@5]. A similar anomalous behaviour of the virgin M-H curve in CeFe$_2$-based pseudobinary alloys has been associated recently with the first order nature of a field induced metamagnetic transition [@16]. The question now arises how to rationalise the interesting magnetic properties of (Fe,Co)Si within the framework already developed for these alloys. Small angle neutron scattering measurements [@6] have suggested the magnetic ordering in (Fe,Co)Si alloys to be of long period helimagnetic in nature. A model to explain such long period helimagnetic order can be based on a competition between a Dzyaloshinski-Moriya (DM) interaction and a Heisenberg type exchange interaction [@5; @6]. The non-centrosymmetric cubic B20 structure of (Fe,Co)Si alloys supports the existence of DM interaction. Can this competition between these two types of interactions in (Fe,Co)Si alloys give rise to a re-entrant spin-glass like behaviour ? DM interaction apparently plays an important role in metallic spin-glasses and re-entrant spin-glasses [@13]. In this context the occurrence of a re-entrant spin-glass like phase in (Fe,Co)Si alloys is not entirely unexpected, especially with the presence of inherent disorder in the (Fe,Co) sublattice. In fact hints of repartition of the magnetic moments in the helix due to alloying effects exist in early neutron studies [@6]. Satellites due to both clockwise and counterclockwise helixes were observed in neutron measurements in zero field cooled samples . After excursion to a high H, the single clockwise helix was stabilized to the field direction with no satellites observed in any other direction [@6]. On reduction of H to zero the helix does not come back to a specific eqilibrium direction. This is in contrast to the case of isostructural ordered compound MnSi where also the helix follows the field but comes back to $<111>$ direction in low H (Ref.6). It was argued that the disorder in (Fe,Co) sublattice caused a local fluctuations of the co-efficient of D-M interaction to produce two kinds of domains consisting of either a clockwise or counterclockwise helix in the zero field cooled state. The local fluctuation of magnetization might play a role of pinning effect of the magnetic impurity preventing the propagation vector from pointing to the equilibrium direction [@6]. The observed “inverted hysteresis loop”, however, does not find a simple explanation within the above framework. Such “inverted hysteresis loops” have been observed in recent years in specific exchange-coupled multilayers such as Co/Pt/Gd/Pt and epitaxial Fe films on W(001) (Ref. 10 and 11). In such materials their thin film structure apparently play an important role and hence it is considered that “inverted hysteresis loop” is probably a phenomenon limited to thin-film type of magnetic materials. However, there is a very recent report of “inverted hysteresis loop” in a bulk magnetic material comprising of cyanide-bridged multi metal complexes [@17]. The observed “inverted hysteresis loop” in this bulk material is explained “by the competition between the sublattice magnetization rotation due to the spin-flip transition and the trapping effect due to the uniaxial magnetic anisotropy” [@17]. While there exists signature as discussed above of spin-flip transition in the present (Fe,Co)Si alloys and also the suggestion that D-M interaction can cause trapping effect for domains especially if spins are canted within the domains [@13], it is a bit premature to import the similar picture here. More experimental information, especially the microscopic ones like neutron scattering, is required to form even a qualitative model to expain the “inverted hysteresis loop” in the present system. We note in Figs.1 (b)-(c) that while the technical saturation point is reached in the M-H curves below T$_C$ for x = 0.15 and 0.35 alloys at fairly low fields (H$_{sat} \approx$1 kOe), M actually continues to increase beyond H$_{sat}$ even up to the highest field of our measurement i.e. 50 kOe. This two stage magnetization process indicates that after the initial low-H alignment, the local spins, which are probably canted, line up slowly with further increase in H beyond H$_{sat}$. We can actually make a reasonable fit of the M-H curve in the regime H$_{sat} <$ H $<$ 50 kOe to a H$^{1/2}$ behaviour. Similar behaviour has also been observed for x = 0.1 and 0.45 alloys. Manyala et al [@7] have earlier reported that magnetoresistance in some of these alloys also varied as H$^{1/2}$ in the H regime beyond technical saturation. This clearly indicates that the behaviour of these alloys is quite different from a conventional ferromagnet even in the high-H regime . In conclusion, our present dc-magnetization measurements in conjunction with the results of earlier neutron studies [@6], suggest that there exists a low-T low-H magnetic state in (Fe, Co)Si alloys which resembles a lot of the re-entrant spin glasses. With the incease in T and H, it transforms to a presently recognized high-H high-T helical FM state. Careful neutron measurements in various (H,T) regimes with different thermomagnetic history will be useful to settle this issue. 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--- author: - Pietro Silvi title: XXX --- Introduction ============ Quantum many-body problems in condensed matter physics are a context of everlasting interest and relentless investigation in physical research. The macroscopic amount of interacting degrees of freedom is such that even the simplest models become extremely hard problems, in exact description as well as in perturbative regime. The analytical and computational complexity of many-body physics is deeply rooted in the foundations of quantum mechanics themselves: the hardness of these problems undergoes a scaling-law with the number of elementary constituents (size) of the system $L$, which is typically much more abrupt than extensive behavior; it usually grows exponentially with $L$. In physical literature a variety of models which have proven to be particularly suitable for analytical study was developed, e.g. due to some peculiar local structure or to some wide symmetry-group invariance; however, the large majority of known non-perturbative Hamiltonians manifest no attitude towards analytical simplification and must be faced head-on with numerical techniques. Efficient simulation methods for condensed matter settings are many, often capable of integrating any bit of theoretical knowledge, then crossing the remaining gap with computation. At the same time, when no previous hint from theory is available, the exact problem turns hard again, and computational costs scale fast with system specifics, so that only very small sizes $L$ are manageable for practical purposes. The family of *variational* algorithms has always been regarded as one of the most natural and promising paths in order to address many-body problems at zero temperature: as ground states of Hamiltonians are minima of the spectrum, searching them through variational procedures seems most appealing. Yet the crucial point of any variational paradigm is the capability of reducing the whole amount of degrees of freedom into a small number of effective, important ones: these must embed all the relevant physics of the target state, while requiring a limited number of numerical resources. Such primary descriptors, or variational parameters (also known as reaction coordinates in some contexts) need to be identified and discriminated from the non-influential ones. This is clearly a delicate issue, especially if no knowledge on the model is available. In other words, the selection a priori of some appropriate basis of variational wavefunctions is the fundamental step to undertake, determining the faithfulness and efficiency (and thus overall success) of any variational algorithm we might want to develop. In this thesis, we will focus on a very general family of variational wavefunctions, whose main peculiarity is that their descriptors/parameters are tailored according to simple linear algebraic relations. The computational power and success of these tools descends from arguments that were born within quantum information framework: entanglement [@Nielsen]. Quantum entanglement is indeed a resource, but it is also a measure of internal correlations in multipartite systems. Once we characterized general entanglement properties of many-body ground states, then by controlling entanglement of a variational trial wavefunction we can exclusively address physical states, and disregard non-physical states, even before the simulation takes place. This is the central concept which Tensor Network architectures are based upon. Historically, the realization and profound understanding of this class of states, was possible only after the introduction of Density Matrix Renormalization Group (DMRG), curiously, an algorithm which is not formulated in variational terms at all. The age of Density Matrix\ Renormalization Group -------------------------- The idea of adapting a Renormalization Group algorithm to a lattice Density Matrix was proposed by Steven R. White, considered undoubtedly one of the founders of the DMRG methods. In his first approaches [@White92] to the technique, he was inspired by a paper of K.W. Wilson [@Wilson75] where a numerical renormalization group paradigm is applied to the Hamiltonian of a Kondo problem. The simple, yet brilliant, idea behind White’s formulation of DMRG was to replace the traditional procedure of renormalization group, which acted extensively in the real space, and thus actually performing a coarse-graining transformation upon the system, with a scheme that applied extensively in the Hilbert space dimension itself, leading to a site-by-site renormalization scheme. In practice, assume that we are describing the state of a given portion of the system (in terms of a density matrix) with a fixed amount of computational resources $D$. Now when another single constituent (a site) is added to the subsystem, the resulting dimension grows linearly with the local dimension $d$ of the new component, $\sim d D$. Then renormalization is performed, allowing us to represent the new subsystem with the same initial amount of resources $D$, cutting the least relevant density matrix-eigenstates out of the description. Obviously, such operations still manifest a group structure, and they are summoned every time the density matrix dimension (and not the real-space size) increases of a given factor $d$: thus DMRG. The great amount of credit and interest gathered by DMRG is surely due to its outstanding successfulness for low-dimensionality quantum systems. In particular, for one-dimensional (open-boundary) systems, DMRG achieved variational precisions (compared to experiment, and theory whenever possible) that challenged other simulation approaches. At the end of the ’90s, it was considered probably the most powerful numerical method to address 1D problems, with practically no knowledge on the model required a priori. In literature, DMRG picture has been exploited in several settings of both physics and quantum chemistry, and numerous variants of its original formulation were proposed [@WhiteOneSite; @SchollDMRG]; in the end the basic idea was proven to be winning, even though within its dimensionality limits. The advent of Matrix Product States ----------------------------------- The DMRG concept was quite renown, but it was in 2004 that the in-depth reason of its success was fully understood: when F. Verstraete, D. Porras and J. I. Cirac started to investigate quantum states built via a DMRG algorithm under a quantum information perspective [@PrimoMPS]. In fact, they realized that DMRG states had a strict equivalence relation with finitely correlated states, i.e. lattice states whose entanglement is upper-bounded by an arbitrary finite value, which does not scale with the system size. Moreover, ground states of short-ranged (non-critical) Hamiltonians, have been known for quite some time to satisfy the so-called *area-law* of entanglement [@Arealaws; @Arealaws2; @Eisarea]. This general rule was developed in quantum information contexts, but carries important physical prescriptions. It claims that the partition entanglement of a non-critical ground state does *not* scale as the volume of the parted spatial region $\sim L^{\#N}$ (which is the typical entanglement scaling for random states), but rather with the parting surface $\sim L^{\#N-1}$, where $\#N$ is the number of spatial dimensions. It is clear that for 1D systems, the correct area law is given by a non-scaling constant $\sim L^{0}$: indeed entanglement of 1D non-critical ground states typically saturates to a finite bound. This means, in turn, that ground states of 1D non-critical Hamiltonians are finitely-correlated states, and therefore, that DMRG procedure can approximate them with arbitrary precision, and their entanglement as well. Moreover, ref. [@PrimoMPS] has shown that finitely-correlated states allow a direct, simple, immediate algebraic representation in terms of a product of matrices, each of these matrices storing all the information related to a single renormalization process. This provides a one-to-one local correspondence between DMRG, and these Matrix Product State (MPS) [@MPSreview; @MPSVidal; @MPScontinuous; @TuttoMPS] which, as they allow an explicit analytic expression, actually form a class of tailored variational wavefunctions. Although not-homogeneously perceived by the condensed-matter physics community, this discovery was definitely a breakthrough, for several reasons. First, a variational formulation of DMRG opened the possibility for new numerical strategies based on finitely-correlated states, so that several minimum-search algorithms could be applied, but still using just the right amount of necessary computational resources. Secondly, the algebraic MPS expression provided faster ways to access physical information, and, at the same time, it allowed innovative problem-solving options even from the analytical point of view [@MPSground]. Finally, it is easy to generalize the MPS concept to suit physical settings other than 1D, and still taking care of appropriate area-laws. This argument lead, for instance, to the design of Product of Entangled Pairs States (PEPS) [@PEPSgo; @PowaPEPS]. Entanglement Renormalization ---------------------------- The understanding of the relationship between DMRG, MPS, and finitely-correlated states provided an unquestionable paradigm for dealing numerically with non-critical 1D system in a fully-contextualized theoretical framework. Despite the fair success of adapting these algorithms to critical problems as well (although in a hand-waving and unnatural way), it was guessed that due to the presence of scale-invariance symmetry, an old-fashioned real-space renormalization group would be more appropriate to simulate strongly correlated systems. Indeed, a state which is locally stationary under the action of a coarse-graining transformation would definitely be scale invariant. At the same time Wilson’s numerical RG had the unpractical feature of suffering loss of short-range detailed structure, identified by translational instability of entanglement. An intriguing proposal to work around this trouble was introduced by G. Vidal in 2007, who fist spoke about Entanglement Renormalization [@MERAzero]. The idea is intuitive, yet very effective: we are still performing a real-space renormalization group, but prior to the renormalization process itself, we apply a quasi-local unitary transformation, whose purpose is to decrease the correlation among regions who are to be renormalized separately. Since the goal of these unitary gates is to absorb, and thus store, entanglement out of the pre-RG state, they are commonly called *disentanglers*. The disentangling operation is then scheduled before every real-space RG operation takes place; in the end, such entanglement renormalization acts at every lenghtscale, since every RG step performs actually a scale transformation. Similarly to DMRG and MPS, these entanglement-RG states have a variational counterpart as well. Precisely, it is possible to define a class of tailored variational wavefunctions, whose descriptors are tied together by linear relations, reproducing exactly the entire set of those states. Such states are thus called *Multiscale Entanglement Renormalization Ansatz* (MERA) [@MERAalgo; @MERAevo; @MERAgauge; @MERAtopo], and manifest a natural attitude towards describing strong correlation and criticality [@QuMERAchan]. MPS, MERA, PEPS, are different classes of variational states sharing some important attributes: they are able to capture interesting physics, yet they require a small, manageable number of parameters, with simple algebraic rules and direct access to relevant physical information. Physicists started to regard them as belonging together to a larger, comprehensive family of tailored variational states, whose entanglement can be directly controlled through the selection of a related graph geometry. This is the concept of Tensor Network states [@TNevenbly; @TNfujii; @TNdenny; @TNchan]. Outline ------- The thesis is organized as follows: - In chapter \[chap:OBCMPS\] we will present an in-depth review on Matrix Product States, in 1D open-boundary conditions settings. We will show how the MPS analytical expression is derived by the DMRG algorithm, show its entanglement bounds and sketch their algebraic manipulation features. We will explain how to achieve physical information on these states in a computationally-fast scheme, and present some protocols to simulate ground states. - In chapter \[chap:PBCMPS\] we will generalize the concept of MPS to periodic boundary conditions systems, and discuss how translational homogeneity of the representation allows us to well-define the thermodynamical limit for MPS. This will be the proper setting to show that MPS manifest natural non-criticality, whose signature is an exponential decay of two-point correlations. We will then present possible generalizations to the PBC case of MPS minimization algorithms, focusing on some tricks of the trade useful to speed-up and stabilize the procedure. - In chapter \[chap:TN\] we will explain how the MPS architecture can be generalized to more complex Tensor Network geometries. We will investigate the entanglement properties of Tensor Network states, and some of their common algebraic features, like efficient contraction schemes or the adaptability to fermionic contexts. We will also present some remarkable subclasses of Tensor Networks, like PEPS and CPS, and discuss on how they relate to each other. - In chapter \[chap:TTNMERA\] we will put our attention on Trees and MERA, two classes of Tensor Networks that share most of their main features. We will show their natural attitude to describe critical states in 1D, identified both by a logarithmic violation of the area law, and more importantly by manifesting power-law decaying correlations. Critical exponents, as well as the TTN/MERA state properties in the thermodynamical limit, are completely characterizable by adopting a completely positive trace-preserving map formalism. We will investigate other general properties of such TN-architectures, e.g. the possibility to construct a parent Hamiltonian. ### Original content Here I will list the original contribution I developed personally, either as brand-new material or as a reinterpretation of previous knowledge, during my Philosophiae Doctorateship. - Section \[sec:Slater\]: analytical MPS representation of Slater Determinants, many-body basis change, and configuration interaction states. - Section \[sec:MPJastrow\]: matrix product representation for correlator product states (Jastrow factors). - Most analytical results of chapter \[chap:TTNMERA\]: scaling properties for TTN, parent Hamiltonians, fluctuations, boundaries, hybrid geometries; and respective generalizations to MERA. Part of this research was published in [@IoTree; @BERA; @MatteoMera]. Matrix Product States {#chap:OBCMPS} ===================== It was in 2004 that computational physicists started to consider Density Matrix Renormalization Group according to a Quantum Information perspective [@PrimoMPS]; they realized that it is possible to understand DMRG in a variational sense, in which the role played by entanglement and quantum correlation is clear. Indeed, as a quantum many-body state achieved by DMRG procedure is uniquely defined by the renormalization transformations (intended as endomorphisms upon the density matrices space) one can regard such transformations as variational elements, and every single choice of those elements defines a state within a set of tailored variational wavefunctions. Nicely enough, it was discovered that such states allow a simple and immediate analytical description, where their many-body wavefunction, wrote upon a product basis of one-body levels, appears just as a product of variational matrices, thus leading to the name of Matrix Product States (MPS). As a matter of fact, such transparent description allowed research to further investigate the properties of these states, leading to a deeper understanding of DMRG as well, and in the end the new knowledge served well the purpose of gaining more computational power in simulations, through a wider range of algebraic manipulations and the adaptability of variational-based algorithms. In the end, it was the Matrix Product State picture that helped to understand the deep physical reason of DMRG successfulness in 1D. Indeed, MPS have proven to be in tight relation with 1D finitely correlated states [@FannesNacht; @AKLT], and in turn this set is known to include ground states of short-range interacting non-critical Hamiltonians. Such argument holds not only for finite, isolated systems, but extends naturally to open and/or thermodynamical limit systems (as long as a zero temperature can be defined), allowing DMRG/MPS to succeed even in these cases. It is important to point out that Matrix Product State methods can be successfully adopted for dealing with fermionic systems, and they naturally avoid the sign problem which, in stead, is a major issue on other variational fermionic algorithms like Montecarlo. Finally, MPS are not merely a numerical tool, they have proven in various contexts to be fundamental to address analytically several condensed matter models [@Karimipour; @TuttoMPS]. Matrix Product State construction\ from DMRG {#sec:dmrgtomps} ---------------------------------- Following the formalism of [@White92; @MPSreview], we start with a one-dimensional lattice, $L$ (length) being the total number of sites, $d$ (local Hilbert dimension) the number of levels per site, and where open boundary conditions (OBC) are chosen for simplicity. Let us assume that we are describing a given quantum state of this system obtained via a DMRG algorithm: $D \ll d^L$ is the maximal number of states allowed for the each renormalization. Now let $\ell < L$ be the site where the last density matrix renormalization was applied in the algorithm (while moving, say, to the right), this means that we know the reduced density matrix of the state $\rho_{\ell}^{L}$, involving all the sites from $\ell$ to the leftmost, but we only have access to its renormalized form: namely in stead of keeping in memory all its $d^{\ell}$ eigenvectors $|L_j\rangle_{\ell}^{L}$ and their relative probability $p_j$ ($\sum_j^{d^{\ell}} p_j = 1$, in decreasing order $p_j \geq p_{j+1}$), only the $D < d^{\ell}$ of such vectors are kept, of course those with highest probability: $$\label{eq:Piribango} \tilde{\rho}_{\ell}^{L} = \sum_{j=1}^{D} \tilde{p}_j \;|L_j\rangle_{\ell}^{L} \langle L_j|, \quad \mbox{where} \quad \tilde{p}_j = \frac{p_j}{\sum_{k = 1}^D p_k}$$ ensures that the new statistic $\tilde{p}_j$ is properly renormalized. The $D$ vectors $|L_j\rangle_{\ell}^{L}$ appearing in are orthogonal by construction, and are normalized on their space of definition (the left part of the system, i.e. sites to the left of $\ell$); these shall be the only relevant vectors in the left part of the system which will contribute to the full analytical expression of the global state. Now, the trick of the trade, is considering that $\tilde{\rho}_{\ell}^{L}$ was obtained from the reduced density matrix of the previous DMRG step $\tilde{\rho}_{\ell-1}^{L} =\sum_{j}^{D} \tilde{q}_j \;|L_j\rangle_{\ell-1}^{L} \langle L_j|$, which of course was renormalized $\sum_{j}^{D} \tilde{q}_j = 1$. This means that, the $D$ states $|L_j\rangle_{\ell-1}^{L}$ joint together with the local levels $|s\rangle$ at site $\ell$, are enough to generate the set of $|L_j\rangle_{\ell}^{L}$: $$\label{eq:wata} |L_j\rangle_{\ell}^{L} = \sum_{k = 1}^D \sum_{s = 1}^d A_{k,j}^{[\ell] s} \; |L_k\rangle_{\ell-1}^{L} \otimes |s\rangle_{\ell}.$$ Here $A_{k,j}^{[\ell] s}$ represents the decomposition over the product basis; it can be either understood as a three-indices tensor (indices being $s$, $k$, and $j$), or, since typically $D \gg d$, a $d$-long array (through $s$) of square $D \times D$ matrices (from $j$ to $k$). Preservation of orthonormality among $|L_j\rangle^L$ states determines a condition upon $A$; indeed, assuming that the local basis $|s\rangle$ is orthonormal by definition, one finds that the transformation must satisfy the equation $$\label{eq:phoneL1} \sum_{k = 1}^D \sum_{s = 1}^d {A^{\star}}_{ki}^{[\ell] s} \; A_{kj}^{[\ell] s} = \delta_{i,j},$$ where the superscript $\vphantom{A}^{\star}$ stands for complex conjugation. Eq.  can be rewritten in an even clearer form once the $A$ are intended as $D$ by $D$ matrices: $$\label{eq:phoneL2} \sum_{s = 1}^d {A^{\dagger}}_{s}^{[\ell]} \cdot A_{s}^{[\ell]} = {{\mathbb 1}},$$ with $\cdot$ being the standard rows-by-columns matrix product. Equation follows directly from the fact that $$\delta_{i,j} = \langle L_i|L_j\rangle_{\ell}^{L} = \sum_{k, m}^D \sum_{s, t}^d {A^{\star}}_{ik}^{[\ell] s} \; A_{mj}^{[\ell] t} \langle L_k|L_m\rangle_{\ell-1}^{L} \langle s|t \rangle_{\ell}.$$ but $\langle s|t \rangle_{\ell} = \delta_{s,t}$ by assumption, and $\langle L_k|L_m\rangle_{\ell-1}^{L} = \delta_{k,m}$ is the inductive hypothesis, thus . Moreover, looking at under a quantum information perspective, we clearly understand that the $A^{[\ell]}_{s}$ actually form a set of Kraus operators for a completely positive trace preserving (CPT) map [@Nielsen]; CPT maps are the most generic transformations mapping density matrices into density matrices, they represent the action of a quantum channel on an open system (for details, see appendix \[app:cptchap\]). On our case, the set of $A_{s}^{[\ell]}$ define exactly the CPT map $\mathcal{M}_{\text{CPT}}$ performing the inverse DMRG transformation $\tilde{\rho}_{\ell}^{L} \longrightarrow \tilde{\rho}_{\ell-1}^{L}$ as follows $$\label{eq:mappuzL} \tilde{\rho}_{\ell-1}^{L} = \mathcal{M}_{\text{CPT}}\left[ \tilde{\rho}_{\ell}^{L} \right] \equiv \sum_{s = 1}^d A_{s}^{[\ell]} \cdot \tilde{\rho}_{\ell}^{L} \cdot {A^{\dagger}}_{s}^{[\ell]}.$$ Let us now go back at ; as DMRG procedure is recursive, one can apply the same argument several times, for instance, until reaching the first site. This leads to $$\label{eq:MPopen} |L_j\rangle_{\ell}^{L} = \sum_{s_1 \ldots s_{\ell} = 1}^d \left( A_{s_1}^{[1]} \cdot A_{s_2}^{[2]} \cdot \ldots \cdot A_{s_{\ell}}^{[\ell]} \right) |s_1\rangle_{1} \otimes |s_2\rangle_{2} \otimes \ldots \otimes |s_{\ell}\rangle_{\ell}$$ the component of $|L_j\rangle_{\ell}^{L}$ over the local homogeneous product basis made of $|s_1 \ldots s_{\ell}\rangle$ is now expressed in terms of a product of matrices. These matrices have a number of rows and columns always bounded by $D$, although in general it is impossible to require for all of them to be $D \times D$ square matrices and satisfy at the same time; typically, as the left boundary grows near, their size shrink, up to the first site, whose $A^{[1]}_{s_1}$ are all one-row matrices. Similarly, $A^{[L]}_{s_L}$ are one-column matrices. In fact, we can associate a *correlation space dimension* $D_{\ell'}$ to every site to the left of $\ell$ (more appropriately: to every *bond* $\ell'$),and state that the matrices $A^{[\ell']}_{s_{\ell'}}$ have common size $D_{\ell'} \times D_{\ell'+1}$. In order for to hold, the following inequality is a necessary condition: $$\label{eq:ConsistenL} D_{\ell'} \leq d \cdot D_{\ell'-1}$$ for $\ell' \leq \ell$, where it is intended that $D_0 = 1$. As provides an upper bound to the correlation dimension, so does the parametric renormalization dimension $D$, typically acting as a simple cutoff. Indeed, in standard DMRG algorithms it is a natural choice to adopt $D_{\ell'} = \min\{d^{\ell'}, D\}$. ### Completing the picture:\ single center site DMRG So far we understood how to represent in a clear, simple analytical way of representing the left block states $|L_j\rangle_{\ell}^{L}$ of our DMRG. In order to complete the picture to include the whole system we first need to identify which specific architecture of DMRG (of those proposed in literature) is being used. For simplicity we start with the case where the DMRG optimization is performed by considering at every step a single center site (system) and the left and right blocks (environment), as in ref. [@WhiteOneSite]. According to such description, the $D$ left environment renormalized states $|L_j\rangle_{\ell-1}^{L}$, joint with the right environment renormalized states $|R_j\rangle_{\ell}^{R}$, and the site levels $|s\rangle_\ell$, generate the DMRG state of the whole system: $$\label{eq:DMfull} |\Psi_{\text{DMRG}}\rangle = \sum_{j,k = 1}^{D} \sum_{s = 1}^{d} C^{[\ell] s}_{j,k} \; |L_j\rangle_{\ell-1}^{L} \otimes |s\rangle \otimes |R_k\rangle_{\ell}^{R},$$ the components tensor $C^{[\ell] s}_{j,k}$ defines uniquely the precise state within the DMRG space. Now, the very argument we used previously to prove that for $|L_j\rangle_{\ell-1}^{L}$ eq.  holds, can be applied in a similar fashion to right environment vectors as well. Precisely, if $\tilde{\rho}_{\ell}^{R} =\sum_{j}^{D} \tilde{p}_j \;|R_j\rangle_{\ell}^{R} \langle R_j|$ is a density matrix obtained by recursive renormalizations starting from the right boundary (site $L$), we have that $$\label{eq:MPopen2} |R_j\rangle_{\ell}^{R} = \sum_{s_{\ell+1} \ldots s_{L} = 1}^d \left( B_{s_{\ell+1}}^{[\ell+1]} \cdot \ldots \cdot B_{s_{L-1}}^{[L-1]} \cdot B_{s_{L}}^{[L]} \right) |s_{\ell+1}\rangle_{\ell+1} \otimes \ldots \otimes |s_{L}\rangle_{L}.$$ Like before, we encounter a product of matrices $B_{s_{\ell'}}^{[\ell']}$ ($\ell' > \ell$); but notice that this time the preservation of orthonormality property for the $|R_j\rangle_{\ell}^{R}$ states goes from the right boundary towards the center site, i.e. propagating toward the left. This means that the matrices $B_{s_{\ell'}}^{[\ell']}$ should satisfy a relation which is different from , namely: $$\label{eq:phoneR2} \sum_{s = 1}^d {B}_{s}^{[\ell']} \cdot {B^{\dagger}}_{s}^{[\ell']} = {{\mathbb 1}}.$$ To satisfy the present equation the following constraint on matrices dimensions (${B}_{s}^{[\ell']}$ being a $D_{\ell'-1} \times D_{\ell'}$ complex matrix) is due: $$D_{\ell'} \leq d \cdot D_{\ell'+1}, \quad \forall \ell'>\ell; \quad D_L = 1.$$ Consistently, the CPT mapping associated to ${B}_{s}^{[\ell']}$ performs the inverse RG transformation, i.e. towards the right $$\label{eq:mappuzR} \tilde{\rho}_{\ell'+1}^{R} = \mathcal{M}_{\ell'}^{R} \left[ \tilde{\rho}_{\ell'}^{R} \right] \equiv \sum_{s = 1}^d {B^{\dagger}}_{s}^{[\ell']} \cdot \tilde{\rho}_{\ell'}^{R} \cdot {B}_{s}^{[\ell']}.$$ After all these considerations, we can put and into , the state $|\Psi_{\text{DMRG}}\rangle$ appears automatically expanded in the natural separable basis: $$\label{eq:PrehistoMPS} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell-1}}^{[\ell-1]} \cdot C_{s_{\ell}}^{[\ell]} \cdot B_{s_{\ell+1}}^{[\ell+1]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where $C_{s_{\ell}}^{[\ell]}$ has been written as an array of matrices as well. Notice that the term within the parentheses is a scalar due to the fact that the $A^{[1]}_{s_1}$ matrices are actually row vectors (one-row matrix) and the $B_{s_{L}}^{[L]}$ are column vectors (one-column matrix). Equation tells us that all the components of $|\Psi\rangle$ over the canonical basis are the product of $L$ matrices, which are local objects and depend only on the state of the site they are associated with. This is the definition of Matrix Product State [@PrimoMPS]. Once we require that matrices $A_{s}^{[\ell']}$ and $B_{s}^{[\ell']}$ respectively satisfy and in order to preserve orthonormality of environment states, the proper normalization of the state $|\Psi\rangle$ becomes an equation involving the element $C_{s_{\ell}}^{[\ell]}$ alone. Indeed we can explicitly calculate the norm of $|\Psi\rangle$ by exploiting its MPS representation , as $$\begin{gathered} \langle \Psi | \Psi \rangle = \sum_{s_1 \ldots s_L = 1}^{d} \left( A_{s_1}^{[1]} \otimes {A^{\star}}_{s_1}^{[1]} \right) \ldots \left( A_{s_{\ell-1}}^{[\ell-1]} \otimes {A^{\star}}_{s_{\ell-1}}^{[\ell-1]} \right) \times \\ \times \left( C_{s_{\ell}}^{[\ell]} \otimes {C^{\star}}_{s_{\ell}}^{[\ell]} \right) \left( B_{s_{\ell+1}}^{[\ell+1]} \otimes {B^{\star}}_{s_{\ell+1}}^{[\ell+1]} \right) \ldots \left( B_{s_L}^{[L]} \otimes {B^{\star}}_{s_L}^{[L]} \right),\end{gathered}$$ which, after some algebraic manipulation, reads $$\begin{gathered} \label{eq:MPSmanip} \langle \Psi | \Psi \rangle = \sum_{s_1 \ldots s_L = 1}^{d} {\text{Tr}}\left[ A_{s_1}^{[1]} A_{s_2}^{[2]} \ldots A_{s_{\ell-1}}^{[\ell-1]} C_{s_{\ell}}^{[\ell]} B_{s_{\ell+1}}^{[\ell+1]} \ldots \right. \\ \left. \ldots B_{s_{L-1}}^{[L-1]} B_{s_L}^{[L]} {B^{\dagger}}_{s_L}^{[L]} {B^{\dagger}}_{s_L-1}^{[L-1]} \ldots {B^{\dagger}}_{s_{\ell+1}}^{[\ell+1]} {C^{\dagger}}_{s_{\ell}}^{[\ell]} A_{s_{\ell-1}}^{[\ell-1]} \ldots {A^{\dagger}}_{s_2}^{[2]} {A^{\dagger}}_{s_1}^{[1]} \right].\end{gathered}$$ And, by exploiting , and cyclicity of the trace, all the $A_{s}^{[\ell']}$ and $B_{s}^{[\ell']}$ matrices disappear from the equation. In the end, we are left with $$1 = \langle \Psi | \Psi \rangle = \sum_{s_{\ell} = 1}^{d} {\text{Tr}}\left[ C_{s_{\ell}}^{[\ell]} {C^{\dagger}}_{s_{\ell}}^{[\ell]} \right] = \sum_{j,k = 1}^{D} \sum_{s = 1}^{d} C_{jk}^{[\ell]s} {C^{\star}}_{jk}^{[\ell]s},$$ the desired normalization condition. We would like to remark that manipulations performed in order to derive are identically suitable if we were to calculate the one-site reduced density matrix $\rho^{1}$ of $|\Psi\rangle$ at site $\ell$, namely $$\rho^{1}_{\ell} = {\text{Tr}}_{\ell^\text{c}} \left[ | \Psi \rangle| \langle \Psi | \right] = \sum_{s, t = 1}^{d} {\text{Tr}}\left[ C_{s}^{[\ell]} {C^{\dagger}}_{t}^{[\ell]} \right] |s\rangle \langle t|,$$ where, clearly, the partial trace spans $\ell^{\text{c}}$, the complementary of $\ell$. Similarly, the reduced density matrices $\tilde{\rho}_{\ell}^{L}$ and $\tilde{\rho}_{\ell}^{R}$ are easily accessible one the representation is at our disposal. In fact, those read $$\tilde{\rho}_{\ell}^{L} = \sum_{j,k,m = 1}^{D} \sum_{s = 1}^{d} C_{jm}^{[\ell]s} {C^{\star}}_{km}^{[\ell]s} |j\rangle\langle k| , \qquad \tilde{\rho}_{\ell}^{R} = \sum_{j,k,m = 1}^{D} \sum_{s = 1}^{d} C_{mj}^{[\ell]s} {C^{\star}}_{mk}^{[\ell]s} |j\rangle\langle k|,$$ and all the other reduced density matrices achieved through the original DMRG algorithm can be generated starting from the previous expressions via and . In practice, the MPS representation provides us a quick access to the whole information of the DMRG algorithm, and at the same time is more immediate and flexible than DMRG itself, proving a useful computational tool as we will see later on. ### Double center site DMRG The original DMRG protocol proposed by White [@White92], and most of the DMRG architectures still in use nowadays adopt a slightly different picture than the one we presented in . The basic idea is to consider the active system block on which to perform the minimization as it were composed by two adjacent sites in stead of just one, coupling to the left and right environments as before. Of course, at fixed renormalization dimension parameter $D$, this procedure is more expensive from a computational point of view, but provides a big gain in algorithm precision and rapid convergence; moreover, it allows to manipulate symmetries in a more natural and flexible fashion, thus improving algorithm stability. $$\label{eq:DM2full} |\Psi_{\text{DMRG}}\rangle = \sum_{j,k = 1}^{D} \sum_{s,t = 1}^{d} T^{s,t}_{j,k} \; |L_j\rangle_{\ell-1}^{L} \otimes |s\rangle_{\ell} \otimes |t\rangle_{\ell+1} \otimes |R_k\rangle_{\ell+1}^{R},$$ In order to recover a complete analytical expression of the form , some manipulation on the components tensor $T^{s,t}_{j,k}$ has to be made. The simplest path to take, is to consider two composite indexes $\alpha$ and $\beta$: $\alpha$ representing the pair $\{j,s\}$, while $\beta$ representing $\{k,t\}$. This allows us to write $T_{\alpha \beta}$ as a matrix from index $\alpha$ to $\beta$, of dimension $dD \times dD$. At this point we perform a Singular Value Decomposition (SVD) upon $T$: $$\label{eq:SVDone} T_{\alpha \beta} = A^{[\ell]}_{\alpha \gamma} \;\lambda_{\gamma}\; B^{[\ell+1]}_{\gamma \beta},$$ where $A^{[\ell]}$ and $B^{[\ell+1]}$ are unitary matrices, and the diagonal matrix of singular values is positive semidefinite, i.e. $\lambda_{\gamma} \geq 0$ $\forall \gamma$. If we write again $A^{[\ell]}$ and $B^{[\ell+1]}$ in the original $j$, $s$ and $k$, $t$ indices it is clear that they satisfy the proper orthonormalization propagation requirements, respectively and . This tells us that can also be interpreted as follows $$\label{eq:Shmidtdeco} |\Psi_{\text{DMRG}}\rangle = \sum_{\gamma = 1}^{dD} \lambda_{\gamma} \; |L_{\gamma}\rangle_{\ell}^{L} \otimes |R_{\gamma} \rangle_{\ell}^{R},$$ where, following the formalism of we substituted $$\begin{aligned} |L_{\gamma} \rangle_{\ell}^{L} &= \sum_{j = 1}^D \sum_{s = 1}^d A_{j,{\gamma}}^{[\ell] s} \; |L_j\rangle_{\ell-1}^{L} \otimes |s\rangle_{\ell} \qquad \mbox{and} \\ |R_{\gamma} \rangle_{\ell}^{R} &= \sum_{k = 1}^D \sum_{t = 1}^d B_{{\gamma},k}^{[\ell+1] t} \; |R_k\rangle_{\ell+1}^{R} \otimes |t\rangle_{\ell+1}. \end{aligned}$$ Indeed, as $|L_{\gamma}\rangle_{\ell}^{L}$ (resp. $|R_{\gamma} \rangle_{\ell}^{R}$) form a set of orthonormal vectors for the left (right) partition of the system, equation actually represents the Schmidt decomposition of $|\Psi_{\text{DMRG}}\rangle$, cut at site $\ell$. The Schmidt coefficients $\lambda_{\gamma}$ must satisfy the normalization condition $1 = \langle \Psi | \Psi \rangle = \sum_{\gamma} \lambda_{\gamma}^2$; they are the positive square roots of the probabilities $\tilde{p}$ in the ($dD$-renormalized) reduced density matrices of either partition of the system. The latter read $$\tilde{\rho}_{\ell}^{L} = \sum_{\gamma = 1}^{dD} \lambda_{\gamma}^{2} |L_{\gamma} \rangle_{\ell}^{L} \langle L_{\gamma}| , \qquad \tilde{\rho}_{\ell}^{R} = \sum_{\gamma = 1}^{dD} \lambda_{\gamma}^{2} |R_{\gamma} \rangle_{\ell}^{R} \langle R_{\gamma}|.$$ In conclusion, the SVD decomposition allows us to recover a matrix product expression, substantially identical to . Precisely $$\label{eq:SchmidtMPS} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell}}^{[\ell]} \cdot \lambda^{[\ell]} \cdot B_{s_{\ell+1}}^{[\ell+1]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where $\lambda^{[\ell]}$ is intended as the diagonal matrix with elements $\lambda^{[\ell]}_{\gamma}$. Normally, in order to press further on with DMRG algorithm, the left (or right) density matrix should be properly renormalized to be $D \times D$ dimensioned; but this is straightforward, by just cutting off the smallest singular values $\lambda_{\gamma}$ until only the $D$ largest of them remain, and renormalize as follows $${\tilde{\lambda}}_{\gamma} = \frac{\lambda_{\gamma}}{\sqrt{\sum_{\eta}^{D} \lambda^2_{\eta}}} \quad \gamma \in \{1..D \} \qquad \longrightarrow \qquad \sum_{\gamma}^{D} {\tilde{\lambda}}_{\gamma}^2 = 1,$$ so that state normalization $\langle \Psi | \Psi \rangle = 1$ is preserved. Even after this cutoff, $A_{s_{\ell}}^{[\ell]}$ will still satisfy the condition : this descends automatically from the fact that if any number of columns are cut out of a unitary matrix, a left-isometric rectangular matrix ($A^{\dagger} A = {{\mathbb 1}}$, but $A A^{\dagger} = P = P^2 = P^{\dagger} \neq {{\mathbb 1}}$) is obtained, thus . Similarly, the $B_{s_{\ell+1}}^{[\ell+1]}$ resulting from the cutoff will still satisfy . It is trivial to make formally match , we can either identify $C_{s}^{[\ell]} = A_{s}^{[\ell]} \lambda^{[\ell]}$, or alternatively $C_{s}^{[\ell+1]} = \lambda^{[\ell]} B_{s}^{[\ell+1]}$ and recover the previous formalism. Similarly, we can manipulate to appear in the latter form, such operation will be clearer once we introduced a state-invariant transformation (gauge) of the MPS representation, which we are going to review in section \[sec:MPSGauge\]. Valence bond picture and\ MPS entanglement {#sec:valencebond} ------------------------- Interestingly enough, it is possible to interpret Matrix Product States in a way [@PrimoMPS; @MPSreview] that clarifies many of their quantum correlation properties, often referred to as *valence bond picture*. The basic idea is to start by considering an auxiliary system space that is actually larger than the proper Hilbert $\mathcal{H} = \mathbb{C}^{d^L}$ of our 1D (open boundary, so far) system; then we project onto the original system by means of a local transformation. Let us associate to any site of our quantum chain a *pair* of $D$-dimensional spins, one per bond formed by that site. the starting state is prepared so that every pair of virtual spins corresponding to the same bond, is initially in a maximally entangled state $|\Phi^{+}\rangle = D^{-\frac{1}{2}}\sum_{\alpha}^{D} | \alpha \alpha \rangle$, known in literature as *entangled bond*. Then apply a local on-site map $$\label{eq:VBmap} \mathcal{A}^{[\ell]} = \sum_{s = 1}^{d} \sum_{j,k = 1}^{D} A^{[\ell]s}_{j,k} |s\rangle_{\ell} \,\langle j,k |^{\text{aux}}_{\ell}$$ to every site $\ell \in \{1 .. L\}$, where $|s\rangle_{\ell}$ is a canonical state in the local physical space at site $\ell$ while $| j,k \rangle^{\text{aux}}_{\ell}$ is a vector of the (double spin) respective auxiliary space. Equation applied on the initial valence bond state $\bigotimes_{\ell} \mathcal{A}^{[\ell]} (\bigotimes_{\ell'} |\Phi^{+}\rangle^{\text{aux}}_{\ell', \ell'+1})$ leads to an expression where auxiliary indexes of neighboring $A^{[\ell]s}_{j,k}$ are contracted. Then, by writing any tensor $A^{[\ell]s}_{j,k}$ as a set of $d$ complex $D \times D$ matrices, the state we are describing is naturally expressed in the matrix product form $$\label{eq:NaturMPS} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot A_{s_{2}}^{[2]} \cdot \ldots \cdot A_{s_{L-1}}^{[L-1]} \cdot A_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle.$$ In general, not only the $\mathcal{A}^{[\ell]}$ operations, but even the auxiliary dimension $D$ of the entangled pair $|\Phi^{+}\rangle$ can be site dependent; this way the $A_{s}^{[L]}$ matrices are $D_{\ell-1} \times D_{\ell}$ dimensioned (where $D_0 = D_L = 1$ to ensure that the complete Matrix Product expression is a scalar quantity). It is important to focus on the fact that, since the $\mathcal{A}^{[\ell]}$ are basically a LOCC transformation (i.e. achievable by means of Local Operations and Classical Communication) its action can only degrade entanglement, thus the entanglement of the resulting state $|\Psi\rangle$ is bound by that of the initial state, which is known and straightforward to calculate. Precisely, consider the entanglement entropy related to a left-right partition of the state $|\Psi\rangle$, say at bond $\{\ell, \ell+1\}$. This is by definition the Von Neumann entropy of the reduced density matrix to the left (or right) part of the system, and it is bounded by the entanglement of the original pair across the bond: $$\mathcal{S}_{\text{VN}}\left( {\rho}_{\ell}^{L} \right) \equiv -{\text{Tr}}\left[ {\rho}_{\ell}^{L} \log {\rho}_{\ell}^{L}\right] \leq \log D_{\ell};$$ where $\log D$ is the entanglement of a maximally entangled spin pair of dimension $D$, like $|\Phi^{+}\rangle$ (to check this, just consider that ${\bar{\rho}}^B_D \equiv {\text{Tr}}[|\Phi^{+}\rangle \langle \Phi^{+} |] = \frac{1}{D} \,{{\mathbb 1}}$, thus $\mathcal{S}_{\text{VN}} ({\bar{\rho}}^B_D) = \log D$). In conclusion, a Matrix Product State, i.e. quantum state on a 1D lattice allowing an analytic representation as in eq. , has well-defined upper bounds on its entanglement. The entropy related to a left-right partition of the system is bounded by the logarithm of $D_{\ell}$, with $D_{\ell}$ being the dimension of the Matrix Product *bondlink* $\ell$ we are breaking. Completeness of Matrix Product\ State representation {#sec:MPScompleteness} ------------------------------- The previous observation involving entanglement in MPS becomes even more meaningful once we will provide a theorem of completeness of MPS representations. Indeed, we are going to prove that, as long as we are NOT imposing a finite bound to the maximal MPS bondlink dimension $D$, *any* 1D finite lattice state can be expressed exactly as an MPS. The argument behind this claim is quite simple indeed. Let us choose a site $\ell$ within the (open boundary) lattice, $1 < \ell < L$. Let $|\Psi\rangle$ be the global quantum state, and let us consider the Schmidt decomposition of $|\Psi\rangle$ where the first subsystem is made by sites $\{1..\ell\}$ and the second by $\{\ell+1 .. L\}$: $$\label{eq:Schm230} |\Psi\rangle = \sum_{\alpha}^{D_\ell} \lambda^{[\ell]}_{\alpha} |L_{\alpha} \rangle^{L}_{\ell} \otimes |R_{\alpha} \rangle^{R}_{\ell}.$$ Following the formalism of previous sections, $|L_{\alpha} \rangle^{L}_{\ell}$ are left block Schmidt vectors and $|R_{\alpha} \rangle^{R}_{\ell}$ the right block ones. $\lambda^{[\ell]}_{\alpha}$ are the Schmidt coefficients ($\sum_{\alpha} {\lambda^{[\ell]}_{\alpha}}^2 = 1$), but now the number $D_{\ell}$ of values the index $\alpha$ can assume is not anymore defined *a priori*; instead it depends of the specifics of the $|\Psi\rangle$, precisely on its partition entanglement across the bond $\{\ell,\ell+1\}$. Similarly, we could adopt the same argument when partitioning the system between sites $\ell-1$ and $\ell$, namely $$|\Psi\rangle = \sum_{\alpha}^{D_{\ell - 1}} \lambda^{[\ell-1]}_{\alpha} |L_{\alpha} \rangle^{L}_{\ell-1} \otimes |R_{\alpha} \rangle^{R}_{\ell-1}.$$ Now, since both descriptions are exact, and the fact that the block $\{1 .. \ell \}$ is actually the composition of block $\{1 .. \ell-1 \}$ with the site $\ell$ alone, we must conclude that the set of product states of the form $|L_{\alpha} \rangle^{L}_{\ell-1} \otimes |s\rangle_\ell$ generate every $|L_{\alpha} \rangle^{L}_{\ell}$ state (completeness argument). In fact, we may define the decomposition tensor $A^{[\ell]}$ as follows $$\label{eq:MconstrL} A_{\alpha, \beta}^{[\ell] s} = \left( \vphantom{\sum} \langle L_{\alpha} |_{\ell-1}^{L} \otimes \langle s|_{\ell} \right) |L_{\beta} \rangle_{\ell}^{L}.$$ so that we can expand $|L_{\alpha} \rangle^{L}_{\ell}$ in equation in the new product basis $$|\Psi\rangle = \sum_{\alpha = 1}^{D_{\ell - 1}} \sum_{\beta = 1}^{D_{\ell}} \sum_{s = 1}^{d} \left( A_{\alpha, \beta}^{[\ell] s} \; \lambda^{[\ell]}_{\beta} \right) |L_{\alpha} \rangle^{L}_{\ell-1} \otimes |s\rangle_\ell \otimes |R_{\alpha} \rangle^{R}_{\ell}.$$ Of course, the completeness argument we used poses a relevant constraint upon dimensions of Schmidt decompositions; in particular as $|L_{\alpha} \rangle^{L}_{\ell}$ are orthogonal, they are linearly independent, and since the $|L_{\alpha} \rangle^{L}_{\ell-1} \otimes |s\rangle_\ell$ basis can generate them, it must be that $D_{\ell} \leq d \cdot D_{\ell-1}$. Then, by construction, if we write $A_{\alpha, \beta}^{[\ell] s}$ as a set of matrices (from $\beta$ to $\alpha$) then it holds $$\label{eq:phoneL3} \begin{aligned} \sum_{s = 1}^d {A^{\dagger}}_{s}^{[\ell]} \cdot A_{s}^{[\ell]} &= {{\mathbb 1}}\\ \sum_{s = 1}^d {A}_{s}^{[\ell]} \cdot \Lambda^{[\ell]} \cdot {A^{\dagger}}_{s}^{[\ell]} &= \Lambda^{[\ell-1]}, \end{aligned}$$ where the positive diagonal matrices $\Lambda^{[\ell]}$ are given by $\Lambda^{[\ell]}_{\alpha, \beta} = \delta_{\alpha, \beta} (\lambda^{[\ell]}_{\beta})^2$, and correspond to the Schmidt-basis reduced density matrices of the partition, i.e. $\Lambda^{[\ell]} = \rho_{\ell}^{L} = \rho_{\ell}^{R}$. The previous equations resume together the orthonormalization preservation relation , and the CPT mapping propagation of reduced density matrices . Similarly to , one can perform the formal expansion into site $\ell$ and reduced environment for the right block of the partition, where we can define $$\label{eq:MconstrR} B_{\alpha, \beta}^{[\ell] s} = \left( \vphantom{\sum} \langle R_{\beta} |_{\ell}^{R} \otimes \langle s|_{\ell} \right) |R_{\alpha}\rangle_{\ell-1}^{R},$$ which allows us to write, provided the completeness constraint upon Schmidt dimensions $D_{\ell-1} \leq d \cdot D_{\ell}$ holds, $$|\Psi\rangle = \sum_{\alpha = 1}^{D_{\ell - 1}} \sum_{\beta = 1}^{D_{\ell}} \sum_{s = 1}^{d} \left( \lambda^{[\ell-1]}_{\alpha} \; B_{\alpha, \beta}^{[\ell] s} \right) |L_{\alpha} \rangle^{L}_{\ell-1} \otimes |s\rangle_\ell \otimes |R_{\alpha} \rangle^{R}_{\ell};$$ and of course, complete positivity relations read $$\label{eq:phoneR3} \begin{aligned} \sum_{s = 1}^d B_{s}^{[\ell]} \cdot {B^{\dagger}}_{s}^{[\ell]} &= {{\mathbb 1}}\\ \sum_{s = 1}^d {B^{\dagger}}_{s}^{[\ell]} \cdot \Lambda^{[\ell-1]} \cdot {B}_{s}^{[\ell]} &= \Lambda^{[\ell]}. \end{aligned}$$ In the end, by applying recursively either the left-block or right-block argument presented in this section, we are allowed to build the analytical MPS representation of the original state. In fact, for any given state $|\Psi\rangle$ and any choice of $\ell$ ($1 \leq \ell < L$), one can formally express it as $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell}}^{[\ell]} \cdot \lambda^{[\ell]} \cdot B_{s_{\ell+1}}^{[\ell+1]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ Where the matrices $A_{s_{\ell'}}^{[\ell']}$ and $B_{s_{\ell'}}^{[\ell']}$ are respectively given by and ; they are $D_{\ell'-1} \times D_{\ell'}$ dimensioned, and are well defined since the Schmidt decomposition exists for any partition of the system. Since the choice for the site $\ell$ to start from, is completely arbitrary, the constraint on Schmidt dimensions holds both left-ways and right-ways for every site in the lattice, namely $$\label{eq:dimconstr} D_{\ell'-1} \leq d\:D_{\ell'} \qquad \mbox{and} \qquad D_{\ell'} \leq d\:D_{\ell'-1} \qquad \forall \ell',\;0 < \ell' \leq L,$$ where, of course, $D_0 = D_L = 1$. This concludes the proof. It is now important to point out a major fact concerning the completeness of MPS representation; since the dimension constraint are quite weak, if the state we are dealing with has no limitations on its entanglement properties (which is the typical case for, say, a random state in the many-body Hilbert) such MPS representation is poorly efficient. Indeed, tells us that the largest correlation dimensions $D_{\text{max}} = \max\{D_{\ell}\}$ are typically reached next to the middle of the 1D chain: precisely we have $$D_{\ell} \leq \min\{ d^\ell, d^{L - \ell} \} \qquad \longrightarrow \quad D_{\text{max}} \leq d^{L/2}.$$ Therefore, in general, the typical dimension (number of rows and columns of $A_{s_{1}}^{[1]}$) of the MPS representation *does* scale with the full size $L$ of the system, and in the worst case scenario it grows exponentially. This is the main reason why, in literature, when speaking of Matrix Product States most of the time one actually refers to the manifold of quantum states allowing an MPS representation for which the maximal bondlink dimension $D$ is finite, does not scale with the system size $L$, and is typically small. By putting together the valence bond picture (introduced in section \[sec:valencebond\]) and the completeness argument, one can conclude that a Matrix Product State representation of bondlink $D$ can describe exactly any state whose partition entanglement is bound by $\log D$. Equivalently, *every finitely-correlated state is a MPS*. Area Law and successfulness of 1D MPS ------------------------------------- After such preliminary considerations, interpreting matrix product states as variational tools turns straightforward. The typical problem we want to address is finding the ground states of a given, typically short-ranged, Hamiltonian upon an OBC system with $L$ sites. Similarly to the DMRG procedure, we choose arbitrarily a maximal bondlink dimension $D$ allowed for the simulation that should lead us to the ground state itself, and regard the elements $A_{s}^{[\ell]}$ in MPS representation as variational tensors/matrices. Then, we adjust variational parameters according to some algorithm (see section \[sec:minimizOBC\]) in order to minimize the energy. Due to the completeness theorem of MPS representations, we know that, for any global system size $L$ it exists a finite $D$ for which the *exact* ground state is representable by a $D$-bondlinked MPS, and such $D$ is related to the estimated entanglement $\varepsilon$ of the state itself, like $\varepsilon \sim \log D$. But now we can exploit some theoretical knowledge involving ground states of many-body systems, known in literature as the area-law of entanglement [@Arealaws; @Arealaws2; @Eisarea]: The partition entanglement in a ground state of a non-critical local Hamiltonian scales with the surface of the partition itself, and not with the parted volume. For 1D non-critical systems, this means that $\varepsilon$ does not scale with the size of the system $L$, but rather saturates to a finite value. This also suggests that the bond dimension $D$ required to achieve good precisions in representing the ground state does *not* scale with $L$. In practice, for many tested models, the $D$ necessary to get an outstanding approximation to the GS is surprisingly small, regardless to system size [@MPSground]. This very argument allows us to address even problems with a large number of sites, and yet deal with them in a quasi-exact fashion. Of course this also explains the great success of DMRG for 1D non-critical systems, a reason which was not yet fully understood in the ’90s. Indeed, the area law argument also suggests that finite-$D$ MPS should also be capable to characterize a 1D problem *directly in the thermodynamical limit* as $\varepsilon$ converges to a finite value (we will discuss this approach in section \[sec:MPSTD\]). A special interest within this framework is raised by critical 1D systems [@QptMps]. They are known for violating the area law of entanglement by a logarithmic (with $L$) correction to the partition entropy, with the proportionality constant given by the central charge $C$ of the model [@CFTHolzhey; @CFTCalab]: $$\varepsilon (\Psi) \equiv \mathcal{S}_{\text{VN}}(\rho_{1..L/2}) \sim \frac{C}{6} \log L + C'$$ and therefore the appropriate $D$ to represent the ground state faithfully, does scale in the end with the system size, according a power-law like behavior $D \propto L^{C/6}$ where the exponent is $C/6$ itself. Now since the large majority of the famous 1D critical models have typically small central charges (e.g. crit. Ising, crit. XXZ, Heisenberg, have $C \leq 1$), even though $D$ scales with $L$, the scaling function is so concave that even in that case we can address efficiently quite large system sizes with good precision. Nevertheless, it is important to remember that for critical 1D systems, their efficient MPS representability depends directly on the central charge, while for non-critical systems it is natural, an automatic consequence of the area law of entanglement. In chapter \[chap:TTNMERA\] we will introduce families of variational states more suitable to address criticality than mere MPS Gauge group of\ Matrix Product State representation {#sec:MPSGauge} ----------------------------------- By now, it should be clear that, given a quantum state on an OBC chain, its exact Matrix Product State representation is in general not unique. The issue is simple: the state components expanded in the canonical basis are composite products of matrices, and the same product can be matrix-factorized in many ways. We will now define and explain the usage of a group of transformations that manipulate the set of matrices in the representation, under which the physical state is invariant: by definition this is the *gauge group* of MPS representation. Let us start again from the state $|\Psi\rangle$, whose MPS representation has bondlink dimension $D$, and is given by $$\label{eq:MPSgau} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot A_{s_{2}}^{[2]} \cdot \ldots \cdot A_{s_{L-1}}^{[L-1]} \cdot A_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where matrices $A_{s_{\ell}}^{[\ell]}$ are $D_{\ell - 1} \times D_{\ell}$ dimensioned ($D_{\ell} \leq D$, $\forall \ell$). For every $\ell < L$, we now define an *invertible* square matrix $X_{\ell}$, of dimension $D_{\ell} \times D_{\ell}$. The expression within parentheses in eq.  is left invariant by $$\begin{gathered} \label{eq:mulgau} A_{s_{1}}^{[1]} \cdot A_{s_{2}}^{[2]} \cdot \ldots \cdot A_{s_{L-1}}^{[L-1]} \cdot A_{s_{L}}^{[L]} =\\= \left( A_{s_{1}}^{[1]} \cdot X_1 \right) \left( X_1^{-1} \cdot A_{s_{2}}^{[2]} \cdot X_2 \right) \ldots \left( X_{L-1}^{-1} \cdot A_{s_{L-1}}^{[L-1]} \cdot X_{L-1} \right) \left( X_{L-1}^{-1} \cdot A_{s_{L}}^{[L]} \right).\end{gathered}$$ But now, any term of the form $( X_{\ell-1}^{-1} \cdot A_{s}^{[\ell]} \cdot X_{\ell} )$ is again a $D_{\ell - 1} \times D_{\ell}$ matrix, and as above we have $d$ of them per site, indexed by $s$. In conclusion, the latter expression in eq.  is again a Matrix Product, where the bondlink dimensions $D_{\ell}$ are preserved site-by-site, and the original Matrices of the representation underwent the (gauge) transformation $$\label{eq:Gaugetransform} A_{s}^{[\ell]} \longrightarrow B_{s}^{[\ell]} \equiv X_{\ell-1}^{-1} \cdot A_{s}^{[\ell]} \cdot X_{\ell} \qquad \forall s \in \{1..d\},$$ while the state is left invariant, i.e. $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( B_{s_{1}}^{[1]} \cdot B_{s_{2}}^{[2]} \cdot \ldots \cdot B_{s_{L-1}}^{[L-1]} \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle.$$ For any nearest-neighboring bond, $X_{\ell}$ defines an allowed transformation as long as its inverse is defined. Therefore, the gauge group of Matrix Product States is equivalent to the direct sum of the groups of Isomorphisms of $D_{\ell}$ dimensioned complex vector spaces $$\mathcal{G}_{\text{MPS}} \equiv \bigoplus_{\ell = 1}^{L-1} \text{Iso}\left( \mathbb{C}^{D_\ell} \right).$$ In order to define properly $\mathcal{G}_{\text{MPS}}$ we did not need to summon the Hilbert structure: the invertibility condition is a rank dimension requirement, *not* a metric constraint. This remark is definitely sensible, since the correlation spaces are fictitious, virtual, and therefore there is no reason for a gauge group to mingle with the physical-space metric properties. Finally, notice that the gauge group we built $\mathcal{G}_{\text{MPS}}$ is identified by the initial choice of site-dependent bondlink dimensions $D_{\ell}$, which we required to be left unaltered from the transformation. Actually, in we could have used any rectangular $D_{\ell} \times D'$ matrix $X_{\ell}$ (with $D' < D_{\ell}$) which is right-invertible, i.e. $$X_{\ell} \cdot X_{\ell}^{-1} = {{\mathbb 1}}, \quad \mbox{but} \quad X_{\ell}^{-1} \cdot X_{\ell} = P = P^2 \neq {{\mathbb 1}},$$ and adopt such $X_{\ell}$ in ; this, of course, leaves the matrix product invariant, but the bondlinks of the representations are altered, their dimensions increased ($B_{s}^{[\ell]}$ now are $D_{\ell -1} \times D'$, and $B_{s}^{[\ell+1]}$ are $D' \times D_{\ell +1}$). However, the state contains the same amount of entanglement as before, but we are spending more resources to describe it: we are working in a non-optimal numerical framework. Moreover, this extension $\mathcal{G}'$ to the previously defined $\mathcal{G}_{\text{MPS}}$ is clearly a group lacking an inverse-element property. For these reasons, in most cases it is interesting to limit the study of MPS gauge features on $\mathcal{G}_{\text{MPS}}$ itself, under which the MPS representation space, given by the $\{D_{\ell}\}_{\ell}$, is stable. The canonical form {#sec:MPScanonical} ------------------ The presence of a gauge group for MPS provides an computational advantage, since freedom and manipulability of our description tools are increased. At the same time, the capability of quickly recognizing state properties, or comparison between states is reduced, as even MPS representations of two identical states may look very different, when their gauges are incompatible. The simplest way to avoid such difficulty is to break the gauge invariance by hand, i.e. by characterizing a representative in the class of equivalence for MPS, which is easy to achieve, recognize, and completely general. This concept realizes in the definition of a *canonical form* for MPS representations. We say that a Matrix Product State, of bond dimension $D$ $$\label{eq:MPS21} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot A_{s_{2}}^{[2]} \cdot \ldots \cdot A_{s_{L-1}}^{[L-1]} \cdot A_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where $A_{s_{\ell}}^{[\ell]}$ are $D_{\ell - 1} \times D_{\ell}$ dimensioned matrices, with open boundary conditions ($D_0 = D_L = 1$), is in the (right-) canonical form if it holds $$\label{eq:phoneRX} \begin{aligned} &\sum_{s = 1}^d A_{s}^{[\ell]} \cdot {A^{\dagger}}_{s}^{[\ell]} = {{\mathbb 1}}\qquad \forall \ell, 1 \leq \ell \leq L \\ &\sum_{s = 1}^d {A^{\dagger}}_{s}^{[\ell]} \cdot \Lambda^{[\ell-1]} \cdot {A}_{s}^{[\ell]} = \Lambda^{[\ell]} \qquad \forall \ell, 1 \leq \ell \leq L \\ &\Lambda^{[0]} = \Lambda^{[L]} = 1, \mbox{ and every $\Lambda^{[\ell]}$ is positive and full rank.} \end{aligned}$$ We recall that the first equation is the CPT-condition to preserve orthonormality among Schmidt vectors, when propagating from the right: for this reason, we will, from now on, refer to this gauge-invariance breaking, , as *right gauge*, for brevity. Given any MPS, it is *always* possible to write a canonical matrix product representation for the same state; the bondlink dimensions are equal or smaller than the original ones. An operational proof of this statement is explained in detail in ref. [@MPSreview], we are now going to sketch the fundamentals, as many of the involved manipulations will be useful later on, and this is the perfect context to introduce them. ### Proof of canonical form generality Let us take an OBC-MPS representation given by $B_{s}^{[\ell]}$ matrices $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( B_{s_{1}}^{[1]} \cdot B_{s_{2}}^{[2]} \cdot \ldots \cdot B_{s_{L-1}}^{[L-1]} \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ we are going to define explicitly a set of rectangular matrices $Y_{\ell}$ and $Z_{\ell}$, with $Y_{\ell} Z_{\ell} = {{\mathbb 1}}$, such that by applying $$\begin{aligned} A_{s}^{[1]} = B_{s}^{[1]} Z_{1}, \qquad A_{s}^{[L]} = Y_{L-1} B_{s}^{[L]},\\ A_{s}^{[\ell]} = Y_{\ell-1} B_{s}^{[\ell]} Z_{\ell} \quad \text{for } 1 < \ell < L, \end{aligned}$$ the resulting matrices $A_{s}^{[\ell]}$ satisfy , and they represent again $|\Psi\rangle$ faithfully via . Moreover, the resulting bondlink dimensions $D_\ell$ will be equal or smaller than those of $B_{s}^{[\ell]}$ representation. Precisely, the full rank condition (\[eq:phoneRX\].c) for $\Lambda^{[\ell]}$ tells us that the resulting $A_{s}^{[\ell]}$ representation uses the *minimal* bondlink dimension, for every bond, necessary to describe $|\Psi\rangle$. The $Y_{\ell}$ and $Z_{\ell}$ represent a gauge transformation followed by a correlation space truncation; constructing them is quite simple. We start from the right of the 1D chain, by performing a Singular Value Decomposition (SVD) of the matrix $B_{j,s_{L}}^{[L]}$ read as if $s_{L}$ were an incoming index, and $j$ an outcoming one: $$B_{j,s_{L}}^{[L]} = \sum_{\beta} U_{j, \beta}^{[L]} \Delta_{\beta}^{[L]} A_{\beta,s_{L}}^{[L]},$$ where $U^{[L]}$ and $A^{[L]}$ are respectively left and right isometric matrices, i.e. $U^{\dagger} U = {{\mathbb 1}}$ and $A A^{\dagger} = {{\mathbb 1}}$, and the diagonal matrix $\Delta^{[L]}$ is positive. Not only, but we can make $\Delta^{[L]}$ strictly positive, by just cutting $\beta$ values for which $\Delta_{\beta}^{[L]} = 0$ out of the sum. If we do, $U^{[L]}$ and $A^{[L]}$ continue to be isometries, as any subset of columns of $U^{[L]}$ (resp. of rows of $A^{[L]}$) is still an orthonormal set. Then we just fix $Y_{L-1}$ and $Z_{L-1}$ matrices as $$Y_{L-1} = {\Delta^{[L]}}^{-1} {U^{[L]}}^{\dagger} \qquad Z_{L-1} = {U^{[L]}} {\Delta^{[L]}}.$$ By construction $Y_{L-1} B_{s}^{[L]} = A_{s}^{[L]}$, which is an isometry and thus in the right gauge. Similarly we define $C^{L-1}_{s} = B_{s}^{[L-1]} Z_{L-1}$, and of course $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( B_{s_{1}}^{[1]} \ldots B_{s_{L-2}}^{[L-2]} \, C_{s_{L-1}}^{[L-1]} \, A_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ is still a faithful representation of the original state $|\Psi\rangle$: the rightmost bondlink $D_{L-1}$ might be decreased after the transformation, but due to the SVD argument we know we disregarded only zero components. In other words we could say that $Z_{L-1} Y_{L-1} = P = P^2 = P^{\dagger}$ is the projector over the actual support of the bondlink space (in $B_{s}^{[\ell]}$ representation). Now we proceed recursively: we consider the composite matrix $C_{j, \alpha}^{[\ell]}$ (with $\ell$ starting from $L-1$ and moving left) whose incoming index $\alpha$ is the pair of indices $\{k, s_{\ell}\}$, and again we perform a SVD $$C_{j,\alpha}^{[\ell]} = \sum_{\beta} U_{j, \beta}^{[\ell]} \Delta_{\beta}^{[\ell]} A_{\beta, \alpha}^{[\ell]}.$$ By construction $A^{[\ell]}$ satisfies the right gauge condition, since $$\sum_{\alpha} A_{\beta, \alpha}^{[\ell]} {A^{\star}}_{\gamma, \alpha}^{[\ell]} = \delta_{\beta, \gamma} \quad \longrightarrow \quad \sum_{s_{\ell}} A_{s_{\ell}}^{[\ell]} {A^{\dagger}}_{s_{\ell}}^{[\ell]} = {{\mathbb 1}}.$$ Then, bondlink $\ell$ space is truncated to the support of $\Delta$, and the (pseudo-) gauge transformation given by $$Y_{\ell-1} = {\Delta^{[\ell]}}^{-1} {U^{[\ell]}}^{\dagger} \qquad Z_{\ell-1} = {U^{[\ell]}} {\Delta^{[\ell]}},$$ which lead to $Y_{\ell-1} C_{s}^{[\ell]} = Y_{\ell-1} B_{s}^{[\ell]} Z_{\ell} = A_{s}^{[L]}$, and redefine $B_{s}^{[\ell-1]} Z_{\ell-1} = C_{s}^{[\ell-1]}$ so that we can apply the procedure again on site $\ell -1$. Once we arrive at the left-end of the chain, $C_{s_{1}}^{[1]}$ is already in the right gauge by assumption of initial state normalization $$\sum_{s_1} C_{s_{1}}^{[1]} {C^{\dagger}}_{s_{1}}^{[1]} = \langle \Psi | \Psi \rangle = 1$$ in conclusion $A_{s_{1}}^{[1]} = C_{s_{1}}^{[1]} = B_{s}^{[\ell]} Z_{\ell} = A_{s}^{[L]}$. This proves the first statement of : converting a complete MPS representation in order to be fully in the right gauge is operatively possible by a recursive application of Singular Value Decompositions. The second statement of is a direct consequence of the CPT mapping argument we presented in previous sections, in particular it corresponds to equation . Finally, the full rank condition follows from the fact that after every SVD steps we truncated the bondlink space (dimension $D_\ell$) to the support of the corresponding reduced density matrix $\Lambda^{[\ell]}$. This issue is argumented in details in [@MPSreview]. As a concluding remark to the present section, we would like to point out that the canonical form we just presented is the right-directed one, i.e. is made so that every MPS block is in the right gauge (and the correlation space used is minimal). Of course, we could similarly define a left-canonical form, where MPS is completely in the left gauge, i.e. , and other statements still hold: $$\label{eq:phoneLX} \begin{aligned} &\sum_{s = 1}^d {A^{\dagger}}_{s}^{[\ell]} \cdot A_{s}^{[\ell]} = {{\mathbb 1}}\qquad \forall \ell, 1 \leq \ell \leq L \\ &\sum_{s = 1}^d {A}_{s}^{[\ell]} \cdot \tilde{\Lambda}^{[\ell]} \cdot {A^{\dagger}}_{s}^{[\ell]} = \tilde{\Lambda}^{[\ell-1]} \qquad \forall \ell, 1 \leq \ell \leq L \\ &\tilde{\Lambda}^{[0]} = \tilde{\Lambda}^{[L]} = 1, \mbox{ and every $\tilde{\Lambda}^{[\ell]}$ is positive and full rank,} \end{aligned}$$ such is the left-canonical form for MPS representations. The demonstration adopted in this section to achieve the canonical form is operational in the sense that is exactly the algorithm we apply in numerical settings: the computational advantage of using canonical MPS, apart from immediate estimation of entanglement, will be clear as soon as we explain how to achieve expectation values onto an MPS state. MPS and Observables {#sec:OBCMpsObs} ------------------- By now, we understood that MPS are outstanding candidates as tools for simulating condensed matter one-dimensional many-body systems. Then, it is fundamental that we realize how to achieve expectation values of observables, in a clear and efficient way. If we are to adopt, say, the global energy as a simulation benchmark, so that our goal becomes achieving the absolute energy minimum, we need first to calculate the expectation value $\langle \Psi | H | \Psi \rangle$ of the Hamiltonian $H$: and operator which is nonlocal, but it is explicitly written as a sum of local (separable) terms. For simplicity we can assume it couples only nearest neighboring sites $$\label{eq:NNhamilt} H = \sum_{\ell = 1}^{L} \sum_{q} g^{\ell}_{q} \;\Theta^{[\ell]}_{q} \;+\; \sum_{\ell = 2}^{L} \sum_{p} h^{\ell}_{p} \;{\Theta'}^{[\ell-1]}_{p} \otimes {\Theta''}^{[\ell]}_{p}.$$ Let us start from getting the expectation value over a MPS of a separable observable $O = \bigotimes_{\ell} \Theta^{[\ell]}$, where every operator $\Theta^{[\ell]}$ can depend on the site $\ell$ on which it acts. We have $$\begin{gathered} \langle \Psi | H | \Psi \rangle = \sum_{s_1 \ldots s_n} \sum_{r_1 \ldots r_n} \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{L}}^{[L]} \right) \times \\ \times \left( {A^{\star}}_{r_{1}}^{[1]} \cdot \ldots \cdot {A^{\star}}_{r_{L}}^{[L]} \right) \langle r_1 \ldots r_L | \bigotimes_{\ell} \Theta^{[\ell]} | s_1 \ldots s_L \rangle.\end{gathered}$$ Now we define the so-called *transfer matrices*, as follows $${{\mathbb E}}_{X}^{[\ell]} \equiv \sum_{s,r = 1}^d \langle r | X | s \rangle \left( A_{s}^{[\ell]} \otimes {A^{\star}}_{r}^{[\ell]} \right),$$ where $X$ is a one-site operator, acting on site $\ell$; transfer matrices ${{\mathbb E}}_{X}^{[\ell]}$ are $D_{\ell-1}^2 \times D_{\ell}^2$ dimensioned. We now calculate the ${{\mathbb E}}_{\Theta_\ell}^{[\ell]}$ for every $\ell$, and the expectation value becomes simply a multiplication of the whole string of transfer matrices $$\label{eq:Transmatstring} \langle \Psi | H | \Psi \rangle = {{\mathbb E}}_{\Theta_1}^{[1]} \cdot {{\mathbb E}}_{\Theta_2}^{[2]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta_L}^{[L]}.$$ The computational cost for acquiring this expectation value scales only *linearly* with the system size (recall that now we treat $L$ and $D$ as independent parameters of the MPS variational ansatz); unfortunately, there is still a harsh dependence on the chosen bondlink dimension $D$. Indeed multiplying two $D^2 \times D^2$ matrices costs $\sim D^6$ elementary operations, yet by adopting some technical tricks we can further improve this scaling law: - We should start performing the multiplication from the right (or left) boundary, as $| Q_{L-1} ) = {{\mathbb E}}_{\Theta}^{[L]}$ is a one-column matrix, i.e. a column vector. And multiplying a $D^2$ dimensioned vector $| Q_{\bullet} )$ for a $D^2 \times D^2$ matrix has an overall $D^4$ cost. - instead of multiplying directly ${{\mathbb E}}_{\Theta}^{[\ell]} | Q_{\ell}) = |Q_{\ell-1})$ one can first calculate $|\Gamma^{[\ell]}_{s}) = [ A_{s}^{[\ell]} \otimes {{\mathbb 1}}] |Q_{\ell})$, followed by $|\Pi^{[\ell]}_{r}) = \sum_{s} \langle r | \Theta^{[\ell]} | s \rangle |\Gamma^{[\ell]}_{s})$ and finally $|Q_{\ell-1}) = \sum_r [ {{\mathbb 1}}\otimes {A^{\star}}_{r}^{[\ell]} ] |\Pi^{[\ell]}_{r})$. These operations requires respectively a number of elementary operations equal to: $dD^3$, $d^2 D^2$, and $dD^3$. In the end, the total computational cost to achieve the MPS-expectation value of a separable observable $O$ scales, with size $L$ and bond dimension $D$, as $$\label{eq:MPScost} \# \mbox{cost} \sim L \left(2 d D^3 + d^2 D^2 \right).$$ Usually, the first term in the parentheses is the leading one (and the other is negligible), since the typical bond dimensions chosen in simulations are sensibly larger than local space dimensions $D \gg d$. The result we got holds in a quite general scenario (provided that $O$ is acts locally); we will now see that if the involved operator has a small support, we can considerably improve this limit by exploiting the gauge group of MPS. ### Local support Observables {#sec:Localsupport} Let us assume that the observable $O$ we are interested with does not involve all the sites within the 1D chain, but only a small connected subset of those, say lattice sites between $\ell_1$ and $\ell_2$ ($1 < \ell_1 \leq \ell_2 < L$). Recalling the previous argument involving transfer matrices, i.e. eq.  we can write $$\label{eq:Transfmatlocal} \langle \Psi | O | \Psi \rangle = \left( {{\mathbb E}}_{{{\mathbb 1}}}^{[1]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell_1-1]} \right) \left( {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} \right) \left( {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell_2+1]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[L]} \right).$$ Notice that on the sites outside $\{\ell_1..\ell_2\}$ the observable $O$ acts trivially, so we are considering the transfer matrix of the local identity operator ${{\mathbb 1}}$ there. Now, as the expectation value in is a physical quantity, i.e. it depends on the properties of the quantum state and *not* on its specific MPS representation: it is a quantity invariant under the action of the MPS gauge group. At the same time the transfer matrices are not gauge invariant, so it is advisable to choose a gauge that reduces the computational cost of . Precisely, we choose a gauge that turns our MPS to look like this $$\begin{gathered} \label{eq:CompaMPS} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell_1-1}}^{[\ell_1-1]} \cdot C_{s_{\ell_1}}^{[\ell_1]} \cdot \ldots \cdot C_{s_{\ell_2}}^{[\ell_2]} \cdot \phantom{A} \right. \\ \left. \phantom{A} \cdot B_{s_{\ell_2+1}}^{[\ell_2+1]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,\end{gathered}$$ where MPS tensors to the left of the $\{\ell_1..\ell_2\}$ support are in the left gauge ($\sum_{s} A_s^{\dagger} A_s = {{\mathbb 1}}$), those to the right of the support are in the right gauge ($\sum_{s} B_s B_s^{\dagger} = {{\mathbb 1}}$), and those in the middle can be in any gauge chosen by the user, with the only constraint that they must satisfy the global state normalization condition. As before, the form can be achieved by means of recursive Singular Value Decompositions that define appropriate gauge transformations (from site 1 to $\ell_1$ to fix the $A$, from site $L$ to $\ell_2$ to fix the $B$), exactly like we did in the section \[sec:MPScanonical\]. Now we focus on the transfer matrices of the outside zone ${{\mathbb E}}_{{{\mathbb 1}}}^{[\ell]}$. Consider site $L$, due to the assumptions we made, it holds $$\label{eq:phoneRTrans} |Q_{L-1}) = {{\mathbb E}}_{{{\mathbb 1}}}^{[L]} = \sum_{j,k} \left( \sum_{s,r}^{d} B^{[L]s}_j \,\delta_{s,r}\, {B^{\star}}_k^{[L]r} \right) |jk) = | \Phi^{+} )$$ where $| \Phi^{+} )$ is the (unnormalized) maximally entangled canonical vector $$| \Phi^{+} ) \equiv \sum_j^{D_{L-1}} |jj) = \sum_{j,k}^{D_{L-1}} \delta_{j,k} |jk).$$ It is east to see that eq.  holds recursively for all $| Q_{\ell} )$, $\ell > \ell_2$, since $$\begin{gathered} | Q_{\ell-1} ) = {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell]} | \Phi^{+} ) = \sum_{j,k}^{D_{\ell-1}} \left( \sum_{s,r}^{d} \sum_{\alpha, \beta}^{D_\ell} B^{[\ell]s}_{j,\alpha} \,\delta_{s,r}\, \delta_{\alpha, \beta}\, {B^{\star}}_{k, \beta}^{[\ell]r} \right) |jk) = \\ \sum_{j,k}^{D_{\ell-1}} \delta_{j,k}\, |jk) = | \Phi^{+} ),\end{gathered}$$ where we used both the fact that the operator acts like identity on site $\ell$ (thus $\delta_{s,r}$), and the recursive hypothesis $| Q_{\ell} ) = | \Phi^{+} )$. An identical argument can be applied to transfer matrices to the left of the support of $O$, where left gauge condition can be exploited to see that $(Q_{\ell}| = (\Phi^{+}|$ for any $\ell \leq \ell_1$. The conclusion simply follows: $$\label{eq:Transfmatcompact} \langle \Psi | O | \Psi \rangle = (\Phi^{+}| {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} | \Phi^{+} ),$$ which means that the number ($\#$cost) of elementary operations we have to perform does *not* even scale with the full size of the system $L$, but merely with the size of the operator support: $$\label{eq:MPScostcompact} \# \mbox{cost} \sim (\ell_2 - \ell_1) \left(2 d D^3 + d^2 D^2 \right).$$ Honestly, we traded the modest effort of performing the SVD, needed to convert the MPS in the proper gauge, to obtain a faster (and non-scaling) computational speed in acquiring finite-range MPS physics. We will see that this result can be partially exploited even when we are to compute expectation values of observables which are not local and not even separable, but allow a natural decomposition into local terms, such are the Hamiltonians of typical short-range interacting models. ### Hamiltonian-like Observables We are now interested defining an operational algorithm that, exploiting MPS properties, computes efficiently the expectation values $\langle \Psi | H | \Psi \rangle$ of an operator $H$ which is formally written as a nearest-neighboring Hamiltonian of the system, i.e. like . For algebraic reasons which shall be clear soon, we rewrite it as $H = \tilde{H}^{\rightarrow}_{0}$, where $$\tilde{H}^{\rightarrow}_{\ell'} = \sum_{\ell = \ell'+1}^{L} \sum_{q} g^{[\ell]}_{q} \;\Theta^{[\ell]}_{q} + \sum_{\ell = \ell'+2}^{L} \sum_{p} h^{[\ell]}_{p} \;{\Theta'}^{[\ell-1]}_{p} \otimes {\Theta''}^{[\ell]}_{p},$$ As before, it is important that we focus on the computational cost of this data acquisition. We learned that working in the proper MPS gauge is instrumental for economy of calculus, thus we already start from a canonical MPS representation (say the right one) $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( B_{s_{1}}^{[1]} \, B_{s_{2}}^{[2]} \ldots B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where all the $B_{s}^{[\ell]}$ are right-gauged, requirement which also guarantees proper state normalization $\langle \Psi | \Psi \rangle = 1$. Again, our scheme to acquire $\langle \Psi | H | \Psi \rangle$ has a recursive formulation: we need to propagate the contraction of our MPS structure and at the same time include every term of $H$. Since this is expensive by definition, we will try to regroup and sum the partially contracted tensors every time we can. Then the starting point, at the right boundary, is defined as follows: $$\begin{aligned} |\xi^{[L-1]}) &= \sum_{j,k}^{D_{L-1}} \left( \sum_q g^{[L]}_q \sum_{r,s}^{d} \langle r | \Theta^{[L]}_q | s \rangle \right) B_{j}^{[L]s} {B^{\star}}_{k}^{[L]r} |jk)\\ |\chi^{[L-1]}_p) &= \sum_{j,k}^{D_{L-1}} \left( \sum_{r,s}^{d} h_p^{[L]} \langle r | {\Theta''}^{[L]}_p | s \rangle \right) B_{j}^{[L]s} {B^{\star}}_{k}^{[L]r} |jk). \end{aligned}$$ Now we propagate towards the left. The idea is that $|\xi^{[\ell]})$ shall contain all the elements of the Hamiltonian who have support in $\{ \ell+1 .. L \}$, while $|\chi^{[\ell]}_p)$ will take care of nonlocal neighboring terms across the bond $\{\ell-1, \ell\}$. Of course, we can exploit the gauge conditions, telling us that $|Q_\ell) = | \Phi^{+} )$. This allows us to calculate every $|\chi^{[\ell]}_p)$ directly $$\label{eq:chi_media} |\chi^{[\ell-1]}_p) = \sum_{j,k}^{D_{L-1}} \sum_{\alpha}^{D_{\ell}} \left( \sum_{r,s}^{d} h_p^{[\ell]} \langle r | {\Theta''}^{[\ell]}_p | s \rangle \right) B_{j,\alpha}^{[\ell]s} {B^{\star}}_{k,\alpha}^{[\ell]r} |jk).$$ Instead $|\xi^{[\ell-1]})$ is obtained via the recursive relation $$\begin{gathered} \label{eq:xi_cumbersome} |\xi^{[\ell-1]}) = \sum_{j,k}^{D_{\ell-1}} \sum_{\alpha}^{D_\ell} \left( \sum_q g^{[\ell]}_q \sum_{r,s}^{d} \langle r | \Theta^{[\ell]}_q | s \rangle \right) B_{j,\alpha}^{[\ell]s} {B^{\star}}_{k,\alpha}^{[\ell]r} |jk) \; +\\+ \sum_{j,k}^{D_{\ell-1}} \sum_{\alpha, \beta}^{D_\ell} \left( \sum_p \sum_{r,s}^{d} \langle r | {\Theta'}^{[\ell]}_p | s \rangle \right) B_{j,\alpha}^{[\ell]s} {B^{\star}}_{k,\beta}^{[\ell]r} |jk) \,(\alpha \beta| \chi^{[\ell]}_p) \; +\\+ \sum_{j,k}^{D_{\ell-1}} \sum_{\alpha, \beta}^{D_\ell} \sum_s^d B_{j,\alpha}^{[\ell]s} {B^{\star}}_{k,\beta}^{[\ell]s} |jk) \,(\alpha \beta| \xi^{[\ell]}_p).\end{gathered}$$ By means of the transfer matrices formalism, we can rewrite the two previous equations in a more compact, and clearer, form $$\label{eq:xi_shorter} \begin{aligned} |\xi^{[\ell-1]}) &= \sum_q g^{[\ell]}_q {{\mathbb E}}_{\Theta_q}^{[\ell]} |\Phi^{+}) + \sum_p {{\mathbb E}}_{\Theta'_p}^{[\ell]} | \chi^{[\ell]}_p) + {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell]} |\xi^{[\ell]}) \\ | \chi^{[\ell-1]}_p) &= h^{[\ell]}_p {{\mathbb E}}_{\Theta''_p}^{[\ell]} |\Phi^{+}). \end{aligned}$$ Acquiring all these data, for every $\ell$ requires an overall computational cost (apart subleading trends) of $$\# \mbox{cost} \sim L \left(\#q + 2\#p + 1 \right) \left(2 d D^3 + d^2 D^2 \right),$$ where $\#q$ and $\#p$ are respectively the number of one body and two body terms in the Hamiltonian expression. As you see, even in this complex scenario, the scaling behavior with $D$ and $L$ of the computational cost remains roughly the same. Now we can conclude that $$\langle \Psi | \tilde{H}^{\rightarrow}_{\ell} | \Psi \rangle = {{\mathbb E}}_{{{\mathbb 1}}}^{[1]} \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[2]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell]} |\xi^{[\ell]}),$$ and in particular $\langle \Psi | H | \Psi \rangle = |\xi^{[0]})$ which is a scalar number, and it is exactly the energy of the state if $H$ is the actual Hamiltonian of the system. The great improvement in computing expectation values we encountered for separable observable is recovered in case of (short-range) Hamiltonian operators: the full computational cost to acquire the energy (which will be later adopted as variational functional) scales only linearly with the system size. If one thinks that (full-search) exact methods typically bear an exponential cost in $L$, it is easy to understand why DMRG/MPS architectures are regarded with great interest. Pictorial representation of Matrix\ Product States Tensor Network ----------------------------------- Through the present chapter, we learned how to deal with MPS: their mathematical properties that allow algebraic manipulation (gauge group), and their physical properties that allow to control quantum entanglement. On the other hand, the equations we encounter start to look cumbersome and confusing, like eq. . To work around this issue, we are now going to provide an alternative way to express MPS representation that is based on diagrams and graph theory rather than standard analytical expressions. This will prove a faster and clearer fashion to represent states, observations, and matrix multiplication; and will become instrumental in later chapters. Let us start back from our definition of MPS. If our system is a 1D-OBC lattice with $L$ sites, and $\{ |s\rangle \}_s$ is the local canonical basis, then a generic state of the system is written as $|\Psi\rangle = \sum_{s_1 \ldots s_L} \mathcal{T}_{s_1 \ldots s_L} |s_1 \ldots s_L\rangle$; the complex tensor (with $L$ indices) $\mathcal{T}_{s_1 \ldots s_L}$ uniquely defines $|\Psi\rangle$. Now, stating that $|\Psi\rangle$ is an MPS (with fixed bondlink $D$), is equivalent to say that $\mathcal{T}_{s_1 \ldots s_L}$ allows the following decomposition $$\label{eq:MPSTN} \mathcal{T}_{s_1 \ldots s_L} = \sum_{ \{ j_\ell = 1 \} }^{D_{\ell} \leq D} A_{j_1}^{[1] s_1} \; A_{j_1, j_2}^{[2] s_2} \; A_{j_2, j_3}^{[3] s_3} \ldots A_{j_{L-2}, j_{L-1}}^{[L-1] s_{L-1}} \; A_{j_{L-1}}^{[L] s_{L}}.$$ As mentioned before, all the $A^{[\ell]}$ elements are three-indices tensors (apart the first $A^{[1]}$ and last $A^{[L]}$ MPS blocks, which have only two indices due to open boundaries), $s_{\ell}$ being the physical index, i.e. related to the local canonical state $|s\rangle_{\ell}$, while the two $j_{\ell}$ being the correlation space indices linking to the two neighboring MPS blocks, namely $A^{[\ell-1]}$ and $A^{[\ell+1]}$. Equation tells us that a MPS is the result of a multiple contraction of (possibly variational) tensors, or more simply a *Tensor Network*. Let us draw eq.  in the following pictorial graph [OBCMps]{} (20, 40)[$A^{[1]}$]{} (81, 40)[$A^{[2]}$]{} (142, 40)[$A^{[3]}$]{} (203, 40)[$A^{[4]}$]{} (316, 40)[$A^{[L]}$]{} where every block (graph vertex) represent a single tensor $A^{[\ell]}$, the legs/links attached to it being the indices $s$ (vertical one), $j_{\ell-1}$ and $j_{\ell-1}$ (horizontal ones). Connecting two tensors though a given link means contracting the product of the two over that index. By these very simple rules, one sees that is recovered, but there is no need anymore to write down either the sums or every single index $s$ or $j$ by its name. Everything is implicit in the pictogram. As we discussed in section \[sec:MPSGauge\], on each of the contracted index/connected links we can insert an isomorphism $X_{\ell}$ together with its inverse $X^{-1}_{\ell}$. Since their global action cancels out during link contraction, they have absolutely no effect on the global tensor $\mathcal{T}$. This is exactly the gauge of Matrix Product States, which we represent like this [MPSGauge]{} (49, 53)[$A^{[\ell]}$]{} (116, 53)[$X_{\ell}$]{} (188, 53)[$X^{-1}_{\ell}$]{} (251, 53)[$A^{[\ell+1]}$]{} (84, 69)[$B^{[\ell]}$]{} (215, 69)[$B^{[\ell+1]}$]{} meaning that $B_{j,k}^{[\ell]s} = \sum_{\alpha}^{D_{\ell}} A_{j,\alpha}^{[\ell]s} X_{\alpha,k}$ and $B_{j,k}^{[\ell+1]s} = \sum_{\alpha}^{D_{\ell}} X^{-1}_{j,\alpha} A_{\alpha,k}^{[\ell]s}$. This operation is definitely equivalent to , once we have defined an isomorphism $X_{\ell}$ for every link $\ell$, $1 \leq \ell < L$. To break by hand the freedom granted the gauge group, we defined two particular gauge choices, namely the left and right gauges. We said that $A^{[\ell]}$ is in the left gauge if $\sum_s {A^{\dagger}}_s^{[\ell]} A_s^{[\ell]} = {{\mathbb 1}}$, i.e. $\sum_{s, \alpha} {A^{\star}}_{j, \alpha}^{[\ell]s} A_{\alpha, k}^{[\ell]s} = \delta_{j,k}$, or equivalently, read in terms of transfer matrices, $(\Phi^{+}| {{\mathbb E}}^{[\ell]}_{{{\mathbb 1}}} = (\Phi^{+}|$; and in pictorial representation, it becomes an equation between graphs: $$\label{eq:phoneLGra} \begin{overpic}[width = 240pt, unit=1pt]{LeftGauge} \put(68, 23){${A^{\star}}^{[\ell]}$} \put(70, 110){$A^{[\ell]}$} \end{overpic}$$ Similarly, the right gauge condition is the left-right specular of this graph equation. Having such equation in graph form lets us to see immediately how to exploit the gauge by substituting pieces of the Tensor Network and thus eliminating tensors. In particular, assume we are to calculate the state square norm $\langle \Psi | \Psi \rangle = \sum_{\{s_{\ell}\}} \mathcal{T}_{s_1 \ldots s_L} \mathcal{T}^{\star}_{s_1 \ldots s_L} =$ [MPSNorm]{} If all the MPS tensors are in the left gauge, we can substitute into the diagram, and tensors start to literally cancel out, from left to right, until only $1 = \langle \Psi | \Psi \rangle $ remains. Moreover, the left gauge condition ensures (reduced density matrix propagation via CPT map), which corresponds to $$\label{eq:mappuzLGra} \begin{overpic}[width = 240pt, unit=1pt]{CPTMap} \put(147, 23){${A^{\star}}^{[\ell]}$} \put(149, 107){$A^{[\ell]}$} \put(207, 67){$\tilde{\rho}_{\ell}^{L}$} \put(37, 67){$\tilde{\rho}_{\ell-1}^{L}$} \end{overpic}$$ In section \[sec:Localsupport\] we saw that having the leftmost MPS tensors in the left gauge and the rightmost in the right gauge provides a great advantage when computing observables having local support. Precisely, let $O = \bigotimes_{\ell = \ell_1}^{\ell_2} \Theta^{[\ell]}$, and we require that $\sum_s {A^{\dagger}}_s^{[\ell]} A_s^{[\ell]} = {{\mathbb 1}}$ for $\ell < \ell_1$ and $\sum_s A_s^{[\ell]} {A^{\dagger}}_s^{[\ell]} = {{\mathbb 1}}$ for $\ell > \ell_2$. Then, the part of the graph outside $\{\ell_1..\ell_2\}$ cancels out thanks to and we are left with [Localsepobs]{} (61, 23)[${A^{\star}}^{[\ell_1]}$]{} (63, 77)[$\Theta^{[\ell_1]}$]{} (63, 132)[$A^{[\ell_1]}$]{} (306, 23)[${A^{\star}}^{[\ell_2]}$]{} (308, 77)[$\Theta^{[\ell_2]}$]{} (308, 132)[$A^{[\ell_2]}$]{} $= (\Phi^{+}| {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta_{\ell_1}}^{[\ell_1]} | \Phi^{+} )$, as we saw in the previous section. To explain the algorithm we use to calculate the expectation value of a nearest-neighbor interacting Hamiltonian $H$, we introduced the relations and , where the transfer vector $|\xi^{[\ell]})$ was defined via a recursive scheme. Despite the unclear look of those equations, it is possible to resume them both in a simple and intuitive graphical equation: $$\label{eq:xi_fig} \begin{overpic}[width = \textwidth, unit=1pt]{RecursHamilt} \put(34, 47){$\xi_{\ell-1}$} \put(367 , 47){$\xi_{\ell}$} \put(103, 77){$A^{[\ell]}$} \put(190, 77){$A^{[\ell]}$} \put(314, 77){$A^{[\ell]}$} \put(103, 45){$R^{[\ell]}_1$} \put(207, 45){$R^{[\ell+1]}_2$} \end{overpic}$$ where we just regrouped together the one-site operators as $R^{[\ell]}_{1} = \sum_q g^{[\ell]}_{q} \;\Theta^{[\ell]}_{q}$, and the two-site ones: $R^{[\ell]}_{2} = \sum_p h^{[\ell]}_{p} \;{\Theta'}^{[\ell-1]}_{p} \otimes {\Theta''}^{[\ell]}_{p}$. Several equations that we will encounter in this thesis involve Tensor Network contraction or decompositions, in most cases they allow a diagrammatic version, granting immediateness, and clarity of understanding. So, where appropriate, we shall provide it for completeness and comfort for the reader. Minimization algorithms {#sec:minimizOBC} ----------------------- We mentioned MPS being powerful variational tools for simulating ground state of 1D many-body systems, and we also managed to give a prescription for evaluating the energy of the matrix product state. It is finally time we adopt such energy $\langle \Psi | H | \Psi \rangle$ as a functional for variational simulation, and describe an algorithm that drives our trial Matrix Product State toward the absolute minimum of this functional. Despite the huge reduction in variational parameters we are left thanks to the MPS representation ($\sim LdD^2$ rather than $\sim d^{L}$), performing full search of the minimum within the whole parameter space at once is still too expensive for practical purposes. Instead, we will follow a scheme similar to original DMRG: the idea is to perform only local or quasi-local variations of the whole state representation, namely variating a limited number of connected MPS blocks while keeping the other fixed. Then we repeat, while choosing each time a different compact subset of MPS blocks to variate, until convergence is eventually reached. Like traditional DMRG algorithms, usually one or two adjacent block are variated at a time, and refrain by sweeping towards the left or the right (bouncing off the boundaries of the system, if our problem is OBC). ### Single variational site In this framework, at every minimization step only one block of the MPS is being treated as variational, say the one related to site $\ell$; the other ones are fixed, and in practice the energy functional itself will depend on them. We also assume that all MPS blocks to the left of $\ell$ are in the left gauge, and those on the right are in the right gauge, so that $$\label{eq:284} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell-1}}^{[\ell-1]} \cdot C_{s_{\ell}}^{[\ell]} \cdot B_{s_{\ell+1}}^{[\ell+1]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ with $A^{[\ell']}$ ($\sum_s {A^{\dagger}}_{s}^{[\ell']}A_{s}^{[\ell']} = {{\mathbb 1}}$) and $B^{[\ell']}$ ($\sum_s B_{s}^{[\ell']} {B^{\dagger}}_{s}^{[\ell']} = {{\mathbb 1}}$) fixed, and we are searching the $C^{[\ell]}$ which minimizes $\langle \Psi | H |\Psi\rangle$. It is always possible to gauge transform an MPS to achieve form by means of repeated SVD as we saw previously, the whole singular part of the decompositions has been embedded inside $C^{[\ell]}$. It is easy to calculate the explicit dependence on $C^{[\ell]}$ of the energy: by means of transfer matrices formalism one can write $$\begin{gathered} \label{eq:cinqueleoni} \langle \Psi | H |\Psi\rangle = ( \Phi^{+} | {{\mathbb E}}^{[\ell]}_{{{\mathbb 1}}} |\xi^{\ell} ) + ( \xi^{\ell-1} | {{\mathbb E}}^{[\ell]}_{{{\mathbb 1}}} |\Phi^{+} ) + \sum_q g^{[\ell]}_q ( \Phi^{+} | {{\mathbb E}}^{[\ell]}_{\Theta} | \Phi^{+} ) + \\ + \sum_p ( \Phi^{+} | {{\mathbb E}}^{[\ell]}_{\Theta'} | \chi^{[\ell]}_p ) + \sum_p h^{[\ell]}_p ( \chi^{[\ell-1]}_p | {{\mathbb E}}^{[\ell]}_{\Theta''} | \Phi^{+} ).\end{gathered}$$ Let us interpret it: the first and second term embed respectively the terms of the Hamiltonian having support to the right and left of $\ell$; the third term are those acting on $\ell$ only, and the two last terms couple $\ell$ with its neighbors. As all the terms contain just one ${{\mathbb E}}^{[\ell]}$, the functional is quadratic in the tensor $C^{[\ell]}$: $\langle \Psi | H |\Psi\rangle = \sum_{\bullet} {C^{\star}}_{m,n}^{r} \mathcal{H}^{jks}_{mnr} {C^{\star}}_{j,k}^{s}$, where the *effective Hamiltonian* $\mathcal{H}$ is hermitian. Also, we should take into account state normalization $\langle \Psi | \Psi\rangle = \sum_{\bullet} {C^{\star}}_{j,k}^{s} C_{j,k}^{s}$, which is a constraint of the problem, and therefore must be inserted in the functional with its appropriate Lagrange multiplier $\varepsilon$. In conclusion, the resulting Lagrangian reads $\mathcal{L}(C, C^{\star}) = \langle \Psi | H |\Psi\rangle - \varepsilon \langle \Psi | \Psi\rangle = \langle \!\langle C | \mathcal{H} - \varepsilon \mathcal{N}| C \rangle \! \rangle = $ $$\label{eq:lagrange_fig} \begin{overpic}[width = 300pt, unit=1pt]{Lagrangean1} \put(-20, 51){\Huge $\mathcal{L}$} \put(181, 51){\Huge $\varepsilon$} \put(80, 53){$\mathcal{H}$} \put(78, 89){$C^{[\ell]}$} \put(76, 17){${C^{\star}}^{[\ell]}$} \put(242, 89){$C^{[\ell]}$} \put(240, 17){${C^{\star}}^{[\ell]}$} \end{overpic}$$ where $| C \rangle \! \rangle$ is intended as a $d D_{\ell} D_{\ell-1} \sim d D^2$ dimensional vector, $\mathcal{H}$ and $\mathcal{N}$ as observables on such space, representing respectively the effective Hamiltonian and effective square norm. Here $\mathcal{H}$ is given by the same five terms of ; in the same order of appearance they read $$\begin{overpic}[width = \textwidth, unit=1pt]{Eff_Hamilt} \put(13, 124){\scriptsize $\mathcal{H}$} \put(113, 124){\scriptsize $\xi_{\ell-1}$} \put(291, 124){\scriptsize $\xi_{\ell}$} \put(105, 37){\scriptsize $R_1^{[\ell]}$} \put(208, 37){\scriptsize $R_2^{[\ell+1]}$} \put(333, 37){\scriptsize $R_2^{[\ell]}$} \end{overpic}$$ with $|\xi^{\ell})$, as before, obtained recursively via , and similarly $(\xi^{\ell-1}|$ arises from the left-right specular equation. You see that, thanks to the gauge condition chosen for the $A^{[\ell']}$ and $B^{[\ell']}$, the effective square norm $\mathcal{N}$ coincides with the identity operator on the $dD^2$-dimensioned effective space, so that $\mathcal{L}(C, C^{\star}) = \langle \!\langle C | \mathcal{H} - \varepsilon {{\mathbb 1}}| C \rangle \! \rangle$ as prescribed by . Finding the minimum of a quadratic Lagrangian is now straightforward since, by exploiting the fact that $C$ and $C^{\star}$ are independent for complex differential calculus, one has $$\label{eq:std_eigenvalue} \frac{\partial \mathcal{L}(C,C^{\star})}{\partial \langle \!\langle C |} = | 0 \rangle \! \rangle \quad \longrightarrow \quad \mathcal{H} | C \rangle \! \rangle = \varepsilon | C \rangle \! \rangle.$$ Therefore we have to deal with a standard eigenvalue problem for $\mathcal{H}$, and among solutions $\mathcal{H} | C \rangle \! \rangle = \varepsilon | C \rangle \! \rangle$ we have to consider the one giving minimal value of $\langle \!\langle C | \mathcal{H} | C \rangle \! \rangle = \varepsilon \langle \!\langle C | C \rangle \! \rangle = \varepsilon$; in other words, we have to find the *lowest eigenvalue* solution of the problem , and we are done. In conclusion, we mapped an eigenproblem for the whole $d^{L}$ dimensioned global system into a $dD^2$ eigenproblem for the local tensor $C^{[\ell]}$. When $D$ is a parameter chosen by the user and not dependent on $L$, solving the local problem requires the same effort for every system size. Also, the great advantage of working in the proper gauge framework is clear now, since if $\mathcal{N}$ were not to coincide with ${{\mathbb 1}}$, we would have to deal with a generalized eigenproblem ($\mathcal{H} | C \rangle \! \rangle = \varepsilon \mathcal{N} | C \rangle \! \rangle$): more expensive and less stable. It is clear that after one has found the minimal $| C_{\text{min}} \rangle \! \rangle$ for the energy of the resulting $|\Psi_2\rangle$ is necessary decreased (or equal) from the initial guess $|\Psi_1 \rangle$ $$\langle \Psi_{2} | H |\Psi_{2}\rangle = \langle \!\langle C_{\text{min}} | \mathcal{H} | C_{\text{min}} \rangle \! \rangle \leq \langle \!\langle C_{\text{guess}} | \mathcal{H} | C_{\text{guess}} \rangle \! \rangle = \langle \Psi_{1} | H |\Psi_{1}\rangle.$$ This is how the algorithm proceeds towards energy minimization, after this step we move to the MPS block to the immediate right $\ell+1$ (or the immediate left $\ell-1$) and repeat. Of course, to complete the iteration step one has to perform the proper gauge transformation so that, after finding $C_{\text{min}}^{[\ell]}$, turns it into left gauge $\rightarrow A^{[\ell]}$ (or right gauge if we are sweeping left) so that immediately holds for site $\ell+1$. But this is easy, just perform a singular value decomposition of $C^{[\ell]s}_{j,k}$ $$C^{[\ell]}_{\alpha,k} = \sum_{\beta} A^{[\ell]}_{\alpha,\beta} \;\lambda_{\beta}\; U_{\beta,k}$$ with $\alpha$ being the composite index $\{j,s\}$, and $A^{\dagger} A = U U^{\dagger} = {{\mathbb 1}}$. The gauge transformation is then $$\begin{aligned} C^{[\ell]s}_{j,k} \quad &\longrightarrow \quad & C^{[\ell]s}_{j,\beta} \;U^{\dagger}_{\beta,k} \;\lambda^{-1}_{k} \quad &= A^{[\ell]s}_{j,k}\\ A^{[\ell+1]s}_{j,k} \quad & \longrightarrow \quad& \lambda_{j} \;U_{j,\beta} \;A^{[\ell+1]s}_{\beta,k} \quad & = C^{[\ell+1]s}_{j,k}, \end{aligned}$$ which concludes the iteration step. The algorithm is usually carried on until some convergence threshold in the energy $\varepsilon$ has been achieved. In most cases this simulation procedure converges surprisingly fast, as very few sweeps are necessary to reach a stable minimum, even if we were to start from a completely random variational MPS. In several simulations where the single-site framework was adopted, computational results are in good agreement with theory and/or experiment, yet this protocol presents some difficulties. The monotonicity of the energy functional at every iteration step, even if it allows fast convergence, hides the possibility of getting stuck in local minima of the variational parameters landscape: in order to work around this issue one has to insert manually artificial fluctuations, as proposed by S. White in DMRG context [@WhiteOneSite]. Similarly, the algorithm encounters trouble when dealing with symmetries (see appendix \[app:symchap\]), where the user is forced to insert symmetry-breaking fluctuations by hand. Despite on how we can solve, with big or small success, these issues in the single-block framework, a very common way to work around them is to recover the original idea that gave birth to DMRG, i.e. dealing with a two-site block minimization at once. ### Double variational site This time we want to variate two adjacent blocks at the same time, say $C^{[\ell]}$ and $C^{[\ell+1]}$, while keeping fixed the other ones. The most clever way to do this is forgetting that $C^{[\ell]}$ and $C^{[\ell+1]}$ are two distinct MPS blocks: we consider them as a single overall tensor $M^{s_{\ell}, s_{\ell+1}}_{j,k} = C^{[\ell]s_{\ell}}_{j,\alpha} \; C^{[\ell]s_{\ell+1}}_{\alpha,k} $ on which the Lagrangian functional is quadratic, and adopt $M$ as our only variational element. Of course, this allows us to momentarily describe *more* entanglement across the bond $\{\ell,\ell+1\}$ than what would be normally allowed by a $D$-bondlink MPS. Therefore, to provide an iterative scheme, we will embed in the algorithm a method for entanglement truncation, so that in the end the original MPS representation is recovered. Then, let us start from our initial guess for the iteration step $$\label{eq:Dublock} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell-1}}^{[\ell-1]} \cdot M_{s_{\ell}, s_{\ell+1}} \cdot B_{s_{\ell+2}}^{[\ell+2]} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle,$$ where every $M_{s_{\ell}, s_{\ell+1}}$ is given by the matrix product $C_{s_{\ell}}^{[\ell]} \cdot C_{s_{\ell+1}}^{[\ell+1]}$ and MPS blocks $A_{s_{\ell'}}^{[\ell']}$ (resp. $B_{s_{\ell'}}^{[\ell']}$) are in the left (right) gauge. As a whole, $M$ is a tensor with four indices, and notice that alongside the sudden increase of allowed entanglement (from $\log D$ to $\log dD$), an increase of variational parameters, with respect to the standard MPS case, comes out: from $2dD^2$ to $d^2 D^2$. As we mentioned the Lagrangian is quadratic in $M$, and thanks to gauge relations for $A$ and $B$ matrices, the effective normalization $\mathcal{N}$ is again the identity operator, since $\langle \Psi | \Psi \rangle = \sum_{j k s r} |M^{s,r}_{j,k}|^2$. Then $$\mathcal{L}(M,M^{\star}) = \langle \!\langle M | \mathcal{H} | M \rangle \! \rangle - \varepsilon \langle \!\langle M | M \rangle \! \rangle,$$ where the effective Hamiltonian $\mathcal{H}$ is given by $$\begin{overpic}[width = \textwidth, unit=1pt]{Eff_Ham2} \put(15, 190){\scriptsize $\mathcal{H}$} \put(128, 190){\scriptsize $\xi_{\ell-1}$} \put(343, 190){\scriptsize $\xi_{\ell+1}$} \put(171, 113){\scriptsize $R_1^{[\ell]}$} \put(296, 113){\scriptsize $R_2^{[\ell+1]}$} \put(184, 37){\scriptsize $R_2^{[\ell+2]}$} \put(310, 37){\scriptsize $R_2^{[\ell]}$} \end{overpic}$$ Graphs 1 and 2 contain the terms of the Hamiltonian having support outside $\{\ell, \ell+1\}$; graphs 3,4 and 5 those of support inside $\{\ell, \ell+1\}$, and the last two represent interaction of the inner sites with the environment. Like for the single site case, the optimal $M$ is found via lowest eigenvalue problem, i.e. smallest $\varepsilon$ allowing $$\mathcal{H} | M \rangle \! \rangle = \varepsilon | M \rangle \! \rangle.$$ Now we have to manipulate the newly found $M$ in order to recover the standard MPS form. To do this we proceed again via singular value decomposition. First we write the tensor $M$ as a matrix $M_{\alpha \beta}$ where the composite index $\alpha \sim \{j,s_{\ell}\}$ refer to the left bondlink index $j$ and the left physical index $s_{\ell}$, while $\beta \sim \{k,s_{\ell+1}\}$ to the right bondlink $k$ and site $s_{\ell+1}$ indices. Now we calculate the SVD as $$M_{\alpha \beta} = \sum_{\gamma}^{dD} A^{[\ell]}_{\alpha \gamma} \; \lambda_{\gamma}\; B^{[\ell+1]}_{\gamma \beta} = \sum_{\gamma}^{dD} A^{[\ell]s_{\ell}}_{j \gamma} \; \lambda_{\gamma}\; B^{[\ell+1] s_{\ell+1}}_{\gamma k}$$ so we can stick it into and obtain again the MPS representation. Still, the bondlink $\ell$ has increased dimension $D_{\ell}^{\text{new}} = \min\{d D_{\ell-1}, d D_{\ell + 1}\} \sim dD$. But now the positive values $\lambda_{\gamma}$ are the Schmidt coefficients of the partition at bond $\ell$ of the state $|\Psi\rangle$, and $\langle \Psi | \Psi \rangle = \sum_{\gamma}^{dD} \lambda_{\gamma}^2$. Therefore, in order to recover the best approximation of this state allowing $\log D$ entanglement, we must truncate the *smallest* Schmidt coefficients $\lambda_{\gamma}$, until only $D$ of them remain. Namely, if $\lambda_{\gamma}$ were sorted in decreasing order ($\lambda_{\gamma} \geq \lambda_{\gamma+1}$), we keep only the first $D$ of them, and renormalize to preserve state norm $$\label{eq:tronky} {\tilde{\lambda}}_{\gamma} = \frac{\lambda_{\gamma}}{\sqrt{\sum_{\eta}^{D} \lambda^2_{\eta}}} \quad \gamma \in \{1..D \} \qquad \longrightarrow \qquad \sum_{\gamma}^{D} {\tilde{\lambda}}_{\gamma}^2 = 1;$$ moreover, $A^{[\ell]}$ and $B^{[\ell]}$ will satisfy respectively the left and right gauge condition even after the truncation. We can now write, assuming we are sweeping towards right, $M_{s_{\ell}, s_{\ell+1}} = A^{[\ell]}_{s_\ell} \cdot C'^{[\ell+1]}_{s_\ell+1}$, where $C'^{[\ell+1] s_\ell+1}_{\gamma, k} = {\tilde{\lambda}}_{\gamma} B^{[\ell+1] s_\ell+1}_{\gamma, k}$ so that $$|\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d \left( A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{\ell-1}}^{[\ell-1]} \cdot A_{s_{\ell}}^{[\ell]} \cdot \overbrace{C_{s_{\ell+1}}^{[\ell+1]} \cdot B_{s_{\ell+2}}^{[\ell+2]}} \cdot \ldots \cdot B_{s_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle.$$ The MPS representation is now ready to perform the next algorithm iteration, just identify the new two-site block $M_{s_{\ell+1},s_{\ell+2}} = C_{s_{\ell+1}}^{[\ell+1]} \cdot B_{s_{\ell+2}}^{[\ell+2]}$. The double-site based algorithm we just described presents two important improvements with respect to the single-site one. First, as two adjacent blocks are being modified at the same time, reconstructing the correct short-range physics runs much faster, and since the Hamiltonian is made of nearest neighboring terms the energy is extremely sensitive to the n-n physics (especially for non-critical systems) thus leading to a faster minimization convergence. Secondly, at the time we perform the truncation , we allow for errors in our description, as we force the state to carry no more entanglement than the MPS representation allows. Therefore, slight fluctuations appear, identified by eventual small increases in energy. This is actually an advantage of the protocol, as fluctuations are a natural way to discourage the algorithm from getting stuck in local minima of the energy landscape. Matrix Product Operators {#sec:MPO} ------------------------ So far, we applied the Matrix Product formalism in order to build multi-indexed tensors $\mathcal{T}_{s_1 \ldots s_L}$, which were components of a target state $|\Psi\rangle$ over a separable (canonical) vector basis $|s_1 \ldots s_L\rangle$; but it is clear that its capabilities extend to *every* algebraic construct which can be expanded in a basis of local elements, regardless their nature. No doubt, applying the Matrix Product concept to describe nonlocal operators seems the most natural goal to pursue. Matrix Product Operators (MPO) were firstly introduced in [@MPDO] and used, for instance, to describe either thermal mixed states, time evolution paradigms for MPS [@MPOMurg] [@Maricarmen] [@Sebadrag], or long range interaction Hamiltonians [@MPOFrowis]. Their open boundary formulation is definitely similar to that of MPS: $$\label{eq:MPO} \hat{\Theta} = \sum_{s_1 \ldots s_L}^d \sum_{r_1 \ldots r_L}^d \left( O_{s_{1}}^{[1]r_{1}} \cdot \ldots \cdot O_{s_{L}}^{[L]r_{L}} \right) | s_1 \ldots s_L \rangle \langle r_1 \ldots r_L |,$$ where, if $|s_{\ell}\rangle$ is the canonical vector basis on site $\ell$, then $|s_{\ell}\rangle \langle r_{\ell}|$ is the canonical local operator basis. The local-basis expansion is then performed over these canonical elements, while adopting Matrix Product-based coefficients. To every site $\ell$, incoming physical index $r_{\ell}$, and outcoming physical index $s_{\ell}$ we associated a $D_{\ell-1} \times D_{\ell}$ matrix $O_{s_{\ell}}^{[\ell]r_{\ell}}$, which sums altogether to a four-indices tensor on every site: $$\begin{overpic}[width = 350pt, unit=1pt]{OBCMPO} \put(20, 40){$O^{[1]}$} \put(81, 40){$O^{[2]}$} \put(142, 40){$O^{[3]}$} \put(203, 40){$O^{[4]}$} \put(316, 40){$O^{[L]}$} \end{overpic}$$ As you can see, blocks $1$ and $L$ of the MPO have only one correlation space link index to be consistent with the OBC setting. As for MPS, Matrix Product Operators are typically prescribed according to a maximal bondlink dimension $D$ ($D_\ell < D$ $\forall \ell$, with $D$ non-scaling with $L$) which makes the expression manageable for practical purposes even for large system sizes $L$. Such $D$ also poses a limit in the entangling capabilities of $\hat{\Theta}$, actually binding the amount of long-range correlation the operator can create. MPOs are outstanding tools when the goal is to apply a transformation $\hat{\Theta}$ to a state $|\Psi\rangle$ whose MPS representation is available. In fact, the resulting state $\hat{\Theta} |\Psi\rangle$ is automatically expressed in Matrix Product form: $$\begin{gathered} \label{eq:centouno} \hat{\Theta} |\Psi\rangle = \sum_{\{s_{\ell}\} \{r_{\ell}\} \{t_{\ell}\}} \left( \vphantom{A_{t_{1}}^{[1]}} O_{s_{1}}^{[1]r_{1}} \cdot \ldots \cdot O_{s_{L}}^{[L]r_{L}} \right) \left( A_{t_{1}}^{[1]} \cdot \ldots \cdot A_{t_{L}}^{[L]} \right) \times \\ \times | s_1 \ldots s_L \rangle \langle r_1 \ldots r_L | t_1 \ldots t_\ell \rangle = \sum_{s_1 \ldots s_L} \left( B_{S_{1}}^{[1]} \cdot \ldots \cdot B_{t_{L}}^{[L]} \right) | s_1 \ldots s_L \rangle\end{gathered}$$ where $$\label{eq:centodue} B_{s}^{[\ell]} = \sum_{r} (A_{s}^{[\ell]} \otimes O_{s}^{[\ell]r}).$$ [MPOS]{} (27, 46)[$O^{[1]}$]{} (27, 107)[$A^{[1]}$]{} (39, 75)[$B^{[1]}$]{} (104, 107)[$A^{[2]}$]{} (104, 46)[$\ldots$]{} Truly, the bondlink dimension of the target MPS $\hat{\Theta} |\Psi\rangle$ is increased to $D' = D_A \cdot D_O$, the product of the original MPS bond $D_{A}$ and that of the MPO $D_{O}$. So, it looks that application of MPO to MPS is definitely expensive in terms of the bondlink. This is true, and nevertheless easy to work around: it is sufficient to reduce the target MPS to the desired $D''$ properly. This is quickly done by following the usual steps: - Choose a bond, say $\{\ell,\ell+1\}$. - Gauge transform the MPS so that blocks to the left (resp. right) of the chosen bond are in left (right) gauge, so that Schmidt coefficients of the partition emerge explicitly - Truncate the smallest Schmidt coefficients and renormalize to one the remaining ones (squared), as . - choose another bond and repeat, until every bondlink has been renormalized to $D''$ or less. The error we intake when renormalizing the state is compatible with the amount of entanglement we are discarding (which is explicitly known by comparing Von-Neumann entropies before and after truncation). Being able to transform MPS into MPS becomes fundamental, for instance, if we want to describe a time-evolution of a system whose starting point is a finitely correlated state: within this paradigm it is very useful to understand how to write an MPO representation of a given Hamiltonian, and how to exponentiate it efficiently. This is a major point of interest of ref. [@MPOMurg]. ### Matrix Product Density Operators A relevant class of operators we are typically interested in, is the family of density matrices, i.e. positive, unity trace, operators. Although it is instructive and useful decomposing such operators into MPO form, it is even more interesting to exploit their positivity (as well as positivity of any partial trace, i.e. degree of freedom reduction) to further decompose their Matrix Product structure. Indeed, if we consider that any $\rho \geq 0$ can be written as $\rho = X X^{\dagger}$ (and, conversely, the whole space of matrices $X$ generate the class of positive operators via $X X^{\dagger}$) one is encouraged to build the MPO decomposition of $X$ rather than $\rho$ itself. Doing so not only eliminates the positivity restraint on the resulting MPO, but also gives us an edge for dealing with state transformations, as the application $T \rho T^{\dagger}$ simplifies into $TX$, which could be a nontrivial numerical improvement. Precisely, in ref. [@MPDO] the Matrix Product Density Operators (MPDO) are properly defined. They are those MPO, according to , whose blocks $O^{[\ell]}$ are given by $$\label{eq:MPDO} O^{[\ell]r}_{s} = \sum_{\tau = 1}^{\tilde{d}_{\ell}} {A^{\star}}^{[\ell]r}_{\tau} \otimes {A}^{[\ell]\tau}_{s},$$ where $\tilde{d}_{\ell}$ is at most $d D_{\ell-1} D_{\ell}$. Decomposition is actually splitting the matrix product layer into two sub-layers stacked together, as $$\label{eq:MPDOfig} \begin{overpic}[width = 350pt, unit=1pt]{MPDO} \put(19, 37){$A^{[1]}$} \put(16, 104){${A^{\star}}^{[1]}$} \put(81, 37){$A^{[2]}$} \put(78, 104){${A^{\star}}^{[2]}$} \put(142, 37){$A^{[3]}$} \put(139, 104){${A^{\star}}^{[3]}$} \put(203, 37){$A^{[4]}$} \put(200, 104){${A^{\star}}^{[4]}$} \put(316, 37){$A^{[L]}$} \put(313, 104){${A^{\star}}^{[L]}$} \end{overpic}$$ where the pentagonal shape of the tensors in the diagram specifies that tensors in the upper layer are up-down specular to those in the lower layer (plus complex conjugation). If we were to give an interpretation to the tensorial index dimension $\tilde{d}_{\ell}$ we could invoke again the valence bond picture: indeed $\log \tilde{d}_{\ell}$ is the maximal allowed entanglement that the system can share with an external degree of freedom coupling expressly to site $\ell$, e.g. a local thermal bath. MPDO are useful tools for addressing one-dimensional open systems, especially where the mixing with external media acts on the bulk itself. It is even possible to formulate master equation problems with matrix product formalism. Example: exact Matrix Product State representation for Slater Determinants {#sec:Slater} -------------------------------------------------------------------------- We would like to conclude this chapter with a simple, yet practical example of the Matrix Product formalism applied analytically to a specific class of many body states: Slater Determinants. In fermionic problems, Slater Determinants are the starting point of most many-body calculations (like Hartree-Fock); they are states where $N$ fermions share no quantum correlations, each one of them filling an orbital which is typically solution of the mean-field Hamiltonian. Nevertheless, since such orbitals are non-necessarily localized in a chosen configuration space, they can still manifest self-correlation entanglement in the separable basis. So it is probably the simplest among non-trivial matrix product decomposition problems. The following construction is somehow related to ref. [@MPOFrowis], but I developed it during my Philosophiae Doctorateship as an independent project, supported by G. Santoro and V. Giovannetti. Let us deal with spinless fermions, for simplicity: the first step we have to perform is to match the physics of this context with the algebraic formulation adopted so far in this chapter. To do this, we can completely forget about physical dimensionality of the problem and boundary conditions: the only initial structure we need is a complete set of $L$ one-body wavefunctions. We also choose a complete ordering for these. They will represent the sites in our 1D OBC (spin) system, placed according to the chosen ordering; the canonical local basis corresponding to $|0\rangle =$ empty level and $|1\rangle =$ filled level. The mapping of a fermionic system into a spin system is made via standard Wigner transformation $$\label{eq:Wigner1} \begin{aligned} | \Omega \rangle \quad &\longrightarrow \quad | 00 \ldots 0 \rangle \\ c_{\ell} \quad &\longrightarrow \quad \sigma^{z}_{1} \otimes \ldots \otimes \sigma^{z}_{\ell-1} \otimes \sigma^{-}_{\ell} \otimes {{\mathbb 1}}\otimes \ldots \otimes {{\mathbb 1}}, \end{aligned}$$ where $| \Omega \rangle$ is the vacuum state, $c_{\ell}$ the destruction operator on level $\ell$, and $\sigma$ being Pauli matrices. Now any state $|\Psi\rangle$ can be expanded in such product basis $|\Psi\rangle = \sum_{s_1 \ldots s_L} \mathcal{T}_{s_1 \ldots s_L} |s_1 \ldots s_L\rangle$, and we are going to apply the matrix product formalism to the components tensor $\mathcal{T}_{s_1 \ldots s_L}$, local dimension $d = 2$. For sake of completeness, let us even write the explicit MPS expansion in the second quantization formalism, $$\label{eq:2quantMPS} |\Psi\rangle_{\text{Fermi}} = \sum_{s_1 \ldots s_L = 1}^2 \left( \vphantom{\sum} A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{L}}^{[L]} \right) (c_{1}^{\dagger})^{s_1} \ldots (c_{L}^{\dagger})^{s_L} |\Omega\rangle.$$ where the construction operators $c_{\ell}^{\dagger}$ are placed in the correct order, and obviously $(c_{\ell}^{\dagger})^{0} = {{\mathbb 1}}$. Simple as that. A Slater determinant state $|\Sigma\rangle$ is defined as follows $$\label{eq:Slaz} |\Sigma\rangle = \tilde{c}_1^{\dagger}\,\tilde{c}_2^{\dagger}\,\ldots \tilde{c}_{N}^{\dagger}\,| \Omega \rangle$$ where $\tilde{c}_{\alpha}^{\dagger}$ fills a one-body orbital which may have a nontrivial expansion over the original one-body levels we chose as basis. Precisely the transformation is given by $$\tilde{c}_{\alpha} = \sum_{\ell = 1}^{L} \phi^{\star}_{\alpha}(\ell)\; c_{\ell},$$ $\phi^{\star}_{\alpha}(\ell)$ being the first-quantization decomposition of the Slater orbital $\alpha$ onto the original wavefunction basis $\ell$. Orthogonality is required among $\alpha$ orbitals, i.e. $\sum_{\ell} \phi_{\alpha}(\ell) \phi^{\star}_{\beta}(\ell) = \delta_{\alpha, \beta}$. As we know from literature, such considerations lead us to write $|\Sigma\rangle$ in its explicit determinant form $$\label{eq:slasla} |\Sigma\rangle = \sum_{0 \leq \ell_1 < \ldots < \ell_{N} \leq L} \left| \begin{array}{cccc} \phi_{1}(\ell_1) & \phi_{1}(\ell_2) & \cdots & \phi_{1}(\ell_N) \\ \phi_{2}(\ell_1) & \phi_{2}(\ell_2) & \cdots & \phi_{2}(\ell_N) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_{N}(\ell_1) & \phi_{N}(\ell_2) & \cdots & \phi_{N}(\ell_N) \\ \end{array} \right| c_{\ell_{1}}^{\dagger} \ldots c_{\ell_{N}}^{\dagger} |\Omega\rangle.$$ To find a MPS representation for $|\Sigma\rangle$, will be instrumental to give a Matrix Product Operator description for Fermi operators $\tilde{c}_{\alpha}$ over delocalized orbitals. Since the vacuum is already in (trivial) MPS form $|0\ldots 0\rangle$ we will then find the MPS structure of $|\Sigma\rangle$ by stacking together MPOs of $\tilde{c}_{\alpha}^{\dagger}$ according to , as we did in and ### MPO for delocalized Fermi operators We are now going to provide a MPO representation of $\tilde{c}_{\alpha}^{\dagger}$ which is compact, elegant, and very general. The only ingredient we need is knowing the expansion of the orbital $\alpha$ in the original one-body wavefunctions basis $\phi_{\alpha}(\ell)$. Then the goal is finding the $B^{[\ell]}$ satisfying $$\label{eq:CreatMPO} \begin{aligned} \tilde{c}_{\alpha}^{\dagger} &\longrightarrow \sum_{s_1 \ldots s_L = 0}^1 \sum_{r_1 \ldots r_L = 0}^1 ( b_0 | \vphantom{\sum} B_{s_{1}}^{[1]r_1} \cdot \ldots \cdot B_{s_{L}}^{[L]r_L} | b_L)\; |s_1 \ldots s_L\rangle \langle r_1 \ldots r_L |\\ &= \sum_{\{s\},\{r\} = 0}^{1} ( b_0 | \vphantom{\sum} B_{s_{1}}^{[1]r_1} \cdot \ldots \cdot B_{s_{L}}^{[L]r_L} | b_L)\; (c_{1}^{\dagger})^{s_1} \ldots (c_{L}^{\dagger})^{s_L} |\Omega\rangle \langle \Omega | \,c_{L}^{r_L} \ldots c_{1}^{r_1}, \end{aligned}$$ where we explicitly set vector boundaries to the matrix product expression (actually $| b_L)$ is a vector and $(b_0|$ a functional) so that we will able to define every $B_{s}^{[\ell]r}$ homogeneously, even those at the furthest sites. In particular we need $D = 2$, and the solution we found is given by $$\label{eq:FermiMPOrecipe} \begin{array}{ccc} {B^{[\ell]}}_{0}^{0} = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) = {{\mathbb 1}}& \qquad & {B^{[\ell]}}_{0}^{1} = \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) = 0 \\ \\ {B^{[\ell]}}_{1}^{0} = \left( \begin{array}{cc} 0 & 0 \\ \phi_{\alpha}(\ell) & 0 \end{array} \right) = \phi_{\alpha}(\ell) \, \sigma^{-} && {B^{[\ell]}}_{1}^{1} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) = \sigma^{z} \\ \\ | b_L ) = \left( \begin{array}{cc} 1 \\ 0 \end{array} \right) = |0) && ( b_0 | = \left( \vphantom{{B^{[\ell]}}_{1}} 0\;\;\;1 \right) = (1|, \end{array}$$ where the information on $\phi_{\alpha}(\ell)$ is used on only one of the 16 elements of the four-indices tensor $B^{[\ell]}$. Apart from that the expression is formally homogeneous in $\ell$, as we wanted. To show that reproduces the correct action of $\tilde{c}_{\alpha}^{\dagger}$ we first notice that when performing the matrix product contraction, the terms which contain one and only one $\sigma^{-}$ are the ones that survive: indeed $(1|0) = (1|\sigma^{z}|0) = 0$, while $(1|\sigma^{-}|0) = 1$, but on the other hand $(\sigma^{-})^2 = 0$. So we can reduce the expression in a simple sum over the site $\ell$ upon which the $\sigma^{-}$ is activated, becoming $$\sum_{\ell} \phi_{\alpha}(\ell) \left[ \sigma_{1}^{z} \otimes \ldots \otimes \sigma_{\ell-1}^{z} \otimes \sigma_{\ell}^{-} \otimes {{\mathbb 1}}_{\ell+1} \otimes {{\mathbb 1}}_{L} \right] \longrightarrow \sum_{\ell} \phi_{\alpha}(\ell)\, c_{\ell}^{\dagger} = \,\tilde{c}_{\alpha}^{\dagger}$$ which proves the equivalence. As an additional remark, it is easy to see that it is possible to *deactivate* the global action of such MPO by just changing a correlation boundary state, say $(b_0|$. Precisely, if we were to set $(b_0| = (0|$ instead of $(1|$, the MPO expression would coincide with the identity ${{\mathbb 1}}$ instead of $\tilde{c}_{\alpha}^{\dagger}$; one can then regard the correlation space boundaries as local switches that control the whole matrix product behavior, even if it is not localized. ### MPO stack to MPS We can now go back to the Slater determinant state $|\Sigma\rangle = \tilde{c}_1^{\dagger}\,\tilde{c}_2^{\dagger}\,\ldots \tilde{c}_{N}^{\dagger}\,| \Omega \rangle$ and adopt the engineering we learned to define its whole MPS exact representation . In particular we start from the vacuum state $|00\ldots 0\rangle$, which is trivially an MPS with $D=0$, and we apply $\tilde{c}_{N}^{\dagger}$ to obtain again an MPS. Then we refrain, by applying in order $\tilde{c}_{N-1}^{\dagger}$, $\tilde{c}_{N-2}^{\dagger} \ldots$ and so forth, up to $\tilde{c}_{1}^{\dagger}$: every step is performed following the prescription of eq.  (although we should never renormalize if we want our description to be exact). In conclusion we have $$\label{eq:Slamprecipe} A^{[\ell]} = \sum_{q_1 \ldots q_N = 0}^{1} \left( {B^{[\ell,N]}}_{q_N}^{0} \otimes \ldots \otimes {B^{[\ell,2]}}_{q_2}^{q_3} \otimes {B^{[\ell,1]}}_{q_1}^{q_2} \right).$$ As you see, our description uses as a whole a total bondlink dimension of $D = 2^{N}$, regardless from $L$. Actually, since every $B^{1}_{0}$ is the null operator, we can also restrict the previous sum to $q_1 \geq q_2 \geq \ldots \geq q_N$, since every term for which any $q_k < q_{k+t}$, with $t>0$, would give zero contribution. In the end it is a sum of merely $N$ terms. Similarly, we define the correlation boundary vectors: $$|b_{L}) = |0)^{\otimes N} = \left( \begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array} \right), \quad \text{and} \quad (b_0| = (1|^{\otimes N} = \left( 0 \; \cdots\; 0 \; 1 \right).$$ Putting these ingredients together leads us to the decomposition of our Slater Determinant $\Sigma$ in the MPS representation, where explicit boundaries of the matrix product expression are present $$\label{eq:Slamp} |\Sigma\rangle = \sum_{s_1 \ldots s_L = 1}^2 (b_0| A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{L}}^{[L]} |b_L)\; (c_{1}^{\dagger})^{s_1} \ldots (c_{L}^{\dagger})^{s_L} |\Omega\rangle.$$ Let us briefly analyze the matrices we built via . It is easy to see that $A^{[\ell]}_{0}$ it is always the identity $A^{[\ell]}_{0} = {{\mathbb 1}}_{D \times D}$, while $A^{[\ell]}_{1}$ contains the information upon orbitals, expanded in the original wavefunctions. To make this clear, we show as an example the cases $N = 2$, for which it holds $A^{[\ell]}_{1} = \phi_1(\ell) [{{\mathbb 1}}\otimes \sigma^{-}] + \phi_2(\ell) [\sigma^- \otimes \sigma^z ]$, i.e. $$N = 2 \quad \longrightarrow \quad A^{[\ell]}_{1} = \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \\ \phi_1(\ell) & 0 & 0 & 0 \\ \phi_2(\ell) & 0 & 0 & 0 \\ 0 & -\phi_2(\ell) & \phi_1(\ell) & 0 \\ \end{array} \right).$$ It is clear that, the only matrix products that lead to nonzero amplitude are those where two excitations $|1\rangle$ are present, Thus the sum reduces to $$|\Sigma_2\rangle = \sum_{\ell_1 < \ell_2} \left( \phi_1(\ell_1)\, \phi_2(\ell_2) - \phi_1(\ell_2)\, \phi_2(\ell_1) \vphantom{\sum} \right) (c_{1}^{\dagger})^{\ell_1} (c_{1}^{\dagger})^{\ell_2} |\Omega\rangle,$$ where we have recovered explicitly the determinant expression. Also, let us write down the case with three orbitals to be filled $N = 3$, in this scenario we and up with $$A_1^{[r]} = \mbox{\begin{footnotesize} $ \left( \begin{array}{cccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \phi_1(r) & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \phi_2(r) & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\phi_2(r) & \phi_1(r) & 0 & 0 & 0 & 0 & 0\\ \phi_3(r) & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\phi_3(r) & 0 & 0 & \phi_1(r) & 0 & 0 & 0\\ 0 & 0 & -\phi_3(r) & 0 & \phi_2(r) & 0 & 0 & 0\\ 0 & 0 & 0 & \phi_3(r) & 0 & -\phi_2(r) & \phi_1(r) & 0 \end{array} \right), $ \end{footnotesize}}$$ the reader is invited to check that the resulting amplitudes are correct. ### Efficiency of the description We want now argument that, if no further information upon the orbitals $\phi_{\alpha}(\ell)$ being filled is exploited, the exact representation we just gave is the most efficient in terms of MPS. By this statement we mean that we are spending the smallest bondlink dimension needed to faithfully reproduce the correct amount of correlation the state can manifest. From section \[sec:valencebond\], we know that a $D$-dimensioned bondlink MPS allows up to $\mathcal{S} \leq \log_2 D$ entanglement, i.e. Von Neumann entropy of a partition (the logarithm base of 2 is chosen as common ground in quantum information theory), thus the optimal is $D = 2^{\mathcal{S}}$. Now, inequality $\mathcal{S} \leq N$ is guaranteed by the existence of an exact MPS representation . But if equality $\mathcal{S} = N$ can be achieved for some choice of $\phi_{\alpha}(\ell)$, we also proved representation optimality. To obtain it, we just adopt a special set of (doubly-periodic) plane-waves $\phi_{\alpha}(\ell) = \exp(4\pi i \alpha \ell / L)$. For simplicity let us perform a half-system partition, and define a new double set of orbitals $\{\phi^{[L]}_{\alpha}(\ell), \phi^{[R]}_{\alpha}(\ell)\}_{\alpha}$ from the previous ones as $$\label{eq:planesplit} \begin{aligned} \phi^{[L]}_{\alpha}(\ell) &= \sqrt{2}\; \Theta(L/2 - \ell) \phi_{\alpha}(\ell) \quad \mbox{and}\\ \phi^{[R]}_{\alpha}(\ell) &= \sqrt{2}\; \Theta(\ell - L/2) \phi_{\alpha}(\ell), \end{aligned}$$ with $\Theta$ being the Heaviside step function. Even though in a general case a new set of wavefunctions generated via would no longer be orthonormal, it is clear that with the specific choice of $L/2$ periodic plane waves, orthonormality is preserved: $\sum_{\ell} \phi^{[L]}_{\alpha}(\ell) {\phi^{\star}}^{[R]}_{\beta}(\ell) = 0$, as the supports are disjoint, and $\sum_{\ell} \phi^{[L]}_{\alpha}(\ell) {\phi^{\star}}^{[L]}_{\beta}(\ell) = \sum_{\ell} \phi^{[R]}_{\alpha}(\ell) {\phi^{\star}}^{[R]}_{\beta}(\ell) = \delta_{\alpha, \beta}$. So we can define Fermi operators corresponding to this new set, satisfying the anticommutation rules $ \{ \tilde{c}_{\alpha, \bullet}, \tilde{c}_{\beta, \bullet} \} = \{ \tilde{c}_{\alpha, L}, \tilde{c}_{\beta, R}^{\dagger} \} = 0$ and $ \{ \tilde{c}_{\alpha, L}, \tilde{c}_{\beta, L}^{\dagger} \} = \{ \tilde{c}_{\alpha, R}, \tilde{c}_{\beta, R}^{\dagger} \} = \delta_{\alpha, \beta}$. It is clear that the original $\tilde{c}$ decompose in the new ones as $\tilde{c}_{\alpha} = 2^{-1/2} (\tilde{c}_{\alpha,L} + \tilde{c}_{\alpha,R})$, thus letting us write the whole Slater determinant state as: $$|\Sigma\rangle = \frac{1}{2^{N/2}} \left( \tilde{c}^{\dagger}_{1,L} + \tilde{c}^{\dagger}_{1,R} \right) \ldots \left( \tilde{c}^{\dagger}_{N,L} + \tilde{c}^{\dagger}_{N,R} \right) |\Omega\rangle,$$ Of this state, we want to calculate the density matrix reduced to half the system, say the left one, so we trace over the right-half degrees of freedom $\rho_{L}^{\Sigma} = {\text{Tr}}_{R}\left[ |\Sigma\rangle\langle\Sigma|\right]$. With this goal, we set $|\Sigma'\rangle = \tilde{c}_{1} |\Sigma\rangle$ and consider: $$\begin{aligned} \rho^{\Sigma}_L &= \frac{1}{2}\; {\text{Tr}}_{R} \left[ ( \tilde{c}^{\dagger}_{1,L} + \tilde{c}^{\dagger}_{1,R} ) \;|\Sigma'\rangle \langle \Sigma'|\; ( \tilde{c}_{1,L} + \tilde{c}_{1,R} ) \right] \\ &= \frac{1}{2} \left( \tilde{c}^{\dagger}_{1,L} {\text{Tr}}_{R} \left[ |\Sigma'\rangle \langle \Sigma'|\right] \tilde{c}_{1,L} + {\text{Tr}}_{R} \left[ |\Sigma'\rangle \langle \Sigma'| \right] \right) \\ &= \frac{1}{2} \;{{\mathbb 1}}_{2 \times 2} \otimes {\text{Tr}}_{R} \left[ |\Sigma'\rangle \langle \Sigma'| \right] = \frac{{{\mathbb 1}}_{2 \times 2}}{2} \otimes \rho^{\Sigma'}_L, \end{aligned}$$ where we used the cyclicity of the trace over right support operators $\tilde{c}_{\alpha,R}$, and clearly $\tilde{c}_{1,R} |\Sigma'\rangle = 0$; then we noticed that $\rho^{\Sigma'}_L$ and $(\tilde{c}^{\dagger}_{1,L} \,\rho^{\Sigma'}_L\, \tilde{c}_{1,L})$ have orthogonal supports. Now we repeat the same argument on $|\Sigma'\rangle$, and proceed by induction. In conclusion, we can claim that $\rho^{\Sigma}_L$ is (isometrically equivalent to) $2^{-N} {{\mathbb 1}}_{2^{N} \times 2^{N}}$, the maximally mixed state on a $2^{N}$ dimensioned space, whose Von Neumann entropy is just $N$. This concludes the proof. An intuitive, but not naive, interpretation of such result can be given in the following terms. As fermions occupying the various orbitals must be mutually uncorrelated due to the Slater determinant state nature, the only possible entanglement the system can manifest under a real-space partition is given by the self-correlation of every orbital, separately accounted. In fact, in the studied case, we presented $N$ uncorrelated completely delocalized orbitals, each one carrying the entanglement of a unit (i.e. the amount of entanglement shared by a spin singlet), so $N$ is naturally the total amount. #### OVERALL REMARKS {#overall-remarks .unnumbered} - The $\sigma^{z}$ matrix in equation is the one and only responsible for establishing the correct anticommutation relations of Fermi statistics. That said, it is straightforward to modify so that the corresponding MPO is describing a *Bose operator* instead: you just need to replace ${B^{[\ell]}}_{1}^{1} = {{\mathbb 1}}$ and leave the rest unchanged (also extensible to abelian anyons by using phase gates ${B^{[\ell]}}_{1}^{1} = e^{i \varphi \sigma^{z}}$). - We mentioned that the present design is modeled on spinless fermions, but actually is naturally extensible to fermion with spins. The only difference is that at the very beginning, when we are selecting a complete basis of orbitals, we need to specify a complete basis of spin-orbitals instead, and then choose a complete ordering. Any ordering is fine and does not compromise the MPS cost in terms of $D$ as long as the particles are uncorrelated and fixed in number. ### Tensor grid representation of one-body\ wavefunction basis change Let us recall that, when we derived the MPO representation for $\tilde{c}_{\alpha}^{\dagger}$, we also mentioned that it is possible to control its overall action by adjusting the left correlation boundary vector $(b_0|$: namely the MPO coincides with $\tilde{c}_{\alpha}^{\dagger}$ if $(b_0|=(1|$, while it is just the identity ${{\mathbb 1}}$ for $(b_1| = (0|$. In other words $$\sum_{\{s\},\{r\} = 0}^{1} ( q | \vphantom{\sum} B_{s_{1}}^{[1]r_1} \cdot \ldots \cdot B_{s_{L}}^{[L]r_L} | b_L)\; {c_{1}^{\dagger}}^{s_1} \ldots {c_{L}^{\dagger}}^{s_L} |\Omega\rangle \langle \Omega | \,c_{L}^{r_L} \ldots c_{1}^{r_1} = \tilde{c}_{\alpha}^{\dagger \,q},$$ with $q \in \{0,1\}$. Also recall that the fermionic orbitals $\phi_{\alpha}(\ell)$ we filled to build the Slater determinant state were an orthonormal set: let us complete it to an orthonormal basis $\{\phi_{\alpha}(\ell)\}_{\alpha}$, with $\alpha \in \{1..L\}$. The dimension must be $L$ by the assumption that the original set of $L$ wavefunctions was complete. For any of those $\phi_{\alpha}(\ell)$ the corresponding MPO is given by . Now we stack together the MPOs, like we did for the Slater state, but instead using only $N$ of them, we stack the complete set, ordered from $\alpha = 1$ on top to $\alpha = L$ at the bottom; moreover, instead of using the standard left correlation boundary vector $(b_0|_\alpha = (1|$ we set a generic $(b_0|_\alpha = (q_{\alpha}|$. It is obvious that the operator arising from this construction is equivalent to $$\tilde{c}_{1}^{\dagger\, q_1} \; \tilde{c}_{2}^{\dagger\, q_2} \; \tilde{c}_{3}^{\dagger\, q_3} \ldots \tilde{c}_{L}^{\dagger\, q_L}.$$ Finally, we apply such operator to the vacuum $|\Omega\rangle$. The meaning of all this construction is that we actually defined an application on the binary strings of $\{q_{\alpha}\}_{\alpha}$ to the real Fermi space, as $$\label{eq:Wigner2} (q_1 \ldots q_L| \longrightarrow \tilde{c}_{1}^{\dagger\, q_1} \; \tilde{c}_{2}^{\dagger\, q_2} \ldots \tilde{c}_{L}^{\dagger\, q_L} |\Omega\rangle.$$ By linearity, this map extends to all the space generated by $(q_1 \ldots q_L|$, which corresponds to the whole correlation bondlink space (as $(q_1 \ldots q_L|$ is its canonical product basis). The map is clearly bijective and thus invertible. But you notice that the inverse of is basically a Wigner transformation from the Fermi space to its spin representation where this time the $\phi_\alpha$ have been chosen as basis of one-body wavenfunctions, so it is formally similar to , but the basis has *changed* (the old one is associated to the $c$, the new one to the $\tilde{c}$). In conclusion, we could use all this MPO stack formalism to represent a many-body state transformation $|\Psi\rangle$ corresponding to a change of the chosen basis of one-body wavefunctions. I.e. assuming that we can expand $$|\Psi\rangle = \sum_{\{s\} = 0}^{1} \mathcal{T}_{s_1 \ldots s_L}^{\text{old}} {c}_{1}^{\dagger\, s_1} \ldots {c}_{L}^{\dagger\, s_L} |\Omega\rangle = \sum_{\{q\} = 0}^{1} \mathcal{T}_{q_1 \ldots q_L}^{\text{new}} \tilde{c}_{1}^{\dagger\, q_1} \ldots \tilde{c}_{L}^{\dagger\, q_L} |\Omega\rangle,$$ then the two components tensors $\mathcal{T}^{\text{old}}$ and $\mathcal{T}^{\text{new}}$ satisfy the equation: $$\label{eq:Grid} \begin{overpic}[width = \textwidth, unit=1pt]{Grid} \put(54, 65){$\mathcal{T}^{\text{old}}$} \put(178, 72){$\mathcal{T}^{\text{new}}$} \put(225, 144.5) {\scriptsize $0$} \put(251, 144.5) {\scriptsize $0$} \put(276, 144.5) {\scriptsize $...$} \put(376, 119) {\scriptsize $0$} \put(376, 71) {\scriptsize $0$} \end{overpic}$$ where the blue tensors in the grid are exactly $B^{[\ell, \alpha]}$ of , with $\ell$ being the coordinate in the horizontal axis, and $\alpha$ the one in the vertical axis (the origin is the lower-left corner). The upper and rightmost edge tensors are trivially $|0\rangle$ and $|0)$. An interesting remark to this result, is that the grid Tensor Network that appear in can be efficiently contracted, despite having several closed loops in its geometry (see section \[sec:Closedloop\]), thanks to particle-conservation symmetry relations. ### Extensions to Configuration Interaction In quantum chemistry settings, the simplest path to move beyond the mere mean field paradigm is adopting Configuration Interaction. In those descriptions, Hartree-Fock solutions are adopted as a canonical vector basis of orbitals for further calculations. According to such viewpoint, one is interested to express correlations by superposing few to several Slater determinant states, which typically *share* some of the HF orbitals as well as *differ* for other ones. If the energy minimization problem were to be performed over the whole space of Slater states the result would be exact, still this would be an extremely hard problem: thus generally the amount of orbitals for which the involved Slaters differ, is kept to a small, manageable number. Having this scheme in mind, we would like to extend our previous Slater MPS (MPO stack) representation to embed also Configuration Interaction states, where different orbital excitations are coherently added. The ultimate ingredient of this perspective would be writing Matrix Product representation for every operator generated by the Fermi ones $\tilde{c}^{\dagger}_{\alpha}$ through sums and multiplications. Of course the related zoology is huge, so we will limit our discussion the simplest nontrivial case. Consider for instance the expression $$\Theta_{2+2} = \alpha \;\tilde{c}^{\dagger}_1 \,\tilde{c}^{\dagger}_2 + \beta \;\tilde{c}^{\dagger}_3 \,\tilde{c}^{\dagger}_4.$$ We want to describe $\Theta_{2+2}$ as a Matrix Product Operator, and as you can guess there is no unique way to perform the extension from the normal Fermi operator case. Depending on whether we focus on the adaptability of the description or the economy on the bondlink dimension we end up with different proposals. ***Standard Guess -*** this path exploits the standard technique to sum coherently Matrix Product objects, and is strongly based on ; thus is highly suitable for further generalization, but at the cost of a sub-optimal bondlink dimension. Let us now adopt $D = 8$ and consider $$\label{eq:blockdj} {B^{[\ell,2+2]}}_{j}^{i} = \sum_k \left( \begin{array}{c|c} {B^{[\ell,1]}}_{k}^{i} \otimes {B^{[\ell,2]}}_{j}^{k} & 0 \\ \hline 0 & {B^{[\ell,3]}}_{k}^{i} \otimes {B^{[\ell,4]}}_{j}^{k} \end{array} \right),$$ where the $B^{[\ell,\alpha]}$ tensors are those defined in for $\tilde{c}^{\dagger}_{\alpha}$. The basic idea behind this construction is to use a correlation space which is the *Cartesian sum* of the two original correlation spaces, and a matrix product object which is the block diagonal composition. Similarly we define the correlation boundary vectors, which contain information on $\alpha$ and $\beta$: $$| b_L ) = \mbox{\scriptsize{$ \left( \begin{array}{c} \alpha \\ 0 \\ 0 \\ 0 \\ \beta \\ 0 \\ 0 \\ 0 \end{array} \right) $}} = \alpha \mbox{\footnotesize{$ \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) $}} \oplus \beta \mbox{\footnotesize{$ \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) $}} = \mbox{\footnotesize{$ \left( \begin{array}{c} \alpha \\ \beta \end{array} \right) $}}\otimes \mbox{\footnotesize{$ \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array} \right) $}},$$ where we used distributivity of the tensor product $\otimes$ with respect to the Cartesian sum $\oplus$. Similarly, $(b_0| = ($$) \otimes ($). ***Cheap Guess -*** this path focuses on keeping the lowest correlation bondlink dimension possible, and actually requires $D = 6$. $${B^{[\ell,2+2]}}_{0}^{0} = {{\mathbb 1}}_{6 \times 6} \qquad \quad {B^{[\ell,2+2]}}_{0}^{1} = 0$$ $${B^{[\ell,2+2]}}_{1}^{0} = \left( \begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\alpha} \phi_1(\ell) & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\beta} \phi_3(\ell) & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\beta}\phi_4(\ell) & 0 & 0 & 0 & 0 & 0 \\ \sqrt{\alpha} \phi_2(\ell) & 0 & 0 & 0 & 0 & 0 \\ 0 & - \sqrt{\alpha} \phi_2(\ell) & -\sqrt{\beta} \phi_4(\ell) & \sqrt{\beta} \phi_3(\ell) & \sqrt{\alpha} \phi_1(\ell) & 0 \end{array} \right)$$ $$\label{eq:cheapyMPO6} \mbox{and}\qquad {B^{[\ell,2+2]}}_{1}^{1} = \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right),$$ while boundaries are as before $|b_L) = |0)$ and $(b_0| = \left( \mbox{\tiny{0 \ldots 0 1}} \right) = (5|$. By multiplying the $B^{[\ell,2+2]}$ matrices it is easy to see that we are reproducing the correct action of the operator, i.e. $$\begin{gathered} \label{eq:sommy1} \sum_{\ell_1 < \ell_2} \left\{ \alpha \left( \vphantom{\sum} \phi_1(\ell_1) \phi_2(\ell_2) - \phi_2(\ell_2) \phi_1(\ell_1) \right) \right. + \\ + \left. \beta \left( \vphantom{\sum} \phi_3(\ell_1) \phi_4(\ell_2) - \phi_4(\ell_2) \phi_3(\ell_1) \right) \right\} \;c^{\dagger}_{\ell_1} \;c^{\dagger}_{\ell_2}.\end{gathered}$$ Like previously, we argued if this Matrix Product representation is optimal in terms of correlation bondlink dimension: we found that a state of the form $\Theta_{2+2}|\Omega\rangle$ has a real-space partition entropy of entanglement *at most* equal to $5/2$. This implies that a faithful MPS description would require a $D \geq \sqrt{32}$, so that $D = 6$ is the smallest allowed integer, and thus optimal. The present proposal presents various options for generalization, although finding the analytical MPO expression for a generic operator which is cheapest in terms of $D$ is definitely a hard task. With this last speculation we conclude this analytical example of Matrix Product formalism for interesting states in condensed matter physics and quantum chemistry. Throughout this chapter we dealt uniquely with open boundary condition problems, and developed a formalism of Matrix Product States based on the OBC framework. Of course, such a description can be adjusted to fit naturally periodic boundary conditions as well, taking care of the correct amount of entanglement. In the next chapter we will introduce a periodic description for finitely-correlated states and thus Matrix Product States, with its proper formulation and tricks of the trade; this will be instrumental in the proper definition of a thermodynamical limit. Periodic and infinite Matrix Product States {#chap:PBCMPS} =========================================== One of the major issues for standard DMRG architectures in 1D problems is dealing with Periodic Boundary Conditions (PBC). It was soon clear that traditional DMRG ideas could not be applied to PBC with the same success and simulation precision, but it was only with the advent of MPS representations that this trouble become clear and argumented. Indeed, while in OBC the DMRG procedures describes all and only the finitely correlated states, i.e. those states whose entanglement is bounded by a finite value (which typically does not scale with system size $L$) in PBC the correspondence is not exact any longer. Nevertheless, finitely correlated states play again a very important role in describing ground states of short-range interacting models, as they manifest the correct entanglement area-law. Indeed, even in PBC finitely correlated states naturally lead to a matrix product representation, but the formulation [@Rossinisupersolid; @Rossinistiff] is slightly different from their OBC counterpart. Valence bond picture for Periodic MPS ------------------------------------- In section \[sec:valencebond\] we introduced the valence bond picture to argument and contextualize MPS with open boundaries; its is straightforward to extend such description to a periodic system. To every site we associate a pair of spins, each one $D$ dimensioned ($D$ chosen by the user, often sensibly larger than the local degree of freedom dimension $d$). We prepare this virtual state so that every pair of neighboring sites share a maximally entangled state through the $D$-dimensioned spins $|\Phi^{+}\rangle = D^{-\frac{1}{2}}\sum_{\alpha}^{D} | \alpha \alpha \rangle$ (entangled bond). Notice the difference with the OBC case, where we had $L$ sites and thus $L-1$ physical bonds: in PBC every site has two neighbors (there is neither first nor last site, or, if you prefer, sites $1$ and $L$ are neighbors), so the amount bonds is $L$. The virtual-to-physical mapping is defined identically to the OBC case: $$\mathcal{A}^{[\ell]} = \sum_{s = 1}^{d} \sum_{j,k = 1}^{D} A^{[\ell]s}_{j,k} |s\rangle_{\ell} \,\langle j,k |^{\text{aux}}_{\ell}.$$ As before, which we are going to apply it to the composite entangled bond state $\bigotimes_{\ell} \mathcal{A}^{[\ell]} (\bigotimes_{\ell'} |\Phi^{+}\rangle^{\text{aux}}_{\ell', \ell'+1})$. Immediately, one can see that the resulting state can be expressed as $$\label{eq:PBCMPS} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ A_{s_{1}}^{[1]} \cdot A_{s_{2}}^{[2]} \cdot A_{s_{3}}^{[3]} \cdot \ldots \cdot A_{s_{L}}^{[L]} \right] | s_1 \ldots s_L \rangle,$$ where the $A_{s_{L}}^{[L]}$ (resp $A_{s_{1}}^{[1]}$) are no longer vectors (dual vectors) in the correlation space, but matrices, $0 \neq D_{L \equiv 0} \leq D$, like for every other site $\ell$. The trace operator in makes the inner matrix product cyclic, so there is no starting nor ending point of the 1D ring. Also let us represent $| \Psi \rangle =$ $$\begin{overpic}[width = \textwidth, unit=1pt]{PBCMps} \put(49, 36){$A^{[1]}$} \put(106, 36){$A^{[2]}$} \put(163, 36){$A^{[3]}$} \put(220, 36){$A^{[4]}$} \put(318, 36){$A^{[L \equiv 0]}$} \end{overpic}$$ diagrammatic version of . If we now are interested in estimating the entanglement of a connected subset of sites, we can use the same argument for OBC and get a similar conclusion. In fact, if we want to part the system in an given interval of sites and its complementary, we need to break two entangled bonds of the virtual state $\bigotimes_{\ell'} |\Phi^{+}\rangle^{\text{aux}}_{\ell', \ell'+1}$. And since the resulting state $|\Psi\rangle$ has entanglement bounded bounded by the first one, we have $$\mathcal{S}_{\text{VN}}\left( {\rho}_{\ell_1 \ldots \ell_2} \right) \equiv -{\text{Tr}}\left[ {\rho}_{\ell_1 \ldots \ell_2} \log {\rho}_{\ell_1 \ldots \ell_2} \right] \leq \log D_{\ell_1} + \log D_{\ell_2} \sim 2 \log D,$$ which is twice as in the OBC case, where we could split the system while breaking just one entangled bond. An interesting point concerning periodic systems is dealing with translational invariance symmetry. As most models have translationally invariant (TI) Hamiltonians $H$, exploiting the expected TI of the ground state becomes fundamental for every simulation method. This is mostly true for PBC, where TI is meaningful and spontaneously broken only in exceptional cases (when ground space degeneracies arise), whereas in OBC the breaking is naturally induced by the presence of boundaries. It is immediate to see that if the tensors in the MPS representation do not depend on the site, i.e. $A^{[\ell]} \longrightarrow A$ regardless from $\ell$, then the state $|\Psi\rangle$ is translationally invariant: $$\begin{gathered} \mathbb{T} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ A_{s_{1}} \cdot A_{s_{2}} \cdot A_{s_{3}} \cdot \ldots \cdot A_{s_{L}} \right]\, \mathbb{T} | s_1 \ldots s_L \rangle = \\ = \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ A_{s_{2}} \cdot A_{s_{3}} \cdot \ldots \cdot A_{s_{L}} \cdot A_{s_{1}} \right] | s_2 \ldots s_L s_1 \rangle = |\Psi\rangle,\end{gathered}$$ where $\mathbb{T}$ is the elementary translation operator. The original state is obtained again by using trace cyclicity and a relabeling of the indices $s$. A more pressing problem is the inverse: given a translational state $\mathbb{T} |\Psi\rangle = |\Psi\rangle$ allowing a periodic MPS representation, does it have also a *homogeneous* representation, i.e. where matrices are not site dependent? We will constructively, and positively, answer such question right away. Translational MPS admit a\ homogeneous description {#sec:MPShomo} -------------------------- Assume we are starting from a site-dependent MPS representation $A^{[\ell]}$ of a state $|\Psi\rangle$ as in , we will build another MPS rep. $B$ where matrices do not depend on the site any longer. Let us write $$\label{eq:howtohomo} B_{s} = L^{- \frac{1}{L}} \left( \begin{array}{ccccc} 0 & A^{[1]}_s &&&\\ & 0 & A^{[2]}_s && \\ && \ldots && \\ &&& 0 & A^{[L-1]} \\ A^{[L]} &&&& 0 \\ \end{array} \right),$$ we will now show that the MPS built with these matrices is equivalent to the original one. In fact $$\begin{gathered} \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ B_{s_{1}} \cdot B_{s_{2}} \cdot \ldots \cdot B_{s_{L}} \right] | s_1 \ldots s_L \rangle = \\ = \frac{1}{L} \sum_{q = 0}^{L-1} \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ A^{[1+q]}_{s_1} \cdot \ldots \cdot A^{[L+q]}_{s_L} \right] | s_1 \ldots s_L \rangle =\end{gathered}$$ $$= \frac{1}{L} \sum_{q = 0}^{L-1} \sum_{s_1 \ldots s_L = 1}^d {\text{Tr}}\left[ A^{[1]}_{s_{1-q}} \cdot \ldots \cdot A^{[L]}_{s_{L-q}} \right] | s_1 \ldots s_L \rangle = \frac{1}{L} \sum_{q = 0}^{L-1} \mathbb{T}^{q} |\Psi\rangle = |\Psi\rangle,$$ because $\mathbb{T} |\Psi\rangle = |\Psi\rangle$ by hypothesis. In conclusion, we succeeded in building a homogeneous representation for a generic finitely correlated state on a 1D PBC ring, but not without expenses. Notice, indeed, that the bondlink dimension we end up with is $D' = \sum_{\ell} D_{\ell} \sim LD$, with $D_{\ell}$ being the original bondlink dimensions of $A^{[\ell]}$; a linear scaling law with the system size $L$ arises. At the same time, we are describing the same amount of entanglement as before, so the $B$ representation is definitely sub-optimal. Unfortunately, this is common ground when dealing with PBC Matrix Product descriptions (for both states and operators). Regarding this issue, ref. [@MPSreview] proposes a trivial example involving the W-state $\sum_q \mathbb{T}^{q} |0\ldots01\rangle$, which has minimal MPS bond dimension of 2, that necessarily increases to $L$ if we want to give a homogeneous MPS description. [HomoMPS]{} (51, 48)[$A^{[q]s}_{j,k}$]{} (99, 94)[$\Delta^{+}$]{} (-3, 48)[$j$]{} (146, 104)[$q'$]{} (146, 58)[$k$]{} (-3, 95)[$q$]{} (68, 7)[$s$]{} (36, 108)[$B$]{} Expectation values for periodic MPS {#sec:PBCObs} ----------------------------------- Often in numerical settings, dealing with periodicity is more tricky and expensive than corresponding OBC versions of the same problems: for MPS this peculiarity manifests immediately in an increase of computational costs needed to acquire expectation values of observables. In section \[sec:OBCMpsObs\] we defined a procedure to evaluate physical quantities of a MPS in OBC with a number of operations that scales nicely, as , with main simulation parameters: system size $L$ and correlation bondlink $D$. In the best case scenario, where the separable observable $O$ acted on an interval of $\ell$ sites, we estimated the cost to scale like $\sim \ell D^3 d$. Scaling laws are not so nice for periodic MPS, and the compactness of operator supports does not help, due to the presence of a global closed loop in the graph (see also section \[sec:Closedloop\]). Precisely, assume that $O = \bigotimes_{\ell = \ell_1}^{\ell_2} \Theta^{[\ell]}$, and we are looking for the expectation value $$\begin{gathered} \langle \Psi | O | \Psi \rangle = \sum_{s_1 \ldots s_n} \sum_{r_1 \ldots r_n} {\text{Tr}}\left[ A_{s_{1}}^{[1]} \cdot \ldots \cdot A_{s_{L}}^{[L]} \right] \times \\ \times {\text{Tr}}\left[ {A^{\star}}_{r_{1}}^{[1]} \cdot \ldots \cdot {A^{\star}}_{r_{L}}^{[L]} \right] \langle r_1 \ldots r_L | \bigotimes_{\ell = \ell_1}^{\ell_2} \Theta^{[\ell]} | s_1 \ldots s_L \rangle.\end{gathered}$$ We can still rewrite this equation in a simpler Matrix Product form thanks to the formalism of transfer matrices, defined identically as before $ {{\mathbb E}}_{X}^{[\ell]} \equiv \sum_{s,r}^d \langle r | X | s \rangle ( A_{s}^{[\ell]} \otimes {A^{\star}}_{r}^{[\ell]} )$. The difference is that this Matrix Product is also cyclic, i.e. $$\label{eq:TranstringPBC} \langle \Psi | O | \Psi \rangle = {\text{Tr}}\left[ {{\mathbb E}}_{\Theta}^{[\ell_1]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta}^{[\ell_2]} \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell_2+1]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell_1-1]} \right].$$ Multiplying two transfer matrices costs $\sim D^6$, an expense that can be reduced to $2dD^5 + d^2D^4$ by calculating in the order $\mathbb{M}_s = (A_s \otimes {{\mathbb 1}}) \mathbb{E}$, then $\mathbb{Q}_r = \sum_{s,r} \langle r | X | s \rangle \mathbb{M}_s$, and finally $\mathbb{E}' = \sum_r ({{\mathbb 1}}\otimes A^{\star}_r) \mathbb{Q}_r$. Unfortunately, this is the only improvement that can be made in general. Equation has no right and left boundary vectors, which were instrumental to remove a $D^2$ scaling power out of the cost. Moreover, the gauge group can be no longer exploited to eliminate terms from the product of matrices; this can be argumented as follows. We would like, for instance, to transform the $\mathbb{E}^{[\ell]}_{{{\mathbb 1}}}$ into the identity so that it disappears from . However, the MPS gauge group transforms the transfer matrix $\mathbb{E}^{[\ell]}_{X}$ according to $$\mathbb{E}^{[\ell]}_{X} \longrightarrow \left( Y_{\ell-1}^{-1} \otimes Y_{\ell-1}^{\star\,-1} \right) \cdot \mathbb{E}^{[\ell]}_{X} \cdot \left( Y_{\ell} \otimes Y_{\ell}^{\star} \right).$$ But the input matrix $\mathbb{E}^{[\ell]}_{X}$ could be entangling, and the transformation is local and invertible, so there is no chance that a generic $\mathbb{E}^{[\ell]}_{X}$ can be mapped into a non-entangling operator (like ${{\mathbb 1}}$) this way. In conclusion, if we want to acquire the *exact* expectation value of a product observable on a PBC MPS, the computational cost is $$\label{eq:PBCMPScost} \# \mbox{cost} \sim L \left(2 d D^5 + d^2 D^4 \right).$$ Honestly, it is absolutely convincing that for large $L$ the system will be less sensitive to finite size effects, thus manifesting an emergent physics which is very similar to OBC physics in the bulk. We could exploit somehow this limit to reduce costs while acquiring controlled errors; nevertheless, it is useful to understand how to work with MPS in the thermodynamical limit before elaborating this idea. Thermodynamical limit MPS {#sec:MPSTD} ------------------------- The chance of extending a Matrix Product State description so that it is actually representing an infinite system $L \to \infty$, follows directly from the notion of MPS homogeneity we discussed in section \[sec:MPShomo\]. Let $A^{s}_{j,k}$ be the elementary tensor block of a PBC homogeneous Matrix Product State. For every system size $L$, this $A$ defines a unique state in the $2^L$ dimensioned Hilbert space, thus forming a sequence of states $|\Psi_L\rangle$. The thermodynamical state is defined through physically relevant quantities, namely expectation values $\langle O \rangle_{\infty}$ of compact support observables which should coincide with the limit of $\langle O \rangle_{L}$ for $L \to \infty$. $$\begin{overpic}[width = 340pt, unit=1pt]{MpsTD} \put(13, 16){$-\infty$} \put(92, 31){$A$} \put(141, 31){$A$} \put(190, 31){$A$} \put(238, 31){$A$} \put(309, 16){$+\infty$} \end{overpic}$$ Of course such expectation value limit must be well-defined in order for the thermodynamical state to be consistent; thus we need to understand under which conditions upon $A$ this uniqueness is achieved. Therefore, consider the expectation value $\Psi ( O )$ of a compact support observable $O$, which acts on $\ell$ adjacent sites $$\begin{gathered} \label{eq:TDlimlim} \Psi_{\infty} ( O ) = \lim_{L \to \infty} \Psi_{L} ( O ) \equiv \lim_{L \to \infty} \frac{\langle \Psi_L | O | \Psi_L \rangle}{\langle \Psi_L | \Psi_L \rangle} = \\ = \lim_{L \to \infty} \frac{{\text{Tr}}\left[ {{\mathbb E}}_{{{\mathbb 1}}} \ldots {{\mathbb E}}_{{{\mathbb 1}}} \,\widetilde{{{\mathbb E}}}_{O}\, {{\mathbb E}}_{{{\mathbb 1}}} \ldots {{\mathbb E}}_{{{\mathbb 1}}} \right]}{ {\text{Tr}}\left[ {{\mathbb E}}_{{{\mathbb 1}}} \ldots {{\mathbb E}}_{{{\mathbb 1}}} \vphantom{\widetilde{{{\mathbb E}}}_{O}} \right] } = \lim_{L \to \infty} \frac{{\text{Tr}}\left[ {{\mathbb E}}_{{{\mathbb 1}}}^{L-\ell} \;\widetilde{{{\mathbb E}}}_{O}\right]}{ {\text{Tr}}\left[ \vphantom{\widetilde{{{\mathbb E}}}_{O}} {{\mathbb E}}_{{{\mathbb 1}}}^{L} \right]}\end{gathered}$$ where $\widetilde{{{\mathbb E}}}_{O}$ is the $\ell$-sites composite transfer matrix of the operator $O$: $$\widetilde{{{\mathbb E}}}_{O} = \sum_{\mbox{{\scriptsize $\begin{array}{c} s_1 \ldots s_{\ell} = 1 \\ r_1 \ldots r_{\ell} = 1 \end{array} $}}}^{d} \!\!\!\!\!\! \langle r_1 \ldots r_{\ell} | O | s_1 \ldots s_{\ell} \rangle\;\, {\text{Tr}}\left[ \left( A_{s_1} \otimes A^{\star}_{r_1} \right) \cdot \ldots \cdot \left( A_{s_{\ell}} \otimes A^{\star}_{r_{\ell}} \right) \right].$$ In eq.  you see that, since we are keeping $A$ fixed, for a generic $L$ the MPS state $| \Psi_L \rangle$ will not be normalized, so we have to introduce manually the square norm in the expression $\langle \Psi_L | \Psi_L \rangle = {\text{Tr}}[ {{\mathbb E}}_{{{\mathbb 1}}}^{L} ]$. It is clear that the Thermodynamical limit state $\Psi_{\infty} (\cdot)$ must not depend on how we perform the limit itself, nor which boundary conditions we used at finite sizes. So we must obtain the same result even starting from an open boundary setting, as long as $A$ still describes the bulk, and distance between the support of $O$ and boundaries diverges. So we will introduce arbitrary correlation-space boundaries $(b_{\text{left}} |$ and $|b_{\text{right}})$ by hand, and write $$\label{eq:TDlim} \Psi_{\infty} ( O ) = \lim_{L \to \infty} \frac{ (b_{\text{left}} | \;{{\mathbb E}}_{{{\mathbb 1}}}^{nL} \;\widetilde{{{\mathbb E}}}_{O}\; {{\mathbb E}}_{{{\mathbb 1}}}^{mL}\; |b_{\text{right}})}{ (b_{\text{left}} | \;{{\mathbb E}}_{{{\mathbb 1}}}^{(n+m)L+\ell}\; |b_{\text{right}}) },$$ we will require that this limit does not depend on $(b_{\text{left}} |$, $|b_{\text{right}})$, $n$ or $m$ ($n$ and $m$ being any two positive integers); it also must coincide with the limit in equation . ***Lemma -*** uniqueness of limit , holds *iff* some spectral requirements upon the transfer matrix of the identity operator ${{\mathbb E}}_{{{\mathbb 1}}}$ are satisfied: - among eigenvalues $\lambda_{\alpha}$ of ${{\mathbb E}}_{{{\mathbb 1}}}$, there is one $\lambda_0$ strictly bigger than all others in modulus, i.e. $|\lambda_{\alpha \neq 0}| < |\lambda_{0}|$, - $\lambda_0$ eigenvalue is simple, meaning that only one related eigenvector $|e_0^{\rightarrow})$ exists i.e. the eigenspace of $\lambda_0$ has dimension 1. Let us prove this statement. The transfer matrix ${{\mathbb E}}_{{{\mathbb 1}}}$ is not necessarily diagonalizable, but as it is on complex field, we can expand it in its generalized eigenvector basis (which is not orthogonal in general), so it appears in the Jordan block form $$\label{eq:etrajordan} {{\mathbb E}}_{{{\mathbb 1}}} = \left( \begin{array}{ccccc|cc} \lambda_0 & 1 &&&&\\ & \lambda_0 & 1 &&&\\ && \cdots & \cdots &&\\ &&& \lambda_0 & 1 &\\ &&&&\lambda_0\\ \hline &&&&& \lambda_1 & \cdots\\ &&&&&& \cdots \end{array} \right)$$ where we highlighted the generalized eigenspace of $\lambda_0$. The conditions we required upon ${{\mathbb E}}_{{{\mathbb 1}}}$ tells us that $|\lambda_{\alpha \neq 0}| < |\lambda_0|$, and that there is only one Jordan block corresponding to $\lambda_0$, as it appears in . Then, given a random vector $|v)$, it is possible to demonstrate that $$\label{eq:totheigen} \lim_{q \to \infty} \frac{{{\mathbb E}}^{q}_{{{\mathbb 1}}} |v)}{ \sqrt{(v| {{{\mathbb E}}^{\dagger}_{{{\mathbb 1}}}}^q {{\mathbb E}}^{q}_{{{\mathbb 1}}} |v) }} = |e^{\rightarrow}_0),$$ with $|e^{\rightarrow}_0)$ normalized $(e^{\rightarrow}_0|e^{\rightarrow}_0) = 1$, provided that $|v)$ has no null component over the generalized eigenspace of $\lambda_0$. To show first expand $|v)$ in the generalized eigenbasis $|e_{\alpha, \partial}^{w})$, where $\alpha$ is the eigenvalue index, $\partial$ refers to the Jordan block where the basis element belongs, and $w$ its position within the block. Then the composite application of several ${{\mathbb E}}^{\dagger}_{{{\mathbb 1}}}$ gives $$\label{eq:Polyexpand} {{\mathbb E}}^{q}_{{{\mathbb 1}}} |v) = \sum_{\alpha} \lambda_{\alpha}^{q}\; \sum_{\partial, w} \mathcal{P}^{[\Delta_{\partial}-w]}_{\lambda_{\alpha}}(q) \; |e_{\alpha, \partial}^{w}),$$ where $\mathcal{P}^{[x]}_{\lambda}(q)$ is a polynomial function of $q$, of degree $x$, and whose coefficients depending on $\lambda$; $\Delta_{\partial}$ is the size of Jordan block $\partial$. When we take the limit of the latter expression, all the components belonging to gen. eigenspaces different from the $\lambda_0$ one vanish, as $(\lambda_{\alpha}/|\lambda_0|)^q \to 0$ for any $\alpha \neq 0$. Then, also components over generalized eigenvectors of $\lambda_0$ which are not the true unique eigenvector $|e^{\rightarrow}_0)$ disappear, since their polynomial multiplier is of lower degree, i.e. $\mathcal{P}^{[\Delta_{0}-w]}_{\lambda_{0}}(q) / |\mathcal{P}^{[\Delta_{0}]}_{\lambda_{0}}(q) |\to 0$ for any $w \neq 0$. Therefore the composite action of ${{\mathbb E}}^{q}_{{{\mathbb 1}}}$ on a generic vector $|v)$ maps it (after normalization) to $|e_0^{\rightarrow})$. With a similar argument, it can be shown that $(v| {{\mathbb E}}^{q}_{{{\mathbb 1}}} \to (e_0^{\leftarrow}|$, where $(e_0^{\leftarrow}|$ is the only ’left-eigenvector’ (eigenfunctional) of ${{\mathbb E}}^{q}_{{{\mathbb 1}}}$. Notice that by construction $(e_0^{\leftarrow}|$ is not necessarily the dual of $|e_0^{\rightarrow})$ via Riesz representation theorem, but the two vectors are not orthogonal either, so that $(e_0^{\leftarrow}|e_0^{\rightarrow}) \neq 0$. After all these considerations, we can extract the desired conclusion, i.e. $$\label{eq:TDexpect} \begin{aligned} \Psi_{\infty} ( O ) &= \lim_{L \to \infty} \frac{{\text{Tr}}\left[ {{\mathbb E}}_{{{\mathbb 1}}}^{L-\ell} \;\widetilde{{{\mathbb E}}}_{O}\right]}{ {\text{Tr}}\left[ \vphantom{\widetilde{{{\mathbb E}}}_{O}} {{\mathbb E}}_{{{\mathbb 1}}}^{L} \right]} = \lim_{L \to \infty} \frac{\sum_v {\text{Tr}}\left[ |v)(v| {{\mathbb E}}_{{{\mathbb 1}}}^{L} \;\widetilde{{{\mathbb E}}}_{O} {{\mathbb E}}_{{{\mathbb 1}}}^{L} \right]}{ \sum_u {\text{Tr}}\left[ |u)(u| \vphantom{\widetilde{{{\mathbb E}}}_{O}} {{\mathbb E}}_{{{\mathbb 1}}}^{2L+\ell} \right]} \\ &= \lim_{L \to \infty} \frac{ (b_{\text{left}} | \;{{\mathbb E}}_{{{\mathbb 1}}}^{nL} \;\widetilde{{{\mathbb E}}}_{O}\; {{\mathbb E}}_{{{\mathbb 1}}}^{mL}\; |b_{\text{right}})}{ (b_{\text{left}} | \;{{\mathbb E}}_{{{\mathbb 1}}}^{(n+m)L+\ell}\; |b_{\text{right}}) }\\ &= \frac{(e_0^{\leftarrow}| \widetilde{{{\mathbb E}}}_{O} |e_0^{\rightarrow})}{(e_0^{\leftarrow}|{{\mathbb E}}^{\ell}_{{{\mathbb 1}}}|e_0^{\rightarrow})} = \lambda_{0}^{-\ell}\;\frac{(e_0^{\leftarrow}| \widetilde{{{\mathbb E}}}_{O} |e_0^{\rightarrow})}{(e_0^{\leftarrow}|e_0^{\rightarrow})}, \end{aligned}$$ where the limit correlation boundaries are defined by $\lambda_0 |e_0^{\rightarrow}) = {{\mathbb E}}_{{{\mathbb 1}}} |e_0^{\rightarrow})$, and $\lambda_0 (e_0^{\leftarrow}| = (e_0^{\leftarrow}| {{\mathbb E}}_{{{\mathbb 1}}}$. As you see, the result is consistent regardless if we are following an OBC or PBC scheme. The reverse implication in the Lemma is trivial by counterexample. The capability of expressing the expectation value for every observable as in , can as well be formulated in terms of density matrices $\rho_{\ell}$. That is, a thermodynamical limit quantum state can be properly defined by the sequence of reduced density matrices $\{\rho_{\ell}\}_{\ell}$ for any finite size $\ell$, having the property that when tracing partially larger-sized ones, we recover smaller-sized ones: $$\label{eq:partrace} \{\rho_{\ell}\}_{\ell} : \longrightarrow \quad {\text{Tr}}_{1..\ell_0}\left[ \vphantom{X^1} \rho_{\ell} \right] = {\text{Tr}}_{\ell-\ell_0 .. \ell} \left[ \vphantom{X^1} \rho_{\ell} \right] = \rho_{\ell - \ell_0} \quad \forall\, \ell_0 < \ell.$$ Here we are implicitly considering translational invariance as well, which is automatically granted by homogeneity of the $A$ in our case. Then, following the prescriptions of the lemma and what we learned from , we can write down the expression for the reduced density matrix $\rho_{\ell}$ of an arbitrary number of sites as $$\label{eq:TDrho} \rho_{\ell} = \!\!\!\!\! \sum_{\mbox{{\scriptsize $\begin{array}{c} s_1 \ldots s_{\ell} = 1 \\ r_1 \ldots r_{\ell} = 1 \end{array} $}}}^{d} \!\!\!\!\!\! \frac{ (e_0^{\leftarrow}| \left( A_{s_1} \otimes A^{\star}_{r_1} \right) \ldots \left( A_{s_{\ell}} \otimes A^{\star}_{r_{\ell}} \right) | e_0^{\rightarrow}) }{\lambda_0^{\ell} (e_0^{\leftarrow}|e_0^{\rightarrow})} \;|s_1 \ldots s_{\ell}\rangle \langle r_1 \ldots r_{\ell}|,$$ where the partial trace property is an automatic consequence of the fact that $\sum_{r,s} \delta_{r,s} (A_{s} \otimes A^{\star}_{r}) |e_0^{\rightarrow}) = \lambda_0 |e_0^{\rightarrow})$. An interesting and physically relevant comparison with Matrix Product Density Operators, we introduced in section \[sec:MPO\], can be made once we gave the pictorial representation of : $$\label{eq:TDrhofig} \begin{overpic}[width = \textwidth, unit=1pt]{TDrho} \put(10, 54){$\rho_\ell = \xi \times$} \put(62, 54){$e_0^{\leftarrow}$} \put(366, 54){$e_0^{\rightarrow}$} \put(139, 24){$A$} \put(138, 83){$A^{\star}$} \put(179, 24){$A$} \put(178, 83){$A^{\star}$} \put(219, 24){$A$} \put(218, 83){$A^{\star}$} \put(294, 24){$A$} \put(293, 83){$A^{\star}$} \end{overpic}$$ where $\xi = [\lambda_0^{\ell} (e_0^{\leftarrow}|e_0^{\rightarrow}) ]^{-1}$ is put for correct state normalization, and correlation boundary vectors are the solutions of the eigenproblem $$\begin{overpic}[width = \textwidth, unit=1pt]{TDedges} \put(10, 41){$\lambda_0 \times$} \put(48, 41){$e_0^{\rightarrow}$} \put(153, 41){$e_0^{\rightarrow}$} \put(105, 68){$A$} \put(104, 12){$A^{\star}$} \put(211, 41){$\lambda_0 \times$} \put(241, 41){$e_0^{\leftarrow}$} \put(307, 41){$e_0^{\leftarrow}$} \put(357, 12){$A^{\star}$} \put(358, 68){$A$} \end{overpic}$$ for maximal modulus eigenvalue $|\lambda_0|$. The resemblance of with is evident, in particular our MP-thermodynamical state is written in a peculiar MPDO form where the mixing dimension $\tilde{d}_{\ell} = 1$ (meaning no external degrees of freedom that couple locally) except for the boundaries where $\tilde{d}_{\ell} = D$. Moreover, the two designs match exactly *iff* $e_0^{\rightarrow}$, read as a matrix (being a two-indices tensor, consider one of the two as incoming matrix index, and the other one outcoming), is *positive*. In this case we can write $e_0^{\rightarrow} = X X^{\dagger}$, and embed the $X$ within the MP-block $A_{s}^{[\ell]} \to B_{s}^{[\ell]} = A_{s} X$ as it was a gauge transformation, and we find precisely the formalism . Truly, we know a peculiar case when both matrices $e_0^{\rightarrow}$ and $e_0^{\leftarrow}$ are necessarily positive: that is when $A$ is either in the left or the right gauge. For instance, let us assume that $A$ is in the left gauge, then the uniqueness condition translates into the requirement that the CPT map is mixing (see appendix \[app:mixin\]). If this requirement is satisfied, then automatically: $\lambda_0 = 1$, $e_0^{\leftarrow} = {{\mathbb 1}}_{D \times D}$ and $e_0^{\rightarrow} = \Lambda$, but this implies that $(e_0^{\leftarrow}|e_0^{\rightarrow}) = (\Phi^{+}|\Lambda) = {\text{Tr}}[\Lambda] = 1$, telling us that $\xi = 1$ $\forall \ell$. So the denominator in equation vanishes, simplifying to $$\label{eq:TDrhogau} \rho_{\ell} = \!\!\!\!\! \sum_{\mbox{{\scriptsize $\begin{array}{c} s_1 \ldots s_{\ell} = 1 \\ r_1 \ldots r_{\ell} = 1 \end{array} $}}}^{d} \!\!\!\!\!\! ( \Phi^{+}| \left( A_{s_1} \otimes A^{\star}_{r_1} \right) \ldots \left( A_{s_{\ell}} \otimes A^{\star}_{r_{\ell}} \right) | \Lambda) \;|s_1 \ldots s_{\ell}\rangle \langle r_1 \ldots r_{\ell}|.$$ Similarly, the argument can be applied to a right-gauged $A$, leading to a version of where correlation boundaries are exchanged. ### Size matters? Being able to address infinite states in a formally exact fashion, while employing a finite amount of resources, sounds really nice and useful for numerical issues. Honestly, in many simulation algorithms and architectures, the computational times often scale non-trivially with system size $L$, which makes approaches to the thermodynamical limit clumsy trials: expensive and imprecise. With Matrix Product States (as well as with other classes of self-similar tensor network structures, like CPS, TTN and MERA, we are going to describe in the next chapters) the infinite problem is perfectly addressed with finite-effort numerics: the amount of calculus scales solely with $D$, the correlation bondlink dimension. A good question, now, is whether a Matrix Product Simulation is actually capable of describing, with a manageable $D$, physical states with a good precision, even at the thermodynamical limit. We understood that a $D$-bondlink MPS can represent exactly the whole class of finitely correlated states, with entanglement bound by $\log D$; this extends to thermodynamical limit as well. Now consider a ground state of a 1D non-critical system: its partition entropy ought to satisfy the area-law of entanglement, stating that as $L$ grows, $\mathcal{S}_{\text{VN}}$ scales like $L^{0}$, going towards a finite value $\varepsilon$ in the TD-limit. Therefore, even the appropriate bondlink dimension $D = 2^{\varepsilon}$ stays finite, meaning that a thermodynamical MPS description can be definitely made. More precisely, in literature there are several classes of quantum states which are physically relevant, and an exact (typically optimal) MPS representation has been found (see e.g. the review [@MPSreview], where MPS for AKLT, Majumdar-Gosh, GHZ, W, and Cluster states are presented). On the contrary, this argument suggests us to think that representing critical ground states, whose $\mathcal{S}_{\text{VN}} \sim (c \log L) / 6$ require infinite bondlink $D$ for a successful description of the thermodynamical limit. This looks to be true, and despite this would be troublesome for numerical implementation, the issue of dealing with infinite-bondlink Matrix Product States has been studied from analytical perspectives, like in refs. [@InfyMPS1; @InfyMPS2] where an equivalence with conformal field theory (CFT) has been established. To provide more clearance on the relationship between Matrix Product representations and criticality in 1D, we are going to discuss about correlations in MPS. Matrix Product States and correlations {#sec:MPScorr} -------------------------------------- One dimensional quantum systems are quite peculiar: they can manifest no quantum phase transition at finite temperature, nor they can exhibit long-range order parameters. Yet, the relationship between criticality and non-criticality of a system is a matter of utmost importance. As quantum entanglement is not a physical observable, the basic way to recognize and identify the presence of a quantum phase transition is through scaling laws for two-point correlations. Let us go back to the thermodynamical limit (TD) MPS, defined homogeneously by the matrices $A_s$, and define the correlation function of two (separate) local observables $\Theta$ and $\Theta'$, acting at arbitrary distance $\ell+1$: $$\mathfrak{C}_{\ell+1}(\Theta, \Theta') \equiv \langle \Theta^{[\ell_0]} \otimes \Theta'^{[\ell_0 + \ell + 1]} \rangle - \langle \Theta^{[\ell_0]} \rangle \langle \Theta'^{[\ell_0 + \ell + 1]} \rangle,$$ since the system is translationally invariant, $\ell_0$ is irrelevant. To workaround normalization issues we will just adopt an $A$ in the left gauge, as we did for , then we end up with $$\label{eq:325} \begin{aligned} \mathfrak{C}_{\ell+1}(\Theta, \Theta') &= (\Phi^{+}| {{\mathbb E}}_{\Theta} \;{{\mathbb E}}_{{{\mathbb 1}}}^{\ell}\; {{\mathbb E}}_{\Theta'} |\Lambda) - (\Phi^{+}| {{\mathbb E}}_{\Theta} |\Lambda)\; (\Phi^{+}| {{\mathbb E}}_{\Theta'} |\Lambda)\\ &= (\Phi^{+}| {{\mathbb E}}_{\Theta} \cdot \left[ {{\mathbb E}}_{{{\mathbb 1}}}^{\ell} - |\Lambda) (\Phi^{+}| \right] \cdot {{\mathbb E}}_{\Theta'} |\Lambda)\\ &= (v^{\text{left}}_{\Theta} | \left[ {{\mathbb E}}_{{{\mathbb 1}}}^{\ell} - |\Lambda) (\Phi^{+}| \right] | v^{\text{right}}_{\Theta'} ). \end{aligned}$$ Let us now study the expression in from an algebraic viewpoint. As we previously stated, both left and right (double) correlation boundary vectors, $|\bullet)$ and $(\bullet|$, can be seen as matrices, whose vector version is the so-called Liouville representation: $$O = \sum_{i,j} \alpha_{i,j} |i\rangle \langle j| \quad \longrightarrow \quad |O) = \sum_{i,j} \alpha_{i,j} |ij),$$ where, in particular ${{\mathbb 1}}\to |\Phi^{+})$. Then ${{\mathbb E}}_{{{\mathbb 1}}}$ is a Complete Positivity, Trace preserving map when applies to the matricial element on its right, as we know from ; similarly, it is a Complete Positive and Unital (i.e. it maps the identity into itself) map when applying to the left. We begin requiring that the CPT map has *mixing* property, meaning that has a single attraction point, that would be $\Lambda$. CPT maps are always contractive, as proven in appendix \[app:CPTspec\], but the relaxation requirement tells us also that ${{\mathbb E}}^{\infty} = |\Lambda)(\Phi^{+}|$ so that we can write $$\label{eq:linfiz} \mathfrak{C}_{\ell+1}(\Theta, \Theta') = (v^{\text{left}}_{\Theta} | \left[ {{\mathbb E}}_{{{\mathbb 1}}}^{\ell} - {{\mathbb E}}_{{{\mathbb 1}}}^{\infty} \right] | v^{\text{right}}_{\Theta'} ),$$ which has a clear physical meaning: the uncorrelated product of expectation values is, as should be, equivalent to the operator product at infinite distance. We can now exploit the expansion for multiple application of a CPT mixing map and write the correlator as: $$\label{eq:linfiz2} \mathfrak{C}_{\ell+1}(\Theta, \Theta') = \sum_{\alpha = 2} \lambda_{\alpha}^{\ell} \;\mathcal{P}^{[\Delta_{\alpha}]}_{\lambda_{\alpha}}(\ell)$$ where $\lambda_{\alpha \geq 2}$ are the eigenvalues of ${{\mathbb E}}_{{{\mathbb 1}}}$ other than 1, and for which it holds $|\lambda_{\alpha \geq 2}| < 1$, while $\mathcal{P}^{[x]}_{\xi}(\ell)$ are polynomial functions of degree $x$ with coefficients depending on $\xi$, and $\Delta_{\alpha}$ is the size of the largest Jordan block belonging to the generalized eigenspace of $\alpha$. Two features of are worthy of remark: - Since every $\lambda_{\alpha}$ has modulus strictly smaller than 1, $\mathfrak{C}_{\ell}$ goes necessarily to $0$ at $\ell \to \infty$, regardless of $\Theta$ and $\Theta'$, which is telling us that the *state manifests no long range order parameter*. Notice that this property depends strictly on the mixing condition employed for ${{\mathbb E}}_{{{\mathbb 1}}}$. - *Correlations decay exponentially*. In particular it is possible to dominate $\mathfrak{C}_{\ell}$ with decreasing exponentials: precisely, let us order the eigenvalues $\{\lambda_{\alpha}\}_{\alpha}$ so that they decrease in modulus. Then $$\label{eq:bibol} \lim_{\ell \to \infty} \frac{|\mathfrak{C}_{\ell+1}(\Theta, \Theta')|}{(|\lambda_2| + \varepsilon)^{\ell}} = 0,$$ for any $\varepsilon \geq 0$; but if $\varepsilon$ is chosen small enough (i.e. $\varepsilon < 1 - |\lambda_2|$) the denominator in decays exponentially, and thus the numerator decays faster. It is still possible for two points correlators of MPS, as , to *resemble* power-law decay rates, by playing with several exponentials with $|\lambda_{\alpha}|$ very close to 1, but only for short ranges, and many eigenvalues are required so $D$ must be chosen appropriately. When the TD-limit MPS state is investigated at long ranges, its ultimate non-critical nature becomes clear, and dominant: ruled by the eigenvalues $\lambda_{\alpha}$ of the Identity transfer matrix ${{\mathbb E}}_{{{\mathbb 1}}}$. Faster expectation values for PBC-MPS at large sizes {#sec:PBCimprove} ---------------------------------------------------- When using Matrix Product States as a variational tailored wavefunction ansatz for classical simulations of quantum systems, is fundamental that we make economy on every computational step of the algorithm. Calculating expectation values of observables, and in particular Hamiltonians, is one of the numerical ingredients which require most computational effort so it is important to optimize its scheme beforehand. In section we acknowledged that acquiring expectation values in periodic Matrix Product States is quite more expensive than in the open boundary case, carrying an overall $D^2$ multiplier to the cost (from $\propto dLD^3$ to $\propto dLD^5$). In a paradoxical way, as we approach the thermodynamical limit $L \to \infty$ the cost drops again to $\propto dLD^3$ (plus solving a fixed point equation, usually subleading), since boundaries of the TD state are not correlated through external channels. Therefore it is natural to think that if the size $L$ of the periodic system we are considering is sensibly large, the outcoming state shall be close enough to the TD-limit, and thus the system will feel little of the periodicity, identified by small amplitudes for finite-size effects. A way to exploit this fact to improve the evaluation algorithm was proposed in ref. [@Pippan], we now sketch the same idea with a slightly different formulation. Assume we want to calculate the composite Transfer Matrix $$\widetilde{{{\mathbb E}}}^{\ell} = {{\mathbb E}}_{\Theta_1}^{[1]} \cdot {{\mathbb E}}_{\Theta_2}^{[2]} \cdot \ldots \cdot {{\mathbb E}}_{\Theta_\ell}^{[\ell]},$$ where we chose a tensor product observable $\bigotimes_{\ell'}^{\ell} \Theta^{[\ell']}_{\ell'}$ for simplicity, but the following arguments apply to an entangling operator as well. We will also state that the whole MPS segment $1 \ldots \ell$, which $\widetilde{{{\mathbb E}}}^{\ell}$ is calculated through, is in the left (or right) gauge, but not necessarily homogeneous. In particular if sites $1 \ldots \ell$ are not the whole system, $\ell < L$, it is always possible to take the singular part to its complementary, and satisfy the gauge condition. We want to achieve $\widetilde{{{\mathbb E}}}^{\ell}$ but without performing singularly any product ${{\mathbb E}}\cdot {{\mathbb E}}$ which costs $\sim D^5$. To this purpose, let us consider the singular value decomposition of $\widetilde{{{\mathbb E}}}^{\ell} = $, i.e. $$\widetilde{{{\mathbb E}}}^{\ell}_{\alpha \beta} = \sum_{\gamma} U_{\alpha \gamma}\, \sigma_{\gamma}\, V_{\beta \gamma} = U \cdot \Delta^{[\sigma]} \cdot V^{\dagger},$$ in a formal sense, where $U^{\dagger} U = V^{\dagger} V = {{\mathbb 1}}$, and $\Delta^{[\sigma]}_{\alpha \beta} = \delta_{\alpha \beta}\, \sigma_{\alpha}$ is diagonal and positive ($\sigma_{\alpha} \geq 0$). Now, *if* the ${{\mathbb E}}_{\Theta_{\ell'}}^{[\ell']}$ were all positive and homogeneous, the singular values $\sigma$ would coincide with the eigenvalues and $U = V$, which leads to $\sigma_{\alpha} = \lambda_{\alpha}^{\ell}$, where $\lambda_{\alpha}^{\ell}$ are the eigenvalues of ${{\mathbb E}}$. But this tells us that the ratio between two singular values $\sigma_{\alpha} / \sigma_{\beta} = (\lambda_{\alpha} / \lambda_{\beta})^{\ell}$ decays as much fast as the segment $\ell$ is long. Telling us that for $\ell$ long enough very few singular values $\sigma$ are relevant before reaching the numerical precision of the calculator. This argument holds for positive ${{\mathbb E}}$, but extends naturally to hermitian matrices, and by linearity and continuity it reasonably works for every matrix, for some $\ell$ large enough, and holds even for dishomogeneous matrix products (proven numerically in [@Pippan]). We could equivalently state that the *range* of $\widetilde{{{\mathbb E}}}^{\ell}$ has actual dimension $p$ smaller than $D^2$, where $p$ is the number of singular values $\sigma$ that are being kept, while the other are discarded as they are of equal or smaller order of magnitude than computational precision. During simulations, $p$ becomes a parameter, and can be kept smaller and smaller as the size $\ell \sim L$ increases. Then the stochastic procedure for calculating $\widetilde{{{\mathbb E}}}^{\ell}$ goes like this: 1. Choose a random matrix $X$, of dimension $D^2 \times p$. If the random number generator is satisfactory, there will be zero probability that one column vector will be linearly dependent on the other $p-1$ ones, due to the fact that those $p-1$ may generate a set of zero probability measure (absolutely continuous with Lebesgue measure). 2. Apply and calculate ${{\mathbb E}}_{\Theta_1}^{[1]} \cdot \ldots {{\mathbb E}}_{\Theta_\ell}^{[\ell]} \cdot X = \widetilde{{{\mathbb E}}}^{\ell} \cdot X = Y $, obviously starting from the right. Every step costs $2dD^3p$ operations. $Y$ is again a $D^2 \times p$ matrix, but the vector columns will span only the range of $\widetilde{{{\mathbb E}}}^{\ell}$ which we will now assume is exactly $p$-dimensional. As before, chances are that all column vectors of $Y$ will be linearly independent, so they will span the whole range $\widetilde{{{\mathbb E}}}^{\ell}$, because dimensions match. 3. Orthonormalize the columns of $Y$, either via a Graham-Schmidt or a QR decomposition. We obtain $Z = Y \cdot T$, with $T$ typically triangular, containing the whole singular part of $Y$. The matrix $Z$ is still $D^2 \times p$ dimensioned, and isometric: $Z^{\dagger} Z = {{\mathbb 1}}$. Moreover, since the columns of $Z$ span $\text{Rng}(\widetilde{{{\mathbb E}}}^{\ell})$, we have $Z \, Z^{\dagger} = P_{\text{Rng}(\widetilde{{{\mathbb E}}}^{\ell})}$ the projector over the range. But then $$\widetilde{{{\mathbb E}}}^{\ell} = P_{\text{Rng}(\widetilde{{{\mathbb E}}}^{\ell})} \cdot \widetilde{{{\mathbb E}}}^{\ell} = Z \cdot Z^{\dagger} \cdot \widetilde{{{\mathbb E}}}^{\ell}$$ 4. Apply and calculate $Z^{\dagger} \cdot {{\mathbb E}}_{\Theta_1}^{[1]} \cdot \ldots {{\mathbb E}}_{\Theta_\ell}^{[\ell]} = Z^{\dagger} \cdot \widetilde{{{\mathbb E}}}^{\ell} = W$, from left to right; the cost is $\sim 2dD^3p$ per step. We are done now, since $$\widetilde{{{\mathbb E}}}^{\ell} = Z \cdot W$$ and we have both $Z$ and $W$ matrices. We do not even have to multiply them together, and instead keep them separated: whenever we will have to use $\widetilde{{{\mathbb E}}}^{\ell}$ as a part of a whole MPS-network contraction, contracting over the $p$-dimensioned index space in the middle will be the last operation to be performed. In conclusion, we can resume these simple steps as follows: $$\label{eq:Pippantech} \begin{overpic}[width = \textwidth, unit=1pt]{Pippan} \put(56, 129){$\widetilde{{{\mathbb E}}}^{\ell}$} \put(107, 129){$X$} \put(199, 129){$Y$} \put(284, 129){$Z$} \put(326, 127){$T$} \put(31, 36){$\widetilde{{{\mathbb E}}}^{\ell}$} \put(213, 36){$\widetilde{{{\mathbb E}}}^{\ell}$} \put(161, 36){$Z^{\star}$} \put(117, 36){$Z$} \put(300, 36){$Z$} \put(345, 36){$W$} \end{overpic}$$ $$\mbox{where $Z$ is isometric, i.e.} \qquad \begin{overpic}[width = 180pt, unit=1pt]{Pippiso} \put(24, 31){\footnotesize $Z^{\star}$} \put(71, 31){\footnotesize $Z$} \end{overpic}$$ If we wish to check explicitly the behavior of singular values of $\widetilde{{{\mathbb E}}}^{\ell}$ we just have to perform a SVD on $W$, because $$\label{eq:trentini} \widetilde{{{\mathbb E}}}^{\ell} = Z \cdot W = Z \cdot (\tilde{U} \cdot \Delta^{[\sigma]}_{W} \cdot V^{\dagger} ) = (Z \cdot \tilde{U}) \cdot \Delta^{[\sigma]}_{W} \cdot V^{\dagger};$$ but $(Z \tilde{U}) (Z \tilde{U})^{\dagger} = {{\mathbb 1}}$, meaning that is a singular value decomposition for $\widetilde{{{\mathbb E}}}^{\ell}$ and singular values of a matrix are uniquely defined; so $\Delta^{[\sigma]}_{W} = \Delta^{[\sigma]}$ and we can see how fast they decay for large $\ell$. By adopting this process, we spent a total amount of elementary computational operations equal to $$\label{eq:Pipancost} \# \mbox{cost} \sim 2Lp \left(2 d D^3 + d^2 D^2 \right),$$ a nice improvement, even because at reasonable lengths ($L \sim 50$), $p$ can be usually chosen orders of magnitude smaller than $D^2$ with practically no loss in simulation precision (see figure \[fig:Rossin02\]). Minimization algorithms\ with periodic MPS ------------------------ For open boundary conditions MPS we presented a fast-converging and numerically manageable algorithm to find the ground state of a generic (short-range interacting) Hamiltonian. In section \[sec:minimizOBC\] we discussed that the essence of such algorithm is minimizing one MPS block at a time, keeping the other fixed; the basic step is composed by a partial contraction of the MPS network with the Hamiltonian operator (effective Hamiltonian), followed by a $dD^2$-dimensioned eigenvalue problem ($d^2D^2$ in the two-blocks simultaneous minimization case). In the periodic boundary case we have more than one naturally available path. If the Hamiltonian is translationally invariant, then a good guess would be using the set of homogeneous MPS as variational wavefunctions, because a translational ground state must exist. This idea would lead to a all-at-once minimization of the MPS state, but unfortunately the Lagrangian would not be quadratic in the MPS homogeneous block and the problem to solve would be way harder than an eigenvalue problem. Moreover, forcing the variational state to be translational would let us not identify easily Hamiltonians bearing a spontaneous translational symmetry breaking. For these reasons, in this section we will instead describe an algorithm that not assumes translationality in the variational state (and thus using dishomogeneous MPS) and minimizes blocks one at a time to preserve a quadratic structure for the Lagrangian [@Rossinisupersolid; @Rossinistiff]. Then let us start again from a nearest neighbor Hamiltonian $H = \sum_{\ell = 1}^{L} R_2^{[\ell]}$ where for comfort we regrouped in $R_2^{[\ell]}$ both one-body and two-body terms, i.e. $$\label{eq:quacchio} R_2^{[\ell]} = \sum_{q} g^{\ell}_{q} \;{{\mathbb 1}}^{[\ell-1]}\ \otimes \Theta^{[\ell]}_{q} + \sum_{r} h^{\ell}_{r} \;{\Theta'}^{[\ell-1]}_{r} \otimes {\Theta''}^{[\ell]}_{r}.$$ Let us assume that we are going to minimize the MPS tensor block $A^{[\ell]}$ associated to site $\ell$. First, we split the Hamiltonian as $H = R_2^{[\ell]} + R_2^{[\ell+1]} + \bar{R}_2$, where $\bar{R}_2$ contains all the terms of that have support in the complementary of site $\ell$. Then we calculate two composite of transfer matrices, which shall be the ingredients of our Lagrangian functional, namely - The transfer matrix $\widetilde{{{\mathbb E}}}_1$ from site $\ell+2$ to $\ell-2$ of the Identity operator, i.e. $\widetilde{{{\mathbb E}}}_1 = {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell+2]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell-2]}$ - The transfer matrix $\widetilde{{{\mathbb E}}}_2$ of $\bar{R}_2$ from site $\ell+1$ to $\ell-1$. Even though $\bar{R}_2$ is not a separable operator it is possible, with some engineering, to calculate $\widetilde{{{\mathbb E}}}_2$ spending quite the same computational cost , apart a non-scaling prefactor. Thanks to the technique , acquiring these transfer matrices is efficient; actually we prefer to store in memory $Z_1$, $W_1$ and $Z_2$, $W_2$ (where $\widetilde{{{\mathbb E}}}_{\alpha} = Z_{\alpha} W_{\alpha}$), since $p \ll D^2$ so keeping $4\,(p \times D^2)$ matrix elements is less expensive than $2\,(D^2 \times D^2)$. Then the Lagrangian for $A^{[\ell]}$ reads: $\mathcal{L}(A^{[\ell]}, {A^{\star}}^{[\ell]}) = \langle \Psi | H |\Psi\rangle - \varepsilon \langle \Psi | \Psi\rangle = \langle \!\langle A^{[\ell]} | \mathcal{H} - \varepsilon\, \mathcal{N} | A^{[\ell]} \rangle \! \rangle = $ $$\begin{overpic}[width = 308pt, unit=1pt]{Lagrangean2} \put(-20, 51){\Huge $\mathcal{L}$} \put(181, 51){\Huge $\varepsilon$} \put(80, 53){$\mathcal{H}$} \put(215, 53){$\mathcal{N}_0$} \put(78, 89){$A^{[\ell]}$} \put(76, 17){${A^{\star}}^{[\ell]}$} \put(242, 89){$A^{[\ell]}$} \put(241, 17){${A^{\star}}^{[\ell]}$} \end{overpic}$$ $\mathcal{N} = \mathcal{N}_0 \otimes {{\mathbb 1}}_{d \times d}$ is the effective square-norm operator, where $\mathcal{N}_0$ is given by $$\begin{overpic}[width = 340pt, unit=1pt]{Eff_PBCNorm} \put(14, 49){\footnotesize $\mathcal{N}_0$} \put(137, 55){\footnotesize $W_1$} \put(300, 55){\footnotesize $Z_1$} \end{overpic}$$ while the effective Hamiltonian $\mathcal{H}$ is obtained as follows $$\begin{overpic}[width = \textwidth, unit=1pt]{Eff_PBCHamilt} \put(14, 128){\footnotesize $\mathcal{H}$} \put(111, 133){\footnotesize $W_1$} \put(163, 132){\footnotesize $R_2^{[\ell]}$} \put(244, 133){\footnotesize $Z_1$} \put(78, 43){\footnotesize $W_1$} \put(154, 42){\footnotesize $R_2^{[\ell+1]}$} \put(211, 43){\footnotesize $Z_1$} \put(280, 43){\footnotesize $W_2$} \put(357, 43){\footnotesize $Z_2$} \end{overpic}$$ We immediately see a difference from the corresponding Lagrangian in the OBC case : the effective (square) norm operator is no longer the identity $\mathcal{N} \neq {{\mathbb 1}}$. If we recall correctly in the OBC case it was a property strictly depending on the choice of a gauge condition for the other MPS blocks. But when the MPS design is Periodic, in general there is no trick with gauge transformations in order to map $\mathcal{N}$ into ${{\mathbb 1}}$. This also means that to find the optimal $A^{[\ell]}$ one has to solve a *generalized eigenvalue problem*, instead of a simple one as in . Precisely, the Euler-Lagrange equation of our problem is: $$\label{eq:gen_eigenvalue} \frac{\partial \mathcal{L}(A^{[\ell]}, {A^{\star}}^{[\ell]}) }{\partial \langle \!\langle A^{[\ell]} |} = | 0 \rangle \! \rangle \quad \longrightarrow \quad \mathcal{H} | A^{[\ell]} \rangle \! \rangle = \varepsilon \,\mathcal{N} | A^{[\ell]} \rangle \! \rangle,$$ whose solution with minimal $\varepsilon$ is the optimal one, since $\varepsilon$ is actually the energy: $\varepsilon = \langle \!\langle A^{[\ell]} |\mathcal{H} | A^{[\ell]} \rangle \! \rangle / \langle \!\langle A^{[\ell]} |\mathcal{N} | A^{[\ell]} \rangle \! \rangle = \langle \Psi | H |\Psi\rangle / \langle \Psi | \Psi\rangle$. After we found the solution, we put the optimized $A^{[\ell]}$ into the MPS and choose another site for the variational paradigm. Then repeat and sweep until convergence is achieved. ![ \[fig:Rossin01\] Offset $E^{*}$ from the ground state energy of a variational periodic MPS a function of the number of algorithm sweeps. The model considered is a 1D spin-$\frac{1}{2}$ XXY ring, with anisotropy $\Delta = 0.5$ and twisted periodic boundary conditions ($\phi$ is the twisting phase). Here a bondlink $D = 18$ was used. The graphic was kindly contributed by D. Rossini [@Rossinistiff]. ](Rossin01){width="260pt"} ### Stabilizing the generalized eigenproblem Dealing with the generalized eigenproblem is no small trouble. Even with the most advanced linear algebra techniques, numerical costs are much greater than those required for addressing standard eigenproblems (for compatible dimensions). For the latter, eigenvector related to minima or maxima of the spectrum are often found quickly thanks power method-inspired procedures. Several algorithms for the generalized eigenproblem, like the Jacobi-Davidson, are also based on power-method principles, but they should be wielded with care. These protocols are very efficient when the inverse matrix of $\mathcal{N}$ is available; and, when it is not, converge faster the more $\mathcal{N}$ is easy to invert. Obviously, an operator $\mathcal{N}$ is not suitable for inversion when there are eigenvalues which are much smaller (closer to zero) than other ones, because numerical methods perceive the relative eigenspaces as if they were a kernel. In this sense, we can relate the ’fast-invertibility’ of a (positive) matrix with the requirement that the relative spreading of its eigenvalues is small, i.e. $(\lambda_{\text{max}} - \lambda_{\text{min}} )/ \lambda_{\text{min}} \ll 1$, and also means that $\mathcal{N}$ is somehow ’close’ to the identity, as the multiples of ${{\mathbb 1}}$ are the only positive operators having relative spread zero. Now, can we perform some gauge transformation that takes $\mathcal{N}$ as close as possible to the identity? We stated that there exist no general solution to this question, although, for large system sizes $\mathcal{L}$ we can argue that the system shall manifest small finite-size effects, and an affinity with the OBC version should be met. In section \[sec:PBCimprove\] we saw that the largest eigenvalues of a composite transfer matrix decay more fast the larger is $L$. In particular, if every ${{\mathbb E}}_{{{\mathbb 1}}}$ was mixing-CPT, we could write $$\label{eq:casino} \widetilde{{{\mathbb E}}}_{{{\mathbb 1}}}^{\ell} \sim |\Lambda_L) (\Lambda_R| + \varepsilon^{\ell} \,O_{\ell}$$ where $\| O_{\ell} \|$ is bounded regardless from $\ell$, and $\varepsilon$ is somehow related to the largest (in modulus) eigenvalue $\lambda_{\alpha}$ of ${{\mathbb E}}$ smaller than 1: $\varepsilon \sim |\lambda_{\alpha}| < 1$. It is clear that for $\ell \to \infty$ the second term in vanishes, but also for $\ell$ finite but large $\varepsilon^{\ell} \,O_{\ell}$ is just a small perturbation to the first, leading, term. $\Lambda_L$ and $\Lambda_R$, read as matrices, are positive thanks to CPT condition. Relying on this concept we perform the following operations. consider $\widetilde{{{\mathbb E}}}_3 = {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell+1]} \cdot \ldots \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell-1]} = {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell+1]} \cdot \widetilde{{{\mathbb E}}}_1 \cdot {{\mathbb E}}_{{{\mathbb 1}}}^{[\ell-1]}$; if $L$ is large and all the fixed MPS-blocks are in the left gauge, then $\widetilde{{{\mathbb E}}}_3$ is written as and in particular $(\Lambda_R| = (\Phi^{+}|$, Then, since $\Lambda_L$ is positive, we can write (via SVD, for example) $\Lambda_L = X X^{\dagger}$, or equivalently $|\Lambda_L) = (X \otimes X^{\star})|\Phi^{+})$. Now we can perform a gauge transformation upon $A^{[\ell-1]}$ so that it adsorbs the $X$ operator: $A_s^{[\ell-1]} \to A_s^{[\ell-1]} \cdot X$. After the transformation, we are left with $$\widetilde{{{\mathbb E}}}_3 \sim |\Phi^{+})(\Phi^{+}| + \varepsilon^{\ell} \,O'_{\ell},$$ which, in turn, makes the effective square-norm operator to read as follows: $$\mathcal{N} \sim {{\mathbb 1}}+ \varepsilon^L \; \widetilde{\mathcal{N}}_{0}^{[L]},$$ whose relative spread of the eigenvalues scales with $\varepsilon^{L}$, where $\varepsilon < 1$. We ended up with an effective normalization operator which is actually the identity apart a small perturbation, which decays exponentially with the length $L$. When we apply the Jacobi-Davidson method in this framework, we find the generalized eigenproblem solution much faster than if we do it naively, as proven in numerical simulations [@Rossinisupersolid; @Rossinistiff]. MPS and Tensor Networks ----------------------- With this last consideration, we conclude our discussion on how Matrix Product States (either in their open or periodic boundary contexts) relate to simulation paradigms as variational tailored wavefunctions, with surprising efficiency even at high precision calculus, and wide manipulation features that let them *overpower* the old-fashioned DMRG design. As you could imagine, though, their application does not limit to numerical settings. MPS are kept in great regard even for analytical calculation purposes. The possibility of building interesting parent Hamiltonians for *any* Matrix Product State [@MPSreview], and their continuous-space enhancement to describe a finitely-correlated quantum field theory [@MPScontinuous; @MPSHolographic], are just two of the many developments achieved in the last ten years. More than anything, MPS have been the main inspiration that led physicists to investigate thoroughly in the simulation capabilities of Tensor Networks in general. PEPS, TTN, MERA, and other well-known Tensor Networks designs, actually gathered interest only after the MPS representation of DMRG states was fully understood. In the following chapters we will try to understand how such Tensor Network-based structures work; how the ideas behind MPS technique can be redesigned in order, for instance, to extend the success of this method even in settings where standard MPS/DMRG fails, like higher-dimensionality systems. ![ \[fig:Rossin02\] Convergence rates of the energy offsets with sweeps, using the same 1D periodic XXY model of figure \[fig:Rossin01\], for various transfer matrices truncation parameter $p$, as explained in section \[sec:PBCimprove\] and equation . Notice that $p = 30$ achieves an excellent precision, even though the original double-bondlink had dimension $D^2 = 324$, thus a factor 10 in algorithm speed. The graphic was kindly contributed by D. Rossini [@Rossinistiff]. ](Rossin02){width="\textwidth"} General features of Tensor Networks {#chap:TN} =================================== Every many-body/multipartite quantum state, when described as amplitude components over a separable product basis, is uniquely represented by a complex Tensor. However, the number of identifiers, of descriptors, one has to express to locate specifically that state, within the whole manifold of system states, is huge: in principle it scales exponentially with the number of elementary constituents of the system, regardless of their nature. Tensor networks (TN) are the trial to express the same state with a number of numerical descriptors (be they variational or parametric) which is small, meaning that they must scale nicely with the system size $L$, and that lead to the original wide amount of descriptors by means of simple *linear* algebraic operations. It is really not fundamental whether the desired analytical state is reproduced exactly as much is important to recover the real-physics features the state should exhibit. It is really impressive to acknowledge how much interesting physics can be generated for an approach that sounds so naive, even in contexts where other analytical or numerical methods are totally clueless. In this chapter, inspired by what we learned about MPS (the father-archetype of TN) we will try to understand what common properties and features the Tensor Network architectures share, and also present some useful examples and comparisons. Definition of Tensor Network state ---------------------------------- Let us start from a generic multipartite system $\mathcal{H}^{\otimes L} = \bigotimes_{\ell = 1}^{L} \mathcal{H}$, where $\mathcal{H}$ is the Hilbert representation of the elementary degree of freedom; here we are taking the constituents to bear a homogeneous representation, as it is the usual physical setting, but this is not really a requirement of the TN description. An orthonormal basis for the single degree of freedom must be chosen as canonical one, which we shall represent as $|s\rangle$; then it is standard procedure to expand any given state $|\Psi\rangle$ of $\mathcal{H}$ over the product canonical basis: $$\label{eq:tenspan} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^{d} \mathcal{T}_{s_1 \ldots s_L} \;|s_1\,s_2 \ldots s_L\rangle$$ where $d$ is the dimension of the elementary $\mathcal{H}$, $L$ the total number of constituents, and $\mathcal{T}_{s_1 \ldots s_L}$ is a complex tensor (i.e. a multi-indexed collection of complex numbers), with $L$ indices, each one allowing $d$ different values. The normalization condition reads $\sum_{\{s_\alpha\}_\alpha}\mathcal{T}_{s_1 \ldots s_L} \mathcal{T}_{s_1 \ldots s_L}^{\star} = 1$. As long as we ordered completely the $L$ degrees of freedom, is meaningful for every dimensionality of the physical system, and for every nature (spin / bosonic / fermionic) of the constituents as well. In fact, the complete ordering allows us to write always a second-quantization version of , as follows $$\label{eq:tenspanII} |\Psi\rangle = \sum_{s_1 \ldots s_L = 1}^{d} \mathcal{T}_{s_1 \ldots s_L} \; (c_{1}^{\dagger})^{s_1} (c_{2}^{\dagger})^{s_2} \ldots (c_{L}^{\dagger})^{s_L} |\Omega\rangle.$$ where $c_{\alpha}^{\dagger}$ are either Bose or Fermi operators, and in each case they satisfy the proper commutation or anticommutation relations. Spin-orbitals $\alpha$ are now the elementary components of the system, their on-site filling being the local canonical basis. Expression represents the more general many-body state, and $\mathcal{T}_{s_1 \ldots s_L}$ contains all its physics and information. At the same time $\mathcal{T}$ is a huge array, with $d^L$ elements, hard to manipulate in every sense. But assume that the state $|\Psi\rangle$ is such that $\mathcal{T}$ can be obtained, via contracting over an additional index $q$, from a pair of tensors $\mathcal{T}'$ and $\mathcal{T}''$, like $$\mathcal{T} = \sum_q \mathcal{T}'_{s_1 \ldots s_{\ell},q} \;\mathcal{T}''_{s_{\ell+1} \ldots s_L,q}.$$ Then we would need a number of descriptors equal to $D (d^{\ell} + d^{L-\ell}) \sim D d^{L/2}$, with a lot less information needed when the number $D$ of allowed values for index $q$ (or index $q$ dimension) is smaller than $d^{L/2}$. The same argument can be applied again and refrained for $\mathcal{T}'$ or $\mathcal{T}''$; every time we split a tensor into smaller tensors partially contracted together. Every time, there is a chance (and typically happens) that we lose description capacity, meaning that the resulting set of states allowing the new decomposition is often than before. But this is not an issue as long as the states we are cutting out of our ansatz are those which are not physically relevant, and we keep those that contain the true physics of the problem we want to study. This is the central point of the Tensor Network paradigm, inspired to MPS. We write $\mathcal{T}$ as product of multiple tensorial objects, where $L$ indices are left open, they are the $d$-dimensioned physical indices, while an arbitrary number of ’fictitious-space’ indices $\{q\}$ (of arbitrary dimension $D_q$) are contracted. Like for MPS, for a given scheme of contraction, which from now on we will call *Network*, there is *always* a choice of virtual links $D_q$ large enough so the whole $\mathcal{H}^{\otimes L}$ is described. But that would be a waste of effort, since it is very unlikely that a physical state would require that very amount of information shared among tensors in the network. Which yields a definition of TN-state that takes into account parametric bounds to our description in terms of $(i)$ the number of tensors, $(ii)$ the number of indices per tensor, and $(iii)$ allowed values per non-physical index: ***Definition of Tensor Network*** - A multipartite state $|\Psi\rangle$ is a Tensor Network state $\{V,k,D\}$, with maximal tensor number $V$, correlation link-number $k$, and link dimension $D$ if - It exists a decomposition of $\mathcal{T}_{s_1 \ldots s_L}$ as a contracted product of tensors: $$\label{eq:quaqua} \mathcal{T}_{s_1 \ldots s_L} = \sum_{q_1 = 1}^{D_1} \ldots \sum_{q_B = 1}^{D_B} \left( \prod_{n = 1}^{V'} T^{[v]\,q_{w(1,n)} \ldots q_{w(b_n,n)}}_{s_{v(1,n)} \ldots s_{v(a_n,n)}} \right),$$ where $V'$ is the total number of elementary tensors $=$ nodes in the network, $B$ is the total number of contracted indices $=$ connected links in the network, Every $s_\alpha$ appears once as tensor index in the expanded expression , and every $q_\alpha$ appears twice - The number of tensors in the decomposition is bounded by $V' \leq V$, the bondlink dimensions are bounded by $D_\alpha \leq D$ for every $\alpha \in \{1..B\}$, and the total amount of indices of a tensor is bound by $a_n + b_n \leq k$ for every $n \in \{1..V \}$. As an immediate consequence, the overall number of complex value descriptors (i.e. variational parameters) required for such representation is $$\# \mbox{elements} \leq \,V \, \left( \vphantom{A^{[1]}} \sup\{ D, d \} \right)^{k} .$$ Although this is a well-formulated definition, there is no doubt that the formalism of is cumbersome and confusing. For most purposes involving Tensor Network states is actually preferable to involve a diagrammatic representation, much similar to the one we adopted for Matrix Product States. In these diagrams, Tensor Network states are represented as graphs: Tensors being vertices, indices being links, either left open if they are physical indices, or connected if they are fictitious. Let us consider the following example: $$\label{eq:Netpic} \begin{overpic}[width = \textwidth, unit=1pt]{Network} \put(47, 98){$\mathcal{T} \longleftrightarrow |\Psi\rangle$} \put(70, 30){$L$} \put(294, 3){$L$} \put(221, 67){\footnotesize $T^{[1]}$} \put(272, 67){\footnotesize $T^{[2]}$} \put(258, 113){\footnotesize $T^{[3]}$} \put(303, 106){\footnotesize $\ldots$} \end{overpic}$$ This network is made out of $V = 8$ tensors, with maximal correlation number $k = 6$; the analytical tailored expression of the corresponding (spin) state is given by $$\begin{gathered} \label{eq:Netugly} |\Psi\rangle = \sum_{\{s_\alpha\} = 1}^{d} \sum_{\{q_\beta \} = 1}^{D_{\beta} \leq D} \left( T^{[1]q_1}_{s_1 s_2 s_3} \; T^{[2]q_1 q_2 q_3}_{s_4 s_5 s_6} \; T^{[3]q_2 q_4 q_5} \; T^{[4]q_3 q_4 q_6} \right. \\ \left. \; T^{[5]q_5 q_7} \; T^{[6]q_6 q_8}_{s_7 s_8 s_9 s_{10}} \; T^{[7]q_7 q_8 q_9} \; T^{[8]q_9}_{s_{11}} \right) |s_1 \ldots s_L\rangle\end{gathered}$$ which is messy, and not immediate as although they represent the same parametric set of states. Most classes of Tensor Network commonly considered in literature, are *scalable* with system size. We intend that the network is obtained by repeating some fixed local pattern of vertices contraction to build a self-similar structure, so it can be adjusted to fit any $L$ bu just adding new tensors according to the same pattern as before. When doing so, it is important that $k$ and $D$ can be kept fixed, and the number $V$ of tensors (recall that $V$ is proportional to the number of variational parameters) scales nicely, e.g. linearly, with $L$. This was the case of Matrix Product States, where precisely $V = L$. $D$ is also occasionally referred to as *refinement parameter* [@EisMERAPEPS], as its value directly influences the capability of the TN ansatz. Entanglement of Tensor Network states {#sec:entanet} ------------------------------------- Quantum entanglement is the primary responsible for argumenting that Tensor Network are actually a good technique to describe physical multipartite states. Ground states are characterized by small correlations, compared to a generic random state in the Hilbert. So probing them with trial wavefunctions that admit a simple description, and yet capable to reproduce just the needed amount of entanglement seems a suitable choice. Tensor Network states have the outstanding feature that their entanglement is perfectly controlled by the network topology itself, as we will show in this section. ***Entanglement bounds of a Tensor Network state*** - Assume $|\Psi\rangle$ is a state which allows a Tensor Network representation, as in . Let us choose any partition of the physical sites $s_j$ into two disjoint subsets $s^{[A]}_j$ and $s^{[B]}_j$. Then the Von Neumann entanglement entropy associated to this partition satisfies the following inequality: $$\label{eq:TNEbound} \mathcal{S}_{\text{VN}} \left( {\rho}^{[A]} \right) \leq \min_{ \text{part.} \to \{ D_\alpha \}} \left( \sum_{\alpha} \log D_\alpha \right)$$ where the minimum is taken over all the partitions of the network graph into two subgraphs, with the condition that the first subgraph embeds all the $A$ sites and the other one the $B$ sites. The $D_\alpha$ are the bondlink dimensions of the links we need to break in order to disconnect the two subgraphs. To make an example, let us consider again a Tensor Network state like , and assume that we are to estimate the entanglement shared between the six leftmost sites $A=\{1..6\}$ and the five rightmost sites $B=\{7..11\}$. Then identify all the possible ways to part the network into two subnetworks, respectively containing $A$ and $B$. Three smart choices are given by: $$\label{eq:Netsplit} \begin{overpic}[width = \textwidth, unit=1pt]{Netsplit} \end{overpic}$$ Finally detect all the network bondlinks $\alpha$ we should break to separate the two subgraphs (violet and green), and sum their $\log D_{\alpha}$ to obtain a bound on the entanglement. In particular, the three graph partitions picted in tell us that $$\mathcal{S}_{\text{VN}}^{[A \leftrightarrow B]} \leq \min\left\{ \log(D_3 D_4 D_5), \log( D_2 D_3), \log( D_7 D_8) \right\}$$ and, since any $D_\alpha \leq D$, we conclude that $\mathcal{S}_{\text{VN}} < D^2$. The reader can easily check that no other graph partition would lead to a tighter bound. Let us prove the statement , by adopting an argument very similar to the valence bond picture for MPS. Consider any network graph partition into two subgraph, in accordance with the lattice sites partition $A \leftrightarrow B$, as above. Then, let us write a starting virtual state $$|\Phi^{+}_{\otimes}\rangle = \bigotimes_{\alpha \in \text{cut}} |\Phi^{+}_{D_\alpha}\rangle \quad \mbox{where} \quad |\Phi^{+}_{p}\rangle = \frac{1}{\sqrt{p}} \sum_{q = 1}^{p} |qq\rangle.$$ Each maximally state $|\Phi^{+}_{D_\alpha}\rangle$ contributes to the entanglement of $|\Phi^{+}_{\otimes}\rangle$ separately, since they lie in different degrees of freedom, and each one contributing with $\log D_{\alpha}$. But now we can find a linear mapping taking the state $|\Phi^{+}_{\otimes}\rangle$ into the original $|\Psi\rangle$ $$\label{eq:locazzin} |\Psi\rangle = \left( M_1^{[A]} \otimes M_2^{[B]} \right)|\Phi^{+}_{\otimes}\rangle$$ the mapping $M_1^{[A]}$ is given by the contraction of the subgraph on $A$, and therefore is linear, and so is $M_1^{[B]}$, telling us that can be viewed as a quantum transformation, not necessarily invertible. But as it is a tensor product, it is also local, and thus can only degrade entanglement, not enhance it. Therefore, the entanglement of $|\Psi\rangle$ must be less than that of $|\Phi^{+}_{\otimes}\rangle$ which is exactly $\sum_{\alpha} \log D_{\alpha}$. The same argument can be repeated for any subgraph partition, thus concluding the proof. Operators and link exchange-statistics {#sec:linkexchange} -------------------------------------- Since we now have a diagrammatic representation for linearly-connected tailored variational wavefunctions, we want now to exploit this idea also to include the action of operators. Especially operators that act locally or at short-ranges appear as new tensorial pieces to add to the network structure, connecting to those physical links that were left open in the TN-state design: for instance, a three-site operator (acting on sites number 2,3 and 4) applied to reads $$\label{eq:Netops} \begin{overpic}[width = 165pt, unit=1pt]{Netops} \put(-70, 60){\large $\Theta_{2,3,4} \;|\Psi\rangle = $} \end{overpic}$$ As was pointed out in refs. [@Corboz; @FOC1; @FOC2], attaining this representation is undoubtedly trivial for a spin or boson system, but we should handle the issue with care when our Tensor Network state is describing a system of fermions. Indeed, when we introduced the second quantization version of Tensor decomposition, ordering the sites was a crucial point. It is clear that changing the ordering of those sites would not only rearrange the components of $\mathcal{T}$ but also change some signs appropriately due to Fermi statistic. We are about to show that this feature can be embedded as an inherent property of the network links, that manifests under crossing (exchanging) of the links themselves. For instance, we will start from a two-site operator $\Theta_{1,2}$; we will also assume for simplicity that this operator preserves *parity*, i.e. $$\begin{gathered} \Theta_{1,2} = \alpha_0 \,{{\mathbb 1}}+ \alpha_1 \,c_1 c_2 + \alpha_2 \,c_1^{\dagger} c_2 + \alpha_3 \,c_1 c_2^{\dagger} + \\ + \alpha_4 \,c_1^{\dagger} c_2^{\dagger} + \alpha_5 \,c_1^{\dagger} c_1 + \alpha_6 \,c_2^{\dagger} c_2 + \alpha_7 \,c_1^{\dagger} c_1 c_2^{\dagger} c_2\end{gathered}$$ requiring parity preservation has the advantage that (quasi-) local operators preserve support locality when representing fermions as spins. Now, assume the tensorial representation of $\Theta_{1,2}$ is known and available, encoded through descriptors $\{ \alpha_0 \ldots \alpha_4\}$. We wonder: how do we express the action of $\Theta_{1,3}$ instead? The standard procedure is swapping the second and third (or equivalently, first and second) sites both before and after performing $\Theta_{1,2}$, as $$\label{eq:CrossX} \begin{overpic}[width = \textwidth, unit=1pt]{CrossX2} \put(35, 66){$\Theta$} \put(68.5, 27){$S$} \put(255, 27){$S$} \put(68.5, 105){$S$} \put(255, 105){$S$} \put(290, 66){$\Theta$} \end{overpic}$$ According to anticommutation rules, it is during the exchanging process that the fermionic statistic should emerge. Indeed the operator $S$ in equation is the standard quantum information Swap gate only for spin and bosonic Tensor Network. For fermionic Tensor Networks it reads instead: $$\label{eq:Swapfermi} \begin{overpic}[width = 140pt, unit=1pt]{Spider} \put(27, 41){$S$} \put(70, 41){$= \quad \left( \begin{array}{cccc} 1 &&&\\ & 0 & 1&\\ & 1 & 0&\\ & & & -1 \\ \end{array} \right)$} \end{overpic}$$ where the antisymmetrization sign $-1$ appears in correspondence to $c^{\dagger}_1 c^{\dagger}_2 |\Omega\rangle = - c^{\dagger}_2 c^{\dagger}_1 |\Omega\rangle$, the doubly occupied canonical state. all the other 2-sites canonical vectors, namely $|\Omega\rangle$, $c^{\dagger}_1 |\Omega\rangle$ and $c^{\dagger}_2 |\Omega\rangle$, being either zero or one particle states, feel no difference from the spin/Bose setting, In conclusion we acknowledged that Tensor Networks ansatze can be successfully applied to fermionic systems with no basic difference in efficiency or computational costs. The only issue we need to take care of, is the exchange of network links within diagrams, that can be rearranged from the original ordering by means of . We could state that the network links satisfy an exchange statistics themselves, that identifies the nature of particles the TN-picture is representing. As an additional remark, let us point out that it is straightforward to generalize all these arguments to include *abelian anyon* (like those of fractional quantum Hall effect) algebras as well. Indeed, let us assume that we are representing, with our trial Tensor Network state, a 2D system, which is where anyons arise. Then if anyonic operators undergo the exchange rule $c^{\dagger}_1 c^{\dagger}_2 = e^{i \phi_0} c^{\dagger}_2 c^{\dagger}_1$, then the the bondlink exchange statistics of the TN representation is replaced by $$S_{\text{anyonic}} = \left( \begin{array}{cccc} 1 &&&\\ & 0 & 1&\\ & 1 & 0&\\ & & & e^{i\phi_0} \\ \end{array} \right).$$ The problem of extending such formulation to include also non-abelian anyons was discussed in ref. [@Corbozanyon]. A similar scheme to deal with fermionic statistics is by using directed-link Tensor Network designs introduced recently, called *Fermi operator circuits* (FOC) [@FOC1; @FOC2]. Gauge group of Tensor Network states {#sec:TNgauge} ------------------------------------ Inspired by our discussion of section \[sec:MPSGauge\] where we defined a group of MPS transformations under which the physical state is invariant, we want to extend this paradigm to any given Tensor Network geometry. As for Matrix Product States, a gauge transformations group provide a clever way to manipulate analytically or numerically tensors within the network structure. The purposes of exploiting this group are many, for example, to perform faster contractions. The generalization of to an arbitrary TN-state is natural. Consider, for instance, a shared (connected at both sides) bondlink $\nu$, corresponding to a virtual degree of freedom, of dimension $D_{\nu}$. Tensors $T^{[\alpha]}$ and $T^{[\beta]}$ are the two nodes in the network sharing link $\nu$. Then we can define the composite tensor emerging from the contraction of the two $$T^{[\alpha \leftrightarrow \beta] \{q_\alpha, q_\beta \}}_{\{s_\alpha, s_\beta\}} = \sum_{q_{\nu} = 1}^{D_{\nu}} T^{[\alpha] \{q_\alpha\},q_\nu}_{\{s_\alpha\}} \; T^{[\beta] \{q_\beta \},q_\nu}_{\{s_\beta\}}$$ We are also assuming that no homogeneity or geometric constraint is requested for the TN to hold, so that the state $|\Psi\rangle$ actually depends on $T^{[\alpha]}$ and $T^{[\beta]}$ only through $T^{[\alpha \leftrightarrow \beta]}$, i.e. even if each of the former two changes but the latter is unaltered, then $|\Psi\rangle$ is also unchanged. Now let us choose an isomorphism $X$ on $\mathbb{C}^{D_{\nu}}$, i.e. an invertible linear application on the $D_{\nu}$-dimensional complex vector space: $X X^{-1} = X^{-1} X = {{\mathbb 1}}_{D_{\nu} \times D_{\nu}}$. Then it is clear that $$\label{eq:TNgauinva} \begin{aligned} T^{[\alpha \leftrightarrow \beta] \{q_\alpha, q_\beta \}}_{\{s_\alpha, s_\beta\}} &= \sum_{r_1 \ldots r_3 = 1}^{D_{\nu}} T^{[\alpha] \{q_\alpha\},r_1}_{\{s_\alpha\}} \; X^{\phantom{-1}}_{r_1 r_2} X^{-1}_{r_3 r_3} T^{[\beta] \{q_\beta \},r_3}_{\{s_\beta\}} \\ &= \sum_{q_{\nu} = 1}^{D_{\nu}} W^{[\alpha] \{q_\alpha\},q_\nu}_{\{s_\alpha\}} \; W^{[\beta] \{q_\beta \},q_\nu}_{\{s_\beta\}}, \end{aligned}$$ where $W^{[\alpha]}$ is the contraction of $T^{[\alpha]}$ with $X$, while $W^{[\beta]}$ is obtained by linking $X^{-1}$ to $T^{[\beta]}$, as in the following diagram $$\label{eq:TNgaupic} \begin{overpic}[width = 230pt, unit=1pt]{TNgauge} \put(47, 100){$T^{[\alpha]}$} \put(91, 80){$X$} \put(100, 123){$W^{[\alpha]}$} \put(116, 18){$W^{[\beta]}$} \put(131, 61){$X^{-1}$} \put(176, 46){$T^{[\beta]}$} \end{overpic}$$ Since $T^{[\alpha \leftrightarrow \beta]}$ is invariant under this transformation, $|\Psi\rangle$ is invariant as well. Rephrasing in other words, we could say that, as $\nu$ is a fictitious, external degree of freedom over which we are performing a (partial) trace, the physical system (set of real degrees of freedom) is insensitive to any local invertible transformation $X$ acting upon $\nu$, which is what does. There is no need to say that transformation has a natural group structure, arising from the fact that the set of Isomorphisms $\{X\}$ is closed under composition. This very argument can be applied and repeated for every closed (doubly connected, i.e. non-physical) link in the network structure. Indeed, we associate an invertible matrix $X_{\nu}$ and a *direction* to every virtual TN-bondlink and transform tensors $T^{[\alpha]} \rightarrow W^{[\alpha]}$ according to $$\label{eq:TNgaugetransform} W^{[\alpha] \{q\},\{p\}}_{\{s\}} = \sum_{\{v\},\{w\}} \left( \prod_j^{k_{\alpha}^{\text{in}}} X^{-1}_{q_j, v_j} \right) \left( \prod_j^{k_{\alpha}^{\text{out}}} X_{p_j, w_j} \right) T^{[\alpha] \{v\},\{w\}}_{\{s\}}$$ where $k_{\alpha}^{\text{in}}$ (resp. $k_{\alpha}^{\text{out}}$) is the number of links whose chosen direction is incoming to (outcoming from) node $\alpha$. Precisely, we are contracting $T^{[\alpha]}$ to $X_{\nu}$ at link $\nu$, if the direction given to $\nu$ is outcoming from node $\alpha$, and to $X^{-1}_{\nu}$ otherwise. As stated, no transformation is allowed on the physical links, i.e.: $$\label{eq:TNgaupic2} \begin{overpic}[width = 260pt, unit=1pt]{TNgauge2} \put(53, 70){$X^{-1}_1$} \put(49, 133){$X_2^{-1}$} \put(104, 92){$T^{[\alpha]}$} \put(184, 105){$W^{[\alpha]}$} \put(126, 10){$\longleftarrow$ physical index} \put(155, 141){$X_3$} \put(178, 57){$X_{4}$} \end{overpic}$$ It is worth mentioning that the group we found, which coincides with the Cartesian sum $\mathcal{G} \equiv \bigoplus_{\nu}^{n} \mbox{Iso}(\mathbb{C}^{D_{\nu}})$ (with $n$ the total number of closed links), is the *most general* state-invariant transformation, provided that the TN-geometry and the bondlink dimensions $D_{\nu}$ are fixed. No closed loop $\Rightarrow$ efficient contraction {#sec:Closedloop} -------------------------------------------------- As we learned how to match a quantum operator with the Tensor Network representation of a state, we need to work out how to achieve expectation values, to get physical information on the state itself. For MPS we argumented the importance of a contraction algorithm which is numerically efficient: as we typically adopt energy as a benchmark for simulation convergence towards the ground state, it is likely that we will need to evaluate $\langle \Psi | H | \Psi \rangle$ many times. An optimal contraction scheme must be modeled on the topology of Tensor Network we are considering. unfortunately, even when such scheme is optimized, its efficiency could still be unsatisfactory for practical purposes, and that obviously depends on the network design. Precisely, even if we have a $L$-scalable Tensor Network structure $\{V(L), k, D \}$ with a number of variational parameters $V$ proportional to $L$, the optimal contraction cost could still scale exponentially with $L$. This is, in particular, the unfortunate case of Product Entangled Pair states, which we will introduce in section \[sec:PEPS\]. However, we can identify a sub-class of Tensor Networks which is still quite general and whose contraction efficiency is guaranteed. We are talking about Tensor Networks that *lack closed loops* in the graph structure. Acquiring an expectation value out of a Tensor Network state $\{V, k, D\}$ without close loops has a computational cost bound by $$\label{eq:Noloopcost} \# \mbox{cost} \sim 2\:V \, D^{2k},$$ which scales linearly with size if $V \propto L$. Here we present an example of the comparison between TN-structures without or with closed loops: $$\label{eq:loop} \begin{overpic}[width = \textwidth, unit=1pt]{Noloop} \end{overpic}$$ where orange dashed lines highlight the minimal loops. The contraction scheme that leads to is definitely intuitive and often close to be the optimal one for a given network. Let us sketch it briefly. Let $\Theta_{\otimes} = \bigotimes \Theta^{[\ell]}_{\ell}$ be the separable product operator whose expectation values $\langle \Psi | \Theta_{\otimes} | \Psi \rangle$ we are interested in. Then, for every node $\mu$ in the network, we define a new ’double’-tensor $$\mathcal{W}^{[\mu] q_1 \ldots q_m}_{p_1 \ldots p_m} = \!\!\!\!\! \sum_{\mbox{{\scriptsize $\begin{array}{c} s_{v(1)} \ldots s_{v(n)}\\ r_{v(1)} \ldots r_{v(n)} \end{array} \!\!\! = 1$}}}^{d} \!\!\!\!\!\! T^{[\mu]q_1 \ldots q_r}_{ s_{v(1)} \ldots s_{v(n)} } T^{\star\:[\mu]p_1 \ldots p_r}_{ r_{v(1)} \ldots r_{v(n)} } \;\prod_{j = 1}^{n} \langle r_{v(j)} | \Theta_{v(j)} | s_{v(j)} \rangle.$$ where $n$ is the number of physical links connected to node $\mu$, which correspond to sites $v(1) \ldots v(n)$, while $m$ is the number of virtual links; of course $m+n \leq k$. The tensor $\mathcal{W}^{[\mu]}$ is obtained by pairing together tensors $T^{[\mu]}$ and its complex conjugate $T^{\star\:[\mu]}$, while contracting every physical link though the action of $\Theta_{\otimes}$; notice that $\mathcal{W}^{[\mu]}$ has correlation number $2m \leq 2k$. Obtaining every $\mathcal{W}^{[\mu]}$ has a computational cost of at most $D^{2k}$. Now, it is easy to see that the $\mathcal{W}^{[\mu]}$ form again a Tensor Network, which is identical to the original one except for having no open (physical) links, which vanished. So it contracts completely into a real number, equal to $\langle \Psi | \Theta_{\otimes} | \Psi \rangle$. The resulting bondlink dimension is the original one squared: $D' = D^2$. We can now exploit that this new network has no closed loop, then we start contracting from *terminal nodes*, i.e. nodes that have correlation number 1 (a finite graph without close loops has always two terminal nodes at least). Contracting a terminal node has a cost in elementary operations that scales like $D^{2k}$, and i have to repeat this operation $\sim V$ times. Thus the overall cost of the full contraction is $\sim V D^{2k}$, and by adding the expense to obtain the double-tensors $\mathcal{W}^{[\mu]}$ we recover . It is important to notice that such procedure was explained disregarding the ordering of sites, so it works for Tensor Network states of spins and bosons, but it is not so trivial for fermions, since link-exchange operators are not separable operators. Nevertheless, it was pointed out in [@VerstraTree] that this very scheme can be generalized to the Fermi case, with equally-scaling efficiency rates, by adopting $\mathbb{Z}_2$-symmetric tensors in the network, with practically no loss of generality (see appendix \[app:symchap\]). ### Peripheral gauge {#sec:peripheral} We gain a remarkable computational speed-up by manipulating with gauges a Tensor Network state without closed loops in its structure. Assume, for instance, that we want to acquire the expectation value of an operator $\Theta$ having support on a restricted set of sites $\{\alpha_{t}\}$. Now, we define the nucleus $\Pi$ as the smallest connected subgraph of the network containing all the external links $\{\alpha_{t}\}$. Then it always exist a gauge for which the number of operations to obtain $\langle \Theta \rangle$ is $$\label{eq:Periphcost} \# \mbox{cost} \sim 2\:\#(\Pi) \, D^{2k},$$ instead of $2V D^{2k}$, where $\#(\Pi)$ is the number of tensors in subgraph $\Pi$. This is definitely an interesting improvement, especially when the support of $\Theta$ is very local with respect to the network geometry, i.e. when $\#(\Pi) \ll V$. This is the same principle that, in OBC-MPS context, lead us to ; and the definition is totally similar. Let us briefly sketch the idea on how to convert the tensors *not* belonging to $\Pi$ into the *peripheral* gauge, that eliminates them automatically (with no need of numerical calculus) when acquiring $\langle \Theta \rangle$. First, let us associate to every tensor in $\Pi^{\text{c}}$, the complementary subgraph of $\Pi$, a distance: the graph distance to the nucleus $\Pi$. Then, starting to the tensors having highest distance, we perform Singular Value Decompositions $$T^{[\alpha]}_{\{q\},\{p,s\}} = \sum_{\gamma} U^{[\alpha]}_{\{q\},\gamma} \;\lambda_{\gamma}\; V^{[\alpha]}_{\gamma,\{p,s\}},$$ where $q$ (resp. $p$) indices are related to links that decrease (increase) distance, i.e. that lead towards (far from) the nucleus $\Pi$, and $s$ are physical link indices. Then we damp all the singular part $ U\cdot\lambda$ of every tensor in $\Pi^{\text{c}}$ into the only linked tensor with shorter distance (this is a gauge transformation mapping $T \to V$). This is done recursively, until all the singular part has been embedded into $\Pi$. And since all the periphery $\Pi^{\text{c}}$ is isometric, it cancels out analytically when contractions with $\langle \Psi |$ are made. It is easy to see that only for non-closed loop Tensor Network a peripheral gauge can be defined regardless of the support of $\Theta$. if $\Pi^{\text{c}}$ contained a loop then there is no gauge that contracts automatically the branch. This is what happened exactly for periodic MPS, whose single loop is sufficient to break down the definition of a peripheral gauge. Energy minimization techniques ------------------------------ Performing the variational step towards the ground state is often the bottleneck of any simulation algorithm. In Tensor Network ansatze, a good procedure of minimization makes the practical difference between better and worse network designs. However, there are some common ideas involving variational algorithms around TN: first, the globular structure of a state allows us to treat various tensors in the network as independent variables, and therefore we are highly encouraged to minimize one (or few) of them at a time, while keeping the other ones fixed. This, of course, is possible as long as we do not require some homogeneity constraint among tensors: we might, for instance, wish to fix some tensors to be identical, e.g. due to the presence of a global symmetry (like homogeneous periodic MPS under Translational invariance); in this case the minimization of various network nodes would be simultaneous. Then we should treat the two cases separately. ***No homogeneity constraint -*** if we are free to adjust every tensor independently of the others, then addressing the problem is simple because the Lagrangian functional is always quadratic on the single tensor we choose to variate $\mathcal{L} = \langle \Psi | H |\Psi\rangle - \varepsilon \langle \Psi | \Psi\rangle$. In practice, we could contract the whole network except for the tensor $T$ of interest, which appears twice in the expression, once as $T$ and once as $T^{\star}$, similarly to the PBC-MPS case. Then $\mathcal{L}(T, T^{\star}) = \langle \!\langle T | \mathcal{H} - \varepsilon \mathcal{N}| T \rangle \! \rangle = $ $$\label{eq:qulan} \begin{overpic}[width = 320pt, unit=1pt]{Quadratic} \put(-15, 64){\Huge $\mathcal{L}$} \put(184, 64){\Huge $\varepsilon$} \put(85, 68){$\mathcal{H}$} \put(90, 121){$T$} \put(88, 16){${T^{\star}}$} \put(262, 121){$T$} \put(260, 16){${T^{\star}}$} \put(257, 68){$\mathcal{N}$} \end{overpic}$$ where the effective Hamiltonian $\mathcal{H}$ is hermitian, and the effective square norm is a positive operator $\mathcal{N}$. This is due to the fact that partially contracting the Tensor Network around an operator $O$ is formally equivalent to $$\mathcal{M}(O) = \sum_{\{s\}} \left( \sum_{\{q\}} \prod M^{\{q\}}_{\{s\}} \right) \cdot O \cdot \left( \sum_{\{p\}} \prod M^{\dagger\, \{p\}}_{\{s\}} \right)$$ but this is a completely positive map (see appendix \[app:cptchap\]), which preserves operator positivity and hermiticity. Then we are allowed to address the Lagrangian problem as a generalized eigenvalue problem $\mathcal{H} |T\rangle = \varepsilon \mathcal{N} |T\rangle$, whose minimal $\varepsilon$ solution defines the optimal $T$. ***With homogeneity constraint -*** when two or more tensors in the network are chosen a priori to be identical, but not fixed, then the problem becomes more complicated as the Lagrangian reads $$\label{eq:nonqulan} \mathcal{L}(T, T^{\star}) = \left( \bigotimes_{j}^{w} \;\langle \!\langle T |_j \right) [ \mathcal{H} - \varepsilon \mathcal{N} ] \left( \bigotimes_{j}^{w} | T \rangle \! \rangle_j \right).$$ where $w$ is the number of times the same variational tensor is repeated in the network structure. The problem can be approached with various numerical methods [@MatteoMera]; those that gathered most interest in literature are the following - *Gradient methods* - this idea involves guessing random variations to the original tensor by using the Lagrangian gradient $\partial \mathcal{L}(T,T^{\star}) / \partial T$ (or the conjugate gradient, which converges usually faster) as favored direction of random walk. - *Linearized problem* - here we try to treat $\mathcal{L}$ as a linear functional $\mathcal{L} = \langle \!\langle Q_{(T,T^{\star})} | T \rangle\!\rangle$ whose solution is found immediately by polar decomposition. Doing so is very cheap and fast-paced but convergence is not guaranteed in general. - *Imaginary time evolution* - Another way of proceeding is starting from a random state and cooling down the system by applying $e^{-\beta H}$ to it, for $\beta \to \infty$ the state thermalizes at zero temperature, i.e. is the ground state. But writing an unperturbative exponential of the Hamiltonian operator $H$ is usually hard task, and not always suitable to match the Tensor Network geometry. This mostly concludes the general features of Tensor Network states that we wanted to discuss in this chapter. Before in-depth investigating the peculiar properties of the famous class of Hierarchical Tensor Networks (Tree networks, Multiscale entanglement renormalization ansatz) we focus on some other archetypal examples that are worth mentioning. Example I:\ Product Entangled Pair States (PEPS) {#sec:PEPS} ------------------------------------ Product Entangled Pair states are the natural generalization at dimensionality $> 1$ of Matrix Product States. The idea behind their formulation is to adapt the valence bond picture of MPS to 2D or 3D systems. We discussed in section \[sec:valencebond\] that an MPS can be seen as an on-site transformation applied to a starting, virtual state made by several maximally entangled pairs $|\Phi^{+}\rangle = \sum_{\alpha}^D |\alpha \alpha \rangle$, (with $D$ is arbitrary): every pair belonging to a physical bond in the lattice. It is clear that this very construction is meaningful for any dimensionality, and for every lattice we can think of as long as nearest neighboring sites are well-defined. For instance, when we apply this framework on a square lattice we obtain: $$\label{eq:PepsVB} \begin{overpic}[width = 300pt, unit=1pt]{PepsVB} \end{overpic}$$ (picture taken from MPQ-Garching) where red dots are physical sites. Every site has four bonds, and every bond carries an entangled pair. Then a linear transformation maps the $D^4$-dimensioned fictitious degree of freedom into the $d$-dimensioned physical one, that defines uniquely the PEPS. Since PEPS are fully parametrized by such local transformation, they are actually a Tensor Network $\{L,k,D\}$, with a number of tensors equal to the total number of sites $L$, correlation number equal to the number of neighbors per site plus 1 for the physical link (e.g. $k = 5$ for a square lattice), and arbitrary bondlink dimension $D$. Their TN representation reads $$\label{eq:PepsTN} \begin{overpic}[width = 270pt, unit=1pt]{2DPEPS} \end{overpic}$$ for a square lattice, where vertical links are physical indices. Red boxes are tensors $A^{[\mu]s}_{q_1,q_2,q_3,q_4}$, where $\mu$ is the site, $s$ the local canonical level index, and $q_1 \ldots q_4$ the four correlation indices. The reason why PEPS gathered so much interest in the latest years [@PowaPEPS; @SchuchPEPS], is that they obey the exact entanglement area law for their geometric dimensionality; e.g. a 2D PEPS satisfy a 2D area law of entanglement. It is quite easy to check it, by using the TN-entanglement bound rules we derived in section \[sec:entanet\]. Consider, for instance, a connected subset of sites in the square lattice, and the entanglement entropy related to the reduced density matrix: we mentioned that this entanglement is bound by the minimal number of links one has to cut to separate the graph into two subgraphs (times $\log D$). Then one sees immediately that this minimal number coincides with the *perimeter* of the subset of sites. But that is the 2D area law: for a region scaling like a surface $\sim \ell^2$, the entanglement, $$\mathcal{S}_{\text{VN}} \sim \ell \log D,$$ scales with the characteristic lenghtscale $\ell$ of the region. This suggests that PEPS are optimal tools to approximate ground states of short-ranged Hamiltonians, even because, similarly to their one-dimensional cousins MPS, they also yield a completeness theorem. PEPS have also a huge drawback when compared to MPS, which is their computational complexity [@HardPEPS]. Namely, if we want to contract *exactly* a PEPS, e.g. for acquiring an expectation value, the number of elementary operators needed scales exponentially with the system size, no matter the contraction scheme. This is due to the presence of an extremely high amount of loops in the network, not an easy trouble to work around. Of course one can deal with the problem approximately, by, for example, renormalizing the width of the resulting composite bondlink during contraction; still, we must be careful that the approximation we are adopting should not break down the area law principle. To this goal, several proposals have been introduced in literature [@PEPSgo; @Maricarmen]. Example II:\ Correlator Product States (CPS) ------------------------------- Correlator product states [@CPStrong; @CPSplaq; @CPScompletegraph; @CPS2D] are a simple, but clever, way to adjust correlations in many body states, by means of a whole abelian algebra made of ranged canonical weight-factor operators. They have been acknowledged and used long since; in some sense, they can be regarded as the completely variational generalization of Jastrow factors. They are called ’states’, but it would be more suitable to regard correlator product elements as operators. Indeed they hardly stand alone as a mere Tensor Network structure, because as they must be finite-support and commuting, they are forbidden to establish global symmetries, even the simplest ones like particle number conservation. On the positive side, they excel in preserving symmetries already present in the state prior to their application as operators, so they are often used in conjunction to a starting ansatz: $$\label{eq:CPS} |\Psi_{\text{CPS}}\rangle = \sum_{s_1 \ldots s_L = 1}^{d} \left( \prod_{\alpha} C^{[\alpha]}_{ s_{v(\alpha,1)} \ldots s_{v(\alpha,k)} } \right) \Phi^{0}_{s_1 \ldots s_L} |s_1 \ldots s_L\rangle.$$ In this representation, $|\Phi^{0}\rangle = \sum \Phi^{0}_{s_1 \ldots s_L} |s_1 \ldots s_L\rangle$ is exactly the starting ansatz state, capable of controlling desired symmetries; e.g. nonzero $\Phi^{0}_{s_1 \ldots s_L}$ components should be only those with the right particle number. It is clear that correlator product factors $C^{[\alpha]}$ can not break such symmetry, since sectors with zero amplitude will stay zero amplitude. Nevertheless, $C^{[\alpha]}$ can build correlations, which is especially useful when $|\Phi^{0}\rangle$ is a somehow uncorrelated trial state, like a Slater determinant (in particular, see the MPS representation for Slater determinants in section \[sec:Slater\]). Notice two facts involving : first, the action of correlator product factors $C^{[\alpha]}$ can be applied in any ordering, because they are all diagonal in the canonical basis, so they must commute as they share a basis of eigenvectors altogether. Secondly, is clearly a Tensor Network, since all the relations between the various $C^{[\alpha]}$ and $\Phi^{0}$ are linear; so we can use arguments treated in this chapter to study CPS (e.g. to guess entanglement bounds). You could argue that such picture does not match exactly the definition of TN we gave in , since the same index is shared by more tensors than merely two, but this is easily worked around by adding triple Kronecker delta nodes in the network $\delta^{[3]}_{i,j,k} = \delta_{i,j} \delta_{j,k}$. Precisely, consider a two-site diagonal operator $\Theta$, defined as $$\Theta = \sum_{s_1, s_2 = 1}^{d} C^{[1\leftrightarrow 2]}_{s_1 s_2} \; |s_1 s_2\rangle \langle s_1 s_2| = \!\!\!\sum_{s_1, s_2, r_1, r_2, k_1 k_2}^{d}\!\!\! \delta^{[3]}_{s_1,r_1,k_1} \delta^{[3]}_{s_2,r_2,k_2} C^{[1\leftrightarrow 2]}_{k_1 k_2} |s_1 s_2\rangle \langle r_1 r_2|$$ then we can exploit its diagonal nature to decompose its operator representation even in the diagrammatic scheme $$\label{eq:DiagOp} \begin{overpic}[width = 280pt, unit=1pt]{DiagOp} \put(52, 40){$\Theta$} \put(201, 40){$C^{[1\leftrightarrow 2]}$} \put(26, 6){$s_1$} \put(26, 81){$r_1$} \put(94, 6){$s_2$} \put(94, 81){$r_2$} \end{overpic}$$ where the black nodes are $\delta^{[3]}$ tensors; so the network structure of is well-defined. Of course, correlation capabilities of CPS depend strongly on the specifics of chosen $C^{[\alpha]}$ factors: - **Involved sites** $(k)$ - any number of sites can be chosen to appear as indices in a single $C^{[\alpha]}$ factor. The amount of variational descriptors and thus the speed of simulation algorithm is extremely sensitive to this parameter (usually scaling exponentially with $k$). - **Range** $(\ell)$ - fixed the number of involved sites per correlator $C^{[\alpha]}$ we can still choose the maximal distance among such sites for which we require the corresponding $C^{[\alpha]}$ to be present in the product. - **Shape** - if we are dimension higher than 1, not only the number of involved sites per factor, but also their shape in the lattice is relevant. For example, a common fashion for square lattice is work with plaquette correlator product [@CPSplaq], where every $C^{[\alpha]}$ involve the $k=4$ vertices of a lattice square (or a rescaled lattice square). At any rate, CPS are are considered successful variational tools, for their ridiculously small number of variational parameters, high manipulability, and the capability of adjusting any single correlation by adding just one suitable factor to the product. Choosing a suitable starting state $|\Phi^{0}\rangle$ is also a delicate issue for the CPS ansatz. For instance, a MPS (with modest bondlink $D$) could be a suitable choice; this lead to the construction of hybrid MPS$\leftrightarrow$CPS Tensor Network designs, also known in literature as *Renormalization Algorithm with Graph Enhancement* (RAGE) [@rage1; @rage2]. ### CPS $\Rightarrow$ MPS/PEPS {#sec:MPJastrow} We would like to point out an interesting fact about CPS. a Correlator Product having a finite range $\ell$ of factors which does not scale with system size $L$ clearly describes a finitely-correlated state: this tells us that finite-range CPS can be put in relation with MPS and PEPS, or, more precisely, with Matrix Product Operators and Product Entangled Pair Operators (PEPO - operatorial version of PEPS). A result of my personal thesis work, is that when $\ell$ is finite is always possible to represent a CPS as an analytical subclass of MPS/PEPS with finite correlation bondlink $D$ being a function of $\ell$. Such construction obviously depends on the involved sites per factor and their shape, so we are here going to present the mapping CPS $\to$ MPS in the simplest setting: 1D, and binary factors $k = 2$, kept at all ranges $\ell'$ from neighbors up to an arbitrary finite bound $2 \leq \ell' \leq \ell$. Let us give this prescription starting from $\ell = 1$ and then increasing $\ell$: $$\label{eq:Jas1} \sum_{s_1 \ldots s_L} \left( \prod_{j}^{\tilde{L}} C^{[1,j]}_{s_j s_{j+1}} \right) |s_1 \ldots s_L \rangle \langle s_1 \ldots s_L |,$$ where we are assuming that $C^{[1,j]}$ may depend on the pair of sites $\{j, j+1\}$ on which it is acting, to keep major variational freedom. Also, boundary conditions are small issue: they only set the upper bound $\tilde{L}$ for the $j$ over which the product is taken (to $L$ for PBC, $L-\ell'$ for OBC). Let us draw the diagrammatic representation of , given by $$\label{eq:Jas1pic} \begin{overpic}[width = \textwidth, unit=1pt]{Jas1} \put(72, 50){$C^{[1,j]}$} \put(82, 92){$O^{[j]}$} \put(185, 50){$C^{[1,j+1]}$} \put(304, 50){$C^{[1,j+2]}$} \end{overpic}$$ As you see, it is possible to embed the correlator within an MPO structure $O^{[j]}$ where the product matrices correspond to the blue boxes in , or equivalently $$O^{[j] r}_{s} = \sum_{s_{j+1}} \delta_{r,s}\; C^{[1,j]}_{s, s_{j+1}} \,|s)(s_{j+1}| = \sum_{\alpha, \beta} \; \delta^{[3]}_{\alpha,r,s}\; C^{[1,j]}_{\alpha, \beta} \;|\alpha)(\beta|.$$ Now let us move to $\ell = 2$, where next-nearest neighboring correlator factors $C^{[2,j]}$are introduced. Then the overall correlator product operator is expressed by: $$\label{eq:Jas2pic} \begin{overpic}[width = \textwidth, unit=1pt]{Jas2} \put(78, 49){\footnotesize$C^{[1,j]}$} \put(78, 92){\footnotesize$C^{[2,j]}$} \put(103, 119){\footnotesize$O^{[j]}$} \end{overpic}$$ It is easy to check that the circuitry of links and delta nodes performs the right index plugging at the correlator factors. Indeed, focus on $C^{[2,j]}$ in the picture, while it is obvious that its left link corresponds to $s_j$, you can similarly follow the right link and verify that it is exactly $s_{j+2}$. So the Matrix Product Operator block $O^{[j]}$ is also correct, as it contains all the inner structure (deltas) and variational information (correlators) needed. Equation has a vertical pattern that is suitable to be copied and repeated, every time we add a layer, it is equivalent to add a new set of product correlators, effectively enhancing the maximal range $\ell$ by one. So if our purpose is to describe the action of $$\label{eq:Jasel} \sum_{s_1 \ldots s_L} \left( \prod_{\ell' = 1}^{\ell} \; \prod_{j}^{\tilde{L}} C^{[\ell',j]}_{s_j s_{j+\ell'}} \right) |s_1 \ldots s_L \rangle \langle s_1 \ldots s_L |,$$ then the corresponding Matrix Product Operator representation would read $$O^{[j] r}_{s} = \sum_{\{\alpha\}, \{\beta\}} \left( \prod_{\ell' = 1}^{\ell} C^{[\ell',j]}_{s,\beta_{\ell'}} \right) \delta^{[3]}_{\alpha_1,s,r} \left( \prod_{\ell' = 1}^{\ell-1} \delta_{\alpha_{\ell'+1}, \beta_{\ell'}} \right) |\alpha_1 \ldots \alpha_{\ell})(\beta_1 \ldots \beta_{\ell}|$$ For better comprehension, let us sketch it for $\ell = 4$: $$\label{eq:Jastrowpic} \begin{overpic}[width = 200pt, unit=1pt]{Jastrow} \put(106, 36){$C^{[1,j]}$} \put(106, 100){$C^{[2,j]}$} \put(106, 164){$C^{[3,j]}$} \put(106, 228){$C^{[4,j]}$} \put(146, 260){$O^{[j]}$} \end{overpic}$$ The total correlation bondlink dimension used for this representation is $d^{\ell}$, it should be convincing that this representation is also optimal (if no further restraints on the $C$ factors are requested), since we are transferring through the MPO bondlink the minimal information to reproduce the factors exactly. In conclusion, we can establish a general entanglement bound on 1D binary Correlator Product States, which is $$\mathcal{S}_{\text{VN}} \leq \ell \log d,$$ characterizing the correct 1D area law of entanglement if the maximal range $\ell$ does not scale with the total length $L$. The scheme we just presented has been originally conceived by me during my doctorateship work, and supported by R. Fazio and F. Becca. Towards hierarchical Tensor Networks ------------------------------------ Tensor Network architectures can be put in tight relation with numerical renormalization groups. In fact, consider a tensor $T^{[\cdot]}$ attached to some of the physical indices; it can be as well interpreted as the action of a linear transformation acting on the local density matrix, mapping it into the virtual links space. As the overall effective dimension is typically reduced, a numerical renormalization is taking place. Tensor Networks entanglement bounds guarantee that the amount of correlation in the TN tailored variational state matches the entanglement that can be built via renormalization process. We want now to analyze and describe detailed properties of another renown class of Tensor Networks, corresponding to the original Wilson’s real-space numerical RG. We are talking about Tree networks, and of course also their recent generalization: Multiscale Entanglement Renormalization Ansatz (MERA). These two network geometries share the intriguing property of embedding a scale-invariance in their pattern, so they are ideal candidates for dealing with critical models. We will classify them together as *hierarchical* Tensor Network states because network nodes are linked according to hierarchical relations. The interest revolving around these methods, and the following in-depth analytical study, are such that it is appropriate to devote an entire chapter to describe their features and peculiarities. Trees and MERA {#chap:TTNMERA} ============== When the Matrix Product State representation as tailored wavefunction paradigm version of the DMRG was realized and understood, it was an important breakthrough. But it was with the advent of Tree Tensor Networks (TTN) [@VidalTree1; @IoTree] and Multiscale Entanglement Renormalization Ansatz (MERA) [@MERAzero; @MERAalgo; @MERAevo; @MERAgauge; @MERAtopo; @QuMERAchan; @BERA] that the computational physicists’ community started talking about Tensor Networks in general. TTN and MERA, similarly to MPS, have a network pattern which is not only very simple and highly adaptive but also self-similar. Nevertheless, the relevant difference between TTN/MERA and MPS is that, while self-similarity in MPS arises every time a new site is added, for (binary) TTN/MERA it happens every time the number of sites is *doubled* (or tripled for ternary TTN/MERA, and so forth). This suggests us that, as we will see, TTN/MERA bear somehow a resemblance to real-space renormalization processes, and also that they should be especially suitable for describing systems in which a scale invariance is emergent. Most of the results provided in this chapter are original work developed by me and was supported by V. Giovannetti, M. Rizzi, S. Montangero and R. Fazio. Real-Space numerical renormalization ------------------------------------ In section \[sec:dmrgtomps\] we argumented thoroughly how a Matrix Product State can be explicitly built from the data of the recursive (standard) density matrix renormalization transformations. Let us follow a similar path now, but with a different starting algorithm in mind. For consistence we will work in 1D. We are again assuming that numerical renormalization process is being performed, but not according to the traditional DMRG framework, where at every step the degree of freedom of a single site is added to the picture, and the joint density matrix is then renormalized, i.e. old block $\cup$ added site $\to$ new block ($D \otimes d \to D$). Instead, here we are assuming that the renormalized degree of freedom of the old block is doubled, or equivalently, coupled with a copy of itself. Now we select a new state within this space (according to whatever benchmark we prefer), describe its density matrix, and then renormalize, i.e. old block $\cup$ another old block $\to$ new block ($D \otimes D \to D$). What inspires this proposal is that during the same time we perform a density matrix renormalization step, 1D lattice sites in the real-space are also renormalized, in a coarse-graining fashion. Let us now recall equation , used in a standard-DMRG, while we are right-propagating the scheme, and keeping $D \times D$ renormalized density matrices $\tilde{\rho}_{\ell} = \sum_{j=1}^{D} p_j \;|L_j\rangle \langle L_j|$. The basic step of the algorithm could be expressed through $A^{[\ell]}$ as $$|L_j\rangle_{\ell}^{L} = \sum_{k = 1}^D \sum_{s = 1}^d A_{k,j}^{[\ell] s} \; |L_k\rangle_{\ell-1}^{L} \otimes |s\rangle_{\ell}.$$ and $A^{[\ell]}$ became the elementary tensor of MPS description. Let us write the corresponding recursive expansion for real-space numerical RG: if no translational invariance assumption is made (for instance, we could be in an OBC setting) we must keep track of every $\tilde{\rho}^{[h]}_{\ell}$, where $h$ tells us how many times the renormalization has been performed already, and $\ell$ carries the information on which original sites the density matrix is actually describing. With these considerations, and defining $\tilde{\rho}^{[h]}_{\ell} = \sum_{j=1}^{D} p_j \;|M_j\rangle^{[h]}_{\ell} \langle M_j|$, we get $$|M_j \rangle_{\ell}^{[h]} = \sum_{k_1, k_2 = 1}^{D_{h-1}} \Lambda_{k_1, k_2}^{[h,\ell]\, j} \;\, |M_{k_1} \rangle_{2\ell-1}^{[h-1]} \otimes |M_{k_1} \rangle_{2\ell-1}^{[h-1]}.$$ where $h$ goes from $0$ to $\bar{h} = \log L$, with $L$ the system size, while $\ell$ ranges from $1$ to $2^{-h}L$ (notice that the number of allowed $\ell$ values halves at every layer $h$ increase, which is exactly the coarse graining). Finally, $|M_j \rangle_{\ell}^{[h]}$ goes from $1$ to the renormalization dimension $D_{h}$, eventually depending on $h$. You should convince yourself that with this procedure, the state is completely and uniquely characterized by the set of $\Lambda^{[h,\ell]}$, which is a three indices complex tensor. Since the global state $|\Psi\rangle$, when written in the canonical separable basis, has components determined by linear contraction relations between the $\Lambda^{[h,\ell]}$, it is a Tensor Network state. Precisely, it reads $$|\Psi\rangle = \left( \bigotimes_{\ell = 1}^{2^{\bar{h} -1}} \Lambda^{[1,\ell]} \right) \cdot \left( \bigotimes_{\ell = 1}^{2^{\bar{h} -2}} \Lambda^{[2,\ell]} \right) \ldots \left( \bigotimes_{\ell = 1}^{2} \Lambda^{[\bar{h}-1,\ell]} \right) |\mathcal{C}\rangle,$$ where we wrote the $\Lambda$ as $D_h^{2} \times D_{h+1}$ matrices. Equivalently, in the diagrammatic Tensor Network representation it appears like this $$\label{eq:TN2pic} \begin{overpic}[width = \textwidth, unit=1pt]{TTN_E2} \put(-2, 27){\footnotesize $\Lambda^{[1,1]}$} \put(16, 55){\footnotesize $\Lambda^{[2,1]}$} \put(115, 54){\footnotesize $\Lambda^{[2,2]}$} \put(63, 85){\footnotesize $\Lambda^{[3,1]}$} \put(170, 116){\footnotesize $\mathcal{C}$} \end{overpic}$$ as you see it is a tree graph, with branching number $b = 2$. The tensor $\mathcal{C}$ standing at the top of the structure is the only one which is topologically different, as it has only two links, and due to its placement is often referred to as *hat* (or *root*) tensor. From equation the coarse graining action performed by renormalizers $\Lambda$ becomes immediate and clear: every *layer* of $\Lambda$ tensors maps an adjacent pair of (eventually already renormalized) sites into a single renormalized site, thus effectively halving the overall size of the system. Once $L$ has been reduced to $2$ we describe it as a simple binary state $|\mathcal{C}\rangle$. Due to hermiticity of every $\tilde{\rho}^{[h]}_{\ell}$ we would like for their respective eigenvectors $|M_j \rangle_{\ell}^{[h]}$ to be an orthonormal set, at every $h$ and $\ell$. As for MPS, where a similar requirement lead to a gauge symmetry breaking, this restraint translates into a condition that every $\Lambda$ must satisfy, namely $$\label{eq:TTNisometr} \delta_{j_1, j_2} = \sum_{k_1, k_2}^{D_{h-1}} \Lambda^{\star\,[h,\ell]\,j_1}_ {k_1, k_2} \Lambda^{[h,\ell]\,j_2}_ {k_1, k_2}, \qquad \forall\;\{h,\ell\},$$ where $\vphantom{A}^{\star}$ stands for complex conjugation. In other words, every $\Lambda$, read as a $D^2_{h-1} \times D_{h}$ matrix, must be (left-) isometric, i.e. $\Lambda^{\dagger} \Lambda = {{\mathbb 1}}$. This is indeed a gauge symmetry breaking. Truly, this is exactly the peripheral gauge we defined in section \[sec:peripheral\], when the nucleus $\Pi$ corresponds to the hat tensor $\mathcal{C}$. We remarked that, for a given Tensor Network with no closed loops (as a tree graph is), it is always possible to find the gauge transformation that maps it into the peripheral gauge, no matter the starting state: this tells us that the isometricity condition for $\Lambda$ carries no loss of generality at all. You can guess that the binary character $b = 2$ of the tree network in is due to the fact that we renormalized just two copies of the old block density matrices into a new one. Of course, mapping an arbitrary number $b$ of copies of the old block into a single one leads to tree Tensor Networks with the corresponding branching number $b$. For example, we could have a ternary tree Tensor Network when $b = 3$: $$\label{eq:TN3pic} \begin{overpic}[width = \textwidth, unit=1pt]{TTN_E3} \put(0, 27){\footnotesize $\Lambda^{[1,1]}$} \put(33, 57){\footnotesize $\Lambda^{[2,1]}$} \put(166, 58){\footnotesize $\Lambda^{[2,2]}$} \put(173, 102){\footnotesize $\mathcal{C}$} \end{overpic}$$ where the total number of layers $\bar{h}$ would be $\bar{h} = \log_3 L$. Throughout this chapter we will present various results, deriving our calculations in accordance to the binary tree Tensor Network (2TTN) design. We want to remark, however, that most of the claims and statements extend naturally to TTN with higher branching number: typically one has just to substitute $2$ with $b$ where appropriate (usually as a logarithm basis) to get the right result. Moreover, notice that we disregarded any assumption on boundary conditions so far. Indeed, unlikely from Matrix Product States, where different boundary conditions lead to substantially different network topologies (no closed loop in OBC, but with closed loop in PBC), in TTN representations of a state, like and , the network topology is insensitive to the presence of a boundary. This, as we will see, will naturally bring a unique and well-built definition of the thermodynamical limit. Tree network entanglement instability and the introduction of MERA {#sec:MERAintro} ------------------------------------------------------------------ If we wish for our real-space self-similar Tree network to be actually capable of describing 1D critical states, we must fist check that the amount of entanglement it can hold satisfies the typical entanglement area-law violation: $$\mathcal{S}_{\text{VN}} (\rho_{\ell}) = \frac{c}{3} \log \ell + \mbox{const.}$$ where $\rho_{\ell}$ is the density matrix of $\ell$ adjacent sites, and in PBC conditions. Also we are assuming that we are approaching the thermodynamical limit, or at least that the region $\ell$ is too small to capture finite size effects: $\ell \ll L$. In order to evaluate upper bounds to the entanglement of a TTN, we are going to employ the Tensor Network entanglement arguments we provided in section \[sec:entanet\]. There we found that the entanglement of a partition $\Gamma_{\ell}$ is bounded by $\sim D^{\chi}$ where $\chi$ is the minimal number of broken links needed to separate the network graph into two subgraphs, respectively containing $\Gamma_{\ell}$ and its complementary $\Gamma_{\ell}^{\text{c}}$. Let us perform the count for binary TTN. First, notice that, since the Tree network design is not translationally-invariant defined, we will expect that $\mathcal{S}_{\text{VN}} (\rho_{\{\ell_1,\ell_1+\ell\}})$ will not depend only on the number of sites $\ell$ in the interval, but also on its placement $\ell_1$. Now we proceed recursively with layers. If $\ell_1$ is odd, then I have to cut no link in the lowest layer, if it is even, i cut one link; either way, i move up one layer and site $\ell_1$ is mapped into renormalized site $\lceil \ell_1 / 2 \rceil$ (where $\lceil x \rceil$ is the smallest integer which is larger than $x$), and we can repeat the procedure. The same argument holds for the other region boundary $\ell_2 = \ell_1 + \ell$, even though the link breaking is needed when $\ell_2$ is odd and not when it is even. In conclusion we have that $\mathcal{S}_{\text{VN}} (\rho_{\{\ell_1,\ell_1+\ell\}}) \leq \# \mbox{cuts} \cdot \log D$, where $$\label{eq:fluctucut} 1 \leq \# \mbox{cuts} \leq 2 \bar{h} = 2 \log_b \ell,$$ depending on $\ell_1$. For clarity, let us show an example of the described procedure when $L \geq 16$ ($\bar{h} \geq 4$), are we are considering $\rho_{3,11}$: $$\label{eq:TN2cut} \begin{overpic}[width = \textwidth, unit=1pt]{TTN_Cut2} \end{overpic}$$ where we have drawn only a branch of the whole Tree network. In this example the entanglement bound is given by $\sim 4 \log D$ (actually by $\sim [3 \log D + \log d]$ since one of the links to be cut is a physical one). Result is quite relevant: it tells us that it is true that in a TTN state one can choose a sequence of subsystems of growing size $\ell$ exhibiting the correct area-law violation scaling, but it is also true that it exists another sequence which is nearly separable. In other words, we can state that the entangling capacity of a TTN *fluctuates* widely with translations, ranging between almost no entanglement and critical entanglement. Such *translational instability* is embedded within the nature of Tree networks themselves, and must be handled with care. In particular, in order to work around this entanglement instability, and representing a variational quantum state showing a more smooth area law violation, two methods are adopted: ***Incoherent translational mixture -*** This path focuses on calculating the translational average every time we wish to acquire an expectation value on the TTN state. In practice, let $\Theta_{\{\ell_\alpha\}_\alpha}$ be an observable having support on sites $\{\ell_\alpha\}_\alpha$, then what we are actually interested in is $$\langle \bar{\Theta} \rangle = \frac{1}{L} \sum_{\ell = 1}^{L} \langle \Theta_{\{\ell + \ell_\alpha\}_\alpha} \rangle.$$ By doing so we always integrate out translational fluctuations, as we are considering solely the zero Fourier mode. This operatorial average procedure is theoretically equivalent to consider a state which is the incoherent mixture of all possible translations of the original TTN state. Namely we define $$\label{eq:incoherentrasl} \bar{\rho}_{\{\ell_\alpha\}_\alpha} = \frac{1}{L} \sum_{\ell = 1}^{L} \rho_{\{\ell+\ell_\alpha\}_\alpha}.$$ for any choice of the support $\{\ell_\alpha\}_\alpha$ and we have represented a state, which is translational by construction: $\bar{\rho}_{\{\ell_\alpha\}_\alpha} = \bar{\rho}_{\{\ell'+\ell_\alpha\}_\alpha}$. Notice that we are *not* considering the coherent superposition of the TTN state translations, because the interference graphs can not be contracted efficiently due to the presence of several closed loops. We can now combine the entanglement argument of Tensor Networks with the concavity property of Von Neumann entropy, and obtain an overall bound on the entanglement for state : $$\label{eq:traslmix} \mathcal{S}_{\text{VN}}\left( \bar{\rho}_{\ell} \right) \leq \log_2 \ell \cdot \log D$$ which satisfies the right logarithmic behavior, and is perfectly smooth under translations. ***Adding new tensor elements: MERA -*** The idea of MERA arose from the request of saving some information on correlations shared by neighboring sites scheduled to be renormalized into different, separate blocks. For instance, in site 7 and site 8, despite being neighbors, are not going to be renormalized together until the hat is reached: this means that most of the entanglement they share is likely to be lost in the renormalization process, and poorly described by the corresponding Tensor Network design. Therefore, the scheme proposed by G. Vidal [@MERAzero] consists into performing a (unitary) operation $X$ coupling these sites, whose purpose is to *disentangle* the two as much as possible before renormalizing them separately. This way, the information concerning the original entanglement of the pair is stored within this operation $X$, and renormalization needs not to concern about it. Disentangling operations and real-space renormalizators are then applied alternately. It is clear that, since unitary disentanglers are linear operations, quantum states achievable by this process are again Tensor Network states. They are known as Multiscale Entanglement Renormalization Ansatz states, and show a hierarchical and real-space self-similar pattern just like TTN states. Here we present a partial frame of a binary MERA Tensor Network $$\label{eq:MERA2pic} \begin{overpic}[width = \textwidth, unit=1pt]{Mera_E2} \put(43, 56){\footnotesize $\Lambda^{[h,\ell]}$} \put(90, 25){\footnotesize $X^{[h,\ell]}$} \end{overpic}$$ where for every layer $h$ and horizontal position $\ell$, tensors are chosen to be unitary/isometric, i.e. $X^{\dagger} X = {{\mathbb 1}}_{D^2 \times D^2}$ and $\Lambda^{\dagger} \Lambda = {{\mathbb 1}}_{D \times D}$. Of course, the corresponding MERA version can be modeled on a Tree Network of arbitrary branching number; in literature, both binary and ternary MERA have been considerably used for simulation purposes. For completeness, let us draw the diagrammatic pattern of a ternary MERA: $$\label{eq:MERA3pic} \begin{overpic}[width = \textwidth, unit=1pt]{Mera_E3} \put(43, 50){\footnotesize $\Lambda^{[h,\ell]}$} \put(58, 32){\footnotesize $X^{[h,\ell]}$} \end{overpic}$$ We will show now that introducing the new Tensor elements $X$ plays a fundamental role in regularizing the entanglement scaling law under translation, even though MERA are again, like TTN, a non-translational network design. Let us assume, as before, that we are to evaluate the entanglement entropy of $\rho_{\{\ell_1, \ell_1+\ell\}}$, estimating an upper bound via the Tensor Network link-cut argument, and proceed recursively layer-by-layer (a MERA full layer is composed by stacking together disentanglers $X^{[h,\bullet]}$ and coarse-grainers $\Lambda^{[h,\bullet]}$). For every layer, and each one of the two region boundaries ($\ell_1$ and $\ell+\ell_1$), I have to cut either one or two links, depending on the region location with respect to the MERA network geometry, as you see from this example $$\label{eq:MERAcut} \begin{overpic}[width = 280pt, unit=1pt]{Meracut} \end{overpic}$$ This leads to a bounding function for the entanglement $\mathcal{S}_{\text{VN}} (\rho_{\ell_1,\ell_1+\ell}) \leq \#\mbox{cuts} \cdot \log D$, where $$\label{eq:MERAfluctucut} 2 \log_b \ell \leq \# \mbox{cuts} \leq 4 \log_b \ell,$$ depending on $\ell_1$. It is clear from that entanglement-bound fluctuations are present even in the MERA case. However, differently from they are hardly a problem: indeed even in the worst case scenario $\mathcal{S}_{\text{VN}} (\rho_{\ell_1,\ell_1+\ell}) \leq 2 \log_b \ell \cdot \log D$: the entanglement is ruled by a logarithmic violation of the 1D area-law, as we wished. This is the main reason why MERA are kept in high regard in 1D critical systems simulations. Despite the enhanced accuracy given by disentangling elements of a MERA, we would like to show, throughout this chapter, that MERA and TTN representations manifest really a common behavior, as they can both easily capture critical properties of strongly-correlated states, e.g. in terms of critical exponents, primary fields, and so forth. Acknowledging that, intuition suggests that renormalizer operations $\Lambda$ are those responsible for keeping track of long-range properties of the TTN/MERA state, while disentanglers $X$ are mostly used to adjust the local variational structure. To avoid translational invariance-breaking issues, we will make large use of translational averages of operators and correlators in the following sections, thus substantially adopting the incoherent mixture paradigm of whenever possible, even for MERA. Similarly, notice that TTN can be definitely seen as a subclass of MERA. They actually correspond to MERA networks where all the disentangling tensors are set equal to the identity, i.e. $X^{[h,\ell]} = {{\mathbb 1}}_{D^2 \times D^2}$ $\forall h,\ell$. One additional remark: we stated that Tree networks are suitable for any boundary condition we might inquire, this is due to the presence of dense graph frontiers in their geometry. On the other hand the MERA Tensor Network has a natural attitude for preferring Periodic boundary conditions. Indeed, as you see from and , each site has a network path through the lowest layer linking it to both its neighbors, recalling somehow the valence bond picture idea for Periodic MPS. Precisely, one can represent the whole 1D binary MERA state as follows: $$\label{eq:MERAPBC} \begin{overpic}[width = 250pt, unit=1pt]{MerPBC} \put(14, 46){$\ell$} \put(46, 102){$h$} \end{overpic}$$ (figure taken from ref. ) where the ring of physical sites stands at the edge of the circle. 3-link nodes are $\Lambda$ tensors, 4-link ones are $X$ tensors, and the tensor in the very center of the diagram is a 4-legs hat tensor $|\mathcal{C}_4\rangle$. You can easily check that is exactly the local pattern of . Of course, a generalization of MERA for OBC can clearly be designed by adjusting its geometry, we will treat this point in section \[sec:BERA\]. Causal cone property {#sec:causalcone} -------------------- A remarkable feature, important both from analytical and computational point of view, shared by TTN and MERA alike is the so-called causal cone property. It is a generalization of the peripheral gauge paradigm, allowing us to contract at no expense whole branches of the Tensor Network, that extends also to MERA geometry, despite having closed loops in their graph, thanks to the isometricity requirements for $X$ and $\Lambda$ tensors. Assume we want to achieve the expectation value over a 1D TTN/MERA state for an observable $\Theta_{\{\ell_1 .. \ell_2 \}}$ whose support is an interval of sites, say $\{\ell_1 .. \ell_2 \}$. Then, by adopting the standard network contraction method, one performs $\langle \Psi | \Theta | \Psi \rangle = \sum_{\{r\},\{s\}} \mathcal{T}^{\star}_{\{r\}} \langle r_{\ell_1} \ldots r_{\ell_2}|\Theta| s_{\ell_1} \ldots s_{\ell_2} \rangle \mathcal{T}_{\{s\}}$. But now, consider the lower layer of disentangler $X^{[1,\ell]}$ (for a MERA), which are unitary by requirement $X^{\dagger} X = {{\mathbb 1}}$. Those $X$ whose (both) physical indices do not belong to the support interval $\{\ell_1 .. \ell_2 \}$ are contracted with their adjoint and automatically vanish. Precisely, we got rid of a whole layer of disentanglers except for those directly connected to the support. The same argument holds for renormalizers $\Lambda$ belonging to the lower layer, which disappear alike, as sketched in this diagram: $$\label{eq:ccone1} \begin{overpic}[width = \textwidth, unit=1pt]{Mapup} \put(0, 145){$a)$} \put(220, 145){$b)$} \end{overpic}$$ As we eliminated an entire (double) layer from the Tensor Network state and its conjugate, we could say that we mapped the original expectation value problem into a new one: $$\langle \Psi | \Theta | \Psi \rangle = \langle \Psi' | \mathcal{A}_{\{\ell_1 .. \ell_2\}}\left(\Theta\right) | \Psi' \rangle$$ where the quantum state $|\Psi'\rangle$ is the original TTN/MERA state without the bottom layer, and the new observable $\mathcal{A}_{\{\ell_1 .. \ell_2\}}\left(\Theta\right)$ is the big composite pink tensor of equation (\[eq:ccone1\].b). Acquiring $\mathcal{A}_{\{\ell_1 .. \ell_2\}}\left(\Theta\right)$ starting from $\Theta$, $X$ and $\Lambda$ tensors is a finite operation, whose complexity scales with the size of the support $\ell_2 - \ell_1$, *not* with the size of the system $L$. Moreover, when mapping $\Theta \to \mathcal{A}_{\{\ell_1 .. \ell_2\}}\left(\Theta\right)$ the actual support typically shrinks: you see how in we map a 5-site observable into a 4-site one. We can obviously refrain this argument, every time mapping up the effective observable in a recursive fashion $$\label{eq:mapup2} \Theta^{[h]} \to \Theta^{[h+1]} = \mathcal{A}^{[h]}_{\{\ell_1^{h} .. \ell_2^{h}\}}(\Theta^{[h]})$$ until we reach the hat tensor $|\mathcal{C}\rangle$, for which it holds $\langle \Psi | \Theta | \Psi \rangle = \langle \mathcal{C} | \Theta^{[\bar{h}-1]} | \mathcal{C}\rangle$. Each mapping contraction has a limited computational cost, and we have to perform it a number of times equal to the total number of layers $\bar{h}$, but we know that $2^{\bar{h}} = L$, the total system size. In conclusion, for binary TTN/MERA the computational cost for evaluating compact-support observables scales as $$\# \mbox{cost} \propto \log_2 L.$$ Obviously, tensors canceled out by isometricity relations can not influence in any way the result of the expectation value, so all the physical properties of sites in $\{\ell_1 \ldots \ell_2 \}$, and so their reduced density matrix, can be determined by a small subset of tensors in the network. Namely, graph nodes we can reach, starting from physical links $\{\ell_1 \ldots \ell_2 \}$, by means of sole vertical network propagation (only moves that increase the layer index are accepted). If we want to use a language formalism borrowed from relativity, those ’influent’ tensors form the *causal cone* of sites $\{\ell_1 \ldots \ell_2 \}$. Here we show you an example for a binary MERA with $L = 16$ sites: $$\label{eq:MERAcc} \begin{overpic}[width = 340pt, unit=1pt]{MERAcc} \put(167, 106){$\mathcal{C}$} \end{overpic}$$ the yellow region is the causal cone of sites $\{4,5,6\}$, it contains only 4 disentanglers $X$, and 5 renormalizators $\Lambda$, out of 25 tensors in the whole network. Causal cone ’horizontal’ sizes (width) are dynamic while moving across the layers, as already pointed out. In practice, given support size and placement of the pre-mapping observable, by simple diagram contraction rules one can determine the size of the mapped observable, which depend also on the chosen network geometry and branching number $b$. Precisely for $b = 2$: $$\mbox{binary TTN} \left[ \begin{aligned} 2 \ell - 1 &\longrightarrow \ell \\ 2 \ell &\longrightarrow \left\{ \begin{array}{cc} \ell &\mbox{if $\ell_1$ odd} \\ \ell+1 &\mbox{if $\ell_1$ even} \end{array} \right. \end{aligned}\right.$$ $$\label{eq:ccsizerulez} \mbox{binary MERA} \left[ \begin{aligned} 2 \ell - 1 &\longrightarrow \ell+1 \\ 2 \ell &\longrightarrow \left\{ \begin{array}{cc} \ell+1 &\mbox{if $\ell_1$ odd} \\ \ell+2 &\mbox{if $\ell_1$ even} \end{array} \right. \end{aligned}\right.$$ We immediately see that for binary TTN geometry, a cone width of 1 is *stable* through the causal path. Accordingly, a size of 3 or greater shrinks in an almost-exponential fashion with layers. Finally, an observable supporting 2 adjacent sites may either be mapped into a one-site observable, or a two-site one, depending on how these sites match the network structure: we will say, then, that 2-sites is a *metastable* size, as it may collapse to 1 but not necessarily, nor immediately. This remains true for higher branching number $b$ TTN states. For a MERA state it can even happen that we have to grow the causal cone width, e.g. if the observable was one-site is mapped into a two-site one. In binary MERA geometry, 3 is the only stable size, while 2 and 4 are metastable. When moving to a ternary MERA geometry (or a higher branching $b$ one), we see that these characteristic widths are smaller: precisely size 2 is stable, sizes 1 and 3 metastable (but 3 becomes unstable if $b > 3$). Ascending and descending maps ----------------------------- In this section we will focus on those maps that propagate operators upwards trough the layers, like the one of equation . According to their nature, we will call them *ascending* maps, and relate them to completely positive maps described in appendix \[app:cptchap\]. For simplicity, let us start with a one-site operator $\Theta_{\ell}$ acting on a binary tree state. We acknowledged that size 1 is a stable causal cone width for this TTN geometry, so the mapped operator $\mathcal{A}_{\{\ell\}}(\Theta)$ will still be one-site. Let us express analytically the action of this mapping, which in turn depends on the parity of $\ell$ $$\label{eq:TNmap1} \mathcal{A}_{\{\ell\}}(\Theta) = \left\{ \begin{aligned} \mathcal{A}_{L}(\Theta) &\equiv \Lambda^{\dagger} (\Theta \otimes {{\mathbb 1}}) \Lambda \qquad \mbox{if $\ell$ odd} \\ \mathcal{A}_{R}(\Theta) &\equiv \Lambda^{\dagger} ({{\mathbb 1}}\otimes \Theta) \Lambda \qquad \mbox{if $\ell$ even} \end{aligned} \right.$$ where the subscript $L/R$ refers to which of the two lower links of the node tensor $\Lambda$ (left or right) is touching site $\ell$. Of course, tensor $\Lambda$ also depends on the layer $h$ and horizontal position $\ell$ that locate its placement within the network, and thus similarly will do maps $\mathcal{A}_{L}$ and $\mathcal{A}_{R}$ (however we shall treat implicitly this dependence to avoid carrying on too many indices). It becomes immediately clear from that both $\mathcal{A}_{R}$ and $\mathcal{A}_{L}$ maps are Completely Positive and Unital. This is due to the fact that we already expressed the Kraus expansion of the mapping, indeed $\Lambda^{i}_{j,k}$ form a Kraus set over $k$ of $D \times D$ matrices $\Lambda^{[L]}_{k} = \sum_{i,j} \Lambda^{i}_{j,k} |i\rangle \langle j|$, and another set over $j$ of matrices $\Lambda^{[R]}_{j} = \sum_{i,k} \Lambda^{i}_{j,k} |i\rangle \langle k|$. Therefore $$\mathcal{A}_{L}(\Theta) = \sum_{k} {\Lambda^{[L]}_k}^{\dagger} \Theta {\Lambda^{[L]}_k} \quad \mbox{and} \quad \mathcal{A}_{R}(\Theta) = \sum_{j} {\Lambda^{[R]}_j}^{\dagger} \Theta {\Lambda^{[R]}_j},$$ but due to isometricity condition on renormalizators, we have $$\sum_{k} {\Lambda^{[L]}_k}^{\dagger} \Lambda^{[L]}_k = \sum_{j} {\Lambda^{[R]}_j}^{\dagger} \Lambda^{[R]}_j = \Lambda^{\dagger} \Lambda = \mathcal{A}_{R}({{\mathbb 1}}) = \mathcal{A}_{L}({{\mathbb 1}}) = {{\mathbb 1}},$$ which is exactly the Kraus set requirement . Interestingly enough, it is relevant that this map should be completely positive; the reason becomes clear one we introduce the formalism of *descending maps* for density matrices. In fact, since $\Theta$ has 1-site support, the reduced density matrix of that site is the sole responsible for determining expectation values: $$\langle \Psi | \Theta | \Psi \rangle = {\text{Tr}}\left[ \Theta \cdot \rho_{\ell} \right].$$ The causal cone argument tells us that this is also equivalent to contracting $\mathcal{A}_{\{\ell\}}(\Theta)$ with the TTN state without the bottom layer, on which $\mathcal{A}_{\{\ell\}}(\Theta)$ acts again as a single site operator, so that $$\label{eq:uptodown} \langle \Psi' | \mathcal{A}_{\{\ell\}}(\Theta) | \Psi' \rangle = {\text{Tr}}\left[ \mathcal{A}_{\{\ell\}}(\Theta) \cdot \rho'_{\ell'} \right] = {\text{Tr}}\left[ \Theta \cdot \mathcal{D}_{\{\ell\}}(\rho'_{\ell'}) \right],$$ where $\rho'_{\ell'}$ is the reduced density matrix of $|\Psi'\rangle$ at site $\ell' = \lceil \ell/2 \rceil$, and $\mathcal{D}_{\{\ell\}}$ is the map adjoint to $\mathcal{A}_{\{\ell\}}$ with respect to the trace scalar product for matrices $(A|B) = {\text{Tr}}[A^{\dagger} B]$. It is clear that since holds for the whole algebra of 1-site observables $\Theta$ it must necessarily be that $$\label{eq:meaningful} \rho_{\ell} = \mathcal{D}_{\{\ell\}}(\rho'_{\ell'}).$$ Luckily $\mathcal{D}_{\{\ell\}}$, being adjoint of a CP-unital map, is a completely-positive trace-preserving (CPT) map, which takes density matrices to density matrices and thus is perfectly meaningful. As pointed out in ref. [@QuMERAchan], applying $\mathcal{D}_{\{\ell\}}$ can be actually seen as the action of a quantum channel [@Nielsen]. The equivalence in can be interpreted as some TTN version of the Heisenberg $\leftrightarrow$ Shrödinger scheme duality: we can map ascending-wise operators, or descending-wise density matrices, the expectation value is achieved either way and the number of contractions is the same. Similarly to $\mathcal{A}_{\{\ell\}}$, $\mathcal{D}_{\{\ell\}}$ also depends on the parity of target site $\ell$, namely $$\label{eq:TNmap2} \begin{aligned} \mathcal{D}_{L}(\rho') &\equiv {\text{Tr}}_2 \left[ \Lambda \,\rho'\, \Lambda^{\dagger} \right] = \sum_{k} {\Lambda^{[L]}_k} \,\rho'\, {\Lambda^{[L]}_k}^{\dagger} \\ \mathcal{D}_{R}(\rho') &\equiv {\text{Tr}}_1 \left[ \Lambda \,\rho'\, \Lambda^{\dagger} \right] = \sum_{j} {\Lambda^{[R]}_j} \,\rho'\, {\Lambda^{[R]}_j}^{\dagger}, \end{aligned}$$ both CPT, and expressed in Kraus formalism. Performing the algebraic contraction of any of the maps defined in and requires a number of elementary operations scaling like $\sim D^{4}$, as you can easily guess from the following diagram representation: $$\label{eq:AscDesc1} \begin{overpic}[width = \textwidth, unit=1pt]{Maps1s} \put(0, 51){\large $\mathcal{A}_L$} \put(99, 51){\large $\mathcal{A}_R$} \put(203, 51){\large $\mathcal{D}_L$} \put(300, 51){\large $\mathcal{D}_R$} \end{overpic}$$ We presented the ascending/descending map formulation for one-site operator/density matrix, but we are not surprised that it extends naturally for any size, according to the causal cone width rules . Clearly, ascending maps will typically reduce the size of the effective operator, and at the same time, descending maps with a given target size will be function of density matrices with smaller size. Let us consider the case of two adjacent sites $\{\ell_1, \ell_1+1\}$, depending whether $\ell_1$ is even or odd we have: $$\label{eq:TNmap3} \begin{aligned} \rho_{\{2\ell-1, 2\ell\}} &= \mathcal{S}(\rho'_{\ell}) = \Lambda \,\rho'_{\ell}\, \Lambda^{\dagger}\\ \rho_{\{2\ell, 2\ell+1\}} &= \mathcal{D}_R \otimes \mathcal{D}_L (\rho'_{\{\ell, \ell+1\}}) \end{aligned}$$ where $\mathcal{S}$ is the CPT map obtained by just multiplying the density matrix by $\Lambda$ and $\Lambda^{\dagger}$; and notice that it holds $\mathcal{D}_L(\cdot) = {\text{Tr}}_R [ \mathcal{S}(\cdot)]$ and $\mathcal{D}_R(\cdot) = {\text{Tr}}_L[\mathcal{S}(\cdot)]$. The map appearing in the bottom line of is separable, so it can only decrease entanglement, and likely it will since it is not an invertible transformation and typically satisfy the mixing property (see appendix \[app:mixin\]). Let us sketch the diagrams for the descending operators of as well: $$\label{eq:TNmap4} \begin{overpic}[width = 330pt, unit=1pt]{Maps2s} \put(20, 65){\large $\mathcal{S}$} \put(115, 65){\large $\mathcal{D}_R \otimes \mathcal{D}_L$} \end{overpic}$$ the ascending ones are simply defined by taking the adjoint of . Due to the fact that TTN has a renormalization flow which is quasi-local at every step, it is easy to check that maps $\mathcal{S}$, $\mathcal{D}_L$ and $\mathcal{D}_R$ are the *only* ingredients we need to build all the maps for any given size. For instance, let us write the generic descending map for the binary TTN, in order to obtain $\rho_{\{\ell_1,\ell_1+\ell\}}$ $$\mbox{odd length} \left[ \begin{aligned} \rho_{\{\ell_1, \ell_1 + 2\ell \}} & = \mathcal{S} \otimes \ldots \otimes \mathcal{S} \otimes \mathcal{D}_L (\rho'_{\{\lceil \ell_1/2 \rceil, \lceil \ell_1/2 \rceil + \ell \}}) \quad & \mbox{for $\ell_1$ odd}\\ \rho_{\{\ell_1, \ell_1 + 2\ell \}} & = \mathcal{D}_R \otimes \mathcal{S} \otimes \ldots \otimes \mathcal{S} (\rho'_{\{\ell_1/2, \ell_1/2 + \ell \}}) \quad & \mbox{for $\ell_1$ even}\\ \end{aligned} \right.$$ $$\label{eq:TNmap5} \mbox{even length} \left[ \begin{aligned} \rho_{\{\ell_1, \ell_1 + 2\ell - 1\}} & = \mathcal{S} \otimes \ldots \otimes \mathcal{S} (\rho'_{\{\lceil \ell_1/2 \rceil, \lceil \ell_1/2 \rceil + \ell \}}) \quad & \mbox{for $\ell_1$ odd}\\ \rho_{\{\ell_1, \ell_1 + 2\ell - 1\}} & = \mathcal{D}_R \otimes \mathcal{S} \otimes \ldots \otimes \\ & \otimes \mathcal{S} \otimes \mathcal{D}_R (\rho'_{\{\ell_1/2, \ell_1/2 + \ell + 1 \}}) \quad & \mbox{for $\ell_1$ even}\\ \end{aligned} \right.$$ We agree that working with all these even versus odd possibilities looks definitely messy and confusing. To work around this issue (even though partially) we suggest to employ the translational average $\leftrightarrow$ incoherent translational mixture formalism. That is what we will discuss in the next section. ***Ascending/descending maps for MERA -*** Before moving on, it should be pointed out that, as we developed an in-depth formalism of completely positive ascending and descending maps for binary tree network, this can be similarly done for other TTN geometries and for MERA as well. Precisely, in MERA geometries, maps also embed the local action of disentanglers $X$ as well as renormalizers $\Lambda$. As an example, let us write the descending maps for 3-site density matrices (recall that 3 is the stable causal cone width) $\mathcal{D}^{\text{MERA}}_{3 \to 3, L}$ and $\mathcal{D}^{\text{MERA}}_{3 \to 3, R}$, respectively reading $$\label{eq:TNmap6} \begin{overpic}[width = 330pt, unit=1pt]{Maps1MERA} \put(160, 60){,} \end{overpic}$$ defined in this fashion so that $$\label{eq:TNmap7} \rho^{\text{MERA}}_{\{\ell_1, \ell_1 + 2\}} = \left\{ \begin{aligned} \mathcal{D}^{\text{MERA}}_{3 \to 3, L} (\rho^{\text{MERA}}_{\{ \ell_1/2 , \ell_1/2 + 2\}}) \quad & \mbox{for $\ell_1$ even} \\ \\ \mathcal{D}^{\text{MERA}}_{3 \to 3, R} (\rho^{\text{MERA}}_{\{ \lfloor \ell_1 \rfloor, \lfloor \ell_1 \rfloor + 2\}}) \quad & \mbox{for $\ell_1$ odd} \end{aligned} \right.$$ Extending the descending map formalism to density matrices of any size is performed similarly to for MERA, but with a main difference: the maps can not be any longer expressed as tensor products of local objects due to the presence of disentanglers. Horizontal homogeneity and\ translationally averaged maps ----------------------------- In TTN/MERA Tensor Network, interesting new properties arise when we adopt the additional assumption that tensors belonging to the same layer $h$ of the network structure (and of the same type $X$, $\Lambda$ for a MERA) are identical, that they are actual copies of the same variational tensor, appearing $L/2^h$ times at different nodes in the network. We will call this requirement *horizontal* homogeneity, because we are still letting the tensors belonging to different layers free to be variationally different. In section \[sec:MPShomo\] we showed how tensor homogeneity in periodic Matrix Product States is a condition strictly related with translational invariance symmetry. Neither for Tree networks nor for MERA, the relationship between homogeneity and translational invariance is so sharp and well-defined, since the network geometry itself is visibly not translational. Nevertheless, horizontal homogeneity is not only the best approximation for emulating a translational invariance, but also, and more importantly, allows us to build a well-defined ascending/descending map formalism for the translational averages $\leftrightarrow$ incoherent translational mixture scheme. This will let us write simplified versions of the (say, descending) map definitions we provided in last section, namely and , as they will depend no longer from the placement index $\ell_1$, but solely on the size $\ell$. Then let $\bar{\rho}^{[\mu]}_{\ell}$ be the one-site density matrix of the TTN, averaged over translations $$\label{eq:transpla} \bar{\rho}^{[\mu]}_{\ell} = \frac{1}{2^{\mu}} \sum_{\ell_1 = 1}^{2^{\mu}} \rho^{[\mu]}_{\{\ell_1, \ell_1+\ell-1 \}},$$ where now the layer index $\mu$ is counted starting from the hat and moving down (i.e. $\mu = \bar{h} - h$), for comfort. This will prove a labeling advantage when we will discuss the thermodynamical limit $\mu \to +\infty$. As before, we start from smaller sizes and then grow with increasing $\ell$, to enhance immediateness and comprehension. Consider then $\bar{\rho}^{[\mu]}_{1}$, by exploiting we can write $$\begin{gathered} \bar{\rho}^{[\mu]}_{1} = \frac{1}{2^{\mu}} \sum_{\ell_1 = 1}^{2^{\mu}} \rho^{[\mu]}_{\{\ell_1\}} = \frac{1}{2^{\mu}} \sum_{\ell_1 = 1}^{2^{\mu}/2} \rho^{[\mu]}_{\{ 2 \ell_1 - 1\}} + \rho^{[\mu]}_{\{ 2 \ell_1\}} \\ = \frac{1}{2^{\mu-1}} \sum_{\ell_1 = 1}^{2^{\mu-1}} \frac{1}{2} \left( \mathcal{D}_{L}^{[\mu]} \left( \rho^{[\mu-1]}_{\{ \ell_1 \}} \right) + \mathcal{D}_{R}^{[\mu]} \left(\rho^{[\mu-1]}_{\{ \ell_1 \}} \right) \right)\end{gathered}$$ where $\mathcal{D}_{L}^{[\mu]}$ and $\mathcal{D}_{L}^{[\mu]}$ are those descending maps defined in when tensor $\Lambda$ belonging to layer $\mu$ is being used. Then by linearity of the descending maps we can re-sum the argument density matrices to obtain again the appropriate translational mixture $$\begin{gathered} \label{eq:avemap1} \bar{\rho}^{[\mu]}_{1} = \frac{1}{2} \,\mathcal{D}_{L}^{[\mu]} \left( \frac{1}{2^{\mu-1}} \sum_{\ell_1 = 1}^{2^{\mu-1}} \rho^{[\mu-1]}_{\{ \ell_1 \}} \right) + \frac{1}{2} \,\mathcal{D}_{L}^{[\mu]} \left( \frac{1}{2^{\mu-1}} \sum_{\ell_1 = 1}^{2^{\mu-1}} \rho^{[\mu-1]}_{\{ \ell_1 \}} \right) = \\ \frac{\mathcal{D}_{L}^{[\mu]} + \mathcal{D}_{R}^{[\mu]}}{2} \left( \bar{\rho}^{[\mu-1]}_{1} \right) = \mathcal{D}^{[\mu]} \left( \bar{\rho}^{[\mu-1]}_{1} \right),\end{gathered}$$ where we defined the average map $\mathcal{D} \equiv (\mathcal{D}_L + \mathcal{D}_R )/2$. This new map is again CPT, as it is the equally-weighted mixture of the two original maps, its Kraus decomposition is given by the union of the two Kraus decompositions of $\mathcal{D}_L$ and $\mathcal{D}_R$ (every matrix multiplied by a $2^{-1/2}$ factor). Eq.  yields to a single layer-to-layer recursive relation $\bar{\rho}^{[\mu]}_{\ell} = \mathcal{D}^{[\mu]} ( \bar{\rho}^{[\mu-1]}_{\ell} )$ instead of the whole family of equations and , thanks to horizontal homogeneity requirement. Let us move to two adjacent sites, a causal cone size which is metastable for TTN geometry: its stability will be regularized when performing translational averages, as we will see. Then $$\begin{gathered} \label{eq:avemap2} \bar{\rho}^{[\mu]}_{2} = \frac{1}{2^{\mu}} \sum_{\ell_1 = 1}^{2^{\mu-1}} \left( \rho^{[\mu]}_{\{2 \ell_1 - 1, 2 \ell_1\}} + \rho^{[\mu]}_{\{2 \ell_1, 2 \ell_1 + 1\}} \right) = \\ \frac{1}{2^{\mu}} \sum_{\ell_1 = 1}^{2^{\mu-1}} \mathcal{S}^{[\mu]} \left( \rho^{[\mu-1]}_{\{\ell_1 \}} \right) + \mathcal{D}_R^{[\mu]} \otimes \mathcal{D}_L^{[\mu]} \left( \rho^{[\mu-1]}_{\{\ell_1, \ell_1 +1 \}} \right) = \\ \frac{1}{2}\, \mathcal{S}^{[\mu]} \left( \bar{\rho}^{[\mu-1]}_{1} \right) + \frac{1}{2}\, \mathcal{D}_R^{[\mu]} \otimes \mathcal{D}_L^{[\mu]} \left( \bar{\rho}^{[\mu-1]}_{2} \right)\end{gathered}$$ Watching this result, one can argue that in order to achieve $\bar{\rho}^{[\mu]}_{2}$ by descending recursive relations, we should need both $\bar{\rho}^{[\mu-1]}_{1}$ and $\bar{\rho}^{[\mu-1]}_{2}$. But indeed, $\bar{\rho}^{[\mu-1]}_{2}$ contains by itself all the information concerning $\bar{\rho}^{[\mu-1]}_{1}$: this is obvious by definition , which tells us that ${\text{Tr}}_L [\bar{\rho}^{[\mu-1]}_{2}] = {\text{Tr}}_R [\bar{\rho}^{[\mu-1]}_{2}] = \bar{\rho}^{[\mu-1]}_{2}$. The present consideration allows us to write $\bar{\rho}^{[\mu]}_{2} = \mathcal{D}_{2 \to 2}^{[\mu]} ( \bar{\rho}^{[\mu-1]}_{2} )$ where $$\label{eq:avemap2bis} \mathcal{D}_{2 \to 2}^{[\mu]}(\cdot) = \frac{1}{2} \mathcal{D}_R^{[\mu]} \otimes \mathcal{D}_L^{[\mu]} (\cdot) + \frac{\cos^2 \theta}{2} \mathcal{S}^{[\mu]} \left({\text{Tr}}_L[\cdot]\right) + \frac{\sin^2 \theta}{2} \mathcal{S}^{[\mu]} \left({\text{Tr}}_R[\cdot]\right).$$ $\cos^2 \theta$ and $\sin^2 \theta$ are just two free real parameters which must be both positive, to guarantee complete positivity of $\mathcal{D}_{2 \to 2}^{[\mu]}$, and whose sum must be 1 to ensure trace preserving property. Then for any real angle $\theta$, eq.  performs the right mapping. By natural extension, we can generalize descending mappings for any density matrix size, which can be always written in terms of $\mathcal{S}$, $\mathcal{D}_L$ and $\mathcal{D}_R$. We report here the result for completeness: $$\label{eq:avemap3} \begin{aligned} \mathcal{D}_{\ell \to 2\ell-1} &= \frac{1}{2} \left( \mathcal{D}_R \otimes \mathcal{S} \otimes \ldots \otimes \mathcal{S} + \mathcal{S} \otimes \ldots \otimes \mathcal{S} \otimes \mathcal{D}_L \right)\\ \mathcal{D}_{\ell \to 2\ell-2} &= \frac{1}{2} \left( \mathcal{D}_R \otimes \mathcal{S} \otimes \ldots \otimes \mathcal{S} \otimes \mathcal{D}_L + \left[\mathcal{S} \otimes \ldots \otimes \mathcal{S}\right] \circ {\text{Tr}}_{\theta} \right), \end{aligned}$$ where ${\text{Tr}}_{\theta}$ means tracing away the rightmost site times $\cos^2 \theta$ plus tracing out the leftmost site times $\sin^2 \theta$ (angle $\theta$ arbitrary), as in . All these maps are convex combinations of the maps previously defined, so they are indeed CPT. Ascending maps can be naturally defined in a translational invariant fashion as well; it is easy to see that the composition relations hold for the adjoint maps $\mathcal{A}_{2\ell-2 \to \ell}$ and $\mathcal{A}_{2\ell-2 \to \ell}$ identically. Complete homogeneity and\ definition of thermodynamical limit ----------------------------------- We mentioned that the main purpose of introducing Tree and MERA Tensor Networks is to simulate strongly-correlated systems. It is clear, though, that this ansatz would be more suitable the more we are capable of reproducing a conformal symmetry, where translational invariance, and more importantly *scale* invariance hold. In previous section we dealt with translational invariance by merging together the horizontal homogeneity requirement and the incoherent translational mixture framework. Now we are going to force scale invariance into the system by assuming complete homogeneity: i.e. requesting that *every* renormalization tensor $\Lambda$ in the network is identical to the others, even throughout the layers. We will see that this assumption will naturally lead to the definition of a thermodynamical limit, in which every physical quantity is perfectly controlled thanks to isometricity relations, and where the scale invariance becomes manifest. This can be explained heuristically, by considering the family of tree graphs. Every tree graph has a self-similar pattern, but only for an infinite tree the self-similarity becomes exact: every branch is topologically identical to each of its sub-branches because they are all infinitely-long. Such symmetry, though, would be broken if one were able to distinguish somehow the graph nodes, thus the assumption of complete tensor homogeneity. Let us then consider the sequence of completely homogeneous TTN states $|\Psi^{[\mu]}(\Lambda)\rangle$, defined solely by a single tensor $\Lambda$, repeated at every network node, and indexed by the total amount of layers $\mu$. We are going to characterize the limit of this sequence for $\mu \to \infty$. As we discussed in section the thermodynamical limit state is defined by the family of density matrices $\bar{\rho}^{\infty}_{\ell}$ for every size $\ell$, with the requirement that elements of this set undergo the correct partial trace relation. In our TTN setting we are first considering translational averages, and then approaching the limit, so that $$\rho^{\infty}_{\ell} = \lim_{\mu \to \infty} \bar{\rho}^{[\mu]}_{\ell}.$$ We will again proceed starting from size 1. Horizontal homogeneity tells us that holds, but at the same time, vertical homogeneity implies that $\mathcal{D}$ does not depend on the layer we are considering. Therefore $$\bar{\rho}^{[\mu]}_{1} = \mathcal{D} \left( \bar{\rho}^{[\mu-1]}_{1} \right) = \mathcal{D}^2 \left( \bar{\rho}^{[\mu-2]}_{\ell} \right) = \ldots = \mathcal{D}^{\mu} \left( \bar{\rho}^{[\text{hat}]}_{1} \right),$$ where $\bar{\rho}^{[\text{hat}]}_{1}$ is the translationally averaged one-site density matrix of the hat tensor. Then we have that $$\rho^{\infty}_{1} = \lim_{\mu \to \infty} \mathcal{D}^{\mu} \left( \bar{\rho}^{[\text{hat}]}_{1} \right),$$ i.e. one-site physical properties of the thermodynamical limit state are given by applying $\mathcal{D}$ to the hat tensor. We will now make an additional assumption, which is indeed a constraint on $\Lambda$ but formulated it in terms of $\mathcal{D}$: we will require that $\mathcal{D}$ is a mixing CPT map (see \[app:mixin\]). This condition on the eigenspace decomposition of $\mathcal{D}$ not only states that $\mathcal{D}$ has a unique fixed point $\rho_{f}$, but also that $\rho_{f}$ is the only attraction pole, so that every state is mapped into $\rho_{f}$ after infinite applications of $\mathcal{D}$. Then $\rho^{\infty}_{1}$ is exactly the fixed point of $\mathcal{D}$: $$\rho^{\infty}_{1} = \mathcal{D} \left( \rho^{\infty}_{1} \right)$$ As you guessed, normalization is kept correctly under control at every finite $\mu$, and thus in the limit of an infinite amount of layers $\mu$. Indeed, we argumented that isometricity of $\Lambda$ is equivalent to the peripheral gauge, for which the state normalization condition is automatically dumped onto the hat tensor alone, $1 = \langle \Psi^{[\mu]}(\Lambda) |\Psi^{[\mu]}(\Lambda)\rangle = \langle \mathcal{C} | \mathcal{C}\rangle$; trace preservation property of descending maps does the rest. Let us proceed to size 2: the two-adjacent sites density matrix in the thermodynamical limit $\rho^{\infty}_{2}$ should satisfy for $\mu \to \infty$, which reads $$\label{eq:dm2td} \rho^{\infty}_{2} = \frac{1}{2}\, \mathcal{S} \left( \rho^{\infty}_{1} \right) + \frac{1}{2}\, \mathcal{D}_R \otimes \mathcal{D}_L \left( \rho^{\infty}_{2} \right).$$ This is a quasi-recursive relation which has also a direct dependence upon $\rho^{\infty}_{1}$. In particular by nesting within itself, we can rewrite the recursive expression into a series, namely $$\label{eq:dm2td2} \rho^{\infty}_{2} = \left[ \sum_{\tau = 0}^{\infty} \frac{1}{2^{\tau+1}} \left( \mathcal{D}_R \otimes \mathcal{D}_L \right)^{\tau} \right] \circ \mathcal{S}(\rho^{\infty}_{1}),$$ where the expression inside the square parentheses is a CPT map, because the positive weights $1/2^{\tau+1}$ sum to 1. At the same time, we want our $\rho^{\infty}_{2}$ to be consistent with equation , which becomes a fixed point equation in the thermodynamical limit: $$\begin{gathered} \label{eq:dm2td3} \rho^{\infty}_{2} = \mathcal{D}_{2 \to 2} \left( \rho^{\infty}_{2} \right) \equiv \frac{1}{2} \,\mathcal{D}_R \otimes \mathcal{D}_L \left( \rho^{\infty}_{2} \right) + \\ + \frac{\cos^2 \theta}{2} \,\mathcal{S} \left( {\text{Tr}}_1 \left[ \rho^{\infty}_{2} \right] \right) + \frac{\sin^2 \theta}{2} \,\mathcal{S} \left( {\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right] \right).\end{gathered}$$ In order to show that and are actually compatible equations (for every $\theta$), it suffices to prove that ${\text{Tr}}_1 [\rho^{\infty}_{2}] = {\text{Tr}}_2 [\rho^{\infty}_{2}] = \rho^{\infty}_{1}$. To show this, simply consider the partial trace, say with respect to the rightmost site, of . Then we get $${\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right] = \frac{1}{2}\,\mathcal{D}_L \left( \rho^{\infty}_1 \right) + \frac{1}{2}\, \mathcal{D}_R \left( {\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right] \right);$$ but $\mathcal{D}_L ( \rho^{\infty}_1 ) = [ 2 \mathcal{D} - \mathcal{D}_R ]( \rho^{\infty}_1 )$ and due to the fixed point property is also equal to $ 2 \rho^{\infty}_1 - \mathcal{D}_R ( \rho^{\infty}_1 )$. Therefore we can write $$\label{eq:bacro} \mathcal{D}_R \left( \rho^{\infty}_{1} - {\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right] \right) = 2 \, \left( \rho^{\infty}_{1} - {\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right] \right),$$ which is an eigenvalue equation. But since a CPT map has spectral radius 1, it can not have 2 as eigenvalue; thus the only solution of is $\rho^{\infty}_{1} = {\text{Tr}}_2 \left[ \rho^{\infty}_{2} \right]$. Similarly, we can trace out the left site in and obtain the other equality: $\rho^{\infty}_{1} = {\text{Tr}}_1 \left[ \rho^{\infty}_{2} \right]$. This guarantees that $\rho^{\infty}_{2}$ is unique and well-defined. Thermodynamical limit density matrices for sizes 1 and 2 are the only ones that require dealing with a fixed point equation to be achieved: all the other $\rho^{\infty}_{\ell \geq 3}$ can be directly calculated by applying a finite number of maps to $\rho^{\infty}_{2}$. Precisely we are referring to size-increasing descending maps of equation , that not only generate the whole family of $\rho^{\infty}_{\ell}$, but ensure that this family satisfies the partial trace requirement. For example: $$\rho^{\infty}_5 = \mathcal{D}_{3 \to 5} \circ \mathcal{D}_{2 \to 3} \left( \rho^{\infty}_{2} \right),$$ and by partially tracing this equation, say on the two rightmost sites, we obtain $$\begin{gathered} {\text{Tr}}_{4,5} \left[ \rho^{\infty}_5 \right] = \mathcal{D}_{2 \to 3} \left( {\text{Tr}}_3 \left[ \mathcal{D}_{2 \to 3} (\rho^{\infty}_2) \right] \right) = \\ \mathcal{D}_{2 \to 3} \circ \mathcal{D}_{2 \to 2} \left( \rho^{\infty}_2 \right) = \mathcal{D}_{2 \to 3}\left( \rho^{\infty}_2 \right) = \rho^{\infty}_3,\end{gathered}$$ which is the right consistency check. In conclusion, we characterized properly and completely the thermodynamical limit TTN, whose uniqueness is ensured by the mixing requirement of maps $\mathcal{D}_{1 \to 1} = \mathcal{D}$ and $\mathcal{D}_{2 \to 2}$. Also notice that in this limit, any residual dependence on the hat tensor $|\mathcal{C}\rangle$ that may linger at finite sizes, vanishes. We can interpret this consideration as a hint that $\Lambda$ is the only responsible for capturing, and keeping track, of all the bulk properties of $|\Psi (\Lambda)\rangle$. We will further argument this claim in the next section, where we will calculate two-point correlators and show that the manifest critical behavior of TTN states is ruled by spectral properties of maps, and thus by $\Lambda$. Correlations and criticality ---------------------------- A critical ground state is a scale-invariant state, whose unmistakable signature is a power-law decay rate, with the distance, of two-point correlation functions. We will now investigate such correlations within a TTN state, to prove its criticality. The translational framework scheme will allow us define correlations depending on the two-point distance alone and not on the location; and then we will drive the results towards the thermodynamical limit, to better match conformal symmetry. Then, we start defining a correlation function similarly to what we did for thermodynamical MPS in section \[sec:MPScorr\], but now we average over translations $$\label{eq:avecordef} \bar{\mathfrak{C}}_{\ell}(\Theta, \Theta') \equiv \frac{1}{L} \,\sum_{\ell_0 = 1}^{L} \langle \Theta_{[\ell_0]} \otimes \Theta'_{[\ell_0 + \ell]} \rangle - \langle \Theta_{[\ell_0]} \rangle \langle \Theta'_{[\ell_0 + \ell + 1]} \rangle,$$ for any pair of single-site observables $\Theta$, $\Theta'$. We start by probing this correlator on the finite but fully homogeneous TTN state $|\Psi^{[\mu]}(\Lambda)\rangle$, with $\mu$ being the total tree graph height. Then it is clear that $$\label{eq:corrextend} \bar{\mathfrak{C}}^{[\mu]}_{\ell}(\Theta, \Theta') = {\text{Tr}}\left[ \left( \Theta \otimes \Theta' \right) \cdot \left( \bar{\sigma}^{[\mu]}_{\ell} - \eta^{[\mu]}_{\ell} \right) \right]$$ which extends by linearity the definition of two-point correlator for any two-site observable $\Gamma$, not only tensor product ones. The pair of two-sites density matrices in expression is defined as: $$\label{eq:sigmaeta} \begin{aligned} \bar{\sigma}^{[\mu]}_{\ell} &= \frac{1}{L} \sum_{\ell_0 = 1}^{L} \sigma^{[\mu]}_{\ell_0, \ell_0+\ell} = \frac{1}{2^{\mu}} \sum_{\ell_0 = 1}^{2^{\mu}} {\text{Tr}}_{\{\ell_0+1 \ldots \ell_0 + \ell - 1\}} \left[ \rho^{[\mu]}_{\{\ell_0, \ell_0+\ell\}} \right] \\ \eta^{[\mu]}_{\ell} &= \frac{1}{2^{\mu}} \sum_{\ell_0 = 1}^{2^{\mu}} \rho^{[\mu]}_{\{\ell_0\}} \otimes \rho^{[\mu]}_{\{\ell_0 + \ell\}} \end{aligned}$$ Notice the difference between these two objects: $\bar{\sigma}^{[\mu]}_{\ell}$ is the reduced density matrix of two sites at distance $\ell$, averaged over translations. $\eta^{[\mu]}_{\ell}$, instead, is the translational average of the product of one-site density matrices, whose sites stand exactly at distance $\ell$. Honestly, $\eta^{[\mu]}_{\ell}$ is a separable density matrix by definition , but not necessarily a tensor product matrix, i.e. $\eta^{[\mu]} \neq \bar{\rho}^{[\mu]} \otimes \bar{\rho}^{[\mu]}$. One can say that $\eta^{[\mu]}$ keeps track of *classical correlations* between sites, but it is free of quantum entanglement. At the same time we have that ${\text{Tr}}_1[\bar{\sigma}^{[\mu]}_{\ell}] = {\text{Tr}}_2[\bar{\sigma}^{[\mu]}_{\ell}] = {\text{Tr}}_1[\eta^{[\mu]}_{\ell}] = {\text{Tr}}_2[\eta^{[\mu]}_{\ell}] = \bar{\rho}^{[\mu]}_{1}$ for any $\ell$. By exploiting the descending translational map formalism of eq. , we can as well develop recursive relations for the quantities $\bar{\sigma}^{[\mu]}_{\ell}$ and $\eta^{[\mu]}_{\ell}$. For simplicity and clarity we will write only the equations for an even distance $\ell$, as follows: $$\bar{\sigma}^{[\mu]}_{2 \ell} = {\cancel{\mathcal{D}}}\left( \bar{\sigma}^{[\mu-1]}_{\ell} \right) \qquad \mbox{and} \qquad \eta^{[\mu]}_{2 \ell} = {\cancel{\mathcal{D}}}\left( \eta^{[\mu-1]}_{\ell} \right),$$ where the 2-sites descending CPT map ${\cancel{\mathcal{D}}}$ is defined as $${\cancel{\mathcal{D}}}= \frac{1}{2} \left( \mathcal{D}_L \otimes \mathcal{D}_L + \mathcal{D}_R \otimes \mathcal{D}_R \right).$$ Notice that the propagation map for the two density matrix elements is the same ${\cancel{\mathcal{D}}}$, although it applies to different arguments, so the difference is nonzero in general. In fact, assume that we are calculating the averaged correlator at a two-point distance $\ell$ which is a power of 2, say $2^{q}$ $$\label{eq:TNcorr1} \bar{\mathfrak{C}}^{[\mu]}_{\ell = 2^{q}}(\Gamma) = {\text{Tr}}\left[ \,\Gamma \cdot {\cancel{\mathcal{D}}}^{\,q} \left( \bar{\sigma}^{[\mu-q]}_{1} - \eta^{[\mu-q]}_{1} \right) \right],$$ where, by definition , we have $\bar{\sigma}^{[\nu]}_{1} = \bar{\rho}^{[\nu]}_{2}$. On the other hand, we have that $\eta^{[\nu]}_{1}$ satisfies a recursive equation formally similar to , but with a different inhomogeneous term $$\label{eq:etavemap} \eta^{[\nu]}_{1} = \frac{1}{2}\, \mathcal{D}_R \otimes \mathcal{D}_L \left( \eta^{[\nu-1]}_{1} \right) + \frac{1}{2}\, \mathcal{D}_R \otimes \mathcal{D}_L \left( \eta^{[\nu-1]}_{0} \right).$$ where for $\eta^{[\nu]}_{0}$ it holds $\eta^{[\nu]}_{0} = {\cancel{\mathcal{D}}}(\eta^{[\nu-1]}_{0})$. After setting up all these ingredients, we can drive equation towards the thermodynamical state easily, where every element gains a well-defined limit. Precisely: $$\label{eq:TNcorr2} \bar{\mathfrak{C}}^{[\infty]}_{2^{q}}(\Gamma) = {\text{Tr}}\left[ \,\Gamma \cdot {\cancel{\mathcal{D}}}^{\,q} \left( \Delta\sigma \right) \right],$$ where $\Delta\sigma$ is a null trace matrix, given by the series $$\label{eq:eta2td2} \Delta\sigma = \left[ \sum_{\tau = 0}^{\infty} \frac{1}{2^{\tau+1}} \left( \mathcal{D}_R \otimes \mathcal{D}_L \right)^{\tau} \right] \circ \left( \mathcal{S}(\rho^{\infty}_{1}) - \mathcal{D}_L \otimes \mathcal{D}_R ( {\cancel{\sigma}}) \right).$$ Here we assumed that the CPT map ${\cancel{\mathcal{D}}}$ satisfies the mixing requirement, with ${\cancel{\sigma}}= \lim_{\mu \to \infty} \eta^{[\mu]}_0$ being its unique fixed point; clearly if ${\cancel{\mathcal{D}}}$ is mixing, then $\mathcal{D}$ is mixing as well, since ${\text{Tr}}_2[{\cancel{\mathcal{D}}}(\cdot)] = {\mathcal{D}}(\cdot)$. It is now worth to analyze result and derive some interesting conclusions involving this correlator. Immediately we notice that when increasing the distance to infinity the correlation function drops to zero as it should, since infinitely distant sites must not be entangled. This can be checked by accepting that ${\cancel{\mathcal{D}}}$ is mixing and using relation ; then we get $$\label{eq:TNcorrzero} \lim_{q \to \infty} \bar{\mathfrak{C}}^{[\infty]}_{2^q}(\Gamma) = \lim_{q \to \infty} {\text{Tr}}\left[ \,\Gamma \cdot {\cancel{\mathcal{D}}}^{\,q} \left( \Delta\sigma \right) \right] = {\text{Tr}}\left[ \,\Gamma \cdot {\cancel{\sigma}}\right]\, {\text{Tr}}[\Delta\sigma] = 0,$$ because $\Delta\sigma$ is a traceless matrix by definition. To investigate how the correlator scales at finite distances, we first switch to the ascending map formalism for ; this is done by just taking the adjoint map ${\cancel{\mathcal{A}}}$ of the superoperator ${\cancel{\mathcal{D}}}$. Then $$\label{eq:TNcorr3} \bar{\mathfrak{C}}^{[\infty]}_{2^{q}}(\Gamma) = {\text{Tr}}\left[ {\cancel{\mathcal{A}}}^{\,q} \left( \Gamma \right) \cdot \Delta\sigma \right],$$ where ${\cancel{\mathcal{A}}}$ is completely positive and unital. The linear map ${\cancel{\mathcal{A}}}$ is not necessarily diagonalizable, nevertheless it exists at least one eigenoperator $\Gamma_{\alpha}$ for each of its eigenvalues $\lambda_{\alpha}$. Then when evaluating the correlator on that eigenoperator we have $$\label{eq:TNcorr4} \bar{\mathfrak{C}}^{[\infty]}_{2^{q}}(\Gamma_{\alpha}) = {\text{Tr}}\left[ {\cancel{\mathcal{A}}}^{\,q} \left( \Gamma_{\alpha} \right) \cdot \Delta\sigma \right] = \lambda^{q}_{\alpha} \,{\text{Tr}}\left[ \Gamma_{\alpha} \cdot \Delta\sigma \right] = \lambda^{q}_{\alpha} \;\bar{\mathfrak{C}}^{[\infty]}_{1}(\Gamma_{\alpha}).$$ Now recall that $q$ is the logarithm of the distance, which leads to $$\label{eq:TNcorr5} \bar{\mathfrak{C}}^{[\infty]}_{\ell}(\Gamma_{\alpha}) = \lambda^{\log_2 \ell}_{\alpha} \;\bar{\mathfrak{C}}^{[\infty]}_{1} = \ell^{\,\xi_{\alpha}} \;\bar{\mathfrak{C}}^{[\infty]}_{1}.$$ We obtained an exact power-law decay for out fixed distance two-point correlator, with the critical exponent $\xi_{\alpha} = \log_2 \lambda_{\alpha}$. This is one of the main results I achieved during my doctorateship study [@IoTree]: an infinite homogeneous tree Tensor Network defines always a critical state, where the two-point critical exponents are the logarithms of the spectrum of ${\cancel{\mathcal{D}}}$, and their respective eigenoperators correspond to primary fields, i.e. exactly-scaling operators. Notice that since $|\lambda_{\alpha}| \leq 1$ as CPT maps are contractive, the critical exponents have always a negative real part $\Re(\xi_{\alpha}) \leq 0$. Thus, although might oscillate, it always decays in modulus, never explodes. For a generic observable $\Gamma$, we can write a formal expression of its correlation function which exploits its expansion in the generalized eigenbasis $\{\alpha, \partial, w\}$ of ${\cancel{\mathcal{D}}}$, like we did for . This allows us to write $$\label{eq:Polypoly} \bar{\mathfrak{C}}^{[\infty]}_{\ell}(\Gamma) = \sum_{\alpha} \ell^{\,\xi_{\alpha}} \cdot \sum_{\partial, w} \mathcal{P}^{[\Delta_{\partial}-w]}_{\lambda_{\alpha}}(\log_2 \ell)$$ Where $\mathcal{P}^{[x]}$ are polynomials of degree $x$, which eventually arise from the Jordan block structure of ${\cancel{\mathcal{D}}}$. As you see, the power-law always dominates the logarithmic part and rules the physical critical behavior of the thermodynamical TTN state at long distances. [Rizz02]{} (0, 110)[$\xi_{\alpha}$]{} (134, -4)[$\alpha$]{} ***Criticality of MERA -*** The results and observations we just presented hold for a MERA geometry as well, although since size 1 is not a stable causal cone width, it is necessary to involve bigger-sized observables and density matrices. Precisely, in a binary MERA, even if we started from a 1+1 (two-point) operator, after applying few ascending maps, we increase its support size, typically mapping it into a 3+3 operator (three adjacent sites, and other three adjacent sites, these two parties standing at arbitrary distance), and after that size becomes stable. According to this framework, critical exponents of 3+3 correlators (and thus also of 1+1 correlators, which are a subclass of the 3+3 ones) are given by the logarithms of the spectrum of $${\cancel{\mathcal{D}}}^{\text{MERA}} = \frac{1}{2} \left( \mathcal{D}^{\text{MERA}}_{3 \to 3, L} \otimes \mathcal{D}^{\text{MERA}}_{3 \to 3, L} + \mathcal{D}^{\text{MERA}}_{3 \to 3, R} \otimes \mathcal{D}^{\text{MERA}}_{3 \to 3, R} \right),$$ where descending maps $\mathcal{D}^{\text{MERA}}_{3 \to 3, L}$ and $\mathcal{D}^{\text{MERA}}_{3 \to 3, R}$ are those defined in . This tells us that homogeneous MERA states are also critical. It is commonly believed that, since MERA have larger stable causal cone sizes, they are more suitable to represent a state where the physics at very short ranges is sensibly different from the critical behavior at mid-to-long ranges: indeed homogeneous TTN states are forced to hold at every lenghtscale, even when $\ell \sim 2$, while for a MERA state density matrices up to size 4 are loosely related related to their long-range physics. [Rizz01]{} (2, 100)[$\xi_{\alpha}$]{} (134, -2)[$\Delta$]{} The importance of translational\ fluctuations in TTN -------------------------------- Addressing translational problems with a non-translational variational ansatz might sound a sub-optimal choice for both analytical study and numerical simulations. Yet, TTN and MERA, as we just showed, manifest natural scaling properties that suit so smoothly critical systems, that are excellent variational candidates. Therefore, we are encouraged to wonder whether the capability of hierarchical Tensor Networks to reproduce a complete conformal symmetry, when $L \to \infty$, goes beyond the mere paradigm of incoherent translational mixture. In other words: it is possible that in the limit of infinite layers $\mu$ (or sites), a tree Tensor Network *recovers* the same translational invariance that is forbidden to achieve at finite sizes? We dealt with this question by considering how the translational fluctuations renormalize and scale in tree networks, and developed a peculiar conclusion: fluctuations are fundamental in TTN: they are necessary to describe interesting strongly-correlated physics, especially in the thermodynamical limit. If fluctuation vanish, the TTN state becomes trivial, and separable. We will now sketch a derivation of this relation between fluctuations and entanglement. In infinite systems, translational invariance is typically addressed as a hierarchy of equations, each one of them referring to a characteristic size $\ell$, and stating that the density matrices of that size are homogeneous in the lattice. Obviously, this implies that the same relation holds for any smaller size $\ell' \leq \ell$, thus the hierarchical relationship. Let us start from a one-site observable $\Theta$, and consider the translational fluctuation upon a finite 1D lattice $$\left(\Delta \Theta^{[\mu]} \right)^2 = \frac{1}{2^{\mu}} \sum_{j = 1}^{2^{\mu}} \langle \Theta_j \rangle^2 - \left( \frac{1}{2^{\mu}} \sum_{j = 1}^{2^{\mu}} \langle \Theta_j \rangle \right)^2.$$ where $L = 2^{\mu}$ is the system size. When we probe such fluctuation upon a homogeneous binary TTN state, the expression becomes $$\begin{gathered} \left(\Delta \Theta^{[\mu]} \right)^2 = {\text{Tr}}\left[ \Theta \otimes \Theta \cdot \left( \eta^{[\mu]}_0 - \bar{\rho}^{[\mu]}_1 \otimes \bar{\rho}^{[\mu]}_1 \right) \right] = \\ = {\text{Tr}}\left[ \Theta \otimes \Theta \cdot \left( {\cancel{\mathcal{D}}}^{\mu} \left(\eta^{[0]}_0 \right) - {\mathcal{D}}^{\mu} \otimes {\mathcal{D}}^{\mu} \left( \bar{\rho}^{[0]}_1 \otimes \bar{\rho}^{[0]}_1 \right) \right) \right],\end{gathered}$$ where we used the fact that ${\text{Tr}}[X]^2 = {\text{Tr}}[X \otimes X]$, while $\eta^{[\mu]}_0$ is the same defined in . Moving to the thermodynamical limit is trivial now, and it reads $$\label{eq:zeropai} {\Delta \Theta}^2_{\infty} = \lim_{\mu \to \infty} \left({\Delta \Theta^{[\mu]}} \right)^2 = {\text{Tr}}\left[ \Theta \otimes \Theta \cdot \left( {\cancel{\sigma}}- \rho^{\infty}_{1} \otimes \rho^{\infty}_{1} \right) \right].$$ with ${\cancel{\sigma}}$ and $\rho^{\infty}_{1}$ being respectively the fixed point of ${\cancel{\mathcal{D}}}$ and ${\mathcal{D}}$, which are uniquely defined once we assume that both maps ${\cancel{\mathcal{D}}}$ and ${\mathcal{D}}$ are mixing. We can now state that the thermodynamical TTN state is ’size-1 translational’ if the quantity we just calculated vanishes; and it can be shown that this happens only if the density matrices ${\cancel{\sigma}}$ and $\rho^{\infty}_{1} \otimes \rho^{\infty}_{1}$ coincide. Indeed, let $A$ and $B$ be any two one-site observables, then $$\begin{gathered} \label{eq:zeroby} {\text{Tr}}\left[ \left( A \otimes B + B \otimes A \right)\cdot \left( {\cancel{\sigma}}- \rho^{\infty}_{1} \otimes \rho^{\infty}_{1} \right) \right] = \\ = {\Delta (A+B)}^2_{\infty} - {\Delta A}^2_{\infty} - {\Delta B}^2_{\infty} = 0,\end{gathered}$$ which must be zero as as we are requiring that every translational fluctuation is negligible. Now let $F$ be the swap operator $F |\alpha\rangle \otimes |\beta\rangle = |\beta\rangle \otimes |\alpha\rangle$, $F = F^{-1} = F^{\dagger}$. Clearly $F(A \otimes B)F = B \otimes A$ and $F(\rho^{\infty}_{1} \otimes \rho^{\infty}_{1} )F = \rho^{\infty}_{1} \otimes \rho^{\infty}_{1}$. Moreover ${\cancel{\mathcal{D}}}_{\nu}$ is left invariant under the action of the swap gate, i.e. $F {\cancel{\mathcal{D}}}_{\nu}( F A F) F = {\cancel{\mathcal{D}}}_{\nu}(A)$. This implies that $F {\cancel{\sigma}}_{\nu}^{f} F$ is a fixed point of ${\cancel{\mathcal{D}}}_{\nu}$, but since it is a mixing map the fixed point must be unique, which leads to ${\cancel{\sigma}}_{\nu}^{f} = F {\cancel{\sigma}}_{\nu}^{f} F$. The previous manipulations with the swap operator allow us to write the following equivalence $${\text{Tr}}\left[ A \otimes B \cdot \left( {\cancel{\sigma}}- \rho^{\infty}_{1} \otimes \rho^{\infty}_{1} \right) \right] = {\text{Tr}}\left[ B \otimes A \cdot \left( {\cancel{\sigma}}- \rho^{\infty}_{1} \otimes \rho^{\infty}_{1} \right) \right].$$ and since the sum of these terms is zero by , they must be both zero separately, for any operator $A$ and $B$. But since tensor product operators generate the whole algebra of 2-site operators we must conclude that $${\cancel{\sigma}}= \rho^{\infty}_{1} \otimes \rho^{\infty}_{1},$$ so that ${\cancel{\mathcal{D}}}_{\nu}$ and ${\mathcal{D}}_{\nu} \otimes {\mathcal{D}}_{\nu}$ must have the same fixed point. When this condition is verified then $\rho^{\infty}_{1}$ is automatically the fixed point of of ${\mathcal{D}}_{L}$ and ${\mathcal{D}}_{R}$ as well, because $$\begin{gathered} \label{eq:ciseci} 0 = \left({\cancel{\mathcal{D}}}- {\mathcal{D}}\otimes {\mathcal{D}}\right)(\rho^{\infty}_{1} \otimes \rho^{\infty}_{1}) = \\ = \frac{1}{4} ({\mathcal{D}}_{L} - {\mathcal{D}}_{R}) (\rho^{\infty}_{1}) \otimes({\mathcal{D}}_{L} - {\mathcal{D}}_{R}) (\rho^{\infty}_{1}).\end{gathered}$$ Therefore, a sufficient and necessary condition for translational invariance to hold at size 1, is that ${\mathcal{D}}_{L}(\rho^{\infty}_{1}) = {\mathcal{D}}_{R}(\rho^{\infty}_{1}) = {\mathcal{D}}(\rho^{\infty}_{1}) = \rho^{\infty}_{1}$. So far, so good. Let us proceed further and require translational invariance at size 2. The same derivation can be applied to two-adjacent sites observables $\Theta_{j,j+1}$, its result is that a common fixed point must be shared by maps ${\mathcal{D}}_{2,R}$ and ${\mathcal{D}}_{2,L}$, where $$\begin{aligned} {\mathcal{D}}_{2,R} &\equiv {\mathcal{D}}_{R} \otimes {\mathcal{D}}_{L}\\ {\mathcal{D}}_{2,L} &\equiv {\mathcal{S}}\circ \left( \cos^2\!\theta\;\, {\text{Tr}}_R + \sin^2\! \theta\;\, {\text{Tr}}_L \right). \end{aligned}$$ It is clear that such fixed point must coincide with $\rho_2^{\infty}$, since by we know that ${\mathcal{D}}_{2 \to 2} = \frac{1}{2} ({\mathcal{D}}_{2,R} + {\mathcal{D}}_{2,L})$, and if we require that ${\mathcal{D}}_{2 \to 2}$ is mixing its unique fixed point $\rho_2^{\infty}$ must be the shared fixed point of ${\mathcal{D}}_{2,R}$ and ${\mathcal{D}}_{2,L}$. At the same time, due to , the fixed point of ${\mathcal{D}}_{2,R}$ is $\rho^{\infty}_{1} \otimes \rho^{\infty}_{1}$, and by the common fixed point property we must have that $${\mathcal{S}}(\rho^{\infty}_{1} ) = \rho^{\infty}_{1} \otimes \rho^{\infty}_{1} = {\mathcal{D}}_{2 \to 2} (\rho^{\infty}_{1} \otimes \rho^{\infty}_{1}) = \rho^{\infty}_{2},$$ meaning that the two-sites reduced density matrix is a separable state, actually a tensor product state. By extension, every reduced density matrix is separable, and the system can manifest no quantum correlations at all, since $\Delta\sigma$ defined in equation would be the null operator. Resuming this whole discussion, we proved that by just requiring that the thermodynamical TTN-state manifests translational invariance property at size 2, we automatically end with a completely factorized state, all entanglement is broken down. In this framework, we could state that translational fluctuations are needed in a tree network if we want to describe strongly correlated physics; they are unavoidable, even in the thermodynamical limit. It is curious to realize that the previous demonstration does *not* hold for a MERA geometry: this could actually be one of the first remarkable arguments for preferring MERA to Trees. Indeed for a MERA topology, according to this theoretical picture, we could hope to reproduce accurately conformal symmetry without implications of triviality for thermodynamical limit entanglement. Parent Hamiltonians of TTN states --------------------------------- Finding ground states of physically-meaningful Hamiltonians has been for decades a maximal interest topic, it is also the very purpose of this whole Tensor Network variational ansatz itself. But even dealing with the reverse problem can be challenging and fruitful: given a quantum (many-body) state $|\Psi\rangle$, can we identify, characterize of even build a non-trivial Hamiltonian for which $|\Psi\rangle$ is the ground state? Of course, the research of such a *parent* Hamiltonian $H$, must be addressed in accordance to some physically sensible constraint: we might for instance require for $H$ to be short-ranged, to be translational, or maybe capable of coupling only a limited number of particles per single interaction term. The more are the requirements, the harder is the problem. In my research work, I focused on Tree Tensor Network as 1D many body states, and analyzed how to explicitly build a *non-trivial, short-ranged and translational* Hamiltonian which is parent for the TTN state, in PBC. In this section we will sketch the construction. Let us start from the formal definition of the Hamiltonian $\mathcal{H}$ we want to achieve, whose elementary terms $H$ have limited size support $\nu \ll L$: $$\label{eq:hamexpand} \mathcal{H} = \sum_{\ell_0 = 1}^{L} H_{\ell_0 + 1, \ldots, \ell_0 + \nu}.$$ In order for this Hamiltonian to be parent for our homogeneous-TTN state $| \Psi^{[\mu]}(\Lambda) \rangle$, we must ensure that its expectation values coincides with the minimum of the spectrum of $\mathcal{H}$, which is also the variational minimum of the expectation values $\langle \mathcal{H} \rangle$ on the whole Hilbert space of states: $$\langle \Psi^{[\mu]}(\Lambda) | \mathcal{H} | \Psi^{[\mu]}(\Lambda) \rangle = \min_{|\Phi\rangle} \left\{ \frac{\langle \Phi |\mathcal{H}| \Phi \rangle}{\langle \Phi | \Phi \rangle} \right\}.$$ At the same time, we could read $\mathcal{H}$ as an unnormalized translational average. This allows us to summon again the incoherent translational mixture formalism for TTN density matrices, which reads $$\begin{gathered} \langle \Psi^{[\mu]}(\Lambda) | \mathcal{H} | \Psi^{[\mu]}(\Lambda) \rangle = \sum_{\ell_0 = 1}^{L} \langle \Psi^{[\mu]}(\Lambda) | H_{\ell_0 + 1, \ldots, \ell_0 + \nu} | \Psi^{[\mu]}(\Lambda) \rangle = \\ = \sum_{\ell_0 = 1}^{L} {\text{Tr}}\left[ H \cdot \rho_{\{\ell_0 + 1, \ldots, \ell_0 + \nu\}}^{[\mu]} \right] = 2^{\mu}\; {\text{Tr}}\left[ H \cdot \bar{\rho}_{\nu}^{[\mu]} \right].\end{gathered}$$ Now, suppose that exists a finite small (= non-scaling) $\nu$ for which $\bar{\rho}_{\nu}^{[\mu]}$ has non-maximal rank. If that is true, then $\bar{\rho}_{\nu}^{[\mu]}$ has some nontrivial kernel, with strictly positive dimension, completely generated by an orthogonal set of vectors $|\kappa_{w}\rangle$. Then we say that the elementary Hamiltonian term $H$ is built as follows $$\label{eq:Parentbuild} H = \sum_{w} \omega_w \,| \kappa_w \rangle \langle \kappa_w |,$$ with arbitrary positive weights $\omega_w \geq 0$. With this prescription, we obtain a Hamiltonian $\mathcal{H}$ which is positive, as it is the sum of positive terms, nontrivial, if at least one $\omega_w$ is strictly greater than zero, and for which it holds $$\langle \Psi^{[\mu]}(\Lambda) | \mathcal{H} | \Psi^{[\mu]}(\Lambda) \rangle = 2^{\mu} \sum_{w} \omega_w \, \langle \kappa_w | \bar{\rho}_{\nu}^{[\mu]} | \kappa_w \rangle = 0,$$ since every $|\kappa_w\rangle$ is in the kernel of $\bar{\rho}_{\nu}^{[\mu]}$. But since $0$ is necessarily the minimum of the spectrum of $\mathcal{H}$, as it must be a positive operator, $| \Psi^{[\mu]}(\Lambda) \rangle$ is clearly a ground state for $\mathcal{H}$. This is the idea of our construction. The central point of our proof, therefore, now becomes to demonstrate that for the (homogeneous) TTN state there is always some finite non-scaling size $\nu$ for which the averaged $\nu$-sites density matrix $\bar{\rho}_{\nu}^{[\mu]}$ has non-full rank. To show this we will exploit the fact that descending maps can grow the size of density matrices, while the corresponding increase in entanglement is well-kept under control by isometricity condition on $\Lambda$ (and $X$ as well, in MERA). We will now discuss dimensionality relations leading to characterization of the smallest size $\nu$ for which $\bar{\rho}_{\nu}^{[\mu]}$ is necessarily non-full rank. This result will obviously depend on the Tree or MERA geometry we are employing. In the following instances, we will consider Trees and MERA having a renormalization dimension $D$ which is equal to the physical local dimension $d$ so that TN-homogeneity is meaningful up to the physical lattice. ***Binary Tree -*** in tree geometries it is not possible to establish any bound upon the entanglement of $\bar{\rho}_{2}^{[\mu]}$, for 2 is a (meta-) stable causal cone width. Let us move to size three: by adopting the formalism we know that $$\label{eq:TNrank1} \begin{aligned} \bar{\rho}_{3}^{[\mu]} &= {\mathcal{D}}_{2 \to 3} \left( \bar{\rho}_{2}^{[\mu-1]} \right) = \frac{1}{2}\left[ {\mathcal{D}}_R \otimes {\mathcal{S}}+ {\mathcal{S}}\otimes {\mathcal{D}}_L \right] \left( \bar{\rho}_{2}^{[\mu-1]} \right)\\ &= \mathcal{I} \otimes {\mathcal{S}}\left[ \frac{1}{2}\, {\mathcal{D}}_R \otimes \mathcal{I} \left( \bar{\rho}_{2}^{[\mu-1]} \right) \right] + {\mathcal{S}}\otimes \mathcal{I} \left[ \frac{1}{2}\, \mathcal{I} \otimes {\mathcal{D}}_L \left( \bar{\rho}_{2}^{[\mu-1]} \right) \right] \end{aligned}$$ where $\mathcal{I}$ is the identical map, taking every operator into itself. Notice that even though the mapping ${\mathcal{S}}$ increases size, as it is the application of an isometry, it is left-invertible, and thus preserves the rank; by natural extension ${\mathcal{S}}\otimes \mathcal{I}$ preserves the rank just as well. Now, the expression within the first square parentheses of is a $d^2 \times d^2$ matrix, whose rank is obviously bound by the amount of rows or columns (whichever the smallest), i.e. $d^2$. Finally, it is clear that the maximal rank of a sum of two matrices $A+B$ cannot overcome the sum of ranks of $A$ and $B$ separately. These considerations tell us that $$\label{eq:TNrank2} \text{Rnk}\left[ \bar{\rho}_{3}^{[\mu]} \right] \leq 2 \,d^2$$ where $\bar{\rho}_{3}^{[\mu]}$ is a $d^3 \times d^3$ matrix, and thus its rank is non-maximal whenever $d^3 > 2\,d^2$, which happens for a local physical dimension of $d \geq 3$. When we are considering a ring of 2-level systems, e.g. a spin-[$\frac{1}{2}$]{} chain, size three is not enough for ensuring non-maximality of the density matrix rank. Let us move to size four then: $$\label{eq:TNrank3} \begin{aligned} \bar{\rho}_{4}^{[\mu]} &= \frac{1}{2} \, {\mathcal{D}}_R \otimes {\mathcal{S}}\otimes {\mathcal{D}}_L \left(\bar{\rho}_{3}^{[\mu-1]} \right) + \frac{1}{2} \, {\mathcal{S}}\otimes {\mathcal{S}}\left(\bar{\rho}_{2}^{[\mu-1]} \right)\\ &= \mathcal{I} \otimes {\mathcal{S}}\otimes \mathcal{I} \left[ \frac{1}{2}\, {\mathcal{D}}_R \otimes \mathcal{I} \otimes {\mathcal{D}}_L \left(\bar{\rho}_{3}^{[\mu-1]} \right)\right] + {\mathcal{S}}\otimes {\mathcal{S}}\left[ \frac{1}{2} \,\bar{\rho}_{2}^{[\mu-1]} \right]. \end{aligned}$$ As previously motivated, both $\mathcal{I} \otimes {\mathcal{S}}\otimes \mathcal{I}$ and ${\mathcal{S}}\otimes {\mathcal{S}}$ maps preserve the rank; and of course the ranks of their respective arguments is bound by their row/column dimension. Which leads to $$\label{eq:TNrank4} \text{Rnk}\left[ \bar{\rho}_{4}^{[\mu]} \right] \leq d^3 + d^2$$ which is always strictly smaller that $d^4$ for any nontrivial dimension $d \geq 2$. In conclusion, for binary trees, it is always possible to write a nontrivial, translational, short-range Hamiltonian, according to the prescription where every term $H$ involves *at most* four (adjacent) sites, which become three for a local dimension $d$ greater than 2. ***Higher branching TTN -*** when the Tree geometry has a branching number $b$ higher than 2, the minimal size $\nu$ of $\bar{\rho}_{\nu}^{[\mu]}$, for which it is possible to ensure non-maximality of the rank, grows. As an example, we can mention that for a ternary tree this size $\nu$ is 5. But the growth rate, as a function of $b$, is somehow irregular, erratic; sometime the local dimension $d$ is influent, sometime it is not. Nevertheless, for each geometry, a finite $\nu$ definitely exists, and it is always equal or less than $2b$. In fact for a $b$-branching tree network it holds $$\label{eq:TNrank5} \text{Rnk}\left[ \bar{\rho}_{2b}^{[\mu]} \right] \leq b \;d^{b+1}$$ which is always less than its row/column dimension $d^{2b}$. This interaction-range bound is not optimal, but it still does not scale with system size. ***MERA -*** The presence of the disentanglers also increases the minimal size of non-full rank $\bar{\rho}_{\nu}^{[\mu]}$. Precisely, for a binary MERA we get $$\label{eq:TNrank6} \text{Rnk}\left[ \bar{\rho}_{5}^{[\mu]} \right] \leq 2\;d^4 \qquad \mbox{and} \qquad \text{Rnk}\left[ \bar{\rho}_{6}^{[\mu]} \right] \leq d^4 + d^5;$$ which tells us that we have to accept an interaction range $\nu = 5$ for a local dimension $d \geq 3$, and move to $\nu = 6$ otherwise. If our MERA geometry is ternary, we have to push to seven sites, since $$\label{eq:TNrank7} \text{Rnk}\left[ \bar{\rho}_{7}^{[\mu]} \right] \leq 3\,d^5.$$ At any rate, the effective range of our nontrivial translational parent Hamiltonian might not be so short, but in the end the construction is always possible in practice. ### Unfrustration and degeneracy It is meaningful to point out some general properties of the parent Hamiltonians generated with the protocol just described. First, we would like to highlight that these Hamiltonians are necessarily frustration-free, meaning that the TTN is ground state of every single interaction term. Indeed, let us consider the expansion . We built every single $H$ term to be a positive operator so that the full Hamiltonian $\mathcal{H}$ would be positive as well. But since the TTN state has zero expectation value of $\mathcal{H}$, it must be $$0 = \langle \Psi^{[\mu]}(\Lambda) | \mathcal{H} | \Psi^{[\mu]}(\Lambda) \rangle = \sum_{\ell_0 = 1}^{L} \langle \Psi^{[\mu]}(\Lambda) | H_{\ell_0 + 1, \ldots, \ell_0 + \nu} | \Psi^{[\mu]}(\Lambda) \rangle.$$ Now, the only way for a sum of positive terms to be zero is that every term is separately zero: $$\langle \Psi^{[\mu]}(\Lambda) | H_{\ell_0 + 1, \ldots, \ell_0 + \nu} | \Psi^{[\mu]}(\Lambda) \rangle = 0 \qquad \forall \,\ell_0,$$ i.e. $\mathcal{H}$ is unfrustrated. This one consideration is curiously related to refs. [@FrustrEisert1; @FrustrEisert2], where it was shown that for structured frustration-free Hamiltonians, it is possible to build analytically a ground state via Tensor Network designs. In some sense, our result is the other face of the same coin. [Histo]{} (-25, 70) (160, -10)[$\langle \mathcal{H} \rangle$]{} Another manifest property of our TTN parent Hamiltonian, and somehow related to the unfrustration, is that the ground space of $\mathcal{H}$ is highly degenerate, and we can characterize it to some extent. To show this, we will consider for simplicity the case of a Binary Tree and a local dimension $d\geq 3$, and assume that the construction is being developed at the thermodynamic limit $\mu = \infty$. We will also request, as additional hypothesis, that $\rho^{\infty}_2$ has full rank. This assumption is typically weak, practically guaranteed in numerical settings, where stochastic noise makes every matrix full rank. Now, via we derive a positive interaction term $H$ having support in the kernel of $\rho^{\infty}_3$. To begin with, let us prove that $H$ is in the Kernel of the ascending map ${\mathcal{A}}_{3 \to 2}$, adjoint to ${\mathcal{D}}_{2 \to 3}$ of eq. : by definition we have $$0 = {\text{Tr}}\left[ H \cdot \rho^{\infty}_3 \right] = {\text{Tr}}\left[ H \cdot {\mathcal{D}}_{2 \to 3} \left( \rho^{\infty}_2 \right) \right] = {\text{Tr}}\left[ {\mathcal{A}}_{3 \to 2}\left(H \right) \cdot \rho^{\infty}_2 \right].$$ where since ${\mathcal{A}}_{3 \to 2}$ map is completely positive and unital, ${\mathcal{A}}_{3 \to 2}\left(H \right)$ is surely a positive operator. $\rho^{\infty}_2$ is also positive, and being full-rank, its smallest eigenvalue $\lambda_0$ is strictly greater than zero; then $$0 = {\text{Tr}}\left[ {\mathcal{A}}_{3 \to 2}\left(H \right) \cdot \rho^{\infty}_2 \right] \geq \lambda_0 \,{\text{Tr}}\left[ {\mathcal{A}}_{3 \to 2}\left(H \right) \right] \geq 0.$$ So it must be that the trace of ${\mathcal{A}}_{3 \to 2}\left(H \right)$ is zero, but the only positive traceless operator is the null operator, thus ${\mathcal{A}}_{3 \to 2}\left(H \right) = 0$. This very argument will let us characterize ground spaces of Hamiltonians $\mathcal{H}$ generated by this interaction term $H$. In fact, let us consider a finite system now, with the same local dimension $d$ and an ever number of sites $L = 2\ell$. we will define our *trial* state as a generic pure state on $\ell$ sites $|\Psi_0\rangle$, that we grow to $2\ell$ sites by means of a single layer of isometries $\Lambda$: the same $\Lambda$ tensor we used to build $H$. Let us sketch the Tensor Network design of this trial state as follows: $$\label{eq:Pluslayer} \begin{overpic}[width = 250pt, unit=1pt]{LayerBT} \put(115, 100){$| \Psi_0 \rangle$} \put(25, 33){$\Lambda$} \put(96, 33){$\Lambda$} \put(218, 33){$\Lambda$} \end{overpic}$$ It is trivial to show that this trial state is definitely a ground state for the Hamiltonian $\mathcal{H} = \sum_{\ell_0 = 1}^{L} H_{\ell_0 + 1 \ldots \ell_0 + 3}$. Indeed, let us write $$\label{eq:Pluslayer2} \langle \mathcal{H} \rangle = {\text{Tr}}[H \cdot \bar{\rho}_{3}^{\downarrow}] = {\text{Tr}}[H \cdot {\mathcal{D}}_{2 \to 3} ( \bar{\rho}_{2}^{\uparrow})] = {\text{Tr}}[{\mathcal{A}}_{3 \to 2}(H) \cdot \bar{\rho}_{2}^{\uparrow}] = 0,$$ where $\bar{\rho}_{\nu}^{\uparrow}$ (resp. $\bar{\rho}_{\nu}^{\downarrow}$) is the reduced density matrix, $\nu$-sites translationally averaged, of the trial state before (after) applying the layer of isometries $\Lambda$. Equation is telling that the trial state is a ground state of the Hamiltonian, regardless from $| \Psi_0 \rangle$. Actually every state that is written in the form is a ground state of the system, and the layer of isometries preserves orthonormality, so we can identify (at least) a set of $d^{\ell}$ orthogonal ground states. In conclusion, the ground space of any $\mathcal{H}$ has a wide degeneracy, namely: $$\label{eq:kerbound} \mbox{dim}\left[ \mbox{Ker}\left(\mathcal{H}\right) \right] \geq d^{L/2},$$ with $L$ the size of the system. The same discussion can be applied to the MERA case, and leads to the same result, although the ground states this time are built by attaching to $|\Psi_0\rangle$ a full MERA layer, with renormalizators $\Lambda$ and disentanglers $X$ together. Open boundary MERA {#sec:BERA} ------------------ In section \[sec:MERAintro\], we mentioned that Tree Tensor Networks are equally suitable to simulate open boundary systems as well as periodic boundary systems. This is mainly due to the fact that there are pairs of adjacent sites that renormalize separately for an arbitrary number of layers, thus actually constituting an inner-boundary when approaching TD-limit (honestly, a thermodynamical TTN is geometrically equivalent to the frontier of the Cantor set). At the same time, we stated that MERA have a natural attitude for periodic topologies, since a single MERA layer couples every pair of neighboring sites. It is easy to see, however, that a MERA state needs only a little adjustment to its network geometry, to take properly into account the presence of an open boundary. This is naturally done by embedding the boundaries, interpreted in the most general setting as a pair of additional degrees of freedom (ancillae), in the MERA picture, while preserving the bulk network pattern, and thus the critical bulk properties. The present idea leads to the following network design: $$\begin{overpic}[width = \textwidth, unit=1pt]{LittleBERA} \put(190, 141){$\mathcal{C}$} \end{overpic}$$ which we refer to as (open) boundary-MERA [@BERA]. As you see from the diagram, during the same algorithm-step when we apply disentanglers, we also allow the boundary ancilla to couple locally with the system. We will add the constraint that this coupling operation is represented by a unitary gate. Such additional requirement is necessary to preserve the causal cone relations we argumented in section \[sec:causalcone\]. Clearly, isometricity and unitarity conditions still hold for renormalizers $\Lambda$ and disentanglers $X$ as before, i.e. $$\begin{overpic}[width = 350pt, unit=1pt]{Berarules} \end{overpic}$$ where the bottom tensors are complex conjugate and up-down reversed versions of the ones standing directly above. Preservation of the causal cone property, together with the fact that the binary MERA pattern is identical to the PBC case in the bulk, tell us that descending map equations still hold when the three involved sites (causal cone width of 3 is still stable) are far from the boundaries, i.e. $$\label{eq:BERAmapbulk} \begin{aligned} \rho^{[\mu]}_{2\ell, 2\ell+1, 2\ell+2} &= \mathcal{D}_{3 \to 3, L} \left( \rho^{[\mu-1]}_{\ell, \ell+1, \ell+2} \right) \qquad \mbox{and} \\ \rho^{[\mu]}_{2\ell+1, 2\ell+2, 2\ell+3} &= \mathcal{D}_{3 \to 3, R} \left( \rho^{[\mu-1]}_{\ell, \ell+1, \ell+2} \right) \end{aligned}$$ where maps $\mathcal{D}^{\text{MERA}}_{3 \to 3}$ are those of eq. . When approaching the boundaries we must accordingly define descending quantum channels that involve the new system-ancilla coupling element. For instance, close to the left boundary we get: $$\label{eq:BERAmapanci} \rho^{[\mu]}_{1,2,3} = \mathcal{K}_L \left( \rho^{[\mu-1]}_{A,1,2} \right) \qquad \mbox{and} \qquad \rho^{[\mu]}_{A,1,2} = \mathcal{B}_L \left( \rho^{[\mu-1]}_{A,1,2} \right),$$ where the subscript $A$ refers to the left-ancilla degree of freedom, and the completely positive trace-preserving maps $\mathcal{K}_L$ and $\mathcal{B}_L$ are given by $$\label{eq:Beranci} \begin{overpic}[width = \textwidth, unit=1pt]{Beranci} \put(3, 60){\large$\mathcal{K}_L = $} \put(220, 60){\large$\mathcal{B}_L = $} \end{overpic}$$ Similarly, the maps $\mathcal{K}_R$ and $\mathcal{B}_R$, ruling the layer-recursive relations of density matrices at the right boundary, are the left-right specular versions of . Notice from that once a causal cone touches a boundary, it sticks to it along its upward propagation. ### Consistency of the thermodynamical limit Before speculating on how the presence of the boundary influences the system, we want to show that as we approach the thermodynamical limit (and consider translationally-averaged quantities), we recover the same physics of the corresponding PBC-MERA. In particular, we will assume that the boundary-MERA is homogeneous, i.e. renormalizers $\Lambda$ are identical, and so disentanglers $X$, and even boundary-ancilla couplings; then move to $\mu \to \infty$. We will sketch the consistency proof for size 3, which extends trivially to any size thanks to the shrinking properties of the causal cone width. Consider the translationally averaged 3-site reduced density matrix $\bar{\rho}^{[\mu]}_3$: $$\bar{\rho}^{[\mu]}_3 = \frac{1}{2^{\mu + 2} - 2} \sum_{\ell_0 = 1}^{2^{\mu + 2} -2} \rho^{\mu}_{\ell_0, \ell_0+1, \ell_0+2}.$$ We are actually not considering the ancillae as part of our system in this framework, but the result would not change even if we did. We can now adopt recursive relations and and write: $$\begin{gathered} \bar{\rho}^{[\mu]}_3 = \frac{1}{2^{\mu + 2} - 2} \left[ \mathcal{K}_L \left( \rho^{[\mu-1]}_{A,1,2} \right) + \mathcal{K}_R \left( \rho^{[\mu-1]}_{L-1,L,A'} \right) \right] + \\ + \left( 1 - \frac{1}{2^{\mu + 1} - 1} \right) \cdot {\mathcal{D}}_{3 \to 3} \left( \bar{\rho}^{[\mu-1]}_3 \right).\end{gathered}$$ In order to identify the thermodynamical limit of this quantity, it is important to notice that the following expression converges to zero in trace-norm as $\mu$ grows: $$\lim_{\mu \to \infty} \left\| \bar{\rho}^{[\mu]}_3 - {\mathcal{D}}_{3 \to 3} \left( \bar{\rho}^{[\mu-1]}_3 \right) \right\| \leq \lim_{\mu \to \infty} \left| \frac{4}{2^{\mu + 2} - 2} \right| = 0,$$ where we used the fact that CPT maps are contractive, that the trace norm of a density matrix is 1, and triangular inequality. The previous equation guarantees that $\bar{\rho}^{[\mu]}_3$ goes in the limit to the fixed point of ${\mathcal{D}}_{3 \to 3}$, as $$\bar{\rho}^{\infty}_3 = \lim_{\mu \to \infty} \bar{\rho}^{[\mu]}_3 = \lim_{\mu \to \infty} {\mathcal{D}}_{3 \to 3} \left( \bar{\rho}^{[\mu-1]}_3 \right) = {\mathcal{D}}_{3 \to 3} \left( \bar{\rho}^{\infty}_3 \right),$$ but the fixed point of ${\mathcal{D}}_{3 \to 3}$ also characterized the thermodynamical limit in the periodic MERA, and since we assume for ${\mathcal{D}}_{3 \to 3}$ to be mixing, it has a unique fixed point, thus the two states must be the same. This proves that PBC and OBC MERA manifest the same averaged physics in the thermodynamical limit, which is the TD-consistency argument we requested. ### Boundary fluctuations and permeation Boundary conformal field theory prescribes a direct relation between critical exponents of two-point correlation functions in the bulk, and one-point fluctuations close to the boundary of an OBC critical system. We are now going to investigate such one-point expectation values as a function of the distance from a boundary, say the left one, and compare these analytical results with our previous acknowledgements involving correlation functions in MERA, namely and . To do this, we consider the expectation value of a 3-site observable applying at a distance $\ell \ll L \sim \infty$ from the, say left, boundary: $$\mathfrak{P}^{\infty}_{\ell} (\Theta) = \lim_{\mu \to \infty} \langle \Theta^{[\mu]}_{\ell, \ell+1, \ell+2} \rangle = \lim_{\mu \to \infty} {\text{Tr}}\left[ \Theta \cdot \rho^{[\mu]}_{\ell, \ell+1, \ell+2} \right],$$ which appears as a function, upon $\ell$, on how the influence of the boundary permeates inside the (infinite) system. We will now assume, for simplicity, that such distance $\ell$ is a power of 2, say $\ell = 2^{\tau}$. Then, by adopting the formalism of , we obtain $$\begin{gathered} \label{eq:Leftdominance} \mathfrak{P}^{\infty}_{\ell = 2^{\tau}} (\Theta) = \lim_{\mu \to \infty} {\text{Tr}}\left[ \Theta \cdot {\mathcal{D}}^{\tau}_{3 \to 3, L} \left( \rho^{[\mu-\tau]}_{1,2,3} \right) \right] = \\ = \lim_{\mu \to \infty} {\text{Tr}}\left[ \Theta \cdot {\mathcal{D}}^{\tau}_{3 \to 3, L} \circ \mathcal{K}_L \left( \rho^{[\mu-\tau-1]}_{A,1,2} \right) \right] = {\text{Tr}}\left[ {\mathcal{A}}_{3 \to 3, L}^{\tau} \left( \Theta \right) \cdot \mathcal{K}_L \left( \rho^{\infty}_{A,1,2} \right) \right],\end{gathered}$$ where $\rho^{\infty}_{A,1,2}$ is the fixed point of $\mathcal{B}_L$, unique if the map is mixing. The only residual dependence on $\tau$ is in the number of times the map ${\mathcal{A}}_{3 \to 3, L}$ is to be applied to the operator (or, equivalently, ${\mathcal{D}}_{3 \to 3, L}$ is to be applied to the boundary density matrix). We can now proceed by adopting an argument similar to , i.e. let us assume that $\Theta_{\alpha,L}$ is an eigenoperator of ${\mathcal{A}}_{3 \to 3, L}$, then the permeation function obeys an exact power-law behavior: $$\mathfrak{P}^{\infty}_{\ell = 2^{\tau}} (\Theta_{\alpha,L}) = \lambda_{\alpha}^{\tau} \;\,\mathfrak{P}^{\infty}_{1} (\Theta_{\alpha,L}) = \ell^{\,\log_2 \lambda_{\alpha}} \;\,\mathfrak{P}^{\infty}_{1} (\Theta_{\alpha,L}),$$ where $\lambda_{\alpha}$ is the relative eigenvalue, $|\lambda_{\alpha}| \leq 1$. It is evident that permeation functions near the boundary are critical indeed, as the ancillary degree of freedom strongly correlates with the system. By expanding a generic observable $\Theta$ in the generalized eigenoperator basis of ${\mathcal{A}}_{3 \to 3, L}$ we can write $$\label{eq:Poyone} \bar{\mathfrak{P}}^{\infty}_{\ell}(\Theta) = \sum_{\alpha} \ell^{\,\zeta_{\alpha}} \cdot \sum_{\partial, w} \mathcal{P}^{[\Delta_{\partial}-w]}_{\lambda_{\alpha}}(\log_2 \ell),$$ with critical exponents $\zeta_{\alpha}$ determined by the spectrum of ${\mathcal{A}}_{3 \to 3,L}$ via logarithmic relation $\zeta_{\alpha} = \log_2 \lambda_{\alpha}$. Now we would require somehow that the previous result does not hold only for $2^{\tau}$ distances, but that it generalizes smoothly to other distance, so that our boundary-MERA ansatz would be able to capture conformal symmetry as closest as possible. In particular, let us now consider distances of the form $\ell = 2^{\tau} - 1$. Then, tells us that the correct recursive relation reads: $$\rho^{[\mu]}_{2^{\tau}-1, 2^{\tau}, 2^{\tau}+1} = {\mathcal{D}}_{3 \to 3, R} \left( \rho^{[\mu-1]}_{2^{\tau-1}-1, 2^{\tau-1}, 2^{\tau-1}+1} \right).$$ Notice that now is the other size-3 descending map (${\mathcal{D}}_{3 \to 3, R}$ and not ${\mathcal{D}}_{3 \to 3, L}$) that rules the recursion, unlike . Thus, the permeation function becomes $$\label{eq:Rightdominance} \mathfrak{P}^{\infty}_{\ell = 2^{\tau} - 1} (\Theta) = {\text{Tr}}\left[ {\mathcal{A}}_{3 \to 3, R}^{\tau - 1} \left( \Theta \right) \cdot \mathcal{K}_L \left( \rho^{\infty}_{A,1,2} \right) \right],$$ stating that now the critical exponents are logarithms of the spectrum of ${\mathcal{A}}_{3 \to 3, R}$, and its eigenoperators $\Theta_{\alpha,R}$ are one-point primary fields. Let us introduce the translational regularity requirement. For instance we demand that critical exponents and primary fields do not depend on the point $\ell$ at which the permeation function is computed. It is easy to see that, when accepting this, it immediately follows that $$\label{eq:splittaben} \begin{aligned} {\mathcal{D}}_{3 \to 3, L} &= {\mathcal{D}}_{3 \to 3, R} = {\mathcal{D}}_{3 \to 3}, \qquad \mbox{thus}\\ {\cancel{\mathcal{D}}}&= {\mathcal{D}}_{3 \to 3} \otimes {\mathcal{D}}_{3 \to 3}. \end{aligned}$$ But now, recall that by , the ${\cancel{\mathcal{D}}}$ was the map ruling the two-point correlation functions in the bulk, two point critical exponents given by $\xi_{\alpha}$ the logarithms of its spectrum. Thanks to we know that two point primary fields in the bulk correspond to the one-point ones at the boundary, and critical exponents satisfy $$\zeta_{\alpha} = \frac{1}{2} \;\xi_{\alpha}.$$ As was pointed out by P. Calabrese [@CFTCalab; @BERA], this is one of the fundamental properties prescribed by boundary conformal field theory. One-point critical exponents at the boundary $\zeta_{\alpha}$ are exactly *half* of the two-point ones $\xi_{\alpha}$ in the bulk. We re-derived independently this feature by solely exploiting geometrical features of boundary-MERA network and smoothness requirements. Hybrid MPS$\,\Leftrightarrow\,$TTN networks ------------------------------------------- When a quantum many-body system draws near to a second-order quantum phase transition, noncritical and critical properties begin to overlap. Often, as system parameters approach the critical region (especially in 1D settings, where phase transitions are allowed only at zero temperature) islands characterized by strong correlation within start to appear, out of a noncritical long-range landscape. Then the characteristic size of these regions themselves increases, until they reach the lenghtscale of the whole system when the critical point is achieved. In order to simulate efficiently this type of quantum behavior with a Tensor Network ansatz, one would like reproduce both power-law like correlation scalings up to a tunable finite distance, and exponential decay rates beyond. A promising candidate for this goal would be a structure which embeds the self-similar geometry of Matrix Product States, which express noncritical character in a natural way as we saw in section \[sec:MPScorr\], when observed at large distances, and resembles a hierarchical Tensor Network, either TTN or MERA which both bear strong correlation capabilities, when observed in proximity of the physical bondlinks. Such idea leads, almost obviously, to the design of a Hybrid MPS$\leftrightarrow$TTN structure, picted as follows: $$\label{eq:HybridTTN} \begin{overpic}[width = 260pt, unit=1pt]{Hybr1} \end{overpic}$$ where the system size $L$ is given by the interplay of two parameters: $n$, the number of Matrix Product blocks, and $\mu$, the full depth of the binary tree ’*curtain*’ hanging from the MPS layer. So that $L = 2^{\mu}\,n$. In order to suit as best as possible translational invariance, despite the presence of TI-breaking tree geometry, we will consider the MPS tensors $A$ to be homogeneous, and the isometries $\Lambda$ of the tree as well. It is clear that, as we want for the system to reach thermodynamical limit but keep strongly-correlated islands at finite size, we will fix the curtain height $\mu$, and increase $n \to \infty$, so that $L$ goes to infinity too. The algebraic ingredients in our analytical study of hybrid networks are again the translationally-averaged density matrices $\bar{\rho}^{[\mu]}_1$, $\bar{\sigma}^{[\mu]}_{\ell}$ and $\eta^{[\mu]}_{\ell}$, respectively defined in and . Recursive relations involving these elements will let us figure out correlation behavior and scaling-laws. ***MPS as starting point -*** In section \[sec:MPSTD\] we learned that the fastest way to keep under control normalization and physical quantities in thermodynamical limit MPS is to choose the appropriate peripheral gauge, say the left one. When $\mu = 0$, the involved density matrices are completely defined by the matrix product properties. Namely $$\label{eq:Hybzero} \begin{aligned} \bar{\rho}_1^{[0]} &= \sum_{r,s = 1}^{d} ( \Phi^{+} | A_s \otimes A^{\star}_r | Q )\; |s\rangle\langle r| \\ \bar{\sigma}_{\ell}^{[0]} &= \sum_{r_1, r_2, s_1, s_2}^{d} ( \Phi^{+} | \left( A_{s_1} \otimes A^{\star}_{r_1} \right) \mathbb{E}^{\ell - 1} \left( A_{s_1} \otimes A^{\star}_{r_1} \right)| Q )\; |s_1 s_2\rangle\langle r_1 r_2| \\ \eta_{\ell}^{[0]} &= \bar{\rho}_1^{(0)} \otimes \bar{\rho}_1^{(0)}, \end{aligned}$$ where the MPS matrices $\{A_s \}$ are chosen so that they satisfy the gauge condition $\sum_{s} A^{\dagger}_s A_s = {{\mathbb 1}}$. and $\mathbb{E} = \sum_{s} A_s \otimes A^{\star}_s$ is the transfer matrix of the Identity operator. Correlation vectors $| Q )$ and $( \Phi^{+} |$ are respectively the right and left fixed points of $\mathbb{E}$, they are both positive when read as matrices, and in particular $( \Phi^{+} | = \sum_{\alpha} (\alpha \alpha|$ corresponds to the identity itself. ***Adding the TTN curtain -*** we can now increase $\mu$ to a finite nonzero value. Recursive relations between density matrices standing different tree layers are as usual given by descending CPT-map formalism. So that $$\begin{aligned} \bar{\rho}_1^{[\nu]} &= {\mathcal{D}}(\bar{\rho}_1^{[\nu-1]})\\ \bar{\sigma}_{2\ell}^{[\nu]} &= {\cancel{\mathcal{D}}}(\bar{\sigma}_{\ell}^{[\nu-1]})\\ \eta_{2\ell}^{[\nu]} &= {\cancel{\mathcal{D}}}(\eta_{\ell}^{[\nu-1]}), \end{aligned}$$ for every $0 < \nu \leq \mu$. It is clear that, while $\bar{\rho}_1^{[\mu]}$ can be always expressed in terms of as ${\mathcal{D}}^{\mu} (\bar{\rho}_1^{[0]} )$, expressions for $\bar{\sigma}_{\ell}^{[\mu]}$ and $\eta_{\ell}^{[\mu]}$ depend whether the distance $\ell$ is smaller or larger than the island size $2^{\mu}$. ***Small scale regime*** ($\ell < 2^{\mu}$) - For simplicity, let us choose $\ell = 2^p$, clearly with $p < \mu$ ($p$ integer). Then the appropriate recursive relation read $$\label{eq:sfilza1} \begin{aligned} \bar{\sigma}_{2^p}^{[\mu]} &= {\cancel{\mathcal{D}}}^p (\bar{\sigma}_{1}^{[\mu-p]}),\\ \eta_{2^p}^{[\mu]} &= {\cancel{\mathcal{D}}}^p (\eta_{1}^{[\mu-p]}), \end{aligned}$$ We still have to exhibit an expression of $\bar{\sigma}_{1}^{[\nu]}$ and $\eta_{1}^{[\nu]}$, for $0 < \nu \leq \mu$. To do this in a clever way we introduce a new density matrix $\theta^{[\nu]}$, defined as $$\label{eq:sfilza0} \theta^{[\nu]} = \frac{1}{2^p n} \sum_{\ell_0 = 1}^{2^p n} \rho_{\{\ell_0, \ell_0+1\}}^{[\nu]} \otimes \rho_{\{\ell_0, \ell_0+1\}}^{[\nu]}$$ Interestingly enough, both $\bar{\sigma}_{1}^{[\nu]}$ and $\eta_{1}^{[\nu]}$ can be extracted from $\theta^{[\nu]}$ by partial trace, precisely $$\bar{\sigma}_{1}^{[\nu]} = {\text{Tr}}_{3,4} \left[ \theta^{[\nu]} \right] \qquad \mbox{and} \qquad \eta_{1}^{[\nu]} = {\text{Tr}}_{2,3}\left[ \theta^{[\nu]} \right].$$ At the same time, $\theta^{[\nu]}$ satisfies the following recursion $$\label{eq:sfilza00} \theta^{[\nu]} = \mathcal{W} \left( \theta^{[\nu-1]} \right) \equiv \frac{ {\mathcal{D}}_R \otimes {\mathcal{D}}_L \otimes {\mathcal{D}}_R \otimes {\mathcal{D}}_L + (S \otimes S)\, {\text{Tr}}_{2,4}}{2} \left[ \theta^{[\nu-1]} \right],$$ starting at the MPS layer from $\theta^{[0]} = \bar{\sigma}_{1}^{[0]} \otimes \bar{\sigma}_{1}^{[0]}$. ***Large scale regime*** ($\ell > 2^{\mu}$) - Things are easier now, provided that we choose, again for simplicity, $\ell = 2^{\mu} \tau$ ($\tau$ integer), with no finite-size effects $\ell \ll L \sim \infty$. The correct expressions are just as follows: $$\begin{aligned} \label{eq:nosfilza} \bar{\sigma}_{2^{\mu} \tau}^{(n)} =&\; {\cancel{\mathcal{D}}}^{\mu} (\bar{\sigma}_{\tau}^{[0]}),\\ \eta_{2^{\mu} \tau}^{(n)} =&\; {\cancel{\mathcal{D}}}^{\mu} (\eta_{\tau}^{[0]}).\end{aligned}$$ We formally achieved every quantity we are interested in. We are now ready to combine together the previous results to investigate correlation properties of Hybrid Tensor Networks. ### Two-point correlation functions We are interested in calculating two-point correlators at fixed distance and translationally averaged on variational states of the form . Let us then recall $$\begin{gathered} \label{eq:avecordef2} \bar{\mathfrak{C}}^{[\mu]}_{\ell}(\Theta, \Theta') \equiv \frac{1}{L} \,\sum_{\ell_0 = 1}^{L} \langle \Theta_{[\ell_0]} \otimes \Theta'_{[\ell_0 + \ell]} \rangle - \langle \Theta_{[\ell_0]} \rangle \langle \Theta'_{[\ell_0 + \ell + 1]} \rangle = \\ = {\text{Tr}}\left[ \left( \Theta \otimes \Theta' \right) \cdot \left( \bar{\sigma}_{\ell}^{[\mu]} - \eta_{\ell}^{[\mu]} \right)\right],\end{gathered}$$ the appropriate correlation function of $\Theta$ and $\Theta'$ at distance $\ell$. Let us focus on the scaling laws of this quantity, in either of the two regimes we mentioned, rather than an exact formal expression for it. ***Small scale regime -*** Again $\ell = 2^p$, with $p < \mu$. By using the recursive relation for $\theta^{[\nu]}$ we can write $$\label{eq:lunghetto} \bar{\mathfrak{C}}^{[\mu]}_{\ell = 2^p}(\Theta, \Theta') = {\text{Tr}}\left[ \vphantom{\sum} \left( \Theta \otimes \Theta' \right) \cdot {\cancel{\mathcal{D}}}^{p} \circ \left( {\text{Tr}}_{3,4} - {\text{Tr}}_{2,3} \right) \left[ \mathcal{W}^{\mu - p} \left( \theta^{[0]} \right) \right] \right].$$ the residual dependence on $p$ (and $\mu$) is left only in the number of times the maps ${\cancel{\mathcal{D}}}$ and $\mathcal{W}$ are to be applied. This allows us to exploit the Jordan block expansion scheme for multiple actions of the same matrix, as we did before, leading us to $$\label{eq:larguzzo} \bar{\mathfrak{C}}^{[\mu]}_{\ell = 2^p}(\Theta, \Theta') = \sum_{\lambda_{{\cancel{\mathcal{D}}}}} \sum_{\lambda_{\mathcal{W}}} \left( \frac{\lambda_{{\cancel{\mathcal{D}}}}}{\lambda_{\mathcal{W}}}\right)^p \cdot \mathcal{P}_{\{\lambda\}}(p)$$ where $\lambda_{{\cancel{\mathcal{D}}}}$ (resp. $\lambda_{\mathcal{W}}$) are eigenvalues of ${\cancel{\mathcal{D}}}$ (of $\mathcal{W}$), and $\mathcal{P}_{\{\lambda\}}$ are finite-degree polynomials. Now recalling that $p$ is the logarithm of the distance $\ell$, it is clear that is dominated by power-law behaving functions, apart logarithmic corrections, whose (quasi-critical) exponents are determined by $$\chi_{\alpha} = \log_2 \left( \frac{\lambda_{{\cancel{\mathcal{D}}}}}{\lambda_{\mathcal{W}}}\right).$$ ***Large scale regime -*** The system behave differently when $\ell = 2^\mu \tau$, for $1 < \tau \ll L$. In fact at these distances eq.  becomes $$\begin{gathered} \bar{\mathfrak{C}}^{[\mu]}_{2^{\mu} \tau}(\Theta, \Theta') = {\text{Tr}}\left[ (\Theta \otimes \Theta') \cdot {\cancel{\mathcal{D}}}^{\mu} \left( \bar{\sigma}^{[0]}_{\tau} - \bar{\eta}^{[0]}_{\tau} \right) \right] = \\ = f(\mu) (\Phi^{+}| \mathbb{E}_{{{\mathbb 1}}}^{\tau} - \mathbb{E}_{{{\mathbb 1}}}^{\infty} | Q).\end{gathered}$$ In the end we are actually calculating a $\tau$-fixed distance correlator upon the MPS layer of the observable ${\cancel{\mathcal{A}}}^{\mu}(\Theta \otimes \Theta')$. But since we chose a homogeneous, left-gauge MPS description in the thermodynamical limit, we can perfectly recover the corresponding result , i.e. $$\label{eq:linfizzi} \mathfrak{C}_{2^{\mu} \tau}(\Theta, \Theta') = \sum_{|\lambda_{\mathbb{E}}|<1} \lambda_{\mathbb{E}}^{\tau} \cdot \mathcal{P}_{\lambda_{\mathbb{E}}}(\tau).$$ In fact, instead of , now $\tau$ is proportional to $\ell$ and we obtain an explicit exponential decay, going to zero at infinite distance since the sum spans only the eigenvalues $\lambda_{\mathbb{E}}$ of $\mathbb{E}_{{\mathbb 1}}$ smaller of 1 in modulus. In conclusion, we discovered that the hybrid Tensor Network state geometry, designed in , manifests a quasi-critical character. At short ranges, strong correlations identified by power-law two-point functions, arise; their exponents characterized by the isometry element $\Lambda$ of the TTN portion. At long ranges, the behavior is evidently noncritical, correlations vanish exponentially and ruled by the MPS block $A$. The number of TTN layers we adopted in the scheme effectively determines the lenghtscale $\ell_0$ of the strong-correlation islands $\ell_0 \sim 2^{\mu}$. ***MPS$\,\leftrightarrow\,$MERA Hybrids -*** The previous discussions were formulated for a binary tree curtain attached to the MPS-basis layer. But it is easy to see that the same results can be generalized to other tree geometries and for MERA as well. So even if we had a network structure of the form $$\label{eq:HybridMera} \begin{overpic}[width = 240pt, unit=1pt]{Hybr2} \end{overpic}$$ we would similarly obtain a quasi critical regime. In fact, descending maps for MERA replace the role of those for TTN, and the scaling laws do not sensibly change. Higher dimensions and comparison\ between TTN/MERA and PEPS --------------------------------- So far in this chapter, we discussed on how tree Tensor Networks and MERA can be used as a suitable and efficient variational for many-body ground states of one dimensional system. But it is clear that their power is not limited to 1D systems only, indeed generalizations of Trees and MERA to higher dimensions is natural and intuitive. The idea remains the same [@VidalTree1]: we may think to perform a real-space renormalization group of the quantum lattice state, expressed in the language of density matrices, for *any* amount of spatial dimensions \#D. By performing this iterated process, we obtain a class of states which is equivalently expressed as a Tensor Network, namely, a tree network in \#D dimensions. For instance, let us consider a 2D square lattice, then the simplest 2D tree geometry we can think of is the one that maps a 4-adjacent sites plaquette into a single renormalized site $(a)$. The renormalization element $\Lambda$ becomes then a 5-link tensor, and read as a $D^4 \times D$ matrix is an isometry $\Lambda^{\dagger} \Lambda = {{\mathbb 1}}$. Alternatively, again dealing with a square lattice, one could prefer to renormalize together pairs of sites which are horizontally adjacent (so that we are actually coarse-graining only one of the two dimensions), and at the following step those who are vertically adjacent $(b)$. Clearly, the latter tree geometry has the advantage on the former that is easier to contract and thus more efficient, for the same refinement parameter $D$. At the same time description capabilities of the second choice (in terms of variational manifold) are reduced. In diagrammatic expression these two TTN designs read respectively: $$\label{eq:Pyramids} \begin{overpic}[width = 270pt, unit=1pt]{2DTree} \put(0, 150){$(a)$} \put(145, 150){$(b)$} \end{overpic}$$ The possibilities are many. The user is encouraged to adopt for her simulation the \#D TREE/MERA geometry that suits most symmetries (see appendix \[app:symchap\]) and other global properties of the problem under study; for example, a plaquette-tree geometry would embed more naturally the square group symmetry. Many of the algebraic properties we discussed in this chapter for 1D TTN/MERA are preserved by their higher-dimensionality versions. The causal cone property is one of these: as TTN remain Tensor Networks without closed loops, transforming them according to the peripheral gauge (with respect to the hat as nucleus) is always possible without loss of generality, no matter the geometry or dimensionality. Thanks to the causal cones, it is clear that even in high-D, the contraction of a Tree network, or MERA, stays efficient: the number of required elementary operations scaling logarithmically with the volume. This is probably the most convincing argument for preferring TTN/MERA to PEPS, from 2D up. Despite the two families of Tensor Network manifest promising description capabilities for physically-relevant ground states, PEPS are computationally complex objects, while hierarchical Tensor Networks keep their high efficiency. It is worth mentioning that it was shown in [@EisMERAPEPS] that starting from 2D up, MERA (and thus Trees) satisfy an entanglement area law, and therefore can be efficiently mapped, through a well-defined formal algorithm, into PEPS. In the end, Trees and MERA are simulation tools hard to ignore, for any physical setting: in 1D they are the answer on how to simulate critical systems in a natural and cheap way; in two or more spatial dimensions, they constitute an intriguing sub-class of finitely correlated states that are efficiently contractible. Conclusions {#conclusions .unnumbered} =========== In this thesis we introduced, developed, and analyzed a large class of variational tailored quantum many-body wavefunctions, called Tensor Network states. These states are meant to be used as variational ansatze for interacting particles problems on a lattice; minimization algorithms adopting these trial functions do not require any a priori knowledge on the model we are studying. Predicting the success of Tensor Network states for a certain setting is based upon arguments borrowed from quantum information theory: primarily entanglement. We discussed profoundly how Tensor Network designs are somehow the variational counterpart of some numerical renormalization group procedure. In fact, the manifold of quantum many-body states that can be constructed via a RG-algorithm, can be equivalently identified by a tailored analytic expression, where the variational descriptors are tied together through simple linear algebraic relations. For these reasons Tensor Network states preserve all the faithfulness and simulation power of numerical renormalization groups, yet the variational picture presents several practical advantages. Two important example are: a more immediate way to access physical information, and a more free numerical manipulability with a consequent computational speed-up. In particular, we showed in chapter \[chap:OBCMPS\] that Matrix Product States arise from White’s DMRG. Since Matrix Product States correspond to finitely-correlated states in 1D, they are allowed to manifest only the correct amount of entanglement of 1D non-critical ground states, and at the same time to be extremely efficient for computation. This explains why DMRG (and MPS-based algorithms alike) are so successful in one-dimensional systems. In chapter \[chap:PBCMPS\] we presented a generalization of MPS formalism that extends to Periodic boundary systems, with the care of keeping the right amount of allowed correlation. This led us to the definition of a homogeneous PBC-MPS representation, which was instrumental in the definition of thermodynamical limit for Matrix Product States. Once we defined an infinite MPS, we could investigate the correlations behavior of these states, and showed that under the assumption of keeping a finite refinement parameter, the target state is distinctly non-critical. Matrix Product States are considered the fundamental template that led to the more general definition of Tensor Network states. Various classes of Tensor Networks manifest properties depending strictly on their geometry. Still, it is possible to determine general features common to every TN-state; those we sketched in chapter \[chap:TN\]. Finally, chapter \[chap:TTNMERA\] contained most of the analytical advancements I contributed. There we discussed about Tree Tensor Networks, and MERA, the latter geometry presented as an enhancement of the former, but still sharing most of its scaling properties. We explained how these TN-variational states arise from a real-space renormalization group technique, an origin which leads unavoidably to scale invariant physics for these states. In fact, we showed that by adopting a CPT map formalism, we can well-define the thermodynamical limit for TTN/MERA states, and this state manifest critical behavior, identified by power-law decay rates of two-point correlations. Acknowledgements {#acknowledgements .unnumbered} ---------------- I want to thank my colleagues D. Rossini, M. Rizzi and S. Montangero for their numerical support, S. Peotta for his stimulating ideas, G. Santoro for his trust, R. Fazio for his protection, and V. Giovannetti for his unfaltering helpfulness. Completely Positive Trace preserving maps {#app:cptchap} ========================================= Here we list some features of Completely Positive Trace preserving (CPT) maps, as they are common and useful tools in quantum information theory, and thus thoroughly studied [@Terhal]. They represent quantum channels, having the property that they map density matrices into density matrices, and often describe the time evolution of quantum states when both coherent and incoherent sources couple with the system. In Tensor Network settings, CPT maps are primarily used as inverse transformation of some numerical renormalization procedure, typically applied to a density matrix, and thus work as propagators for renormalized density matrices within the network structure. Definition ---------- A homomorphism between matrix spaces $\mathcal{M}: \mathbb{C}^{n \times n} \to \mathbb{C}^{m \times m}$ is said to be a CPT map if satisfies the following requirements 1. $\mathcal{M}$ is positive: $\forall\, A \geq 0, A \in \mathbb{C}^{n \times n} \Rightarrow \mathcal{M}(A) \geq 0$ 2. $\mathcal{M}$ is completely positive, meaning that its identity extension on any larger space is still positive. Let us define $\mathcal{I}_k \otimes \mathcal{M}: \mathbb{C}^{k \times k} \otimes \mathbb{C}^{n \times n} \to \mathbb{C}^{k \times k} \otimes \mathbb{C}^{m \times m}$ defined on separable matrices $[\mathcal{I}_k \otimes \mathcal{M}] (B \otimes A) = B \otimes \mathcal{M}(A)$. Then the complete positivity reads: $\forall\, C \geq 0,C \in \mathbb{C}^{k \times k} \otimes \mathbb{C}^{n \times n} \Rightarrow [\mathcal{I}_k \otimes \mathcal{M}](C) \geq 0$, for every extension $k$. 3. $\mathcal{M}$ preserves the trace: ${\text{Tr}}[\mathcal{M}(A)] = {\text{Tr}}[A]$. Of course, a matrix $A$ is positive if its spectrum lies in $\mathbb{R}^{+}$, or also $\langle \psi | A |\psi \rangle \geq 0$ $\forall |\psi\rangle$ if we are in a Hilbert space. Choi’s theorem [@Choi] states that a mapping between matrices $\mathcal{M}$ is completely positive *iff* it exists a set of Kraus operators $\{V_s\}_s$ for $\mathcal{M}$, namely a set of matrices $V_s \in \mathbb{C}^{m \times n}$ which satisfy: $$\label{eq:Kraus} \sum_s V_s^{\dagger} V_s = {{\mathbb 1}}_{n \times n} \qquad \mbox{and} \qquad \sum_s V_s A V_s^{\dagger} = \mathcal{M}(A) \quad \forall\,A.$$ For later reference, it is also convenient to define also the map adjoint of $\mathcal{M}$ with respect to the trace scalar product between matrices $(A|B) = {\text{Tr}}[A^{\dagger} B]$; by definition $\mathcal{M}^{\text{adj}}: \mathbb{C}^{m \times m} \to \mathbb{C}^{n \times n}$ so that $$(A|\mathcal{M}^{\text{adj}}(B)) = (\mathcal{M}( A)|B) = {\text{Tr}}[ (\mathcal{M} (A))^{\dagger} B] = \sum_s {\text{Tr}}[V_s A^{\dagger} V_s^{\dagger} B].$$ Then, by using cyclicity of the trace, you immediately see that $\mathcal{M}^{\text{adj}}$ must read: $$\mathcal{M}^{\text{adj}}(B) = \sum_s V^{\dagger}_s B V_s \quad \forall\,B.$$ Which tells us that $\mathcal{M}^{\text{adj}}$ is again completely positive, and *unital*, i.e. it maps the identity operator onto itself, thanks to , but not necessarily trace preserving. Spectral properties {#app:CPTspec} ------------------- When CPT maps $\mathcal{M}$ are endomorphisms ($n = m$), they can be expressed as square matrices, and expanded in a basis of eigenoperators and generalized eigenoperators as usual. However, spectrum and eigenmatrices of a CPT map are bound to undergo certain properties: - *Every eigenoperator $O_{\lambda}$ with eigenvalue $\lambda \neq 1$, must have null trace*. This is clear from the fact that ${\text{Tr}}[O_{\lambda}] = {\text{Tr}}[\mathcal{M}(O_{\lambda})] = {\text{Tr}}[ \lambda O_{\lambda}] = \lambda \,{\text{Tr}}[O_{\lambda}]$, and since $\lambda \neq 1$ its only solution is ${\text{Tr}}[O_{\lambda}] = 0$. Also, it follows that such a $O_{\lambda}$ can not be positive, because the only positive traceless matrix is the null matrix. The traceless requirement trivially extends to generalized eigenoperators as well. - *The spectrum of $\mathcal{M}$ must be symmetric with respect to the real axis*. Indeed, $\mathcal{M}(A^{\dagger}) = \sum_s V_s A^{\dagger} V_s^{\dagger} = \sum_s ( V_s A V_s^{\dagger})^{\dagger} = (\mathcal{M}(A))^{\dagger}$. Then, if $O_{\lambda}$ is eigenoperator with eigenvalue $\lambda$, we have $\mathcal{M}(O_{\lambda}^{\dagger}) = (\mathcal{M}(O_{\lambda}))^{\dagger} = (\lambda O_{\lambda})^{\dagger} = \lambda^{\star} O_{\lambda}^{\dagger}$, so also $\lambda^{\star}$ is in the spectrum. In particular, a hermitian eigenoperator has necessarily a real eigenvalue. This also tells us that the spectrum of $\mathcal{M}$ and $\mathcal{M}^{\text{adj}}$ coincide. - *1 always belongs to the spectrum of $\mathcal{M}$*. By absurd if $\mathcal{M}$ had no eigenvalue 1, then its entire basis of generalized eigenoperators would be traceless, but that would necessarily be an incomplete set because the identity operator ${{\mathbb 1}}$ can not be generated. Another way to see this is that ${{\mathbb 1}}$ is always eigenoperator of $\mathcal{M}^{\text{adj}}$, and 1 its eigenvalue; but spectra of $\mathcal{M}^{\text{adj}}$ and $\mathcal{M}$ are equivalent, so 1 is also eigenvalue for $\mathcal{M}$. - *The generalized eigenspace of $\lambda = 1$ coincides with the strict eigenspace* In fact if we assume that there is a Jordan block of dimension 2 or greater, we can define the generalized eigenoperator $O'_{1}$ for which $\mathcal{M}^{\text{adj}}(O'_{1}) = O'_{1} + {{\mathbb 1}}$. but the equation $\alpha = {\text{Tr}}[O'_{1}] = {\text{Tr}}[O'_{1}] + {\text{Tr}}[{{\mathbb 1}}] = \alpha + n$ has no solution. - *The spectral radius is 1*, or equivalently, CPT maps are contractive. To prove this, we are going to show that if $\mathcal{M}^{\text{adj}}$ had an eigenvalue $\lambda$ greater in modulus than 1, it could not be a positive map. Indeed let $O_\lambda$ be the related eigenoperator, and consider $O_{+} = O_\lambda + O_{\lambda}^{\dagger}$, which is hermitian, but traceless and thus not positive, meaning that it exists a (normalized) vector $|\psi\rangle$ for which $\langle \psi | O_+ |\psi\rangle = w < 0$. Now, the operator $({{\mathbb 1}}+ \varepsilon\, O_{+})$ is definitely positive for $0 < \varepsilon < \| O_{+} \|$. Therefore ${[\mathcal{M}^{\text{adj}}]}^q({{\mathbb 1}}+ \varepsilon\, O_{\lambda}) = {{\mathbb 1}}+ \varepsilon ( \lambda^{q} O_{\lambda} + \lambda^{\star\,q} O^{\dagger}_{\lambda}) = \Gamma_q$ should be positive for every $q$, but $$\label{eq:aquat} \langle \psi | \Gamma_q | \psi \rangle = 1 + \varepsilon |\lambda|^{q} \left( w \cos(\phi q) + z \sin(\phi q) \right),$$ where $\phi = \arg(\lambda)$. This expression either oscillates with exponentially increasing amplitude, with period $2\pi / \phi$, or it is monotonically, and exponentially, decreasing if $\phi = 0$, since $w < 0$. Either way, sooner or later we encounter some integer $q$ for which is negative, telling us $\mathcal{M}^{\text{adj}}$ mapped a positive operator into a non-positive one, which is the desired absurd. This final remark on contractivity of CPT maps is particularly important, it tells us that within the space of matrices, there is a proper subspace which is attractive, i.e. that every operator is driven towards it by multiple applications of the map $\mathcal{M}$, its distance from such set exponentially decreasing. Also we know that the attraction subspace is generated by generalized eigenoperators of eigenvalues $\lambda$ of modulus $|\lambda|=1$. Obviously, the case that gathers greatest interest from this point of view is when this subspace is one-dimensional, so that the map contracts the projective space into a point. We will see that this additional request is also related to spectral properties of $\mathcal{M}$. The Mixing requirement {#app:mixin} ---------------------- A CPT map $\mathcal{M}$ is said to be *mixing* when infinite applications of $\mathcal{M}^q \equiv \mathcal{M} \circ \ldots \circ \mathcal{M}$ contract the whole space of matrices (modulus the trace) into a unique point $\Lambda$, i.e. $$\label{eq:mixing} \lim_{q \to \infty} \mathcal{M}^q (A) = \Lambda \; {\text{Tr}}[A] \qquad \forall\,A.$$ It is clear that when this condition holds $\Lambda$ is obviously a fixed point of the map, since $$\mathcal{M}(\Lambda) = \mathcal{M} \circ \lim_{q \to \infty} \mathcal{M}^q ({{\mathbb 1}}/n) = \lim_{q \to \infty} \mathcal{M}^{q+1} ({{\mathbb 1}}/n) = \Lambda.$$ which immediately tells us that $\Lambda$ is positive and ${\text{Tr}}[\Lambda] = 1$, so $\Lambda$ is a density matrix. The mixing condition has consequences on the spectral periphery of $\mathcal{M}$, which are also related to well-known physical properties [@mixing]: - ***Ergodicity** - The eigenvalue 1 of $\mathcal{M}$ is simple*, meaning that the related eigenspace is one dimensional, i.e. $\mathcal{M}$ has a unique fixed point. When this statement fails, it is clear that it exists a whole manifold (at least one dimensional) of operators $O_{\mathbb{M}}$, with ${\text{Tr}}[O_{\mathbb{M}}]=1$, which are all fixed points of the map. Thus eq.  breaks down since $\mathcal{M}^{\infty}(O_{\mathbb{M}}) = O_{\mathbb{M}}$ and $O_{\mathbb{M}}$ not unique. - ***Relaxation** - $\mathcal{M}$ has no modulus $|\lambda_{\alpha}| = 1$ eigenvalue $\lambda_{\alpha}$, except for 1 itself.* This tells us that every component of a given observable $O$, expanded in the generalized eigenbasis, other than $\Lambda$, decays exponentially with a rate governed by the second greatest modulus eigenvalue $\lambda_2$: $|\lambda_{\alpha > 2}| \leq |\lambda_{2}| < 1$. By contradiction, if an eigenvalue $\bar{\lambda} = e^{i\varphi}$ existed, one can construct a whole set of operators that *rotate* infinitely around the fixed point $\Lambda$, like $$\mathcal{M}^{q}(\Lambda + \alpha O_{\bar{\lambda}}) = \Lambda + \alpha \,e^{iq\varphi}\, O_{\bar{\lambda}}$$ and the sequence in $q$ has no limit. When the map $\mathcal{M}$ is mixing, expanding the action of several, but finite, applications of $\mathcal{M}$ in the generalized eigenbasis $O_{\alpha, \partial, w}$ ($\alpha$ - eigenvalue index, $\partial$ - Jordan block index, $w$ - position within the block) is particularly useful, and it reads $$\label{eq:APolyexpand} \mathcal{M}^{q}(A) = \Lambda {\text{Tr}}[A] + \sum_{\alpha = 2} \lambda_{\alpha}^q \sum_{\partial, w} \mathcal{P}^{[\Delta_{\partial}-w]}_{\lambda_{\alpha}}(q) \; O_{\alpha, \partial, w},$$ with $\mathcal{P}^{[x]}_{\lambda_{\alpha}}(q)$ is a polynomial function of degree $x$, with coefficients depending on $A$ and $\lambda_{\alpha}$; obviously all the sum terms vanish at $q \to \infty$. Symmetries in Tensor Networks {#app:symchap} ============================= Exploiting symmetries in numerical simulations of quantum problems is one of the best and most valuable techniques we can adopt to drastically improve the efficiency of computation, with no actual loss of accuracy. When the Hamiltonian of the many-body problem is invariant under a group of unitary transformations, then an immediate characterization of the ground states, under the action of the group itself, emerges. Even in the rare cases when a discrete symmetry is spontaneously broken, it is always possible, thanks to the wave-like formulation of quantum mechanics, to identify a symmetry-invariant ground state. In variational contexts, where the efficiency of every protocol is extremely sensitive to the number of effective parameters, the total amount of such descriptors is heavily reduced every time a symmetry constraint is embedded in the framework. Of course, the capability of upholding a symmetry in a given variational ansatz is not guaranteed a priori: the manifold of states belonging to the variational family must be able to recognize, distinguish, and capture the physics of the symmetry group as a whole. Tensor Networks, as we discussed thoroughly, are variational counterparts of numerical renormalization groups. Since RG-processes naturally manifest a notion of locality preservation, TN-states seem most suitable to reproduce symmetries which are global, but act locally as an uncorrelated product of on-site transformations. The strategies to implant symmetries within Matrix Product States had been known from the DMRG era [@SchollDMRG; @Singh1]; generalizations to other Tensor Network geometries were studied in depth by S. Singh *et al.* [@Singh2; @Singh3]. It was shown that the overall effect of inserting a symmetry constraint in a TN-ansatz is that one can easily isolate residual variational degrees of freedom out of structure factors and selection rules. In practice, every tensor decomposes, into a variational and a structural part (or *fragment*), which are again tied together by network linking relations. Pointwise symmetries and\ representations {#sec:irrep} ------------------------- We will now define the class of symmetry transformation groups we will take into account through the following sections, and then review some basic, but useful, results of representation theory. Let $\mathcal{G}$ be our compact symmetry group; it can be a finite group as well as a Lie group: typical candidates are $\mathbb{Z}_n$, O$(n)$, SO$(n)$, U$(n)$, SU$(n)$ but also dihedral groups are common. We identify $\{U_g\}_{g \in \mathcal{G}}$ as its unitary representation on the one-site $d$-dimensioned Hilbert space: $U_{g_1 g_2} = U_{g_1} \cdot U_{g_2}$ Then the overall transformations are given by tensor products of on-site unitaries $$\label{eq:pointwise} U^{\otimes L}_g = \bigotimes_{\ell = 1}^{L} U^{[\ell]}_{g},$$ where $L$ is the total number of sites, and $U^{[\ell]}_{g}$ does not depend on the site $\ell$ on which it acts, as we are requiring homogeneity of the local representation. In literature, these transformation are often referred to either as *local* symmetries, as $U^{\otimes L}_g$ is a product of local terms, or as *global* symmetries, as $U^{\otimes L}_g$ has global support. To avoid misunderstandings, we prefer to call transformations of the form , *pointwise symmetries*, as they are an overall operation acting on every site singularly. Pointwise symmetries are extremely relevant from a physical point of view: they can uphold most of the extensive constraints of a physical framework, like total particle conservation, parity, or spin conservation. Trivially, the $U^{\otimes L}_g$ form a group and are an actual representation of $\mathcal{G}$, although it is hardly an irreducible representation, even when the original $U_{g}$ is irreducible. Before moving further, let us briefly recall some basic principles of matricial group representation theory; in particular, we summon the natural expansion in irreducible components and subspaces. Precisely, let $\mathbb{V}$ be any given vector space, on complex field $\mathbb{C}$, and $\{W_g\}_{g \in \mathcal{G}}$ a unitary representation of $\mathcal{G}$ on $\mathbb{V}$. Then $\mathbb{V}$ decomposes naturally as the Cartesian sum of irreducible subspaces $\mathbb{V}^{[c]}$: $$\label{eq:irrepdeco} \mathbb{V} \approx \bigoplus_{c} \left[ \bigoplus_{\partial = 1}^{\bar{\partial}_c} \mathbb{V}^{[c]} \right] \approx \bigoplus_{c} \left( \mathbb{D}^{[c]} \otimes \mathbb{V}^{[c]} \right),$$ where the $\mathbb{V}^{[c]}$ are the smallest invariant subspaces of $\mathbb{V}$ under the action of $U_{\mathcal{G}}$. In , $c$ is a scalar integer labeling which of the various irreducible representations (irreps) of $\mathcal{G}$ is related to the given subspace $\mathbb{V}^{[c]}$, which, in turn, determines the dimension $\bar{m}_c$ of $\mathbb{V}^{[c]}$. Index $c$ is commonly called *charge*, or *sector*; for comfort, we will always label with $c = 0$ the trivial representation $V^{[0]}_g = 1$, of dimension $\bar{m}_0 = 1$, which always exists, regardless of $\mathcal{G}$. Moreover, in a generic representation, the same $c$-charged irrep can appear an arbitrary number of times $\bar{\partial}_c$; this (integer positive) $\bar{\partial}_c$ number actually defines the *degeneracy* of sector $c$ within representation $W_{\mathcal{G}}$. In conclusion we can build a complete basis for $\mathbb{V}$, in accordance to expansion , listed as $|c,\partial_c,m_c\rangle$, where $\partial_c \in \{1 .. \bar{\partial}_c \}$ labels degeneracy-space vectors, while $m_c \in \{1 .. \bar{m}_c \}$ labels irrep-space vectors. Then the action of $W_{g}$ is respectful of this decomposition: $$\begin{aligned} W_{g} |\varphi \rangle &= W_{g} \left( \sum_{c} \sum_{\partial_c = 1}^{\bar{\partial}_c} \sum_{m_c = 1}^{\bar{m}_c} \varphi_{c,\partial_c, m_c} |c,\partial_c,m_c\rangle \right) \\ &= \sum_{c} \sum_{\partial_c = 1}^{\bar{\partial}_c} \sum_{m'_c = 1}^{\bar{m}_c} \left( \sum_{m_c = 1}^{\bar{m}_c} \left[V^{[c]}_g \right]_{m'_c}^{m_c} \;\varphi_{c,\partial_c, m_c} \right) |c,\partial_c,m'_c\rangle, \end{aligned}$$ or more simply, by exploiting the direct sum formalism of we can write $$\label{eq:irrepdeco2} W_{g} = \bigoplus_{c} \left( {{\mathbb 1}}_{\bar{\partial}_c \times \bar{\partial}_c} \otimes V^{[c]}_g \right),$$ where ${{\mathbb 1}}$ is the identity operator, and the unitary matrix $V^{[c]}_g$ is the $c$-charge irrep of the group element $g$. An important remark is the following: undoubtedly, the set of allowed $c$ values in expansion and is a peculiar property of the representation $W$ we chose, and similarly also the degeneracy numbers $\bar{\partial}_c$. Instead, the irrep dimensions $\bar{m}_c$ depend only the group $\mathcal{G}$ itself (after irreps and charges $c$ have been associated once and for all), not specifically on $W$. In particular, let us recall that if $\mathcal{G}$ is an *abelian group*, then all its irreps have always dimension one, i.e. $\bar{m}_c = 1$ regardless from $c$. Network geometry preservation ----------------------------- We will now select an arbitrary network geometry, and focus on the manifold of Tensor Network states $\{ |\Psi_{\text{TN}}\rangle \}$ which can be generated through that geometry. A preliminary argument that must be introduced, is that $\{ |\Psi_{\text{TN}}\rangle \}$ is closed under the action of any pointwise symmetry group: $$| \Psi' \rangle = U_{g}^{\otimes L} |\Psi_{\text{TN}}\rangle \quad \Longrightarrow \quad | \Psi' \rangle \in \{ |\Psi_{\text{TN}}\rangle \},$$ telling us that the target state $| \Psi' \rangle$ admits an exact tensor network representation (with the same graph design). This follows trivially from the fact that the operation $U_{g}^{\otimes L}$ is a tensor product of local and invertible transformations. Therefore the target state $| \Psi' \rangle$ must yield the same entanglement properties of the original state $|\Psi_{\text{TN}}\rangle$. Another way to see the equivalence, from an algebraic viewpoint, is that every of the local $U_g$ transformations can be adsorbed into the nearest linked tensor, thus effectively recovering the original network structure (and unaltered bondlink dimensions $D$). Zero charge Tensor Network states {#app:zerocharge} --------------------------------- The first, and simplest, step to undertake if we want match Tensor Network ansatze and symmetries is to provide a characterization of TN-states which are invariant under the pointwise symmetry group, i.e. those $|\Psi^{[0]}_{\text{TN}}\rangle$ for which it holds $$|\Psi^{[0]}_{\text{TN}}\rangle = U_{g}^{\otimes L} |\Psi^{[0]}_{\text{TN}}\rangle \qquad \forall\,g \in \mathcal{G}.$$ It is clear that, in order for the unitary symmetry group to have no effect on them, these $|\Psi^{[0]}_{\text{TN}}\rangle$ must strictly belong to the ($\bar{\partial}_{0}$-dimensioned) sector $c = 0$ of $U_{g}$. Therefore, we will call states $|\Psi^{[0]}_{\text{TN}}\rangle$ belonging to this sub-manifold, zero charge Tensor Network states. Clearly, if all tensors of $|\Psi^{[0]}_{\text{TN}}\rangle$ are invariant under the action of $U_g$ (applied at the physical links solely), the state $|\Psi^{[0]}_{\text{TN}}\rangle$ is invariant. This condition is sufficient, but not necessary. Indeed the application of $U_g$ could map the tensors into a *gauge-transformed version* of the original ones, and the state would be unaltered anyway. In practice, tensors $T$ of $|\Psi^{[0]}_{\text{TN}}\rangle$ must be such that, for every $g \in \mathcal{G}$, applying $U_{g}^{\otimes L}$ is equivalent to performing a gauge transformation. For obvious reasons the selected gauges must form a group (intended as a subgroup of the whole gauge group), representing $\mathcal{G}$. But since gauge transformations of Tensor Networks are made of local, independent isomorphisms upon doubly-connected links, every link is implicitly carrying a group representation separately. In conclusion, we may associate to every non-physical link $\alpha$ a unitary representation $W_{(\alpha)}$ of $\mathcal{G}$ and a direction, and tensors $T$ must be symmetric, i.e. invariant under the action of $\mathcal{G}$ on *all* of their indices (with their respective representation and direction): $$\label{eq:TNsymtransform} T^{\{q\},\{p\}}_{\{s\}} \! = \!\!\! \sum_{\{r\},\{v\},\{w\}} \! \left( \prod_{\alpha}^{\bar{\alpha}_{\text{phys}}} \left[U_g\right]_{s_{\alpha}}^{r_{\alpha}} \right) \left( \prod_{\alpha}^{\bar{\alpha}_{\text{in}}} \left[W^{\dagger}_{(\alpha),g}\right]_{q_{\alpha}}^{v_{\alpha}} \right) \left( \prod_{\alpha}^{\bar{\alpha}_{\text{out}}} \left[W_{(\alpha),g}\right]_{p_{\alpha}}^{w_{\alpha}} \right) T^{\{v\},\{w\}}_{\{r\}}$$ for every $g \in \mathcal{G}$, where tensor $T$ is connected to $\bar{\alpha}_{\text{phys}}$ physical links, $\bar{\alpha}_{\text{in}}$ incoming virtual links, and $\bar{\alpha}_{\text{out}}$ outcoming virtual links. The representation $U$ is the only one fixed by the physics of the problem, while the $W_{(\alpha)}$ are arbitrary, chosen by the user. We are strongly encouraged to model the $W_{(\alpha)}$ on the problem we are investigating, especially if we are implementing a simulation algorithm. It is instructive to rewrite equation as follows: $$\label{eq:Symgauge} \begin{overpic}[width = \textwidth, unit=1pt]{Symgauge} \put(77, 83){$T$} \put(286, 91){$T$} \put(230, 80){\footnotesize{$W^{\dagger}_{(1)g}$}} \put(246, 138){\footnotesize{$W^{\dagger}_{(2)g}$}} \put(318, 153){\footnotesize{$W_{(3)g}$}} \put(330, 90){\footnotesize{$W_{(4)g}$}} \put(291, 35){\footnotesize{$U_{g}$}} \end{overpic}$$ The diagrammatic formulation of Tensor Network formalism will help us to understand why equation is the standard requirement for a zero charge TN-state. Precisely, consider the following example: $$\label{eq:Symequiv} \begin{overpic}[width = \textwidth, unit=1pt]{Symequiv} \end{overpic}$$ where a three-sites state is involved, on which we are applying the pointwise transformation $U^{\otimes 3}_g$ for a given $g \in \mathcal{G}$. The first equality in is just creation of pairs operator-antioperator, which is freely allowed since it is a gauge transformation (as discussed in section \[sec:TNgauge\]). By choice, we are picking exactly the unitary operators $W_{(\alpha)g}$ (green circle) and $W_{(\alpha)g}^{-1} = W_{(\alpha)g}^{\dagger}$ (orange circle), with the selected representation for each virtual link $\alpha$. Now we apply the tensor symmetricity requirement of eq. , which automatically gives the second equality. Then the state is invariant under the action of the whole pointwise symmetry group $U^{\otimes 3}_{\mathcal{G}}$. We showed an example with a given network geometry, but it is obvious that this argument applies equivalently to every Tensor Network state whose tensors satisfy , . A peculiar ingredient in this framework is selecting the virtual representations $W_{(\alpha)}$ of the symmetry group. It is true, as we stated, that for any choice of those representations the resulting TN-state would be zero charged; but at the same time, the description capabilities of the symmetric Tensor Network ansatz may depend on such choice, and typically will. Symmetric tensors fragmentation {#sec:fragment} ------------------------------- The prescription of using symmetric tensors in order to generate invariant network states, can be interpreted [@grouptheory] as a generalization of Schur’s lemma to a linear operand (tensor) with an an arbitrary number of indices. Precisely, consider the case investigated by Schur, where an operator $O$ commutes with a unitary representation $W$ of a compact symmetry group $\mathcal{G}$; the $O$ can have nontrivial support only on the degeneracy space, i.e. $$\label{eq:schur} [O, W_{\mathcal{G}}] = 0 \quad \Longrightarrow \quad O = \bigoplus_{c} \left( O^{[c]} \otimes {{\mathbb 1}}_{\bar{m}_c \times \bar{m}_c} \right),$$ using the same irrep subspace decomposition of . It is easy to check that operators in this form are the only ones commuting with every $W_g$ of equation . Then, as it was discussed in refs. [@Singh2; @grouptheory] a similar argument applies to every symmetric tensor, whichever its amount of indices might be. Ultimately, symmetric tensors decompose in such a way that the degrees of freedom which are not fixed by symmetry (variational DOF) are isolated, and separated from the symmetry constraints (structural DOF). This decomposition takes place at every node of the network structure, substantially *fragmenting* a single tensor $T$ into a fully-variational tensor $R$, and one (or more) structural-tensor $S$; thus actually splitting the original network into a pair of connected superimposed graphs. We will now explain how this symmetric tensor fragmentation scheme is performed in practice, by considering tensors with a limited correlation number (up to four attached links). We will proceed step-by-step starting from the simplest cases, and sketching diagrams whenever possible for clarity and comfort. ***Bondlink fragmentation -*** Before considering fragmentation of tensors, it is useful to understand how symmetry relations decompose network bondlinks themselves. Indeed, we mentioned that to every link $\alpha$ we associated a representation $W_{(\alpha)}$ of $\mathcal{G}$, as well as a direction which helps us to discriminate between the application of the direct unitary transformation $W_{(\alpha)g}$ and its inverse $W_{(\alpha)g}^{\dagger}$ (according to this picture, we may think open physical links as *outgoing* links). Then, every value $j$ of the link $\alpha$, intended as a virtual state $|j)_{\alpha}$ decomposes according to the irrep subspace expansion: $|j)_{\alpha} \rightarrow |c, \partial_c, m_c)_{\alpha}$. Therefore, the network link literally splits into: $$\label{eq:Fraglink} \begin{overpic}[width = 270pt, unit=1pt]{Fraglink} \put(36, 73){link $\alpha$} \put(-4, 60){$j$} \put(271, 60){$c$} \put(271, 106){$\partial_c$} \put(271, 15){$m_c$} \end{overpic}$$ where the red fragment-link carries the charge index $c$, the green one holds the degeneracy index $\partial_c$, and the blue one keeps track of the irrep basis vector label $m_c$. As we stated, the number $\bar{m}_c$ of allowed values for the index $m_c$ depends on the value of $c$, in other words the blue link has a *dimension* depending on the value of the red link. Similarly, the degeneracy number $\bar{\partial}^{(\alpha)}_c$ depends on both $c$ and the representation chosen on link $\alpha$. This is the purpose of the wavy arrows in , to remember that the value of the red link influences the dimension of the blue and green links tied to it. The total bondlink dimension becomes then $$D_{\alpha} = \sum_{c \;\in\; \text{irreps}(\mathcal{G})} \bar{m}_c \times \bar{\partial}_c^{(\alpha)},$$ where we can sum over all irreducible representations of $\mathcal{G}$. with $c$ being the charge. To control $D_{\alpha}$ and keep it finite, even when there are infinite independent irreps for $\mathcal{G}$, it is sufficient to set the $\bar{\partial}_c^{(\alpha)}$ so that only a finite number of them are nonzero. To make a practical example, let us assume that $\mathcal{G}$ is SO(3), then $\bar{m}_c = 2c + 1$ with $c \in \mathbb{N}$. Assume we are performing a renormalization process, keeping track of pointwise spin rotation symmetry; and three spin-1 sites were already renormalized into the given link $\alpha$. Then, by spin-sum rules, we might want to describe for $W_{(\alpha), \mathcal{G}}$: one singlet, three triplets, two quintuplets, and one 7-plet, so that $\bar{\partial}_0 = 1$, $\bar{\partial}_1 = 3$, $\bar{\partial}_3 = 2$ and $\bar{\partial}_4 = 1$ ($\bar{\partial}_c = 0$ for $c \geq 5$). Clearly, if we take these degeneracies, we are keeping all the state information, and not actually renormalizing anything: indeed the total bondlink dimension $D$ is 27, equal to $d^3$. As we stated, if the symmetry group $\mathcal{G}$ we are considering is abelian, $\bar{m}_c$ is always 1, regardless from $c$ or the link $\alpha$; so the blue link-fragment allows only one value, and therefore is futile. Indeed in abelian symmetry frameworks, typically only the red and green sub-links appear, as they are the only ones needed. ***One-link tensors -*** a tensor having only one index behaves like a vector. This means that it is symmetric only if $\mathcal{G}$ acts trivially on it. This implies that its support must be restricted to the $c = 0$ sector, and $j = \partial_0$ labels completely the space $|0, \partial_0, 1)$, which is fully degenerate ***Two-links tensors -*** This is the case considered in Schur’s lemma. Equation can be read in the following terms: a two-link symmetric tensor $T$, written in the basis $|j\rangle\langle k|\rightarrow|c,\partial_c, m_c\rangle\langle q, \partial'_q, m'_q|$ must preserve both charge and irrep label $m$. Therefore $T$ naturally decomposes as $$\label{eq:twofrag} T_{k}^{j} = (R^c_{q})^{\partial_c}_{\partial'_q} \times (S^c_{q})^{m_c}_{m'_q} \qquad \mbox{with} \qquad (S^c_{q})^{m_c}_{m'_q} = \delta_{c,q}\; \delta_{m,m'}$$ where $R$ is the variational fragment of $T$, and $S$, the structural one, is equal to the identity ${{\mathbb 1}}$. Let us sketch fragmentation diagrams for one-link and two-link tensors: $$\label{eq:Frag12} \begin{overpic}[width = \textwidth, unit=1pt]{Frag12} \put(0, 110){a)} \put(200, 110){b)} \put(21, 81){$T$} \put(105, 78){$R$} \put(140, 89){$\delta_{c,0}$} \put(222, 57){$T$} \put(306, 57){$R$} \put(353, 57){\footnotesize $\delta^{[3]}$} \end{overpic}$$ Notice that in the case of one-link symmetric tensor, we disregarded the irrep vector sub-link (blue one) since the only relevant charge is $c = 0$ which is the trivial irrep, and thus 1-dimensioned. The peculiar double-delta form for $S$ in actually depends on the fact that we are considering one ingoing link and one outgoing, which is the most common setting. Other choices of directing links (say two incoming or two outcoming links) lead to different structure tensors: $(S^c_{q})^{m_c}_{m'_q} = \delta_{c,q}\; Q_{m}^{m'}$. The best way to recover these setups is by contracting a 3-link symmetric tensor (we will discuss it shortly) with a 1-link one. The two-leg symmetric tensor case is the first where we encounter an actual *reduction of variational parameters*, by employing symmetries, while preserving the original total bondlink dimensions $D_{\alpha}$, and thus entanglement features as well. Indeed consider respectively $T$ and $R$ of (\[eq:Frag12\].b): $$T \longleftarrow D_{\alpha} D_{\beta} \;\mbox{ descriptors}, \qquad R \longleftarrow \bar{D}_{\alpha, \beta}^{\text{II}} \equiv \sum_c \bar{\partial}^{[\alpha]}_c \bar{\partial}^{[\beta]}_c \;\mbox{ descriptors}.$$ Then, since $D_{\alpha} = \sum_{c} \bar{\partial}^{[\alpha]}_c \bar{m}_c$, we typically end up with an effective amount of parameters $\bar{D}_{\alpha, \beta}^{\text{II}} \ll D_{\alpha} D_{\beta} \sim D^2$ way smaller than the original one, especially when the *active* sectors ($c$ values for which $\bar{\partial}^{[\alpha]}_c > 0$) are many. [Frag3]{} (48, 106)[$T$]{} (211, 113)[$R$]{} (297, 112)[$S$]{} (222, 170)[$\partial_c$]{} (282, 170)[$m_c$]{} (254, 159)[$c$]{} (206, 62)[$a$]{} (171, 62)[$\partial'_{a}$]{} (216, 31)[$m'_{a}$]{} (286, 31)[$\partial''_{b}$]{} (309, 56)[$b$]{} (342, 56)[$m''_b$]{} ***Three-links tensors -*** The tensor product of two irreps, with charges $a$ and $b$ respectively, is still a representation of $\mathcal{G}$; and thus can be decomposed in a direct sum of irreps according to and : $$\label{eq:mfedco} \mathbb{V}^{[a]} \otimes \mathbb{V}^{[b]} \approx \bigoplus_c \left[ \bigoplus_{\partial_c = 1}^{\mu_{a, b}^c} \mathbb{V}^{[c]} \right]$$ where $\mu_{a, b}^c$ is the number of copies of $\mathbb{V}^{[c]}$ appearing in the tensor product representation. Here it is comfortable to assume that the group $\mathcal{G}$ is multiplicity free, i.e. that $\mu_{a, b}^c$ can be either zero or 1, no matter the sectors. This is quite a typical case for physically relevant symmetries: SO(3) and SU(2) which take into account rotational invariance, as well as every abelian group that keeps track of particle number and parity conservation, are multiplicity free symmetries (but not SU(3), for example). In this framework, the Wigner-Eckart theorem provides a remarkable fragmentation law for a three-leg tensor $T$: $$\label{eq:3frag} T^{\,i}_{j,k} = ( R^{\,c}_{a,b} )^{\partial_c}_{\partial'_a, \partial''_b} \times ( S^{\,c}_{a,b} )^{m_c}_{m'_a, m''_b},$$ where the structural tensor fragment $S$ contains the Clebsh-Gordan coefficients for irreps of $\mathcal{G}$, which are well defined thanks to and the multiplicity freedom requirement: $$( S^{\,c}_{a,b} )^{m_c}_{m'_a, m''_b} = \langle c, m_c | a, m'_a; b, m''_b \rangle.$$ An analogous decomposition with different structural tensors $S$ holds for other direction configuration of the links connected to $T$. Figure \[fig:Frag3\] shows the diagram for , the case we considered. We want to check, in this three-link tensor scenario, the effective reduction of variational descriptors caused by enforcing symmetry relations. So, let us compare the amount of parameters: $$T \longleftarrow D_{\alpha} D_{\beta} D_{\gamma} \sim D^3 \,, \quad R \longleftarrow \bar{D}^{\text{III}} \equiv \sum_a \sum_b \sum_{c \in \{a \oplus b\}}\bar{\partial}^{[\alpha]}_a\, \bar{\partial}^{[\beta]}_b\, \bar{\partial}^{[\gamma]}_c,$$ where the innermost sum spans the sole sectors $c$ that are achievable by fusing together charges $a$ and $b$, via . It is clear that $\bar{D}^{\text{III}}$ is very small compared to $D^3$; it is more likely to scale like $D^2$ rather than $D^3$, especially if the symmetry $\mathcal{G}$ is abelian. ***Four-links tensors -*** The tensor product of three irreps $\mathbb{V}^{[a]}$, $\mathbb{V}^{[b]}$ and $\mathbb{V}^{[c]}$ may contain several copies of the same irrep $\mathbb{V}^{[q]}$, even when the symmetry group $\mathcal{G}$ is multiplicity free. However, we can workaround this issue by fusing together two charges, say $a$ and $b$, beforehand, operation which is well-defined through Clebsh-Gordan sum rules. [Frag4]{} (48, 116)[$T$]{} (206, 100)[$R$]{} (284, 83)[$S_0$]{} (316, 143)[$S_0$]{} (177, 47)[$a$]{} (146, 55)[$\partial'_a$]{} (162, 19)[$m'_a$]{} (258, 42)[$b$]{} (241, 17)[$\partial''_b$]{} (279, 17)[$m''_b$]{} (333, 46)[$c$]{} (330, 17)[$\partial'''_c$]{} (358, 52)[$m'''_c$]{} (270, 130)[$e$]{} (298, 112)[$m^{+}_e$]{} (263, 167)[$q$]{} (241, 187)[$\partial_q$]{} (280, 191)[$m_q$]{} Let us focus on a directing configuration having one ingoing link, and three outgoing ones. Let then $e = a \oplus b$, where $\oplus$ stands for charge fusion, and clearly $q = e \oplus c$. We have to take into account all the allowed intermediate charges, i.e. those $e$ for which $$\mu^{q}_{e,c} \times \mu^{e}_{a,b} \neq 0.$$ Adopting the intermediate charge scheme is sufficient to label and address separately the different copies of the same irrep $\mathbb{V}^{[q]}$. Then the symmetric tensor $T$ fragments as follows: $$T^{\,i}_{j,k,l} = \sum_e \left[ ( R^{\,q,e}_{a,b,c} )^{\partial_q}_{\partial'_a, \partial''_b, \partial'''_c} \times ( S^{\,q,e}_{a,b,c} )^{m_q}_{m'_a, m''_b, m'''_c} \right],$$ but, at the same time, we know that the structural fragment is obtained by two consecutive fusions, so in the end $$( S^{\,q,e}_{a,b,c} )^{m_c}_{m'_a, m''_b, m'''_c} = \sum_{m^{+}_{e}} \; \langle e, m^{+}_e| a, m'_a, b; m''_b \rangle \; \langle q, m_q| e, m^{+}_e; c, m'''_c \rangle,$$ i.e. $S$ further decomposes into a pair of sub-fragmented structure tensors $S_0$, coinciding with the Clebsh-Gordan tensor we used for the three-links case, and we must contract over the $m^{+}_e$ levels related to the $e$-charge irrep. The fragmentation scheme we just introduced is picted in figure \[fig:Frag4\]. [Fragbranch]{} (0, 107)[a)]{} (150, 107)[b)]{} (61, 105)[$q$]{} (217, 105)[$q$]{} (0, 12)[$a$]{} (159, 12)[$a$]{} (74, 12)[$b$]{} (222, 12)[$b$]{} (134, 12)[$c$]{} (294, 12)[$c$]{} (45, 61)[$e$]{} (249, 61)[$f$]{} Of course, we could alternatively decompose $T$ by fusing together $b$ and $c$ at a first step, and then fuse $a$ with $f = b \oplus c$, as in figure \[fig:branch\].b. In this case, we have different variational $\bar{R}$ and structural $\bar{S}$ tensor fragments, according to: $$T^{\,i}_{j,k,l} = \sum_f \left\{ ( \bar{R}^{\,q,f}_{a,b,c} )^{\partial_q}_{\partial'_a, \partial''_b, \partial'''_c} \times \sum_{m^{-}_f} \left[ ( S^{\,q}_{0\,a,f} )^{m_q}_{m'_a, m^{-}_f} \times ( S^{\,f}_{0\,b,c} )^{m^{-}_f}_{m''_b, m'''_c} \right] \right\},$$ which is the right-left specular of the diagram \[fig:Frag4\]. The two intermediate fusion schemes we showed are in strict connection. Indeed, if we consider the partial fusion basis $|e, m^{+}_e \rangle$ and $|f, m^{-}_f \rangle$ they are related [@Singh2] by the 6-index tensor $F$, (e.g. for the group $\mathcal{G} =$ SU(2), $F$ coincides with Wigner’s 6-j symbols), so that: $$\bar{S}^{q,f}_{a,b,c} = \sum_e ( F^{a,b,c}_d )^{e}_f \times S^{q,e}_{a,b,c}.$$ At the same time, since the global tensor $T$ is uniquely defined and does not depend on the choice of intermediate fusion basis, the variational fragments $R$ and $\bar{R}$ must transform accordingly: $$\bar{R}^{q,e}_{a,b,c} = \sum_e ( F^{a,b,c}_d )^{\star\,e}_f \times R^{q,e}_{a,b,c}.$$ Theoretically, we could also study the scenario where $a$ and $c$ fuse together into an intermediate charge $h$, which fuses with $b$ afterwards. But in order to investigate this framework, we should exchange the order of links, say $a$ and $b$, before the first fusion takes place. As we discussed in section \[sec:linkexchange\], when swapping Tensor Network indices is needed, the inner nature of the degrees of freedom we are describing (i.e. whether they are spins, fermions, bosons or even anyons) manifests as an exchange-statistic of links themselves. Therefore, the resulting fragmentation picture will depend on the statistic obeyed by the particles we are describing . Generalization of the equations we encountered in this section to other tensor topologies can be done by hand following the same intermediate charge rules. This concludes our discussion regarding Tensor Network fragmentation due to symmetry relations. Finite charge Tensor Network states ----------------------------------- The fragmentation schemes we just analyzed for different tensor topologies, derived from the argument that all the tensors in the network ought to be symmetric to make the global state $|\Psi_{\text{TN}}\rangle$ invariant under the application of $U^{\otimes L}_{\mathcal{G}}$. Although this is an intriguing context, it is quite limited for practical purposes of studying physical settings. Indeed, in most variational problems, we are actually interested in working with an arbitrary, fixed, finite symmetry charge $q$. For instance, assume our model Hamiltonian $\mathcal{H}$ manifests a pointwise U(1) invariance, i.e. undergoes a particle conservation law: this means that global states, expanded in irreps subspaces as $|c, \partial_c, m_c\rangle$, belonging to different sectors $c$ are not coupled by $\mathcal{H}$. Then we might wish to achieve the ground state of $\mathcal{H}$ *restricted* to a given charge, i.e. number of particles; this is definitely a physical question. Analogously, we could be interested in describing the lowest energy state of a SU(2)-invariant $\mathcal{H}$, on a spin-$\frac{1}{2}$ lattice, with total spin, say, $\frac{7}{2}$. Achieving a *charge-selective ground state* is no trivial task from a variational point of view. It can not be performed blindly, by starting from the correct sector and hoping that the symmetry-invariant dynamics will prevent other sectors to be explored. This approach typically fails, as numerical errors will inevitably introduce other-sectors fluctuations, which will ultimately break the symmetry, and push the algorithm towards the *absolute* ground state. The right way to deal with this charge-selective quantum problem is by forcing the variational wavefunctions to belong to the right sector. Here is were Tensor Networks succeed: we will be able to recover fragmentation rules we learned in the previous section, and apply them even for finite charge $c$ TN-states, with $c$ chosen by the user. Therefore, let $|\Psi_{\text{TN}}\rangle$ be our Tensor Network state, for a given graph geometry, bondlink dimensions $D$, and consequent entanglement bounds. We will require that $|\Psi_{\text{TN}}^{[q]}\rangle$, when expanded in irrep-subspace basis $|c, \partial_c, m_c\rangle$ of the pointwise symmetry group $U^{\otimes L}_{\mathcal{G}}$, has nonzero components only for a given charge $q$: $$\label{eq:chargeselect} |\Psi_{\text{TN}}^{[q]}\rangle = \sum_{c} \sum_{\partial_c}^{\bar{\partial}_c} \sum_{m_c}^{\bar{m}_c} \left( \delta_{c,q} \; \mathcal{T}^{[q]}_{\partial_{q}, m_{q} } \right)\; |c, \partial_c, m_c \rangle.$$ Let us recall that $\partial_c$ are degeneracy indices, while $m_c$ are irrep basis indices. So when a symmetry transformation occurs, it interferes with the $m_c$ but leaves the $\partial_c$ unaltered, separately for every sector $c$. Then, our fixed-charge TN-state transforms as $$\label{eq:chargeselect2} U_g^{\otimes L} |\Psi_{\text{TN}}^{[q]}\rangle = \sum_{\partial_q}^{\bar{\partial}_q} \sum_{m_q}^{\bar{m}_q} \left( \sum_{m'_q}^{\bar{m}_q} ( V^{[q]}_g )_{m_q, m'_q} \; \mathcal{T}^{[q]}_{\partial_{q}, m'_{q} } \right) |q, \partial_q, m_q \rangle$$ with $V^{[q]}_g$ being the irrep unitary matrix, charge $q$, group element $g$. Now, we will give a prescription on the Tensor Network itself that will allow *only* states in the form $|\Psi_{\text{TN}}^{[q]}\rangle$, i.e. , to be variationally generated. [Chargefix]{} (16, 133)[$C$]{} (26, 159)[$\delta_{c,q}$]{} (80, 163)[$c$]{} (80, 127)[$m_c$]{} (118, 145)[source]{} (92, 65)[Tensor Network]{} (74, 65)[$\downarrow$]{} (182, 65)[$\downarrow$]{} (8, 13)[$s_1$]{} (50, 13)[$s_2$]{} (252, 13)[$s_L$]{} 1. Choose a single tensor in the network, a node in the graph. We will refer to this tensor as *source node*. 2. Direct the graph, i.e. associate a direction to every network link. Also, we require that every tensor in the network has *at least one incoming link*, except for the source node, which instead must have no incoming links. Such a directing scheme always exists, as long as the graph is fully connected. Moreover, if there are no closed loops, this scheme is unique. Physical links are meant as outgoing. [Chargefix2]{} (16, 133)[$C$]{} 3. Associate a representation $W_{(\alpha)}$ of $\mathcal{G}$ to every link $\alpha$. 4. Add a single one-leg tensor to the network (*selector node*), which connects to the source node. It is easy to see that TN-entanglement bounds are unaltered by this graph geometry adjustment. The link we just added is directed from the selector node to the source node. The representation of $\mathcal{G}$ associated to the new link is the $q$-charge irrep $V^{[q]}_{\mathcal{G}}$. 5. Symmetrize every tensor in the network, except for the selector node (but including the source tensor), according to directions and representations $W_{(\alpha)}$ of the links touched. This means that every tensor $T$ decomposes into its variational $R$ and structural $S$ fragments, as we discussed in section \[sec:fragment\]. 6. Fix the charge selector tensor to be $T_{c, m_c} = \delta_{c,q}\,C_{m_q}$, $C$ arbitrary. This concludes the prescription. The Tensor Network state then reads as in figure \[fig:chgfix\]. The charge selector tensor can not be symmetric (unless $C_{m_q} = 0$, or $q = 0$), as its only connected link has a nontrivial representation $V^{[q]}$. And in fact, $|\Psi_{\text{TN}}^{[q]}\rangle$ will not be $U^{\otimes L}_{\mathcal{G}}$ invariant since not every tensor is symmetric (actually, all but one are). Then, consider $\mathcal{Z}$, the contraction of all the tensors in the final network except for the selector node. Clearly $\mathcal{Z}$ is symmetric, as contraction of symmetric tensors; i.e. it holds $$\mathcal{Z}^{m_q}_{s_1 \ldots s_L} = \sum_{m'_q}^{\bar{m}_q} \sum_{\{r\}}^d ( V^{[q]\,\star}_{g} )_{m'_q, m_q} \left( \prod_{\ell} ( U_{g} )_{s_{\ell}, r_{\ell}} \right) \mathcal{Z}^{m'_q}_{r_1 \ldots r_L} \qquad \forall g \in \mathcal{G},$$ or, in matricial form, $U^{\otimes L}_{g} \cdot \mathcal{Z} \cdot V^{[q]\,\dagger}_{g} = \mathcal{Z}$. In conclusion, when we apply the pointwise symmetry group to the Tensor Network state, we obtain $$\begin{gathered} \label{eq:cfpass} U^{\otimes L}_{g} |\Psi_{\text{TN}}^{[q]}\rangle = U^{\otimes L}_{g} \sum_{s_1 \ldots s_L}^d \sum_{m_q}^{\bar{m}_q} \left( \mathcal{Z}^{m_q}_{s_1 \ldots s_L} C_{m_q} \right) |s_1 \ldots s_L\rangle = \\ = \sum_{s_1 \ldots s_L}^d \sum_{m_q}^{\bar{m}_q} \mathcal{Z}^{m_q}_{s_1 \ldots s_L} \left( \sum_{m'_q}^{\bar{m}_q} ( V^{[q]}_g )_{m_q, m'_q} \;C_{m'_q} \right) |s_1 \ldots s_L\rangle,\end{gathered}$$ which is formally equivalent to , thus proving that our prescription is sound. The reason why we needed to direct the graph so that it had a single source of directions (the source node) is to spread the information about $q$ to the whole network. If this is not the case, then one can identify regions of the network insensitive to $q$ thus actually behaving like symmetry-invariant zones: e.g. a party of sites which are always empty. Although the resulting state would still be a $q$-charge Tensor Network, it would be far more trivial. With the construction we just introduced, we are finally able to understand and exploit the strict relationship that ties symmetries and Tensor Networks through representation theory. In these sections we developed selection rules and manipulation techniques to embed symmetries into Tensor Network variational ansatze, allowing us to address charge-specific problems, and to meet a drastic speed-up in computational time. Example: Symmetries in MPS -------------------------- We would like to conclude this appendix chapter by applying the symmetry arguments and techniques upon a most common template in the family of Tensor Network, namely on Matrix Product States, the variational counterpart of DMRG algorithms. Methods for dealing with symmetries within the Density Matrix Renormalization Group framework were already known before the acknowledgement of Tensor Network states [@SchollDMRG], still, the in-depth understanding of both MPS representations and symmetric Tensor Network states, allows us to build a formulation for finite charge MPS which is compact, elegant, and efficient. Here we will work with open boundary conditions MPS, as the no-closed loop geometry encounters less accidents, and show the MPS-fragmentation scheme respectively for an abelian symmetry group $\mathcal{G}$, and then for a non-abelian one. Generalization to PBC is not trivial but possible nevertheless. ### MPS with pointwise U(1) The abelian symmetry U(1) has infinite one-dimensional ($\bar{m}_c = 1$, $\forall\,c$) non-equivalent representations labeled by integer numbers $c \in \mathbb{Z}$. U(1) is used to take care of particle conservation, when the Hamiltonian $\mathcal{H}$ has only terms that preserve particle number; indeed its fusion rule $\oplus$ corresponds to the simple sum of two integer numbers, i.e. $c \oplus c' \equiv c + c'$. A natural way to choose the source node (defined in the previous section), in order to characterize MPS states with an arbitrary particle number $q$, is to choose one of the edge blocks, say the one at right boundary. MPS tensors have three connection links, so we can use three-leg fragmentation rule to split a block into structural and variational part. Also, recall that since $\bar{m}_c = 1$, we have no need for blue (irrep vector) links. Ultimately, the resulting fragmented-MPS reads: $$\label{eq:Mpssym} \begin{overpic}[width = \textwidth, unit=1pt]{Mpssym} \put(17, 94){$R^{[1]}$} \put(19, 43){$S$} \put(64, 58){\footnotesize $c_1$} \put(64, 106){\footnotesize $\partial_{c_1}^{[1]}$} \put(106, 94){$R^{[2]}$} \put(108, 43){$S$} \put(153, 58){\footnotesize $c_2$} \put(153, 106){\footnotesize $\partial_{c_2}^{[2]}$} \put(195, 94){$R^{[3]}$} \put(197, 43){$S$} \put(307, 94){$R^{[L]}$} \put(309, 43){$S$} \put(335, 79){\footnotesize $c_L$} \put(360, 67){$Q$} \put(29, 6){\footnotesize $s_1$} \put(118, 6){\footnotesize $s_2$} \put(207, 6){\footnotesize $s_3$} \put(319, 6){\footnotesize $s_L$} \end{overpic}$$ where the structural tensors are homogeneously defined $S^{s_j}_{c_{j-1}, c_j} = \delta_{c_j, c_{j-1} + s_j}$. The yellow tensor $Q$ is the charge selector node, properly connected to the source node $A^{[L]}$; its purpose is to select the global sector: $Q_{c_L} = \delta_{c_L,q}$ with a total charge $q$ chosen by the user. The tensor fragments $R^{[j]}$ are completely variational, and we can freely manipulate their parameters, for instance, to lower the total energy, without constraints: we will always remain forcefully in the correct $q$ sector due to the presence of $S$ fragments. An intriguing feature of this abelian symmetric-MPS is that it is always operationally possible to gauge-transform it into the left (or right) gauge, while preserving the fragmentation scheme . The basic idea is to perform a singular value decomposition of $R^{[j]}$ tensors separately for every $c_{j}$: $$( R^{[j]}_{c_j} )^{\partial_{c_j}^{[j]}}_{c_{j-1}, \partial_{c_{j-1}}^{[j-1]}} = \sum_{\beta_{c_j}} = ( U^{[j]}_{c_j} )^{\beta_{c_j}}_{c_{j-1}, \partial_{c_{j-1}}^{[j-1]}} \;\lambda_{\beta_{c_j}}\; ( V^{[j]\,\dagger}_{c_j} )^{\partial_{c_j}^{[j]}}_{\beta_{c_j}}.$$ This is equivalent to performing an SVD of a block diagonal matrix, by actually singular value decomposing every diagonal block separately. This is not only formally meaningful, but also cheaper in terms of computational time. In the end we can recover all the engineering we developed in section \[sec:minimizOBC\], which strongly exploited left and right gauges, and further enhance its computational power by embedding symmetries. ### MPS with pointwise SU(2) The Heisenberg model $\vec{\sigma}_i \cdot \vec{\sigma}_j$ is the archetype of an SU(2)-invariant lattice Hamiltonian. Being a continuous symmetry, SU(2) can not be spontaneously broken, so its ground state has to be a total spin 0 (provided it is possible by fusion rules). Nevertheless, we might be interested to describe either this ground state, or maybe the lowest energy level at fixed total spin $q$. SU(2) irrep charges $c$ are typically labeled by integer and half-integer positive numbers, i.e. $c \in \mathbb{N}/2$, and corresponding irrep (blue link) dimension $\bar{m}_c = 2c + 1$. The MPS fragmentation scheme [@Diasym] then reads: $$\label{eq:Mpssym2} \begin{overpic}[width = \textwidth, unit=1pt]{Mpssym2} \put(17, 94){$R^{[1]}$} \put(19, 42){$S$} \put(64, 58){\footnotesize $c_1$} \put(64, 26){\footnotesize $m^{[1]}_{c_1}$} \put(64, 106){\footnotesize $\partial_{c_1}^{[1]}$} \put(106, 94){$R^{[2]}$} \put(108, 42){$S$} \put(153, 58){\footnotesize $c_2$} \put(153, 26){\footnotesize $m^{[2]}_{c_2}$} \put(153, 106){\footnotesize $\partial_{c_2}^{[2]}$} \put(195, 94){$R^{[3]}$} \put(197, 42){$S$} \put(307, 94){$R^{[L]}$} \put(309, 42){$S$} \put(335, 79){\footnotesize $c_L$} \put(360, 67){$Q$} \put(363, 37){$Y$} \put(29, 6){\footnotesize $s_1$} \put(118, 6){\footnotesize $s_2$} \put(207, 6){\footnotesize $s_3$} \put(319, 6){\footnotesize $s_L$} \end{overpic}$$ where structure fragments $S$ are Clebsh-Gordan coefficients. 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--- abstract: 'We demonstrate an efficient photon number detector for visible wavelengths using a silicon avalanche photodiode. Under sub-nanosecond gating, the device is able to resolve up to four photons in an incident optical pulse. The detection efficiency at 600 nm is measured to be 73.8%, corresponding to an avalanche probability of 91.1% of the absorbed photons, with a dark count probability below $1.1 \times 10^{-6}$ per gate. With this performance and operation close to room temperature, fast-gated silicon avalanche photodiodes are ideal for optical quantum information processing that requires single-shot photon number detection.' author: - 'O. Thomas' - 'Z. L. Yuan' - 'J. F. Dynes' - 'A. W. Sharpe' - 'A. J. Shields' title: Efficient photon number detection with silicon avalanche photodiodes --- Many applications require low-level light detection, such as biomedical imaging, time-of-flight ranging, astronomy and scientific research. In particular, highly efficient and low noise single photon detectors are a prerequisite for quantum information processing based on photonic qubits.[@buller10] For some advanced functions, these detectors must be able to resolve the number of photons in an optical pulse.[@kok07; @simon07; @briegel98] For over a decade, silicon avalanche photodiodes[@dautet93; @cova04] (Si-APD) have been the detector of choice for single photon detection at visible wavelengths due to their practicality, high quantum efficiency (QE) and low dark count noise. Their spectral response also makes them compatible with a number of quantum light sources.[@shields07] However, Si-APDs are believed to be threshold devices that detect only the presence or absence of photons, but not the photon number. Multiplexing in temporal[@achilles03] or spatial domain[@godik04; @yamamoto06; @akiba09] can be used to approximate a photon number detector (PND), but these devices suffer from cross-talk, high dark count rates as well as reduced efficiency due to the geometrical fill-factor. Alternative technologies, such as visible light photon counters[@petroff87; @kim99] or transition edge sensors,[@lita08] are capable of efficient PND, but the requirement of cryogenic cooling limits their prospect for practical use. For single photon detection, APDs are usually biased with a dc voltage above their breakdown, such that a single photo-excited carrier can stimulate a macroscopic current by means of avalanche multiplication. In this Geiger-mode, the avalanche is allowed to develop to a level that saturates the device, resulting in high detection efficiencies but a current that is independent of the number of photons initiating the avalanche. Recently, a practical scheme for PND has been demonstrated for InGaAs APDs operating at telecom wavelengths.[@kardynal08] This is implemented by gating the device with sub-nanosecond voltage pulses,[@yuan07] which are sufficiently short so as for photo-induced avalanches *not* to saturate the device. However, these APDs are insensitive to photons at wavelengths that are currently most commonly used for wider applications in applied physics. To bridge this gap, we demonstrate here an efficient PND using a fast-gated Si-APD. Operating close to room temperature we obtain a detection efficiency of 73.8%, giving an avalanche detection probability of 91.1% for absorbed photons. The device is able to resolve avalanches due to 0, 1, 2, 3 and 4 photons. This performance makes the detection scheme highly attractive for *single-shot* PND that is urgently required for quantum information processing. ![Photo- and dark currents vs dc bias. For the photocurrent measurement, the device is illuminated with 0.4 $\mu$W at 600 nm. []{data-label="fig:iv"}](fig0){width=".85\columnwidth"} For this study we used a Si-APD that was pigtailed with a multi-mode fiber for optical coupling. Cooled thermo-electrically to the operation temperature of –30 $^\circ$C, its breakdown voltage is measured to be 41.7 V at a dark current of 10 $\mu$A (Fig. \[fig:iv\]). By measuring the photosensitivity of the APD at dc biases far below the breakdown voltage, for which the gain is unity (Fig. \[fig:iv\]), the external $QE$ of the complete system was determined to be $81\pm1$% at a wavelength of 600 nm. This represents the highest obtainable single photon detection efficiency ($\eta$) with this device at this wavelength. The quantum efficiency can be improved, to a value close to unity, by optimizing the optical coupling, reducing reflection at the front surface and increasing the thickness of the absorbing layer in the APD. The surface microstructure can also be engineered to enhance the absorption.[@wu01] ![(a) Schematic of the Si-APD biasing and measurement electronics; (b) Gating voltage, $V_{apd}$, supplied to the Si-APD; (c) Output voltage signal from the Si-APD; (d) Self-differencer voltage output, $V_{sd}$. []{data-label="fig:sd"}](fig1){width=".88\columnwidth"} A schematic of the set-up used to characterize the Si-APD for single photon detection is shown in Fig. \[fig:sd\](a). A bias voltage consisting of a fixed dc component, $V_{dc}$, and a square wave with a frequency of 1 GHz, $V_{ac}$, \[Fig. \[fig:sd\](b)\] is applied to the APD so as to periodically produce an excess bias above the breakdown voltage ($V_{br}$), stimulating the detector into a single-photon sensitive state. $V_{ac}\sim10$ V was used throughout the experiment. Photon-induced avalanches cause a current through the APD that is sensed as a voltage across the 50 $\Omega$ resistor. The avalanches are subsequently quenched during the low voltage part of the detector bias cycle, restricting the duration of the avalanche to be less than 500 ps. The alternating bias induces a strong background signal \[Fig. \[fig:sd\](c)\] across the 50 $\Omega$ resistor due to the rapid charging and discharging of the APD. In order to detect photon-induced avalanches, this capacitive component is removed off the APD response with a self-differencing circuit,[@yuan07] which subtracts the gated APD output from that in the preceding period. This has the effect of cancelling the periodic background, leaving only the response to avalanches. The output signal of the self-differencer is amplified by a 20-dB broadband amplifier \[not shown in Fig. \[fig:sd\](a)\] before measurements by an oscilloscope or a time-correlated photon counting system. Figure \[fig:sd\](d) presents a typical output from the device captured on an oscilloscope. We show later that the output pulses of different amplitude can be attributed to avalanches stimulated by different numbers of incident photons. ![(a) Avalanche detection probability ($P_\eta$) for photo-electrons and single photon detection efficiency ($\eta$) as a function of the applied dc bias, $V_{dc}$. Inset shows the dc photosensitivity measured under dark (grey line) and illuminated (black line) conditions; (b) Dark count ($P_D$) and afterpulse ($P_A$) probabilities plotted as a function of the APD dc bias.[]{data-label="fig:performance"}](fig2){width=".96\columnwidth"} The single photon detection efficiency ($\eta$) was first measured as a function of the applied dc bias using the time-resolved single photon counting technique.[@yuan07] The optical excitation pulses at 600 nm, obtained using sum frequency generation between a 1550 nm pulsed laser and a 980 nm continuous-wave pump, were synchronized at 1/64 of the gate frequency (1 GHz/64) and attenuated to an average flux of $\mu$=0.033 photons/pulse using a calibrated optical power meter. In determining $\eta$, the dark and afterpulse counts were carefully removed. As shown in Fig. \[fig:performance\](a), $\eta$ increases with the applied bias to a maximum value of 73.8%. Using the measured $QE = 81\%$, we are able to obtain an avalanche detection probability ($P_{\eta}=\eta / {QE}$) of 91.1%, *i.e.* the probability that an absorbed photon produces a detectable avalanche. This suggests that very high detection efficiencies ($\eta>$90%) may be achieved using our detection scheme with an APD when the external quantum efficiency is optimized. ![(a) Probability distribution (symbols) for avalanche amplitudes measured for a incident photon flux of $\mu = 5.16$. The solid line is a fit using the sum of individual Gaussians (dashed lines) to represent photon number states. (b),(c) and (d) Probability distributions for avalanche amplitudes measured for different photon fluxes of $\mu$ = 0.21, 1.82, and 4.00, respectively.[]{data-label="fig:pnr"}](fig3){width=".95\columnwidth"} As usual, a trade-off is present between the detection efficiency and error rate. Figure \[fig:performance\](b) shows that the probability of dark counts, *i.e.*, false signals occurring spontaneously in the absence of illumination, increases with the dc bias. However, the measured dark count probability remains relatively low ($<1.1 \times 10^{-6}$/gate) even for the highest detection efficiencies. APDs also exhibit ‘afterpulses’ \[also shown in Fig. \[fig:performance\](b)\], which are spurious counts arising from avalanches triggered by re-emission of trapped avalanche charges. Afterpulsing is negligible for a dc voltage of 47.6 V or below. The afterpulse probability increases sharply when further raising the dc bias. This rapid increase in the afterpulse rate occurs because the APD quenching voltage approaches the breakdown condition. This gives rise to a large number of residual unquenched avalanche carriers, which can stimulate secondary additional avalanches. By using a square wave with higher amplitude than was possible here, we expect to be able to reduce significantly the afterpulse probability, even for avalanche detection probabilities ($P_\eta$) approaching unity. For the present gating conditions, an afterpulse probability of 7.5% for a corresponding avalanche detection probability of 91.1% is already very attractive for applications. To evaluate the PND capability, a fast oscilloscope is used to sample the distribution of output voltage pulses. Figure \[fig:pnr\](a) shows the measured probability distribution (symbols), acquired from around $5 \times 10^6$ samples for an incident photon flux $\mu = 5.16$, revealing that the APD produces signals of distinctly different amplitudes. These are ascribed to avalanches stimulated by different numbers of photons. The peak at 0 mV reflects the background noise from gates in which no photon was detected. The feature around 22.4 mV is due to the absorption of single photons and those at 43.1 mV, 63.8 mV and 85.5 mV correspond to $N=$ 2, 3 and 4 photons respectively. A good fit (solid line) to the measured data is obtained by modeling each photon number state with a Gaussian (dashed line), assuming the widths of which were scaled by $N^{0.5}$ to reflect the statistical broadening due to avalanche noise. The occurrence probabilities for each photon number is found to be in good agreement with the Poissonian statistics of the light source for a detected flux of 3.7. The assignment of detected photon number is further confirmed by the illumination flux dependence. Figures \[fig:pnr\](b–d) shows the evolution of the avalanche voltage distribution, measured for different incident photon fluxes. As expected, for $\mu$ = 0.21, signals due to the detection of single photons forms a distinctive peak beside the dominant 0-photon ‘noise’ peak. Signals from 2 or more photons are not clearly visible. As the photon flux increases, the contribution due to 2- and higher photon events increases. The evolution of the area of the photon number peaks is found to be consistent with Poissonian statistics for all incident fluxes. ![ Avalanche probability distributions for a detected flux of 0.92. The dashed curves depict number states corresponding to *N* = 0 - 2, assuming Poissonian statistics of the source, were normalized to reflect a uniform photon distribution (dotted curves). Discrimination levels (thin solid lines) were placed at the crossing between adjacent peaks.[]{data-label="fig:error"}](fig4){width=".7\columnwidth"} We now consider the PND capability of the device for the Bell-state measurements[@zukowski93] used in quantum information processing, which require that we distinguish between photon number states with *N* = 0, 1 and $\geq2$. Figure \[fig:error\] shows the output pulse distribution obtained for a moderate detected flux of 0.92, giving rise to significant contributions from only 0, 1 and 2-photon events. The error in determining the photon number from the measured avalanche voltage, $\varepsilon_N$, was calculated from the numerical overlap between the Gaussians used to fit the photon number states (dashed curves), normalized to reflect a uniform photon number distribution (dotted curves). To retain the detection efficiency we placed single discrimination levels at the crossing between successive photon numbers, for which we obtain photon number detection errors of $\varepsilon_0$ = 0.2 %, $\varepsilon_1$ = 12.2 % and $\varepsilon_2$ = 6.95 %. Finally we comment upon the origin of the PND in Si-APDs in which the avalanches are quenched before saturation. The excess noise of the avalanches, quantified as $\langle V^2\rangle / \langle V \rangle ^2$ with $V$ as the avalanche amplitude, is found to be $1.175$ for the 1-photon peak in Fig. \[fig:error\]. This value is smaller than 2, which is the theoretical minimum expected for macroscopic avalanche multiplication.[@kim99] Such a reduction in the excess noise could arise from two mechanisms: local saturation intrinsic low-noise avalanche multiplication. Further experiments are required to clarify this issue although it has recently been shown that InGaAs APDs can resolve the photon number for tightly-focused optical excitation.[@yuan10] In conclusion, Si-APDs under short excess bias pulses can be used for photon number detection with high detection efficiency in a practical, thermo-electrically cooled device. The detector is attractive for quantum information applications requiring single-shot photon number detection at high repetition rates. 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--- abstract: 'We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a nontrivial singular continuous part. A new local Borg–Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan’s condition.' address: - ' Department of Mathematics, Rice University, Houston, TX 77005, USA' - 'Ocean University of China, Qingdao 266100, Shandong, China and Rice University, Houston, TX 77005, USA' - ' Department of Mathematics, Rice University, Houston, TX 77005, USA' author: - David Damanik - Licheng Fang - Selim Sukhtaiev title: Zero Measure and Singular Continuous Spectra for Quantum Graphs --- [^1] [^2] [^3] Introduction ============ Overview -------- The spectral theory of Schrödinger operators with irregular potentials has been of great interest in mathematical physics since the 1950’s. A large number of models have been treated rigorously in the setting of the one-dimensional Laplacian perturbed by an irregular potential. As far as multidimensional phenomena are concerned, only a small fraction of the expected results has been proved. In this paper we focus on structures of intermediate dimensionality – continuum metric graphs. Specifically, we study the Kirchhoff Laplacian on aperiodic infinite volume graphs and demonstrate that they exhibit nontrivial spectral behavior. The first example of a quantum graph with “exotic" spectrum is due to Simon, cf. [@S96] where an infinite combinatorial graph with singular continuous spectrum was constructed. Typically, interesting spectral phenomena (e.g., Anderson localization) occur due to irregular lower order perturbations of a fixed second order operator, which is often the Laplacian. In the setting of graphs, however, it is natural to consider another type of perturbation – by geometry. For example, the graph in Figure \[Fig1\] may be viewed as a geometric perturbation, by inserting diamond tiles, of the half line. The latter has purely absolutely continuous spectrum $[0,\infty)$, while the former may exhibit all kinds of spectra depending on how the tiles are inserted. This paper concerns aperiodic geometric perturbations (of more general graphs) leading to zero Lebesgue measure spectra, a scenario of interest in the modeling of quasicrystals, [@DL2; @LSS]. In the past two decades aperiodic Schrödinger operators on continuum and combinatorial graphs have attracted a significant amount of attention. The main interest has been around the relation between the geometry of graphs and their spectra. Let us mention some relevant results. In [@Br07] Breuer constructed a sparse combinatorial tree graph with singular continuous spectrum. This result was extended to the continuum setting by Breuer and Frank in [@BF], where it was also shown that the singular continuous spectrum is in fact generic. Next, regular trees provided some insight into the Anderson model. For instance, delocalization in the regime of low disorder on the Bethe lattice was established by Klein in [@K3]. A new spectral behavior for the tree Anderson Hamiltonian near spectral edges was discovered by Aizenman and Warzel, [@AW11; @AW13]. Dynamical localization for radial trees with disordered branching numbers and edge lengths was established by Damanik, Fillman, and Sukhtaiev in [@DFS; @DS]. Periodicity of radial trees in the presence of absolutely continuous spectrum was shown by Exner, Seifert, and Stollmann in [@ESS]. Grigorchuk, Lenz, and Nagnibeda recently proved zero measure spectrum for [*discrete*]{} Laplacians on certain aperiodic graphs, cf. [@GLN]. To the best of our knowledge, continuum Kirchhoff Laplacians on [*nontrivial*]{}[^4] graphs with zero measure spectrum and nonempty singular continuous part have not yet been discussed in the literature. Our goal is to address this issue by combining a recent work of Kostenko and Nicolussi [@KN] (see also [@BKe; @BL]) and the theory of ergodic Schrödinger operators [@CL; @D2; @DF]. \(1) at (1,0); (2) at (2,0.7); (3) at (2,-0.7); (4) at (3,0); (5) at (4,0); (6) at (5,0.7); (7) at (5,-0.7); (8) at (6,0); (9) at (6,0); (10) at (7,0.7); (11) at (7,-0.7); (12) at (8,0); (13) at (9,0); (1) to (2); (1) to (3); (2) to (4); (3) to (4); (4) to (5); (5) to (6); (5) to (7); (6) to (8); (7) to (8); (9) to (10); (9) to (11); (10) to (12); (11) to (12); (12) to (13); (9.8,0) circle (.7pt); (9.4,0) circle (.7pt); (9.6,0) circle (.7pt); [Fig1]{} Setup ----- Let $\Gamma=({{\mathcal V}},{{\mathcal E}})$ be a connected graph with root $o\in{{\mathcal V}}$. For $v\in{{\mathcal V}}$, let $|v|$ denote the combinatorial length of the shortest path connecting $v$ and $o$. The combinatorial sphere of radius $n\geq0$ is given by $S_n:=\{v\in{{\mathcal V}}: |v|=n\}$, the cardinality of this sphere is denoted by $s_n:=|S_n|$. In this paper we focus on graphs $\Gamma$ satisfying the following: Two vertices $u,v\in{{\mathcal V}}$ are adjacent if and only if $$(u,v)\in (S_0\times S_1)\bigcup_{n\geq 1}S_n\times (S_{n-1}\cup S_{n+1}),$$ see, e.g., Figure \[Fig1\] where $s_0=1$, $s_1=2$, $s_3=1$. We equip $\Gamma$ with a metric by assigning each edge $e\in{{\mathcal E}}$ length $1$. A natural orientation is determined by the growth of spheres $S_n$. The Kirchhoff Laplacian on $\Gamma$ is defined by $$\begin{aligned} \begin{split}{\label}{new1.1} &{{\mathbb{H}}}:{\operatorname{dom}}({{\mathbb{H}}})\subset L^2(\Gamma)\rightarrow L^2(\Gamma), {{\mathbb{H}}}u:= -u'',\ u\in{\operatorname{dom}}({{\mathbb{H}}}),\\ &{\operatorname{dom}}({{\mathbb{H}}})=\{f\in {\widehat}H^2(\Gamma)\cap C(\Gamma): \sum\limits_{e \in \mathcal E : v \in e}\partial_{\nu}^{e}f(v)=0, v \in \mathcal{V}\}, \end{split} \end{aligned}$$ where ${\widehat}H^2(\Gamma):=\bigoplus_{e\in{{\mathcal E}}}H^2(e)$ is the $L^2$ based Sobolev space, cf. [@BK]. The main goal of this paper is to prove that the spectrum of ${{\mathbb{H}}}$ is a zero Lebesgue measure set whenever the sequence of sphere numbers satisfies certain dynamical conditions. Due to the large number of symmetries of $\Gamma$, the operator ${{\mathbb{H}}}$ can be written, see [@KN Theorem 3.5], as the direct sum of self-adjoint realizations of the following Sturm-Liouville differential expression $$\begin{aligned} \begin{split} &\tau = -\frac{1}{\mu(x)}\left(\frac{d}{dx}\mu(x)\frac{d}{dx}\right),{\label}{n1.1n}\\ &\mu(x):=\sum_{n=0}^{\infty}{s}_n {s}_{n+1}\chi_{[n, {n+1})}(x), x\in {{\mathbb{R}}}_+. \end{split} \end{aligned}$$ The self-adjoint realizations $\tau$ in question are of two spectrally relevant types. The operators of the first type have compact resolvent and, hence, only discrete spectrum. They do enter the direct sum decomposition of ${{\mathbb{H}}}$ infinitely many times. However, there is only a finite collection of mutually non-unitarily equivalent operators of the first type. Therefore, their total spectral contribution is worth only a zero Lebesgue measure set (as it is countable). Of course, some points of this set end up being eigenvalues of ${{\mathbb{H}}}$ of infinite multiplicity. Since they could be isolated from the rest of the spectrum of ${{\mathbb{H}}}$, we do not claim that $\sigma({{\mathbb{H}}})$ is a (generalized) Cantor set. The second type of self-adjoint realizations is given by $\tau$ acting in $L^2(0,\infty)$ subject to the Neumann condition at $0$. Showing zero measure spectrum for this operator is the main technical issue addressed in this work. To that end we utilize Kotani theory for $\tau$ and the base dynamical system $(\Omega, T)$ given by a subshift over the admissible values of the sphere numbers. Assuming that this subshift satisfies Boshernitzan’s condition we obtain that the spectrum is given by the set ${{\mathcal Z}}$ on which the Lyapunov exponent vanishes, see Section \[section4\]. This step, in particular, requires a Sch’nol-type result for $\tau$, see Lemma \[lemma2.2\], [@BMLS]. Then we argue that $|{{\mathcal Z}}|>0$ implies periodicity of the sequence of sphere numbers, contrary to the construction of $\Omega$. This gives that the spectrum of the half-line operator is a zero measure set, see Theorem \[theorem4.5\], which in turn yields the main assertion. A key to this argument is a new version of the celebrated Borg–Marchenko result, [@GL; @GS; @M; @S], establishing a one-to-one correspondence between the Weyl–Titchmarsh functions and the potentials. A rather general version of this result was obtained in [@EGNT1]; however, the operators of the form are notably excluded from consideration, see [@EGNT1 Hypothesis 6.1]. In this work, we prove a local version of Borg–Marchenko result that allows us to recover arbitrary finite blocks of sphere numbers from the asymptotic behavior of the $m$-functions for large non-real values of the spectral parameter, see Theorem \[thm1.4\]. It is worth mentioning that Breuer and Frank, [@BF Proposition 12], obtained the Borg–Marchnko result for a somewhat relevant class of operators. Our method, which itself stems from Bennewitz’s work [@Ben], can easily be adapted to yield [@BF Proposition 12]. Main Results ============ Fix a finite set ${{\mathcal J}}\subset{{\mathbb{N}}}$. Let ${{\mathcal J}}^{{{\mathbb{Z}}}}$ be equipped with the following metric $${\label}{3.21} d(\omega, {\widetilde }\omega):=\sum_{n\in{{\mathbb{Z}}}}\frac{1-\delta_{\omega_n, {\widetilde }\omega_n}}{2^{|n|+1}},\ \omega, {\widetilde }\omega\in{{\mathcal J}}^{{{\mathbb{Z}}}}.$$ Let $T: {{\mathcal J}}^{{{\mathbb{Z}}}}\rightarrow{{\mathcal J}}^{{{\mathbb{Z}}}}$ denote the left shift, i.e., $[T(\omega)]_n:=\omega_{n+1}$, $\omega\in{{\mathcal J}}^{{{\mathbb{Z}}}}$. A $T$-invariant, closed (with respect to $d$) subset $\Omega\subset {{\mathcal J}}^{{{\mathbb{Z}}}}$ is called a [*subshift*]{} over ${{\mathcal J}}$. This paper is concerned with a special class (which is yet very general[^5]) of subshifts satisfying Boshernitzan’s condition $(B)$. Let us recall the relevant definitions. We say that $(\Omega, T)$ is [*minimal*]{} if every orbit $\{T^n\omega: \omega\in\Omega\}$ is dense. A minimal subshift is called [*aperiodic*]{} if one of its elements is not periodic, that is, $T^p \omega = \omega$ for some $p \in {{\mathbb{Z}}}$ implies $p = 0$. It is then easy to see that all elements of $\Omega$ are aperiodic. The set of words corresponding to $\Omega$ is defined by $${{\mathcal W}}:=\{\omega_k\cdots \omega_{k+n-1}:k\in\mathbb{Z},\ n\in\mathbb{N},\ \omega\in\Omega\}.$$ Each word $w\in{{\mathcal W}}$ of length $|w|\in{{\mathbb{N}}}$ determines the cylinder set $$V_{w}:=\{\omega\in\Omega: \omega_1\cdots \omega_{|w|}=w \}.$$ For a $T$-invariant probability measure $\nu$ on $\Omega$, let us define the following quantity, $$\eta_{\nu}(n):=\min\{\nu(V_{w}): w\in {{\mathcal W}}, |w|=n \}.$$ A minimal subshift $(\Omega, T)$ is said to satisfy Boshernitzan’s condition (B) if there is an ergodic probability measure $\nu$ on $\Omega$ such that $${\label}{BC} \limsup\limits_{n\rightarrow\infty} n\,\eta_{\nu}(n)>0.$$ In the theorem below we denote $\Omega\upharpoonright_{{{\mathbb{Z}}}_+}:=\{ \{s_n\}_{n=0}^{\infty}: \{s_n\}_{n\in{{\mathbb{Z}}}}\in\Omega\}$. [main]{} Suppose that $(\Omega, T)$ is a minimal aperiodic subshift over a finite set ${{\mathcal J}}\subset {{\mathbb{N}}}$, $|{{\mathcal J}}|\geq 2$ satisfying Boshernitzan’s condition . Then for every $\{s_n\}_{n=0}^{\infty}\in\Omega\upharpoonright_{{{\mathbb{Z}}}_+}$, the spectrum of the Kirchhoff Laplacian ${{\mathbb{H}}}$ defined in is a zero Lebesgue measure set. The Kirchhoff Laplacian ${{\mathbb{H}}}$ is unitarily equivalent, cf. [@KN Theorem 3.5], to the following direct sum $${\label}{dec} H^+\oplus\bigoplus_{n\geq 1} \oplus _{j=1}^{s_{n}-1} H_n^1 \oplus\bigoplus_{n\geq 1}\oplus_{j=1}^{(s_{n}-1)(s_{n+1}-1)}H_n^2,$$ where $H^+, H^1_n, H^2_n$ are the self-adjoint realizations of on ${{\mathbb{R}}}_+=(0,\infty)$, $J_n:=(n-1, n+1),$ and $I_n:=(n, n+1)$, correspondingly, defined next. The half-line operator is given by $$\begin{aligned} \begin{split}{\label}{nn1.1n} &H^+:{\operatorname{dom}}(H^+)\subset L^2({{\mathbb{R}}}_+; \mu)\rightarrow L^2({{\mathbb{R}}}_+; \mu), H^+ u:= \tau u,\ u\in{\operatorname{dom}}(H^+),\\ &{\operatorname{dom}}(H^+)=\{f\in L^2({{\mathbb{R}}}_+, \mu): f, \mu f'\in AC({{\mathbb{R}}}_+), \tau f\in L^2({{\mathbb{R}}}_+; \mu), f'(0)=0\}. \end{split} \end{aligned}$$ The operators $H_n^1, H_n^2$, defined on finite intervals, are given by $$\begin{aligned} {\label}{9.1} &H^1_n:{\operatorname{dom}}(H^1_n)\subset L^2(J_n; \mu)\rightarrow L^2(J_n; \mu), H^1_n u:= \tau u,\ u\in{\operatorname{dom}}(H^1_n),\\ &{\operatorname{dom}}(H^1_n)=\left\{f\in L^2(J_n, \mu)\Big| \begin{matrix} f, \mu f'\in AC(J_n), \tau f\in L^2(J_n; \mu),\\ f(n-1)=f(n+1)=0 \end{matrix}\right\}, \end{aligned}$$ and $$\begin{aligned} {\label}{9.2} &H_n^2:{\operatorname{dom}}(H_n^2)\subset L^2(I_n; \mu)\rightarrow L^2(I_n; \mu), H_n^2 u:= \tau u,\ u\in{\operatorname{dom}}(H_n^2),\\ &{\operatorname{dom}}(H_n^2)=\left\{f\in L^2(I_n, \mu)\Big| \begin{matrix} f, \mu f'\in AC(I_n), \tau f\in L^2(I_n; \mu),\\ f(n)=f(n+1)=0 \end{matrix}\right\}. \end{aligned}$$ Since $s_n$ takes finitely many values, the sequence of operators $\{H^1_n\}_{n\in{{\mathbb{Z}}}_+}$ contains only finitely many mutually non-unitarily equivalent operators, each of which has only discrete spectrum (in fact the eigenvalues are given by the roots of some trigonometric transcendental equation, see [@Ber]). The union $\Sigma_1$ of these spectra is a countable set. Next, every operator $H_n^2$ is unitarily equivalent to the Dirichlet Laplacian on $(0,1)$, its spectrum is discrete and denoted by $\Sigma_2$. Then employing we get $$\sigma({{\mathbb{H}}}) = \sigma(H^+) \cup \Sigma_1\cup\Sigma_2.$$ Thus, to prove the assertion it is enough to show that $|\sigma(H^+)|=0$. To that end, we extend the sequence $\{s_n\}_{n\geq 0}$ to a two-sided sequence belonging to the subshift $\Omega$, and introduce the corresponding self-adjoint operator on ${{\mathbb{R}}}$ as follows $$\begin{aligned} \begin{split} &H :{\operatorname{dom}}(H)\subset L^2({{\mathbb{R}}}; \mu )\rightarrow L^2({{\mathbb{R}}}; \mu ), H u:= \tau u,\ u\in{\operatorname{dom}}(H ),\\ &{\operatorname{dom}}(H )=\{f\in L^2({{\mathbb{R}}}, \mu ): f, \mu f'\in AC({{\mathbb{R}}}), \tau f\in L^2({{\mathbb{R}}}; \mu )\}, \end{split}\end{aligned}$$ where the function $\mu$ is defined for the two-sided sequence as in . Now, by [@T09 Theorem 9.11], we have $\sigma_{ess}(H^+)\subset \sigma_{ess}(H)$[^6]. By Theorem \[theorem4.5\], the Lebesgue measure of $\sigma_{ess}(H)$ is equal to zero (in fact, the whole spectrum is a fixed zero measure generalized Cantor set for every choice of the sequence of the sphere numbers from $\Omega$). Thus, we have $|\sigma(H^+)|\leq|\sigma_{ess}(H^+)|+|\sigma_{disc}(H^+)|=0$. Hence, one has $$|\sigma({{\mathbb{H}}})|\leq |\sigma(H^+)|+|\Sigma_1|+|\Sigma_2|=0,$$ as asserted. As pointed out in [@DL1; @DL2], many subshifts of interest satisfy condition (B). In fact, one can construct examples of such subshifts rather explicitly. Let us provide two examples. Let $N\geq 2$ and let ${{\mathcal J}}=\{\gamma_k; 1\leq k\leq N\}\subset {{\mathbb{N}}}$ be a set containing at least two distinct elements. Fix $\alpha\in{{\mathbb{R}}}\setminus{{\mathbb{Q}}}$ and a rational partition of the unit circle $0=a_0<a_1<...<a_N=1$, $a_k\in{{\mathbb{Q}}}$. Pick any $\omega\in[0,1)$ and define $$s_n(\omega):= \sum_{k=1}^{N}\gamma_k \chi_{[a_{k-1},a_k)}(n\alpha+\omega), n\in{{\mathbb{Z}}}.$$ Then the Kirchhoff Laplacian ${{\mathbb{H}}}$ given by such a sequence of sphere numbers has zero measure spectrum. Indeed, in this case the subshift $\Omega:=\overline{\{s(\omega): \omega\in[0,1)\}}$ satisfies Boshernitzan’s condition , see e.g. [@DL2 Theorem 10]. [ex2.3]{} Assume that the sequence of the sphere numbers $s\in \{1,2\}^{{{\mathbb{Z}}}_+}$ is invariant under the action of the Fibonacci substitution ${{\mathcal S}}(1)=12$, ${{\mathcal S}}(2)=1$, e.g. $s=12112121...$, see Figure \[Fig1\]. Then the Kirchhoff Laplacian ${{\mathbb{H}}}$ given by such a sequence of sphere numbers has zero measure spectrum. Here, $$\Omega:=\big\{\omega\in{{\mathcal J}}^{{{\mathbb{Z}}}}:\text{\ every subword of $\omega$ is a subword of $s$}\big\}.$$ It satisfies Boshernitzan’s condition by [@DL2 Theorem 4]. The sequences in both examples generate minimal subshifts satisfying (B), see [@D1; @D2; @DL1; @DL2] for details. The assertion of Theorem \[main\] also holds in the discrete setting, i.e., when ${{\mathbb{H}}}$ is given by the combinatorial graph Laplacian. Up to a countable set, the spectrum of the discrete Hamiltonian is given by the spectrum of a certain Jacobi matrix, see [@BKe]. Of course, the Borg–Marchenko and Sch’nol-type results become trivial at this level. Elements of Weyl–Titchmarsh Theory ================================== In this section we discuss Weyl–Titchmarsh theory for $\tau$, cf. , in order to prepare the necessary ingredients for Kotani theory. Specifically we prove a local version of the Borg–Marchenko uniqueness result in Theorem \[thm1.4\], cf. [@Ben; @B; @EGNT1; @GL; @GS; @M; @S], and utilize it to construct the restriction maps in Theorem \[thm3.6\]. In order to simplify the notation, cf. , we set $${\label}{1.1n} \mu(x):=\sum_{n=0}^{\infty}{s}_n \chi_{[n, {n+1})}(x), x\in {{\mathbb{R}}}_+,$$ where $\{s_n\}_{n\in{{\mathbb{Z}}}_+}\subset {{\mathcal J}}$ and ${{\mathcal J}}\subset{{\mathbb{N}}}$ is a finite set of cardinality at least two. The differential expression $\tau$ is defined as in . The Dirichlet and Neumann solutions of the differential equation $${\label}{1.1} \tau u-zu=0, z\in{{\mathbb{C}}},$$ are denoted by $\varphi(z, \cdot), \theta(z, \cdot)$, correspondingly, i.e., $$\varphi(z,0)=\mu(0)\theta'(z,0)=0,\ \mu(0)\varphi'(z,0)=\theta(z,0)=1.$$ [1.1nn]{} The conditions $f, \mu f'\in AC_{loc}({{\mathbb{R}}}_+)$ imply $$s_{n-1}f'(n^-)=s_{n}f(n^+), n\in{{\mathbb{N}}}.$$ Throughout this paper $\sqrt{z}$ denotes the branch of the square root corresponding to ${\operatorname{Re }}\sqrt{{{\bf i}}}>0$ and $\arg(z)\in(-\pi, \pi]$. In particular, ${\operatorname{Re }}(\sqrt{-z})>0$ whenever $z\in{{\mathbb{C}}}\setminus [0, \infty)$, and ${\operatorname{Re }}(\sqrt{-z})=0$ if $z\in[0,\infty)$. Borg–Marchenko Theorem For Sturm–Liouville Operators in Impedance Form ---------------------------------------------------------------------- The following lemma concerns the asymptotic behavior of the Dirichlet and Neumann solutions of for large non-real spectral parameter. [lemma1.1]{} For every $x\in{{\mathbb{R}}}_+\setminus{{\mathbb{N}}}$ one has $$\begin{aligned} \begin{split}&\varphi(z,x)=\frac{c(\lfloor x\rfloor)e^{\sqrt{-z}x}}{2\sqrt{-z}}(1+o(1)),{\label}{1.2}\\ &\varphi'(z,x)=\frac{c(\lfloor x\rfloor)e^{\sqrt{-z}x}}{2}(1+o(1)), \end{split} \end{aligned}$$ $$\begin{aligned} \begin{split}{\label}{1.2nn} &\theta(z,x)=\frac{c(\lfloor x\rfloor)e^{\sqrt{-z}x}}{2}(1+o(1)),\\ &\theta'(z,x)=\frac{c(\lfloor x\rfloor)\sqrt{-z}e^{\sqrt{-z}x}}{2}(1+o(1)), \end{split} \end{aligned}$$ as $z\rightarrow\infty$, $z\in{{\mathbb{C}}}\setminus[0,\infty)$, where $${\label}{1.2nnn} c(\lfloor x\rfloor):=\frac{1}{s_0}\prod_{k=1}^{\lfloor x\rfloor}\frac{s_k+s_{k-1}}{2s_k}$$ Furthermore, for each $j\in{{\mathbb{N}}}$ one has $${\label}{1.3} \varphi(z,x^+)={{\mathcal O}}\left(\frac{1}{\sqrt{z}}\right),\, \varphi'(z,x^+)={{\mathcal O}}\left(1\right),$$ uniformly for $x\in[0,j)$ as $z\rightarrow\infty$, $z\in[0,\infty)$. To simplify the notation we denote $\alpha_k:=\frac{s_{k-1}}{s_k}$, and declare that all asymptotic formulas in this proof are for $z\rightarrow\infty$, $z\in{{\mathbb{C}}}\setminus[0,\infty)$. Without loss we may assume that $s_0=1$. We will prove by induction in $\lfloor x\rfloor$. If $\lfloor x\rfloor=0$, then $\tau u-zu=0$ is given by the free equation $-u''(x)-zu(x)=0$, $x\in(0,1)$ and $$\begin{aligned} &\varphi(z,x^+)=\frac{\sinh(\sqrt{-z}x)}{\sqrt{-z}}=\frac{e^{\sqrt{-z}x}}{2\sqrt{-z}}(1+o(1)),\\ &\varphi'(z,x^+)=\cosh\sqrt{-z}x=\frac{e^{\sqrt{-z}x}}{2}(1+o(1)), \end{aligned}$$ as asserted. Suppose that both equations in hold for $\lfloor x\rfloor=k$, $k\in {{\mathbb{Z}}}_+$. Our first objective is to derive asymptotic formulas for $\varphi(z, (k+1)^+)$ and $\varphi'(z, (k+1)^+)$. Since $${\label}{n1.4} \left(\cosh\sqrt{-z}x; \frac{\sinh\sqrt{-z}x}{\sqrt{-z}}\right)$$ is a fundamental system for the differential equation considered on $(k, k+1)$, and, by Remark \[1.1nn\], $${\label}{nn1.4} \varphi(z,(k+1)^+)=\varphi(z, (k+1)^-),\ \varphi'(z,(k+1)^+)=\alpha_{k+1}\varphi'(z, (k+1)^-),$$ the vector $(\varphi, \varphi')^{\top}(z,(k+1)^+)$ may be recovered using the fundamental matrix corresponding to and the matrix of vertex conditions corresponding to as follows $$\begin{aligned} &\begin{bmatrix} \varphi(z, (k+1)^+)\\ \varphi'(z, (k+1)^+) \end{bmatrix} = \begin{bmatrix} \cosh\frac{\sqrt{-z}}{2}& \frac{\sinh\frac{\sqrt{-z}}{2}}{\sqrt{-z}}\\ {\alpha_{k+1}} {\sqrt{-z}\sinh\frac{\sqrt{-z}}{2}}&{\alpha_{k+1}}\cosh\frac{\sqrt{-z}}{2} \end{bmatrix} \begin{bmatrix} \varphi(z, k+1/2)\\ \varphi'(z, k+1/2) \end{bmatrix}. \end{aligned}$$ Combining this and the induction hypothesis we get $$\begin{aligned} \begin{split}{\label}{1.4} &\varphi(z, (k+1)^+)=\cosh\left(\frac{\sqrt{-z}}{2}\right)\varphi\left(z, k+\frac12\right)+ \frac{\sinh\frac{\sqrt{-z}}{2}}{\sqrt{-z}}\varphi'\left(z, k+\frac12\right)\\ &\quad=\left(\frac{e^{\frac{\sqrt{-z}}2}}{2}\frac{e^{\sqrt{-z}(k+1/2)}}{2\sqrt{-z}}+\frac{e^{\frac{\sqrt{-z}}2}}{2\sqrt{-z}}\frac{e^{\sqrt{-z}(k+1/2)}}{2}\right)c_k(1+o(1))\\ &\quad= \frac{c_ke^{\sqrt{-z}(k+1)}}{2\sqrt{-z}}(1+o(1)). \end{split} \end{aligned}$$ Likewise, $$\begin{aligned} \begin{split}{\label}{1.5} &\varphi'(z, (k+1)^+)={\alpha_{k+1}}{\sqrt{-z}}\,\sinh\left(\frac{\sqrt{-z}}{2}\right)\varphi\left(z, k+\frac12\right)\\ &\hspace{4cm}+{\alpha_{k+1}}\cosh\left(\frac{\sqrt{-z}}{2}\right)\varphi'\left(z, k+\frac12\right)\\ &\quad={\alpha_{k+1}} c_k\left(\frac{\sqrt{-z}e^{\frac{\sqrt{-z}}2}}{2}\frac{e^{\sqrt{-z}(k+1/2)}}{2\sqrt{-z}}+\frac{e^{\frac{\sqrt{-z}}2}}{2}\frac{e^{\sqrt{-z}(k+1/2)}}{2}\right)(1+o(1))\\ &\quad= \frac{{\alpha_{k+1}}c_ke^{\sqrt{-z}(k+1)}}{2}(1+o(1)). \end{split} \end{aligned}$$ Using the fundamental matrix corresponding to as before we get $$\begin{aligned} &\begin{bmatrix} \varphi(z, x)\\ \varphi'(z, x) \end{bmatrix} = \begin{bmatrix} \cosh{\sqrt{-z}(x-k)}& \frac{\sinh\sqrt{-z}(x-k)}{\sqrt{-z}}\\ {\sqrt{-z}\sinh\sqrt{-z}(x-k)}& \cosh\sqrt{-z}(x-k) \end{bmatrix} \begin{bmatrix} \varphi(z, k^+)\\ \varphi'(z, k^+) \end{bmatrix}, \end{aligned}$$ for every $x\in(k+1,k+2)$. Then combining this and the asymptotic formulas , we arrive at $$\begin{aligned} \begin{split} &\varphi(z, x)=\cosh(\sqrt{-z}(x-k))\varphi\left(z, k^+\right)+ \frac{\sinh{\sqrt{-z}(x-k)}}{\sqrt{-z}}\varphi'\left(z, k^+\right)\\ &\quad=\left(\frac{e^{\sqrt{-z}(x-k)}}{2}\frac{e^{\sqrt{-z}k}}{2\sqrt{-z}}+\frac{e^{\sqrt{-z}(x-k)}}{2\sqrt{-z}}\frac{{\alpha_{k+1}}e^{\sqrt{-z}k}}{2}\right)c_k(1+o(1))\\ &\quad= \frac{c_ke^{\sqrt{-z}x}}{2\sqrt{-z}}\left(\frac{1+{\alpha_{k+1}}}{2}\right)(1+o(1))=\frac{c_{k+1}e^{\sqrt{-z}x}}{2\sqrt{-z}}(1+o(1)), \end{split} \end{aligned}$$ and $$\begin{aligned} \begin{split} &\varphi'(z, x)= {\sqrt{-z}}\,\sinh(\sqrt{-z}(x-k))\varphi(z, k^+)+ \cosh(\sqrt{-z}(x-k))\varphi'\left(z, k^+\right)\\ &\quad= c_k\left(\frac{\sqrt{-z}e^{\sqrt{-z}(x-k)}}{2}\frac{e^{\sqrt{-z}k}}{2\sqrt{-z}}+\frac{e^{\sqrt{-z}(x-k)}}{2}\frac{{\alpha_{k+1}}e^{\sqrt{-z}k}}{2}\right)(1+o(1))\\ &\quad= \frac{c_ke^{\sqrt{-z}x}}{2}\left(\frac{1+{\alpha_{k+1}}}{2}\right)(1+o(1))=\frac{c_{k+1}e^{\sqrt{-z}x}}{2}(1+o(1)). \end{split} \end{aligned}$$ Therefore holds as asserted. The asymptotic formulas in can be shown similarly. All asymptotic formulas in this proof are for $z\rightarrow\infty$, $z\in[0,\infty)$. We will prove by induction in $j$. If $j=1$, then $$\begin{aligned} &\varphi(z,x^+)=\frac{\sin(\sqrt{z}x)}{\sqrt{z}},\varphi'(z,x^+)=\cos\sqrt{z}x, \end{aligned}$$ and thus holds. Assuming that holds for some $j\in{{\mathbb{N}}}$, we will prove that it also holds for $j+1$. Since $\varphi$ satisfies the vertex conditions, cf. , and since $\cos(\sqrt{z}x), \frac{\sin(\sqrt{z}x)}{\sqrt{z}}$ is a fundamental system for the differential equation considered on $(j-1,j)$, we get $$\begin{aligned} &\begin{bmatrix} \varphi(z, j^+)\\ \varphi'(z, j^+) \end{bmatrix} = \begin{bmatrix} \cos\frac{\sqrt{z}}{2}& \frac{\sin\frac{\sqrt{z}}{2}}{\sqrt{z}}\\ {-{\alpha_{j}}\sqrt{z}\sin\frac{\sqrt{z}}{2}}&{\alpha_{j}}\cos\frac{\sqrt{z}}{2} \end{bmatrix} \begin{bmatrix} \varphi(z, j-1/2)\\ \varphi'(z, j-1/2) \end{bmatrix}. \end{aligned}$$ Using this and the induction hypothesis we obtain $$\begin{aligned} \begin{split} &\varphi(z, j^+)=\cos\left(\frac{\sqrt{z}}{2}\right)\varphi\left(z, j-\frac12\right)+ \frac{\sin\frac{\sqrt{z}}{2}}{\sqrt{z}}\varphi'\left(z, j-\frac12\right)\\ &\quad=\cos\frac{\sqrt{z}}{2}{{\mathcal O}}\left(\frac{1}{\sqrt{z}}\right)+ \frac{\sin\frac{\sqrt{z}}{2}}{\sqrt{z}}{{\mathcal O}}\left(1\right)={{\mathcal O}}\left(\frac{1}{\sqrt{z}}\right) \end{split} \end{aligned}$$ and $$\begin{aligned} \begin{split} &\varphi'(z, j^+)=-\alpha_j{\sqrt{z}}\,{\sin\left(\frac{\sqrt{z}}{2}\right)}\varphi\left(z, j-\frac12\right)+\alpha_j\cos\left(\frac{\sqrt{z}}{2}\right)\varphi'\left(z, j-\frac12\right)\\ &\quad=-\alpha_j{\sqrt{z}}\,{\sin\left(\frac{\sqrt{z}}{2}\right)}{{\mathcal O}}\left(\frac{1}{\sqrt{z}}\right)+\alpha_j\cos\left(\frac{\sqrt{z}}{2}\right){{\mathcal O}}\left(1\right)={{\mathcal O}}\left(1\right). \end{split} \end{aligned}$$ Combining these asymptotic formulas and $$\begin{aligned} &\begin{bmatrix} \varphi(z, x)\\ \varphi'(z, x) \end{bmatrix} = \begin{bmatrix} \cos\sqrt{z}(x-j)& \frac{\sin\sqrt{z}(x-j)}{\sqrt{z}}\\ {- \sqrt{z}\sin\sqrt{z}(x-j)}& \cos\sqrt{z}(x-j) \end{bmatrix} \begin{bmatrix} \varphi(z, j^+)\\ \varphi'(z, j^+) \end{bmatrix}, \end{aligned}$$ $x\in (j, j+1)$ one infers with $j$ replaced by $j+1$ as required. Choosing $s_k\equiv 1$ leads to the free Laplacian in which case , are consistent with the explicit solutions to the unperturbed problem. Let us recall the definition of the Weyl–Titchmarsh function $m$ corresponding to $\tau$. First, for $z\in{{\mathbb{C}}}\setminus{{\mathbb{R}}}$ there exists a unique (up to a scalar multiple) square integrable solution $\psi(z,\cdot)\in L^2({{\mathbb{R}}}_+; \mu)$ of . Since $z$ is non-real, we have $\varphi(z,\cdot)\not\in L^2({{\mathbb{R}}}_+; \mu)$ (otherwise $z$ would be an eigenvalue of the Dirichlet realization of $\tau$). Thus there exists a non-zero $m(z)\in{{\mathbb{C}}}$ such that $${\label}{1.5n} \psi(z,x)= \theta(z,x)+m(z)\varphi(z,x).$$ The Weyl–Titchmarsh function $m$ is the mapping $z\mapsto m(z)$. It is analytic in ${{\mathbb{C}}}_+:=\{z\in{{\mathbb{C}}}: {\operatorname{Im}}z>0\}$ and, in fact, it is a Herglotz function, i.e., ${\operatorname{Im}}m(z)>0$, $z\in{{\mathbb{C}}}_+$. For every $x\in{{\mathbb{R}}}_+$, $\alpha\in(-\pi,\pi)\setminus\{0\}$ one has $${\label}{1.7} \varphi(z,x)\psi(z,x)\rightarrow 0,$$ as $z\rightarrow\infty$ along the ray $\arg(z)=\alpha$. Our strategy is to view the left-hand side of as the diagonal Green function. In order to show that the latter vanishes asymptotically for high energies we will employ and the following representation of $m$, see e.g. [@T09 Chapter 9], $$\begin{aligned} &m(z)=d+\int_{{{\mathbb{R}}}}\left(\frac{1}{\lambda-z}-\frac{\lambda}{1+\lambda^2}\right) d\nu(\lambda), \\ &d= {\operatorname{Re }}(m({{\bf i}})),\ \int_{{{\mathbb{R}}}}\frac{1}{1+\lambda^2}d\nu(\lambda)={\operatorname{Im}}(m({{\bf i}}))<\infty.{\label}{1.8} \end{aligned}$$ The Green function of $H^+$, cf. , that is, the integral kernel of $(H^+-z)^{-1}$, admits two representations, $${\label}{1.9} G(z,x,x)=\varphi(z,x)\psi(z,x), x\in{{\mathbb{R}}}_+,$$ and $$G(z,x,x)=\int_{{{\mathbb{R}}}}\frac{\varphi^2(\lambda, x)}{z-\lambda}d\nu(\lambda), x\in{{\mathbb{R}}}_+.$$ Then utilizing , the second equation in , the inequality (cf. [@T09 (3.114)]) $$\left|\frac{1}{\lambda-z}\right|\leq \frac{1}{1+|\lambda|}\frac{1+|z|}{{\operatorname{Im}}(z)},$$ and the dominated convergence theorem, we get $G(z, x,x)\rightarrow0$ as $z\rightarrow\infty$, $\arg(z)=\alpha$. Hence by we get . The proof of the following theorem is motivated by [@Ben], where the Borg–Marchenko Theorem is established for Schrödinger operators $-\frac{d^2}{dx^2}+V(x)$. The main ingredients of Bennewitz’s proof are certain asymptotic formulas for the Dirichlet and Neumann solutions $\varphi(z,x), \theta(z,x)$ and for the Green function $G(z,x,x)$. In our setting these are given by , and , respectively. [thm1.4]{} Let $s, {\widetilde }s\in{{\mathcal J}}^{{{\mathbb{Z}}}_+}$ and let $m, {\widetilde }m$ be the corresponding Weyl–Titchmarsh functions. Then $${\label}{1.9n} s_j={\widetilde }s_j,\ 0\leq j\leq k,$$ if and only if for every $\alpha\in (-\pi,\pi)\setminus\{0\}$, one has $${\label}{1.9nn} |m(z)-{\widetilde }m(z)|=o\left(\exp\{-2\left(k+1/2\right){\operatorname{Re }}\sqrt{-z}\}\right),$$ as $z\rightarrow\infty$ along the ray $\arg(z)=\alpha$. Let $\varphi(z, \cdot)$, ${\widetilde }\varphi(z, \cdot)$, $\theta(z, \cdot)$, ${\widetilde }\theta(z, \cdot)$ be the Dirichlet and Neumann solutions corresponding to $s$ and ${\widetilde }s$ respectively. Then for every $x\in{{\mathbb{R}}}_+\setminus{{\mathbb{N}}}$ the first equation in yields $${\label}{1.10} \frac{\varphi(z, x)}{{\widetilde }\varphi(z, x)}\rightarrow\frac{c(\lfloor x\rfloor)}{{\widetilde }c(\lfloor x\rfloor)},\text{\ as\ } z\rightarrow\infty,z\in{{\mathbb{C}}}\setminus[0,\infty),$$ with $c,{\widetilde }c$ as in . Using the fact that $\frac{c(\lfloor x\rfloor)}{{\widetilde }c(\lfloor x\rfloor)}$ is a nonzero constant with respect to $z$ and , we obtain $${\widetilde }\varphi (z, x)\psi(z,x)\rightarrow0,\ \varphi (z, x){\widetilde }\psi(z,x)\rightarrow0,$$ as $z\rightarrow\infty$ along an arbitrary non-real ray. Therefore for every $x\in{{\mathbb{R}}}_+\setminus{{\mathbb{N}}}$, yields $$\begin{aligned} {\label}{nn1.12} {\widetilde }\varphi (z, x)\theta(z,x)-\varphi (z, x){\widetilde }\theta(z,x) +(m(z)-{\widetilde }m(z))\varphi (z, x){\widetilde }\varphi (z, x)\rightarrow 0, \end{aligned}$$ as $z\rightarrow\infty$ along an arbitrary non-real ray. We are ready to prove the“if" part. Combining , and , and fixing $x\in[0,k]\setminus{{\mathbb{N}}}$, we obtain $${\label}{n1.12} {\widetilde }\varphi (z, x)\theta(z,x)-\varphi (z, x){\widetilde }\theta(z,x)\rightarrow 0,$$ as $z\rightarrow\infty$ along every non-real ray. Let us introduce $$f(z):= {\widetilde }\varphi (z, x)\theta(z,x)-\varphi (z, x){\widetilde }\theta(z,x).$$ This is an entire function whose growth rate is at most $1/2$, cf. , , and it is bounded along every non-real ray. Then, by the Phragmén–Lindelöf principle, $f$ is bounded on ${{\mathbb{C}}}$. Thus it is constant and gives $f(z)=0$ for all $z\in{{\mathbb{C}}}$. That is, for all $x\in[0,k]\setminus{{\mathbb{N}}}$, $z\in{{\mathbb{C}}}$, one has $$\frac{{\widetilde }\theta(z,x)}{{\widetilde }\varphi (z, x)}=\frac{ \theta(z,x)}{\varphi (z, x)}, x\in[0,k]\setminus{{\mathbb{N}}}.$$ Taking the derivative with respect to $x$ and recalling that the Wronskian is conserved, we get $$\frac{1}{{\widetilde }\mu (x^+){\widetilde }\varphi ^2(z, x^+)}=\frac{1}{\mu(x^+)\varphi ^2(z, x^+)}.$$ Hence, for each $0\leq j\leq k$ $${\label}{1.12} {{\widetilde }s_j {\widetilde }\varphi ^2(z, x^+)}={s_j \varphi ^2(z, x^+)},$$ and upon (right-)differentiating with respect to $x$ $${\label}{1.13} {{\widetilde }s_j {\widetilde }\varphi (z, x^+)}={s_j \varphi (z, x^+)}.$$ Then $\eqref{1.12}$ and $\eqref{1.13}$ give $$\left(\frac{{\widetilde }s_j}{s_j}\right)^2=\frac{{\widetilde }s_j}{s_j} \text{\ and\ }{\widetilde }s_{j}=s_j,\ 0\leq j\leq k,$$ as asserted. Next, we prove the “only if" part. Using we get $$\varphi(z,x)={\widetilde }\varphi(z,x),\ \theta(z,x)={\widetilde }\theta(z,x), x\in[0, k+1), z\in{{\mathbb{C}}}.$$ Then with $x\in [0, k+1/2]\setminus{{\mathbb{N}}}$ yields $$|(m(z)-{\widetilde }m(z))\varphi (z, x){\widetilde }\varphi (z, x)|=o(1),$$ $z\rightarrow\infty$ along every non-real ray. Combining this and the first formula in we get . Continuity of $m$-functions With Respect to Jumps. -------------------------------------------------- [lemma3.5]{} Fix a sequence $\{s(n)\}_{n=1}^{\infty}\subset{{\mathcal J}}^{{{\mathbb{Z}}}_+}$. If $s(n)\rightarrow s$ as $n\rightarrow\infty$ with respect to the metric $d$, then $m_n(z)\rightarrow m(z)$ as $n\rightarrow\infty$ uniformly on compact subsets of ${{\mathbb{C}}}_+$. The proof is based on the fact that the Weyl circles (more precisely their centers and radii) are continuous with respect to $z$ and eventually constant with respect to $n$. Let $m(z,b, \alpha)\in{{\mathbb{C}}}$ be such that the function $$\chi(z,x):=\theta(z,x)+m(z,b,\alpha) \varphi (z,x),\ x\in(0,b)$$ satisfies the following condition at $b>0$, $$\chi(z,b)\cos\alpha+\mu (b)\chi(z,b) \sin\alpha =0.$$ Then $$C(z,b):=\{m(z,b,\alpha): \alpha\in[0,\pi] \},$$ is a circle in ${{\mathbb{C}}}$ centered at $$O(z,b):=-\frac{W(\varphi(z,\cdot), \overline{\theta(z,\cdot)})(b)}{W(\theta(z,\cdot), \overline{\theta(z,\cdot)})(b)},$$ with radius $${\label}{1.15} r(z,b):=\frac{1}{|W(\theta(z,\cdot),\overline{\theta(z,\cdot)})(b)|}\rightarrow0, b\rightarrow\infty.$$ The corresponding disk is denoted by $D(z,b)$. Then $D(z,b')\subset D(z,b)$ whenever $b'>b>0$, and $m(z)\in\cap_{b>0}D(z,b)$. Since $m_n, m$ are analytic in ${{\mathbb{C}}}_+$ it is enough to prove that $m_n$ converges to $m$ pointwise and that $m_n$ are uniformly bounded on compact subsets of ${{\mathbb{C}}}_+$. To prove the former, fix $\varepsilon>0$. Then by we have $r(z,b)<\varepsilon$ for $b\geq b(\varepsilon)$ for some $b(\varepsilon)>0$. Next, there exists $N(\varepsilon)>0$ such that $${\label}{1.18} s_k(n)=s_k, 0\leq k\leq \lfloor b(\varepsilon) \rfloor,$$ for all $n\geq N(\varepsilon)$. Denoting the Weyl disk, Dirichlet and Neumann solutions corresponding to $s(n)$ by $D_n$, $\varphi_n, \theta_n$ respectively, we note that yields $$\varphi_n(z,x)=\phi(z,x), \theta(z,x)=\theta_n(z,x), x\in (0, b(\varepsilon)),\ n\geq N(\varepsilon).$$ Hence, $D_n(z,b(\varepsilon))=D(z,b(\varepsilon))$ $n\geq N(\varepsilon)$, and since $m_n(z)\in D_n(z,b(\varepsilon))$, we get $${\label}{1.19} m_n(z),m(z)\in D(z,b(\varepsilon)), n\geq N(\varepsilon).$$ Therefore, $$|m_n(z)-m(z)|\leq 2\varepsilon, n\geq N(\varepsilon).$$ Next, utilizing with $\varepsilon=1$ we get $$|m_n(z)-m(z)|\leq 2 r(z,b(1)), n\geq N(1).$$ Since $r(z,b(1))$ and $m(z)$ are uniformly bounded on compacts, we infer that the sequence $\{m_n\}_{n\geq1}$ is uniformly bounded on compact subsets of ${{\mathbb{C}}}_+$. Restriction Maps ---------------- Throughout this section $m_{\pm}(s,z)$, $\varphi_{\pm}(s,\cdot)$, $\theta_{\pm}(s,\cdot)$ denote the Weyl–Titchmarsh functions, Dirichlet, and Neumann solutions corresponding to the sequence $s\in{{\mathcal J}}^{{{\mathbb{Z}}}}$. Let ${{\mathcal R}}_{\pm}$ denote the restriction operators ${{\mathcal R}}_{\pm}s:=\{s_n\}_{n\in{{\mathbb{Z}}}_{\pm}}$. For a set ${{\mathcal Z}}\subset {{\mathbb{R}}}$ we define $${{\mathcal D}}({{\mathcal Z}}):=\{ s\in {{\mathcal J}}^{{{\mathbb{Z}}}}: m_-(s, E+i 0)=-\overline{m_+(s, E+i0)},\ \text{for Lebesgue a.e.}\, E\in{{\mathcal Z}}\}.$$ [thm3.6]{} Suppose that ${{\mathcal Z}}$ has positive Lebesgue measure. Then - The mapping $${{\mathcal R}}: {{\mathcal R}}_{-}({{\mathcal D}}({{\mathcal Z}}))\rightarrow{{\mathcal R}}_{+}({{\mathcal D}}({{\mathcal Z}})), \ {{\mathcal R}}: {{\mathcal R}}_{-}(s)\mapsto {{\mathcal R}}_{+}(s), s\in{{\mathcal D}}({{\mathcal Z}}),$$ is well-defined (i.e. single valued). - ${{\mathcal R}}$ is bijective and uniformly continuous. - ${{\mathcal D}}({{\mathcal Z}})$ is $T$ invariant and closed with respect to the metric $d$, cf. . The restrictions ${{\mathcal R}}_{\pm}(s)$ uniquely determine the Weyl–Titchmarsh functions $m_{\pm} (s,\cdot)$ by a Borg–Marchenko type result Theorem \[thm1.4\]. Recalling that every Herglotz function is determined by its boundary values on a set of positive Lebesgue measure, we get that the Weyl–Titchmarsh functions $m_{-} (s,\cdot)$ and $m_{+} (s,\cdot)$ are in one-to-one correspondence since $$m_-(s, E+i 0)=-\overline{m_+(s, E+i0)},\ a.e.\ E\in{{\mathcal Z}},\ |{{\mathcal Z}}|>0.$$ $$\begin{tikzcd} & s \arrow{dl}{}\arrow{dr}{} \\ {{\mathcal R}}_-(s) \arrow{rr}{{{\mathcal R}}}\arrow{d}{\text{Theorem \ref{thm1.4}}} && \arrow{ll} {{\mathcal R}}_+(s)\arrow{d}{\text{Theorem \ref{thm1.4}}}\\ m_- (s,\cdot) \arrow{u}{} \arrow{rr}{|{{\mathcal Z}}|>0} && \arrow{u}\arrow{ll} m_+(s, \cdot) \end{tikzcd}$$ This proves that ${{\mathcal R}}$ is well-defined and bijective. In order to show that it is uniformly continuous it is enough to check that ${{\mathcal D}}({{\mathcal Z}})$ is closed (hence, compact). Indeed, in this case ${{\mathcal R}}_{\pm}$ are bijective, continuous mappings between compact metric spaces, thus they are homeomorphic and ${{\mathcal R}}$ is uniformly continuous, see the top part of the diagram. The closedness of ${{\mathcal D}}({{\mathcal Z}})$ follows from Lemma \[lemma3.5\] and [@Ko85 Lemma 5]. The $T$-invariance of ${{\mathcal D}}({{\mathcal Z}})$ can be deduced from the following relation between $m_{\pm}(s,z)$ and $m_{\pm}(Ts,z)$, $$\begin{aligned} m_{\pm}(Ts, z)&=\frac{\mu(Ts, 0^{\pm})\psi'_{\pm}(Ts, 0^{\pm})}{\psi_{\pm}(Ts, 0^{\pm})}=\frac{\mu(s, 1^{\pm})\psi'_{\pm}(s, 1^{\pm})}{\psi_{\pm}(s, 1^{\pm})}\\ &=\frac{\cosh{\sqrt{-z}}\mu(s, 0^{\pm})\psi'_{\pm}(s, 0^{\pm})+ \frac{\sinh\sqrt{-z}}{\sqrt{-z}}\psi_{\pm}(s, 0^{\pm})}{{\sqrt{-z}\sinh\sqrt{-z}}\mu(s, 0^{\pm})\psi'_{\pm}(s, 0^{\pm})+ \cosh\sqrt{-z}\psi_{\pm}(s, 0^{\pm})}\\ &=\frac{m_{\pm}(s,z)\cosh{\sqrt{-z}}+ \frac{\sinh\sqrt{-z}}{\sqrt{-z}}}{m_{\pm}(s,z){\sqrt{-z}\sinh\sqrt{-z}}+ \cosh\sqrt{-z} }. \end{aligned}$$ Ergodic Sturm–Liouville Operators in Impedance Form =================================================== [section4]{} This section concerns the spectral analysis of the full-line version of the operator with a dynamically defined sequence of sphere numbers $\{s_n\}_{n\in {{\mathbb{Z}}}}$. First, we show that the spectrum is given by the zero set of the Lyapunov exponent whenever the underlining cocycles are uniform, see Lemma \[lemma4.3\]. Further, assuming Boshernitzan’s condition and using Kotani theory we derive the main result of this section stating that the spectrum is given by a generalized Cantor set of zero Lebesgue measure. Let $\Omega$ be a compact metric space, let $T:\Omega\rightarrow\Omega$ be a homeomorphism, and suppose that $(\Omega, T)$[^7] is uniquely ergodic with the unique $T-$invariant probability measure $\nu$. For a non-constant measurable function $f:\Omega\rightarrow{{\mathcal J}}$, let $$\mu_{\omega}(x):=\sum_{n=-\infty}^{\infty}f(T^n\omega) f(T^{n+1}\omega)\chi_{[n, {n+1})}(x), x\in {{\mathbb{R}}}, \omega\in\Omega.$$ Let $\tau_{\omega}$ be defined as in with $\mu=\mu_{\omega}$ and introduce a self-adjoint operator $$\begin{aligned} \begin{split}{\label}{3.25} &H_{\omega}:{\operatorname{dom}}(H_{\omega})\subset L^2({{\mathbb{R}}}; \mu_{\omega})\rightarrow L^2({{\mathbb{R}}}; \mu_{\omega}), H_{\omega} u:= \tau_{\omega} u,\ u\in{\operatorname{dom}}(H_{\omega}),\\ &{\operatorname{dom}}(H_{\omega})=\{u\in L^2({{\mathbb{R}}}, \mu_{\omega}): u, \mu_{\omega} u'\in AC({{\mathbb{R}}}), \tau_{\omega} u\in L^2({{\mathbb{R}}}; \mu_{\omega})\}. \end{split} \end{aligned}$$ For $\nu$ almost every $\omega\in\Omega$, the spectrum of $H_{\omega}$ is given be a deterministic set $\Sigma\subset{{\mathbb{R}}}$. Moreover, by [@CL Proposition V.2.4 and Remark V.2.5] there exists $\Sigma_{\bullet}\subset {{\mathbb{R}}}$ such that for $\nu-$a.e. $\omega$ one has $$\sigma_{\bullet}(H_{\omega})=\Sigma_{\bullet},\ \bullet\in\{\text{ac, sc, pp, disc}\}.$$ A much finer spectral analysis of the ergodic family $H_{\omega}, \omega\in\Omega$ is possible through the dynamical approach on which we focus next. First, notice that $${\label}{n4.1} \tau_{\omega}u=Eu,\ u, \mu_{\omega}u\in AC_{loc}({{\mathbb{R}}}_+)$$ holds if and only for every $n\in{{\mathbb{Z}}}$, one has $-u''(x)=Eu(x), x\in(n-1, n)$ and $$\begin{aligned} &\begin{bmatrix} {u}(n^+)\\ {u}'(n^+) \end{bmatrix} = \begin{bmatrix} \cos\sqrt{E}& \frac{\sin\sqrt{E}}{{\sqrt{E}}}\\ {-{\frac{f(T^{n-1}\omega)\sqrt{E}\sin\sqrt{E}}{f(T^{n+1}\omega)}}}&{\frac{f(T^{n-1}\omega)\cos\sqrt{E}}{f(T^{n+1}\omega)}} \end{bmatrix} \begin{bmatrix} {u}((n-1)^+)\\ {u}'((n-1)^+) \end{bmatrix}. \end{aligned}$$ The latter can be rewritten as follows, $$\begin{aligned} \begin{bmatrix} f(T^{n+1}\omega){u}(n^+)\\ {u}'( n^+) \end{bmatrix}& = \begin{bmatrix} \frac{f(T^{n+1}\omega)\cos\sqrt{E}}{f(T^{n-1}\omega)} & \frac{f(T^{n+1}\omega)\sin\sqrt{E}}{{\sqrt{E}}}\\ {-{\frac{\sqrt{E}\sin\sqrt{E}}{f(T^{n+1}\omega)}}}&{\frac{f(T^{n-1}\omega)\cos\sqrt{E}}{f(T^{n+1}\omega)}} \end{bmatrix} \begin{bmatrix} f(T^{n-1}\omega){u}((n-1)^+)\\ {u}'((n-1)^+) \end{bmatrix}\\ &=\mathcal{M}^E(T^{n-1}\omega)\begin{bmatrix} f(T^{n-1}\omega){u}((n-1)^+)\\ {u}'((n-1)^+) \end{bmatrix}, \end{aligned}$$ where the mapping ${{\mathcal M}}^E:\Omega\rightarrow \text{SL}(2,{{\mathbb{R}}})$ is given by $${\label}{4.2aa} \mathcal{M}^E(\omega):=\begin{bmatrix} \frac{f(T^2\omega)\cos\sqrt{E}}{f(\omega)} & \frac{f(T^2\omega)\sin\sqrt{E}}{{\sqrt{E}}}\\ {-{\frac{\sqrt{E}\sin\sqrt{E}}{f(T^2\omega)}}}&{\frac{f(\omega)\cos\sqrt{E}}{f(T^2\omega)}} \end{bmatrix}.$$ The spectral properties of $H_{\omega}$ will be described using an SL$(2, {{\mathbb{R}}})$ cocycle $(\omega, n)\mapsto {{\mathcal M}}^E_n(\omega)$ over $T$ given by $${\label}{4.2a} \mathcal{M}_n^E(\omega)=\begin{cases} {\mathcal{M}^E\left(T^{n-1}\omega \right)\times \cdots \times \mathcal{M}^E(\omega),} & {n \geq 1}, \\ {I_{2},} & {n=0}, \\ {(\mathcal{M}^E\left(T^{n}\omega \right))^{-1}\times \cdots \times \mathcal{M}^E(T^{-1}\omega)^{-1},} & {n \leq-1}.\end{cases}$$ The Lyapunov exponent of this cocycle is defined by $${\label}{n1.20} L(E):=\lim_{n\rightarrow\infty}\frac{1}{n}\int_{\Omega}\log\|\mathcal{M}_n^E(\omega)\|d\nu(\omega)\geq0,$$ by Kingman’s Subadditive Ergodic Theorem, one has $${\label}{nn1.20} L(E)=\lim\limits_{n\rightarrow\infty} \frac1n\log\|M^E_n(\omega)\|,$$ for $\nu$-almost every $\omega$. We say that the function ${{\mathcal M}}^E$ is [*uniform*]{} if the limit exists for all $\omega\in\Omega$ and the convergence is uniform. Dynamical Description of The Spectrum ------------------------------------- The main goal of this section is to describe the spectrum of $H_{\omega}$ in terms of the cocycle $(\omega, n)\mapsto M^E_n(\omega)$. To that end, we first define a set of energies corresponding to uniformly hyperbolic cocycles, $${{\mathcal U}}{{\mathcal H}}=\left\{E \in \mathbb{R} \Big| \begin{matrix} \text{there exist\ } \lambda>1, C>0 \text { such that } \\ \left\|\mathcal{M}_n^E(\omega)\right\| \geq C \lambda^{|n|} \text{\ for all\ } \omega \in \Omega, n\in\mathbb{Z} \end{matrix} \right\}.$$ We note that ${{\mathcal U}}{{\mathcal H}}$ is open by [@DFLY Corollary 2.2]. With the Lyapunov exponent $L(E)$ defined in , denote $$\mathcal{Z}:=\left\{E\in{{\mathbb{R}}}: L(E)=0\right\},\ \mathcal{NUH}:=\left\{E \in \mathbb{R} : L(E)>0\right\} \backslash \mathcal{UH}.$$ Then we arrive at the following disjoint partition of the real line $$\mathbb{R}=\mathcal{Z}\sqcup \mathcal{NUH}\sqcup \mathcal{UH}.$$ One has $${\label}{n1.21} \bigcup_{\omega\in\Omega}\sigma(H_{\omega})={{\mathbb{R}}}\setminus{{\mathcal U}}{{\mathcal H}}.$$ Assume that $E_0\in{{\mathcal U}}{{\mathcal H}}$ and fix $\omega\in\Omega$. Since ${{\mathcal U}}{{\mathcal H}}$ is open there is an open interval ${{\mathcal I}}\subset {{\mathcal U}}{{\mathcal H}}$ containing $E_0$. For each $E\in{{\mathcal I}}$ any solution of the equation $\tau_{\omega}u=Eu$ grows exponentially fast on at least one half-line. By [@BdMS Theorem 1.1][^8], for spectrally almost every $\lambda\in{{\mathbb{R}}}$ there is an $L^2$-subexponentially bounded solution of $\tau_{\omega}f=\lambda f$. Since ${{\mathcal I}}$ contains no such $\lambda$ we conclude that $E_0\not\in\sigma(H_{\omega})$ That is, $$\bigcup_{\omega\in\Omega}\sigma(H_{\omega})\subset{{\mathbb{R}}}\setminus{{\mathcal U}}{{\mathcal H}}.$$ Next, assume that $E\in{{\mathbb{R}}}\setminus{{\mathcal U}}{{\mathcal H}}$. Then the cocycle $(T,{{\mathcal M}}^E)$ admits a bounded orbit, i.e., there exist $\omega\in\Omega$ and $v\in \{x\in {{\mathbb{R}}}^2: \|x\|=1 \}$ such that $$\|{{\mathcal M}}^E_n(\omega)v\|\leq 1,$$ for all $n\in{{\mathbb{Z}}}$, see e.g. [@DFLY Theorem 1.2]. Therefore, the equation $\tau_{\omega}u=Eu$ admits a bounded solution $u_{0}$ satisfying all vertex conditions. Thus by Lemma \[lemma2.2\] we have $E\in\sigma(H_{\omega})$. Assuming uniformity of the cocycle, we will show the spectrum of every $H_{\omega}$ is given by the right-hand side of , which in turn reduces to ${{\mathcal Z}}$. To that end, we first prove a Sch’nol-type result in the deterministic setting. [lemma2.2]{} Let $\tau,\mu$ be defined as in for a two sided sequence $\{s_n\}_{n\in{{\mathbb{Z}}}}\subset {{\mathcal J}}$, and let $H$ be the corresponding self-adjoint operator in $L^2({{\mathbb{R}}})$ $($a two sided version of $)$. Suppose that $w$ is a subexponentially bounded solution $$\tau w=Ew, E\in{{\mathbb{R}}}, w, \mu_{\omega}w\in AC_{loc}({{\mathbb{R}}}),$$ that is, for every $\kappa>0$ there exists $C>0$ such that $${\label}{new5} |w(x)|\leq C e^{\kappa x}\text{\ for all\ } x\in{{\mathbb{R}}}.$$ Then $E\in\sigma(H)$. Our strategy is to construct a Weyl sequence for $E$. First, we note that the quadratic form of $H$ (acting in $L^2({{\mathbb{R}}}_+, \mu)$) is given by $$\begin{aligned} &\mathfrak h[u,v]:=\langle u',v'\rangle_{L^2({{\mathbb{R}}}_+, \mu)},{\operatorname{dom}}(\mathfrak h)= H^1({{\mathbb{R}}}). \end{aligned}$$ Indeed, for every $u\in {\operatorname{dom}}(\mathfrak h)$ and every compactly supported $v\in {\operatorname{dom}}(H)$, ${\operatorname{supp}}(v)\in(-K,K)$, we have $$\begin{aligned} &\langle u,Hv\rangle_{L^2({{\mathbb{R}}}_+,\mu)}=-\langle u,(\mu v')'\rangle_{L^2({{\mathbb{R}}}_+,dx)}\\ &=-\sum_{j=-K+1}^{K}\int_{{j-1}}^{{j}} s_j s_{j-1} \overline{u(x)}v''(x) \, dx=\sum_{j=-K+1}^{K}\int_{{j-1}}^{{j}} s_j s_{j-1} \overline{u'(x)}v'(x)dx\\ &\quad+\sum_{j=-K+1}^{K}s_js_{j+1}\overline{u(j^+)} v'(j^+)-s_js_{j-1}\overline{u(j^-)} v'(j^-){\nonumber}\\ &=\langle u',v'\rangle_{L^2({{\mathbb{R}}}, \mu)}+\sum_{j=-K+1}^{K}(s_js_{j+1}v'(j^+)-s_js_{j-1}v'(j^-))\overline{u(j^+)} =\langle u',v'\rangle_{L^2({{\mathbb{R}}}, \mu)}. \end{aligned}$$ For every $n\in{{\mathbb{N}}}$, let $\varphi_n\in C^{\infty}({{\mathbb{R}}})$ be a mollifier taking values in $[0,1]$ and satisfying $$\varphi_n(x):=\begin{cases} 1& 0\leq |x|\leq n, \\ 0&|x|\geq n+1; \end{cases} \text{\ \ and\ \ }\sup_{n\geq 1}\|\varphi'_n\|_{L^{\infty}({{\mathbb{R}}}, dx)}<\infty.$$ We claim that $$\left\{\frac{w\varphi_n}{\|w\chi_{(-n,n)}\|}_{L^2({{\mathbb{R}}}, \mu)}\right\}_{n\geq 1}\subset {\operatorname{dom}}(\mathfrak h)$$ is a Weyl sequence for $E$. First, we note that the $L^2({{\mathbb{R}}}, \mu)$ norm of all elements of this sequence is at least one. Next, fix $u\in {\operatorname{dom}}(\mathfrak h)$, $\|u\|_{H^1({{\mathbb{R}}})}\leq 1$, and denote $I_n:=(n, n+1)\cup (-n-1,-n)$. Then one has $$\begin{aligned} \begin{split}{\label}{new4} &{\mathfrak h[u,\varphi_n w]-E\langle u,\varphi_n w\rangle_{L^2({{\mathbb{R}}}, \mu)}}\\ &={\langle u',\varphi_n w'\rangle_{L^2({{\mathbb{R}}}, \mu)}+\langle u,\varphi_n' w'\rangle_{L^2({{\mathbb{R}}}, \mu)}-E\langle u,\varphi_n w\rangle_{L^2({{\mathbb{R}}}, \mu)}}\\ &= \int_{-n}^{n} \overline{u'(x)}w'(x)\mu(x)dx-E\int_{-n}^{n} \overline{u(x)}w(x)\mu(x)dx+\langle u',\chi_{I_n}\varphi_n w'\rangle_{L^2({{\mathbb{R}}}, \mu)}\\ &\quad-E\langle u,\chi_{I_n}\varphi_n w\rangle_{L^2({{\mathbb{R}}}, \mu)}+\langle u,\varphi'_n\chi_{I_n} w'\rangle_{L^2({{\mathbb{R}}}, \mu)}\\ &= \langle u, (\tau w-Ew)\chi_{(-n, n)}\rangle_{L^2({{\mathbb{R}}}, \mu)}\\ &+s_ns_{n-1}\overline{u(n^-)}w'(n^-)-s_{-n}s_{-n+1}\overline{u(-n^+)}w'(-n^+)+\langle u',\chi_{I_n}\varphi_n w'\rangle_{L^2({{\mathbb{R}}}, \mu)}\\ &\quad-E\langle u\chi_{I_n},\varphi_n w\rangle_{L^2({{\mathbb{R}}}, \mu)}+\langle u, \chi_{I_n}\varphi'_nw'\rangle_{L^2({{\mathbb{R}}}, \mu)}. \end{split} \end{aligned}$$ Sobolev–type inequalities, see, for example, [@K80 IV.1.2], yield $$\begin{aligned} \begin{split} &|u(x_0)|\lesssim \|u\|_{H^1(I_n)}, x_0\in\{-n,n\},\\ &|w'(x_0)|\lesssim \|w\|_{H^1(I_n)}\lesssim \|w\|_{L^2(I_n, dx)}, {\label}{new2} \end{split} \end{aligned}$$ where we used $-w''(x)=Ew(x)$, $x\in I_n$ to get the last inequality in . Combining , , and the Cauchy–Schwarz inequality, we get $$\begin{aligned} &\frac{|{\mathfrak h[u,\varphi_n w]-E\langle u,\varphi_n w\rangle_{L^2({{\mathbb{R}}}, \mu)}}|}{\|w\chi_{(-n,n)}\|_{L^2({{\mathbb{R}}}, \mu)}}\lesssim\frac{\|u\|_{H^1(I_n)} \|w\|_{H^1(I_n)}}{\|w\chi_{(-n,n)}\|_{L^2({{\mathbb{R}}}, dx)}}\\ &\qquad\lesssim \frac{\|w\chi_{(n, n+1)\cup (-n-1,-n)}\|_{L^2({{\mathbb{R}}}, dx)}}{\|w\chi_{(-n,n)}\|_{L^2({{\mathbb{R}}}, dx)}}.{\label}{new1} \end{aligned}$$ The right-hand side of tends to zero as $n\rightarrow\infty$, for otherwise the norm $$\|w\chi_{(n, n+1)\cup (-n-1,-n)}\|_{L^2({{\mathbb{R}}}, dx)}$$ would grow exponentially, contradicting . [lemma4.3]{} Suppose that ${{\mathcal M}}^E$ is uniform for all $E\in{{\mathbb{R}}}$. Then $${\label}{n1.22} \sigma(H_{\omega})= {{\mathcal Z}}\text{\ for all\ }\omega\in\Omega.$$ By assumption we have ${{\mathcal N}}{{\mathcal U}}{{\mathcal H}}=\emptyset$. Therefore, due to it is enough to show that ${{\mathcal Z}}\subset \sigma(H_{\omega})$ for all $\omega\in\Omega$. The latter follows form the fact that every solution of $\tau_{\omega}u=Eu$ is subexponentially bounded for every $E\in{{\mathcal Z}}$, $\omega\in\Omega$ (since ${{\mathcal M}}^E$ is uniform) and Lemma \[lemma2.2\]. Next, we switch to Kotani’s description of $\Sigma_{ac}$. Recall that the essential closure of a set $S\subset {{\mathbb{R}}}$ is given by $$\overline{S}^{ess}:=\{E\in{{\mathbb{R}}}: |S\cap(E-\varepsilon, E+\varepsilon)|>0\text{ for every}\ \varepsilon>0\}.$$ [theorem4.4]{}*(Ishii–Pastur–Kotani)* The almost sure absolutely continuous spectrum of $H_{\omega}$ is given by the essential closure of the zero set of the Lyapunov exponent, that is, $\Sigma_{ac}=\overline{{{\mathcal Z}}}^{ess}$. Moreover, for $\nu$-almost every $\omega$, one has $${\label}{4.9} m_-(\omega, E+i 0)=-\overline{m_+(\omega, E+i0)},\ \emph{Leb-a.e.}\ E\in{{\mathcal Z}}.$$ Kotani’s original proof is formally recorded for Schrödinger operators with ergodic $L^1_{loc}({{\mathbb{R}}})$ potentials, see [@Ko]. However, it extends directly to the ergodic family $H_{\omega}$ defined in . Let us indicate the only two nontrivial adjustments to be made in Kotani’s text. First, since we are dealing with Sturm–Liouville operators in impedance form, the derivatives $\psi'_{\lambda}, \varphi'_{\lambda}$ of the Dirichlet and Neumann solutions (here we use Kotani’s notation, see [@Ko $\S$1]) should be replaced by the quasi-derivatives $\mu_{\omega}\psi'_{\lambda}, \mu_{\omega}\varphi'_{\lambda}$ which are locally absolutely continuous. Second, the uniform boundedness of the Weyl–Titchmarsh functions $m_{\pm}$ discussed in [@Ko Lemma 1.2] follows (exactly as in [@LSS Lemma 6.2]) from the continuity of $m-$functions with respect to $\omega$ which has been established in Lemma \[lemma3.5\]. Cantor Spectrum For the Full Line Model --------------------------------------- In this section we specialize the dynamical system $(\Omega, T)$ to the class of minimal subshifts over ${{\mathcal J}}$ satisfying the Boshernitzan condition (B), and $f(\omega):=\omega_0$. [theorem4.5]{} Let $(\Omega,T)$ be an aperiodic, minimal subshift over ${{\mathcal J}}$ satisfying Boshernitzan’s condition . Then there exists a generalized Cantor set $\Sigma\subset{{\mathbb{R}}}$ of Lebesgue measure zero such that $$\sigma(H_{\omega})=\Sigma\text{\ for all\ }\omega\in\Omega.$$ Since $(\Omega,T)$ satisfies (B) and since ${{\mathcal M}}^E$ is locally constant we get that it is uniform for every $E\in{{\mathbb{R}}}$ by [@DL1 Theorem 1]. Therefore, Lemma \[lemma4.3\] is applicable in the present setting and one has $\sigma(H_{\omega})={{\mathcal Z}}$ for all $\omega\in\Omega$. We claim that ${{\mathcal Z}}$ has zero Lebesgue measure. Seeking contradiction let us assume that $|{{\mathcal Z}}|>0$. By Theorem \[theorem4.4\] the assertion holds for $\omega\in{\widetilde }\Omega\subset\Omega$, $\nu({\widetilde }\Omega)=1$. In other words, ${\widetilde }\Omega\subset{{\mathcal D}}({{\mathcal Z}})$. Then combining Theorem \[thm3.6\] and $|{{\mathcal Z}}|>0$, we infer that the mapping $${{\mathcal R}}: \{f(T^n\omega) f(T^{n+1}\omega)\}_{n<0}\mapsto \{f(T^n\omega) f(T^{n+1}\omega)\}_{n\geq0},\ \omega\in{\widetilde }\Omega,$$ is well-defined and continuous with respect to the discrete metric $d$, cf. . Pick an $\omega\in{\widetilde }\Omega$ and denote $s_n:=f(T^n\omega)$ for $n\in{{\mathbb{Z}}}$. By uniform continuity of ${{\mathcal R}}$ there exists $k=k({{\mathcal J}})\in{{\mathbb{N}}}$ (which is $\omega$-independent) such that the collection $\{s_{-k}s_{-k+1}; ...; s_{-1}s_{0}\}$ uniquely determines $s_{0}s_{1}$. Shifting and repeating this argument, we deduce that $\{s_{-k}s_{-k+1}; ...; s_{-1}s_{0}\}$ determines the entire sequence $\{s_{n}s_{n+1}\}_{n\in{{\mathbb{Z}}}}$. Thus $$\#\{\{f(T^n\omega) f(T^{n+1}\omega)\}_{n\in{{\mathbb{Z}}}}: \omega\in\Omega \}\leq |{{\mathcal J}}|^{k+1}<\infty.$$ In particular, the sequence $\{s_{n}s_{n+1}\}_{n\in{{\mathbb{Z}}}}$ is periodic with some period $p$. Then since ${{\mathcal J}}$ is finite, the sequence $\{s_{n}\}_{n\in{{\mathbb{Z}}}}$ is periodic with period $q\leq 2|{{\mathcal J}}|^2p$ (in general $p<q$, e.g.: $...141222\ 141222\ ...$ here $p=3$ and $q=6$). This contradicts the fact the $\Omega$ is an aperiodic subshift. Therefore, $|\Sigma|=|{{\mathcal Z}}|=0$, in particular the closed set $\Sigma$ does not contain intervals, hence, it is nowhere dense. Moreover, $\nu$ almost surely, the discrete spectrum spectrum $\sigma_{\emph{disc}}(H_{\omega})$ is empty, cf. [@CL Proposition V.2.8], so $\Sigma$ does not have isolated points, hence, it is a generalized Cantor set as asserted. Spectral Decomposition of the Two-sided Fibonacci Graph ======================================================= In this section we focus on the two-sided version of the graph $\Gamma$ corresponding to an element $\omega$ of the Fibonacci subshift $\Omega$ introduced in Example \[ex2.3\], see Figure \[Fig2\]. \(1) at (1,0); (2) at (2,0.7); (3) at (2,-0.7); (4) at (3,0); (5) at (4,0); (6) at (5,0.7); (7) at (5,-0.7); (8) at (6,0); (9) at (6,0); (10) at (7,0.7); (11) at (7,-0.7); (12) at (8,0); (13) at (9,0); (1) to (2); (1) to (3); (2) to (4); (3) to (4); (4) to (5); (5) to (6); (5) to (7); (6) to (8); (7) to (8); (9) to (10); (9) to (11); (10) to (12); (11) to (12); (12) to (13); (0,0) circle (.7pt); (.2,0) circle (.7pt); (.4,0) circle (.7pt); (9.8,0) circle (.7pt); (9.4,0) circle (.7pt); (9.6,0) circle (.7pt); [Fig2]{} First, we discuss the spectral properties of Fibonacci hamiltonians. Let us recall relevant notions and fix notation. Let $s\in\{1,2\}^{{{\mathbb{Z}}}_+}$ be the one-sided sequence invariant under the Fibonacci substitution ${{\mathcal S}}$, cf. Example \[ex2.3\], and let $\omega(s)\in\Omega$ be a two-sided sequence which matches $s$ on ${{\mathbb{Z}}}_+$. Let $F_n$ be the $n$-th element of the Fibonacci sequence with $F_0:=1$, $F_1:=2$ (note that $F_n=|{{\mathcal S}}^n(1)|$ ). Setting $f(\omega):=\omega_0$ in we denote $M(n,E):={{\mathcal M}}^E_{F_n}(\omega(s))$, $n\geq0$, cf. . The key to the spectral analysis of Fibonacci hamiltonians is the following recurrence relation $${\label}{7.5} M(n+1,E)=M(n-1, E)M(n,E), n\geq 3.$$ We stress that the transfer matrix depends on the window (...) of size [*three*]{}. Consequently, $M(n, E)$ depends on the first $F_n+2$ elements of $s$. Thus, the recurrent relation $${\label}{100} {{\mathcal S}}^{n+1}(1)={{\mathcal S}}^{n}(1){{\mathcal S}}^{n-1}(1),$$ does not automatically yield for [*all*]{} $n\geq 1$. However, it does imply that the extra two elements required in the definition of $M(n, E)$ are $n$-independent and, in fact, given by$$s_{F_n+1}=1,\ s_{F_n+2}=2,\ n\geq 2.$$ Now, this observation together with do yield . [thm5.1]{} Let $\Omega$ be the Fibonacci subshift. Then $\sigma_{pp}(H_{\omega})=\emptyset$ for all $\omega\in\Omega$. In fact, the spectrum of $H_{\omega}$ is purely singular continuous. Define $$x_n(E):=\frac{1}{2}\text{tr}(M(n,E)).$$ Then together with the Cayley–Hamilton theorem yield $${\label}{7.6} x_{n+1}(E)=2x_{n}(E)x_{n-1}(E)-x_{n-2}(E), n\geq 4.$$ Let $$B:=\big\{E\in{{\mathbb{R}}}: \sup_{n\in{{\mathbb{Z}}}_+}|x_n(E)|<+\infty \big\}.$$It is enough to prove that $\Sigma\subset B$ (see Theorem \[theorem4.5\] for definition of $\Sigma$) and that $B$ contains no eigenvalues of $H_{\omega(s)}$. Let $H_n$ denote the periodic hamiltonian corresponding to the sequence of sphere numbers obtained by two-sided periodic extension of the patch ${{\mathcal S}}^n(1)$. We claim that $${\label}{7.1} \Sigma\subset \bigcap_{n\geq 0}\overline{\bigcup_{k\geq n} \sigma(H_k) }\subset B.$$ Let us prove the first inclusion in . Fix $n\geq 0$ and suppose that $$E\not\in \overline{\bigcup_{k\geq n} \sigma(H_k) },$$ then $$\varkappa:=\sup_{k\geq n}\|(H_k-E)^{{-1}}\|_{{{\mathcal B}}(L^2({{\mathbb{R}}}))}<+\infty.$$ Thus, for every compactly supported $\psi\in{\operatorname{dom}}(H_{\omega(s)})$ we have $\psi \in{\operatorname{dom}}(H_k)$ and $$\|H_{k}\psi-E\psi \|_{L^2({{\mathbb{R}}})}\geq\varkappa^{-1}\|\psi\|_{L^2({{\mathbb{R}}})},$$ for sufficiently large $k$. For such $k$ we have $\|H_{k}\psi-E\psi \|_{L^2({{\mathbb{R}}})}=\|H_{\omega(s)}\psi-E\psi \|_{L^2({{\mathbb{R}}})}$, and infer $${\label}{5.1new} \|H_{\omega(s)}\psi-E\psi \|_{L^2({{\mathbb{R}}})}\geq\varkappa^{-1}\|\psi\|_{L^2({{\mathbb{R}}})}.$$ Since the set of compactly supported functions from ${\operatorname{dom}}(H_{\omega(s)})$ is a core of $H_{\omega(s)}$, the inequality holds for all $\psi\in{\operatorname{dom}}(H_{\omega(s)})$. Furthermore, since $H_{\omega(s)}$ is self-adjoint we get $E\not\in \sigma(H_{\omega(s)})=\Sigma$ which proves the first inclusion in . As was noted in the proof of [@DFG Proposition 6.3], the assertion [@S05 Proposition 12.8.6] together with yield the following criterion: $E\not\in B$ if and only if there exists $n$ such that $${\label}{7.2} |x_{n+1}(E)|>1, |x_{n}(E)|>1, |x_{n+1}(E)x_{n}(E)|>|x_{n-1}(E)|,$$ and in this case $${\label}{7.3} |x_k(E)|>1, k\geq n.$$ To prove the second inclusion in let us fix $E\not\in B$. Since $x_j(E)$ is continuous with respect to $E$, the inequalities in hold in a small neighborhood of $E$. Thus also holds in the same neighborhood (uniformly for $k\geq n$!). Then by Floquet–Bloch theory $E\not\in \overline{\bigcup_{k\geq n} \sigma(H_k)}$. The fact that $\tau_{\omega(s)}u=Eu$ has no square integrable solution for $E\in B$ follows form the standard argument invoking Gordon’s lemma and the boundedness of $x_n(E)$, see, for example, [@Su Proposition 2]. Our next objective is to obtain the complete spectral decomposition of the Kirchhoff Laplacian corresponding to the two-sided Fibonacci Graph $\Gamma$. To that end, we first note that the decomposition result of Kostenko and Nicolussi [@KN Theorem 3.5] can be extended to the two-sided setting. Let $\{s_n\}_{n\in{{\mathbb{Z}}}}\subset{{\mathbb{N}}}$ be an arbitrary sequence of natural numbers. For each $n\in{{\mathbb{Z}}}$ let $S_n$ denote a set of $s_n$ distinct vertices. Let $\Gamma$ be the graph with the set of vertices ${{\mathcal V}}:=\cup_{n\in{{\mathbb{Z}}}} S_n$ and such that two vertices $u,v\in{{\mathcal V}}$ are adjacent if and only if $$(u,v)\in\bigcup_{n\in {{\mathbb{Z}}}}S_n\times (S_{n-1}\cup S_{n+1}),$$ see, e.g., Figure \[Fig2\]. [prop5.2]{} The Kirchhoff Laplacian ${{\mathbb{H}}}$ on the graph $\Gamma$ introduced above is unitarily equivalent to $$H\bigoplus_{n\in{{\mathbb{Z}}}}\oplus _{j=1}^{s_{n}-1} H_n^1 \oplus\bigoplus_{n\in{{\mathbb{Z}}}}\oplus_{j=1}^{(s_{n}-1)(s_{n+1}-1)}H_n^2,$$ where $H_n^1, H_n^2$ are as in , , respectively, and $H$ is the self-adjoint Sturm–Liouville operator acting in $L^2({{\mathbb{R}}}, \mu)$ and corresponding to $$\begin{aligned} \begin{split} &\tau = -\frac{1}{\mu(x)}\left(\frac{d}{dx}\mu(x)\frac{d}{dx}\right),\\ &\mu(x):=\sum_{n=-\infty}^{\infty}{s}_n {s}_{n+1}\chi_{[n, {n+1})}(x), x\in {{\mathbb{R}}}. \end{split}\end{aligned}$$ The proof of this assertion is similar to that of [@KN Theorem 3.5]. Let us highlight the main ingredient of the proof by example. Let ${{\mathbb{H}}}$ be the Kirchhoff Laplacian on the two-side graph $\Gamma$ introduced above and corresponding to an element $\omega$ of the Fibonacci subshift $\Omega\subset\{1,2\}^{{{\mathbb{Z}}}}$. Such a graph consists of tiles and line segments as shown on Figure \[Fig2\]. One has $${\label}{10} L^2(\Gamma)={{\mathcal F}}_{sym}\oplus ({{\mathcal F}}_{sym})^{\perp},$$ where ${{\mathcal F}}_{sym}$ consists of $L^2(\Gamma)$ functions that are horizontally symmetric on each tile (for precise definition see [@KN (2.11)] with “$n\geq 0$" replaced by “$n\in{{\mathbb{Z}}}$", and $s_n\in\{1,2\}$). Then the operator block of ${{\mathbb{H}}}$ corresponding to ${{\mathcal F}}_{sym}$ is unitarily equivalent to the full line Strum–Liouville operator $H$. 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[**111**]{} (1987), 409–415. G. Teschl, [*Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators*]{}, 2nd ed., Amer. Math. Soc., Providence, 2014. [^1]: D.D. was supported in part by NSF grant DMS–1700131. [^2]: L.F. was supported by NSFC (No. 11571327) and the Joint PhD. Scholarship Program of Ocean University of China. [^3]: S.S. was supported in part by an AMS-Simons travel grant, 2017-2019 [^4]: noncompact and connected [^5]: the scope of generality is comprehensively discussed in [@DL2] [^6]: in fact, “$\subset$” can be replaced by “$=$" by standard arguments utilizing ergodicity/minimality of $(\Omega,T)$, cf., e.g., [@DF], [@CL] [^7]: at this point $(\Omega, T)$ is not assumed to be a minimal subshift satisfying (B) [^8]: our operator is of the form described in [@BdMS Section 3.2], hence, the referenced result is applicable

--- abstract: 'Spectra of the magnetic white dwarf binary AM Her were obtained with the Far Ultraviolet Spectroscopic Explorer (FUSE) during three consecutive spacecraft orbits. These observations were split into 19 spectra of about 5 min duration (0.025P binary phase) partially covering the binary orbit. We report the phase-related changes in the far ultraviolet continuum light curve and the emission lines, noting particularly the behavior of O VI. We discuss the fluxes and velocities of the narrow and broad O VI emissions. We find the FUV light curve has maximum amplitude at $\sim$1000Å, although at shorter wavelengths the continuum may be strongly affected by overlapping Lyman lines. Weak, narrow Lyman absorption lines are present. Their velocities don’t appear to vary over the observed orbital phases, and their mean value is consistent with the systemic velocity.' author: - 'J.B. Hutchings' - 'A.W. Fullerton' - 'A.P. Cowley, P.C. Schmidtke' title: 'FUSE 900–1200Å Spectroscopy of AM Her[^1] ' --- Introduction ============ AM Her, originally classified as an irregular variable and associated with the X-ray source 3U 1809+50, was discovered in 1976 to be a short period binary related to the cataclysmic variables (CVs) (Szkody & Brownlee 1997, Cowley & Crampton 1977, Priedhorsky 1977). Through its 3.1-hr orbital period, AM Her’s brightness varies continuously, with short-term flickering superimposed on an orbital light curve with a range of $\Delta$$V\sim0.7$ mag. Independently, Tapia (1977) found the same orbital period from polarization variations indicating that the white dwarf in the system has a very strong magnetic field ($B\sim 2\times10^8$ gauss) and rotates synchronously with the orbit. The magnetic field constrains the mass lost from the secondary star to flow along a stream directly onto the white dwarf rather than into an accretion disk as in the non-magnetic CVs. The optical spectrum shows strong emission lines of H, He I, and He II, along with weaker lines of other ions such as N III, C III, C II, and Ca II, all showing large velocity variations. The strong lines show both broad and narrow components which clearly arise in different regions since their phasing and velocity amplitudes differ considerably (e.g. Crampton & Cowley 1977, Crosa et al. 1981, and many more recent studies). TiO bands from the M4 V secondary star are detected when the system is in its faint (“low") state (e.g. Young, Schneider, & Shectman 1981). Even early observations showed light curves and times of minima in $V$, $U$, and X-ray bands to be quite different (e.g. see Fig. 6 of Crampton & Cowley, 1977). Subsequent observations in other wavelengths have revealed additional complex patterns of orbital behavior, much of which can be interpreted as a result of the strong magnetic field. Gänsicke et al.(2001) attribute the $V$-band light curve to cyclotron emission arising near the accreting magnetic pole, the small $B$-band variation to the accretion stream, and the hard X-ray variation to the changing view of the hot plasma at the shock where stream impacts the white dwarf. Because times of minima depend on the wavelength of observation, there is some uncertainty about what ephemeris to use. In this paper we adopt the well-determined period given by Heise & Verbunt (1988) and a phase based on our time of FUV minimum light. In other published work often the adopted T$_0$ is the time of minimum $V$ light which corresponds approximately to the superior conjunction of the white dwarf (e.g. Gänsicke et al. 1998, Southwell et al. 1995). Although AM Her usually remains in its ‘high’ state ($V\sim$13 mag), from time to time it drops into a ‘low’ state ($V$ below 15 mag) which may last weeks or months (e.g. Mattei 1980). A plot of the long-term light curve from 1977 to 1998 is shown in Fig. 1 of Hessman, Gänsicke, & Mattei (2000), which nicely illustrates the high and low states of AM Her. Although the system underwent a rather prolonged low state a few years ago (e.g. de Martino et al. 1998), the FUSE data described in the present paper were obtained during a normal ‘high’ state, with magnitude close to $V\sim13.3$, as is shown by the on-line AAVSO database. AM Her has become the proptotype of the class of magnetic CVs, commonly called “polars", in which the rotation period of the white dwarf is locked to the orbital period. An excellent summary of the properties of these systems is given by Warner (1995). FUSE Observations ================= Observations of AM Her were made with the Far Ultraviolet Spectroscopic Explorer (FUSE) on 2000, June 12-13 using the large science aperture and recording the data in time-tag mode. The observations covered three FUSE orbits, and the archival processing (Calfuse 1.6.9) produced three spectra - one for each FUSE orbital window. We also re-extracted the spectra with Calfuse 1.9.9, with the time-tagged data split into 19 individual spectra of duration 0.0033 days each (4.75 min, or 0.025P in binary orbital phase). Because of earth occultations, there are gaps in the phase coverage so that only about half the orbit is well sampled. This includes phases $\Phi_{UVmin}$=0.28-0.52 and 0.84-0.97, using an ephemeris based on the period given by Heise & Verbunt (1988) and T$_0$ corresponding to the time of minimum far-ultraviolet light (as given in §4). However, the observed phases do cover two main portions of the orbit, about 180$^{\circ}$ apart, with some phase overlap. The spectra have sufficient signal and time resolution to obtain phase-related information. FUSE Spectrum of AM Her ======================= Figure 1 shows two spectra obtained from the entire second and third FUSE orbits. These spectra cover the phases near minimum and maximum FUV light. Stellar emission features which are present include O VI 1032, 1038Å, He II 1085 Å, N III 992Å, C III 977, 1175Å, and S IV 1073Å. These lines are labelled in Fig. 1, and their rest wavelengths are marked. Note the continuum differences and the changes in the emission line structure between the two spectra. Measurements of these lines are given in Table 1. The HJDs in the table have been converted from the MJDs provided in the FUSE data header, and the heliocentric time correction has been added. Heliocentric velocity corrections are included in the standard FUSE reductions. Most of these lines were previously observed with $ORFEUS~II$ and Hopkins Ultraviolet Telescope (HUT), but with lower spectral resolution (Mauche & Raymond 1998, Greeley et al. 1999). The strong narrow emission lines in the summed spectra are mainly airglow, except for the narrow component in the O VI lines. O VI shows both a broad and a narrow emission component, similar to the structure seen for the strongest lines in the optical and ultraviolet regions. There is also broad emission from the stronger Lyman lines. Gänsicke et al. (1998) point out that in the UV only the highest excitation lines (N V and Si IV) show this broad plus narrow line structure, while the lower ionization features of C II, C III, Si II, and Si III show only the broad component. In the FUV we similarly find that the C III lines (and probably the Si IV lines, which are weaker) have only the broad emission line, although the profiles change significantly with binary phase. Narrow Lyman absorption lines are seen below 950Å, especially noticeable in the spectrum at maximum FUV light. These were not observed in the $ORFEUS~II$ spectra, probably because of their lower resolution. There is only weak evidence for any H$_2$ absorption, in strong contrast to the supersoft X-ray binary QR And in which Hutchings et al. (2001) found probable circumbinary as well as interstellar H$_2$ absorption. Airglow emissions lie in the middle of the N III 992Å and He II 1085Å emission lines, making it difficult to make clean measurements of these features. While there are some airglow lines near the O VI lines, we can make use of the doublet to isolate these fairly well. The strong C III 1175Å line is free of airglow and shows very different profile changes from O VI, as described in §5 below. [llcccc]{} 8.4956 &0.281 &2.25$\times10^{-13}$ &$-$27, $-$36 &214, — & 3008.4989 &0.306 &2.62$\times10^{-13}$ &$-$33, $-$18 & 69, 63 & 4308.5023 &0.333 &2.94$\times10^{-13}$ &$-$38, $-$47 &159, $-$27 & 4258.5056 &0.358 &3.09$\times10^{-13}$ &$-$41, $-$53 &252, 97 & 3058.5089 &0.384 &3.21$\times10^{-13}$ &$-$44, $-$47 &66, 251 & 6108.5122 &0.410 &3.03$\times10^{-13}$ &$-$47, $-$44 &267, 245 & 3558.5155 &0.435 &3.18$\times10^{-13}$ &$-$53, $-$44 &110, 126 & 5308.5676 &0.839 &1.55$\times10^{-13}$ &66, 57 &$-$247, $-$189 & $-$1808.5709 &0.865 &1.45$\times10^{-13}$ &95, 71 &$-$294, $-$348 & $-$2308.5742 &0.890 &1.31$\times10^{-13}$ &69, 66 &$-$172, $-$333 & $-$2258.5775 &0.916 &1.06$\times10^{-13}$ &66, 83 &$-$201, $-$119 & $-$2808.5808 &0.942 &1.03$\times10^{-13}$ &89, 83 &$-$178, $-$134 & $-$2608.5841 &0.967 &0.94$\times10^{-13}$ &81, 80 &$-$152, $-$206 & $-$2558.6385 &0.389 &3.04$\times10^{-13}$ &$-$44, $-$50 &113, 216 & 1508.6418 &0.415 &3.00$\times10^{-13}$ &$-$53, $-$41 &113, 57 & 2508.6451 &0.440 &3.20$\times10^{-13}$ &$-$41, $-$50 &101, 112 & 1008.6484 &0.466 &3.50$\times10^{-13}$ &$-$35, $-$38 &127, — & 2008.6517 &0.492 &3.27$\times10^{-13}$ &$-$33, $-$10 &211, 222 & 2508.6550 &0.517 &3.38$\times10^{-13}$ &$-$33, $-$33 &142, 199 & 300 FUV Continuum Changes and Adopted Phases ======================================== There are continuum changes across the whole FUSE range. We have defined a FUV phase, using the period of Heise & Verbunt (1988) and the time of minimum FUV light from our FUSE data. The continuum light curve used for our ephemeris is based on the total signal from the LiF1a channel, after removal of the strong airglow emissions. This channel has the best guiding and signal levels in the FUSE data. As discussed below, there are differences within the FUSE band, so we also measured the continuum over several ranges of wavelengths (see Figure 2, lower panel). Although all the continuum flux datasets are well fit by a sine curve, we have used the fit for 980-1090Å to define the time of minimum FUV flux and hence our zero phase ($\Phi_{FUVmin}$=0.0). Table 2 shows the parameters of the fits for various wavelength ranges. We adopt: $\Phi_{FUVmin}=0$ = HJD 2451708.4594(11) + 0.128927041(5)E Comparing this to other phase conventions used in the literature, we find $\Phi_{mag}$ = $\Phi_{FUV}$ + 0.11 (based on the ephemeris given by Heise & Verbunt 1988). Similarly orbital phases, where zero phase is the superior conjunction of the white dwarf, are $\Phi_{orb}$ = $\Phi_{mag}$ + 0.37 = $\Phi_{FUV}$ + 0.48 (using information from Southwell et al. 1995 and Gänsicke et al. 1998). The problem with these alternative zero points is that they are defined by older data, so here we prefer to use only our FUV data to avoid possible error accumulation over many years. The phasing for our FUV light curve is very similar to that observed in other ultraviolet and far-ultraviolet studies. Using IUE data, Gänsicke, Beuermann, & de Martino (1995) found the flux at $\lambda$1460 brightest at $\Phi_{mag}\sim0.6$ and faintest at $\Phi_{mag}\sim$0.1 (see their Fig. 5). From HST data, Gänsicke et al. (1998) observed similar phasing for three different wavelength regions between 1150Å and 1427Å, with the amplitude increasing towards shorter wavelengths (see their Fig. 3). Greeley et al. (1999), using HUT data, found the peak brightness at the Lyman limit occurred at $\Phi_{mag}\sim0.6$, and Mauche & Raymond (1998) found the flux at 1010Å also peaked at magnetic phase $\sim$0.6. Formally, our peak occurs at $\Phi_{mag}=0.61$. Thus, all of these independent data sets are consistent in the phasing of the UV and FUV light curves. However, in our FUSE data the continuum shows a deeper minimum over the wavelength range 950-1050Å than outside that range, with the deepest minimum occuring at $\sim$1000Å. There the total flux range is a factor six, compared with three or less elsewhere. This deep minimum occurs over several FUSE detector channels, and thus is unlikely to be a detector, data processing, or tracking artifact. Figure 2 shows our light curves from the 1000Å region and a longer wavelength region (1090-1180Å). Similarly, Gänsicke et al (1998) found a trend for the amplitude of the modulation to be largest at their shortest wavelength region which was centered on 1158Å, but their HST observations did not go into the shorter wavelength region covered by the FUSE data. Mauche & Raymond (1998) reported a flux range of a factor $\sim$3.3 at 1010Å from their $ORFEUS~II$ data, and they did not note any differences at other wavelengths. They fit the continuum with a two-temperature model, but there are deviations from it at wavelengths shorter and longer than 1000Å. Hence, the effect of a larger variation at $\sim$1000Å may be present in their data as well, or there may be some change in the FUV flux between their observations and ours. They have modeled the light curves with cyclotron components in the visible region and with a localized hotspot on the white dwarf surface in the FUV. Our wavelength-dependent light curve is not seen in the broad-line wings when we take ratios of the two spectra. This is demonstrated in Figure 3, where the broad regions around the strong emission lines show the shallower minima seen in the continuum below 950Å and above 1050Å. This suggests that there are very wide wings associated with the strong emissions which fully overlap in the higher Lyman lines. These features are visible in the ratio of any two of the three FUSE orbits, all of which have different binary phase coverage, but it is strongest in the ratio shown in Figure 3, covering the largest continuum changes. If the ratio in the shortest wavelengths is affected by overlapping Lyman lines, it is possible that the real continuum variation remains high at all wavelengths below about 1000Å. Such a scenario is indicated schematically in Fig. 3 by the dotted line. This line is not calculated, but merely drawn in to show a real continuum that would be feasible with broad-line blanketing, and compatible with the white dwarf hot-spot model. Emission Line Measurements and Changes ====================================== The O VI lines are higher ionization lines than any emission observed in the HST or optical ranges, and thus they are potentially of unique interest. They have been observed previously in two visits by $ORFEUS$ (in 1993 and 1996). The 1996 data are superior and were taken when AM Her was in a high state (see Mauche & Raymond 1998). The behavior of the O VI lines is similar to the He II line at 4686Å in having sharp and broad components with very different velocity amplitudes and phasing (see e.g. Cowley and Crampton 1977). Line velocities were measured by fitting a parabola to the entire feature, and also by taking the flux centroid of the emission line. Values given are the mean of the separate measures. The broad component of the O VI doublet shows a velocity semi-amplitude K$\sim$190 km s$^{-1}$ with the minimum velocity at $\Phi_{FUV}\sim0.9$ (or approximately $\Phi_{mag}\sim0.0$), whereas the narrow component shows K of only 62 km s$^{-1}$ with phasing about half a cycle later. These values can be directly compared to the measurements made by Mauche & Raymond who had fewer spectra but more evenly distributed around the orbit. Their amplitudes for the broad and narrow components of O VI were K=412 and 57 km s$^{-1}$, respectively, with phasing very similar to what we observe after taking into account the difference between FUV and magnetic phases. Additionally, there is a broad component of L$\beta$, easily seen on either side of the airglow emission. While blended with the airglow emission, it appears to have a similar line profile and to move with the broad O VI lines (see Fig. 4). This broad emission component is also present in the other strong Lyman lines where it is too weak to measure. Figure 2 includes our flux and velocity measurements of L$\beta$. The broad L$\beta$ line shows greater fractional change than the broad O VI, so that their ratio changes markedly at phases $\Phi_{FUV}$=0.4 to 0.5. We also measured the flux and velocity of the C III emission line at 1175Å. This line is broad and has no narrow component significantly different from the noise level. We have also summed the spectra in groups of three to increase the signal-to-noise in the line profile. Its overall profile changes strongly, as shown in Figure 5. Using the C III velocities from all 19 spectra (to retain the maximum phase resolution), we obtained the sine-curve fit given in Table 2. Both the semi-amplitude (K=311 km s$^{-1}$) and the mean value (+50 km s$^{-1}$) are significantly different from the fit found for the broad O VI lines. Gänsicke et al. found C III 1176Å to have a semi-amplitude of 540 km s$^{-1}$ with the minimum velocity occurring about 0.1P later than we observe. However, their HST data had more complete phase coverage, so these two C III lines probably behave similarly. The other emission lines were weaker and contaminated with airglow emissions, so we were not able to measure them. However, Fig. 3 suggests the presence of other possible emission features if the dips are all caused by line broad line emission that has a lower flux variation than the continuum. This idea is reinforced by the fact that these wavelengths correspond to those of possible emission lines. Along the bottom of this figure we have marked the positions of the identified lines we measured, as well as the stronger S IV line positions - all of which lie at dips in the ratio plot. Thus, we conclude that several weak S IV emission features may be present that do not show up clearly in either the individual or summed spectra. While the phasing of FUV continuum are in good agreement with those of Mauche & Raymond and with the UV data of Gänsicke et al., the velocity amplitudes of the broad lines differs, as described above. Our amplitudes are less well-defined because of our more limited phase coverage, but they do appear to cover the velocity extremes fairly well. [llcccr]{}      & & & & & 980–1090Å  &8.4594$\pm$0.0011 &(0.0) & 0.65$\pm$0.05 &2.17$\times10^{-12}$ &0.05 994–1020Å  &8.4585$\pm$0.0011 &0.993$\pm$0.009 &1.20$\pm$0.05 &1.84$\times10^{-13}$ &0.08 1090–1180Å &8.4619$\pm$0.0011 &0.019$\pm$0.009 &0.61$\pm$0.05 &2.17$\times10^{-13}$ &0.05    & & & & & O VI broad &8.4479$\pm$0.0041 &0.910$\pm$0.03 &190$\pm$15 &$-$30$\pm$14 &79 O VI peak &8.3828$\pm$0.0014 &0.406$\pm$0.01 &62$\pm$2 &16$\pm$2 &9 C III broad &8.4416$\pm$0.005 &0.862$\pm$0.04 &311$\pm$38 &50$\pm$31 &125 Absorption Lines ================ As noted earlier, H$_2$ absorption is weak, indicating there is little H$_2$ along the line of sight. In the supersoft binary system QR And where H$_2$ was particularly strong, we presented evidence that this absorption was at least partially due to circumbinary material. By comparison, this suggests that little cool gas is leaving the AM Her system. However, we do see a clear set of narrow H absorption lines (see Figure 1). Within the precision of the FUSE wavelength scale at the shortest wavelengths, these H absorptions all show the same velocity and do not change with time. Their mean velocity is $-36\pm$8 km s$^{-1}$, which can be compared to the systemic velocity of $-19$ km s$^{-1}$ (Young, Schneider, & Shectman 1981). We note however, that our Calfuse 1.9.9 extractions gave different results in the region than the version 1.6.9 pipeline data. We regard ours as better as a) the version 1.6.9 radial velocities range systematically with wavelength while the later version 1.9.9 values do not. The FUSE wavelength scale precision is not well established at the shortward limit, but it is possible that there is a systematic velocity error of order $\sim$20 km s$^{-1}$. The Lyman absorption lines between 918 and 950Å were measured for equivalent width. Noting again a difference between our extractions and the pipeline, we find the absorptions have a mean EW of 0.26$\pm$0.04Å. There is no significant change with binary phase. Thus, the absorption appears to arise outside the system, and may be normal interstellar gas. There are several weak to moderate absorption features seen in all spectra in the wavelength range 1036 to 1041Å, where we have good signal and are interested in contamination of the O VI emission. We identify the strongest as the C II doublet at 1036.3, 1037.0Å. Other weak absorptions match the H$_2$ spectrum at the same velocity (close to zero, subject to the FUSE wavelength scale uncertainty), and also O I. We do not see any changes in these absorptions with binary phase. They are weak enough that they do not affect the O VI line measures significantly. Summary ======= A general model of the system has been discussed and developed by a number of authors (e.g. Gänsicke et al. 1998 and Mauche & Raymond 1998 are recent examples). There is mass transfer along magnetic field lines within the system, principally on to one pole of the white dwarf, at least in the system high state. This pole is hot and causes the X-ray and UV flux variations as the system rotates. The optical-NIR light changes are caused by a combination of the temperature gradients on both the donor star and the white dwarf. The broad emission lines arise along the path of the gas stream, while the narrow emission lines appear to be located near the cusp of the donor star’s Roche lobe. The FUSE data are consistent with this picture, while providing some new details and aspects of the complex system. Our principal results are as follows. 1\. The broad O VI emission lines have the same phasing as the other broad emissions, but appear to have a different velocity amplitude from that seen in 1996 in the ORFEUS data. Changes in other broad-line amplitudes have been noted (e.g. by Crosa et al. 1981 for He II 4686Å), and presumably indicate variations in the gas stream ionisation and kinematics, as the mass transfer rate varies. 2\. The narrow O VI emission lines also behave much like the other narrow emissions. Our data agree with the ORFEUS results in velocity, but the higher FUSE resolution gives a very different separation of flux between the broad and narrow components. While we do not cover the entire orbit, we find the narrow line to have less that 1/4 the broad line flux at all times and that the narrow line is very weak at FUV light minimum. The velocities of the narrow lines seem more stable over the history of various investigations and support the standard interpretation that they arise at the Roche lobe cusp. The range of ionization seen (from Ca II to O VI) is very large, and it is clear that some mechanisms other than photoionization are involved. If we examine the velocity amplitudes for all the lines, there is a trend with ionization potential running from values near 100 km s$^{-1}$ for Ca II and H I to about 60 km s$^{-1}$ for O VI and N V. He I, C II, He II, Si IV, N III are part of the intermediate sequence. As proposed by Gänsicke et al., this suggests an ionization gradient moving away from the Roche lobe cusp towards the system center of mass. 3\. The FUV light curve shows strong changes over the FUSE wavelength range. This is compatible with the model of a hot spot on the white dwarf if a significant part of the FUV flux arises as broad line emission from all the lines detected and where the higher Lyman lines overlap each other. 4\. We find sharp H I Lyman absorptions with a constant radial velocity of $-$36 km s$^{-1}$. Weak H$_2$ absorption appears to be present, which does not change with binary phase, and has the same velocity as other interstellar absorptions. AM Her’s H$_2$ absorption is much weaker than that in the X-ray binary QR And, which is at a similar distance from us. The authors acknowledge use of the AAVSO web site and thank the many variable star observers for their contributions to this dataset. APC also acknowledges her support from NASA for this work. 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Press [^1]: Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-3298

--- abstract: 'This paper presents our segmentation system developed for the MLP 2017 shared tasks on cross-lingual word segmentation and morpheme segmentation. We model both word and morpheme segmentation as character-level sequence labelling tasks. The prevalent bidirectional recurrent neural network with conditional random fields as the output interface is adapted as the baseline system, which is further improved via ensemble decoding. Our universal system is applied to and extensively evaluated on all the official data sets without any language-specific adjustment. The official evaluation results indicate that the proposed model achieves outstanding accuracies both for word and morpheme segmentation on all the languages in various types when compared to the other participating systems.' author: - | Yan Shao\ Department of Linguistics and Philology, Uppsala University\ [[email protected]]{} bibliography: - 'acl2017.bib' title: 'Cross-lingual Word Segmentation and Morpheme Segmentation as Sequence Labelling' --- Introduction ============ In natural language processing, word segmentation and morpheme segmentation are the initial steps to identify basic linguistic units, namely words and morphemes, for further analysis in higher-level tasks. Word segmentation can be very non-trivial, especially for languages without explicit indicators for word boundaries, such as Chinese, Japanese and Vietnamese. For morphologically rich languages like Turkish, words are further segmented into morphemes, such as stems, prefixes and suffixes for morphological analysis. Similar to non-trivial word segmentation, there are no clear boundaries between morphemes in the surface forms of words. Both word segmentation and morpheme segmentation can be viewed as identifying valid boundaries between consecutive characters. Word segmentation is often formalised as a character-based sequence labelling problem to predict position tags [@xue2003chinese; @udpipe]. Standard machine algorithms, such as Maximum Entropy [@berger1996maximum; @low2005maximum], Conditional Random Fields (CRF) [@lafferty2001crf; @peng2004chinese] and neural networks [@chen2015long] are applied for the task in previous research. Additionally, a number of word-based approaches have also been proposed [@zhang2007chinese; @cai2016neural]. For morpheme segmentation, apart from unsupervised methods [@creutz2007unsupervised; @poon2009unsupervised], model the task as sequence labelling, similarly to character-based word segmentation. They use CRF to predict position tags given words as sequences of characters. Furthermore, instead of employing traditional statistical models, propose and apply several recurrent neural network architectures to avoid heavy feature engineering. Considering the similarities between word segmentation and morpheme segmentation, we present a universal neural sequence labelling model that is capable of solving both segmentation tasks in this paper. Our baseline model is an adaptation of a bidirectional recurrent neural network (RNN) using conditional random fields (CRF) as the output interface for sentence-level optimisation (BiRNN-CRF). BiRNN-CRF achieves state-of-the-art accuracies on various sequence labelling tasks [@huang2015bidirectional; @maend]. We modify the conventional position tags used for word segmentation so that they are also applicable to morpheme segmentation. Furthermore, a simple ensemble decoding technique is implemented to obtain additional improvements over the baseline model. Our system is fully data-driven and language-independent. It is extensively evaluated on the MLP 2017 shared task data sets. Segmentation Model ================== Baseline Model -------------- [UTF8]{}[gbsn]{} Our baseline model is shown in Figure \[fig:1\]. We adopt the concatenated 3-gram model introduced in as the vector representation of input characters. The pivot character in a given context is represented as the concatenation of the context-free vector along with the local bigram and trigram vectors. All the vectors are initialised randomly and separately. Utilising the concatenated n-grams ensures that the same character has different yet closely related vector representations in different contexts, which is an effective way to encode contextual features. We use a single vector to represent all the characters that appeared only once in the training set while training. This vector is later used as the representation for unknown characters in the development and test sets. The same representation scheme is also applied to local bigrams and trigrams. The character vectors are passed to the forward and backward recurrent layers. Gated recurrent units (GRU) [@cho2014properties] are employed as the basic recurrent cell to capture long term dependencies and global information. Compared to the more prevalent long-short term memory cells (LSTM) [@hochreiter1997long], GRU has similar functionalities but fewer parameters [@chung2014empirical]. Dropout [@srivastava2014dropout] is applied to the input vectors as character representations and the outputs of the bidirectional recurrent layers. A first-order chain CRF layer is added on top of the recurrent layers to incorporate the transition information between consecutive tags, which ensures that the optimal sequence of tags over the entire sentence is obtained. The optimal sequence can be obtained efficiently via the Viterbi algorithm both for training and decoding. The time complexity is linear with respect to sentence length. For the chain CRF interface while decoding, the final sequence of the position tags $y$ is obtained via the conditional scores $S(y_i|x_i)$ and the transition scores $T(y_i, y_j)$ given the input sequence $x$. In the baseline model, the optimal sequence is computed with respect to the scores returned by a single model: $$y^* = \operatorname*{argmax}_{y \in L(x)} p(y|x; S, T)$$ There is a post processing step to retrieve segmented units with respect to the predicted tags, which varies from different segmentation tasks and different formats of data sets. Tag Set ------- For word segmentation, we use four position tags B, I, E, and S to indicate a character positioned at the beginning (B), inside (I), or at the end (E) of a word, or occurring as a single-character word (S). We extend this tag set for morpheme segmentation by adding an extra tag X to represent the word boundaries. Figure \[fig:2\] illustrates the input characters and the boundary tags to be predicted in morpheme segmentation. Unlike previous work of and , our model performs morpheme segmentation at the sentence level rather than the word level to incorporate information beyond word boundaries. Ensemble Decoding ----------------- We use a simple ensemble averaging technique to mitigate the deviations caused by random weight initialisation of the neural network and improve the baseline. For ensemble decoding, both the conditional scores $S(y_i|x_i)$ and the transition scores $T(y_i, y_j)$ are averaged over four models with identical parameter settings but trained independently with different random seeds: $$y^* = \operatorname*{argmax}_{y \in L(x)} p(y|x; \bar{\{S\}}, \bar{\{T\}})$$ Implementation -------------- Our neural segmenter is implemented using the TensorFlow 1.2.0 library [@abadi2016tensorflow]. The bucket model is applied so that the training and tagging speed of our neural network on GPU devices can be drastically improved. The training time is proportional to the size of the training set. We provide an open-source implementation of our method.[^1] Table \[tab:1\] shows the adopted hyper-parameters. We use one set of parameters for both tasks on all the provided data sets. The weights of the neural networks, including the character vectors, are initialised using the scheme introduced in . The network is trained with the error back-propagation algorithm. The vector representations of input characters are fine-tuned during training by back-propagating gradients. Adagrad [@duchi2011adaptive] with mini-batches is employed for optimisation with the initial learning rate $ \eta_0 = 0.1$, which is updated with a decay rate $\rho = 0.05$ as $\eta_t = \frac{\eta_0}{\rho (t - 1) + 1} $, where $t$ is the index of the current epoch. To increase the efficiency and reduce memory demand both for training and decoding, we chop the sentences longer than 300 characters. For decoding, the chopped sentences are recovered after being processed. The model is optimised according to its performance on the development sets. F1-score with respect to the basic segmented unit is employed to measure the performance of the model after each epoch during training. In our experiments, the models are trained for 30 epochs. To ensure that the weights are well optimised, we only adopt the best epoch after the model is trained at least for 5 epochs. Experiments =========== Data Sets --------- There are in total 10 data sets provided in the MLP 2017 shared tasks. Traditional Chinese, Japanese and Vietnamese are for the word segmentation task, while Basque, Farsi, Filipino, Finnish, Kazakh, Marathi and Uyghur are for morpheme segmentation. The provided languages vary substantially both in typology and written form. The sizes of the data sets are also different. The detailed information can be found in Table \[tab:2\]. In our experiments, we only use the official training data to build separate segmentation models for each language without utilising any external resources. For Vietnamese, we use the space-delimited units as the basic elements for boundary prediction. For all the rest, no language-specific modification or adjustment is made. Experimental Results -------------------- Diff 1 Diff 2 ------------ ------- ----- ------- -------- -------- ------ ------ ------ ------ ------ ------- -- -- -- Dataset Train Dev Test P R F P R F F F Chinese 2,029 250 250 84.2 87.1 85.7 85.4 87.8 86.6 +0.9 +9.7 Japanese 1,600 200 200 96.1 97.8 96.9 96.6 97.7 97.2 +0.3 +3.4 Vietnamese 3,000 500 500 90.9 92.8 91.8 92.0 92.8 92.4 +0.6 - Basque 599 100 100 81.5 77.2 79.3 82.4 80.8 81.6 +2.3 +28.2 Farsi 500 100 100 77.6 74.0 75.8 77.9 76.2 77.0 +1.2 +17.0 Filipino 1,999 200 200 91.2 93.0 92.1 92.0 92.4 92.2 +0.1 - Finnish 3,537 750 762 89.8 90.5 90.2 90.9 90.5 90.7 +0.5 +26.5 Kazakh 7,298 999 1,000 97.0 97.1 97.1 97.5 97.5 97.5 +0.4 - Marathi 5,098 450 450 95.1 93.3 94.2 95.1 93.8 94.4 +0.2 - Uyghur 3,999 500 501 65.1 61.5 63.3 67.5 61.4 64.3 +0.1 -0.8 Uyghur\* 3,999 500 501 96.8 96.8 96.8 97.1 97.3 97.2 +0.4 +32.7 For word segmentation, word level precision, recall and F1-score are employed as the evaluation metrics. For morpheme segmentation, precision, recall and F1-score are calculated only with respect to the identified prefixes and suffixes. The official experimental results are shown in Table \[tab:2\]. The results of both the single and ensemble models are presented. The F1-scores of the single model as Baseline are in comparison to the ensemble model as well as the best performing systems among the other participants of the shared task. In general, the BiRNN-CRF model is effective for both word segmentation and morpheme segmentation. Our baseline model is substantially better than the rest of the participating systems on all the languages, especially for morpheme segmentation. Due to some encoding issues, our system is relatively under-performing on Uyghur referring to the official scores. We fixed the problem and report the corrected scores. The ensemble decoding is beneficial across all the data sets, but the overall improvement is rather marginal, especially if the baseline accuracy is very high. It is nonetheless helpful if the training sets are small as in the cases of Basque and Farsi. For word segmentation, we can see that very high accuracy is obtained on Japanese in spite of the relatively small training set. The writing system of Japanese is a combination of hiragana, katakana and Chinese characters (kanji). The switching of different types of characters can be an indicator for word boundaries in a sentence. As opposed to Japanese, Chinese and Vietnamese only contain one type of characters. The identification of word boundaries depends more heavily on the context. Additionally, the Chinese data set is composed of sentences from web search in different genres and the percentage of out-of-vocabulary words is high in the test set, which makes the segmentation task more challenging. For morpheme segmentation, the size of the training set has the biggest impact on accuracy. The evaluation scores on Basque and Farsi are therefore drastically lower than the others. Unlike Chinese and Japanese characters and the space-delimited units in Vietnamese, individual characters by themselves in the languages for morpheme segmentation are less informative and the character vocabulary size is much smaller. The types of prefixes and suffixes to be identified are very limited and less ambiguous. As long as the data set is standardised and properly tokenised, very high accuracies can be achieved across languages in different writing systems given sufficient training data. Conclusions =========== This paper presents our segmentation system for the MLP 2017 shared task on word segmentation and morpheme segmentation. Viewing both word and morpheme segmentation as character level sequence labelling tasks, we adapt the BiRNN-CRF model that has been applied to various sequence labelling tasks previously. Regardless of the vast variety of the data sets, we employ a universal model that uses a single set of hyper-parameters on all the languages without any task and language-specific adaptations. The evaluation results indicate that our model is effective for both segmentation tasks. In general, the proposed model achieves relatively high accuracies across all the languages for both tasks if sufficient amounts of training data are provided. However, as both word segmentation and morpheme segmentation are at the very low levels of the complete natural language processing framework, the segmentation errors propagate further to higher level tasks. Thus, it is still very valuable to develop systems with higher performances in the future. In addition, we will explore the possibility of adapting the proposed model to low-resource languages using cross-lingual approaches if sufficient amount of training data is not available. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge the computational resources provided by CSC in Helsinki and Sigma2 in Oslo through NeIC-NLPL (www.nlpl.eu). [^1]: https://github.com/yanshao9798/segmenter

--- abstract: 'In this paper, we study Higgs bundles and flat bundles over non-Kähler manifolds. Suppose $(X,\omega)$ is a compact Hermitian manifold of dimension $n$ satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$, we prove that every semi-stable Higgs bundle $(E,\bar{\partial}_{E},\theta)$ over $(X,\omega)$ with vanishing first and second Chern numbers is an extension of Higgs-Hermitian flat bundles. Moreover, suppose $(X,\omega)$ also satisfies $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for every $[\eta]\in H^{0,1}(X)$, we prove that there is an equivalence of categories between the category of poly-stable Higgs bundles with vanishing Chern numbers and the category of semi-simple flat bundles. At last, in rank $2$ case, we show that there is an one-to-one correspondence between the moduli space of semi-stable Higgs bundles with vanishing Chern numbers and the moduli space of flat bundles.' address: - | Changpeng Pan\ School of Mathematical Sciences\ University of Science and Technology of China\ Hefei, 230026,P.R. China\ - | Chuanjing Zhang\ School of Mathematical Sciences\ University of Science and Technology of China\ Hefei, 230026,P.R. China\ - | Xi Zhang\ School of Mathematical Sciences\ University of Science and Technology of China\ Hefei, 230026,P.R. China\ author: - 'Changpeng Pan, Chuanjing Zhang and Xi Zhang' title: 'Semi-stable Higgs bundles and flat bundles over non-Kähler manifolds' --- Introduction ============ Higgs bundles were introduced by Hitchin [@H] and developed by Simpson [@S1; @S2]. It carries a rich structure and plays an important role in many areas including gauge theory, Kähler and hyperkähler geometry, group representations, and nonabelian Hodge theory. The results by Corlette [@Cor], Donaldson [@Don3], Hitchin [@H] and Simpson [@S2], provide a correspondence between the moduli space for poly-stable Higgs bundles with vanishing Chern numbers and the moduli space for semi-simple flat bundles over compact Kähler manifolds. It is nature to ask whether this correspondence holds for non-Kähler manifolds. Biswas [@B] showed us that this is not true for general non-Kähler manifolds. He constructed a class of non-Kähler manifolds such that the correspondence is invalid. But it is still a question whether it is true on non-Kähler manifolds which satisfy some suitable conditions or not. Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, $(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle over $X$, i.e. $(E,\bar{\partial}_{E})$ is a holomorphic vector bundle and $\theta \in \Omega^{1, 0}(\mbox{End}E)$ is a holomorphic section satisfying $\theta \wedge \theta =0 $. Suppose the Hermitian metric $\omega $ satisfies the Gauduchon and Astheno-Kähler conditions, i.e. $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$, then the first and second Chern numbers of holomorphic vector bundles are well-defined. Let $\mathcal{C}_{DR}^{s}$ (resp. $\mathcal{C}_{DR}$) denote the quotient set of semi-simple flat bundles (resp. flat bundles) module an equivalent relation, and $\mathcal{C}_{Dol}^{s}$ (resp. $\mathcal{C}_{Dol}$) denote the quotient set of polystable (resp. semi-stable) Higgs bundles with the vanishing first and second Chern numbers module an equivalent relation. The Donaldson-Uhlenbeck-Yau theorem [@NarSe; @Don2; @UhYau] states that every poly-stable holomorphic vector bundle over a Kähler manifold admits a Hermitian-Einstein metric. This theorem was generalized to the Higgs bundles case by Hitchin [@H] and Simpson [@S1], and is also valid for compact Gauduchon manifolds [@bu; @LY; @MA]. So we know that there is an injective map from $\mathcal{C}_{Dol}^{s}$ to $\mathcal{C}_{DR}^{s}$. It is natural to construct an injective map from $\mathcal{C}_{DR}^{s}$ to $\mathcal{C}_{Dol}^{s}$ on some non-Kähler manifolds. Let $D$ be a flat connection on the bundle $V$. Given a Hermitian metric $H$, $D$ decomposes uniquely as $ D=D_{H}+\psi_{H}$, where $D_{H}$ is a unitary connection and $\psi_{H}\in \Omega^{1} (X, \mbox{End}(V))$ is self-adjoint. Define $$D_{H}''=D_{H}^{0, 1} +\psi_{H}^{1, 0}, \quad D_{H}'=D_{H}^{1, 0} + \psi_{H}^{0, 1}, \quad G_{H}=(D_{H}'')^{2}.$$ We say a Hermitian metric $H$ is harmonic if it satisfies: $$\label{h1} \sqrt{-1}\Lambda _{\omega} G_{H}=0.$$ When $(X, \omega )$ is Kähler, (\[h1\]) is equivalent to $D_{H}^{\ast }\psi_{H}=0$, Donaldson [@Don3] and Corlette [@Cor] proved the existence of harmonic metric on the semi-simple flat bundle by using the heat flow method. In this paper, by using the continuous method, we extend Corlette and Donaldson’s result to some non-Kähler case, see section 3 for details. Then, we construct an injective map from $\mathcal{C}_{DR}^{s}$ to $\mathcal{C}_{Dol}^{s}$ and prove the following theorem. \[thm:2-1\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$ and $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$. Then there is an equivalence of categories between the category of poly-stable Higgs bundles with vanishing Chern numbers and the category of semi-simple flat bundles. Assume that the metric $\omega$ satisfies strongly Gauduchon and astheno-Kähler conditions, then the conditions in Theorem \[thm:2-1\] are satisfied. For example, in dimension $3$, there are some nontrivial examples in [@OUV Proposition 2.1] which satisfy our conditions. It is an interesting question whether there is a more general correspondence between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$ or not. When the base manifold is a smooth projective variety, Simpson [@S2] proved that every semi-stable Higgs bundle with vanishing Chern numbers is an extension of stable Higgs bundles with vanishing Chern numbers, then he showed there is a correspondence between the category of semi-stable Higgs bundles with vanishing Chern numbers and the category of flat bundles over smooth projective varieties. In this paper, by using an argument in geometric analysis, we prove that every semi-stable Higgs bundle with the vanishing first and second Chern numbers is an extension of Higgs-Hermitian flat bundles over some non-Kähler manifolds. \[thm:1\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$. Let $\mathfrak{E}=(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle over $X$. Then $\mathfrak{E}$ is semi-stable with $ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]=0$ if and only if $\mathfrak{E}$ admits a filtration $$0=\mathfrak{E}_{0}\subset \mathfrak{E}_{1}\subset\cdots\subset \mathfrak{E}_{l}=\mathfrak{E}$$ by Higgs subbundles such that the quotients $\mathfrak{Q}_{k}=\mathfrak{E}_{k+1}/\mathfrak{E}_{k}$ are Higgs-Hermitian flat. By using the Hermitian-Yang-Mills flow, Nie and the third author [@NZ1] generalized the above Simpson’s result to the Kähler case. In the non-Kähler case, we combine the continuous method and the heat flow method to prove the above theorem, see section $5$ for details. As an application, when $rank(E)=2$ or the length $l$ of the above filtration is two, we establish a correspondence between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$ on some non-Kähler manifolds, i.e. we obtain the following theorem. \[thm:2-2\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$ and $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$. Then there is an one-to-one map between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$ in rank $2$ case. We also find a nature one-to-one map between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$ for high rank case, but as pointed in Remark \[rmk:7-1\], we can see this map only exists in Kähler case, i.e. we prove the following theorem. \[thm:2-3\] Let $(X,\omega)$ be a Kähler manifold of dimension $n$. Then there is an one-to-one map between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$. Actually, using Theorem \[thm:1\] ([@NZ1] for Kähler case) together with Simpson’s argument in [@S2], one can get a correspondence between $\mathcal{C}_{Dol}$ and $\mathcal{C}_{DR}$ over Kähler manifolds. However, Simpson’s argument [@S2] is totally algebraic and highly abstract. We construct the correspondence map more directly, so we write down here. It should be point out that, by using Nie and third author’s result [@NZ1], Deng also give a constructing proof of this correspondence in his thesis [@D]. The idea of our construction should be coincide, but there are some small differences in the details. Deng’s proof relies on Simpson’s $\partial\bar{\partial}$-lemma, but we solve elliptic equation (\[eq:7-2\]) instead, see Section 7 for details. This paper is organized as follows. In Section 2, we introduce some basic concepts and results about Higgs bundles and flat bundles over non-Kähler manifolds. In Section 3, we prove Theorem \[thm:2-1\]. In Section 4, we obtain an $\epsilon$-regularity theorem about the modified Yang-Mills-Higgs flow. In Section 5, we give a proof of Theorem \[thm:1\]. In Section 6 and Section 7, we prove Theorem \[thm:2-2\] and Theorem \[thm:2-3\]. [**Acknowledgement:**]{} The authors would like to thank Professor Jixiang Fu and Professor Xiangwen Zhang to point out the examples satisfying our assumptions in Theorem \[thm:2-1\] and Theorem \[thm:2-2\]. The authors are partially supported by NSF in China No.11625106, 11571332 and 11721101. The second author is also supported by NSF in China No.11801535, the China Postdoctoral Science Foundation (No.2018M642515) and the Fundamental Research Funds for the Central Universities. Preliminaries ============= Higgs bundles ------------- Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$ and suppose that $\partial\bar{\partial}\omega^{n-1}=0$ and $\partial\bar{\partial}\omega^{n-2}=0$. Let $(E,\bar{\partial}_{E},\theta)$ be a rank $r$ Higgs bundle over $X$, $H$ be a Hermitian metric on $E$. Define the Hitchin-Simpson connection by $D_{H,\theta}=D_{H}+\theta+\theta^{*H}$, where $D_{H}$ is the Chern connection and $\theta^{*H}$ is the adjoint of $\theta$ with respect to $H$. The related curvature of $D_{H,\theta}$ is $$F_{H,\theta}=F_{H}+[\theta,\theta^{*H}]+\partial_{H}\theta+\bar{\partial}_{E}\theta^{*H}.$$ The Chern characters $ch_{k}(E,\bar{\partial}_{E},H)\in\mathcal{A}^{k,k}(X)$ are defined by $$ch_{k}(E,\bar{\partial}_{E},H)=\frac{1}{k!}{\mbox{\rm tr\,}}((\frac{\sqrt{-1}}{2\pi}F_{H})^{k}).$$ Define the first and second Chern numbers by $$ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=\int_{X}ch_{1}(E,\bar{\partial}_{E},H)\wedge\frac{\omega^{n-1}}{(n-1)!}$$ and $$ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]=\int_{X}ch_{2}(E,\bar{\partial}_{E},H)\wedge\frac{\omega^{n-2}}{(n-2)!}.$$ Donaldson [@Don2 Proposition 6] proved that, given two metrics $H_{1}$ and $H_{2}$ on $E$, there exists $R_{k-1}(H_1,H_2)\in A^{k-1,k-1}(X)$ such that $$ch_k(E,H_1)-ch_k(E,H_2)=\sqrt{-1}\bar{\partial}\partial R_{k-1}(H_1, H_2),$$ for every $1\leq k\leq \min{\{r,n\}}$. Since $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$, we know the Chern numbers are well-defined and independent of the Hermitian metric on $E$. By the direct calculations, we have $$ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=\int_{X}\frac{\sqrt{-1}}{2\pi}{\mbox{\rm tr\,}}(F_{H,\theta})\wedge\frac{\omega^{n-1}}{(n-1)!}$$ and $$\begin{split} &\frac{-1}{8\pi^{2}}\int_{X}{\mbox{\rm tr\,}}(F_{H,\theta}\wedge F_{H,\theta})\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&\frac{-1}{8\pi^{2}}\int_{X}\{{\mbox{\rm tr\,}}(F_{H}\wedge F_{H})+2{\mbox{\rm tr\,}}(F_{H}\wedge[\theta,\theta^{*H}])\\ &+{\mbox{\rm tr\,}}([\theta,\theta^{*H}]\wedge[\theta,\theta^{*H}])+2{\mbox{\rm tr\,}}(\partial_{H}\theta\wedge\bar{\partial}_{E}\theta^{*H})\}\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&\frac{-1}{8\pi^{2}}\int_{X}\{{\mbox{\rm tr\,}}(F_{H}\wedge F_{H})+2{\mbox{\rm tr\,}}(F_{H}\wedge[\theta,\theta^{*H}])\\ &-2\partial\bar{\partial}{\mbox{\rm tr\,}}(\theta\wedge\theta^{*H})+2{\mbox{\rm tr\,}}(\theta\wedge[F_{H},\theta^{*H}])\}\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]. \end{split}$$ For any torsion free coherent Higgs sheaf $(\mathcal{F},\theta_{\mathcal{F}})$, define the $\omega$-degree and $\omega$-slope by $$\deg_{\omega}(\mathcal{F})=\int_{X}ch_{1}(\det{\mathcal{F}})\wedge\frac{\omega^{n-1}}{(n-1)!}$$ and $$\mu_{\omega}(\mathcal{F})=\frac{\deg_{\omega}(\mathcal{F})}{\mbox{rank}(\mathcal{F})}.$$ We call $(E,\bar{\partial}_{E},\theta)$ is stable (semi-stable) if $\mu_{\omega}(\mathcal{F})<(\leq)\mu_{\omega}(E)$ for every proper coherent sub-Higgs sheaf $\mathcal{F}\subset E$. A Higgs bundle $(E,\bar{\partial}_{E},\theta)$ is said to be admitting an approximate Hermitian-Einstein structure, if for every $\delta >0$, there exists a Hermitian metric $H_{\delta}$ such that $$\sup_X|\sqrt{-1}\Lambda_{\omega }(F_{H_{\delta}}+[\theta,\theta^{*{H_{\delta}}}])-\lambda \cdot \textmd{Id}_E|_{H_{\delta}}<\delta.$$ Let’s consider the following perturbed equation on $(X, \omega)$: $$\label{eq} \sqrt{-1}\Lambda_{\omega } (F_{H}+[\theta,\theta^{*H}])-\lambda \cdot \textmd{Id}_E+\epsilon \log (K^{-1}H)=0,$$ where $K$ is a fixed Hermitian metric on $E$. Making use of the continuous method in [@UhYau] and applying the Fredholmness of the elliptic operators, one can see that the above perturbed equation can be solved for any $\epsilon \in (0, 1]$. Let $H_{\epsilon}$ be the solution of the equation (\[eq\]), we can conclude that [@NZ Lemma 2.2] $$\frac{1}{2}\sqrt{-1}\Lambda_{\omega }\partial \overline{\partial }(|\log (K^{-1}H_{\epsilon})|^2_{K})\geq \epsilon |\log (K^{-1}H_{\epsilon})|^2_{K}-|\Lambda_{\omega }F_{K, \theta}|_{K}|\log (K^{-1}H_{\epsilon})|_{K},$$ and then $$\label{t2} \sup_{X} \epsilon |\log (K^{-1}H_{\epsilon})|_{K}\leq \sup_{M}|\Lambda_{\omega }F_{K, \theta}|_{K}.$$ If the Higgs bundle $(E,\bar{\partial}_{E},\theta)$ is semi-stable, we have [@NZ Theorem 3.2] $$\sup_{X} \epsilon |\log (K^{-1}H_{\epsilon})|_{K}=\sup_{X} \epsilon |\log (K^{-1}H_{\epsilon})|_{H_{\epsilon}} \rightarrow 0,$$ as $\epsilon \rightarrow 0$. Furthermore, we have the following theorem: \[thm:3\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=0$. Let $(E,\bar{\partial}_{E},\theta)$ be a Higgs bundle over $X$, then $(E,\bar{\partial}_{E},\theta)$ is semi-stable if and only if it admits an approximate Hermitian-Einstein structure. For any Hermitian metric $H$ on $E$, applying Lemma 2.7 in [@S2], we obtain $$\label{eqn:2} |\sqrt{-1}\Lambda_\omega[\theta, \theta^{\ast H}]|_{H}=|[\theta , \theta^{\ast H}]|_{H, \omega}\geq a_1|\theta |_{H, \omega}^2-a_2|\theta |_{K, \omega}^2,$$ where $a_1$ and $a_2$ are positive constants depending only on $r$ and $n$. By choosing complex coordinates $\{z^{1}, \cdots , z^{n}\}$ on the considered point, we deduce: $$\label{t1} \begin{split} \sqrt{-1}\Lambda_{\omega }\partial \bar{\partial }|\theta |_{H_{\epsilon}, \omega }^{2} =&|\nabla \theta |_{H_{\epsilon}, \omega }^{2} +Re \langle g^{\alpha \bar{\beta}}(\nabla_{\bar{\partial}_{\beta }}\nabla_{\partial_{\alpha } }-\nabla_{\partial_{\alpha } }\nabla_{\bar{\partial}_{\beta }})\theta , \theta\rangle \\ &+ 2Re \langle g^{\alpha \bar{\beta}}\nabla_{\partial_{\alpha } }\nabla_{\bar{\partial}_{\beta }} \theta , \theta \rangle \\ =& Re\langle-[\sqrt{-1}\Lambda_{\omega}F_{H_{\epsilon}},\theta],\theta\rangle-Re\langle\theta\sharp R,\theta\rangle +|\nabla\theta|_{H_{\epsilon},\omega }^{2}\\ & + 2Re \langle g^{\alpha \bar{\beta}}\nabla_{\partial_{\alpha } }\nabla_{\overline{\partial}_{\beta }} \theta , \theta \rangle \\ \geq & Re \langle [\sqrt{-1}\Lambda_{\omega}[\theta , \theta^{\ast H_{\epsilon }}],\theta ] , \theta \rangle+Re \langle [\epsilon \log (K^{-1}H_{\epsilon}),\theta ] , \theta \rangle\\ &+|\nabla\theta |_{H_{\epsilon}, \omega }^{2}-\check{C}_{1}|\theta |_{H_{\epsilon}, \omega }^{2}-\check{C}_{2}|\nabla\theta |_{H_{\epsilon}, \omega }|\theta |_{H_{\epsilon}, \omega }, \\ \end{split}$$ where we have used that $H_{\epsilon}$ satisfies the perturbed equation (\[eq\]) and $\overline{\partial}_{E}\theta =0$, $\check{C}_{1}$ and $\check{C}_{2}$ are positive constants depending only on the geometry of $(X, \omega )$. Of course (\[eqn:2\]) implies: $$\label{eqn:3} \begin{split} Re \langle [\sqrt{-1}\Lambda_{\omega}[\theta , \theta^{\ast H_{\epsilon }}], \theta ] , \theta \rangle &= |\sqrt{-1}\Lambda_\omega[\theta, \theta^{\ast H_{\epsilon}}]|_{H_{\epsilon}, \omega }^{2}\\ &\geq \frac{a_1^2}{2}|\theta |_{H_{\epsilon }, \omega}^4-a_2^2|\theta |_{K, \omega}^4. \end{split}$$ Combining this and (\[t1\]), (\[t2\]) gives us that $$\label{t3} \sqrt{-1}\Lambda_{\omega }\partial \overline{\partial }|\theta |_{H_{\epsilon}, \omega }^{2} \geq \frac{1}{2}|\nabla\theta |_{H_{\epsilon}, \omega }^{2}+\frac{a_1^2}{2}|\theta |_{H_{\epsilon }, \omega}^4-\check{C}_{3}|\theta |_{H_{\epsilon}, \omega }^{2}-a_2^2|\theta |_{K, \omega}^4,$$ where $\check{C}_{3}$ is a positive constant depending only on $\sup_{X}|\Lambda_{\omega }F_{K, \theta}|_{K}$ and the geometry of $(X, \omega )$. Then the maximum principle means: $$\label{t4} \sup_{X}|\theta |_{H_{\epsilon}, \omega }^{2} \leq \check{C}_{4},$$ where $\check{C}_{4}$ is a positive constant depending only on $\sup_{X}|\theta |_{K, \omega}$, $\sup_{X}|\Lambda_{\omega }F_{K, \theta}|_{K}$ and the geometry of $(X, \omega )$. Flat bundles ------------ Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, $(V,D)$ be a flat bundle over $X$. We call $(V,D)$ is simple if it has no proper flat subbundle, otherwise call it non-simple. We say $(V,D)$ is semi-simple if it is a direct sum of simple flat bundles. For any metric $H$ on $V$, there is a unique decomposition $$D=D_{H}+\psi_{H},$$ where $D_{H}$ is a unitary connection and $\psi_{H}\in \Omega^{1}(\mbox{End}(V))$ is self-adjoint. Denote by $\partial_{H}$ (resp. $\bar{\partial}_{H}$) the $(1,0)$-part (resp. $(0,1)$-part) of $D_{H}$. Define $$D_{H}^{''}=\bar{\partial}_{H}+\psi_{H}^{1,0},\ \ D^{'}_{H}=\partial_{H}+\psi_{H}^{0,1}$$ and $$G_{H}=(D_{H}^{''})^{2}=\bar{\partial}_{H}^{2}+\bar{\partial}_{H}\psi_{H}^{1,0}+\psi_{H}^{1,0}\wedge\psi_{H}^{1,0}.$$ Suppose $G_{H}=0$, then $(V,\bar{\partial}_{H},\psi_{H}^{1,0})$ will be a Higgs bundle and $H$ is a Higgs-Hermitian flat metric. \[lem:x1\] Assume $\bar{\partial}\partial\omega^{n-2}=0$, then $\sqrt{-1}\Lambda_{\omega}G_{H}=0\iff G_{H}=0$. By the flatness of $D$, we have $$\label{dd:1} \left\{\begin{split} &\bar{\partial}_{H}^{2}+\psi_{H}^{0,1}\wedge\psi_{H}^{0,1}=0,\\ &\bar{\partial}_{H}\psi_{H}^{1,0}+\partial_{H}\psi_{H}^{0,1}=0,\\ &\partial_{H}\psi_{H}^{1,0}=0,\\ &\partial_{H}\bar{\partial}_{H}+\bar{\partial}_{H}\partial_{H}+[\psi_{H}^{1,0},\psi_{H}^{0,1}]=0. \end{split}\right.$$ The Riemann bilinear relations assert that $$\int_{X}{\mbox{\rm tr\,}}(G_{H}\wedge G_{H})\wedge\frac{\omega^{n-2}}{(n-2)!} =\int_{X}(|G_{H}|^{2}-|\Lambda_{\omega}G_{H}|^{2})\frac{\omega^{n}}{n!}.$$ On the other hand, by (\[dd:1\]), we can show $$\begin{split} &\int_{X}{\mbox{\rm tr\,}}(G_{H}\wedge G_{H})\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&\int_{X}\{-2{\mbox{\rm tr\,}}(\psi_{H}^{1,0}\wedge\psi_{H}^{1,0}\wedge\psi_{H}^{0,1}\wedge\psi_{H}^{0,1})-{\mbox{\rm tr\,}}(\bar{\partial}_{H}\psi_{H}^{1,0}\wedge\partial_{H}\psi_{H}^{0,1})\}\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&\int_{X}\bar{\partial}\partial {\mbox{\rm tr\,}}(\psi_{H}^{1,0}\wedge\psi_{H}^{0,1})\wedge\frac{\omega^{n-2}}{(n-2)!}\\ =&0. \end{split}$$ A metric $H$ on a flat bundle is called harmonic if it satisfies $\sqrt{-1}\Lambda_{\omega}G_{H}=0$. For any two Hermitian metrics $H$ and $K$, let $h=K^{-1}H$ and $D_{K}^{c}=D_{K}^{''}-D_{K}^{'}$. Then $$D_{H}^{''}-D_{K}^{''}=\frac{1}{2}h^{-1}D_{K}^{c}h$$ and $$\label{dd:2} \begin{split} \sqrt{-1}\Lambda_{\omega}(G_{H}-G_{K})=\frac{1}{4}\sqrt{-1}\Lambda_{\omega}D(h^{-1}D_{K}^{c}h). \end{split}$$ Set $s=\log(h)$, then by a similar discussion with [@NZ] (or [@ZZZ]), we have \[d1:3\] $$\begin{split} \langle\sqrt{-1}\Lambda_{\omega}D(h^{-1}D_{K}^{c}h),s\rangle_{K}=-\langle\Theta(s)(Ds),Ds\rangle_{K}+\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|s|_{K}^{2}, \end{split}$$ where $$\Theta(x,y)= \left\{\begin{split} &\frac{e^{y-x}-1}{y-x}, &x\neq y;\\ &\ \ \ \ 1, &x=y. \end{split}\right.$$ In an open dense subset $U$ of $X$, choosing a suitable orthonormal basis $\{e_{1},\cdots,e_{r}\}$ of $V$ with respect to the metric $K$, we can assume $$h=\sum_{\alpha=1}^{r}e^{\lambda_{\alpha}}e_{\alpha}\otimes e^{\alpha}$$ and $$s=\sum_{\alpha=1}^{r}\lambda_{\alpha}e_{\alpha}\otimes e^{\alpha}.$$ Let $D=d+A$, then $D_{K}=d+\frac{1}{2}(A-\bar{A}^{T})$ and $\psi_{K}=\frac{1}{2}(A+\bar{A}^{T})$. Denote the $(1,0)$-part and $(0,1)$-part of $A$ by $B$ and $C$. Computing directly shows $$\begin{split} h^{-1}D_{K}^{c}h&=\sum_{\alpha=1}^{r}d^{c}\lambda_{\alpha}e_{\alpha}\otimes e^{\alpha}+\sum_{\alpha\neq\beta}(e^{\lambda_{\beta}-\lambda_{\alpha}}-1)(\overline{C_{\alpha}^{\beta}}-\overline{B_{\alpha}^{\beta}})e_{\alpha}\otimes e^{\beta},\\ Ds&=\sum_{\alpha=1}^{r}d\lambda_{\alpha}e_{\alpha}\otimes e^{\alpha}+\sum_{\alpha\neq\beta}(\lambda_{\beta}-\lambda_{\alpha})(B_{\beta}^{\alpha}+C_{\beta}^{\alpha})e_{\alpha}\otimes e^{\beta}. \end{split}$$ Then $$\begin{split} \sqrt{-1}\Lambda_{\omega}d{\mbox{\rm tr\,}}(h^{-1}D_{K}^{c}hs)=&\sum_{\alpha=1}^{r}\sqrt{-1}\Lambda_{\omega}d(\lambda_{\alpha}d^{c}\lambda_{\alpha})\\ =&\sum_{\alpha=1}^{r}2\sqrt{-1}\Lambda_{\omega}\lambda_{\alpha}\partial\bar{\partial}\lambda_{\alpha}+\sum_{\alpha=1}^{r}2|\partial\lambda_{\alpha}|^{2}\\ =&\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|s|_{K}^{2} \end{split}$$ and $$\begin{split} &\sqrt{-1}\Lambda_{\omega}{\mbox{\rm tr\,}}(h^{-1}D_{K}^{c}h\wedge Ds)\\ =&\sum_{\alpha=1}^{r}\sqrt{-1}\Lambda_{\omega}(d^{c}\lambda_{\alpha}\wedge d\lambda_{\alpha})\\ &+\sum_{\alpha\neq\beta}(e^{\lambda_{\beta}-\lambda_{\alpha}}-1)(\lambda_{\alpha}-\lambda_{\beta})\sqrt{-1}\Lambda_{\omega}(\overline{C_{\alpha}^{\beta}}-\overline{B_{\alpha}^{\beta}})\wedge(B^{\beta}_{\alpha}+C_{\alpha}^{\beta})\\ =&-\sum_{\alpha=1}^{r}|d\lambda_{\alpha}|^{2}-\sum_{\alpha\neq\beta}\frac{(e^{\lambda_{\beta}-\lambda_{\alpha}}-1)}{(\lambda_{\beta}-\lambda_{\alpha})}(\lambda_{\beta}-\lambda_{\alpha})^{2}(|B_{\alpha}^{\beta}|^{2}+|C_{\alpha}^{\beta}|^{2})\\ =&-\sum_{\alpha,\beta}\Theta(\lambda_{\alpha},\lambda_{\beta})|(Ds)_{\alpha}^{\beta}|^{2}. \end{split}$$ Combining all of the above, we complete this proof. \[pp:0\] $$\sqrt{-1}\Lambda_{\omega}dd^{c}\log({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))\geq -4(|\sqrt{-1}\Lambda_{\omega}G_{H}|_{H}+|\sqrt{-1}\Lambda_{\omega}G_{K}|_{K})$$ and $\sup_{X}|\log(h)|_{K}\leq \grave{C}_{1}\|\log(h)\|_{L^{2}(X,K)}+\grave{C}_{2}$. Clearly (\[dd:2\]) yields $$\begin{split} h(\sqrt{-1}\Lambda_{\omega}(G_{H}-G_{K}))=&\frac{1}{4}\sqrt{-1}\Lambda_{\omega}DD^{c}_{K}h-\frac{1}{4}\sqrt{-1}\Lambda_{\omega}D(h)h^{-1}D_{K}^{c}(h). \end{split}$$ Taking the trace on both sides, we see $$\begin{split} &\frac{1}{4}\sqrt{-1}\Lambda_{\omega}dd^{c}{\mbox{\rm tr\,}}(h)-\frac{1}{4}|Dh\cdot h^{-1/2}|_{K}^{2}\\ \geq &-{\mbox{\rm tr\,}}(h)(|\sqrt{-1}\Lambda_{\omega}G_{H}|_{H}+|\sqrt{-1}\Lambda_{\omega}G_{K}|_{K}). \end{split}$$ On the other hand, it holds that $$\begin{split} &({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))\sqrt{-1}\Lambda_{\omega}dd^{c}\log({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))\\ =&-\frac{|{\mbox{\rm tr\,}}(Dh)+{\mbox{\rm tr\,}}(Dh^{-1})|^{2}}{{\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1})}+\sqrt{-1}\Lambda_{\omega}dd^{c}({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1})). \end{split}$$ According to Young’s inequality, we have $$\begin{split} \frac{|{\mbox{\rm tr\,}}(Dh)+{\mbox{\rm tr\,}}(Dh^{-1})|^{2}}{{\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1})}\leq &\frac{(1+\varepsilon)|{\mbox{\rm tr\,}}(Dh)|^{2}+(1+\frac{1}{\varepsilon})|{\mbox{\rm tr\,}}(Dh^{-1})|^{2}}{{\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1})}. \end{split}$$ Take $\varepsilon={\mbox{\rm tr\,}}(h^{-1})/{\mbox{\rm tr\,}}(h)$, then $$\begin{split} \frac{|{\mbox{\rm tr\,}}(Dh)+{\mbox{\rm tr\,}}(Dh^{-1})|^{2}}{{\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1})}\leq&|Dh\cdot h^{-1/2}|_{K}^{2}+|Dh^{-1}\cdot h^{1/2}|_{H}^{2} . \end{split}$$ So $$\sqrt{-1}\Lambda_{\omega}dd^{c}\log({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))\geq -4(|\sqrt{-1}\Lambda_{\omega}G_{H}|_{H}+|\sqrt{-1}\Lambda_{\omega}G_{K}|_{K}).$$ Noting $$\log\big(\frac{1}{2r}({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))\big)\leq |\log(h)|_{K}\leq r^{1/2}\log({\mbox{\rm tr\,}}(h)+{\mbox{\rm tr\,}}(h^{-1}))$$ and applying Moser’s iteration, we finish the proof. For a Higgs bundle $(E,\bar{\partial}_{E},\theta)$ and a Hermitian metric $H$ on $E$, set $D_{E}^{''}=\bar{\partial}_{E}+\theta$ and $D_{H}^{'}=\partial_{H}+\theta^{*H}$, sometimes we omit the subscript for simplicity. Then $(D_{E}^{''})^{2}=0$ and $(D_{H}^{'})^{2}=0$. The Dolbeault cohomology groups are defined by $$H^{i}_{Dol}(E)=\frac{\mbox{Ker} (D_{E}^{''}:\mathcal{A}^{i}(E)\rightarrow\mathcal{A}^{i+1}(E))}{\mbox{Im} (D_{E}^{''}:\mathcal{A}^{i-1}(E)\rightarrow\mathcal{A}^{i}(E))}.$$ For a flat bundle $(V,D)$, the de Rham cohomology groups are defined by $$H^{i}_{DR}(E)=\frac{\mbox{Ker} (D:\mathcal{A}^{i}(E)\rightarrow\mathcal{A}^{i+1}(E))}{\mbox{Im} (D:\mathcal{A}^{i-1}(E)\rightarrow\mathcal{A}^{i}(E))}.$$ \[lem:l1\] Suppose $(E,\bar{\partial}_{E},\theta, H)$ is a Higgs bundle over $X$ with $F_{H,\theta}=(D_{H,\theta})^{2}=0$. If $\partial\bar{\partial}\omega^{n-1}=0$, then $H_{DR}^{0}(X,E)= H_{Dol}^{0}(X,E)$. For any $e\in H^{0}_{DR}(X,E)$, we have $$\label{x:2} \sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|e|_{H}^{2}=|D_{E}^{''}e|_{H}^{2}+|D^{'}_{H}e|_{H}^{2}.$$ Integrating both sides of this equation derives $e\in H^{0}_{Dol}(X,E)$. Similarly, if $e\in H^{0}_{Dol}(X,E)$, we get $$\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|e|_{H}^{2}=|\partial_{H}e|_{H}^{2}+|\theta^{*}(e)|_{H}^{2}.$$ Hence $e\in H_{DR}^{0}(X,E)$. Proof of Theorem \[thm:2-1\] ============================ We first study the existence of harmonic Hermitian metric by using the continuous method. Let $(V,D)$ be a flat bundle over $X$ and $H$ be a Hermitian metric on $V$. Set $$\mbox{Herm}(V,H)=\{\eta\in\mbox{End}(V)|\ \eta^{*H}=\eta\}$$ and $$\mbox{Herm}^{+}(V,H)=\{\rho\in Herm(V,H)|\ \rho\ is\ \mbox{positive}\ \mbox{definite}\}.$$ Assume that $(X,\omega)$ satisfies $\partial\bar{\partial}\omega^{n-1}=0$ and $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any $[\eta]\in H^{0,1}(X)$. Let $K$ be a fixed background metric on $V$. Consider the following perturbed equation $$\label{dd:3} L_{\epsilon}(h)=4\sqrt{-1}\Lambda_{\omega}G_{K}+\sqrt{-1}\Lambda_{\omega}D(h^{-1}D_{K}^{c}h)-\epsilon\log(h)=0,$$ where $h\in Herm^{+}(V,K)$ and $\epsilon\in(0,1]$. By choosing a suitable background metric, we can assume ${\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K})=0$ and the equation (\[dd:3\]) has a solution for $\epsilon=1$. Indeed, let $\varphi$ be a function satisfying $$\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}\varphi=-\frac{2}{r}{\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K}),$$ then ${\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{e^{\varphi}K})=0$. Define $h_{1}=\exp(4\sqrt{-1}\Lambda_{\omega}G_{e^{\varphi}K})$ and choose $\tilde{K}=e^{\varphi}Kh_{1}^{-1}$ as the fixed background metric, then $$\begin{split} {\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{\tilde{K}})&={\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{e^{\varphi}K})+\frac{1}{2}\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}\log\det(h_{1}^{-1})\\ &={\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{e^{\varphi}K})-2\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}{\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{e^{\varphi}K})\\ &=0 \end{split}$$ and $$4\sqrt{-1}\Lambda_{\omega}G_{\tilde{K}}+\sqrt{-1}\Lambda_{\omega}D(h_{1}^{-1}D_{\tilde{K}}^{c}h_{1})-\log(h_{1})=0.$$ Let ${\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K})=0$. If $h$ is a solution of equation (\[dd:3\]) for $\epsilon\in(0,1]$. Then $\det(h)=1$. Taking the trace on both sides of equation (\[dd:3\]) yields $$2\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}\log\det(h)-\epsilon\log\det(h)=0.$$ Due to the maximum principle, we have $\log\det(h)=0$ and $\det(h)=1$. For any $0<\delta<1$, define $$J_{\delta}=\{\epsilon\in[\delta,1]|\ \text{there is a smooth solution to}\ L_{\epsilon}(h)=0 \}$$ and $$J=\{\epsilon\in(0,1]|\ \text{there is a smooth solution to}\ L_{\epsilon}(h)=0 \}.$$ We will use the continuity method to show that $J_{\delta}=[\delta,1]$ for any $0<\delta<1$, hence $J=(0,1]$. It is obvious that $J_{\delta}\neq\emptyset$. By the Fredholmness of an elliptic operator over a compact manifold and the implicit function theorem, we know that $J_{\delta}$ is open. Next, we conclude that $J_{\delta}$ is closed. \[pp:1\] Let $h$ be the solution of $L_{\epsilon}(h)=0$ for $\epsilon\in[\delta,1]$. Then $$\begin{split} \sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|\log(h)|_{K}^{2}\geq \epsilon|\log(h)|_{K}^{2}-4|\sqrt{-1}\Lambda_{\omega}G_{K}|_{K}|\log(h)|_{K}, \end{split}$$ and $\sup_{X}|\log(h)|_{K}\leq\frac{4}{\epsilon}\sup_{X}|\sqrt{-1}\Lambda_{\omega}G_{K}|_{K}\leq C_{12}$, where $C_{12}$ is a constant depending only on $\delta$. This lemma comes from Proposition \[d1:3\] and the maximum principle. Using the above $C^{0}$-estimate, we can derive the $C^{1}$-estimate and $L_{2}^{p}$-estimate by Donaldson’s arguments in [@Don2 Lemma 19]. But we will give another proof by the maximum principle. \[pp:2\] Let $h$ be the solution of $L_{\epsilon}(h)=0$ for $\epsilon\in[\delta,1]$. Then $$\sup_{X}|\psi_{H}^{1,0}|_{H}\leq C_{13},$$ where $C_{13}$ is constant depending only on the bound of $\sup_{X}|\log{h}|_{K}$. After a straightforward calculation, we can show $$\begin{split} \sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|\psi_{H}^{1,0}|_{H}^{2}\geq &|\nabla \psi_{H}^{1,0}|_{H}^{2}+|[\psi_{H}^{1,0},\psi_{H}^{0,1}]|_{H}^{2}+2|\psi_{H}^{1,0}\wedge\psi_{H}^{1,0}|_{H}^{2}\\ &-2Re\langle\partial_{H}(\sqrt{-1}\Lambda_{\omega}G_{H}),\psi_{H}^{1,0}\rangle_{H}-\breve{C}|\nabla\psi_{H}^{1,0}|_{H}|\psi_{H}^{1,0}|_{H}-|R||\psi_{H}^{1,0}|_{H}^{2}, \end{split}$$ where $\breve{C}$ is a constant depending only on $d\omega$ and the torsion of Chern connection, $R$ is the curvature of Chern connection. According to the equation (\[dd:3\]), we obtain $$\begin{split} \sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|\psi_{H}^{1,0}|_{H}^{2}\geq &-\breve{C}_{1}|\psi_{H}^{1,0}|_{H}^{2}-\frac{\epsilon}{2}|\partial_{H}(\log(h))|_{H}|\psi_{H}^{1,0}|_{H}. \end{split}$$ Notice that $$|\partial_{H}(\log(h))|_{H}\leq \breve{C}_{2}|h^{-1}\partial_{H}h|_{H}\leq \breve{C}_{3}|\psi_{H}^{1,0}|_{H}+\breve{C}_{4},$$ where $\breve{C}_{2},\breve{C}_{3}$ and $\breve{C}_{4}$ are the constants depending only on $C_{12}$. So $$\begin{split} \sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}|\psi_{H}^{1,0}|_{H}^{2}\geq &-\breve{C}_{5}|\psi_{H}^{1,0}|_{H}^{2}-\breve{C}_{6}|\psi_{H}^{1,0}|_{H}, \end{split}$$ where $\breve{C}_{5}$ and $\breve{C}_{6}$ are the constants depending only on $C_{12}$. On the other hand, we have $$\begin{split} \sqrt{-1}\Lambda_{\omega}dd^{c}{\mbox{\rm tr\,}}(h)=&\sqrt{-1}\Lambda_{\omega}{\mbox{\rm tr\,}}(Dh\wedge h^{-1}D_{K}^{c}h)+\sqrt{-1}\Lambda_{\omega}{\mbox{\rm tr\,}}(hD(h^{-1}D_{K}^{c}h))\\ =&|Dh\cdot h^{-1/2}|_{K}^{2}-4{\mbox{\rm tr\,}}(h\sqrt{-1}\Lambda_{\omega}G_{K})+\epsilon {\mbox{\rm tr\,}}(h\log(h)), \end{split}$$ and $$|Dh\cdot h^{-1/2}|_{K}^{2}\geq \breve{C}_{7}|Dh\cdot h^{-1}|_{H}^{2}\geq \breve{C}_{8}|\psi_{H}^{1,0}|_{H}^{2}-\breve{C}_{9}.$$ Hence it follows that $$\begin{split} \sqrt{-1}\Lambda_{\omega}dd^{c}{\mbox{\rm tr\,}}(h)\geq &\breve{C}_{8}|\psi_{H}^{1,0}|_{H}^{2}-\breve{C}_{10}. \end{split}$$ Let $f=|\psi_{H}^{1,0}|_{H}^{2}+A{\mbox{\rm tr\,}}(h)$, where $A$ is large enough such that $$\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}f\geq \breve{C}_{11}|\psi_{H}^{1,0}|_{H}^{2}-\breve{C}_{12},$$ where $\breve{C}_{11}$ and $\breve{C}_{12}$ are positive constants depending only on $C_{12}$. Assume $f(p)=\max\limits_{X}f$, then $$0\geq\breve{C}_{11}|\psi_{H}^{1,0}|_{H}^{2}(p)-\breve{C}_{12}.$$ Therefore, one can see that $$\max_{X}|\psi_{H}^{1,0}|^{2}_{H}\leq C_{13},$$ where $C_{13}$ is a constant depending only on $C_{12}$. Combining Lemma \[pp:1\], Lemma \[pp:2\] and the $L_{2}^{p}$-estimate of elliptic operator, we immediately know $h_{\epsilon}$ is uniformly bounded in $L^{p}_{2}$ for $\epsilon\in[\delta,1]$. Thus $J_{\delta}$ is closed for $[\delta,1]$ and $J=(0,1]$. \[lem:1\] Let $h_{\epsilon}$ be a solution of equation (\[dd:3\]), $s_{\epsilon}=\log(h_{\epsilon})$. Then $$\begin{split} &-\int_{X}4{\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K}s_{\epsilon})\frac{\omega^{n}}{n!}+\int_{X}\langle\Theta(s_{\epsilon})(Ds_{\epsilon}),Ds_{\epsilon}\rangle_{K}\frac{\omega^{n}}{n!}+\epsilon\|s_{\epsilon}\|_{L^{2}}^{2}=0. \end{split}$$ This follows directly from Proposition \[d1:3\]. \[thm:x1\] Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=0$ and $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$. Let $(V,D)$ be a simple flat bundle. Then there exists a harmonic metric $H$ on $V$. Let $h_{\epsilon}$ be the solution of equation (\[dd:3\]) for any $\epsilon\in(0,1]$. Suppose $(V,D)$ is simple, we conclude that there is a constant $\breve{C}_{13}$ independent of $\epsilon$ satisfying $\|\log(h_{\epsilon})\|_{L^{2}}<\breve{C}_{13}$. Then Proposition \[pp:0\] implies that $\|\log(h_{\epsilon})\|_{L^{\infty}}$ are uniformly bounded. By a similar argument with Lemma \[pp:2\], we can get the uniform $C^{1}$-estimate and also the uniform $L_{2}^{p}$-estimate. After choosing a subsequence of $\epsilon$, we can obtain a limiting metric satisfying $\sqrt{-1}\Lambda_{\omega}G_{H}=0$. Suppose $\overline{\lim\limits_{\epsilon\rightarrow 0}}\|\log(h_{\epsilon})\|_{L^{2}}=+\infty$. Then there is a sequence $\epsilon_{j}\rightarrow 0$, such that $$\|\log(h_{\epsilon_{j}})\|_{L^{2}} \rightarrow +\infty.$$ Set $$s_{j}=\log(h_{\epsilon_{j}}),\ \ l_{j}=\|s_{j}\|_{L^{2}},\ \ u_{j}=l_{j}^{-1}s_{j},$$ then $${\mbox{\rm tr\,}}(u_{j})=0, \ \ \|u_{j}\|_{L^{2}}=1,\ \ \|u_{j}\|_{L^{\infty}}\leq C^{*}.$$ Based on Lemma \[lem:1\], we have $$-\int_{X}4{\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K}u_{j})\frac{\omega^{n}}{n!}+l_{j}\int_{X}\langle\Theta(l_{j}u_{j})(Du_{j}),Du_{j}\rangle_{K}\frac{\omega^{n}}{n!}+\epsilon_{j}l_{j}=0.$$ According to the argument in [@NZ page 637], if $\Upsilon:R\times R \rightarrow R$ is a positive smooth function such that $\Upsilon(\lambda_{1},\lambda_{2})<(\lambda_{1}-\lambda_{2})^{-1}$ whenever $\lambda_{1}>\lambda_{2}$, it holds that $$\begin{split} &-\int_{X}4{\mbox{\rm tr\,}}(\sqrt{-1}\Lambda_{\omega}G_{K}u_{j})\frac{\omega^{n}}{n!}+\int_{X}\langle\Upsilon(u_{j})(Du_{j}),Du_{j}\rangle_{K}\frac{\omega^{n}}{n!}+\epsilon_{j}l_{j}\leq 0,\ \ j\gg 0. \end{split}$$ Then we know $u_{j}$ is uniformly bounded in $L_{1}^{2}$. By choosing a subsequence which is also denoted by $u_{j}$, we deduce $u_{j}\rightarrow u_{\infty}$ weakly in $L_{1}^{2}$ as $j\rightarrow +\infty$. Then $${\mbox{\rm tr\,}}(u_{\infty})=0, \ \ \|u_{\infty}\|_{L^{2}}=1,\ \ \|u_{\infty}\|_{L^{\infty}}\leq C^{*}.$$ Following Simpson’s argument in [@S1 Lemma 5.5], we know the eigenvalues of $u_{\infty}$ are constants almost everywhere. Let $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{l}$ denote the distinct eigenvalues. Since ${\mbox{\rm tr\,}}(u_{\infty})=0$ and $\|u_{\infty}\|_{L^{2}}=1$, we have $l\geq 2$. Define $$p_{\alpha}=\left\{\begin{split} 1,\quad & \text{$x\leq \lambda_{\alpha}$}, \\ 0,\quad & \text{$x\geq\lambda_{\alpha+1}$}. \end{split}\right.$$ Let $\pi_{\alpha}=p_{\alpha}(u_{\infty})$. From [@S1 Lemma 5.6], one can see that - $\pi_{\alpha}\in L_{1}^{2}$; - $\pi_{\alpha}^{2}=\pi_{\alpha}^{*}=\pi_{\alpha}$; - $(Id-\pi_{\alpha})D\pi_{\alpha}=0$. Set $V_{\alpha}=\pi_{\alpha}(V)$. Since $(Id-\pi_{\alpha})D^{0,1}\pi_{\alpha}=0$, the Uhlenbeck-Yau’s regularity theorem of $L_{1}^{2}$-subbundle [@UhYau] states that $V_{\alpha}$ is smooth outside an analytic subset $\Sigma$ which is of co-dimension at least $2$. Then $V_{\alpha}$ is a flat subbundle of $V$ on $X\setminus\Sigma$. Since $X\setminus \Sigma$ is connected, we can extend $V_{\alpha}$ to the whole $X$ as a flat subbundle of $V$, which is contradict with the simpleness of $V$. Suppose $(E,\bar{\partial}_{E},\theta)$ is a polystable Higgs bundle with vanishing Chern numbers, then there is a Hermitian metric $H$ on $E$, such that $D_{H,\theta}$ is flat. Conversely, assume $(V,D)$ is a semi-simple flat bundle. By Theorem \[thm:x1\] and Lemme \[lem:x1\], there is a metric $H$ such that $G_{H}=0$. So $(V,\bar{\partial}_{H},\psi_{H}^{1,0})$ is a polystable Higgs bundle with vanishing Chern numbers. Let $(E,\bar{\partial}_{E},\theta, D, H)$ be a vector bundle with a Higgs structure and a flat structure, $H$ be a Higgs-Hermitian flat metric. From Lemma \[lem:l1\], we know $H^{0}_{DR}(X,E)=H^{0}_{Dol}(X,E)$. Hence there is an equivalence of categories between the category of poly-stable Higgs bundles with vanishing Chern numbers and the category of semi-simple flat bundles. Modified Yang-Mills-Higgs flow ============================== Suppose $(X, g)$ is a compact Hermitian manifold of dimension $n$ with the associated Kähler form $\omega$ satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$. Let $(E,H_{0})$ be a Hermitian vector bundle over $X$, ${\mathcal A}_{H_{0}}$ be the space of connections on $E$ compatible with the metric $H_{0}$, and ${\mathcal A}^{1,1}_{H_{0}}$ be the space of unitary integrable connections on $E$. Set $$\mathcal{B}_{E,H_{0}}=\{(A,\phi)\in\mathcal{A}^{1,1}_{0}\times\Omega^{1,0}(End(E))| \bar{\partial}_{A}\phi=0,\phi\wedge\phi=0\}.$$ The element $(A,\phi)\in\mathcal{B}_{E,H_{0}}$ is called a Higgs pair which determines a Higgs bundle $(E, D^{0,1}_{A}, \phi )$. Define the Yang-Mills-Higgs functional on the space of Higgs pairs $\mathcal{B}_{E,H_{0}}$ by $$YMH(A,\phi)=\int_{X}|F_{A}+[\phi,\phi^{*H_{0}}]|_{H_{0},\omega}^{2}+2|\partial_{A}\phi|_{H_{0},\omega}^{2}dv_{g},$$ where $dv_g$ is the volume form. Consider the following modified Yang-Mills-Higgs flow (it is the gradient flow of Yang-Mills-Higgs functional in Kähler manifolds case): $$\label{mymh} \left\{ \begin{split} \frac{\partial A(t)}{\partial t}&=-\sqrt{-1}(\partial_{A(t)}-\bar{\partial}_{A(t)})\Lambda_{\omega}(F_{A(t)}+[\phi(t),\phi^{*H_{0}}(t)]), \\ \frac{\partial\phi(t)}{\partial t}&=-[\sqrt{-1}\Lambda_{\omega}(F_{A(t)}+[\phi(t),\phi^{*H_{0}}(t)]),\phi(t)], \\ A(0)&=A_{0},\ \ \phi(0)=\phi_{0}. \end{split}\right.$$ As that in [@LZ], we can obtain the long time existence and uniqueness of solution to the modified Yang-Mills-Higgs flow. Indeed, let $H(t)$ be the solution of the following Hermitian-Yang-Mills flow on the Higgs bundle $(E,\bar{\partial}_{A_{0}},\phi_{0})$ $$\label{HYMF} \left\{ \begin{split} &H^{-1}(t)\frac{\partial H(t)}{\partial t}=-2(\sqrt{-1}\Lambda_{\omega}(F_{H(t)}+[\phi_{0},\phi_{0}^{*H(t)}])-\lambda Id), \\ &H(0)=H_{0}, \end{split} \right.$$ where $\lambda=\frac{2\pi}{\mbox{Vol}(X,\omega)}\mu_{\omega}(E,\bar{\partial}_{A_{0}}).$ There is a family of complex gauge transformations $\sigma(t)$ satisfying $\sigma^{*H_{0}}(t)\sigma(t)=H_{0}^{-1}H(t)$, such that $(A(t),\phi(t))=\sigma(t)(A_{0},\phi_{0})$ is a solution of the modified Yang-Mills-Higgs flow. For simplicity, we denote $(A(t),\phi(t))$, $H(t)$ and $\sigma(t)$ by $(A,\phi)$, $H$ and $\sigma$. It is easy to check that $$\begin{split} \bar{\partial}_{A}\phi^{*H_{0}}&=\sigma\circ\bar{\partial}_{A_{0}}\phi_{0}^{*H}\circ \sigma^{-1},\\ \partial_{A}\phi&=\sigma\circ\partial_{A_{0}}\phi_{0}\circ \sigma^{-1},\\ F_{A}+[\phi,\phi^{*H_{0}}]&=\sigma\circ(F_{H}+[\phi_{0},\phi_{0}^{*H}])\circ \sigma^{-1},\\ D_{A,\phi}(\Lambda_{\omega}(F_{A}+[\phi,\phi^{*H_{0}}]))&=\sigma\circ(D_{H,\phi_{0}}(\Lambda_{\omega}(F_{H}+[\phi_{0},\phi_{0}^{*H}])))\circ \sigma^{-1}. \end{split}$$ Along the flow, we have $$\label{re1} \begin{split} &YMH(A(t),\phi(t))\\ =&\int_{X}|F_{A(t)}+[\phi(t),\phi^{*H_{0}}(t)]|_{H_{0}}^{2}+2|\partial_{A(t)}\phi(t)|_{H_{0}}^{2}dv_{g}\\ =&\int_{X}|F_{H(t)}+[\phi_{0},\phi_{0}^{*H(t)}]|_{H(t)}^{2}+2|\partial_{A_{0}}\phi_{0}|_{H(t)}^{2}dv_{g}\\ =&\int_{X}|\Lambda_{\omega}(F_{H(t)}+[\phi_{0},\phi_{0}^{*H(t)}])|_{H(t)}^{2}dv_{g}-8\pi^{2}ch_{2}(E,\bar{\partial}_{A_{0}})[\omega^{n-2}]\\ =&\int_{X}|\sqrt{-1}\Lambda_{\omega}(F_{H(t)}+[\phi_{0},\phi_{0}^{*H(t)}])-\lambda Id_{E}|_{H(t)}^{2}dv_{g}\\ &+\lambda^{2}\mbox{rank}(E)\mbox{Vol}(X,\omega)-8\pi^{2}ch_{2}(E,\bar{\partial}_{A_{0}})[\omega^{n-2}]\\ =&\int_{X}|\sqrt{-1}\Lambda_{\omega}(F_{A(t)}+[\phi(t),\phi^{*H_{0}}(t)])-\lambda Id_{E}|_{H_{0}}^{2}dv_{g}\\ &+\lambda^{2}\mbox{rank}(E)\mbox{Vol}(X,\omega)-8\pi^{2}ch_{2}(E,\bar{\partial}_{A_{0}})[\omega^{n-2}]. \end{split}$$ Furthermore, if $H(t)$ is a solution of the Hermitian-Yang-Mills flow, by the same discussion as that in [@S1 Lemma 6.1], we get ([@Jac1 Lemma 1] or [@ZZZ Proposition 2.1]) $$\label{mc1} (\frac{\partial}{\partial t}-2\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial})|\Psi (t)|_{H(t)}^{2}=-2|D_{H(t),\phi_{0}}\Psi (t)|_{H(t)}^{2},$$ where we have set $\Psi (t)=\sqrt{-1}\Lambda_{\omega}(F_{H(t)}+[\phi_{0},\phi_{0}^{*H(t)}])-\lambda Id$ for simplicity. Based on (\[re1\]) and (\[mc1\]), we establish the following proposition. \[p\] Let $(A,\phi)$ be a solution of the modified Yang-Mills-Higgs flow (\[mymh\]), then $ \sup_{X}|\sqrt{-1}\Lambda_{\omega}(F_{A}+[\phi,\phi^{*H_{0}}])-\lambda Id|_{H_{0}} $ is uniformly bounded and $YMH(t)$ is decreasing along the flow. It is well known that there are two connections on the tangent bundle $TX$: Chern connection and Levi-Civita connection. They are coincide on Kähler manifolds, but they are different in non-Kähler manifolds case. In the rest of this section, we use $\nabla_{A}$ to denote the Chern connection. Notice that for a smooth function $f$ on a non-Kähler manifold, there holds $$\Delta f=2\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial} f+\langle df,V\rangle,$$ where $V=*(\sqrt{-1}(\bar{\partial}-\partial)\omega^{n-1})$ and $\Delta $ is the Beltrami-Laplacian. As that in [@LZ Section 2.2], the following result comes from a direct computation. \[p:1\] Let $(A,\phi)$ be a solution of the modified Yang-Mills-Higgs flow, then $$(2\sqrt{-1}\Lambda_{\omega}\partial\bar{\partial}-\frac{\partial}{\partial t})|\phi|^{2}\geq2|\nabla_{A}\phi|^{2}+C_{1}(|\phi|^{2}+1)^{2}-C_{2}(|\phi|^{2}+1)-C_{3}|\nabla_{A}\phi||\phi|,$$ where the constants $C_{1},C_{2},C_{3}$ depend only on $\sup_{X}|\phi_{0}|_{H_{0}}$ and the geometry of $(X,\omega)$. So we get $\sup_{X}|\phi|^{2}\leq \max\{\sup_{X}|\phi_{0}|^{2},C_{2}/C_{1}+C_{3}^{2}/4C_{1}\}$, and $$\begin{split} &(\Delta-\frac{\partial}{\partial t})|\nabla_{A}\phi|^{2}-2|\nabla_{A}\nabla_{A}\phi|^{2}\\ \geq&-C_{4}(|F_{A}|+|Rm|+|Ric|+|\phi|^{2})|\nabla_{A}\phi|^{2}\\ &-C_{5}(|\nabla Ric|+|d\omega||Rm|+|d\omega||Ric|+|d\omega||F_{A}|)|\phi||\nabla_{A}\phi|\\ &-C_{6}|d\omega||\nabla_{A}\phi||\nabla_{A}\nabla_{A}\phi|, \end{split}$$ where the constants $C_{4},C_{5},C_{6}$ depend only on the dimension $n$, and $$\begin{split} &(\Delta-\frac{\partial}{\partial t})(|F_{A}+[\phi,\phi^{*}]|^{2}+2|\partial_{A}\phi|^{2})-2|\nabla_{A}(F_{A}+[\phi,\phi^{*}])|^{2}-4|\nabla_{A}(\partial_{A}\phi)|^{2}\\ \geq&-C_{7}(|F_{A}+[\phi,\phi^{*}]|+|\nabla_{A}\phi|+|\phi|^{2}+|Rm|+|d\omega|+|d\omega|^{2})(|F_{A}+[\phi,\phi^{*}]|^{2} +2|\partial_{A}\phi|^{2})\\ &-C_{8}|d\omega|(|F_{A}+[\phi,\phi^{*}]|+2|\partial_{A}\phi|)(|\nabla_{A}(F_{A}+[\phi,\phi^{*}])|+2|\nabla_{A}(\partial_{A}\phi)|), \end{split}$$ where the constants $C_{7},C_{8}$ depend only on the dimension $n$. Monotonicity inequality and $\epsilon$-regularity ------------------------------------------------- Regard $X$ as a compact Riemannian manifold of dimension $2n$. For any $x_{0}\in X$, choose normal geodesic coordinates $\{x^{i}\}_{i=1}^{2n}$ in the geodesic ball $B_{r}(x_{0})$($r\leq i_X$, where $i_X$ is the injective radius of $X$) such that $x_{0}=(0,\cdots,0)$. Then it follows that $$\label{a} |g_{ij}(x)-\delta_{ij}|\leq C|x|^{2},\ \ |\partial_{k}g_{ij}(x)|\leq C|x|,\ \ \forall x\in B_{r}(x_0),$$ where $C$ is a positive constant depending only on $x_{0}$. Let $u=(x,t)\in X\times\mathbb{R}$. For any $u_{0}=(x_{0},t_{0})\in X\times\mathbb{R}^{+}$, set $$\begin{split} &S_{r}(u_{0})=X\times\{t=t_{0}-r^{2}\},\\ &T_{r}(u_{0})=X\times [t_{0}-4r^{2},t_{0}-r^{2}],\\ &P_{r}(u_{0})=B_{r}(x_{0})\times [t_{0}-r^{2},t_{0}+r^{2}]. \end{split}$$ For simplicity, we denote $S_{r}(0,0),T_{r}(0,0),P_{r}(0,0)$ by $S_{r},T_{r},P_{r}$. \[p:2\] Let $(A,\phi)$ be a solution of the modified Yang-Mills-Higgs flow with the initial value $(A_{0},\phi_{0})$. For any $u_{0}=(x_{0},t_{0})\in X\times\mathbb{R}^{+}$, $r\in(0,\min\{\sqrt{t_{0}},i_{X}/2\})$, we have $$\int_{P_{r}(u_{0})}|\nabla_{A}\phi|^{2}dv_{g}dt\leq C_{9}r^{2n},$$ where the constant $C_{9}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Choose $f\in C^{\infty}_{0}(B_{2r}(x_{0}))$ satisfying $0\leq f\leq 1$, $f\equiv 1$ on $B_{r}(x_{0})$, $|d f|\leq 4/r$ and $|\Delta f|\leq 16/r^{2}$ on $B_{2r}(x_{0})\setminus B_{r}(x_{0})$. According to Proposition \[p:1\], we obtain $$\begin{split} &(\Delta-\frac{\partial}{\partial t})f^{2}|\phi|^{2}\\ =&2|df|^{2}|\phi|^{2}+2f|\phi|^{2}\Delta f+4f\langle d f,d|\phi|^{2}\rangle+f^{2}(\Delta-\frac{\partial}{\partial t})|\phi|^{2}\\ \geq&2|df|^{2}|\phi|^{2}+2f|\phi|^{2}\Delta f-8f|df||\phi||\nabla_{A}\phi|+2f^{2}|\nabla_{A}\phi|^{2}\\ &-C_{2}f^{2}(|\phi|^{2}+1)-C_{3}f^{2}|\phi||\nabla_{A}\phi|-f^{2}|V||\phi||\nabla_{A}\phi|\\ \geq& f^{2}|\nabla_{A} \phi|^{2}-\bar{C}_{1}-\bar{C}_{2}(|df|^{2}+|\Delta f|), \end{split}$$ where the constants $\bar{C}_{1},\bar{C}_{2}$ depend only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. One can conclude this proposition by integrating both sides of the inequality over $X\times[t_{0}-r^{2},t_{0}+r^{2}]$. The fundamental solution of (backward) heat equation with singularity at $u_{0}=(x_{0}, t_{0})$ is $$G_{u_{0}}(x,t)=G_{(x_{0},t_{0})}(x,t)=\frac{1}{(4\pi(t_{0}-t))^{2n}}\exp\Big(-\frac{|x-x_{0}|^{2}}{4(t_{0}-t)}\Big),\ \ t\leq t_{0}.$$ For simplicity, denote $G_{(0,0)}(x,t)$ by $G(x,t)$. Given $0<R\leq i_{X}$, take $f\in\ C^{\infty}_{0}(B_{R})$ satisfying $0\leq f\leq 1$, $f\equiv 1$ on $B_{R/2}$ and $|\nabla f|\leq 2/R$ on $B_{R}\setminus B_{R/2}$. Let $(A,\phi)$ be a solution of the modified Yang-Mills-Higgs flow with initial value $(A_{0},\phi_{0})$ and $A^{'}=A+\phi+\phi^{*}$. For any $(x,t)\in X\times [0,+\infty)$, set $$e(A,\phi)(x,t)=|F_{A^{'}}|^{2}$$ and $$\Phi(r)=r^{2}\int_{T_{r}(u_{0})}e(A,\phi)f^{2}G_{u_{0}}dv_{g}dt.$$ \[thm:4\] Let $(A,\phi)$ be a solution of the modified Yang-Mills-Higgs flow. For any $u_{0}=(x_{0},t_{0})\in X\times [0,T]$ and $0<r_{1}\leq r_{2}\leq \min\{R/2,\sqrt{t_{0}}/2\}$, we have $$\begin{split} \Phi(r_{1})\leq &C_{10}\exp(C_{10}(r_{2}-r_{1}))\Phi(r_{2})+C_{10}(r_{2}^{2}-r_{1}^{2})YMH(A_{0},\phi_{0})\\ &+C_{10}R^{2-2n}\int_{P_{R}(u_{0})}e(A,\phi)dv_{g}dt, \end{split}$$ where the constant $C_{10}$ depends only on the geometry of $(X,\omega)$. Choose normal geodesic coordinates $\{x^{i}\}_{i=1}^{2n}$ in the geodesic ball $B_{R}(x_{0})$. Let $x=r\tilde{x}$, $t=t_{0}+r^{2}\tilde{t}$. There holds that $$\begin{split} \Phi(r)&=r^{2}\int_{T_{r}(u_{0})}e(A,\phi)f^{2}G_{u_{0}}dv_{g}dt\\ &=r^{2}\int_{t_{0}-4r^{2}}^{t_{0}-r^{2}}\int_{\mathbb{R}^{2n}}e(A,\phi)(x,t)f^{2}(x)G_{u_{0}}(x,t)\sqrt{\det{(g_{ij})}}(x)dxdt\\ &=r^{4}\int_{-4}^{-1}\int_{\mathbb{R}^{2n}}e(A,\phi)(r\tilde{x},t_{0}+r^{2}\tilde{t})f^{2}(r\tilde{x})G(\tilde{x},\tilde{t})\sqrt{\det{(g_{ij})}}(r\tilde{x})d\tilde{x}d\tilde{t}. \end{split}$$ Then one can see that $$\begin{split} &\frac{d\Phi(r)}{dr}=4r^{3}\int_{-4}^{-1}\int_{\mathbb{R}^{2n}}e(A,\phi)(r\tilde{x},t_{0}+r^{2}\tilde{t})f^{2}(r\tilde{x})G(\tilde{x},\tilde{t})\sqrt{\det{(g_{ij})}}(r\tilde{x})d\tilde{x}d\tilde{t}\\ &+r^{3}\int_{-4}^{-1}\int_{\mathbb{R}^{2n}}\{x^{i}\partial_{i}e(A,\phi)(r\tilde{x},t_{0}+r^{2}\tilde{t})\}f^{2}(r\tilde{x})G(\tilde{x},\tilde{t})\sqrt{\det{(g_{ij})}}(r\tilde{x})d\tilde{x}d\tilde{t}\\ &+r^{3}\int_{-4}^{-1}\int_{\mathbb{R}^{2n}}\{2(t-t_{0})\partial_{t}e(A,\phi)(r\tilde{x},t_{0}+r^{2}\tilde{t})\}f^{2}(r\tilde{x})G(\tilde{x},\tilde{t})\sqrt{\det{(g_{ij})}}(r\tilde{x})d\tilde{x}d\tilde{t}\\ &+r^{4}\int_{-4}^{-1}\int_{\mathbb{R}^{2n}}e(A,\phi)(r\tilde{x},t_{0}+r^{2}\tilde{t})\frac{d}{dr}\{f^{2}(r\tilde{x})\sqrt{\det{(g_{ij})}}(r\tilde{x})\}G(\tilde{x},\tilde{t})d\tilde{x}d\tilde{t}\\ =&I_{1}+I_{2}+I_{3}+I_{4}. \end{split}$$ First, we have $$\label{eqn:31} \begin{split} I_{2}&=r\int_{T_{r}(u_{0})}\{x^{i}\partial_{i}e(A,\phi)(x,t)\}f^{2}(x)G_{u_{0}}(x,t)dv_{g}dt\\ &=2rRe\int_{T_{r}(u_{0})}\langle x^{i}\nabla_{i}F_{A^{'}},F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt\\ &=2rRe\int_{T_{r}(u_{0})}\langle x^{i}\nabla_{i}F_{A^{'}}(\partial_{j},\partial_{k})dx^{j}\wedge dx^{k}, F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ From the Bianchi identity $$D_{A^{'}}F_{A^{'}}=D_{A}F_{A^{'}}+[\phi+\phi^{*}, F_{A^{'}}]=0,$$ it follows that $$\label{eqn:32} \begin{split} \nabla_{i}F_{A^{'}}(\partial_{j},\partial_{k})=&-[\phi_{i}+\phi^{*}_{i},F_{A^{'},jk}]+\nabla_{j}F_{A^{'}}(\partial_{i},\partial_{k})+[\phi_{j}+\phi^{*}_{j},F_{A^{'},ik}]\\ &-\nabla_{k}F_{A^{'}}(\partial_{i},\partial_{j})-[\phi_{k}+\phi^{*}_{k}, F_{A^{'},ij}], \end{split}$$ and $$\label{eqn:33} \begin{split} &\langle x^{i}\nabla_{j}F_{A^{'}}(\partial_{i},\partial_{k})dx^{j}\wedge dx^{k}, F_{A^{'}}\rangle\\ =&\langle x^{i}D_{A}(F_{A^{'},ik}dx^{k})-x^{i}(F_{A^{'}}(\nabla_{j}\partial_{i},\partial_{k})+F_{A^{'}}(\partial_{i},\nabla_{j}\partial_{k}))dx^{j}\wedge dx^{k},F_{A^{'}}\rangle\\ =&\langle D_{A}(x^{i}F_{A^{'},ik}dx^{k})-x^{i}F_{A^{'}}(\nabla_{j}\partial_{i},\partial_{k})dx^{j}\wedge dx^{k},F_{A^{'}}\rangle-|F_{A^{'}}|^{2}, \end{split}$$ and $$\label{eqn:34} \begin{split} &-\langle x^{i}\nabla_{k}F_{A^{'}}(\partial_{i},\partial_{j})dx^{j}\wedge dx^{k}, F_{A^{'}}\rangle\\ =&\langle x^{i}D_{A}(F_{A^{'},ij}dx^{j})+x^{i}(F_{A^{'}}(\nabla_{k}\partial_{i},\partial_{j})+F_{A^{'}}(\partial_{i},\nabla_{k}\partial_{j}))dx^{j}\wedge dx^{k},F_{A^{'}}\rangle\\ =&\langle D_{A}(x^{i}F_{A^{'},ij}dx^{j})+x^{i}F_{A^{'}}(\nabla_{k}\partial_{i},\partial_{j})dx^{j}\wedge dx^{k},F_{A^{'}}\rangle-|F_{A^{'}}|^{2}. \end{split}$$ Under the condition $\partial\bar{\partial}\omega^{n-2}=\partial\bar{\partial}\omega^{n-1}=0$, Demailly[@D] proved $$\label{z:1} \partial_{A}^{*}=\sqrt{-1}[\Lambda_{\omega},\bar{\partial}]-\tau^{*},\ \ \bar{\partial}_{A}^{*}=-\sqrt{-1}[\Lambda_{\omega},\partial_{A}]-\bar{\tau}^{*},$$ where $\tau=[\Lambda_{\omega},\partial\omega]$. Letting $x\odot F_{A^{'}}=x^{i}F_{A^{'},ij}dx^{j}$, we derive $$\label{eqn:35} \begin{split} &Re\langle x\odot F_{A^{'}},D_{A}^{*}F_{A^{'}}\rangle \\ =&-Re\langle x\odot F_{A^{'}},\frac{\partial A}{\partial t}\rangle +Re\langle x\odot F_{A^{'}},\sqrt{-1}\Lambda_{\omega}(\bar{\partial}_{A}-\partial_{A})[\phi,\phi^{*}])\rangle \\ &+Re\langle x\odot F_{A^{'}},\sqrt{-1}\Lambda_{\omega}([F_{A},\phi]-[F_{A},\phi^{*}])\rangle \\ &-Re\langle x\odot F_{A^{'}},(\tau^{*}+\bar{\tau}^{*})F_{A^{'}}\rangle. \end{split}$$ It is easy to check $$\label{eqn:36} \begin{split} &\langle x^{i}[\phi_{i}+\phi^{*}_{i},F_{A^{'}}],F_{A^{'}}\rangle=0. \end{split}$$ Because of $(F_{A}+[\phi,\phi^{*}])^{*}=-(F_{A}+[\phi,\phi^{*}]),(\partial_{A}\phi+\bar{\partial}_{A}\phi^{*})^{*}=\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}$, we know $$\label{eqn:37} \begin{split} &Re\langle[\phi+\phi^{*},x\odot F_{A^{'}}],F_{A^{'}}\rangle\\ =&Re\langle[\phi+\phi^{*},x\odot(F_{A}+[\phi,\phi^{*}])],\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}\rangle\\ &+Re\langle[\phi+\phi^{*},x\odot(\partial_{A}\phi+\bar{\partial}_{A}\phi^{*})],F_{A}+[\phi,\phi^{*}]\rangle, \end{split}$$ where $$\label{eqn:38(1)} \begin{split} \langle[\phi,x\odot (F_{A}+[\phi,\phi^{*}])],\partial_{A}\phi\rangle=&\langle x\odot (F_{A}+[\phi,\phi^{*}]),\sqrt{-1}\Lambda_{\omega}\partial_{A}[\phi,\phi^{*}]\rangle,\\ \langle[\phi^{*},x\odot (F_{A}+[\phi,\phi^{*}])],\bar{\partial}_{A}\phi^{*}\rangle=&-\langle x\odot (F_{A}+[\phi,\phi^{*}]),\sqrt{-1}\Lambda_{\omega}\bar{\partial}_{A}[\phi,\phi^{*}]\rangle\\ \end{split}$$ and $$\label{eqn:38(2)} \begin{split} &\langle[\phi^{*},x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*})],F_{A}+[\phi,\phi^{*}]\rangle\\ =&-\langle x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}),\sqrt{-1}\Lambda_{\omega}[F_{A}+[\phi,\phi^{*}],\phi]\rangle\\ &+\langle x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}),[\sqrt{-1}\Lambda_{\omega}(F_{A}+[\phi,\phi^{*}]),\phi]\rangle,\\ &\langle[\phi,x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*})],F_{A}+[\phi,\phi^{*}]\rangle\\ =&\langle x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}),\sqrt{-1}\Lambda_{\omega}[F_{A}+[\phi,\phi^{*}],\phi^{*}]\rangle\\ &-\langle x\odot (\partial_{A}\phi+\bar{\partial}_{A}\phi^{*}),[\sqrt{-1}\Lambda_{\omega}(F_{A}+[\phi,\phi^{*}]),\phi^{*}]\rangle. \end{split}$$ So $$\label{eqn:39} \begin{split} &Re\langle[\phi+\phi^{*},x\odot F_{A^{'}}],F_{A^{'}}\rangle\\ =&-Re\langle x\odot F_{A^{'}},\sqrt{-1}\Lambda_{\omega}(\bar{\partial}_{A}-\partial_{A})[\phi,\phi^{*}])\rangle \\ &-Re\langle x\odot F_{A^{'}},\sqrt{-1}\Lambda_{\omega}([F_{A},\phi]-[F_{A},\phi^{*}])\rangle \\ &-Re\langle x\odot F_{A^{'}},\frac{\partial\phi}{\partial t}+\frac{\partial\phi^{*}}{\partial t}\rangle. \end{split}$$ Combining (\[eqn:32\]), (\[eqn:33\]), (\[eqn:34\]), (\[eqn:35\]), (\[eqn:36\]) and (\[eqn:39\]), and using $\phi\wedge\phi=0$, we deduce $$\label{310} \begin{split} I_{1}+I_{2}=&-4rRe\int_{T_{r}(u_{0})}\langle d(f^{2}G_{u_{0}})\wedge x\odot F_{A^{'}},F_{A^{'}}\rangle dv_{g}dt\\ &-4rRe\int_{T_{r}(u_{0})}\langle x\odot F_{A^{'}},\frac{\partial A^{'}}{\partial t}\rangle f^{2}G_{u_{0}}dv_{g}dt \\ &+4rRe\int_{T_{r}(u_{0})}\langle x^{i}F_{A^{'}}(\nabla_{k}\partial_{i},\partial_{j})dx^{j}\wedge dx^{k},F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt\\ &-4rRe\int_{T_{r}(u_{0})}\langle x\odot F_{A^{'}},(\tau^{*}+\bar{\tau}^{*})F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ Noting that $$\begin{split} \langle F_{A^{'}},\Big[\phi,\frac{\partial A^{'}}{\partial t}\Big]\rangle=&\langle \partial_{A}\phi,\Big[\phi,\frac{\partial A^{'}}{\partial t}\Big]\rangle+\langle F_{A}+[\phi,\phi^{*}],\Big[\phi,\frac{\partial A^{'}}{\partial t}\Big]\rangle\\ =&\langle\sqrt{-1}\Lambda_{\omega}[\partial_{A}\phi,\phi^{*}],\frac{\partial A^{'}}{\partial t}\rangle+\langle\sqrt{-1}\Lambda_{\omega}[F_{A}+[\phi,\phi^{*}],\phi^{*}],\frac{\partial A^{'}}{\partial t}\rangle\\ &-\langle[\sqrt{-1}\Lambda_{\omega}(F_{A}+[\phi,\phi^{*}]),\phi^{*}],\frac{\partial A^{'}}{\partial t}\rangle, \end{split}$$ $$\begin{split} \langle F_{A^{'}},\Big[\phi^{*},\frac{\partial A^{'}}{\partial t}\Big]\rangle=&-\langle\sqrt{-1}\Lambda_{\omega}[\bar{\partial}_{A}\phi^{*},\phi],\frac{\partial A^{'}}{\partial t}\rangle-\langle\sqrt{-1}\Lambda_{\omega}[F_{A}+[\phi,\phi^{*}],\phi],\frac{\partial A^{'}}{\partial t}\rangle\\ &+\langle[\sqrt{-1}\Lambda_{\omega}(F_{A}+[\phi,\phi^{*}]),\phi],\frac{\partial A^{'}}{\partial t}\rangle, \end{split}$$ and applying $\phi\wedge\phi=0$ again, one can get $$\begin{split} &Re\langle D_{A}^{*}F_{A^{'}},\frac{\partial A^{'}}{\partial t}\rangle+Re\langle F_{A^{'}},\Big[\phi+\phi^{*},\frac{\partial A^{'}}{\partial t}\Big]\rangle\\ =&-\Big|\frac{\partial A}{\partial t}\Big|^{2}-2\Big|\frac{\partial \phi}{\partial t}\Big|^{2}-Re\langle(\tau^{*}+\bar{\tau}^{*})F_{A^{'}},\frac{\partial A^{'}}{\partial t}\rangle. \end{split}$$ The flow and Stokes formula imply that $$\label{eqn:311} \begin{split} I_{3}=&2r\int_{T_{r}(u_{0})}(t-t_{0})\partial_{t}e(A,\phi)f^{2}G_{u_{0}}dv_{g}dt\\ =&-4rRe\int_{T_{r}(u_{0})}(t-t_{0})\Big(\Big|\frac{\partial A}{\partial t}\Big|^{2}+2\Big|\frac{\partial \phi}{\partial t}\Big|^{2}\Big)dv_{g}dt\\ &-4rRe\int_{T_{r}(u_{0})}(t-t_{0})\langle F_{A^{'}},d(f^{2}G_{u_{0}})\wedge\frac{\partial A^{'}}{\partial t}\rangle dv_{g}dt\\ &-4rRe\int_{T_{r}(u_{0})}(t-t_{0})\langle(\tau^{*}+\bar{\tau}^{*})F_{A^{'}},\frac{\partial A^{'}}{\partial t}\rangle f^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ Let $x\cdot F_{A^{'}}=\frac{1}{2}x^{i}g^{ij}F_{A^{'},jk}dx^{k}$, and notice that $\partial_{i}G_{u_{0}}=\frac{x^{i}G_{u_{0}}}{2(t-t_{0})}$, then $$\label{eqn:312} \begin{split} &I_{1}+I_{2}+I_{3}\\ =&4rRe\int_{T_{r}(u_{0})}\frac{1}{|t-t_{0}|}\Big||t-t_{0}|\frac{\partial A^{'}}{\partial t}-x\odot F_{A^{'}}\Big|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &+4rRe\int_{T_{r}(u_{0})}\frac{1}{|t-t_{0}|}\langle x\cdot F_{A^{'}}-x\odot F_{A^{'}},x\odot F_{A^{'}}-|t-t_{0}|\frac{\partial A^{'}}{\partial t}\rangle f^{2}G_{u_{0}}dv_{g}dt \\ &+4rRe\int_{T_{r}(u_{0})}\langle 2f^{-1}\nabla f\llcorner F_{A^{'}},|t-t_{0}|\frac{\partial A}{\partial t}-x\odot F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt\\ &+4rRe\int_{T_{r}(u_{0})}\langle x^{i}F_{A^{'}}(\nabla_{k}\partial_{i},\partial_{j})dx^{j}\wedge dx^{k},F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt\\ &+4rRe\int_{T_{r}(u_{0})}\langle |t-t_{0}|\frac{\partial A^{'}}{\partial t}-x\odot F_{A^{'}},(\tau^{*}+\bar{\tau}^{*})F_{A^{'}}\rangle f^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ Calculating straightforward, we obtain $$\label{eqn:313} \begin{split} I_{4}=&r\int_{T_{r}(u_{0})}e(A,\phi)x^{i}\partial_{i}(f^{2}\sqrt{\det(g_{ij})})G_{u_{0}}dxdt\\ =&r\int_{T_{r}(u_{0})}e(A,\phi)2fx^{i}\partial_{i}fG_{u_{0}}dv_{g}dt\\ &+\frac{r}{2}\int_{T_{r}(u_{0})}e(A,\phi)x^{i}{\mbox{\rm tr\,}}(g^{-1}\partial_{i}g)f^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ Based on (\[a\]) and $|\Gamma_{jk}^{i}|\leq C|x|$, there exists constants $\tilde{C}_{1}$ and $\tilde{C}_{2}$, such that $$\label{eqn:314} \begin{split} &|x\cdot F_{A^{'}}-x\odot F_{A^{'}}|^{2}=\Big|\frac{1}{2}x^{i}(\delta_{j}^{i}-g^{ij})F_{A^{'},jl}dx^{l}\Big|^{2}\leq \tilde{C}_{1}|x|^{6}|F_{A^{'}}|^{2},\\ &\langle x^{i}F_{A^{'}}(\nabla_{j}\partial_{i},\partial_{k})dx^{j}\wedge dx^{k},F_{A^{'}}\rangle\leq \tilde{C}_{1}|x|^{2}|F_{A^{'}}|^{2},\\ &{\mbox{\rm tr\,}}(g^{-1}\partial_{k}g)\leq \tilde{C}_{1}|x|, \end{split}$$ and $|(\tau^{*}+\bar{\tau}^{*})F_{A^{'}}|^{2}\leq \tilde{C}_{2}|F_{A^{'}}|^{2}$. Due to (\[eqn:312\]) and (\[eqn:313\]), it holds $$\begin{split} \frac{d\Phi(r)}{dr}\geq& r\int_{T_{r}(u_{0})}\frac{1}{|t-t_{0}|}\Big||t-t_{0}|\frac{\partial A^{'}}{\partial t}-x\odot F_{A^{'}}\Big|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &-\tilde{C}_{3}r\int_{T_{r}(u_{0})}\frac{|x|^{6}}{|t-t_{0}|}|F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &-r\int_{T_{r}(u_{0})}|t-t_{0}||2f^{-1}\nabla\llcorner F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &-\tilde{C}_{3}r\int_{T_{r}(u_{0})}|t-t_{0}||F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &-\tilde{C}_{3}r\int_{T_{r}(u_{0})}|x|^{2}|F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &-2r\int_{T_{r}(u_{0})}|x||\nabla f||f||F_{A^{'}}|^{2}G_{u_{0}}dv_{g}dt. \end{split}$$ According to Chen-Struwe’s arguments in [@CS], we know there exists a constant $\tilde{C}_{4}>0$ such that $$\begin{split} &r^{-1}|t-t_{0}||x|^{6}G_{u_{0}}\leq \tilde{C}_{4}(1+G_{u_{0}}),\\ &r^{-1}|x|^{2}G_{u_{0}}\leq \tilde{C}_{4}(1+G_{u_{0}}) \end{split}$$ on $T_{r}(u_{0})$. Hence it follows that $$\begin{split} &-\tilde{C}_{3}r\int_{T_{r}(u_{0})}\Big(\frac{|x|^{6}}{|t-t_{0}|}+|t-t_{0}|+|x|^{2}\Big)|F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt\\ &\geq -\tilde{C}_{5}\Phi(r)-\tilde{C}_{5}rYMH(A_{0},\phi_{0}), \end{split}$$ where the constant $\tilde{C}_{5}$ depends only on $\tilde{C}_{3}$ and $\tilde{C}_{4}$. As that shown in [@NZ2](page 15), we have $$\begin{split} -r\int_{T_{r}(u_{0})}|t-t_{0}||2f^{-1}\nabla f\llcorner F_{A^{'}}|^{2}f^{2}G_{u_{0}}dv_{g}dt&\geq-\frac{C(n)r}{R^{2n}}\int_{P_{R}(u_{0})}|F_{A^{'}}|^{2}dv_{g}dt,\\ -2r\int_{T_{r}(u_{0})}|x||\nabla f||f||F_{A^{'}}|^{2}G_{u_{0}}dv_{g}dt&\geq-\frac{C(n)r}{R^{2n}}\int_{P_{R}(u_{0})}|F_{A^{'}}|^{2}dv_{g}dt. \end{split}$$ From above, it can be seen that $$\frac{d\Phi(r)}{dr}\geq -\tilde{C}_{6}\Phi(r)-\tilde{C}_{6}rYMH(A_{0},\phi_{0})-\frac{\tilde{C}_{6}r}{R^{2n}}\int_{P_{R}(u_{0})}|F_{A^{'}}|^{2}dv_{g}dt.$$ Then integrating two sides of this inequality concludes this proof. By the monotonicity inequality, we can derive the $\epsilon$-regularity theorem. \[thm:5\] Let $(A(t),\phi(t))$ be a solution of the modified Yang-Mills-Higgs flow. There exist positive constants $\epsilon_{0},\delta_{0}<1/4$, such that if $$R^{2-2n}\int_{P_{R}(u_{0})}e(A,\phi)dv_{g}dt<\epsilon_{0}$$ holds for some $0<R\leq \min\{i_{X}/2,\sqrt{t_{0}}/2\}$, then for any $\delta\in(0,\delta_{0})$, we have $$\sup_{P_{\delta R}(u_{0})}e(A,\phi)\leq\frac{16}{(\delta R)^{4}}$$ and $$\sup_{P_{\delta R}(u_{0})}|\nabla_{A}\phi|^{2}\leq C_{11},$$ where $C_{11}$ is a positive constant depending only on $R$, $\delta_{0}$, $\sup_{X}|\phi_{0}|_{H_{0}}$ and the geometry of $(X,\omega)$. Regard $X$ as a real manifold. The proof is very similar to the one of Theorem 2.6 in [@LZ1]. For any $\delta\in(0,1/4]$, define $$\begin{split} f(r)=(2\delta R-r)^{4}\sup_{P_{r}(x_{0},t_{0})}e(A,\phi). \end{split}$$ Since $f(r)$ is continuous and $f(2\delta R)=0$, we know that $f(r)$ attains its maximum at some point $r_{0}\in[0,2\delta R)$. Suppose $(x_{1},t_{1})\in\bar{P}_{r_{0}}(x_{0},t_{0})$ is a point such that $$e(A,\phi)(x_{1},t_{1})=\sup_{P_{r}(x_{0},t_{0})}e(A,\phi).$$ We claim that when $\epsilon_{0},\delta_{0}$ are small enough, $f(r_{0})\leq 16$. Otherwise, set $$\rho_{0}=(2\delta R-r_{0})f(r_{0})^{-1/4}=e(A,\phi)(x_{1},t_{1})^{-1/4}<\delta R-\frac{r_{0}}{2}.$$ Rescaling the Riemannian metric $\tilde{g}=\rho_{0}^{-2}g$ and $t=t_{1}+\rho_{0}^{2}\tilde{t}$, we get $$\begin{split} &|F_{A}+[\phi,\phi^{*H_{0}}]|_{\tilde{g}}^{2}=\rho_{0}^{4}|F_{A}+[\phi,\phi^{*H_{0}}]|_{g}^{2},\\ &|\partial_{A}\phi|_{\tilde{g}}^{2}=\rho_{0}^{4}|\partial_{A}\phi|_{g}^{2},\ \ |\nabla_{A}\phi|_{\tilde{g}}^{2}=\rho_{0}^{4}|\nabla_{A}\phi|_{g}^{2}. \end{split}$$ Setting $$\begin{split} &e_{\rho_{0}}(x,\tilde{t})=|F_{A}+[\phi,\phi^{*H_{0}}]|_{\tilde{g}}^{2}+2|\partial_{A}\phi|_{\tilde{g}}^{2}=\rho_{0}^{4}e(A,\phi)(x,t_{1}+\rho_{0}^{2}\tilde{t}),\\ &b_{\rho_{0}}(x,\tilde{t})=|\nabla_{A}\phi|_{\tilde{g}}^{2}=\rho_{0}^{4}|\nabla_{A}\phi|_{g}^{2}(x,t_{1}+\rho_{0}^{2}\tilde{t}),\\ &\tilde{P}_{\tilde{r}}(x_{1},0)=B_{\rho_{0}\tilde{r}}(x_{1})\times[-\tilde{r}^{2},\tilde{r}^{2}], \end{split}$$ we have $e_{\rho_{0}}(x_{1},0)=\rho_{0}^{4}e(A,\phi)(x_{1},t_{1})=1$, and $$\begin{split} \sup_{\tilde{P}_{1}(x_{1},0)}e_{\rho_{0}}&=\rho_{0}^{4}\sup_{P_{\rho_{0}}(x_{1},t_{1})}e(A,\phi)\leq \rho_{0}^{4}\sup_{P_{\delta R+r_{0}/2}(x_{0},t_{0})}e(A,\phi)\\ &\leq \rho_{0}^{4}f(\delta R+r_{0}/2)(\delta R-r_{0}/2)^{-4}\leq 16. \end{split}$$ Thus $$\label{eqn:315} \begin{split} |F_{A}+[\phi,\phi^{*H_{0}}]|_{\tilde{g}}^{2}+2|\partial_{A}\phi|_{\tilde{g}}^{2}\leq 16,\ \ \ \ on\ \ \tilde{P}_{1}(x_{1},0). \end{split}$$ Combining this together with Proposition \[p:1\] yields that $$\begin{split} &(\Delta_{\tilde{g}}-\frac{\partial}{\partial \tilde{t}})(b_{\rho_{0}}+\rho_{0}^{4})=\rho_{0}^{6}(\Delta_{g}-\frac{\partial}{\partial t})(|\nabla_{A}\phi|_{g}^{2}+1)\\ \geq &2\rho_{0}^{6}|\nabla_{A}\nabla_{A}\phi|_{g}^{2}-C_{4}\rho_{0}^{6}(|F_{A}|_{g}+|Rm|_{g}+|Ric|_{g}+|\phi|_{g}^{2})|\nabla_{A}\phi|_{g}^{2}\\ &-C_{5}\rho_{0}^{6}(|\nabla Ric|_{g}+|d\omega|_{g}|Rm|_{g}+|d\omega|_{g}|Ric|_{g}+|d\omega|_{g}|F_{A}|_{g})|\phi|_{g}|\nabla_{A}\phi|_{g}\\ &-C_{6}\rho_{0}^{6}|d\omega|_{g}|\nabla_{A}\phi|_{g}|\nabla_{A}\nabla_{A}\phi|_{g}\\ \geq&-\tilde{C}_{1}(b_{\rho_{0}}+\rho_{0}^{4}) \end{split}$$ on $\tilde{P}_{1}(x_{1},0)$, where the constant $\tilde{C}_{1}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Then by the parabolic mean value inequality and Proposition \[p:2\], we observe $$\label{eqn:316} \begin{split} \sup_{\tilde{P}_{1/2}(x_{1},0)}(b_{\rho_{0}}+\rho_{0}^{4})&\leq C\int_{\tilde{P}_{1}(x_{1},0)}(b_{\rho_{0}}+\rho_{0}^{4})dv_{\tilde{g}}d\tilde{t}\\ &=C\rho_{0}^{2-2n}\int_{P_{\rho_{0}}(x_{1},t_{1})}(|\nabla_{A}\phi|_{g}^{2}+1)dv_{g}dt\\ &\leq C^{*}\rho_{0}^{2}\leq \tilde{C}_{2}, \end{split}$$ where the constant $\tilde{C}_{2}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Similarly, combining (\[eqn:316\]), (\[eqn:315\]) and Proposition \[p:1\], one knows that $$(\Delta_{\tilde{g}}-\frac{\partial}{\partial \tilde{t}})e_{\rho_{0}}\geq -\tilde{C}_{3}e_{\rho_{0}}$$ on $\tilde{P}_{1/2}(x_{1},0)$, where the constant $\tilde{C}_{3}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Using the parabolic mean value inequality again, we can see that $$\begin{split} 1=e_{\rho_{0}}(x_{1},0)\leq\sup_{\tilde{P}_{1/4}(x_{1},0)}e_{\rho_{0}}&\leq C\int_{\tilde{P}_{1/2}(x_{1},0)}e_{\rho_{0}}dv_{\tilde{g}}d\tilde{t}\\ &\leq \tilde{C}_{4}\rho_{0}^{2-2n}\int_{P_{\rho_{0}}(x_{1},t_{1})}e(A,\phi)dv_{g}dt,\\ \end{split}$$ where the constant $\tilde{C}_{4}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Choose normal geodesic coordinates centred at $x_{1}$, and a smooth cut-off function $f\in C^{\infty}_{0}(B_{R/2}(x_{1}))$ such that $0\leq f\leq 1$, $f\equiv 1$ on $B_{R/4}(x_{1})$, $|d f|\leq 8/R$ on $B_{R/2}(x_{1})\setminus B_{R/4}(x_{1})$. Taking $r_{1}=\rho_{0}$ and $r_{2}=\delta_{0} R$, applying the monotonicity inequality, we obtain $$\begin{split} &\rho_{0}^{2-2n}\int_{P_{\rho_{0}}(x_{1},t_{1})}e(A,\phi)dv_{g}dt\\ \leq& C\rho_{0}^{2}\int_{P_{\rho_{0}}(x_{1},t_{1})}e(A,\phi)G_{(x_{1},t_{1}+2\rho_{0}^{2})}f^{2}dv_{g}dt\\ \leq& C\rho_{0}^{2}\int_{T_{\rho_{0}}(x_{1},t_{1}+2\rho_{0}^{2})}e(A,\phi)G_{(x_{1},t_{1}+2\rho_{0}^{2})}f^{2}dv_{g}dt\\ \leq& C_{*}r_{2}^{2}\int_{T_{r_{2}}(x_{1},t_{1}+2\rho_{0}^{2})}e(A,\phi)G_{(x_{1},t_{1}+2\rho_{0}^{2})}f^{2}dv_{g}dt+C_{*}\delta_{0}^{2}R^{2}YMH(A_{0},\phi_{0})\\ &+C_{*}(R/2)^{2-2n}\int_{P_{R/2}(x_{1},t_{1})}e(A,\phi)dv_{g}dt\\ \leq& C_{*}\delta_{0}^{2-2n}R^{2-2n}\int_{P_{R}(x_{0},t_{0})}e(A,\phi)dv_{g}dt+C_{*}\delta_{0}^{2}R^{2}YMH(A_{0},\phi_{0})\\ \leq& \tilde{C}_{5}(\delta_{0}^{2-2n}\epsilon_{0}+\delta_{0}^{2}R^{2}YMH(A_{0},\phi_{0})), \end{split}$$ where the constant $\tilde{C}_{5}$ depends only on the geometry of $(X,\omega)$ and $\sup_{X}|\phi_{0}|_{H_{0}}$. Choose $\epsilon_{0},\delta_{0}$ properly such that $\tilde{C}_{4}\tilde{C}_{5}(\delta_{0}^{2-2n}\epsilon_{0}+\delta_{0}^{2}R^{2}YMH(A_{0},\phi_{0}))< 1$, and then a contradiction occurs. So $f(r_{0})\leq 16$, which implies $$\sup_{P_{\delta R}(u_{0})}e(A,\phi)\leq 16/(\delta R)^{4}.$$ Let $r=3\delta R/2$, then $$\sup_{P_{3\delta R/2}(u_{0})}e(A,\phi)=f(3\delta R/2)(\delta R/2)^{-1/4}\leq 256/(\delta R)^{4}.$$ On $P_{3\delta R/2}(u_{0})$, from Proposition \[p:1\], Proposition \[p:2\] and the parabolic mean value inequality, we derive $$\begin{split} \sup_{P_{\delta R}(u_{0})}(|\nabla_{A}\phi|_{g}^{2}+1)&\leq C\int_{P_{3\delta R/2}(u_{0})}(|\nabla_{A}\phi|_{g}^{2}+1)dv_{g}dt\\ &\leq \tilde{C}_{6}, \end{split}$$ where the constant $\tilde{C}_{6}$ depends only on $R$, $\delta_{0}$, $\sup_{X}|\phi_{0}|_{H_{0}}$ and the geometry of $(X,\omega)$. Proof of Theorem \[thm:1\] ========================== Let $(X,\omega)$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$. Let $(E,\bar{\partial}_{E},\theta)$ be a semi-stable Higgs bundle with $ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]=0$, then $(E,\bar{\partial}_{E},\theta)$ is approximate Higgs-Hermitian flat. By Theorem \[thm:3\] and $ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=0$, we know that there is a family of Hermitian metrics $\{H_{\epsilon}\}_{\epsilon>0}$, satisfying $$\sup_{X}|\sqrt{-1}\Lambda_{\omega}F_{H_{\epsilon},\theta}|_{H_{\epsilon}}\leq \epsilon.$$ Clearly $ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]=0$ means $$0=\int_{X}{\mbox{\rm tr\,}}(F_{H_{\epsilon},\theta}\wedge F_{H_{\epsilon},\theta})\wedge\frac{\omega^{n-2}}{(n-2)!}=\int_{X}(|F_{H_{\epsilon},\theta}|^{2}-|\Lambda_{\omega}F_{H_{\epsilon},\theta}|^{2})\frac{\omega^{n}}{n!}.$$ Consider the following Donaldson flow $$\left\{ \begin{split} &H_{\epsilon}^{-1}(t)\frac{\partial H_{\epsilon}(t)}{\partial t}=-2(\sqrt{-1}\Lambda_{\omega}(F_{H_{\epsilon}(t)}+[\theta,\theta^{*H_{\epsilon}(t)}])-\lambda Id), \\ &H_{\epsilon}(0)=H_{\epsilon}. \end{split} \right.$$ It is gauge equivalent to $$\left\{ \begin{split} \frac{\partial A_{\epsilon}(t)}{\partial t}&=-\sqrt{-1}(\partial_{A_{\epsilon}(t)}-\bar{\partial}_{A_{\epsilon}(t)})\Lambda_{\omega}(F_{A_{\epsilon}(t)}+[\phi_{\epsilon}(t),\phi_{\epsilon}^{*H_{\epsilon}}(t)]), \\ \frac{\partial\phi_{\epsilon}(t)}{\partial t}&=-[\sqrt{-1}\Lambda_{\omega}(F_{A_{\epsilon}(t)}+[\phi_{\epsilon}(t),\phi_{\epsilon}^{*H_{\epsilon}}(t)]),\phi_{\epsilon}(t)], \\ A_{\epsilon}(0)&=D_{H_{\epsilon},\bar{\partial}_{E}},\ \ \phi_{\epsilon}(0)=\theta. \end{split} \right.$$ According to Proposition \[p\], we observe $$\begin{split} \int_{X}|F_{A_{\epsilon}(t),\phi_{\epsilon}(t)}|_{H_{\epsilon}}^{2}\frac{\omega^{n}}{n!}&=\int_{X}|F_{H_{\epsilon}(t),\theta}|_{H_{\epsilon}(t)}^{2}\frac{\omega^{n}}{n!}\\ &=\int_{X}|\Lambda_{\omega}F_{H_{\epsilon}(t),\theta}|_{H_{\epsilon}(t)}^{2}\frac{\omega^{n}}{n!}\leq \int_{X}|\Lambda_{\omega}F_{H_{\epsilon},\theta}|_{H_{\epsilon}}^{2}\frac{\omega^{n}}{n!}. \end{split}$$ Since $\sup_{X}|\theta|_{H_{\epsilon},\omega}^{2}\leq \check{C}_{4}$, the constants $\tilde{C}_{4}$ and $\tilde{C}_{5}$ in the proof of Theorem \[thm:5\] depend only on the geometry of $(X,\omega)$ and $\check{C}_{4}$. Assume $\epsilon<1$, then there is a constant $\check{C}_{5}$, such that $$\int_{X}|F_{H_{\epsilon},\theta}|_{H_{\epsilon},\omega}^{2}\frac{\omega^{n}}{n!}\leq \check{C}_{5}.$$ We can fix a pair of positive constants $\epsilon_{0},\delta_{0}$ in Theorem \[thm:5\] satisfying $\tilde{C}_{4}\tilde{C}_{5}(\delta_{0}^{2-2n}\epsilon_{0}+\check{C}_{5}\delta_{0}^{2}(i_{X}/2)^{2})<1$. Take $R=\frac{i_{X}}{2}$. When $\epsilon\leq \min\{\sqrt{\frac{2^{3-2n}\epsilon_{0}}{i_{X}^{4-2n}Vol(X,\omega)}},1\}$, for any $u_{0}\in X\times(i_{X}^{2},\infty)$, we have $$\begin{aligned} R^{2-2n}\int_{P_{R}(u_{0})}|F_{A_{\epsilon}(t),\phi_{\epsilon}(t)}|_{H_{\epsilon}}^{2}\frac{\omega^{n}}{n!}\leq \epsilon_{0}.\end{aligned}$$ By Theorem \[thm:5\], for any $\delta\in(0,\delta_{0})$, it holds that $\sup_{P_{\delta R}(u_{0})}e(A_{\epsilon}(t),\phi_{\epsilon}(t))\leq\frac{16}{(\delta R)^{4}}$ and $\sup_{P_{\delta R}(u_{0})}|\nabla_{A_{\epsilon}(t)}\phi_{\epsilon}(t)|_{H_{\epsilon}}^{2}\leq C_{11}.$ Therefore we deduce $$\label{d:1} \sup_{X\times(i_{X}^{2},\infty)}e(A_{\epsilon}(t),\phi_{\epsilon}(t))\leq \hat{C}_{1}$$ and $$\label{d:2} \sup_{X\times(i_{X}^{2},\infty)}|\nabla_{A_{\epsilon}(t)}\phi_{\epsilon}(t)|^{2}\leq \hat{C}_{2},$$ where the constants $\hat{C}_{1},\hat{C}_{2}$ are independent of $\epsilon$. Combining (\[d:1\]), (\[d:2\]) and Proposition \[p:1\], we can find a constant $\check{C}_{6}$ independent of $\epsilon$ and $t$, such that $$(\Delta-\frac{\partial}{\partial t})e(A_{\epsilon}(t),\phi_{\epsilon}(t))\geq- \check{C}_{6} e(A_{\epsilon}(t),\phi_{\epsilon}(t)).$$ Applying the mean value inequality of parabolic equation, one can see that there is a constant $\check{C}_{7}$ independent of $\epsilon$ and $t$, such that for $t>i_{X}^{2}+1$, there holds $$\sup_{X}e(A_{\epsilon}(t),\phi_{\epsilon}(t))\leq \check{C}_{7}\int_{X}e(A_{\epsilon}(t-1),\phi_{\epsilon}(t-1))\frac{\omega^{n}}{n!}\leq \check{C}_{7}\int_{X}|\Lambda_{\omega}F_{H_{\epsilon},\theta}|_{H_{\epsilon}}^{2}\frac{\omega^{n}}{n!}.$$ Take $t=i_{X}^{2}+2$, then we know $\{H_{\epsilon}(i_{X}^{2}+2)\}$ is a family of approximate Higgs-Hermitian flat metrics. Replacing the Kähler condition by the Gauduchon condition, one can easily get the following lemma by a similar argument as that in [@NZ1 Proposition 2.2]. \[l:1\] Let $X$ be a compact Hermitian manifold of dimension $n$, satisfying $\partial\bar{\partial}\omega^{n-1}=\partial\bar{\partial}\omega^{n-2}=0$. Let $(E,\bar{\partial}_{E},\theta)$ be an approximate Higgs-Hermitian flat Higgs bundle. If $s$ is a non-trivial $\theta$-invariant holomorphic section of $E$, then $s$ has no zeros. The proof is completely similar to the one of [@NZ1] in Kähler case. Anyway, we will present its proof here for the readers’ convenience. The sufficiency is easy to show by directly calculating. Here we only prove the necessity. Suppose $(E,\bar{\partial}_{E},\theta)$ is strictly semi-stable. There is a Higgs coherent subsheaf $S$ of minimal rank $p$ with $\deg_{\omega}(S)=0$. Thus $S$ is stable. Replacing $S$ by its double dual $S^{**}$, we can assume $S$ is reflexive. In the following, we prove that $S$ is a Higgs subbundle. Consider the following exact sequence of Higgs sheaves: $$0\rightarrow (S,\theta_{S})\rightarrow(E,\theta)\rightarrow(Q,\theta_{Q})\rightarrow 0.$$ Let $\{H_{\epsilon}\}$ be the approximate Higgs-Hermitian flat metrics. We can define the bundle isomorphisms $$f_{\epsilon}:S\oplus Q\rightarrow E$$ on $X\setminus Z$, where $Z$ is the singularity set of $S$. Then the pull-back holomorphic structure and Higgs field on $S\oplus Q$ are $$\begin{split} f_{\epsilon}^{*}(\bar{\partial}_{E})=\left(\begin{array}{cc} \bar{\partial}_{S} & \gamma_{\epsilon} \\ 0 & \bar{\partial}_{Q} \end{array}\right),\ \ f_{\epsilon}^{*}(\theta)=\left(\begin{array}{cc} \theta_{S} & \zeta_{\epsilon} \\ 0 & \theta_{Q} \end{array}\right). \end{split}$$ On the basis of Gauss-Codazzi equation proved in [@NZ1], we know $$f_{\epsilon}^{*}(F_{H_{\epsilon},\theta})^{1,1}=\left(\begin{split} &F_{H_{S,\epsilon},\theta_{S}}^{1,1}-\gamma_{\epsilon}\wedge\gamma_{\epsilon}^{*}+\zeta_{\epsilon}\wedge\zeta_{\epsilon}^{*} & D_{S\otimes Q^{*}}^{1,0}\gamma_{\epsilon}+\zeta_{\epsilon}\wedge\theta_{Q}^{*}+\theta_{S}^{*}\wedge\zeta_{\epsilon} \\ &D_{S^{*}\otimes Q}^{0,1}\gamma_{\epsilon}^{*}+\zeta_{\epsilon}^{*}\wedge\theta_{S}+\theta_{Q}\wedge\zeta_{\epsilon}^{*} &F_{H_{Q,\epsilon},\theta_{Q}}^{1,1}-\gamma_{\epsilon}^{*}\wedge\gamma_{\epsilon}+\zeta_{\epsilon}^{*}\wedge\zeta_{\epsilon} \end{split}\right).$$ Since ${\mbox{\rm tr\,}}([\theta_{S},\theta_{S}^{*}])=0$ and $\frac{\sqrt{-1}}{2\pi}{\mbox{\rm tr\,}}(-\gamma_{\epsilon}\wedge\gamma_{\epsilon}^{*}+\zeta_{\epsilon}\wedge\zeta_{\epsilon}^{*})\geq 0$, we conclude $$\label{tt:1} \frac{\sqrt{-1}}{2\pi}{\mbox{\rm tr\,}}(F_{H_{S,\epsilon}})\leq\frac{\sqrt{-1}}{2\pi}{\mbox{\rm tr\,}}(F_{H_{\epsilon},\theta}^{1,1}|_{S})\leq \delta(\epsilon)\omega$$ on $X\setminus Z$, where $\delta(\epsilon)=C(n,r)\sup\limits_{X}|F_{H_{\epsilon},\theta}|_{H_{\epsilon}}\rightarrow 0$ as $\epsilon\rightarrow 0$. Consider the determinant line bundle $\det(S)=(\wedge^{p}S)^{**}$ equipped with the possibly singular metric $\hat{H}_{\epsilon}=\det(H_{S,\epsilon})$ on $X$. From (\[tt:1\]), it follows that $-\frac{\sqrt{-1}}{2\pi}F_{\hat{H}_{\epsilon}}+\delta(\epsilon)\omega\geq 0$ in the sense of current on $X$. Then $$\begin{split} 0&\leq\int_{X}\big(-\frac{\sqrt{-1}}{2\pi}F_{\hat{H}_{\epsilon}}+\delta(\epsilon)\omega\big)\wedge\frac{\omega^{n-1}}{(n-1)!}\\ &=-\deg_{\omega}(\det S)+\delta(\epsilon)\int_{X}\frac{\omega^{n}}{n!}\\ &=\delta(\epsilon)\int_{X}\frac{\omega^{n}}{n!}. \end{split}$$ Therefore, subsequently, $\frac{\sqrt{-1}}{2\pi}F_{\hat{H}_{S,\epsilon}}$ converges to zero weakly in the sense of current. We can also show that $ch_{2}(\det{S})[\omega^{n-2}]=0$. This means $\det(S)$ is Hermitian flat. Furthermore, because $(E,\bar{\partial}_{E},\theta)$ is approximate Higgs-Hermitian flat, $(\wedge^{p}E\otimes(\det S)^{-1},\theta_{p})$ is also approximate Higgs-Hermitian flat, where $\theta_{p}$ is the induced Higgs field on $\wedge^{p}E\otimes(\det S)^{-1}$. A Higgs morphism $\det(S)\rightarrow \wedge^{p}E$ can be seen as a $\theta_{p}$-invariant section of $\wedge^{p}E\otimes(\det S)^{-1}$. Lemma \[l:1\] tells us that the non-zero Higgs morphism $\det(S)\rightarrow \wedge^{p}E$ is injective. Then by Lemma 1.20 in [@DPM], we know that $S$ is a Higgs subbundle of $E$. From the exact sequence of Higgs bundles, one can see that $$ch_{1}(S)+ch_{1}(Q)=ch_{1}(E),\ \ ch_{2}(S)+ch_{2}(Q)=ch_{2}(E).$$ But Chern-Weil theory implies that $ch_{2}(S)[\omega^{n-2}]\geq0$ and $ch_{2}(Q)[\omega^{n-2}]\geq0$. So there must hold $ch_{2}(S)[\omega^{n-2}]=ch_{2}(Q)[\omega^{n-2}]=0$. From above, it follows that $S$ is Higgs-Hermitian flat and $Q$ is Higgs semi-stable with $ch_{1}(Q)[\omega^{n-1}]=ch_{2}(Q)[\omega^{n-2}]=0$. Set $(E_{1},\theta_{E_{1}})=(S,\theta_{S})$ and then establish the desired filtration by induction on the rank. Proof of Theorem \[thm:2-2\] ============================ First, under the assumption of Theorem \[thm:2-2\], we can prove \[nl:1\] Suppose $(E,\bar{\partial}_{E},\theta,H)$ is a Higgs bundle with metric $H$ satisfying $F_{H,\theta}=0$. Then $$H^{1}_{DR}(X,E)\simeq H^{1}_{Dol}(X,E).$$ We will show that for any $[\beta]\in H_{Dol}^{1}(X,E)$, there is a representative $\tilde{\beta}\in[\beta]$, such that $D_{H,\theta}\tilde{\beta}=0$. Consider the following equation $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\kappa.$$ Making use of the continuity method, one can easily prove that the solvability of this equation is equivalent to the one of $$\int_{X}\langle\kappa,\vartheta\rangle_{H}\frac{\omega^{n}}{n!}=0,$$ for any $\vartheta\in\Gamma(X,E)$ satisfying $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$. By the assumption $\int_{X}\partial [\eta]\wedge\frac{\omega^{n-1}}{(n-1)!}=0$ for any Dolbeault class $[\eta]\in H^{0,1}(X)$, we know $$\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta,\vartheta\rangle_{H}\frac{\omega^{n}}{n!} =\int_{X}\sqrt{-1}\partial\langle\beta^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)!} =0.$$ Suppose $\gamma\in\Gamma(X,E)$ is a solution of $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta.$$ Let $\tilde{\beta}=\beta+D^{''}_{E}\gamma$, then $\sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}=0$. According to (\[z:1\]), one can easily check that $$\sqrt{-1}[\Lambda_{\omega},D_{E}^{''}]=(D_{H}^{'})^{*}+\tau^{*},\ \ -\sqrt{-1}[\Lambda_{\omega},D_{H}^{'}]=(D_{E}^{''})^{*}+\bar{\tau}^{*}.$$ A simple computation gives $$\label{eq:61} \begin{split} 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},F_{H,\theta}]\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}_{E}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{''}_{E}]D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}|D^{'}_{H}\tilde{\beta}|^{2}_{H,\omega}\frac{\omega^{n}}{n!}+\int_{X}\langle\tau^{*}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}. \end{split}$$ From Proposition 4.1 in [@MZ], we can see that for any $\alpha\in\Omega^{2}(E)$ with $\Lambda_{\omega}\alpha=0$, there holds $$(\tau+\bar{\tau}^{*})\alpha=\frac{-*(d(\omega^{n-2})\wedge(\alpha^{1,1}-\alpha^{2,0}-\alpha^{0,2}))}{(n-2)!}.$$ Let $\alpha=D^{'}_{H}\tilde{\beta}$. Then using Stokes formula and $D^{''}_{E}\tilde{\beta}=0$, we derive $$\begin{split} &\int_{X}\langle\tau^{*}D^{'}_{H}\tilde{\beta},\tilde{\beta}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ =&\int_{X}\langle\frac{-*(\bar{\partial}(\omega^{n-2})\wedge(\partial_{H}\tilde{\beta}^{0,1}+\theta^{*}(\tilde{\beta}^{1,0})-\partial_{H}\tilde{\beta}^{1,0}))}{(n-2)!},\tilde{\beta}^{0,1}+\tilde{\beta}^{1,0}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ =&\frac{-1}{(n-2)!}\int_{X} \langle\bar{\partial}\omega^{n-2}\wedge(\partial_{H}\tilde{\beta}^{0,1}+\theta^{*}(\tilde{\beta}^{1,0})-\partial_{H}\tilde{\beta}^{1,0}),\tilde{\beta}^{0,1}+\tilde{\beta}^{1,0}\rangle_{H}\\ =&\frac{-1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\partial_{H}\tilde{\beta}^{0,1},\tilde{\beta}^{0,1}\rangle_{H}\\ &-\frac{1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\theta^{*}(\tilde{\beta}^{1,0}),\tilde{\beta}^{0,1}\rangle_{H}\\ &+\frac{1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\partial_{H}\tilde{\beta}^{1,0},\tilde{\beta}^{1,0}\rangle_{H}\\ =&\frac{1}{(n-2)!}\int_{X}\bar{\partial}\omega^{n-2}\wedge \langle\tilde{\beta}^{1,0},\bar{\partial}_{E}\tilde{\beta}^{1,0}+\theta(\tilde{\beta}^{0,1})\rangle_{H}\\ =&0. \end{split}$$ Hence $D^{'}_{H}\tilde{\beta}=0$, and $[\tilde{\beta}]\in H^{1}_{DR}(E)$. Conversely, take $[\beta_{1}]\in H^{1}_{DR}(E)$. For any $\vartheta\in\Gamma(X,E)$ satisfying $D^{''}_{E}\vartheta=D_{H}^{'}\vartheta=0$, we have $$\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta_{1},\vartheta\rangle_{H}\frac{\omega^{n}}{n!} =\int_{X}\sqrt{-1}\partial\langle\beta_{1}^{0,1},\vartheta\rangle_{H}\wedge\frac{\omega^{n-1}}{(n-1)!} =0.$$ Assume $\gamma_{1}\in\Gamma(X,E)$ is a solution of $$\sqrt{-1}\Lambda_{\omega}D^{'}_{H}D^{''}_{E}\gamma_{1}=-\sqrt{-1}\Lambda_{\omega}D^{'}_{H}\beta_{1}.$$ Let $\tilde{\beta}_{1}=\beta_{1}+D\gamma_{1}$, then $\sqrt{-1}\Lambda_{\omega}D_{E}^{''}\tilde{\beta}_{1}=0$ and $\sqrt{-1}\Lambda_{\omega}D_{H}^{'}\tilde{\beta}_{1}=0$. Similar to (\[eq:61\]), we obtain $$\begin{split} 0=&\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},DD^{c}_{H}+D_{H}^{c}D]\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ =&\int_{X}\langle\sqrt{-1}\Lambda_{\omega}DD^{c}_{H}\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}. \end{split}$$ So $$\label{new:1} \begin{split} -2\int_{X}|D^{''}_{E}\tilde{\beta}_{1}|^{2}_{H,\omega}\frac{\omega^{n}}{n!}-2\int_{X}\langle\bar{\tau}^{*}D^{''}_{E}\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}=0, \end{split}$$ and $$\label{new:2} \begin{split} -2\int_{X}|D^{'}_{E}\tilde{\beta}_{1}|^{2}_{H,\omega}\frac{\omega^{n}}{n!}-2\int_{X}\langle\tau^{*}D^{'}_{E}\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}=0. \end{split}$$ Set $A=\int_{X}\langle\bar{\tau}^{*}D^{''}_{E}\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}+\int_{X}\langle\tau^{*}D^{'}_{E}\tilde{\beta}_{1},\tilde{\beta}_{1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}$. Then $$\begin{split} A=&-\frac{1}{(n-2)!}\int_{X}\partial\omega^{n-2}\wedge \langle\bar{\partial}_{E}\tilde{\beta}_{1}^{1,0},\tilde{\beta}_{1}^{1,0}\rangle_{H} -\frac{1}{(n-2)!}\int_{X}\partial\omega^{n-2}\wedge \langle\theta(\tilde{\beta}_{1}^{0,1}),\tilde{\beta}^{1,0}\rangle_{H}\\ &+\frac{1}{(n-2)!}\int_{X}\partial\omega^{n-2}\wedge \langle\bar{\partial}_{E}\tilde{\beta}_{1}^{0,1},\tilde{\beta}_{1}^{0,1}\rangle_{H} -\frac{1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\partial_{H}\tilde{\beta}^{0,1},\tilde{\beta}^{0,1}\rangle_{H}\\ &-\frac{1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\theta^{*}(\tilde{\beta}^{1,0}),\tilde{\beta}^{0,1}\rangle_{H} +\frac{1}{(n-2)!}\int_{X} \bar{\partial}\omega^{n-2}\wedge \langle\partial_{H}\tilde{\beta}^{1,0},\tilde{\beta}^{1,0}\rangle_{H}. \end{split}$$ It is easy to check that $\bar{A}=-A$, so it is imaginary. Adding (\[new:1\]) and (\[new:2\]), we get $A=0$ and $$\int_{X}|D^{''}_{E}\tilde{\beta}_{1}|^{2}_{H}\frac{\omega^{n}}{n!}+\int_{X}|D^{'}_{E}\tilde{\beta}_{1}|^{2}_{H}\frac{\omega^{n}}{n!}=0.$$ Hence $[\tilde{\beta}_{1}]\in H^{1}_{Dol}(X,E)$. From Higgs bundle to flat bundle -------------------------------- First, we construct a map $j:\mathcal{C}_{Dol}\rightarrow \mathcal{C}_{DR}$ in rank $2$ case. Let $(E,\bar{\partial}_{E},\theta)$ be a semi-stable Higgs bundle with vanishing Chern numbers. Now we discuss the following two cases that may occur. [**Case 1.**]{} $(E,\bar{\partial}_{E},\theta)$ is polystable. By Theorem \[thm:2-1\], there is a flat connection on $E$ denoted by $D_{E}$ and $(E,D_{E})$ is semi-simple. Define $j$ mapping from the equivalent class of $(E,\bar{\partial}_{E},\theta)$ to the equivalent class of $(E,D_{E})$. [**Case 2.**]{} $(E,\bar{\partial}_{E},\theta)$ is strictly semi-stable. By Theorem \[thm:1\], there is a filtration $$0\subset S\subset E.$$ It induces an exact sequence of Higgs bundles $$\label{exact:1} 0\rightarrow (S,\bar{\partial}_{S},\theta_{S})\rightarrow (E,\bar{\partial}_{E},\theta)\rightarrow (Q,\bar{\partial}_{Q},\theta_{Q})\rightarrow 0,$$ where $(S,\theta_{S})$ and $(Q,\theta_{Q})$ are stable Higgs bundles with vanishing Chern numbers. The Hitchin-Simpson connections on $S$ and $Q$ defined by the Hermitian-Einstein metrics are flat. For simplicity, denote them by $D_{S}$ and $D_{Q}$. In general, there is no such a metric on $E$. We want to find a nature flat connection on $E$. For the exact sequence (\[exact:1\]), it determines a Higgs extension class in $H^{1}_{Dol}(Q^{*}\otimes S)$. Choose a $C^{\infty}$ splitting $f:S\oplus Q\rightarrow E$. The pull-back of $\bar{\partial}_{E}$ and $\theta$ on $S\oplus Q$ can be written as: $$f^{*}(\bar{\partial}_{E})=\left(\begin{split} &\bar{\partial}_{S} &\gamma \\ &0 &\bar{\partial}_{Q} \end{split}\right),\ \ f^{*}(\theta)=\left(\begin{split} &\theta_{S} &\xi \\ &0 &\theta_{Q} \end{split}\right),$$ where $\gamma\in\Omega^{0,1}(Q^{*}\otimes S)$ is the second fundamental form and $\xi\in\Omega^{1,0}(Q^{*}\otimes S)$. Set $\beta=\gamma+\xi$, one can check that $D^{''}_{Q^{*}\otimes S}\beta=0$. The Higgs extension class can be presented by $\beta$. It is well-known that there is an one-to-one correspondence between the equivalence class of Higgs extensions of $(Q,\theta_{Q})$ by $(S,\theta_{S})$ and the elements in $H^{1}_{Dol}(Q^{*}\otimes S)$. Let $H_{Q^{*}\otimes S}$ be the Higgs-Hermitian flat metric on $Q^{*}\otimes S$, $D_{Q^{*}\otimes S}$ be the related flat connection. Because of Lemma \[nl:1\], there is a $\tilde{\beta}=\beta+D^{''}\gamma\in[\beta]$ satisfying $D_{Q^{*}\otimes S}\tilde{\beta}=0$. Define $\tilde{f}:S\oplus Q\rightarrow E$ to be $$\tilde{f}=f\circ \left(\begin{split} &Id_{S} &\gamma \\ &0 &Id_{Q} \end{split}\right),$$ then $$\tilde{f}^{*}(D_{E}^{''})=\left(\begin{split} &D_{S}^{''} &\tilde{\beta} \\ &0 &D_{Q}^{''} \end{split}\right).$$ Define the flat connection $D_{E}$ on $E$ by $$\tilde{f}^{*}(D_{E})=\left(\begin{split} &D_{S} &\tilde{\beta} \\ &0 &D_{Q} \end{split}\right),$$ then $(E,D_{E})$ is flat and $$0\rightarrow (S,D_{H_{s},\theta_{S}})\rightarrow (E,D_{E})\rightarrow (Q,D_{H_{Q},\theta})\rightarrow 0$$ is an exact sequence of flat bundles. Define $j$ mapping from the equivalent class of $(E,\bar{\partial}_{E},\theta)$ to the equivalent class of $(E,D_{E})$. We claim this map is well-defined. Suppose we have another filtration $$0\subset \tilde{S}\subset E.$$ The induced exact sequence is $$0\rightarrow (\tilde{S},\bar{\partial}_{\tilde{S}},\theta_{\tilde{S}})\rightarrow (E,\bar{\partial}_{E},\theta)\rightarrow (\tilde{Q},\bar{\partial}_{\tilde{Q}},\theta_{\tilde{Q}})\rightarrow 0.$$ Choose a suitable splitting $\tilde{g}:\tilde{S}\oplus \tilde{Q}\rightarrow E$. The pull-back of $D_{E}^{''}$ is $$\tilde{g}^{*}(D_{E}^{''})=\left(\begin{split} &D_{\tilde{S}}^{''} &\tilde{\rho} \\ &0 &D_{\tilde{Q}}^{''} \end{split}\right),$$ and $D^{'}\tilde{\rho}=0$. Define a flat connection on $E$ by $$\tilde{g}^{*}(\tilde{D}_{E})=\left(\begin{split} &D_{\tilde{S}} &\tilde{\rho} \\ &0 &D_{\tilde{Q}} \end{split}\right).$$ Now we need to show $(E,D_{E})\simeq(E,\tilde{D}_{E})$. First, we have a Higgs isomorphism $$P=\tilde{g}^{-1}\circ \tilde{f}:(S\oplus Q,\left(\begin{split} &D_{S}^{''} &\tilde{\beta} \\ &0 &D_{Q}^{''} \end{split}\right))\rightarrow (\tilde{S}\oplus \tilde{Q},\left(\begin{split} &D_{\tilde{S}}^{''} &\tilde{\rho} \\ &0 &D_{\tilde{Q}}^{''} \end{split}\right)).$$ That is, $$P\circ\left(\begin{split} &D_{S}^{''} &\tilde{\beta} \\ &0 &D_{Q}^{''} \end{split}\right)=\left(\begin{split} &D_{\tilde{S}}^{''} &\tilde{\rho} \\ &0 &D_{\tilde{Q}}^{''} \end{split}\right)\circ P.$$ Let $$P=\left(\begin{split} &P_{1}^{1} &P_{1}^{2} \\ &P_{2}^{1} &P_{2}^{2} \end{split}\right),$$ then $$\left(\begin{split} &P_{1}^{1}\circ D_{S}^{''} &P_{1}^{1}\circ\tilde{\beta}+P_{1}^{2}\circ D_{Q}^{''} \\ &P_{2}^{1}\circ D_{S}^{''} &P_{2}^{1}\circ\tilde{\beta}+P_{2}^{2}\circ D_{Q}^{''} \end{split}\right)=\left(\begin{split} &D_{\tilde{S}}^{''}\circ P_{1}^{1}+\tilde{\rho}\circ P_{2}^{1} &D_{\tilde{S}}^{''}\circ P_{1}^{2}+\tilde{\rho}\circ P_{2}^{2} \\ &\qquad D_{\tilde{Q}}^{''}\circ P_{2}^{1} &D_{\tilde{Q}}^{''}\circ P_{2}^{2}\qquad \end{split}\right).$$ Comparing both sides of this equation, we get $$\label{neweq:1} \left\{\begin{split} &D^{''}(P_{2}^{1})=0;\\ &D^{''}(P_{1}^{1})+\tilde{\rho}\circ P_{2}^{1}=0;\\ &D^{''}(P_{2}^{2})-P_{2}^{1}\circ\tilde{\beta}=0;\\ &D^{''}(P_{1}^{2})+\tilde{\rho}\circ P_{2}^{2}-P_{1}^{1}\circ\tilde{\beta}=0. \end{split}\right.$$ Using $D^{''}(P_{2}^{1})=0$, we know $\sqrt{-1}\Lambda_{\omega}D^{'}D^{''}(P_{2}^{1})=0$. This means $D^{'}(P_{2}^{1})=0$. By $D^{'}\tilde{\rho}=D^{'}\tilde{\beta}=0$, one can obtain $$\sqrt{-1}\Lambda_{\omega}D^{'}D^{''}(P_{1}^{1})=0, \ \sqrt{-1}\Lambda_{\omega}D^{'}D^{''}(P_{2}^{2})=0.$$ So $D^{'}(P_{1}^{1})=D^{''}(P_{1}^{1})=0$ and $D^{'}(P_{2}^{2})=D^{''}(P_{2}^{2})=0$. From equation (\[neweq:1\]) and $D^{'}\tilde{\rho}=D^{'}\tilde{\beta}=0$, we also have $\sqrt{-1}\Lambda_{\omega}D^{'}D^{''}(P_{1}^{2})=0$. Then $D^{'}(P_{1}^{2})=0$ and $D^{''}(P_{1}^{2})=0$. Together with all of the above, we see $$P\circ\left(\begin{split} &D_{S}^{'} &0 \\ &0 &D_{Q}^{'} \end{split}\right)=\left(\begin{split} &D_{\tilde{S}}^{'} &0 \\ &0 &D_{\tilde{Q}}^{'} \end{split}\right)\circ P.$$ Hence, $(E,D_{E})\simeq(E,\tilde{D}_{E})$. From flat bundle to Higgs bundle -------------------------------- Now we construct a map $i:\mathcal{C}_{DR}\rightarrow \mathcal{C}_{Dol}$ in rank $2$ case. Suppose $(V,D)$ is a flat bundle with $\mbox{rank}(V)=2$. There are also two cases to consider. [**Case 1.**]{} $(V,D)$ is semi-simple. By Theorem \[thm:2-1\], there is a Higgs structure on $V$ denoted by $(\bar{\partial}_{V},\theta_{V})$ and $(V,\bar{\partial}_{V},\theta_{V})$ is polystable with vanishing Chern numbers. Define $i$ mapping from the equivalent class of $(V,D)$ to the equivalent class of $(V,\bar{\partial}_{V},\theta_{V})$. [**Case 2.**]{} $(V,D)$ is not semi-simple. Then there is a flat subbundle $(S,D_{S})$ of $(V,D)$, and we have the following exact sequence of flat bundles $$0\rightarrow (S,D_{S})\rightarrow (V,D)\rightarrow (Q,D_{Q})\rightarrow 0.$$ It is obvious that $(S,D_{S})$ and $(Q,D_{Q})$ are simple flat bundles. According to Theorem \[thm:2-1\], we can get Higgs structures on $S$ and $Q$ by choosing good metric, denoted by $(\bar{\partial}_{S},\theta_{S})$ and $(\bar{\partial}_{Q},\theta_{Q})$. For such an exact sequence of flat bundles, it determines a flat extension class in $H^{1}_{DR}(Q^{*}\otimes S)$. Choose a $C^{\infty}$ splitting $f:S\oplus Q\rightarrow V$. The pull-back of $D$ can be expressed as $$f^{*}(D)=\left(\begin{split} &D_{S} &\beta \\ &0 &D_{Q} \end{split}\right),$$ and one can check that $D_{Q^{*}\otimes S}\beta=0$. The extension class can be presented by $\beta$. Let $H_{Q^{*}\otimes S}$ be the metric on $Q^{*}\otimes S$ satisfying $G_{H_{Q^{*}\otimes S}}=0$, $(\bar{\partial}_{Q^{*}\otimes S},\theta_{Q^{*}\otimes S})$ be the related Higgs structure. By Lemma \[nl:1\], there is a $\tilde{\beta}=\beta+D\gamma\in [\beta]$ satisfying $D_{Q^{*}\otimes S}^{''}\tilde{\beta}=0$. Define $\tilde{f}:S\oplus Q\rightarrow E$ to be $$\tilde{f}=f\circ \left(\begin{split} &Id_{S} &\gamma \\ &0 &Id_{Q} \end{split}\right),$$ then $$\tilde{f}^{*}(D_{E})=\left(\begin{split} &D_{S} &\tilde{\beta} \\ &0 &D_{Q} \end{split}\right).$$ Define the holomorphic structure $\bar{\partial}_{V}$ and Higgs field $\theta_{V}$ on $V$ by $$\tilde{f}^{*}(\bar{\partial}_{V})=\left(\begin{split} &\bar{\partial}_{S} &\tilde{\beta}^{0,1} \\ &0 &\bar{\partial}_{Q} \end{split}\right) \quad \mbox{and} \quad \tilde{f}^{*}(\theta_{V})=\left(\begin{split} &\theta_{S} &\tilde{\beta}^{1,0} \\ &0 &\theta_{Q} \end{split}\right).$$ Then $(V,\bar{\partial}_{V},\theta_{V})$ is a semi-stable Higgs bundle with vanishing Chern numbers and $$0\rightarrow (S,\bar{\partial}_{S},\theta_{S})\rightarrow (V,\bar{\partial}_{V},\theta_{V})\rightarrow (Q,\bar{\partial}_{Q},\theta_{Q})\rightarrow 0$$ is an exact sequence of Higgs bundles. Define $i$ mapping from the equivalent class of $(V,D)$ to the equivalent class of $(V,\bar{\partial}_{V},\theta_{V})$, and we can also show that this definition is well-defined, which we omit here. From above, we define two maps $i$ and $j$. It is easy to see $i\circ j=Id_{\mathcal{C}_{Dol}}$ and $j\circ i=Id_{\mathcal{C}_{DR}}$. So $i$ and $j$ are one-to-one maps, this finishes the proof. Proof of Theorem \[thm:2-3\] ============================ In this section, we assume that $(X,\omega)$ is a Kähler manifold. Let $(E,\bar{\partial}_{E},\theta)$ be a semi-stable Higgs bundle over $X$ with $ch_{1}(E,\bar{\partial}_{E})[\omega^{n-1}]=ch_{2}(E,\bar{\partial}_{E})[\omega^{n-2}]=0$. Then there is a filtration $$0=E_{0}\subset E_{1}\subset\cdots\subset E_{l}=E,$$ such that $Q_{i}=E_{i}/E_{i-1}$ associated with the induced Higgs field is a stable Higgs bundle and $ch_{1}(Q_{i})[\omega^{n-1}]=ch_{2}(Q_{i})[\omega^{n-2}]=0$, $i=1,\cdots,l$. Take a smooth splitting $f:\oplus_{i=1}^{l}Q_{i}\rightarrow E$. The pull back of $D_{E}^{''}$ can be written as $$\begin{split} f^{*}(D_{E}^{''})=\left(\begin{array}{ccccc} D_{Q_{1}}^{''} &\cdots & \beta_{1}^{l} \\ \vdots &\ddots & \vdots\\ 0 & \cdots & D_{Q_{l}}^{''} \end{array}\right). \end{split}$$ The condition $(D_{E}^{''})^{2}=0$ implies $$\label{eq:7-1} \left\{\begin{split} &(D_{Q_{i}}^{''})^{2}=0, \ \ i=1,\cdots,l;\\ &D^{''}(\beta_{i}^{i+1})=0,\ \ i=1,\cdots,l-1;\\ &D^{''}(\beta_{i}^{j})+\sum_{k=i+1}^{j-1}\beta_{i}^{k}\wedge\beta_{k}^{j}=0,\ \ 1\leq i\leq j-2\leq l-2. \end{split}\right.$$ \[l:7-1\] There exists a smooth splitting $\tilde{f}:\oplus_{i=1}^{l}Q_{i}\rightarrow E$, such that $\tilde{\beta}_{i}^{j}$ satisfies the equation (\[eq:7-1\]) and $D^{'}\tilde{\beta}_{i}^{j}=0$ for $1\leq i<j\leq l$. Let $\tilde{\beta}_{i}^{i+1}=\beta_{i}^{i+1}+D^{''}\gamma_{i}^{i+1}$, where $\gamma_{i}^{i+1}\in \Gamma(X,Q_{i+1}^{*}\otimes Q_{i})$ satisfies $$\sqrt{-1}\Lambda_{\omega}D^{'}D^{''}\gamma_{i}^{i+1}=-\sqrt{-1}\Lambda_{\omega}D^{'}\beta_{i}^{i+1}.$$ Then by (\[eq:61\]), $D^{'}\tilde{\beta}_{i}^{i+1}=0$. Let $h_{i}^{i+1}:\oplus_{k=1}^{l}Q_{k}\rightarrow \oplus_{k=1}^{l}Q_{k}$ and $h_{i}^{i+1}=\oplus_{k=1}^{l}Id_{Q_{k}}\oplus\gamma_{i}^{i+1}$. Define $f_{1}=f\circ h_{1}^{2}\circ\cdots \circ h_{l-1}^{l}$. Then the straightforward computations show us $$\begin{split} f_{1}^{*}(D_{E}^{''})=\left(\begin{array}{ccccc} D_{Q_{1}}^{''} & \tilde{\beta}_{1}^{2} & \cdots &\cdots & * \\ 0 & D_{Q_{2}}^{''} & \tilde{\beta}_{2}^{3} & \cdots & * \\ \vdots & \ddots&\ddots &\ddots & \vdots\\ 0 & \cdots &0 & D_{Q_{l-1}}^{''} & \tilde{\beta}_{l-1}^{l}\\ 0 & \cdots& \cdots & 0 & D_{Q_{l}}^{''} \end{array}\right). \end{split}$$ Inductively, suppose we construct a splitting $f_{p}$, such that $\beta_{i}^{j}$ satisfies the equation (\[eq:7-1\]) and $D^{'}\beta_{i}^{j}=0$ for $1\leq j-i\leq p$. Let $\tilde{\beta}_{i}^{i+p+1}=\beta_{i}^{i+p+1}+D^{''}\gamma_{i}^{i+p+1}$, where $\gamma_{i}^{i+p+1}\in \Gamma(X,Q_{i+p+1}^{*}\otimes Q_{i})$ satisfies $$\label{eq:7-2} \sqrt{-1}\Lambda_{\omega}D^{'}D^{''}\gamma_{i}^{i+p+1}=-\sqrt{-1}\Lambda_{\omega}D^{'}\beta_{i}^{i+p+1}.$$ This equation can be solved in Kähler manifolds case. Calculating directly deduces $$\begin{split} 0&=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},(D^{''}+D^{'})^{2}]\tilde{\beta}_{i}^{i+p+1},\tilde{\beta}_{i}^{i+p+1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}\Lambda_{\omega}D^{''}D^{'}\tilde{\beta}_{i}^{i+p+1},\tilde{\beta}_{i}^{i+p+1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}\langle\sqrt{-1}[\Lambda_{\omega},D^{''}]D^{'}\tilde{\beta}_{i}^{i+p+1},\tilde{\beta}_{i}^{i+p+1}\rangle_{H,\omega}\frac{\omega^{n}}{n!}\\ &=\int_{X}|D^{'}\tilde{\beta}_{i}^{i+p+1}|^{2}_{H,\omega}\frac{\omega^{n}}{n!}, \end{split}$$ where in the second equality we have used the condition $D^{''}\tilde{\beta}_{i}^{i+p+1}=D^{''}\beta_{i}^{i+p+1}=-\sum_{k=i+1}^{i+p}\beta_{i}^{k}\wedge \beta_{k}^{i+p+1}$ and $D^{'}\beta_{i}^{j}=0$ for $1\leq j-i\leq p$. Thus $D^{'}\tilde{\beta}_{i}^{i+p+1}=0$. Set $h_{i}^{i+p+1}=\oplus_{k=1}^{l}Id_{Q_{k}}\oplus\gamma_{i}^{i+p+1}$. Define $f_{p+1}=f_{p}\circ h_{1}^{2+p}\circ\cdots \circ h_{l-p-1}^{l}$. Then $f_{p+1}$ is a splitting satisfying equation (\[eq:7-1\]) and $D^{'}\beta_{i}^{j}=0$ for $1\leq j-i\leq p+1$. Let $\tilde{f}=f_{l-1}$, then $\tilde{f}$ is a splitting satisfying the conditions in this lemma. \[rmk:7-1\] In non-Kähler manifolds case, the equation (\[eq:7-2\]) may have no solutions because the right hand side of the equation may not satisfy the integrability condition. According to Lemma \[l:7-1\], we can define a flat connection on $E$ by $$\begin{split} \tilde{f}^{*}(D_{E})=\left(\begin{array}{ccccc} D_{Q_{1}} & \tilde{\beta}_{1}^{2} & \cdots &\cdots & \tilde{\beta}_{1}^{l} \\ 0 & D_{Q_{2}} & \tilde{\beta}_{2}^{3} & \cdots & \tilde{\beta}_{2}^{l} \\ \vdots & \ddots&\ddots &\ddots & \vdots\\ 0 & \cdots &0 & D_{Q_{l-1}} & \tilde{\beta}_{l-1}^{l}\\ 0 & \cdots& \cdots & 0 & D_{Q_{l}} \end{array}\right). \end{split}$$ Then we can define a map $j:\mathcal{C}_{Dol}\rightarrow \mathcal{C}_{DR}$. Next, we will show this map is well-defined. Suppose we have another filtration $$0=\tilde{E}_{0}\subset \tilde{E}_{1}\subset\cdots\subset \tilde{E}_{l}=E.$$ Let $\tilde{Q}_{i}=\tilde{E}_{i}/\tilde{E}_{i-1}$ and choose a suitable splitting $\tilde{g}:\oplus_{i=1}^{l}\tilde{Q}_{i}\rightarrow E$, such that it satisfies the equations in Lemma \[l:7-1\]. Then we can define a flat connection $\tilde{D}_{E}$ on $E$. Assume the pull-back of $D_{E}^{''}$ is $$\begin{split} \tilde{g}^{*}(D_{E}^{''})=\left(\begin{array}{ccccc} D_{\tilde{Q}_{1}}^{''} &\cdots & \tilde{\rho}_{1}^{l} \\ \vdots &\ddots & \vdots\\ 0 & \cdots & D_{\tilde{Q}_{l}}^{''} \end{array}\right). \end{split}$$ Set $P=\tilde{g}^{-1}\circ \tilde{f}=(P_{i}^{j})$, where $P_{i}^{j}\in \Gamma(X,Q_{j}^{*}\otimes \tilde{Q}_{i})$. Then $$P\circ\left(\begin{array}{ccccc} D_{Q_{1}}^{''} &\cdots & \tilde{\beta}_{1}^{l} \\ \vdots &\ddots & \vdots\\ 0 & \cdots & D_{Q_{l}}^{''} \end{array}\right)=\left(\begin{array}{ccccc} D_{\tilde{Q}_{1}}^{''} &\cdots & \tilde{\rho}_{1}^{l} \\ \vdots &\ddots & \vdots\\ 0 & \cdots & D_{\tilde{Q}_{l}}^{''} \end{array}\right)\circ P.$$ This means $$\left\{\begin{split} &D^{''}(P_{l}^{1})=0;\\ &D^{''}(P_{i}^{1})+\sum_{k=i+1}^{l}\tilde{\rho}_{i}^{k}\circ P_{k}^{1}=0,\ \ 1\leq i\leq l-1;\\ &D^{''}(P_{l}^{j})-\sum_{k=1}^{j-1}P_{l}^{k}\circ\tilde{\beta}_{k}^{j}=0,\ \ 2\leq j<l;\\ &D^{''}( P_{i}^{j})+\sum_{k=i+1}^{l}\tilde{\rho}_{i}^{k}\circ P_{k}^{j}-\sum_{k=1}^{j-1}P_{i}^{k}\circ\tilde{\beta}_{k}^{j}=0,\ \ 1\leq i\leq l-1,\ 2\leq j\leq l. \end{split}\right.$$ Using $D^{''}(P_{l}^{1})=0$, similar to the argument in Section 6, we can show $D^{'}(P_{l}^{1})=0$. By induction, we can prove $D^{'}P_{i}^{j}=0$ for $1\leq i,j\leq l$, which implies $(E,D_{E})\simeq (E,\tilde{D}_{E})$. 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--- abstract: 'A mapping study of IRAS 05553+1631 was performed with $^{12}$CO J=3-2 and $^{13}$CO J=2-1 lines observed by the KOSMA 3 m-telescope. A core with a size of 0.65 pc and with a LTE mass of 120 M$_\odot$ was defined by the mapping with $^{13}$CO J=2-1 line. We have identified a bipolar outflow with $^{12}$CO J=3-2. For accuracy in the calculation of outflow parameters, overcoming the projection effect is important. We propose a new method to directly calculate the inclination-angle $\theta$. We establish two basic equations with the help of outflow contour diagram and finally obtain the “angle function” and the “angle equation” to derive $\theta$. We apply our method to the outflow of IRAS 05553+1631, finding that $\theta_{blue}$ is 73$^\circ$ and $\theta_{red}$ is 78$^\circ$. Compared to the parameters initially estimated under an assumption of 45$^\circ$ inclination-angle, the newly derived parameters are changed with different factors. For instance, the timescales for the blue and the red lobes are reduced by 0.31 and 0.21, respectively. Larger influences apply to mechanical luminosity, driving force, and mass-loss rate. The comparisons between parameters before and after the correction show that the effect of the inclination-angle cannot be neglected.' author: - | Shuo Kong$^{1}$[^1], Yuefang Wu$^{2}$[^2]\ $^{1}$School of Physics, Peking University, Beijing 100871, China.\ $^{2}$Department of Astronomy, School of Physics, Peking University, Beijing 100871, China. date: Accepted by the Monthly Notices of the Royal Astronomical Society title: 'Inclination-angle of the outflow in IRAS 05553+1631: A method to correct the projection effect' --- \[firstpage\] stars: formation – ISM: jets and outflows. Introduction ============ Massive star formation (MSF) has attracted much attention. It has enormous impact on the natal interstellar medium (ISM) and on the evolution of Galaxy. MSF is believed to originate in dense molecular cores (DMCs) that can be traced by CO and its isotopologues. With the help of mapping, we can reveal the structure of molecular cores and investigate the MSF activity taking place within them. High velocity outflows have also been intensively studied. First uncovered in 1976 [@b19; @b8], molecular outflows toward massive young stellar objects (YSOs) have attracted much attention [@b16]. Perhaps tracing the earliest stage [@b9] of star formation, molecular outflows are critical in the debate about two mechanisms for MSF: massive stars forming through accretion-disk-outflow or via collision-coalescence [@b22; @b23]. The properties of outflows reveal the mass-loss phase before the main sequence. What we observe in the sky is not the real outflow but its two-dimensional projection. Therefore, it is essential to know the inclination-angle $\theta$ between the outflow axis and our line-of-sight. For instance, it will introduce a factor of 1/cos$\theta$ to the momentum. So far the determination of the inclination-angle has been somewhat neglected though some previous researchers have made a few attempts by modeling [@b3; @b10]. Until now, authors usually took assumptions for the inclination-angle. For instance, @b7 hypothesized 45$^\circ$ while @b6 supposed 60$^\circ$. The outflow parameters derived from these assumptions can only be meaningful statistically, but they are not accurate. In addition, work has been done to study the collimation of outflows [@b1; @b15]. However, the projection effect should also be included. In order to directly calculate the inclination-angle and to eliminate or reduce outflow parameter uncertainties, we propose a new method in this paper. In the next section, we will describe our observations of IRAS 05553+1631. In section \[sec:results\], the results will be presented. We will introduce our method and discuss some of its properties in section \[sec:discussion\]. Section \[sec:summary\] is a brief summary of our work. Observation =========== The observations were performed with the KOSMA 3 m submillimeter telescope at the Gornergrat Observatory in Switzerland in 2002 to 2004. The dual-channel-SIS receiver was used. We observed simultaneously transitions in the 230 GHz tuning range ($^{13}$CO J=2-1) and in the 345 GHz tuning range ($^{12}$CO J=3-2). The beam sizes were 120“ at 230 GHz and 80” at 345 GHz. The beam efficiencies for 230 GHz is 0.68 and for 345GHz is 0.72. The integration time for each point was about 1.5 min for both $^{12}$CO J=3-2 and $^{13}$CO J=2-1. The system temperatures were about 160 K for $^{13}$CO J=2-1 and 235 K for $^{12}$CO J=3-2. The spectral resolutions were 0.22 km/s and 0.29 km/s for $^{13}$CO J=2-1 and $^{12}$CO J=3-2, respectively. We used the position-switch mode in the observations. The map step size is 1 arcmin. Results {#sec:results} ======= Both $^{13}$CO J=2-1 and $^{12}$CO J=3-2 emission were detected and mapped. The typical $^{13}$CO spectrum is shown in the left panel of Figure \[fig:13CO\]. There is a single component in this molecular region. Table \[tab:observed\] summarizes the parameters. Column (1) is the IRAS source name. Columns (2) to (5) are the positions of the source, including both the 1950 coordinates and the 2000 coordinates. Columns (6) to (8) list the $^{13}$CO J=2-1 spectrum parameters derived from Gaussian fitting. Columns (9) to (11) exhibit the IRAS flux parameters. According to @b14, IRAS sources that satisfy Log(F$_{25}$/F$_{12}$)$>$0.57, Log($F_{60}$/F$_{12}$)$>$1.30 and that peak at 100 $\mu m$ are promising candidates for UC H[II]{} regions. From Table \[tab:observed\] we can see that the first two criteria are satisfied. We confirm that IRAS 05553+1631 has its maximum flux at 100 $\mu m$. The results show that IRAS 05553+1631 is a candidate for UC H[II]{} region. The right panel of figure \[fig:13CO\] shows the core contours. The grey-scale background is MSX 8.28 $\mu m$ emission. The three MSX sources are named as S$_1$, S$_2$, and S$_3$, respectively. Spatially, S$_1$ and S$_2$ are both possibly associated with the core. We also made use of the Two Micron All Sky Survey (2MASS) data. S$_1$ has one 2MASS counterpart 05581473+1631070 whose magnitudes in the J, H, and K$_s$ bands are 13.103, 10.752, and 9.041, respectively; S$_2$ has one 2MASS counterpart 05581574+1631373 whose magnitudes in the J, H, and K$_s$ bands are 12.571, 11.886, and 11.565, respectively. S$_1$ is the reddest source and more likely to be correlated with the molecular core. Figure \[fig:12CO\] presents the $^{12}$CO J=3-2 results. In the P-V diagrams (c) and (d), the wide wings are obvious. The velocity ranges of the high-velocity gas were determined as -8 km s$^{-1}$ to 3.1 km s$^{-1}$ and 8.5 km s$^{-1}$ to 13 km s$^{-1}$ (hereafter refered to as HV-ranges) for blue and red lobes, respectively. The HV-ranges were integrated into the outflow contour diagram (Figure \[fig:outflow\]). The outermost blue and red contours are 50 per cent of the blue and red peak intensities, respectively. The contours show a bipolar structure. @b13 has identified YSO G192.16 as the driving source. Both their work (see Figure 1 of their paper) and our result (Figure \[fig:outflow\]) show that the blue lobe is more extended from G192.16 than the red one. G192.16 is in good alignment between the two outflow lobes, and we adopt G192.16 as the driving centre. The core parameters are summarized in Table \[tab:core\]. Column (2) lists the 1.4 kpc distance calculated using the Galactic rotation model [@b2] while @b13 gave a distance of 2.0 kpc. Column (3) gives the velocity range for the $^{13}$CO J=2-1 line. Column (4) displays the core radius which is half the mean value of two linear lengths of the half maximum contour. Columns (5) to (8) are derived from the radiation transfer equation. A ratio of 8.9$\times$10$^5$ for \[H$_2$/$^{13}$CO\] was adopted to calculate the H$_2$ column density (column 8). In column (9), we give the H$_2$ density calculated from the column density and the diameter (which is twice the radius). In the calculation of LTE mass which is shown in column (10), the mean atomic weight factor of 1.36 was introduced [@b6]. In column (11), the virial mass was calculated following @b5. Table \[tab:outflow\] summarizes the outflow parameters. Column (1) indicates the outflow lobes. Column (2) lists the outflow sizes. The thick contour was covered by an ellipse whose major axis is the linear length from west to east and whose minor axis is the linear length from north to south. The outflow sizes are the lengths of semimajor axes. Column (3) lists the masses for blueshifted and redshifted gas, respectively. The beam averaged column density for the $^{12}$CO J=3-2 transition is $$\overline N=4.8\times10^{12}\frac{(T_{ex}+0.92)}{exp(-33.2/T_{ex})}\int\frac{T_a^*}{\eta_b}\frac{\tau}{[1-exp(-\tau)]}\mathrm{d}v$$ where T$_{ex}$ is the core excitation temperature. The integration is over the HV-range. The $^{12}$CO J=3-2 transition is usually optically thick even in the line wings [@b12], therefore we adopted a mean optical depth $\overline{\tau}$ of 4 [@b6]. For the mass calculation, an \[H$_2$/CO\] ratio of 10$^4$ was taken. Momentum and energy are displayed in columns (4) and (5). The adopted velocity $v$ is the line-of-sight velocity divided by cos$\theta$ where $\theta$ is the inclination-angle. Column (6) shows the dynamical timescales which were estimated as R/$v$ where R is the distance from the driving centre to the peak of high-velocity gas. The mechanical luminosity, driving force, and mass-loss rate are listed in columns (7) to (9), respectively. They were calculated as E/t, P/t, and P/(tV$_w$), respectively [@b17], where V$_w$ is the wind velocity assumed to be 100 km/s. In the calculation, we first assumed $\theta$=45$^\circ$, the calculation will be improved with our method (see section \[sec:outflow\]). Discussion {#sec:discussion} ========== Core ---- Table \[tab:core\] shows that the core has a size $\sim$0.65 pc and that M$_{LTE}$ is equal to 120 $M_{\odot}$. The $^{13}$CO line width is 2.5 km s$^{-1}$ which is larger than those of low-mass cores [@b11]. These results suggest that the core is massive. According to virial theory, a core is gravitationally bound when its mass is larger than its virial mass. In our results, this is not satisfied, which is possibly due to the \[H$_2$/$^{13}$CO\] ratio adopted or the optically thick assumption for $^{13}$CO J=2-1 emission; alternatively, the core may be unbound. Outflow {#sec:outflow} ------- In our calculation for outflow parameters, 45$^\circ$ was assumed as the inclination-angle. Actually, almost all the outflow parameters depend on the angle. For instance, the velocity of outflow gas is the projection along our line-of-sight. Therefore a factor of 1/cos$\theta$ should be introduced. The same goes for momentum, timescale, etc., but sometimes with different factors. The inclination-angle also affects the collimation factor. Thus, accurate determination of $\theta$ is important. Unfortunately, until now, people assume a value for $\theta$ [@b7; @b6; @b18]. In the following, we propose a new method to improve this situation. ### The inclination-angle $\theta$ Our method to derive $\theta$ requires knowledge of the outflow geometry, so before the calculation, it is necessary to discuss something about the shapes of outflows. Generally, without impact from the surrounding materials, the molecular outflows are likely to be axially symmetric. This could be inferred from the symmetry of the outflow contour diagrams (@b7, Figure 2; @b17, Figure 5c; @b1, Figure 13 NGC 2071) and from modeling [@b20; @b10; @b21]. In our results, the P-V diagram (Figure \[fig:12CO\]) and the outflow contours (Figure \[fig:outflow\]) show good symmetry, especially the blue lobe. In addition, the ellipse-like outflow contours indicate a projection of a conical outflow (see Section \[sec:more\]). Since a cone is an ideal form for an axially symmetric outflow (see Section \[sec:comparison\], Paragraph 2), we assume that the shape of the outflow is a cone. We also assume that high velocity components form cone-shell like structure whose emission intensity decreases from inside to outside. The cone could be called a “cone onion”. When we observe, what we see is the outflow projection on the sky. Thus, if the outflow axis is at an angle $\theta$ to our line-of-sight, the contours will appear as ellipses (Figure \[fig:cone\]; the contours could also be parabola or hyperbola, see section \[sec:more\]). Figure \[fig:model\] presents the calculation of the inclination-angle $\theta$. In Figure \[fig:model\](a), one can see the driving centre and an elliptical outflow contour. From Figure \[fig:model\](b), a relation of the four angles: $\beta_1$, $\beta_2$, $\theta_1$, and $\theta$ can be expressed by equation (\[equ:4angle\]) $$\label{equ:4angle} \frac{\tan\theta_1}{\tan\theta}=\frac{\tan\beta_1}{\tan\beta_2}\approx\frac{\beta_1}{\beta_2}$$ where $\beta_1$ and $\beta_2$ are the angle distances in the contour-diagram ($\beta_1$ accounts for the angle between driving centre and point D; $\beta_2$ is the angle between driving centre and axis point A). $\beta_1$ and $\beta_2$ are usually very small (about several arcmin), thus the small angle approximation is valid. $\theta_1$ and $\theta$ are unknown. Then, we have to find another equation for either $\theta_1$ or $\theta$. Notice that the length of line AB does not change as a function of $\theta$. In Figure \[fig:model\](a), we have $$\label{equ:a} \overline{AB}=\overline{AO}\tan\alpha$$ where $\alpha$ is the angle between OA and OB. Overlines indicate length. If the outflow contour is not symmetric about line AO, we take $\alpha$ as half the angle $\angle BOC$. We have already assumed the shape of cone, so the opening angle $\theta$-$\theta_1$ is the same for the entire cone. Therefore in Figure \[fig:model\](b) we have $$\label{equ:b} \overline{AO_1}=\overline{AO_2}\sin\theta$$ $$\label{equ:c} \overline{AB}=\overline{AO_2}\tan(\theta-\theta_1)$$ where O$_1$ and O$_2$ are the same points as O in the plane of the sky. $\overline{AO_1}$ is equal to $\overline{AO}$. Thus, combining equations (\[equ:a\]), (\[equ:b\]), (\[equ:c\]), and substitute $\overline{AO_1}$ with $\overline{AO}$ we have $$\label{equ:alpha} \tan\alpha=\frac{\tan(\theta-\theta_1)}{\sin\theta}$$ Putting together equation (\[equ:4angle\]) and equation (\[equ:alpha\]), we can solve the inclination-angle $\theta$ by eliminating $\theta_1$. Since $\beta_1$, $\beta_2$, and $\alpha$ can be derived from the outflow contour, we try to further manipulate the two equations to obtain a better expression. We define the “angle constants” $$P=\frac{\beta_1}{\beta_2}$$ and $$Q=\tan\alpha$$ Combining equations (\[equ:4angle\]) and (\[equ:alpha\]) and eliminating $\theta_1$, we have $$\frac{1-P}{Q}\frac{1}{\cos\theta}-P\tan\theta\tan\theta=1$$ Defining the “angle function” $$\label{angle function} A(\theta)=\frac{1-P}{Q}\frac{1}{\cos\theta}-P\tan\theta\tan\theta,$$ after determining P and Q, we obtain the inclination-angle by solving the “angle equation” $$\label{angle equation} A(\theta)=1$$ ### Comparison {#sec:comparison} We utilize our method on IRAS 05553+1631. Figure \[fig:theta\] shows the derivation process. In general, the shapes and positions in Figure \[fig:theta\] are similar to those in our model (Figure \[fig:model\](a)). Table \[tab:comparison\] presents the comparison between newly derived and old outflow parameters. We find the inclination-angles to be 73$^\circ$ and 78$^\circ$ for blue and red lobes, respectively. This suggests that the two outflow lobes are in good alignment. In Table \[tab:comparison\], compared with the former results, the newly derived parameters have rather large corrections. For the blue lobe, the momentum is enlarged by a factor of (cos45$^\circ$/cos73$^\circ\approx$) 2.4, as the velocity is corrected by 1/cos$\theta$. The correction for energy is proportional to that of P$^2$ which is about 5.8. The timescale is reduced by a factor of (cot73$^\circ$/cot45$^\circ\approx$) 0.31. The correction factors for mechanical luminosity, driving force, and mass-loss rate are 19, 7.7, and 7.7, respectively. For the red lobe, the analysis is similar, but the factors are different. These results suggest that the correction from inclination-angle cannot be neglected and our method is practical. From the process of our method one can see that the cone assumption ensure the viability of equation (\[equ:alpha\]). If not, the equation would be an approximation. Let $\theta_0$=$\theta$-$\theta_1$, the Taylor expansion of tan$\theta$ around $\theta_0$ is (keeping the first order) tan$\theta$=tan$\theta_0$+$\frac{\Delta\theta}{(cos\theta_0)^2}$. For the blue outflow lobe in our case, tan$\theta_0$=0.143, if $\Delta\theta$=1$^\circ$ (0.017 radians), then the second term of the Taylor expansion would be $\sim$0.018, the uncertainty is about 12 per cent. For the red outflow lobe, the uncertainty is about 4 per cent. ### More properties {#sec:more} Geometrically, when a cone is cut by a plane, the resulting curve can be an ellipse (including a circle), a parabola, or a hyperbola, depending on the angle between the cone-axis and the plane. For an ellipse, the larger the inclination-angle is, the larger its eccentricity will be, and vice versa. When $\theta$ approaches 90$^\circ$ we obtain a parabola or a hyperbola. Theoretically, all of the curves can be utilized for the calculation. But the ellipse has the advantage that it is usually much more conspicuous and easy to manipulate. The parabola and the hyperbola, on the contrary, can be confusing because of the large inclination-angle. The outflow edge from the driving source has very weak emission and the contours exhibit a fan-like structure. The centre and the curve will be ambiguous. The opposite extreme is when the inclination-angle becomes zero and the ellipse changes to a circle. Fortunately, the angle equation (\[angle equation\]) still works. In the angle function (\[angle function\]), the coefficient of the first term $\frac{1-P}{Q}$ is ($\frac{1-\frac{\overline{OD}}{\overline{OA}}}{\frac{\overline{AB}}{\overline{OA}}}$=$\frac{\overline{OA}-\overline{OD}}{\overline{AB}}$=$\frac{\overline{AD}}{\overline{AB}}$). As $\theta$ approaches zero, the projection shape becomes a circle and $\frac{\overline{AD}}{\overline{AB}}\rightarrow$1. Meanwhile, cos$\theta$ $\rightarrow$1 and the second term vanishes. Thus A($\theta\rightarrow$0)$\rightarrow$1. The angle equation (\[angle equation\]) is tenable in this extreme case. However, when $\theta$ approaches zero, use of this method could be problematic if the resolution is low. Another property is the collimation. In our view, it can be manifested by the opening angle $\theta$-$\theta_1$. Larger $\theta$-$\theta_1$ means lower collimation, and vice versa. Thus it is reasonable to define a collimation factor as cot($\theta$-$\theta_1$) (=(tan$\alpha$sin$\theta$)$^{-1}$). In IRAS 05553+1631, the factors are 7.0 and 1.6 for blue and red lobes, respectively. One thing should be mentioned, the ellipses we used for IRAS 05553+1631 are 90 per cent contours (Figure \[fig:theta\]) as stronger emission could reduce the errors. If we chose the 50 per cent contours, the factor would change. Additionally, the method highly depends on the accuracy with which the driving centre is located. One question is the identification, and the second is the spatial resolution. The identification involves personal judgement, which is accompanied by considerable uncertainty in some cases. @b4 showed that 6.7 GHz methanol maser and relevant 24 $\mu m$ emission usually coincide with each other and they are good tracers to the driving centre, this may offer much help for the identification. The resolution is usually low for single-dish observations. Using high-resolution telescopes and interferometers can improve this. In the angle function (\[angle function\]), there are two parameters P and Q. Since we can always choose an ellipse excluding the driving centre, it is reasonable to assume 0$<$P$<$1. Figure \[fig:Q\] illustrates the value of Q, which shows the projection on x-z plane. Lines AO$_1$ and AD’ represent the projections of two conditions for the plane of the sky: (1) AO$_1$ perpendicular to O$_2$E (solid, hereafter named c1); (2) AD’ not perpendicular to O$_2$E (dashed, hereafter named c2). Condition c2 is denoted by prime. $\overline{AE}$ equals the $\overline{AB}$ of Figure \[fig:model\](a). Thus, it is easy to see that the maximum of Q is $\frac{\overline{AE}}{\overline{AO_1}}$ which is just 1/cos($\theta$-$\theta_1$). Since ($\theta$-$\theta_1$) cannot be large and an angle of 60$^\circ$ can only bring Q=2, we consider 0$<$Q$<$2. When P=0, the second term of the angle function (\[angle function\]) vanishes and $\theta$=arccos(1/Q). In order to have solution, Q must be larger than 1. In fact, in this case Q reaches its maximum (see Figure \[fig:Q\]). The result is 0$<\theta<$71$^\circ$. When P=1, there is no solution. Figure \[fig:curves\] shows the variation of the angle function as a function of P and Q. With a fixed Q, the function decreases as P increases. With a fixed P, the function decreases when Q increases. Usually within (0$^\circ$,90$^\circ$), there is one solution. When Q approaches 1 (Figure \[fig:curves\](d)), in order to have solution, P must be closer to 0. When P approaches 1, the same happens to Q. The variation of angle function is sensitive to the values of P and Q. Summary {#sec:summary} ======= We mapped IRAS 05553+1631 with $^{12}$CO J=3-2 and $^{13}$CO J=2-1 lines. A core was identified from $^{13}$CO J=2-1 observations. It has a size of 0.65 pc and LTE mass of 120 M$_{\odot}$ which is lower than the virial mass of 850 M$_{\odot}$. $^{12}$CO J=3-2 mapping revealed a bipolar outflow. Its parameters were initially estimated under the assumption of a 45$^\circ$ inclination-angle. A new method to directly calculate the inclination-angle $\theta$ was proposed, and was utilized for the bipolar outflow of IRAS 05553+1631. We found that $\theta_{blue}$ is 73$^\circ$ and $\theta_{red}$ is 78$^\circ$. Parameters with the new $\theta$s were compared with the former ones. For the blue lobe, the momentum was enlarged from 82 M$_\odot$ km s$^{-1}$ to 200 M$_\odot$ km s$^{-1}$ by a factor of 2.4 while the timescale was reduced from 8.8$\times$10$^4$ yrs to 2.7$\times$10$^4$ yrs by a factor of 0.31. The enlarging factors for energy, mechanical luminosity, driving force, and mass-loss rate are 5.8, 19, 7.7, and 7.7, respectively. For the red lobe, the momentum was enlarged from 11 M$_\odot$ km s$^{-1}$ to 36 M$_\odot$ km s$^{-1}$ by a factor of 3.4 while the timescale was reduced from 6.3$\times$10$^4$ yrs to 1.3$\times$10$^4$ yrs by a factor of 0.21. The enlarging factors for energy, mechanical luminosity, driving force, and mass-loss rate are 12, 55, 16, and 16, respectively. The results show that a selection of parameters were influenced by the inclination-angle $\theta$. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful for Dr. Martin Miller for the assistance of observations. We also thank Tie Liu, Zhiyuan Ren, and Xueying Tang for the helpful discussions. Thank Hongping Du for the language checking. Thank Prof. P. Goldsmith for the constructive suggestions and thank Bella Lock for the helpful work. This project is supported by grant 10733030 and 10873019 of NSFC. [99]{} Bally J., Lada C. J., 1983, ApJ, 265, 824B Bonnell I.A., Bate M.R., Zinnecker H., 1998, MNRAS, 298, 93B Brand J. and Blitz L., 1993, A&A, 275, 67 Cabrit S., Bertout C., 1986, ApJ, 307, 313C Cyganowski C.J., Brogan C.L., Hunter T.R., Churchwell E., 2009, ApJ, 702, 1615 Estalella R., Mauersberger R., Torrelles J.M., Anglada G., Gomez J.F., Lopez R., Muders D., 1993, ApJ, 419, 698 Garden R.P., Hayashi M., Gatley I., Hasegawa T., Kaifu N., 1991, ApJ, 374, 540 Goldsmith P. F., Snell R. L., Hemeon-Heyer M., Langer W. D., 1984, ApJ, 286, 599 Kwan J., Scoville N., 1976, ApJ, 210L, 39K Lada C.J., 1985, ARA&A, 23, 267L Meixner M., Ueta T., Bobrowsky M., Speck A., 2002, ApJ, 571, 936M Meyers-Rice B.A., Lada C.J., 1991, ApJ, 368, 445M Myers P.C., Linke R.A., Benson P.J., 1983, ApJ, 264, 517 Plambeck R.L., Snell R.L., Loren R.B., 1983, ApJ, 266, 321P Shepherd D.S., Watson A.M., Sargent A.I., Churchwell E., 1998, ApJ, 507, 861 Snell R.L., Loren R.B., Plambeck R.L., 1980, ApJ, 239L, 17S Wolfire M.G., Cassinelli J.P., 1987, ApJ, 319, 850W Wood D.O.S., Churchwell E., 1989, ApJ, 340,265 Wu Y., Huang M., 1998, ChPhL, 15, 388W Wu Y., Wei Y., Zhao M., Shi Y., Yu W., Qin S., Huang M., 2004, A&A, 426, 503 Wu Y., Zhang Q., Chen H., Yang C., Wei Y., and Ho P.T.P., 2005, AJ, 129, 330 Zhu L., Wu Y., Wei Y., 2006, ChJAA, 6, 61Z Zuckerman B., Kuiper T.B.H., Rodriguez Kuiper E.N., 1976, ApJ, 209L, 137Z 0.5mm [ccccccccccc]{}\ & & & & & & & & & &\ \ 05553+1631 & 05 55 18.0 & 16 31 00 & 05 58 11.5 & 16 31 14 & 5.5 & 3.1 &2.5 & 1.70 & 2.52 & 528\ \  \  \  \  \  \ 1.0mm [ccccccccccc]{}\ & & & & & & & & & &\ \ 05553+1631 &1.4 &(3.5,7.3) &0.65 &25.8 &0.22 &6.8 &6.0 &3.0 &120 &850\ \  \  \  \  \  \ 1.0mm [ccc ccc ccc]{}\ & & & & & & & &\ \ blue & 0.65 & 6.2 & 82 & 11 & 8.8 & 10 & 9.3 & 9.3\ red & 0.60 & 1.9 & 11 &    0.60 & 6.3 &    0.80 & 1.7 & 1.7\ \  \  \  \  \  \ 1.0mm [ccc ccc ccc c]{}\ & & & & & & & & &\ \ new blue & 73 & 0.65 & 6.2 & 200   & 64 & 2.7 & 190 & 74 & 74\ new red & 78 & 0.60 & 1.9 & 36 &    7.0 & 1.3 &    43 & 27 & 27\ \ old blue & 45 & 0.65 & 6.2 & 82 & 11 & 8.8 & 10 & 9.3 & 9.3\ old red & 45 & 0.60 & 1.9 & 11 &    0.60 & 6.3 &    0.80 & 1.7 & 1.7\ \ {width="150mm"} {width="160mm"} ![Outflow contour diagram. The east blue contours represent the blue outflow lobe while the west red contours represent the red lobe. The peaks for the blue and red lobe are 14.6 K km s$^{-1}$ and 4.3 K km s$^{-1}$. The triangles show the positions of the three MSX sources. The square stands for the position of the $^{13}$CO core centre. The black solid circle presents the position of G192.16.[]{data-label="fig:outflow"}](05553_2.eps){width="80mm"} {width="130mm"} {width="120mm"} {width="100mm"} ![The illustration of parameter Q. The utilization of letters follows Figure \[fig:model\]. Condition c2 is denoted by prime.[]{data-label="fig:Q"}](Q_maximum.eps){width="80mm"} {width="160mm"} \[lastpage\] [^1]: E-mail: [email protected] [^2]: E-mail: [email protected]

**MATRIX PRODUCT STATE REPRESENTATIONS** D. PEREZ-GARCIA *Max Planck Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany* *Departamento de Análisis Matemático, Universidad Complutense de Madrid, 28040 Madrid, Spain* F. VERSTRAETE *Institute for Quantum Information, Caltech, Pasadena, US* *Fakultät für Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria.* M.M. WOLF and J.I. CIRAC *Max Planck Institut für Quantenoptik, Hans-Kopfermann-Str. 1, Garching, D-85748, Germany* Introduction and Overview ========================= The notorious complexity of quantum many-body systems stems to a large extent from the exponential growth of the underlying Hilbert space which allows for highly entangled quantum states. Whereas this is a blessing for *quantum information theory*—it facilitates exponential speed-ups in quantum simulation and quantum computing—it is often more a curse for *condensed matter theory* where the complexity of such systems make them hardly tractable by classical means. Fortunately, physical interactions are *local* such that states arising for instance as ground states from such interactions are not uniformly distributed in Hilbert space. Hence, it is desirable to have a representation of quantum many-body states whose correlations are generated in a ‘local’ manner. Despite the fact that it is hard to make this picture rigorous, there is indeed a representation which comes close to this idea—the *matrix product state* (MPS) representation. In fact, this representation lies at the heart of the power of the *density matrix renormalization group* (DMRG) method and it is the basis for a large number of recent developments in quantum information as well as in condensed matter theory. This work gives a detailed investigation of the MPS representation with a particular focus on the freedom in the representation and on canonical forms. The core of our work is a generalization of the results on finitely correlated states in [@FaNaWe92] to finite systems with and without translational invariance. We will mainly discuss exact MPS representations throughout and just briefly review results on approximations in Sec.\[sec:classical-sim\]. In order to provide a more complete picture of the representation and its use we will also briefly review and extend various recent results based on MPS, their parent Hamiltonians and their generation. The following gives an overview of the article and sketches the obtained results: - Sec.\[Sec:prelim\] will introduce the basic notions, provide some examples and give an overview over the relations between MPS and the valence bond picture on the one hand and frustration free Hamiltonians and finitely correlated states on the other. - In Sec.\[Sec:canonical\] we will determine the freedom in the MPS representation, derive canonical forms and provide efficient ways for obtaining them. Cases with and without translational invariance are distinguished. In the former cases we show that there is always a translational invariant representation and derive a canonical decompositions of states into superpositions of ‘ergodic’ and periodic states (as in [@FaNaWe92]). - Sec.\[Sec:Hamiltonians\] investigates a standard scheme which constructs for any MPS a local Hamiltonian, which has the MPS as exact ground state. We prove uniqueness of the ground state (for the generic case) without referring to the thermodynamic limit, discuss degeneracies (spontaneous symmetry breaking) based on the canonical decomposition and review results on uniform bounds to the energy gap. - In Sec.\[sec:generation\] we will review the connections between MPS and sequential generation of multipartite entangled states. In particular we will show that MPS of sufficiently small bond dimension are feasible to generate in a lab. - In Sec.\[sec:classical-sim\] we will review the results that show how MPS efficiently approximate many important states in nature; in particular, ground states of 1D local Hamiltonians. We will also show how the MPS formalism is crucial to understand the need of a large amount of entanglement in a quantum computer in order to have a exponential speed-up with respect to a classical one. Definitions and Preliminaries {#Sec:prelim} ============================= MPS and the valence bond picture -------------------------------- We will throughout consider pure quantum states $|\psi\rangle\in \mathbb{C}^{\otimes d^N}$ characterizing a system of $N$ sites each of which corresponds to a $d$-dimensional Hilbert space. A very useful and intuitive way of thinking about MPS is the following valence bond construction: consider the $N$ parties (’spins’) aligned on a ring and assign two virtual spins of dimension $D$ to each of them. Assume that every pair of neighboring virtual spins which correspond to different sites are initially in an (unnormalized) maximally entangled state $|I\rangle=\sum_{\alpha=1}^{D} |\alpha,\alpha\rangle$ often referred to as entangled *bond*. Then apply a map $$\label{curlyA}{\cal A}=\sum_{i=1}^d\sum_{\alpha,\beta=1}^{D} A_{i,\alpha,\beta} |i\rangle\langle\alpha,\beta|$$ to each of the $N$ sites. Here and in the following Greek indices correspond to the virtual systems. By writing $A_{i}$ for the $D\times D$ matrix with elements $A_{i,\alpha,\beta}$ we get that the coefficients of the final state when expressed in terms of a product basis are given by a matrix product $\operatorname{tr}\left[A_{i_1}A_{i_2}\cdots A_{i_N}\right]$. In general the dimension of the entangled state $|I\rangle$ and the map ${\cal A}$ can both be site-dependent and we write $A_i^{[k]}$ for the $D_k\times D_{k+1}$ matrix corresponding to site $k\in\{1,\ldots,N\}$. States obtained in this way have then the form $$\label{MPS0} |\psi\rangle=\sum_{i_1,\ldots,i_N=1}^d \operatorname{tr}\left[A_{i_1}^{[1]}A_{i_2}^{[2]}\cdots A_{i_N}^{[N]}\right]|i_1,i_2,\ldots,i_N\rangle\;,$$ and are called *matrix product states* [@MPSorigin]. As shown in [@Vi03] *every* state can be represented in this way if only the bond dimensions $D_k$ are sufficiently large. Hence, Eq.(\[MPS0\]) is a representation of states rather than the characterization of a specific class. However, typically states are referred to as MPS if they have a MPS-representation with small $D=\max_k D_k$ which (in the case of a sequence of states) does in particular not grow with $N$. Note that $\psi$ in Eq.(\[MPS0\]) is in general not normalized and that its MPS representation is not unique. Normalization as well as other expectation values of product operators can be obtained from $$\begin{aligned} \nonumber \langle\psi|\bigotimes_{k=1}^N S_k|\psi\rangle &=&\operatorname{tr}\left[\prod_{k=1}^N E_{S_k}^{[k]}\right]\;,\quad\text{with}\\ E_S^{[k]}&\equiv&\sum_{i,j=1}^d \langle i|S|j\rangle \overline{A}_i^{[k]}\otimes A_j^{[k]}\label{E}\;.\end{aligned}$$ \[htbp\] Finitely correlated states -------------------------- The present work is inspired by the papers on *finitely correlated states* (FCS) which in turn generalize the findings of Affleck, Kennedy, Lieb and Tasaki (AKLT) [@AKLT]. In fact, many of the results we derive are extensions of the FCS formalism to finite and/or non-translational invariant systems. For this reason we will briefly review the work on FCS. A FCS is a translational invariant state on an infinite spin chain which is constructed from a completely positive and trace preserving map $\mathbb{E}:{\cal B}({\cal H}_A)\rightarrow {\cal B}({\cal H}_A\otimes {\cal H}_B)$ and a corresponding fixed point density operator $\Lambda=\operatorname{tr}_B[\mathbb{E}(\Lambda)]$. Here ${\cal H}_B=\mathbb{C}^d$ is the Hilbert space corresponding to one site in the chain and ${\cal H}_A=\mathbb{C}^D$ is an ancillary system. An $n$-partite reduced density matrix $\rho_n$ of the FCS is then obtained by repeated application of $\mathbb{E}$ to the ancillary system (initially in $\Lambda$) followed by tracing out the ancilla, i.e., $$\rho_n=\operatorname{tr}_A\big[\mathbb{E}^n(\Lambda)\big]\;.$$ An important instance are *purely generated* FCS where $\mathbb{E}(x)=V^\dag x V$ is given by a partial isometry $V$. The latter can be easily related to the $A$’s in the matrix product representation via $V=\sum_{i=1}^d\sum_{\alpha,\beta=1}^D A_{i,\alpha,\beta} |\alpha\rangle\langle\beta i|$. Expressed in terms of the matrices $A_i$ the isometry condition and the fixed point relation read $$\label{FCSA} \sum_{i=1}^d A_iA_i^\dag=\mathbbm{1}\;,\qquad \sum_{i=1}^d A_i^\dag\Lambda A_i =\Lambda\;,$$ which already anticipates the type of canonical forms for MPS discussed below. As shown in [@FaNaWe92-2] purely generated FCS are weakly dense within the set of all translational invariant states on the infinite spin chain. Moreover, a FCS is *ergodic*, i.e., an extreme point within all translational invariant states, iff the map ${\mathcal{E}}(x)=\sum_iA_ixA_i^\dag$ has a non-degenerate eigenvalue 1 (i.e., $\mathbbm{1}$ and $\Lambda$ are the only fixed points in Eq.(\[FCSA\])). Every FCS has a unique decomposition into such ergodic FCS which in turn can be decomposed into $p$ $p$-periodic states each of which corresponds to a root of unity $\exp(\frac{2\pi i}p m)$, $m=0,\ldots,p-1$ in the spectrum of ${\mathcal{E}}$. A FCS is pure iff it is purely generated and 1 is the only eigenvalue of ${\mathcal{E}}$ of modulus 1. In this case the state is *exponentially clustering*, i.e., the connected two-point correlation functions decay exponentially $$\label{eq:corr} \langle S_i\otimes\mathbbm{1}^{\otimes l-1}\otimes S_{i+l}\rangle-\langle S_i\rangle\langle S_{i+l}\rangle={\cal O}\big(|\nu_2|^{l-1}\big)\;,$$ where $\nu_2$ ($|\nu_2|<1$) is the second largest eigenvalue of ${\mathcal{E}}$. Frustration free Hamiltonians ----------------------------- Consider a translational invariant Hamiltonian on a ring of $N$ $d$-dimensional quantum systems $$H=\sum_{i=1}^N \tau^i\big(h\big)\;,$$ where $\tau$ is the translation operator with periodic boundary conditions, i.e., $\tau\big(\bigotimes_{i=1}^N x_i\big)=\bigotimes_{i=1}^N x_{i+1}$ where sites $N+1$ and $1$ are identified. The interaction is called $L$-*local* if $h$ acts non-trivially only on $L$ neighboring sites, and it is said to be *frustration free* with respect to its ground state $\phi_0$ if the latter minimizes the energy locally in the sense that $\langle \phi_0|H|\phi_0\rangle=\inf_\phi \langle \phi|H|\phi\rangle=N \inf_\phi \langle \phi|h|\phi\rangle$. As proven in [@Hast1] all gapped Hamiltonians can be approximated by frustration free ones if one allows for enlarging the interaction range $L$ up to ${\cal O}(\log N)$. For every MPS and FCS $\psi$ one can easily find frustration free Hamiltonians such that $\psi$ is their exact ground state. Moreover, these *parent* Hamiltonains are $L$-local with $L\sim 2 \log D/\log d$ and they allow for a detailed analysis of the ground state degeneracy (Sec.\[sec:Huniqueness\]) and the energy gap above the ground state (Sec.\[Sec:Hgap\]). Typically, these Hamiltonians are, however, not exactly solvable, i.e., information about the excitations might be hard to obtain. Examples -------- 1. [*AKLT*]{}: The father of all matrix product states is the ground state of the AKLT-Hamiltonian $$\label{HAKLT} H=\sum_i \vec{S}_i\vec{S}_{i+1}+\frac13\Big(\vec{S}_i\vec{S}_{i+1}\Big)^2\;,$$ where $\vec{S}$ is the vector of spin-1 operators (i.e., d=3). Its MPS representation is given by $\{A_i\}=\big\{\sigma^z,\sqrt{2}\sigma^+,-\sqrt{2}\sigma^-\big\}$ where the $\sigma$’s are the Pauli matrices. 2. [*Majumdar-Gosh*]{}: The Hamiltonian $$\label{HMG} H=\sum_i 2\vec{\sigma}_i\vec\sigma_{i+1} + \vec{\sigma}_i\vec\sigma_{i+2}\;$$ is such that every ground state is a superposition of two 2-periodic states given by products of singlets on neighboring sites. The equal weight superposition of these states is translational invariant and has an MPS representation $$A_1=\left(\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0 \\ \end{array}\right)\;,\quad A_2=\left(\begin{array}{ccc} 0 & 0& 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array}\right)\;.$$ 3. [*GHZ states*]{} of the form $|\psi\rangle=|++\ldots +\rangle+|--\ldots -\rangle$ have an MPS representation $A_\pm=\mathbbm{1}\pm\sigma^z$. Anti-ferromagnetic GHZ states would correspond to $A_\pm=\sigma^\pm$. 4. [*Cluster states*]{} are unique ground states of the three-body interactions $\sum_i \sigma^z_i\sigma^x_{i+1}\sigma^z_{i+2}$ and represented by the matrices $$A_1= \left(\begin{array}{cc} 0 & 0 \\ 1 & 1 \\ \end{array}\right)\;,\quad A_2=\left(\begin{array}{cc} 1 & -1 \\ 0 & 0 \\ \end{array}\right)\;.$$ 5. [*W-states*]{} can for instance appear as ground states of the ferromagnetic XX model with strong transversal magnetic field. A W-state is an equal superposition of all translates of $|100\ldots00\rangle$. For a simple MPS representation choose $\{A_1^{[k]},A_2^{[k]}\}$ equal to $\{\sigma^+,\mathbbm{1}\}$ for all $k< N$ and $\{\sigma^+\sigma^x,\sigma^x\}$ for $k=N$. Although the state itself is translational invariant there is no MPS representation with $D=2$ having this symmetry. The canonical form {#Sec:canonical} ================== The general aim of this section will be to answer the following questions about the MPS representation of a given pure state: \[quest.can.1\] Which is the freedom in the representation? \[quest.can.2\] Is there any [*canonical*]{} representation? \[quest.can.3\] If so, how to get it? We will distinguish two cases. The general case, or the case of open boundary conditions (OBC) and the case in which one has the additional properties of translational invariance (TI) and periodic boundary conditions (PBC). Open boundary conditions {#sec:Open} ------------------------ A MPS is said to be written with open boundary conditions (OBC) if the first and last matrices are vectors, that is, if it has the form $$\label{eq.vidal} |\psi\rangle=\sum_{i_1,\ldots,i_N} A^{[1]}_{i_1}A^{[2]}_{i_2}\cdots A^{[N-1]}_{i_{N-1}}A^{[N]}_{i_N}|i_1\cdots i_N\rangle,$$ where $A^{[m]}_i$ are $D_{m}\times D_{m+1}$ matrices with $D_1=D_{N+1}=1$. Moreover, if $D=\max_m D_m$ we say that the MPS has [*(bond) dimension*]{} $D$. The following is shown in [@Vi03]: \[thm:OBC-Vidal\] Any state $\psi\in\mathbb{C}^{d\otimes N}$ has an OBC-MPS representation of the form Eq.(\[eq.vidal\]) with bond dimension $D\leq d^{\lfloor N/2\rfloor}$ and 1. $\sum_i A^{[m]}_iA^{[m]\dagger}_i=\mathbbm {1}_{D_m}$ for all $1\le m\le N$. 2. $\sum_i A^{[m]\dagger}_i\Lambda^{[m-1]} A^{[m]}_i=\Lambda^{[m]}, $ for all $1\le m\le N$, 3. $\Lambda^{[0]}=\Lambda^{[N]}=1$ and each $\Lambda^{[m]}$ is a $D_{m+1}\times D_{m+1}$ diagonal matrix which is positive, full rank and with $\operatorname{tr}{\Lambda^{[m]}}=1$. \[Thm:completeness\] Thm.\[Thm:completeness\] is proven by successive singular value decompositions (SVD), i.e., Schmidt decompositions in $\psi$, and the *gauge conditions* [*1.-3.*]{} can be imposed by exploiting the simple observation that $A_i^{[m]}A_j^{[m+1]}=(A_i^{[m]}X) (X^{-1}A_j^{[m+1]})$. If [*1.-3.*]{} are satisfied for a MPS representation, then we say that the MPS with OBC is in [*the canonical form*]{}. From the way it has been obtained one immediately sees that: - it is unique (up to permutations and degeneracies in the Schmidt Decomposition), - $\Lambda^{[m]}$ is the diagonal matrix of the non-zero eigenvalues of the reduced density operator $\rho_m=\operatorname{tr}_{m+1,\ldots,N}|\psi\rangle\langle\psi|$, - any state for which $\max_m \operatorname{rank}(\rho_m)\le D$ can be written as a MPS of bond dimension D. This answers questions \[quest.can.2\] and \[quest.can.3\]. Question \[quest.can.1\] will be answered with the next theorem which shows that the entire freedom in any OBC-MPS representation is given by ‘local’ matrix multiplications. \[free-OBC\] Let us take a OBC-MPS representation $$|\psi\rangle=\sum_{i_1,\ldots,i_N} B^{[1]}_{i_1}B^{[2]}_{i_2}\cdots B^{[N-1]}_{i_{N-1}}B^{[N]}_{i_N}|i_1\cdots i_N\rangle\;.$$ Then, there exist (in general non-square) matrices $Y_j$, $Z_j$ with $Y_jZ_j=\mathbbm{1}$ such that, if we define $$\begin{aligned} \nonumber A^{[1]}_i&=B^{[1]}_iZ_1,\quad A^{[N]}_i =Y_{N-1}B^{[N]}_i \\ A^{[m]}_i &=Y_{m-1}B^{[m]}_iZ_m,\ \text{for}\ 1<m<N \label{eq.lem.II.9} $$ the canonical form is given by $$\label{Eq:canonical2} |\psi\rangle=\sum_{i_1,\ldots,i_N} A^{[1]}_{i_1}A^{[2]}_{i_2}\cdots A^{[N-1]}_{i_{N-1}}A^{[N]}_{i_N}|i_1\cdots i_N\rangle.$$ [**Proof.**]{} We will prove the theorem in three steps. ***[STEP 1.]{}*** First we will find the matrices $A^{[j]}_i$ verifying relation (\[eq.lem.II.9\]) but just with the property $\sum_i A^{[j]}_iA^{[j]\dagger}_i=\mathbbm{1}.$ To this end we start from the right by doing SVD: $B^{[N]}_{\alpha,i}=\sum_{\beta}U^{[N-1]}_{\alpha,\beta}\Delta^{[N-1]}_{\beta} A^{[N]}_{\beta,i}$, with $U^{[N-1]},A^{[N]}$ unitaries and $\Delta^{[N-1]}$ diagonal. That is $B^{[N]}_i=Z_{N-1}A_i^{[N]}$, with $Z_{N-1}=U^{[N-1]}\Delta^{[N-1]}$. Clearly $\sum_iA_i^{[N]}A_i^{[N]\dagger}=\mathbbm{1}$ and $Z_{N-1}$ has a left inverse. Now we call $\tilde{B}^{[N-1]}_i=B^{[N-1]}_iZ_{N-1}$ and make another SVD: $\tilde{B}^{[N-1]}_{\alpha,i,\beta}=\sum_\gamma U^{[N-2]}_{\alpha,\gamma}\Delta^{[N-2]}_\gamma A^{[N-1]}_{\gamma,i,\beta}$. That is $$B^{[N-1]}_iZ_{N-1}=\tilde{B}^{[N-1]}_i=Z_{N-2}A_i^{[N-1]},$$ where $\sum_iA_i^{[N-1]}A_i^{[N-1]\dagger}=\mathbbm{1}$ and $Z_{N-2}=U^{[N-2]}\Delta^{[N-2]}$ has left inverse. We can go on getting relations (\[eq.lem.II.9\]) to the last step, where one simply defines $A^{[1]}_i=B^{[1]}_iZ_1$. From the construction one gets Eq.(\[Eq:canonical2\]) and that $\sum_i A^{[j]}_iA^{[j]\dagger}_i=\mathbbm{1}$ for every $1<j\le N$. The case $j=1$ comes simply from the normalization of the state: $$1={\langle}\psi|\psi{\rangle}=\sum_{i_1,\ldots,i_N} A^{[1]}_{i_1}\cdots A^{[N]}_{i_N}A^{[N] \dagger}_{i_N}\cdots A^{[1] \dagger}_{i_1}=\sum_{i_1}A^{[1]}_{i_1}A^{[1] \dagger}_{i_1},$$ where in the last equality we have used that $\sum_{i_j}A^{[j]}_{i_j}A^{[j] \dagger}_{i_1}={\mathbbm{1}}$ for $1<j\le N$. ***[STEP 2.]{}*** Now we can assume that the $B$’s verify $\sum_i B^{[j]}_iB^{[j]\dagger}_i=\mathbbm{1}$. Diagonalizing $\sum_i B^{[1]\dagger}_iB^{[1]}_i$ we get a unitary $V^{[1]}$ and a positive diagonal matrix $\Lambda^{[1]}$ such that $\sum_i B^{[1]\dagger}_iB^{[1]}_i=V^{[1]}\Lambda^{[1]}V^{[1]\dagger}$. Calling $A^{[1]}_i=B^{[1]}_iV^{[1]}$ we have both $\sum_iA_i^{[1]}A_i^{[1]\dagger}=\mathbbm{1}$ and $\sum_iA_i^{[1]\dagger}A_i^{[1]}=\Lambda^{[1]}$. Now we diagonalize $\sum_i B^{[2]\dagger}_iV^{[1]}\Lambda^{[1]}V^{[1]\dagger}B^{[2]}_i=V^{[2]}\Lambda^{[2]}V^{[2]\dagger}$ and define $A^{[2]}_i=V^{[1]\dagger}B^{[2]}_iV^{[2]}$ to have both $\sum_iA_i^{[2]}A_i^{[2]\dagger}=\mathbbm{1}$ and $\sum_iA_i^{[2]\dagger}\Lambda^{[1]}A_i^{[2]}=\Lambda^{[2]}$. We keep on with this procedure to the very last step where we simply define $A_i^{[N]}=V^{[N-1]\dagger}B^{[N]}_i$. $\sum_iA_i^{[N]}A_i^{[N]\dagger}=\mathbbm{1}$ is trivially verified and $\sum_iA_i^{[N]\dagger}\Lambda^{[N-1]}A_i^{[N]}=\Lambda^{[N]}=1$ comes, as above, from the normalization of the state. Moreover, by construction we have the relation (\[eq.lem.II.9\]) and Eq.(\[Eq:canonical2\]). ***[STEP 3.]{}*** At this point we have matrices ${Y}_j, {Z}_j$ with ${Y}_jZ_j={\mathbbm{1}}$ such that, if we define ${A}_i^{[j]}$ by (\[eq.lem.II.9\]), we get ${D}_j\times {D}_{j+1}$ matrices verifying the conditions 1, 2 and 3 of Theorem \[thm:OBC-Vidal\] with the possible exception that the matrices $\Lambda^{[j]}$ are not full rank. Now we will show that we can redefine $Y_j, Z_j$ (and hence $D_j, A_i^{[j]}$) to guarantee also this [*full rank*]{} condition. We do it by induction. Let us assume that $\Lambda^{[j-1]}$ is full rank and the $D_{j+1}\times D_{j+1}$ positive diagonal matrix $\Lambda^{[j]}$ is not. Then, calling $$\widetilde{D}_{j+1}=\operatorname{rank}(\Lambda^{[j]})\;, \quad P_j=\left({\mathbbm{1}}_{\widetilde{D}_{j+1}}\big|0_{D_{j+1}-\widetilde{D}_{j+1}}\right),$$ we are finished if we update $Z_j$ as $Z_jP_j^{\dagger}$, $Y_j$ as $P_jY_j$ (and hence ${D}_{j+1}$ as $\tilde{D}_{j+1}$, $A_i^{[j]}$ as $A_i^{[j]}P_j^{\dagger}$, $A_i^{[j+1]}$ as $P_jA_i^{[j+1]}$ and $\Lambda^{[j]}$ as $P_j\Lambda^{[j]}P_j^{\dagger}$). The only non-trivial part is to prove that $A^{[j]}_{i_j}A^{[j+1]}_{i_{j+1}}=A^{[j]}_{i_j}P_j^{\dagger}P_jA^{[j+1]}_{i_{j+1}}$. For that, calling $C={\mathbbm{1}}_{D_{j+1}}-P_j^{\dagger}P_j$, it is enough to show that $A^{[j]}_iC=0$. Since $${\mathbbm{1}}_{D_{j+1}}-P_j^{\dagger}P_j=\left(\begin{array}{cc} 0_{\widetilde{D}_{j+1}} & 0 \\ 0 & {\mathbbm{1}}_{D_{j+1}-\widetilde{D}_{j+1}} \\ \end{array}\right),$$ we have $$0=C\Lambda^{[j]}C=\sum_iC A^{[j]\dagger}_i\Lambda^{[j-1]}A^{[j]}_i C.$$ Since $\Lambda^{[j-1]}$ is positive and full rank we get $A^{[j]}_iC=0$. ${\hfill\fbox\\\medskip }$ Periodic boundary conditions and translational invariance --------------------------------------------------------- Clearly, if the $A's$ in the MPS in Eq.(\[MPS0\]) are the same, i.e., site-independent ($A_i^{[m]}=A_i$), then the state is translationally invariant (TI) with periodic boundary conditions (PBC). We will in the following first show that the converse is also true, i.e., that every TI state has a TI MPS representation. Then we will derive canonical forms having this symmetry, discuss their properties and show how to obtain them. An important point along these lines will be a canonical decomposition of TI states into superpositions of TI MPS states which may in turn be written as superpositions of periodic states. This decomposition closely follows the ideas of [@FaNaWe92] and will later, when constructing parent Hamiltonians, give rise to discrete symmetry-breaking. ### Site independent matrices Before starting with the questions \[quest.can.1\], \[quest.can.2\] and \[quest.can.3\], we will see that we can use TI and PBC to assume the matrices in the MPS representation to be site independent. That is, if the state is TI, then there is also a TI representation as MPS.[^1] [*Every*]{} TI pure state with PBC on a finite chain has a MPS representation with site-independent matrices $A_i^{[m]}=A_i$, i.e.,$$\label{eq.1} |\psi\rangle=\sum_{i_1,\ldots,i_N} \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N\rangle\;.$$ If we start from an OBC MPS representation, to get site-independent matrices one has (in general) to increase the bond dimension from $D$ to $ND$ (note the $N$-dependence). [[**Proof.**]{} We start with an OBC representation of the state with site-dependent $A_i^{[m]}$ and consider the matrices (for $0\le i\le d-1$) $$B_i=N^{-\frac1N}\begin{pmatrix} 0 & A^{[1]}_i & & & \\ & 0 & A^{[2]}_i & & \\ & & \cdots & & \\ & & & 0 & A^{[N-1]}_i \\ A^{[N]}_i & & & & 0 \end{pmatrix}.$$ This leads to $$\sum_{i_1,\ldots,i_N=0}^{d-1}\operatorname{tr}(B_{i_1}\cdots B_{i_N})|i_1,\ldots,i_N\rangle=$$ $$=\frac1N\sum_{j=0}^{N-1}\sum_{i_1,\ldots,i_N=0}^{d-1}\operatorname{tr}(A^{[1]}_{i_{1+j}}\cdots A^{[N]}_{i_{N+j}}) |i_1,\ldots,i_N\rangle,$$ where $i_{j}=i_{j-N}$ if $j>N$. Due to TI of $\psi$ this yields exactly Eq.(\[eq.1\]). ${\hfill\fbox\\\medskip }$]{} To explicitly show the $N$-dependence of the above construction we consider the particular case of the $W$-state $|10\ldots0{\rangle}+|01\ldots 0{\rangle}+\cdots+|0\ldots 01{\rangle}$. In this case the minimal bond dimension is $2$ as a MPS with OBC. However, if we want site-independent matrices, it is not difficult to show that one needs bigger matrices. In fact, we conjecture that the size of the matrices has to grow with $N$ (Appendix \[open.problems\]). From now on we suppose that we are dealing with a MPS of the form in Eq.(\[eq.1\]) with the matrices $A_i$ of size $D\times D$. In cases where we want to emphasize the site-independence of the matrices, we say the state is TI represented or simply a TI MPS. ### MPS and CP maps There is a close relation (and we will repeatedly use it) between a TI MPS and the completely positive map ${\mathcal{E}}$ acting on the space of $D\times D$ matrices given by $$\label{eq.E} {\mathcal{E}}(X)=\sum_iA_iXA_i^{\dagger}.$$ One can always assume without loss of generality that the cp map ${\mathcal{E}}$ has spectral radius equal to $1$ which implies by [@EvansH-Krohn78 Theorem 2.5] that *${\mathcal{E}}$ has a positive fixed point*. As in the FCS case stated in Eq.(\[eq:corr\]) the second largest eigenvalue of ${\mathcal{E}}$ determines the correlation length of the state and as we will see below the eigenvalues of magnitude one are closely related to the terms in the canonical decomposition of the state. Note that ${\mathcal{E}}$ and $E_\mathbbm{1}=\sum_i A_i\otimes \bar{A}_i$ have the same spectrum as they are related via $$\label{E2E} \langle \beta_1|{\mathcal{E}}(|\alpha_1\rangle\langle \alpha_2|)|\beta_2\rangle =\langle\beta_1,\beta_2|E_\mathbbm{1}|\alpha_1,\alpha_2\rangle\;.$$ Since the Kraus operators of the cp map ${\mathcal{E}}$ are uniquely determined up to unitaries, it implies that ${\mathcal{E}}$ uniquely determines the MPS up to local unitaries in the physical system. This is used in [@renorm] to find the fixed points of a renormalization group procedure on quantum states. There it is made explicit in the case of qubits, where a complete classification of the cp-maps is known. To be able to characterize the fixed points in the general case one has to find the reverse relation between MPS and cp maps. That is, given a MPS, which are the possible ${\mathcal{E}}$ that can arise from different matrices in the MPS representation? It is clear that a complete solution to question \[quest.can.1\] will give us the answer. However, though we will below provide the answer in the generic case, this is far from being completely general. As a simple example of how different the cp-maps ${\mathcal{E}}$ can be for the [*same*]{} MPS, let us take an arbitrary cp-map ${\mathcal{E}}(X)=\sum_iA_iXA_i^{\dagger}$ and consider the associate MPS for the case of $2$ particles: $|\psi\rangle=\sum_{i_1,i_2} \operatorname{tr}(A_{i_1}A_{i_2})|i_1 i_2\rangle$. Now translational invariance means permutational invariance and hence it is not difficult to show that there exist diagonal matrices $D_i$ such that $|\psi\rangle=\sum_{i_1,i_2} \operatorname{tr}(D_{i_1}D_{i_2})|i_1 i_2\rangle$. This defines a new cp-map $\tilde{{\mathcal{E}}}(X)=\sum_iD_iXD_i^{\dagger}$ with diagonal Kraus operators, for which e.g many of the additivity conjectures are true [@diagonal-krauss]. ### The canonical representation In this section we will show that one can always decompose the matrices of a TI-MPS to a [*canonical*]{} form. Subsequently we will discuss a generic condition based on which the next section will answer question \[quest.can.2\] concerning the uniqueness of the canonical form. \[Th:TIcanonical\] Given a TI state on a finite ring, we can [*always*]{} decompose the matrices $A_i$ of any of its TI MPS representations as $$A_i=\left(\begin{array}{ccc} \lambda_1 A_i^1 & 0 & 0 \\ 0 & \lambda_2 A_i^2 & 0 \\ 0 & 0 & \cdots \\ \end{array}\right),$$ where $1\ge \lambda_j>0$ for every $j$ and the matrices $A_i^j$ in each block verify the conditions: 1. $\sum_i A_i^jA_i^{j\dagger}=\mathbbm {1}$. 2. $\sum_i A_i^{j\dagger}\Lambda^j A_i^j=\Lambda^j,$ for some diagonal positive and full-rank matrices $\Lambda^j$. 3. $\mathbbm {1}$ is the only fixed point of the operator ${\mathcal{E}}_j(X)=\sum_i A_i^jXA_i^{j\dagger}$. If we start with a TI MPS representation with bond dimension $D$, the bond dimension of the above [*canonical form*]{} is $\le D$. [**Proof.**]{} We assume w.l.o.g. that the spectral radius of ${\mathcal{E}}$ is $1$ (this is where the $\lambda_j$ appear) and we denote by $X$ a positive fixed point of ${\mathcal{E}}$. If $X$ is invertible, then calling $B_i=X^{-\frac{1}{2}}A_i X^{\frac{1}{2}}$ we have $\sum_i B_iB_i^{\dagger}=\mathbbm{1}$ and hence condition 1. If $X$ is not invertible and we write $X=\sum_\alpha \lambda_{\alpha} |\alpha{\rangle}{\langle}\alpha|$, and we call $P_{R}$ the projection onto the subspace $R$ spanned by the $|\alpha{\rangle}$’s, then we have that $A_iP_R=P_RA_iP_R$ for every $i$. To see this, it is enough to show that $A_i|\alpha{\rangle}\in R$ for every $i,|\alpha{\rangle}$. If this does not happen for some $j,|\beta{\rangle}$, then $\sum_\alpha \lambda_{\alpha} |\alpha{\rangle}{\langle}\alpha|- \lambda_\beta A_j|\beta{\rangle}{\langle}\beta|A_j^{\dagger}\not \ge 0$. But, since $\sum_\alpha \lambda_{\alpha} |\alpha{\rangle}{\langle}\alpha|=\sum_i \sum_\alpha \lambda_{\alpha} A_i|\alpha{\rangle}{\langle}\alpha|A_i^{\dagger}$, we have obtained that $$\sum_{(i,\alpha)\not =(j,\beta)} \lambda_{\alpha} A_i|\alpha{\rangle}{\langle}\alpha|A_i^{\dagger}\not \ge 0,$$ which is the desired contradiction. If we call $R^{\perp}$ the orthogonal subspace of $R$, we can decompose our state as $$\begin{aligned} |\psi{\rangle}&=&\sum_{i_1,\ldots,i_N} \operatorname{tr}_R(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N\rangle+\nonumber\\ &&+\sum_{i_1,\ldots,i_N} \operatorname{tr}_{R^{\perp}}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N\rangle.\nonumber\end{aligned}$$ On the one hand $\operatorname{tr}_R(A_{i_1}\cdots A_{i_N})$ is given by $$\operatorname{tr}(P_RA_{i_1}\cdots A_{i_N}P_R)= \operatorname{tr}(P_RA_{i_1}P_R\cdots P_RA_{i_N}P_R)$$ which corresponds to a MPS with matrices $B_i=P_RA_iP_R$ of size $\dim(R)\times \dim(R)$. On the other hand $$\begin{aligned} \nonumber \operatorname{tr}_{R^{\perp}}(A_{i_1}\cdots A_{i_N})&=&\operatorname{tr}(P_{R^{\perp}}A_{i_1}\cdots A_{i_N}P_{R^{\perp}})\\ \nonumber &=& \operatorname{tr}(P_{R^{\perp}}A_{i_1}P_{R^{\perp}}\cdots P_{R^{\perp}}A_{i_N}P_{R^{\perp}})\end{aligned}$$ since $A_{i}P_{R^{\perp}}=P_RA_{i_N}P_{R^{\perp}}+P_{R^{\perp}}A_{i_N}P_{R^{\perp}}$ and the $P_R$ in the first summand goes through all the matrices $A_{i_j}$ to finally cancel with $P_{R^{\perp}}$. Then we have also matrices $C_i=P_{R^{\perp}}A_{i_N}P_{R^{\perp}}$ of size $\dim(R^{\perp})\times \dim(R^{\perp})$ such that we can write out original state with the following $D\times D$ matrices $$\left(\begin{array}{cc} B_i & 0 \\ 0 & C_i \\ \end{array}\right).$$ For each one of these blocks we reason similarly and we end up with block-shaped matrices with the property that each block satisfies $1$ in the Theorem. Let us now assume that for one of the blocks, the map $X\mapsto \sum_i B_iXB_i^{\dagger}$ has a fixed point $X\not =\mathbbm{1}$. We can suppose $X$ self-adjoint and then diagonalize it $X=\sum_\alpha \lambda_\alpha |\alpha{\rangle}{\langle}\alpha|$ with $\lambda_1\ge\cdots \ge \lambda_n$. Obviously, $\mathbbm{1}-\frac{1}{\lambda_1}X$ is a positive fixed point that is not full rank, and this allows us (reasoning as above) to decompose further the block $B_i$ in subblocks until finally every block satisfies both properties 1 and 3 in the Theorem. By the same arguments we can ensure that the only fixed point of the dual map $X\mapsto \sum_i B_i^{\dagger}XB_i$ of each block is also positive and full rank, and so, by choosing an adequate unitary $U$ and changing $B_i$ to $UB_iU^{\dagger}$, we can diagonalize this fixed point to make it a diagonal positive full-rank matrix $\Lambda$, which finishes the proof of the Theorem. ${\hfill\fbox\\\medskip }$ Note that Thm.\[Th:TIcanonical\] gives rise to a decomposition of the state into a superposition of TI MPS each of which has only one block in its canonical form and a respective cp-map ${\mathcal{E}}_j$ with a non-degenerate eigenvalue 1 (due to the uniqueness of the fixed point). The following argument shows that in cases where ${\mathcal{E}}_j$ has other eigenvalues of magnitude one further decomposition into a superposition of periodic states is possible. Examples of states with such periodic decompositions (for $p=2$) are the anti-ferromagnetic GHZ state and the Majumdar-Gosh state. \[Th:periodic\] Consider any TI state $\psi\in\mathbb{C}^{d\otimes N}$which has only one block in its canonical TI MPS representation (Thm.\[Th:TIcanonical\]) with respective $D\times D$ matrices $\{A_i\}$. If ${\mathcal{E}}(X)=\sum_iA_iXA_i^\dagger$ has $p$ eigenvalues of modulus one, then if $p$ is a factor of $N$ the state can be written as a superposition of $p$ $p$-periodic states each of which has a MPS representation with bond dimension $D$. If $p$ is no factor, then $\psi=0$. [[**Proof.**]{} The theorem is a consequence of the spectral properties of the cp map ${\mathcal{E}}$, which were proven in [@FaNaWe92]. There it is shown that if the identity is the only fixed point of ${\mathcal{E}}$, then there exists a $p\in\mathbb{N}$ such that $\{\omega^k\}_{k=1\ldots p}$ with $\omega=\exp \frac{2\pi i}p$ are all eigenvalues of ${\mathcal{E}}$ with modulus 1. Moreover, there is a unitary $U=\sum_{k=1}^p \omega^k P_k$, where $\{P_k\}$ is a set of orthogonal projectors with $\sum_kP_k=\mathbbm{1}$ such that ${\mathcal{E}}(X P_k)={\mathcal{E}}(X)P_{k-1}$ for all $D\times D$ matrixes $X$ (and cyclic index $k$). It is straightforward to show that the latter implies that $$\forall j,k:\ \ A_jP_k=P_{k-1}A_j\;.$$ Exploiting this together with the decomposition of the trace $\operatorname{tr}[\ldots]=\sum_k \operatorname{tr}[P_k\ldots P_k]$ leads to a decomposition of the state $|\psi\rangle=\sum_{k=1}^p |\psi_k\rangle$ where each of the states $|\psi_k\rangle$ in the superposition has a MPS representation with site-dependent matrices $A_{i_j}=P_{k+j-1}A_{i_j}P_{k+j}$. Hence, each $|\psi_k\rangle$ is $p$-periodic and, since $P_kP_l=\delta_{k,l}P_k$, non-zero only if $p$ is a factor of $N$. ${\hfill\fbox\\\medskip }$]{} ### Generic cases Before proceeding we have to introduce two *generic* conditions on which many of the following results are based on. The first condition is related to *injectivity* of the map $$\label{eq:injectivity} \Gamma_L:X\mapsto\sum_{i_1,\ldots,i_L=1}^d \operatorname{tr}\big[XA_{i_1}\cdots A_{i_L}\big]\;|i_1\ldots i_L\rangle.$$ Note that $\Gamma_L$ is injective iff the set of matrices $\{A_{i_1}\cdots A_{i_L}: 1 \le i_1,\ldots,i_L\le d\}$ spans the entire space of $D\times D$ matrices. Moreover, if $\sum_i A_iA_i^\dagger=\mathbbm{1}$ then evidently injectivity of $\Gamma_L$ implies injectivity of $\Gamma_{L'}$ for all $L'\geq L$. To see the relation to ‘generic’ cases consider $d$ randomly chosen matrices $A_i$. The dimension of the span of their products $A_{i_1}\cdots A_{i_L}$ is expected to grow as $d^L$ up to the point where it reaches $D^2$. That is, for generic cases we expect to have injectivity for $L\geq\frac{2\ln D}{\ln d}$. This intuition can easily be verified numerically and rigorously proven at least for $d=D=2$. However, in order not to rely on the vague notion of ‘generic’ cases we introduce the following:\ [**Condition C1:**]{} *There is a finite number $L_0\in\mathbb{N}$ such that $\Gamma_{L_0}$ is injective.*\ We continue by deriving some of the implications of condition C1 on the TI canonical form: \[prop-inj\] Consider a TI state represented in canonical MPS form (Thm.\[Th:TIcanonical\]). If condition C1 is satisfied for $L_0<N$, then 1. we have only one block in the canonical representation. 2. if we divide the chain in two blocks of consecutive spins $[1\ldots R],[R+1\ldots N]$, both of them with at least $L_0$ spins, then the rank of the reduced density operator $\rho_{[1..R]}$ is exactly $D^2$. [[**Proof.**]{} The first assert is evident, since any $X$ which has only entries in the off-diagonal blocks would lead to $\Gamma_{L_0}(X)=0$. To see the other implication we take our $D$-MPS $$|\psi\rangle=\sum_{i_1,\ldots,i_N} \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N\rangle,$$ and introduce a resolution of the identity $$\sum_{\alpha,\beta=1}^D\sum_{i_1,\ldots,i_N} {\langle}\alpha|A_{i_1}\cdots A_{i_R}|\beta{\rangle}{\langle}\beta|A_{i_{R+1}}\cdots A_{i_N}|\alpha{\rangle}|i_1\cdots i_N\rangle=\vspace*{-20pt}$$ $$=\sum_{\alpha,\beta=1}^D |\Phi_{\alpha,\beta}{\rangle}|\Psi_{\alpha,\beta}{\rangle},\quad\text{where}\vspace*{-10pt}$$ $$\begin{aligned} \nonumber |\Phi_{\alpha,\beta}{\rangle}&=&\sum_{i_1,\ldots,i_R} {\langle}\alpha|A_{i_1}\cdots A_{i_R}|\beta{\rangle}|i_1\cdots i_R\rangle,\\ |\Psi_{\alpha,\beta}{\rangle}&=&\sum_{i_1,\ldots,i_R} {\langle}\beta|A_{i_R}\cdots A_{i_N}|\alpha{\rangle}|i_{R+1}1\cdots i_N\rangle.\nonumber\end{aligned}$$ It is then sufficient to prove that both $\{|\Phi_{\alpha,\beta}{\rangle}\}$ and $\{|\Psi_{\alpha,\beta}{\rangle}\}$ are sets of linearly independent vectors. But this is a consequence of C1: Let us take complex numbers $c_{\alpha,\beta}$ such that $\sum_{\alpha,\beta} c_{\alpha,\beta} |\Phi_{\alpha,\beta}{\rangle}=0$ (the same reasoning for the $|\Psi_{\alpha,\beta}{\rangle}$). This is exactly $$\Gamma_R\left[\sum_{\alpha,\beta}c_{\alpha,\beta}|\beta{\rangle}{\langle}\alpha|\right]=0.$$ By C1 we have that $\sum_{\alpha,\beta}c_{\alpha,\beta}|\beta{\rangle}{\langle}\alpha|=0$ and hence $c_{\alpha,\beta}=0$ for every $\alpha,\beta$. ${\hfill\fbox\\\medskip }$]{} Now we will introduce a second condition for which we assume w.l.o.g. the spectral radius of ${\mathcal{E}}$ to be one:\ [**Condition C2:**]{} *The map ${\mathcal{E}}$ has only one eigenvalue of magnitude one.*\ Again this is satisfied for ‘generic’ cases as the set of cp maps with eigenvalues which are degenerated in magnitude is certainly of measure zero. It is shown in [@FaNaWe92] that this condition is [*essentially*]{} equivalent (Appendix \[open.problems\]) to condition C1. In particular C2 also implies that there is just one block in the TI canonical representation (Thm.\[Th:TIcanonical\]) of the MPS $|\psi{\rangle}=\sum_{i_1,\ldots,i_N} \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N\rangle$. Moreover, condition C2 implies that, for sufficiently large $N$, we can approximate $E_{\mathbbm{1}}^N$ (which corresponds to ${\mathcal{E}}^N$ via Eq.(\[E2E\])) by $|R{\rangle}{\langle}L|$; where $|R{\rangle}$ corresponds to the fixed point of ${\mathcal{E}}$ (that is, the identity), and ${\langle}L|$ to the fixed point $\Lambda$ of the dual map. Introducing a resolution of the identity as above, we have that $|\psi{\rangle}=\sum_{\alpha,\beta}|\Psi_{\alpha,\beta}{\rangle}|\Psi_{\beta,\alpha}{\rangle}$, with $|\Psi_{\alpha,\beta}{\rangle}=\sum_{i_1,\ldots,i_{\frac{N}{2}}} {\langle}\alpha|A_{i_1}\cdots A_{i_{\frac{N}{2}}}|\beta{\rangle}|i_1\cdots i_{\frac{N}{2}}\rangle$. But now $${\langle}\Psi_{\alpha,\beta}|\Psi_{\alpha',\beta'}{\rangle}= {\langle}\alpha'|{\mathcal{E}}^{\frac{N}{2}}(|\beta'{\rangle}{\langle}\beta|)|\alpha{\rangle}=\lambda_{\alpha}\delta_{\alpha,\alpha'}\delta_{\beta,\beta'},$$ up to corrections of the order $|\nu_2|^{N/2}$ (where $\nu_2$ is the second largest eigenvalue of ${\mathcal{E}}$). This implies that with increasing $N$ $$|\psi{\rangle}=\sum_{\alpha,\beta}\sqrt{\lambda_\alpha \lambda_\beta} \frac{|\Psi_{\alpha,\beta}{\rangle}}{\sqrt{\lambda_\alpha}}\frac{|\Psi_{\beta,\alpha}{\rangle}}{\sqrt{\lambda_\beta}}$$ becomes the Schmidt decomposition associated to half of the chain. Hence we have proved the following. Consider a TI MPS state. In the generic case (condition C2), the eigenvalues of its reduced density operator with respect to half of the chain converge with increasing $N$ to the diagonal matrix $\Lambda\otimes \Lambda$ with $\Lambda$ from the TI canonical form (Thm.\[Th:TIcanonical\]). ### Uniqueness We will prove in this section that the TI canonical form in Thm.\[Th:TIcanonical\] is unique in the generic case. \[thm-uniq\] Let $$|\psi{\rangle}=\sum_{i_1,\ldots,i_N=0}^{d-1} \operatorname{tr}(B_{i_1}\cdots B_{i_N})|i_1\cdots i_N{\rangle}$$ be a TI canonical $D$-MPS such that (i) condition C1 holds, (ii) the OBC canonical representation of $|\psi{\rangle}$ is unique, and (iii) $N>2L_0+D^4$ (a condition polynomial in $D$). Then, if $|\psi{\rangle}$ admits another TI canonical $D$-MPS representation $$|\psi{\rangle}=\sum_{i_1,\ldots,i_N=0}^{d-1} \operatorname{tr}(C_{i_1}\cdots C_{i_N})|i_1\cdots i_N{\rangle},$$ there exists a unitary matrix $U$ such that $B_i=UC_iU^{\dagger}$ for every $i$ (which implies in the case where $\Lambda$ is non-degenerate that $B_i=C_i$ up to permutations and phases). To prove it we need a pair of lemmas. \[lem-same-matr\] Let $T,S$ be linear maps defined on the same vector spaces and suppose that there exist vectors $Y_1,\ldots,Y_n$ such that - $T(Y_k)=S(Y_{k+1})$ for every $1\le k\le n-1$, - $Y_1,\ldots,Y_{n-1}$ are linearly independent, - $Y_n=\sum_{k=1}^{n-1}\lambda_k Y_k$. Consider a solution $x\neq 0$ of the equation $\lambda_1 x^{n-1} +\cdots + \lambda_{n-1}x=1,$ and define $$\begin{aligned} \mu_1&=\lambda_1x\\ \mu_2&=\lambda_1x^2+\lambda_2x\\ \cdots\\ \mu_{n-1}&=\lambda_1 x^{n-1} +\cdots + \lambda_{n-1}x=1.\end{aligned}$$ Then, if $Y=\sum_{k=1}^{n-1}\mu_k Y_k$, we have that $Y\not =0$ and $T(Y)=\frac{1}{x}S(Y)$. [[**Proof.**]{} Clearly $$T(Y)=\sum_{k=1}^{n-1}\mu_kT(Y_k)=\sum_{k=1}^{n-1}\mu_kS(Y_{k+1})=$$ $$=S\left(\sum_{k=1}^{n-2}\mu_kY_{k+1}+\sum_{k=1}^{n-1}\lambda_kY_{k}\right)=$$ $$=S\left(\lambda_1Y_1+ (\lambda_2+\mu_1)Y_2+\cdots +(\lambda_{n-1}+\mu_{n-2})Y_{n-1}\right),$$ and this last expression is exactly $\frac{1}{x}S(Y)$ by the definition of the $\mu$’s. Moreover, since $\mu_{n-1}=1$ and $Y_1,\ldots,Y_{n-1}$ are linearly independent we have that $Y\not = 0$. ${\hfill\fbox\\\medskip }$]{} The following lemma is a consequence of [@HornJohnson91 Theorem 4.4.14]. \[lem-horn\] If $B,C$ are square matrices of the same size $n\times n$, the space of solutions of the matrix equation $$W(C\otimes \mathbbm{1})=(B\otimes \mathbbm{1})W$$ is $S\otimes M_n$, where $S$ is the space of solutions of the equation $XC=BX$. We can prove now Theorem \[thm-uniq\]. [**Proof.**]{} By Proposition \[prop-inj\] we know that the matrices $A^{[j]}_i$ in the canonical OBC representation of $|\psi{\rangle}$ are of dimension $D^2\times D^2$ for any $L_0\le j\le N-L_0$ (in particular there are at least $D^4$ of such $j$’s). From the TI MPS representation of $|\psi{\rangle}$ we can obtain an alternative OBC representation by noticing that $$|\psi{\rangle}=\sum_{i_1,\ldots,i_N=0}^{d-1} b^{[1]}_{i_1}(B_{i_2}\otimes \mathbbm{1})\cdots (B_{i_N}\otimes \mathbbm{1})b^{[N]}_{i_N}|i_1\cdots i_N{\rangle},$$ where $b^{[1]}_i$ is the vector $1\times D^2$ that contains all the rows of $B_i$, that is, $$b^{[1]}_i =(B_i(1,1),B_i(1,2),\ldots,B_i(1,D), B_i(2,1),\ldots),$$ and $b^{[N]}_i$ is the vector $D^2\times 1$ that contains all the columns of $B_i$, that is, $$(b^{[N]}_i)^T= (B_i(1,1),B_i(2,1),\ldots,B_i(D,1), B_i(1,2),\ldots).$$ Doing the same with the $C$’s we have also $$|\psi{\rangle}=\sum_{i_1,\ldots,i_N=0}^{d-1} c^{[1]}_{i_1}(C_{i_2}\otimes \mathbbm{1})\cdots (C_{i_N}\otimes \mathbbm{1})c^{[N]}_{i_N}|i_1\cdots i_N{\rangle}.$$ Using now Theorem \[free-OBC\] and the fact that between $L_0$ and $N-L_0$ both $A^{[j]}_i$, $(B_i\otimes \mathbbm{1})$ and $(C_i\otimes \mathbbm{1})$ are $D^2\times D^2$ matrices, we can conclude that there exist invertible $D^2\times D^2$ matrices $W_1\ldots W_{D^4}$ such that $W_k(C_i\otimes \mathbbm{1})=(B_i\otimes \mathbbm{1})W_{k+1}$ for every $1\le k\le D^4-1$. Now take $n$ such that $W_1,\ldots, W_{n-1}$ are linearly independent but $W_n=\sum_{k=1}^{n-1}\lambda_k W_k$. Let us define $x$ and $\mu_1,\ldots,\mu_{n-1}$ as in Lemma \[lem-same-matr\] and $W=\sum_{k=1}^{n-1}\mu_k W_k$. By Lemma \[lem-same-matr\] we have $W\not =0$ and $W(C_i\otimes \mathbbm{1})=(\frac{1}{x}B_i\otimes \mathbbm{1})W$ for every $i$. Now, Lemma \[lem-horn\] implies that there exist $R\not =0$ such that $RC_i=\frac{1}{x}B_iR$ for every $i$. We can use now that $\Lambda=\sum_i B_i^{\dagger}\Lambda B_i$ to prove that $\sum_iC_i^{\dagger}R^{\dagger}\Lambda RC_i=\frac{1}{|x|^2}R^{\dagger}\Lambda R$. Since the completely positive map $X\mapsto \sum_iC_i^{\dagger}XC_i$ is trace preserving (and $R^{\dagger}\Lambda R\not =0$) one has that $|x|^2= 1$. Now, from $\mathbbm{1}=\sum_i C_iC_i^{\dagger}$, we obtain that $\sum_{i}B_iRR^{\dagger}B_i^{\dagger}=RR^{\dagger}$. Since the $B_i$’s have only one box (Proposition \[prop-inj\]) we conclude that $RR^{\dagger}=\mathbbm{1}$ so that $R$ is a unitary. ${\hfill\fbox\\\medskip }$ ### Obtaining the canonical form In the previous section we have implicitly used the “freedom" that one has in the choice of the matrices in the generic case. In this section we will make this explicit (answering question \[quest.can.1\]) and will use it to show how to obtain efficiently the canonical form (answering question \[quest.can.3\]). Let us take a TI state $|\psi{\rangle}\in\mathbb{C}^{d\otimes N}$ such that the rank of all the reduced density operators is bounded by $D^2$. Clearly it can be stored using a MPS with OBC in $Nd$ $D^2\times D^2$ matrices. If we are in the generic case and this state has a canonical form verifying condition C1, it would be very convenient to have a way of obtaining it, since it allows us to store the state using only $d$ $D\times D$ matrices! In this section we will show how the techniques developed so far allow us to do it by solving an independent of $N$ system of $O(D^8)$ quadratic equations with $O(D^4)$ unknowns. We will assume that the problem has a solution, that is, the state has a TI canonical form with condition C1. We will also assume that we are in the generic case in the sense that the OBC canonical form is unique (no degeneracy in the Schmidt Decomposition). Then, the algorithm to find it reads as follows: We start with the $D^2\times D^2$ matrices $A^{[L_0+1]}_i,\ldots,A^{[L_0+D^4]}_i$ of the OBC canonical form. We solve the following system (S) of quadratic equations in the unknows $Y_j, Z_{j+1}, B_i$, $j=L_0+1,\ldots,L_0+D^4$, $i=1,\ldots,d$ ($Y_j, Z_j$ are $D^2\times D^2$ matrices and $B_i$ $D\times D$ matrices). $$\begin{aligned} B_i\otimes \mathbbm{1}&= Y_j A_j^{[j]} Z_{j+1} &\forall \, i,j\\ Y_jZ_j&=\mathbbm{1} &\forall \, j\\ \sum_i B_iB_i^{\dagger}&=\mathbbm{1}\end{aligned}$$ and we have the following. Consider any TI state $|\psi{\rangle}$ with unique OBC canonical form and such that the rank of each reduced density operator (of successive spins) is bounded by $D^2$. 1. If there is a TI MPS representation verifying condition C1, then the above systems (S) of quadratic equations has a solution. 2. Any solution of (S) gives us a TI $D$-MPS representation of $|\psi{\rangle}$, that is related to *the* canonical one by unitaries ($A_i=UB_iU^{\dagger}$). [[**Proof.**]{} We have seen in the previous Section that the canonical representation is a solution for (S). Now, if $B_i$’s are the solution of (S) and $A_i$’s are the matrices of the canonical representation, we have, reasoning as in the proof of Theorem \[thm-uniq\], that there exists an $R\not = 0$ and an $x\not =0$ such that $RB_i=\frac{1}{x}A_iR$ for every $i$. Using that $\sum_iB_iB_i^{\dagger}=\mathbbm{1}$, that $\Lambda=\sum_i A_i^{\dagger}\Lambda A_i$ and that $\mathbbm{1}$ is the only fixed point of $X\mapsto \sum_iA_iXA_i^{\dagger}$ we can conclude, as in the proof of Theorem \[thm-uniq\], that $R$ is unitary. ${\hfill\fbox\\\medskip }$]{} Parent Hamiltonians {#Sec:Hamiltonians} =================== This section pretends to extend the results of the seminal paper [@FaNaWe92] to the case of a finite chain. That is, we will study when a certain MPS is the *unique* ground state of certain *gapped* local hamiltonian. However, since we deal with a finite chain, the arguments given in [@FaNaWe92] for the “uniqueness" part are no longer valid, and we have to find a different approach. As in the previous section we will start with the case of OBC and then move to the case of TI and PBC. In the “gap" part we will simply sketch the original proof given in [@FaNaWe92]. Uniqueness {#sec:Huniqueness} ---------- ### Uniqueness of the ground state under condition C1 in the case of OBC Let us take a MPS with OBC given in the canonical form $|\psi\rangle=\sum_{i_1,\ldots,i_N} A^{[1]}_{i_1}\cdots A^{[N]}_{i_N}|i_1\cdots i_N\rangle$. Let us assume that we can group the spins in blocks of consecutive ones in such a way that, in the regrouped MPS $|\psi\rangle=\sum_{i_1,\ldots,i_M} B^{[1]}_{i_1}\cdots B^{[M]}_{i_M}|i_1\cdots i_M\rangle$, every set of matrices $B^{[j]}_{i_j}$ verifies condition C1, that is, generates the corresponding space of matrices. If we call $h_{j,j+1}$ the projector onto the orthogonal subspace of $$\{\sum_{i_j,i_{j+1}}\operatorname{tr}(XB^{[j]}B^{[j+1]}): X \text{ arbitrary}\},$$ then $|\psi\rangle$ is the unique ground state of the local Hamiltonian $H=\sum_j h_{j,j+1}$. [**Proof.**]{} Any ground state $|\phi{\rangle}$ of $H$ verifies that $h_{j,j+1}\otimes {\mathbbm{1}}|\phi{\rangle}=0$ for every $j$, that is $$\label{thm:unique-OBC}|\phi{\rangle}= \sum_{i_1,\ldots,i_M} \operatorname{tr}(X^j_{I(j,j+1)}B^{[j]}_{i_j}B^{[j+1]}_{i_{j+1}})|i_1\ldots i_M{\rangle},$$ where $I(j,j+1)$ is the set of indices $i_1\ldots i_{j-1}, i_{j+2} \ldots i_M$. Mixing (\[thm:unique-OBC\]) for $j$ and $j+1$ and using condition C1 for $B^{[j+1]}$ gives $$X^j_{I(j,j+1)}B^{[j]}_{i_j}= B^{[j+2]}_{i_{j+2}} X^{j+1}_{I(j+1,j+2)}.$$ Using now that $\sum_{i_j} B^{[j]}_{i_j} B^{[j]\, \dagger}_{i_j}={\mathbbm{1}}$ and calling $Y^j_{I(j,\ldots,j+2)}=\sum_{i_j}X^{j+1}_{I(j+1,j+2)} B^{[j]\, \dagger}_{i_j}$ we get $$|\phi{\rangle}= \sum_{i_1,\ldots,i_M} \operatorname{tr}(Y^j_{I(j,\ldots,j+2)}B^{[j]}_{i_j}\cdots B^{[j+2]}_{i_{j+2}})|i_1\ldots i_M{\rangle}.$$ Using the trivial fact that blocking again preserves condition C1 one can easily finish the argument by induction. We just notice that in the last step one obtains $$|\phi{\rangle}=\sum_{i_1,\ldots,i_M} XB^{[1]}_{i_1}\cdots B^{[M]}_{i_M}|i_1\cdots i_M\rangle,$$ where $X$ is just a number that, by normalization, has to be $1$, giving $|\phi{\rangle}=|\psi{\rangle}$ and hence the result. ${\hfill\fbox\\\medskip }$ ### Uniqueness of the ground state under condition C1 with TI and PBC To obtain the analogue result in the case of TI and PBC one can apply the same argument. However, since we do not have any more vectors in the first and last positions, we need to refine the reasoning of the last step. Moreover, using the symmetry we have now, one can decrease a bit the interaction length of the Hamiltonian, from $2L_0$ to $L_0+1$. Let us be a bit more concrete. Given our ring of $N$ $d$-dimensional quantum systems, $L\le N$ and a subspace $S$ of ${{\mathbb{C}}^d}^{\otimes L}$, we denote $H_S=\sum_{i=1}^N\tau^i(h_S)$, where $h_S$ is the projection onto $S^{\perp}$ [^2]. If we start with a MPS $|\psi{\rangle}=\sum_{i_1,\ldots,i_N}\operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N{\rangle}$ with property C1, we will consider $L>L_0$ and, as before, the subspace ${\mathcal{G}}_L^A$ (or simply ${\mathcal{G}}_L$) formed by the elements $\sum_{i_1,\ldots,i_L}\operatorname{tr}(XA_{i_1}\cdots A_{i_L})|i_1\cdots i_L{\rangle}$. It is clear that $H_{{\mathcal{G}}_L}|\psi{\rangle}=0$ and that $H_{{\mathcal{G}}_L}$ is frustration free. Moreover, if $N\ge 2L_0$ and $L>L_0$, then \[uniqueGS\] $|\psi{\rangle}$ is the only ground state of $H_{{\mathcal{G}}_L}$. [[**Proof.**]{} Reasoning as in the case of OBC one can easily see that any ground state $|\phi{\rangle}$ of $H_{{\mathcal{G}}_L}$ is in ${{\mathcal{G}}}_N$, that is, has the form $|\phi{\rangle}=\sum_{i_1,\ldots,i_{N}}\operatorname{tr}(XA_{i_1}\cdots A_{i_{N}})|i_1\cdots i_{N}{\rangle}$. Since there is no distinguished first position, $|\phi{\rangle}$ can also be written $$|\phi{\rangle}=\sum_{i_1,\ldots,i_{N}}\operatorname{tr}(A_{i_1}\cdots A_{i_{L_0}}YA_{i_{L_0+1}}\cdots A_{i_{N}})|i_1\cdots i_{N}{\rangle}.$$ By condition C1, $XA_{i_1}\cdots A_{i_{L_0}}=A_{i_1}\cdots A_{i_{L_0}}Y$ for every $i_1,\ldots,i_{L_0}$. But, also by condition C1, $A_{i_1}\cdots A_{i_{L_0}}$ generates the whole space of $D\times D$ matrices. Hence $X=Y$ which, in addition, commutes with $A_{i_1}\cdots A_{i_{L_0}}$ for every $i_1,\ldots,i_{L_0}$. This means that $X$ commutes with every matrix and hence $X=\lambda\mathbbm{1}$ and $|\phi{\rangle}=|\psi{\rangle}$. ${\hfill\fbox\\\medskip }$]{} ### Non-uniqueness in the case of two or more blocks In the absence of condition C1 in our MPS, we cannot guarantee uniqueness for the ground state of the parent Hamiltonian. There are two different properties that can lead to degeneracy. One is the existence of a periodic decomposition (Theorem \[Th:periodic\]), that can happen even in the case of one block. This is the case of the Majumdar-Gosh model (Section \[Sec:prelim\]). The other property is the existence of more than one block in the canonical form (Thm.\[Th:TIcanonical\]). As we will see below, this leads to a stronger version of degeneracy that is closely related to the number of blocks. In particular, we are going to show that whenever we have more than one block, the MPS is *never* the unique ground state of a frustration free local Hamiltonian (Thm \[2blocks.1\]). In addition, there exists one local Hamiltonian which has the MPS as ground state and with ground space degeneracy equal to the number of blocks (Thm \[2blocks.2\]). That is, a number of blocks greater than one can correspond to a spontaneously broken symmetry [@NachtergaeleSym]. In all this section we need to assume that we have condition C1 in each block. For a detailed discussion of the *reasonability* of this hypothesis see Appendix \[open.problems\]. Let us take a TI MPS $|\phi{\rangle}$ with $b (\ge 2)$ blocks $A^1_i,\ldots, A_i^b$ in the canonical form, $\operatorname{size}(A^1_i)\ge \cdots\ge \operatorname{size}(A^b_i)$, and condition C1 in each block. Let us call $L_0=\max_j\{L_0^{A^j}\}$, which will be logarithmic in $N$ for the *generic* case. Clearly $|\phi{\rangle}=\sum_{j=1}^b|\phi_{A^j}{\rangle}$, where $$|\phi_{A^j}{\rangle}=\lambda_j^N\sum_{i_1,\ldots,i_N} \operatorname{tr}(A^j_{i_1}\cdots A^j_{i_N})|i_1\cdots i_N\rangle.$$ Moreover, w.l.o.g. we can assume that the states $|\phi_{A^j}{\rangle}$ are pairwise different. The following lemmas will take care of the technical part of the section. \[lem1\] Given any $D\times D$ matrices $C\not =0$ and $X$ there exist matrices $R_i, S_i$ such that $X=\sum_i R_iCS_i$. [[**Proof.**]{} By the polar decomposition, it is easy to find matrices $E,F,G, H$ (with $G,H$ invertible) such that $ECF=|1{\rangle}{\langle}1|$ and $GXH=\sum_{i=1}^m|i{\rangle}{\langle}i|$. Clearly $\sum_{i=1}^m|i{\rangle}{\langle}i|=\sum_{i=1}^m P_i|1{\rangle}{\langle}1|P_i,$ where $P_i$ is the matrix obtained from the identity by permuting the first and the $i$-th row. So $R_i=G^{-1}P_iE$ and $S_i=FP_iH^{-1}$. ${\hfill\fbox\\\medskip }$]{} \[lem:direct-sum\] If $L\ge 3(b-1)(L_0+1)$, the sum $\bigoplus_{j=1}^b \mathcal{G}_L^{A^j}$ is direct. [**Proof.**]{} We group the spins in $3(b-1)$ blocks of at least $L_0+1$ spins each and then use induction. First the case $b=2$. We assume on the contrary that there exist $X, Y\not =0$ such that $$\sum_{i_{1},i_2,i_3} \operatorname{tr}(A^1_{i_1}XA^1_{i_2}A^1_{i_3})|i_{1}i_2 i_3\rangle=\sum_{i_{1},i_2,i_3} \operatorname{tr}(A^2_{i_1}YA^2_{i_2}A^2_{i_3})|i_{1}i_2 i_3\rangle.$$ If we consider now an arbitrary matrix $Z$, by C1 and Lemma \[lem1\], there exist complex numbers $\mu^i_{i_1}, \rho^i_{i_2}$ such that $Z=\sum_i\sum_{i_1,i_2}\mu^i_{i_1} \rho^i_{i_{2}}A^1_{i_1}XA^1_{i_{2}}.$ Calling $W=\sum_i\sum_{i_1,i_2}\mu^i_{i_1} \rho^i_{i_{2}}A^2_{i_1}YA^2_{i_{2}}$ we have that $\sum_{i_3}\operatorname{tr}(ZA^1_{i_3})|i_3{\rangle}=\sum_{i_3}\operatorname{tr}(WA^2_{i_3})|i_3{\rangle}$. This means that ${\mathcal{G}}^{A^1}_{L_0+1}\subset{\mathcal{G}}^{A^2}_{L_0+1}$. Since $\operatorname{size}(A^1_i)\ge \operatorname{size}(A^2_i)$, this implies that $\operatorname{size}(A^1_i)= \operatorname{size}(A^2_i)$ and that ${\mathcal{G}}^{A^1}_{L_0+1}={\mathcal{G}}^{A^2}_{L_0+1}$. But now, taking the local Hamiltonian $H_{{\mathcal{G}}_{L_0+1}^{A^1}}=H_{{\mathcal{G}}_{L_0+1}^{A^2}}$, by Theorem \[uniqueGS\], both $|\phi_{A^1}{\rangle}$ and $|\phi_{A^2}{\rangle}$ should be its *only* ground state; which is the desired contradiction. Now the induction step. Let us start with $\sum_{j=1}^{b+1}w^j=0$, where $${\mathcal{G}}^{A^j}_{3b(L_0+1)}\ni w_j=\sum_{i_1,\ldots,i_{3b}}\operatorname{tr}(A^j_{i_1}W^jA^j_{i_2}\cdots A^j_{i_{3b}})|i_1\cdots i_{3b}{\rangle}.$$ We want to prove that $W^j=0$ for every $j$. So let us assume the opposite, take $j$ such that $W^j\not =0$ and call $\tilde{w}_j=\left({\mathbbm{1}}_{[12]}\otimes h_{{\mathcal{G}}^{A^{b+1}}_{[3]}}\otimes {\mathbbm{1}}_{[4\ldots 3b]}\right)(w_j)$. We have that $\sum_{j=1}^b\tilde{w}_j=0$, and, by the induction hypothesis, each $\tilde{w}_j=0$. Now $$\tilde{w}_j=\sum_{i_1,\ldots,i_{3b}}\operatorname{tr}(A^j_{i_1}W^jA^j_{i_2}X^j_{i_3}A^j_{i_4}\cdots A^j_{i_{3b}})|i_1\cdots i_{3b}{\rangle},$$ with $X^j_{i_3}\not = 0$ for some $i_3$ (Theorem \[uniqueGS\]). Moreover, we can use condition C1 and Lemma \[lem1\] to get complex numbers $\mu^i_{i_1}, \rho^i_{i_2}$ such that ${\mathbbm{1}}=\sum_i\sum_{i_1,i_2}\mu^i_{i_1} \rho^i_{i_{2}}A^j_{i_1}W^jA^j_{i_{2}}.$ Hence, $0=\sum_{i_4,\ldots, i_{3b}}\operatorname{tr}(X_{i_3}^jA^j_{i_4}\cdots A^j_{i_{3b}})|i_4\cdots i_{3b}{\rangle}$ for every $i_3$, which implies by C1 the contradiction $X^j_{i_3}=0$. ${\hfill\fbox\\\medskip }$ Finally the results, \[2blocks.1\] If $N\ge 3(b-1)(L_0+1)+L$ and $H=\sum_i\tau^i(h)$ is any translationally invariant frustration free $L$-local Hamiltonian on our ring of $N$ spins that has $|\phi{\rangle}$ as a ground state (that is, $H|\phi{\rangle}=0$), then $|\phi_{A^j}{\rangle}$ is also a ground state of $H$ for every $j$. In particular $H$ has more than one ground state. [[**Proof.**]{} One has $0=(h\otimes {\mathbbm{1}})|\phi{\rangle}=\sum_j(h\otimes {\mathbbm{1}})|\phi_{A^j}{\rangle}$. Since $(h\otimes {\mathbbm{1}})|\phi_{A^j}{\rangle}\in \left({\mathbb{C}}^d\right)^{\otimes L}\otimes {\mathcal{G}}^{A^j}_{N-L}$, we can use Lemma \[lem:direct-sum\] to get the desired conclusion: $(h\otimes {\mathbbm{1}})|\phi_{A^j}{\rangle}=0$ for every $j$. ${\hfill\fbox\\\medskip }$]{} \[2blocks.2\] There exists a local Hamiltonian $H$ acting on $L\ge 3(b-1)(L_0+1)+1$ spins such that its ground space is exactly $\ker(H)=\operatorname{span}\{|\phi_{A^j}{\rangle}\}_{1\le j\le b}$. [**Proof.**]{} The hamiltonian will be $H_S$ with $S=\bigoplus_j \mathcal{G}^{A^j}_{L}$, and $L\ge 3(b-1)(L_0+1)+1$. For $m\ge L$, $${\mathbb{C}}^d\otimes\left(\bigoplus_j \mathcal{G}^{A^j}_{m}\right)\cap \left(\bigoplus_j \mathcal{G}^{A^j}_{m}\right)\otimes {\mathbb{C}}^d=\bigoplus_j \mathcal{G}^{A^j}_{m+1}.$$ In fact, if $|\phi{\rangle}\in {\mathbb{C}}^d\otimes(\bigoplus_j \mathcal{G}^{A^j}_{m})\cap (\bigoplus_j \mathcal{G}^{A^j}_{m})\otimes {\mathbb{C}}^d$, we have simultaneously that $$\begin{aligned} |\psi{\rangle}&=\sum_j \sum_{i_1,\ldots,i_{m+1}}\operatorname{tr}(A^j_{i_{m+1}}C^j_{i_1}A^j_{i_2}\cdots A^j_{i_m}) |i_1\cdots i_{m+1}{\rangle}\quad \text{ and}\\ |\psi{\rangle}&=\sum_j\sum_{i_1,\ldots,i_{m+1}}\operatorname{tr}(D^j_{i_{m+1}}A^j_{i_1}A^j_{i_2}\cdots A^j_{i_m})|i_1\cdots i_{m+1}{\rangle}\;.\end{aligned}$$ Lemma \[lem:direct-sum\] and condition C1 allows us to identify for every $j, i_1, i_{m+1}$ $$A^j_{i_{m+1}}C^j_{i_1}=D^j_{i_{m+1}}A^j_{i_1}.$$ Calling $E^j=\sum_{i_1}C^j_{i_1}A^j_{i_1}$ and using that $\sum_{i_1}A^j_{i_1}A^{j\, \dagger}_{i_1}={\mathbbm{1}}$, we get $A^j_{i_{m+1}}C^j_{i_1}=A^j_{i_{m+1}}E^jA^j_{i_1},$ which implies that $|\phi{\rangle}\in \bigoplus_j \mathcal{G}^{A^j}_{m+1}.$ Then one can easily follow the lines of the proof of Theorem \[uniqueGS\] (assuming $N\ge L+L_0$) to conclude that $\ker(H_S)= \operatorname{span}\{|\phi_{A^j}{\rangle}\}_{1\le j\le b}$. ${\hfill\fbox\\\medskip }$ Energy gap {#Sec:Hgap} ---------- If the ground state energy is zero (which can always be achieved by a suitable offset), the *energy gap* $\gamma$ above the ground space is the largest constant for which $$H^2\geq\gamma H\;.$$ If in addition $H=\sum_i \tau^i(h)$ is frustration free and has interaction length $l$, by taking any $p\ge l$ and grouping the spins in blocks of $p$, one can define an associated $2$-local interaction in the regrouped chain by $\tilde{h}_{i,i+1}=H_{[pi+1,\ldots, (i+2)p]}$ ( $=\sum_{j=pi}^{p(i+2)-l}\tau^j(h)$). The [*new*]{} hamiltonian $\tilde{H}$ verifies $$H_{[1,\ldots, mp]}\le \tilde{H}_{[1\ldots m]}=\sum_{i=0}^{m-1}\tau^i(\tilde{h})\le 2H_{[1,\ldots, mp]}.$$ Moreover, calling $P$ to the projection onto $\ker(\tilde{h})$, there exists a constant $\gamma_{2p}$ (that is exactly the spectral gap of $H_{[1\ldots,2p]}$) such that $\tilde{h}\ge \gamma_{2p}({\mathbbm{1}}-P)$. Therefore, to study the existence of an energy gap in a local TI frustration free Hamiltonian, it is enough to study the case of a nearest neighbor interaction $\hat{H}=\sum_i\tau^i(P_{i,i+1})$ where $P$ is a projector. In this situation, Knabe [@Knabe] gave a sufficient condition to assure the existence of a gap, namely that the gap $\epsilon_n$ of $\hat{H}_{[1\ldots, n+1]}$ is bigger than $\frac{1}{n}$ for some $n$. In the particular case of the parent hamiltonian of an MPS, a much more refined argument was provided in [@FaNaWe92] to prove the existence of a gap under condition C1. The idea reads as follows. Clearly $\hat{H}^2\ge \hat{H}+ \sum_{i} P_{i,i+1}P_{i+1, i+2}+ P_{i,i+2}P_{i+1, i+1}$. By the proof of Theorem \[uniqueGS\], $$P_{i,i+1}={\mathbbm{1}}_{[1\ldots pi]}\otimes ({\mathbbm{1}}-P_{{\mathcal{G}}_{2p}}) \otimes {\mathbbm{1}}_{[p(i+2)+1\ldots N]}.$$ Then a technical argument proves that $P_{i,i+1}P_{i+1, i+2}+ P_{i,i+2}P_{i+1, i+1}\ge -O(\nu_2^p)( P_{i,i+1}+ P_{i+1, i+2})$, where $\nu_2$ is the second largest eigenvalue of ${\mathcal{E}}$. This gives $\hat{H}^2\ge(1-O(\nu_2^p))\hat{H}$, which concludes the argument. Generation of MPS {#sec:generation} ================= The MPS formalism is particularly suited for the description of sequential schemes for the generation of multipartite states. Consider for instance a chain of spins in a pure product state. Two possible *sequential* ways of preparing a more general state on the spin chain are either to let an ancillary particle (the head of a Turing machine) interact sequentially with all the spins or to make them interact themselves in a sequential manner: first spin 1 with 2 then 2 with 3 an so on. Clearly, many physical setups for the generation of multipartite states are of such sequential nature: time-bin photons leaking out of an atom-cavity system, atoms passing a microwave cavity or laser pulses propagating through atomic ensembles. We will see in the following that the MPS formalism provides the natural language for describing such schemes. This section reviews and extends the results obtained in [@SSVCW]. A detailed application of the formalism to particular physical systems can be found in [@SSVCW; @SSVCW2]. Sequential generation with ancilla ---------------------------------- Consider a spin chain which is initially in a product state $|0\rangle^{\otimes N}\in {\cal H}_{\cal B}^{\otimes N}$ with ${\cal H}_{\cal B}\simeq\mathbb{C}^d$ and an additional ancillary system in the state $ |\varphi_I\rangle\in{\cal H}_{\cal A}\simeq \mathbb{C}^D$. Let $\sum_{i,\alpha,\beta} A^{[k]}_{i,\alpha,\beta} |\alpha,i\rangle\langle\beta,0|$ be a general stochastic operation on ${\cal H}_{\cal A}\otimes {\cal H}_{\cal B}$ applied to the ancillary system and the $k$’th site of the chain. This operation could for instance correspond to one branch of a measurement or to a unitary interaction in which case $\sum_i A^{[k]\dag}_iA^{[k]}_i=\mathbbm{1}$. If we let the ancillary system interact sequentially with all $N$ sites and afterwards measure the ancilla in the state $|\varphi_F\rangle$, then the remaining state on ${\cal H}_{\cal B}^{\otimes N}$ is (up to normalization) clearly given by the MPS $$\label{OBCMPSgen} |\psi\rangle=\sum_{i_1,\ldots,i_N=1}^d \langle\varphi_F|A^{[N]}_{i_N}\cdots A^{[2]}_{i_{2}}A^{[1]}_{i_1}|\varphi_I\rangle|i_N\cdots i_1\rangle.$$ By imposing different constraints on the allowed operations (and thus on the $A$’s) we can distinguish the following types of sequential generation schemes for pure multipartite states: 1. *Probabilistic schemes*: arbitrary stochastic operations with a $D$-dimensional ancilla are allowed. 2. *Deterministic schemes*: the interactions must be unitary and the $D$-dimensional ancilla must decouple in the last step (without measurement). 3. *Deterministic transition schemes* (for $d=2$): Here we consider an enlarged ancillary system ${\cal H}_{\cal A}=\mathbb{C}^D\oplus\mathbb{C}^D\simeq \mathbb{C}^D\otimes\mathbb{C}^2$ (corresponding e.g. to $D$ ‘excited’ and $D$ ‘ground state’ levels) and a fixed interaction of the form $$\begin{aligned} |\varphi\rangle|1\rangle|0\rangle_{\cal B} &\mapsto& |\varphi\rangle|0\rangle|1\rangle_{\cal B}\nonumber\;,\\ |\varphi\rangle|0\rangle|0\rangle_{\cal B} &\mapsto& |\varphi\rangle|0\rangle|0\rangle_{\cal B}\nonumber\;. \end{aligned}$$ Moreover, we allow for arbitrary unitaries on $\cal H_A$ in every step and require the ancilla to decouple in the last step. \[Thm:seqwith\] The three sets of multipartite states which can be generated by the above sequential schemes with ancilla are all equal to the set of states with OBC MPS representation with maximal bond dimension $D$. Note that the proof of this statement in [@SSVCW] (based on subsequent singular value decompositions) also provides a *recipe* for the generation of any given state (with minimal resources). This idea has been recently exploited in [@Kike] to analyze the resources needed for sequential quantum cloning. Sequential generation without ancilla {#Sec:gen2} ------------------------------------- Let us now consider sequential generation schemes without ancilla. The initial state is again a product $|0\rangle^{\otimes N} \in \mathbb{C}^{d\otimes N}$ and we perform first an operation affecting the sites 1 and 2, then 2 and 3 up to one between $N-1$ and $N$. Again we may distinguish between probabilistic and deterministic schemes and as before both classes coincide with a certain set of MPS: \[Sequential generation without ancilla\] The sets of pure states which can be generated by a sequential scheme without ancilla either deterministically or probabilistically are both equal to the class of states having an OBC MPS representation with bond dimension $D\leq d$. [**Proof.**]{} Let us denote the map acting on site $k$ and $k+1$ by $U^{[k]}$. Then for $k<N-1$ we can straight forward identify the matrices in the MPS representation by $A^{[k]}_{i,\alpha,\beta}=\langle i, \beta |U^{[k]}|\alpha,0\rangle$ where for $k=1$ we have $\alpha=0$, i.e., $A^{[1]}_i$ are vectors. From the last map with coefficients $\langle i,j|U^{[N-1]}|\alpha,0\rangle$ we obtain the $A$’s by a singular value decomposition (in $i|\alpha,j$), such that $A_i^{[N]}$ is again a set of vectors. Hence, all states generated in this way are OBC MPS with $D\leq d$. Conversely, we can generate every such MPS deterministically in a sequential manner without ancilla. To do this we exploit Thm.\[Thm:seqwith\] and use the site $k+1$ as ‘ancilla’ for the $k$’th step (i.e., the application of a unitary $U^{[k]}$) followed by a swap between site $k+1$ and $k+2$. $N-1$ of these steps are sufficient since the last step in the proof for the deterministic part in Thm.\[Thm:seqwith\] is just a swap between the ancilla and site $N$. ${\hfill\fbox\\\medskip }$ Classical simulation of quantum systems {#sec:classical-sim} ======================================= We saw in Section \[Sec:prelim\] that if a quantum many-body state has a MPS representation with sufficiently small bond dimension $D$, then we can efficiently store it on a classical computer and calculate expectation values. The practical relevance of MPS representations in the context of classical simulation of quantum systems stems then from two facts: (i) Many of the states arising in condensed matter theory (of one-dimensional systems) or quantum information theory either have such a small-$D$ MPS representation or are well approximated by one; and (ii) One can efficiently obtain such approximating MPS. The following section will briefly review the most important results obtained along these lines. Properties of ground states of spin chains ------------------------------------------ The main motivation for introducing the class of MPS was to find a class of wavefunctions that capture the physics needed to describe the low-energy sector of local quantum Hamiltonians. Once such a class is identified and expectation values of all states in the class can be computed efficiently, it is possible to use the corresponding states in a variational method. The very successful renormalization group methods, first developed by Wilson [@Wilson] and later refined by White [@White], are precisely such variational methods within the class of matrix product states [@Ostlund; @vonDelft; @VPC04]. In the case of 1-D systems (i.e. spin chains) with local interactions, the low-energy states indeed exhibit some remarkable properties. First, ground states have by definition extremal local properties (as they minimize the energy), and hence their local properties determine their global ones. Let us consider any local Hamiltonian of $N$ spins that exhibits the property that there is a unique ground state $|\psi_{ex}\rangle$ and that the gap is $\Delta(N)$. Let us furthermore consider the case when $\Delta(N)$ decays not faster than an inverse polynomial in $N$ (this condition is satisfied for all gapped systems and for all known critical translationally invariant systems). Then let us assume that there exists a state $|\psi_{appr}\rangle$ that reproduces well the local properties of all nearest neighbor reduced density operators: $\|\rho_{appr}-\rho_{ex}\|\leq \delta$. Then it follows that the global overlap is bounded by $$\||\psi_{ex}\rangle-|\psi_{appr}\rangle\|^2\leq \frac{N\delta}{\Delta(N)}.$$ This is remarkable as it shows that it is enough to reproduce the local properties well to guarantee that also the global properties are reproduced accurately: for a constant global accuracy $\epsilon$, it is enough to reproduce the local properties well to an accuracy $\delta$ that scales as an inverse polynomial in the number of spins. This is very relevant in the context of variational simulation methods: if the energy is well reproduced and if the computational effort to get a better accuracy in the energy only scales polynomially in the number of spins, then a scalable numerical method can be constructed that reproduces all global properties well (here scalable means essentially a polynomial method) [^3]. Second, there is very few entanglement present in ground states of spin chains, even in the case of a critical system. The relevant quantity here is to study area-laws: if one considers the reduced density operator $\rho_L$ of a contiguous block of $L$ spins in the ground state of a spin chain with $N\gg L$ spins, how does the entropy of that block scale with $L$? This question was first studied in the context of black-hole entropy [@Bombelli-Srednicki-Bousso] and has recently attracted a lot of attention [@arealaw; @Ortiz; @Cardy]. Ground states of local Hamiltonians of spins seem to have the property that the entropy is not an extensive property but that the leading term in the entropy only scales as the boundary of the block (hence the name area-law), which means a constant in the case of a 1-D system [@Cardy; @Jin]: $$S^\alpha(\rho_L)\simeq \frac{c}{6}\left(1+\frac{1}{\alpha}\right)\log(\xi)\label{arealaw}$$ Here $S_\alpha$ is the Renyi entropy $$S^\alpha(\rho)=\frac{1}{1-\alpha}\log\left({\rm Tr}\rho^\alpha\right),$$ $c$ is the central charge[^4] and $\xi$ the correlation length. This has a very interesting physical meaning: it shows that most of the entanglement must be concentrated around the boundary, and therefore there is much less entanglement than would be present in a random quantum state (where the entropy would be extensive and scale like $L$). The area law (\[arealaw\]) is mildly violated in the case of 1-D critical spin systems where $\xi$ has to be replaced with $L$, but even in that case the amount of entanglement is still exponentially smaller than the amount present in a random state. This is very encouraging, as one may exploit the lack of entanglement to simulate these systems classically. Indeed, we already proved that MPS obey the same property [@Ortiz]. The existence of an area law for the scaling of entropy is intimately connected to the fact that typical quantum spin systems exhibit a finite correlation length. In fact, it has been recently proven [@hastings1] that all connected correlation functions between two blocks in a gapped system have to decay exponentially as a function of the distance of the blocks. Let us therefore consider a 1-D gapped quantum spin system with correlation length $\xi_{corr}$. Due to the finite correlation length, it is expected that the reduced density operator $\rho_{AB}$ obtained when tracing out a block $C$ of length $l_{AB}\gg \xi_{corr}$ is equal to $$\rho_{AB}\simeq \rho_A\otimes\rho_B\label{hasg}$$ up to exponentially small corrections [^5]. The original ground state $|\psi_{ABC}\rangle$ is a purification of this mixed state, but it is of course also possible to find another purification of the form $|\psi_{AC_l}\rangle\otimes|\psi_{BC_r}\rangle$ (up to exponentially small corrections) with no correlations whatsoever between $A$ and $B$; here $C_l$ and $C_r$ together span the original block $C$. Since different purifications are unitarily equivalent, there exists a unitary operation $U_C$ on the block $C$ that completely disentangles the left from the right part: $$I_A\otimes U_{C}\otimes I_C|\psi_{ABC}\rangle\simeq|\psi_{AC_l}\rangle\otimes|\psi_{BC_r}\rangle.$$ This implies that there exists a tensor $A_{i,\alpha,\beta}$ with indices $1\leq i,\alpha,\beta\leq D$ (where $D$ is the dimension of the Hilbert space of $C$) and states $|\psi^A_\alpha\rangle,|\psi_i^C\rangle,|\psi^B_\beta\rangle$ defined on the Hilbert spaces belonging to $A,B,C$ such that $$|\psi_{ABC}\rangle\simeq \sum_{i,\alpha,\beta}A_{i,\alpha,\beta}|\psi^A_\alpha\rangle|\psi^C_i\rangle|\psi^B_\beta\rangle.$$ Applying this argument recursively leads to a matrix product state description of the state and gives a strong hint that ground states of gapped Hamiltonians are well represented by MPS. It turns out that this is even true for critical systems. MPS as a class of variational wavefunctions {#Sec:approx} ------------------------------------------- Here we will review the main results of [@VeCi05], which give analytical bounds for the approximation of a state by a MPS that justify the choice of MPS as a reasonable class of variational wavefunctions. We consider an arbitrary state $|\psi\rangle$ and denote by $$\{\mu^{[k]i}\}, i=1..N_k=d^{\min(k,N-k)}$$ the eigenvalues of the reduced density operators $$\rho_k=\rm{Tr}_{k+1,k+2,\ldots ,N}|\psi\rangle\langle\psi|,$$ sorted in decreasing order. There exists a MPS $|\psi_D\rangle$ with bond dimension $D$ such that $$\||\psi\rangle-|\psi_D\rangle\|^2\leq 2\sum_{k=1}^{N-1}\epsilon_k(D)$$ where $\epsilon_k(D)=\sum_{i=D+1}^{N_k}\mu^{[k]i}$. This shows that for systems for which the $\epsilon_k(D)$ decay fast in $D$, there exist MPS with *small* $D$ which will not only reproduce well the local correlations (such as energy) but also all the nonlocal properties (such as correlation length). The next result relates the derived bound to the Renyi entropies of the reduced density operators. Given a density matrix $\rho$, we denote as before $\epsilon(D)=\sum_{i=D+1}^\infty\lambda_i$ with $\lambda_i$ the nonincreasingly ordered eigenvalues of $\rho$. Then we have If $0<\alpha<1$, then $\log(\epsilon(D))\leq \frac{1-\alpha}{\alpha}\left(S^\alpha(\rho)-\log\frac{D}{1-\alpha}\right)$. The two results together allow us to investigate the computational effort needed to represent critical systems, arguable the hardest ones to simulate [^6], as MPS. The key fact here is the area-law (\[arealaw\]), which for critical systems reads $$S^\alpha(\rho_L)\simeq\frac{c}{6} \left(1+\frac{1}{\alpha}\right)\log(L)\label{infg}$$ for all $\alpha>0$ ($c$ the central charge). Let us therefore consider the Hamiltonian associated to a critical system, but restricted to $2L$ sites. The entropy of a half chain (we consider the ground state $|\psi_{ex}\rangle$ of the finite system) will typically scale as in eq. (\[infg\]) but with an extra term that scales like $1/L$. Suppose we want to get that $\||\psi_{ex}\rangle-|\psi_D\rangle\|^2\leq\epsilon_0/L$ with $\epsilon_0$ independent of $L$ [^7]. If we call $D_L$ the minimal $D$ needed to obtain this precision for a chain of length $2L$, the previous two results combined yield $$D_L\leq cst\left(\frac{L^2}{(1-\alpha)\epsilon_0}\right)^{\frac{\alpha}{1-\alpha}} L^{\frac{c+\bar{c}}{12}\frac{1+\alpha}{\alpha}}.$$ This shows that $D$ only has to scale polynomially in $L$ to keep the accuracy $\epsilon_0/L$ fixed; in other words, there exists an efficient scalable representation for ground states of critical systems (and hence also of noncritical systems) in terms of MPS. This is a very strong result, as it shows that one can represent ground states of spin chains with only polynomial effort (as opposed to the exponential effort if one would do e.g. exact diagonalization). The above result was derived from a certain scaling of the block Renyi entropy with $\alpha<1$. In fact, for $\alpha\geq 1$, i.e., in particular for the von Neumann entropy ($\alpha=1$), even a saturating block entropy does in general not allow to conclude that states are efficiently approximable by MPS [@entropyMPS]. Conversely, a linear scaling for the von Neumann entropy, as generated by particular time evolutions, rules out such an approximability [@entropyMPS]. Variational algorithms ---------------------- Numerical renormalization group methods have since long been known to be able to simulate spin systems, but it is only recently that the underlying structure of matrix product states has been exploited. Both NRG, developed by K. Wilson [@Wilson] in the ’70s, and DMRG, developed by S. White [@White] in the ’90s, can indeed be reformulated as variational methods within the class of MPS [@VPC04; @vonDelft]. The main question is how to find the MPS that minimizes the energy for a given spin chain Hamiltonian: given a Hamiltonian $\mathcal{H}$ acting on nearest neighbours, we want to find the MPS $|\psi\rangle$ such that the energy $$\frac{\langle\psi|\mathcal{H}|\psi\rangle}{\langle\psi|\psi\rangle}$$ is minimized. If $|\psi\rangle$ is a TI MPS, then this is a highly complex optimization problem. The main trick to turn this problem into a tractable one is to break the translational symmetry and having site-dependent matrices $A_i^{[k]}$ for the different spins $k$; indeed, then the functional to be minimized is a multiquadratic function of all variables, and then one can use the standard technique of alternating least squares to do the minimization [@VPC04]. This works both in the case of open and closed boundary conditions, and in practice the convergence of the method is excellent. The computational effort of this optimization scales as $D^4 d^2$ in memory and similarly in time (for a single site optimization). But what about the theoretical worst case computational complexity of finding this optimal MPS? It has been observed that DMRG converges exponentially fast to the ground state with a relaxation time proportional to the inverse of the gap $\Delta$ of the system [@DMRG1]. For translationally invariant critical systems, this gap seems to close only polynomially. As we have proven that $D$ only have to scale polynomially too, the computational effort for finding ground states of 1-D quantum systems is polynomial ($P$). This statement is true under the following conditions: 1) the $\alpha$-entropy of blocks in the exact ground state grow at most logarithmically with the size of the block for some $\alpha<1$; 2) the gap of the system scales at most polynomially with the system size; 3) given a gap that obeys condition 2, there exists an efficient DMRG-like algorithm that converges to the global minimum. As the variational MPS approach [@VPC04] is essentially an alternating least squares method of solving a non-convex problem, there is a priori no guarantee that it will converge to the global optimum, although the occurrence of local minima seems to be unlikely [@DMRG1]. But still, this has not been proven and the worst-case complexity could well be NP-hard, as multiquadratic cost functions have been shown to lead to NP-hard problems [@Nemirovski]. [^8] Classical simulation of quantum circuits ---------------------------------------- In the standard circuit model of quantum computation a set of unitary one and two-qubit gates is applied to a number of qubits, which are initially in a pure product state and measured separately in the end. In the cluster state model a multipartite state is prepared in the beginning and the computation is performed by applying subsequent single-qubit measurements. For both computational models the MPS formalism provides a simple way of understanding why a large amount of entanglement is crucial for obtaining an exponential speed-up with respect to classical computations. The fact that quantum computations (of the mentioned type) which contain too little entanglement can be simulated classically is a simple consequence of the following observations (see also [@Jozsa]): 1. Let $|\psi\rangle\in \mathbb{C}^{\otimes d^N}$ have a MPS representation with maximal bond dimension $D$. By Eq.(\[E\]) the expectation values of factorizing observables are determined by a product of $N$ $D^2\times D^2$ matrices. Hence, their calculation as well as the sampling of the respective measurement outcomes and the storage of the state requires only polynomial resources in $N$ and $D$. 2. A gate applied to two neighboring sites increases the respective bond dimension (Schmidt rank) at most by a factor $d^2$ and the MPS representation can be updated with $poly(D)$ resources. 3. A gate applied to two qubits, which are $l$ sites apart, can be replaced by $2l-1$ gates acting on adjacent qubits. In this way we can replace a circuit in which each qubit line is crossed or involved by at most $L$ two-qubit gates by one where each qubit is affected only by at most $4 L$ nearest neighbor gates. A trivial implication of 1. is that measurement based quantum computation (such as cluster state computation) requires more than one spatial dimension: \[Thm:ClusterComputation\] Consider a sequence of states with increasing particle number $N$ for which the MPS representation has maximal bond dimension $D=O(poly(N))$. Then every measurement based computation on these states can be simulated classically with $O(poly(N))$ resources. This is in particular true for a one-dimensional cluster state, for which $D=2$ is independent of $N$. Similarly, if the initial states are build by a constant number of nearest neighbor interactions on a two-dimensional $n\times \log n$ lattice, the measurements can be simulated with $poly(n)$ resources. Concerning the standard circuit model for quantum computation on pure states the points 1.-3. lead to: \[Thm:CircuitComputation\] If at every stage the $N$-partite state in a polynomial time circuit quantum computation has a MPS representation with $D=O(poly(N))$, then the computation can be simulated classically with $poly(N)$ resources. This is in particular true if every qubit-line in the circuit is crossed or involved by at most $O(log N)$ two-qubit gates. For more details on the classical simulation of quantum circuits we refer to [@Jozsa] and [@YoranShort] for measurement based computation as well as for the relation between contracting tensor networks and the tree width of network graphs. The general problem of contracting tensor networks was shown to be $\sharp P$ complete in [@PEPScomplexity]. In [@FCcluster] a simple interpretation of measurement based quantum computation in terms of the valence bond formalism is provided, which led to new resources in [@newresources]. 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A [**70**]{}, 060302(R) (2004). D. Gross, J.Eisert, arXiv:quant-ph/0609149 (2006). An open problem {#open.problems} =============== The main open problem we left open is the [*reasonability*]{} of assuming condition C1 in the blocks of the canonical form. Since condition C2 is known to imply C1 for sufficiently large $L_0$ [@FaNaWe92] and we do have C2 in each block by construction, it only remains to show how [*big*]{} $L_0$ can be. (We are neglecting complex eigenvalues of ${\mathcal{E}}$ of modulus $1$. The reason is that, by Thm.\[Th:periodic\], they can be avoided by grouping the spins into blocks or simply considering $N$ prime.) In the generic case this $L_0$ can be taken $L_0=\lfloor\frac{2\ln{D}}{\ln{d}}\rfloor+1$. However there are cases in which $L_0\sim O(D^2)$. For instance $A_0=\sum_{i=1}^D|i+1{\rangle}{\langle}i|$, $A_1=|2{\rangle}{\langle}D|$. Our conjecture is that this is exactly the worst case. \[Conj1\] There exists a function $f(D)$ such that for any sequence of $D\times D$ matrices $A_i$ verifying C1 for some $L_0$, $L_0$ can be taken $f(D)$. \[Conj2\] $f(D)\sim O(D^2)$. We have been able to prove both in a particular (but generic) case: \[prop:appendix\] If $A_0$ is invertible, then $L_0$ can be taken $D^2$. [[**Proof.**]{} Take an $r$ and call $S_r\subset \{0,\ldots,d-1\}^r$ to a maximal set of indices for which $\{A_{i_1}\cdots A_{i_r}\}_{(i_1,\ldots,i_r)\in S}$ is linearly independent. It is then sufficient to prove that the cardinalities $\#S_{r+1}>\#S_r$, whenever $\#S_r<D^2$ . If not, using that $A_0$ is invertible, we can assume that $S_{r+1}=S_r\times\{0\}$. Now $A_{i_1}\cdots A_{i_{r+2}}=$ $$A_{i_1}\sum_{(j_2,\ldots,j_{r+1})\in S_r} c^{j_2,\ldots,j_{r+1}}_{i_2,\ldots, i_{r+2}}A_{j_{2}}\cdots A_{j_{r+1}}A_0=$$ $$\sum_{(j_2,\ldots,j_{r+1})\in S_r} c^{j_2,\ldots,j_{r+1}}_{i_2,\ldots, i_{r+2}}\sum_{(k_1,\ldots,k_{r})\in S_r} c^{k_1,\ldots,k_{r}}_{i_1,j_2\ldots, j_{r+1}}A_{k_1}\cdots A_{k_r} A_0^2,$$ which implies that one can take $S_{r+2}=S_r\times\{0\}\times\{0\}$ and hence $\#S_{r+2}=\#S_{r+1}=\#S_r$. We can continue the reasoning and prove that, in fact, $\#S_{k}=\#S_r$ for every $k\ge r$; which is the desired contradiction. ${\hfill\fbox\\\medskip }$]{} A particular case of this proposition occurs when one of the matrices in the canonical decomposition is hermitian. To see that one can group the spins in blocks of two. The new matrices are $\tilde{A}_{ij}=A_iA_j$, and the canonical condition $\sum_i A_i^2={\mathbbm{1}}$ reads $\sum_i \tilde{A}_{ii}=1$. Now, by doing a unitary operation in the physical system on can assume that $\tilde{A}_0=\sum_i \tilde{A}_{ii} ={\mathbbm{1}}$ (up to normalization) and therefore we are in the conditions of Proposition \[prop:appendix\]. The conjectures, if true, can be used to prove a couple of interesting results, one concerning the MPS representation of the W-state, and the other concerning the approximation by MPS of ground states of gapped hamiltonians W state ------- The $W$-state can be written with matrices $2\times 2$ in the OBC representation. However, this representation does not respect the symmetry of the state. What if we ask instead for representations of this kind? Let us start with a permutational invariant one, that is, one with diagonal matrices. By the permutational invariance of $|W_N{\rangle}$, it can be written as a sum $\sum_{n=1}^{N+1} \delta_n |x_n{\rangle}^{\otimes N}$. Calling $\alpha_n$ to a $N$-th root of $\delta_n$ and $A_i=\sum_n \alpha_n {\langle}i|x_n{\rangle}|n{\rangle}{\langle}n|$, $$|W_N{\rangle}=\sum_{i_1,\ldots, i_N=1}^2 \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N{\rangle}.$$ with the matrices $A_i$ diagonal and of size $(N+1)\times (N+1)$. Of course we would be interested in a representation with smaller matrices. The surprising consequence of Conjecture \[Conj2\] is that this is impossible even if we ask for a TI representation; that is, one with site-independent matrices (not necessarily diagonal): If we assume Conjecture \[Conj2\] and $|W_N{\rangle}=\sum_{i_1,\ldots, i_N=1}^2 \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N{\rangle}$ for $D\times D$ matrices $A_i$, then $D\succeq O\left(N^{1/3}\right)$. [**Proof.**]{} By Conjecture \[Conj2\], we have condition C1 by blocks with $L_0=O(D^2)$. Let us assume that $\frac{N}{2}\ge 3(b-1)(L_0+1)$, where (as usual) $b$ is the number of different blocks in the canonical form. We can break the chain in two parts, each one having at least $3(b-1)(L_0+1)$ spins and decompose the state as $|W_N{\rangle}=\sum_{\alpha,\beta=1}^D |\Phi_{\alpha,\beta}{\rangle}|\Psi_{\alpha,\beta}{\rangle},$ where $$|\Phi_{\alpha,\beta}{\rangle}=\sum_{i_1,\ldots,i_R} {\langle}\alpha|A_{i_1}\cdots A_{i_R}|\beta{\rangle}|i_1\cdots i_R\rangle$$ $$|\Psi_{\alpha,\beta}{\rangle}=\sum_{i_1,\ldots,i_R} {\langle}\beta|A_{i_R}\cdots A_{i_N}|\alpha{\rangle}|i_{R+1}\cdots i_N\rangle.$$ If we call $S_j$ to the set of positions of the $j$-th block in the canonical form of $|W_N{\rangle}$ and $S=\cup S_j$, we claim that the sets $\{|\Phi_{\alpha,\beta}{\rangle}\}_{(\alpha,\beta)\in S}$ and $\{|\Phi_{\alpha,\beta}{\rangle}\}_{(\alpha,\beta)\in S}$ are both linearly independent. This implies that the rank of the reduced density operator after tracing out the particles $1,\ldots, R$ is $\ge \sum_{i=1}^bD_i^2$, being $D_i\times D_i$ the size of the $i$-th block. We know that in the case of $|W_N{\rangle}$, this rank is $2$, so we get exactly two blocks of size $1\times 1$. It is trivial now to see that this is impossible. Therefore, $\frac{N}{2}\le 3(b-1)(L_0+1)$ and hence the result. So it only remains to prove the claim. For $(\alpha,\beta)\in S_j$ let us take complex numbers $c_{\alpha,\beta}$ such that $\sum_{\alpha,\beta} c_{\alpha,\beta} |\Phi_{\alpha,\beta}{\rangle}=0$ ; which is exactly $$\sum_{j=1}^b\operatorname{tr}(\left[\sum_{(\alpha,\beta)\in S_j}c_{\alpha,\beta}|\beta{\rangle}{\langle}\alpha|\right]A^j_{i_1}\cdots A^j_{i_R})|i_1\cdots i_R{\rangle}=0.$$ By Lemma \[lem:direct-sum\] the sum in $j$ is direct, so each summand is $0$. Finally, by condition C1 by blocks $\sum_{(\alpha,\beta)\in S_j}c_{\alpha,\beta}|\beta{\rangle}{\langle}\alpha|=0$ for every $j$ and hence $c_{\alpha,\beta}=0$ for every $(\alpha,\beta)\in S_j$. ${\hfill\fbox\\\medskip }$ Approximation of ground states by MPS ------------------------------------- As we saw in Sec.\[sec:classical-sim\], one of the big open questions in condensed matter theory is to mathematically explain the high accuracy of DMRG. Since DMRG can be seen as a variational method in the class of MPS, it is crucial to prove that any ground state of a gapped local Hamiltonian can be efficiently approximated by a MPS of low bond dimension $D$. Despite the important recent advances in this direction (see Sec.\[sec:classical-sim\]) the problem is still unsolved. An important step has been done by Hastigns, who has recently reduced this problem to the case in which the Hamiltonian is frustration free [@Hast1]. This highlights the importance of the following dichotomy If Conjecture \[Conj2\] holds and $H_N=\sum_{i=1}^N \tau^i(h)$ is a local TI Hamiltonian which is frustration free for every $N$, then the bond dimension $D$ of any of its [*exact*]{} ground states, viewed as a MPS, is: 1. either independent of $N$ 2. or $>O(N^{\frac{1}{3}})$ [[**Proof.**]{} Let us take the canonical decomposition of a ground state $|\psi{\rangle}$ of $H$ acting on $N$ particles $$|\psi{\rangle}=\sum_{i_1,\ldots, i_N=1}^d \operatorname{tr}(A_{i_1}\cdots A_{i_N})|i_1\cdots i_N{\rangle}.$$ If (ii) does not hold, by Conjecture \[Conj2\], $N>3(b-1)(L_0+1)+L$ ($L$ the interaction length of $H$). Now, by Lemma \[lem:direct-sum\], the products of the last $N-L$ matrices generate the whole space of block diagonal matrices. This immediately implies that the same matrices $A_i$ give us a ground state of $H_{N'}$ for any $N'> N$. That is, $D$ is independent of $N$. ${\hfill\fbox\\\medskip }$]{} [^1]: In a similar way other symmetries can be restored in the representation. For example if the state is reflection symmetric then we can find a representation with $A_i=A_i^T$ and if it is real in some basis then we can choose one with real $A_i$. Both representations are easily obtained by doubling the bond dimension $D$. For the encoding of other symmetries in the $A$’s we refer to [@FaNaWe92; @NachtergaeleSym; @Ortiz]. [^2]: For all the reasonings it is enough to consider $\tilde{h}:{{\mathbb{C}}^d}^{\otimes L}{\longrightarrow}{{\mathbb{C}}^d}^{\otimes L}$ positive such that $\ker{\tilde{h}}=S$. We take $\tilde{h}=P_{S^{\perp}}$ for simplicity. [^3]: Of course this does not apply to global quantities, like entropy, where one needs exponential accuracy in order to have closeness [^4]: We note that Eq.(\[arealaw\]) has been proven for critical spin chains (in particular the XX-model with transverse magnetical field [@Jin]) which are related to a conformal field theory. A general result is still lacking. [^5]: Strictly speaking, Hastings theorem does not imply the validity of equation (\[hasg\]), as it was shown in [@datahiding] that orthogonal states exist whose correlation functions are exponentially close to each other; although it would be very surprising that ground states would exhibit that property, this prohibits to turn the present argument into a rigorous one. [^6]: For non–critical systems, the renormalization group flow is expected to increase the Renyi entropies in the UV direction. The corresponding fixed point corresponds to a critical system whose entropy thus upper bounds that of the non–critical one. [^7]: We choose the $1/L$ dependence such as to assure that the absolute error in extensive observables does not grow. [^8]: In fact, it can be shown [@Eisert] that a variant of DMRG leads to NP-hard instances in intermediate steps. However, one has to note that (i) such a worst case instance might be avoided by starting from a different initial point and (ii) convergence to the optimum at the end of the day does not necessarily require finding the optimum in every intermediate step.

--- abstract: 'We present a new, simple renormalization group method of investigating groundstate properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows to investigate the properties unavailable through mean field approximations, such as one- and two-particle reduced density matrices of the groundstate. As an example, we model a weakly interacting 1D Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the groundstate one-particle density matrix. We have also obtained the Hall-Post lower bounds for the groundstate energy of the gas. All results have been obtained by the straightforward numerical diagonalization of the Hamiltonian matrix.' author: - Roman Werpachowski - Jerzy Kijowski date: '19th March, 2007' title: A simple renormalization group approximation of the groundstate properties of interacting bosonic systems --- Introduction {#sec:introduction} ============ The numerical investigation of the groundstate properties of a multiparticle interacting bosonic system is a much harder task than in the case of a single particle system. The naive approach consists in choosing a large enough finite Hilbert space basis and the numerical diagonalization of the resulting Hamiltonian matrix. However, the necessary basis size grows exponentially with the number of particles, which makes this simple method inadequate for the treatment of large systems. To avoid this problem, many approximations have been invented, such as the Gross-Pitaevskii (GP) mean field approach [@pitaevskii2; @pitstr], the Density Matrix Renormalization Group (DMRG) method [@white1; @schollwock] or, in the case of strong interactions, the Thomas-Fermi approximation [@pitstr] and the Tonks-Girardeau model [@tonks; @girardeau]. For fermionic systems, the Exact Diagonalization-Ab Initio (EDABI) method has been implemented [@spalek1; @spalek2; @spalek3]. One should also note the exceptional case of a full analytical solution in one dimension by Lieb and Liniger [@lieb]. From this solution, two- and three-pair correlation functions of an interacting one-dimensional (1D) Bose gas have been derived [@shlyapnikov; @shlyapnikov2; @shlyapnikov3]. In this paper we present a new approach, which is in the flavour of Renormalization Group methods but is conceptually simple and easy to implement. Our method amends the problem of unmanageable basis size by reducing the number of particles in the system and renormalizing the Hamiltonian. We approximate one- and two-particle properties of the large system with the same properties of the smaller system. In contrast to mean field methods, our approach allows to calculate such quantities as one- or two-particle reduced density matrices (1-RDM and 2-RDM) of the groundstate. In Section \[sec:method\], we describe how our method works. Section \[sec:example\] contains an example application of the method to the problem of one-dimensional interacting Bose gas in a harmonic trap. The results are summarized in Section \[sec:summary\]. Hamiltonian renormalization and the approximation of groundstate properties {#sec:method} =========================================================================== We present our approximation in the case of a system with two-body interactions. It can be easily generalized to the general case of $n$-body interactions. Consider a 1D Hamiltonian describing a system of $N$ scalar (zero spin) bosons, $$\hat{H}_N = \sum_{k=1}^N \left( - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x_k^2} + V_1(x_k) \right) + \sum_{k=1}^N \sum_{k'=1}^{k-1} V_2(x_k, x_{k'}) \ ,$$ where $N$ is the number of particles, $m$ is the particle mass, $V_1(x)$ is the external one-particle potential and $V_2(x,x') = V_2(x',x)$ is the two-particle interaction potential. Investigating such a system, we are often interested only in one- or two-particle properties of the groundstate. One way to calculate them is to obtain an approximation of the 1-RDM or 2-RDM of the groundstate. We approximate the system by replacing it with a much smaller system containing $N' \ll N$ scalar bosons. The smaller system is governed by a renormalized version of the original Hamiltonian $\hat{H}_N$, that is $$\label{eq:HNprim} \begin{split} \hat{H}_{N'} &= \frac{N}{N'} \Biggl[ \sum_{k=1}^{N'} \left( - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x_k^2} + V_1(x_k) \right)\\ &\quad+ \frac{N-1}{N'-1} \sum_{k=1}^{N'} \sum_{k'=1}^{k-1} V_2(x_k, x_{k'}) \Biggr] \ . \end{split}$$ We calculate the properties of interest from the groundstate of $\hat{H}_{N'}$, thus avoiding the insurmountable problem of diagonalizing the Hamiltonian of the large system, $\hat{H}_N$. The results for increasing values of $N'$ will converge to the values of the corresponding properties of the $N$-particle system. We will now justify our procedure. Let $\Psi$ and $\rho$ be a state of the large system and its $N'$-RDM, respectively. It is easy to show that their mean energies, measured by the respective Hamiltonians, are equal, $$\label{eq:aven} \mathrm{Tr} \left( \hat{H}_{N'} \rho \right) = \left\langle \Psi \left| \hat{H}_N \right| \Psi \right\rangle \ .$$ Hence, when the mean energy of $\Psi$ becomes lower, moving closer to the mean energy of the groundstate $\Psi_0$ of $\hat{H}_N$, the mean energy of $\rho$, also becomes lower and moves closer to the mean energy of the (pure state) density matrix $\rho'_0$ of the groundstate $\Psi'_0$ of $\hat{H}_{N'}$. Because of the variational principle, the density matrix $\rho'_0$ is an approximation of the reduced density matrix (RDM) $\rho_0$ of the groundstate $\Psi_0$. The one- or two-particle properties of $\rho_0$ (i.e. of $\Psi_0$), like the probability density, are approximated by the same properties of $\rho'_0$ (i.e. of $\Psi'_0$). Since $N' \ll N$, it is much easier to calculate numerically the groundstate $\Psi'_0$ than the groundstate $\Psi_0$, and to investigate the one- or two- particle properties of $\Psi_0$ by investigating the same properties of $\Psi'_0$. The main source of error in our method is the fact that the variational search for the groundstate converges to the $N'$-particle groundstate $\Psi_0'$, not to the RDM of the $N$-particle groundstate, $\rho_0$. This is because for bosons, not every $N'$-particle density matrix is an RDM of an $N$-particle state. A better strategy would be to perform the variational search not in the whole $N'$-particle Hilbert space but in the smaller space of $N'$-particle density matrices which are RDMs of $N$-particle states. However, the problem of identifying this space, the so-called $N$-representability problem [@coleman; @ruskai; @gidofalvi], remains unsolved. Therefore, we have to perform our calculations for a sequence of $N'$. The energy of the $N'$-particle groundstate is a lower bound of the energy of the investigated $N$-particle groundstate [@hallpost]. When $N'$ increases, the groundstate energy increases and approaches the groundstate energy of the $N$-particle groundstate. Due to the variational principle, this means that the $N'$-particle groundstates approximate the $N$-particle groundstate increasingly well, in the sense that the one- and two-particle properties calculated from these groundstates converge to the corresponding properties of the $N$-particle groundstate. In the general case of $n$-particle interactions, the renormalization goes as follows: an $n$-particle interaction potential term $\sum_{k_1=1}^{N} \sum_{k_2=1}^{k_1-1} \dotsm \sum_{k_2=1}^{k_1-1} \dotsm \sum_{k_n = 1}^{k_{n-1}-1} V_n(x_1,\dotsc,x_n)$, symmetrical with respect to permutations of coordinates $x_k$, is replaced by the term $\frac{N(N-1) \cdot \dotsm \cdot (N-n+1)}{N'(N'-1) \cdot \dotsm \cdot (N'-n+1)}$ $ \sum_{k_1=1}^{N'} \sum_{k_2=1}^{k_1-1} \dotsm \sum_{k_n = 1}^{k_{n-1}-1} V_n(x_1,\dotsc,x_n)$. Equation  is true also in this general case. A simple example {#sec:example} ================ Investigated system ------------------- In our example, we consider a system of $N=100$ scalar bosons with a dimensionless Hamiltonian $$\hat{H}_N = \sum_{k=1}^N \left( - \frac{1}{2} \frac{\partial^2}{\partial x_k^2} + \frac{1}{2} x^2_k \right) + \lambda \sum_{k=1}^N \sum_{k'=1}^{k-1} \delta(x_k - x_{k'}) \ ,$$ where $\delta(x-x')$ is the Dirac $\delta$ function. This interaction potential is often used to describe cold bosons forming a Bose-Einstein condensate (BEC) in a trap, when only the $s$-wave scattering occurs [@pitstr]. Our example concerns positive $\lambda$, which lead to the repulsive interaction. We have approximated numerically the 1-RDM and 2-RDM of the groundstate for two values of interaction strength $\lambda$. The procedure begins with the calculation of a finite matrix of the renormalized Hamiltonian $\hat{H}_{N'}$ in a finite basis composed of the noninteracting Hamiltonian ($\lambda = 0$) eigenstates, permanents [^1] of $N'$ one-particle Hermite functions $\mathcal{H}_k$, $$\label{eq:Hermite} \mathcal{H}_k(x) = \frac{1}{\sqrt{k! 2^k \sqrt{\pi}}} H_k(x) \exp\left( - \frac{x^2}{2} \right) \ ,$$ where $H_k(x)$ is $k$-th Hermite polynomial. The basis contains all eigenstates with (nonrenormalized) energies lower than a cutoff energy $L + N'/2$, i.e. permanents of functions $\mathcal{H}_{k_n}$, $n=1,\dotsc,N'$ such that $\sum_{n=1}^{N'} k_n < L$. Then, the groundstate is calculated with the help of an iterative Lanczos-type numerical procedure [@arpack]. From the groundstate we obtain the 1-RDM and the 2-RDM, with trace normalized to unity. Using them, we can calculate any one- or two-particle property of the groundstate. For given $N'$, the basis cutoff $L$ is chosen to be large enough so that calculated properties do not change upon further increase of $L$. In the case of 1-RDM, $\rho_1$, we compare the diagonal part, $\rho_1(x,x)$, with the probability density calculated by minimizing numerically the GP energy functional [@pitstr] of our system, $$\label{eq:GPen} \begin{split} E[\Psi] &= \frac{N}{2} \int_{-\infty}^\infty \Biggl[ {{{\Psi}^\ast}}(x) \left( - \frac{\partial^2}{\partial x^2} + x^2 \right) \Psi(x)\\ &\quad+ (N-1) \lambda {\left|\Psi(x)\right|}^4 \Biggr] \mathrm{d} x \ . \end{split}$$ The minimization is performed by expanding the wavefunction $\Psi(x)$ in the finite basis of the first twenty Hermite functions , inserting the expansion into  and minimizing numerically the resulting functional of the expansion coefficients. We present numerical results for two values of $\lambda$, $10^{-2}$ and $5 \cdot 10^{-2}$. Both values are in the regime of weak interactions, where the minimization of the GP energy functional provides a good approximation of the groundstate. Groundstate energy ------------------ First, we provide the data on the groundstate energy $E_0$. In the noninteracting case $\lambda = 0$, $E_0$ is precisely known and equals 50. Table \[tab:en\] lists three different approximations of $E_0$ for two nonzero values of $\lambda$. In the second and third column of Table \[tab:en\], two different upper bounds for $E_0$ are listed: the one obtained from the GP functional, $E_\mathrm{GP}$, and the variational bound, $E_\mathrm{Gauss}$, calculated as a minimal mean value of $\hat{H}_N$ in a state $\Psi_\sigma$, a product of $N$ Gaussian one-particle wavefunctions with a common variational parameter $\sigma$, $$\Psi_\sigma(x_1,\dotsc,x_N) = \prod_{k=1}^N \frac{1}{\sqrt[4]{2\pi\sigma^2}} \exp \left( - \frac{x^2_k}{4 \sigma^2} \right) \ ,$$ i.e. $E_\mathrm{Gauss} = \min_{\sigma \in \mathbb{R}} \langle \Psi_\sigma | \hat{H}_N | \Psi_\sigma \rangle$. Relatively small differences between $E_\mathrm{GP}$ and $E_\mathrm{Gauss}$ indicate that the groundstates are close to Gaussian. As expected, the GP approximation provides a better estimation of the groundstate energy than the Gaussian *Ansatz*. Our method provides an estimation of the true groundstate energy $E_0$ as the so-called Hall-Post [@hallpost] lower bound, $E_\mathrm{HP}$. Values of $E_\mathrm{HP}$ are listed in the fourth column of Table \[tab:en\]. $\lambda$ $E_\mathrm{GP}$ $E_\mathrm{Gauss}$ $E_\mathrm{HP}$ ------------------- ----------------- -------------------- ----------------- $10^{-2}$ 68.8 68.9 68.2 $5 \cdot 10^{-2}$ 130.9 132.3 121.4 : Bounds for the groundstate energy $E_0$. The first two, $E_\mathrm{GP}$ and $E_\mathrm{Gauss}$, are the upper bounds calculated from the GP approximation and the variational method with a Gaussian wavefunction, respectively. The last one, $E_\mathrm{HP}$, is the Hall-Post lower bound, which is provided by our method as an estimate of the true groundstate energy.[]{data-label="tab:en"} They were calculated by NLSQ (nonlinear least squares) fitting of the groundstate energy for fixed $N'$ (in our calculations, we have used results for $N'=8$, so as to make $E_\mathrm{HP}$ as high as possible) as function of $L$ to a power law $$E(L) = E_\mathrm{HP} + B L^C \ , \quad C < 0$$ and taking the $L \to \infty$ limit, obtaining $E_\mathrm{HP}$ as the answer. It has been necessary to follow this procedure, since raw numerical results vary with $L$, even for $L$ large enough so that the 1-RDM does not change. The relative asymptotic standard error of the fitting procedure is below 0.01% for both values of $\lambda$. The bounds on $E_0$ ($E_\mathrm{HP}$ and $E_\mathrm{GP}$) presented above are quite close. The relative uncertainty with which $E_0$ is determined by them is below 1% for $\lambda = 10^{-2}$ and around 8% for $\lambda = 5 \cdot 10^{-2}$. (We have calculated the relative uncertainty as the ratio of the difference between the upper and the lower bound to the lower bound.) The fact that it is small supports the applicability of our approximation, as it means that the true groundstate energy is also close to the obtained lower bound. On the other hand, if the reduced density matrix of the $N'$-particle groundstate $\Psi_0'$ is to be a good approximation of the reduced density matrix of the true groundstate, the mean energy of $\Psi_0'$—i.e. the Hall-Post lower bound—must be close to the true groundstate energy of the system. Our results fulfill this condition. Density matrices ---------------- For $\lambda = 10^{-2}$, we obtain identical one-particle probability densities from our method and from the GP approximation, as shown on Fig. \[fig:5\_50\]. ![Comparison of the probability densities obtained from our method (for $N'=5$ and $L=50$) and from the GP approximation, for $\lambda = 10^{-2}$. The curves overlap perfectly, indicating convergence. The probability density of the noninteracting ($\lambda = 0$) groundstate is shown too. One can clearly see the difference between the interacting system and the noninteracting one, with the repulsive interaction pushing away the bosons in the trap.[]{data-label="fig:5_50"}](figure1) The accuracy of our approximation is confirmed by Fig. \[fig:conv\_1e-2\], which shows that different values of $N'$ and $L$ yield identical probability densities. ![Probability densities for $\lambda = 10^{-2}$ calculated for increasing $N'$ ($L=56,50,60,40,40$ respectively). The curves overlap perfectly, indicating convergence.[]{data-label="fig:conv_1e-2"}](figure2) A magnified section of this plot is shown in Fig. \[fig:conv\_1e-2m\]. ![Probability densities for $\lambda = 10^{-2}$ calculated for increasing $N'$ ($L=56,50,60,40,40$ respectively), shown in the range $x \in [-0.25,0.25]$. The plot has been maginified in order to show the details of the convergence.[]{data-label="fig:conv_1e-2m"}](figure3) We will use the convergence with increasing $N'$ as a benchmark of the accuracy of our method, treating our numerical results as correct if they stabilize quickly. For each $N'$, we take the results for $L$ large enough so that they do not change upon further increase of $L$. A similar convergence occurs for the antidiagonal part of the 1-RDM, $\rho_1(x,-x)$. The GP approximation, however, cannot provide us with the knowledge about the nondiagonal parts of the 1-RDM. The merit of our method is that we can calculate $\rho_1(x,y)$ for any $(x,y)$. For $\lambda = 10^{-2}$, we obtain numerically $$\rho_1(x,y) \approx \rho_1\left(\sqrt{x^2 + y^2} \right) \ ,$$ which is clearly shown by the contour plot of $\rho_1(x,y)$ in Fig. \[fig:r1\_1e-2\_cnt\]. ![Contour plot of $\rho_1(x,y)$ for $\lambda = 10^{-2}$, calculated for $N'=5$ and $L=50$. Isolines display a radial symmetry.[]{data-label="fig:r1_1e-2_cnt"}](figure4) The convergence of the diagonal part of 1-RDM, $\rho_1(x,x)$ and of the diagonal part of 2-RDM, $\rho_2(x,x,x,x)$ (the probability of finding both particles in the same position $x$), is compared in Fig. \[fig:1e-2\_dd\_cmp\]. It is clear that the convergence, with increasing $N'$, of the second function is much slower. The consequence of this is that using only such simple diagonalization techiques as we did, which limit us to $N' < 10$, we cannot estimate the 2-RDM, even for $\lambda$ as small as $10^{-2}$. ![On the left plot, the probability density $\rho_1(x,x)$ for $\lambda = 10^{-2}$ is shown for increasing $N'$ ($L=56,50,60$, respectively). The curves converge quickly. On the right plot, the diagonal part of the 2-RDM, $\rho_2(x,x,x,x)$ is plotted (for the same $\lambda$) for increasing $N'$ ($L=56,50,60,40,40$, respectively). Even for $N'$ equal 7 or 8, the curves do not converge.[]{data-label="fig:1e-2_dd_cmp"}](figure5) For $\lambda = 5 \cdot 10^{-2}$, we obtain the convergence of the probability densities as easily, as for $\lambda = 10^{-2}$ (see Fig. \[fig:conv\_5e-2\]), although it is slightly slower (not visible on the plot). ![Convergence of the probability densities $\rho_1(x,x)$ for $\lambda = 5 \cdot 10^{-2}$ and increasing values of $N'$ ($L = 55, 50, 60, 40, 40$, respectively). The curves overlap, indicating convergence.[]{data-label="fig:conv_5e-2"}](figure6) A magnified section of this plot is shown in Fig. \[fig:conv\_5e-2m\]. ![Probability densities for $\lambda = 5 \cdot 10^{-2}$ calculated for increasing $N'$ ($L=55,50,60,40,40$ respectively), shown in the range $x \in [-0.25,0.25]$. The plot has been magnified in order to show the details of the convergence.[]{data-label="fig:conv_5e-2m"}](figure7) A similar convergence occurs for the antidiagonal part of the 1-RDM, $\rho_1(x,-x)$. However, the probability density differs slightly from the one obtained from the GP functional, as seen in Fig. \[fig:prbdens\_5e-2\]. ![Probability density for $\lambda = 5 \cdot 10^{-2}$, as calculated with our method ($N'=8$ and $L=40$) and from the minimization of the GP functional. A slight difference between the two curves can be seen in the middle of the plot, indicating that interactions are strong enough so that GP approximation’s results differ from ours.[]{data-label="fig:prbdens_5e-2"}](figure8) Contrary to the case of $\lambda = 10^{-2}$, the contour plot of the 1-RDM for $\lambda = 5 \cdot 10^{-2}$ is no longer radially symmetric, as can be seen on Fig. \[fig:r1\_5e-2\_cntr\]. ![Contour plot of $\rho_1(x,y)$ for $\lambda = 5\cdot 10^{-2}$, calculated for $N'=8$ and $L=40$. Isolines do not display the radial symmetry, unlike for $\lambda = 10^{-2}$.[]{data-label="fig:r1_5e-2_cntr"}](figure9) It differs noticeably from the one (shown in Fig. \[fig:hart\]) we would obtain from the mean field method, using the formula $$\rho_1(x,y) \approx \Psi_\mathrm{GP}(x) \Psi_\mathrm{GP}(y) \ ,$$ where $\Psi_\mathrm{GP}$ is the real wavefunction which minimizes the GP energy functional. This difference is the most striking result in this Section, and indicates that our method gives more accurate results than mean-field approximations. ![Contour plot of $\rho_1(x,y)$ calculated for $\lambda = 5\cdot 10^{-2}$ using GP approximation. It is clearly visible that this plot is more symmetrical than the one in Fig. \[fig:r1\_5e-2\_cntr\].[]{data-label="fig:hart"}](figure10) For even higher value of $\lambda$, $10^{-1}$, we did not obtain fast enough convergence of neither $\rho_1(x,x)$ (see Fig. \[fig:convcmp\]) ![On the left plot, the probability density $\rho_1(x,x)$ for $\lambda = 10^{-2}$ is shown for increasing $N'$ ($L=56,50,60$, respectively). On the right plot, the the same quantity is shown for $\lambda = 10^{-1}$ ($L=55,50,60$, respectively). The convergence for the higher value of $\lambda$ is visibly slower.[]{data-label="fig:convcmp"}](figure11) nor, especially, $\rho_1(x,-x)$ (see Fig. \[fig:anticonv\]). This prevented us from investigating the full 1-RDM for this interaction strength. ![Antidiagonal part of 1-RDM for $\lambda = 10^{-1}$ for increasing $N'$ ($L=55,50,60,39,40$, respectively). Convergence is too slow for the results to be meaningful.[]{data-label="fig:anticonv"}](figure12) We conclude that $\lambda = 10^{-1}$ is outside the range of our approximation in its present form. To reach this interaction strength, we would have to calculate the density matrices of the $N'$-particle groundstate for $N'$ higher than those treatable with the simple numerical diagonalization algorithm used by us. Summary {#sec:summary} ======= We have presented a method of investigating one- and two-particle reduced density matrices of the groundstate of an interacting systems with a large number of bosonic particles. The method approximates it with a smaller, renormalized system. It is conceptually simple and easy to implement numerically. The results it provides would be, for high enough interaction strengths (e.g. $\lambda = 5 \cdot 10^{-2}$ in our example), impossible to calculate using mean-field methods, such as the GP approximation. We have provided an example application of our method to the problem of one-dimensional interacting Bose gas in a harmonic trap, obtaining accurate approximations of the quantity unobtainable from mean field methods, namely the full one-particle density matrix. The results are precise and accurately describe the large system, which is proven by the fact the the results converge quickly with increasing $N'$. The GP approximation does not give as accurate picture of the groundstate one-particle density matrix as our approach. Additionally, the Hall-Post lower bounds for the groundstate energy have been calculated. Relatively small difference between them and the upper bounds (GP and Gaussian) also supports the applicability of our method. Even using simple numerical procedures, our method gives access to properties which were previously not as accurately described by mean-field methods. To investigate the two-particle density matrix, or to perform simulations of systems with higher interaction strengths, would require the use of more refined approaches to the calculation of the groundstate of the renormalized Hamiltonian, for example the DMRG method [@white1; @schollwock] or the variational optimization of the basis wavefunctions [@spalek1; @spalek2; @spalek3]. The authors want to thank prof. Iwo Białynicki-Birula and prof. Kazimierz Rzażewski for helpful discussions and advice. [18]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ** (, , ). , , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , , **, . [^1]: A permanent is the bosonic counterpart of the Slater determinant. For example, for two particles the wavefunction $\Psi(x_1,x_2) \sim \psi_a(x_1) \psi_b(x_2) + \psi_b(x_1) \psi_a(x_2)$ is the permanent of one-particle wavefunctions $\psi_a$ and $\psi_b$.

--- abstract: '[We prove gauge-invariant uniqueness theorems with respect to maximal and normal coactions for $C^*$-algebras associated to product systems of $C^*$-correspondences. Our techniques of proof are developed in the abstract context of Fell bundles. We employ inner coactions to prove an essential-inner uniqueness theorem for Fell bundles. As application, we characterise injectivity of homomorphisms on Nica’s Toeplitz algebra ${\mathcal T}(G, P)$ of a quasi-lattice ordered group $(G, P)$ in the presence of a finite non-trivial set of lower bounds for all non-trivial elements in $P$.]{}' address: - 'School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, 85287' - 'Department of Mathematics, University of Oslo, PO BOX 1053 Blindern, N- 0316 Oslo, Norway.' - 'School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona, 85287' author: - 'S. Kaliszewski' - 'Nadia S. Larsen' - John Quigg date: 'August 24, 2011. Revised April 5, 2012' title: 'Inner coactions, Fell bundles, and abstract uniqueness theorems' --- [^1] Introduction {#intro} ============ Starting with the early constructions of $C^*$-algebras associated to generating families of operators on Hilbert space such as isometries or partial isometries, possibly subject to certain relations, a question of interest arose as to whether the $C^*$-algebra was unique. Coburn’s theorem asserts that the $C^*$-algebra generated by a non-unitary isometry on Hilbert space is unique up to isomorphism, [@Co]. In [@Cu], Cuntz constructed large classes of $C^*$-algebras, both simple and non-simple, generated by families of isometries satisfying certain relations, and proved that two tuples of isometries on Hilbert space fulfilling the same relation generate isomorphic $C^*$-algebras. In a remarkable generalization, Nica introduced the notion of a quasi-lattice ordered group $(G, P)$ and constructed a Toeplitz $C^*$-algebra ${\mathcal T}(G, P)$ and a universal $C^*$-algebra $C^*(G, P)$, [@N]. He obtained both analogues of Coburn’s theorem, and results relating to Cuntz’s uniqueness theorems in the particular case of the quasi-lattice ordered group $({{\mathbb{F}}}_n, {{\mathbb{F}}}_n^+)$ consisting of the free group and the free semigroup on $n$ generators. Laca and Raeburn [@LacR1] discovered a semigroup crossed product structure of $C^*(G, P)$, and used it to prove faithfulness results for representations of this algebra in the presence of an amenability hypothesis. Our starting point is two-fold. For one thing, we noticed that the analysis of the gauge-invariant uniqueness property from [@CLSV] involved two crucial ingredients of nonabelian duality, namely *maximal* and *normal* coactions. The second motivating fact was that the quasi-lattice ordered group $({{\mathbb{F}}}_n, {{\mathbb{F}}}_n^+)$ belongs to the class of those $(G, P)$ for which, as Nica showed, ${\mathcal T}(G, P)$ contains ${\mathcal K}(l^2(P))$. We could see that for such pairs $(G, P)$, the ideal ${\mathcal K}(l^2(P))$ of ${\mathcal T}(G, P)$ contains a family of projections that determines an *inner* coaction. Our thrust in this paper is to show how the general theory of coactions gives uniqueness theorems for $C^*$-algebras of Fell bundles in a systematic manner. We apply these results in familiar contexts with sharpened or new characterizations of uniqueness as outcomes. The first gauge-invariant uniqueness type result was proved by an Huef and Raeburn in [@anHR]. Here we obtain a gauge-invariant uniqueness result in the context of Fell bundles. Since the proofs of the abstract gauge-invariant uniqueness results for $C^*$-algebras of Fell bundles are painless, albeit non-trivial, we chose to place this material in an appendix. The other type of abstract uniqueness results we prove emerges from inner coactions. The first main application is to establish a gauge-invariant uniqueness property for the Cuntz-Nica-Pimsner algebra ${\operatorname{\mathcal{NO}}_{\!X}}$ of Sims and Yeend from [@SY] by highlighting the feature observed in [@CLSV] that it carries a maximal coaction. If $(G, P)$ is a quasi-lattice ordered group and $X$ is a compactly aligned product system over $P$ of $C^*$-correspondences over a $C^*$-algebra $A$, then Sims and Yeend’s $C^*$-algebra ${\operatorname{\mathcal{NO}}_{\!X}}$ is universal for Cuntz-Nica-Pimsner covariant representations of $X$. When $X$ is $\tilde{\phi}$-injective, ${\operatorname{\mathcal{NO}}_{\!X}}$ has the desired property of admitting an injective universal Cuntz-Nica-Pimsner covariant representation. For product systems, ${\operatorname{\mathcal{NO}}_{\!X}}$ is the appropriate candidate for the Cuntz-Pimsner algebra ${\mathcal O}_Y$ associated in [@Ka] to a single $C^*$-correspondence $Y$, in a generalization of Pimsner’s work from [@P]. The gauge-invariant uniqueness property for ${\operatorname{\mathcal{NO}}_{\!X}}$ proved in [@CLSV] (see Corollaries 4.11 and 4.12) is equivalent to asking for the canonical maximal coaction on ${\operatorname{\mathcal{NO}}_{\!X}}$ to be normal. In our treatment here we look at the gauge-invariant uniqueness property in two separate classes, that of $C^*$-algebras with maximal coactions, and of $C^*$-algebras with normal coactions. Thereby we are in the context of coactions and can streamline the proofs by using specific techniques. We obtain gauge-invariant uniqueness theorems for ${\operatorname{\mathcal{NO}}_{\!X}}$, seen in the category of maximal coactions, and for the co-universal algebra ${\operatorname{\mathcal{NO}}_{\!X}}^{r}$ identified in [@CLSV] and viewed in the category of normal coactions. As a bonus for sorting out abstract gauge-invariant uniqueness results for Fell bundles, we also obtain a gauge-invariant uniqueness theorem for the Toeplitz-like extension of ${\operatorname{\mathcal{NO}}_{\!X}}$. This is the universal $C^*$-algebra for Nica covariant Toeplitz representations of the compactly aligned product system $X$; this algebra was denoted ${{\mathcal T}_{\rm cov}}(X)$ in [@F], but we shall follow [@BanHLR], see their Remark 5.3, and use the notation ${\operatorname{\mathcal{NT}}}(X)$. Faithfulness of representations of ${\mathcal T}(G, P)$ was characterized by Laca and Raeburn for all amenable quasi-lattice ordered groups $(G, P)$, [@LacR1]. In coaction terminology, $(G, P)$ amenable means that the canonical maximal coaction on $C^*(G, P)$ is also normal. Here we exploit the fact that ${\mathcal T}(G, P)$ has a natural normal coaction. For a quasi-lattice ordered group $(G, P)$ with the property that there is a finite set $F$ of elements in $P\setminus\{e\}$ such that every non-trivial element in $P$ has a lower bound in $F$, we characterize directly injectivity of homomorphisms from ${\mathcal T}(G, P)$ to a $C^*$-algebra $B$. The crucial observation is that existence of $F$ not only characterizes the fact that ${\mathcal K}(l^2(P))$ is included in ${\mathcal T}(G, P)$, as proved by Nica in [@N Proposition 6.3], but that it also characterizes existence of an inner coaction on the ideal ${\mathcal K}(l^2(P))$. With this card at hand, we can apply our abstract essential-inner uniqueness result, i.e. Corollary \[cor:eiut\]. For the pair $({{\mathbb{F}}}_n, {{\mathbb{F}}}_n^+)$, which clearly admits a finite set of lower bounds for elements in ${{\mathbb{F}}}_n^+$, our Theorem \[thm:describe-inner-Nica-alg\] thus provides a characterization of faithful representations of ${\mathcal T}({{\mathbb{F}}}_n, {{\mathbb{F}}}_n^+)$ without reference to the amenability of the pair, a property that is by no means trivial to verify. The organization of the paper is as follows: after a preliminary section in which we recall terminology and facts about coactions, quasi-lattice ordered groups and $C^*$-algebras of product systems, in section \[NO\] we present gauge-invariant uniqueness theorems for the Nica-Toeplitz algebra, the Cuntz-Nica-Pimsner algebra and the co-universal $C^*$-algebra of a class of compactly aligned product systems $X$. In section \[section:inner\] we prove the abstract inner-uniqueness and essential-inner uniqueness results. In section \[subsect:Nica-alg-semigp-cp\] we place the representation of $C^*(G, P)$ arising from the Toeplitz representation of $P$ in the framework of nonabelian duality. Section \[section:fessub\] contains the essential-inner uniqueness theorem for ${\mathcal T}(G, P)$, namely Theorem \[thm:describe-inner-Nica-alg\], and a converse to it, Theorem \[thm:inner-to-FES\]. The appendix collects the promised gauge-invariant uniqueness results for Fell bundles. Preliminaries {#prelim} ============= Throughout, $G$ will be a discrete group. If $A$ is a $C^*$-algebra and $\delta:A\to A\otimes C^*(G)$ is a coaction, we will just say “$(A,\delta)$ is a coaction”. For the theory of coactions we refer to [@BE Appendix A], and for discrete coactions in particular we refer to [@EKQ; @QuiggDiscrete]. For maximalizations and normalizations of coactions we refer to [@clda; @nordfjordeid]. If $(A,\delta)$ is a (full[^2]) coaction of $G$, we will let ${\mathcal A}$ denote the associated Fell bundle, and similarly for other capital letters. If $\pi:(A,\delta)\to (B,\varepsilon)$ is a morphism of coactions, we write $\tilde{\pi}:{\mathcal A}\to {\mathcal B}$ for the corresponding homomorphism of Fell bundles. Note that $\pi$ is surjective if and only if $\{\pi(A_s):s\in G\}$ generates $B$. Also note that if $(A,\delta)$ and $(B,\varepsilon)$ are coactions, then a homomorphism $\pi:A\to B$ is $\delta-\varepsilon$ equivariant if and only if $\pi(A_s)\subset B_s$ for all $s\in G$ (because equivariance can be checked on the generators $a_s\in A_s$ for $s\in G$). A morphism $\pi:(B,\varepsilon)\to (A,\delta)$ of coactions is a *maximalization* of $(A,\delta)$ if $(B,\varepsilon)$ is maximal and $\pi\times G:B\times_\varepsilon G\to A\times_\delta G$ is an isomorphism. Sometimes we call $(B,\varepsilon)$ itself a maximalization of $(A,\delta)$. Maximalizations of $(A,\delta)$ always exist, and all are uniquely isomorphic. Choosing one for every coaction, we get a *maximalization functor* that sends $(A,\delta)$ to the maximalization $$q_A^m:(A^m,\delta^m)\to (A,\delta),$$ and sends a morphism $\pi:(A,\delta)\to (B,\epsilon)$ to the unique morphism $\pi^m$, called the *maximalization* of $\pi$, making the diagram $$\xymatrix@C+30pt{ (A^m,\delta^m) \ar@{-->}[r]^-{\pi^m}_-{!} \ar[d]_{q_A^m} &(B^m,\epsilon^m) \ar[d]^{q_B^m} \\ (A,\delta) \ar[r]_-\pi &(B,\epsilon) }$$ commute. A parallel theory exists for normalizations: $\pi:(A,\delta)\to (B,\varepsilon)$ is a *normalization* of $(A,\delta)$ if $(B,\varepsilon)$ is normal and $\pi\times G:A\times_\delta G\to B\times_\varepsilon G$ is an isomorphism. We sometimes call $(B,\varepsilon)$ itself a normalization of $(A,\delta)$. Normalizations of $(A,\delta)$ always exist, and all are uniquely isomorphic. Choosing one for every coaction, we get a *normalization functor* that sends $(A,\delta)$ to the normalization $$q_A^n:(A,\delta)\to (A^n,\delta^n),$$ and sends a morphism $\pi:(A,\delta)\to (B,\epsilon)$ to the unique morphism $\pi^n$, called the *normalization* of $\pi$, making the diagram $$\xymatrix@C+30pt{ (A,\delta) \ar[r]^-\pi \ar[d]_{q_A^n} &(B,\epsilon) \ar[d]^{q_B^n} \\ (A^n,\delta^n) \ar@{-->}[r]_-{\pi^n}^-{!} &(B^n,\epsilon^n) }$$ commute. Maximalizations and normalizations are automatically surjective. Moreover, if $\pi:(A,\delta)\to (B,\varepsilon)$ is either a maximalization or a normalization, then $\pi$ maps each *spectral subspace* $A_s:=\{a\in A:\delta(a)=a\otimes s\}$ isometrically onto the corresponding subspace $B_s$, and in particular maps the *fixed-point algebra* $A^\delta:=A_e$ isomorphically onto $B^\epsilon$. If $(A,\delta)$ is normal, then the maximalization $q_A^m:(A^m,\delta^m)\to (A,\delta)$ is also a normalization of $(A^m,\delta^m)$, and similarly if $(A,\delta)$ is maximal then the normalization $q_A^n:(A,\delta)\to (A^n,\delta^n)$ is also a maximalization of $(A^n,\delta^n)$. For every Fell bundle $p:{\mathcal A}\to G$, the (full) cross-sectional algebra $C^*({\mathcal A})$ carries a maximal coaction $\delta_{\mathcal A}$, determined on ${\mathcal A}$ by $\delta_{\mathcal A}(a)=a\otimes p(a)$, the reduced cross-sectional algebra $C^*_r({\mathcal A})$ carries a normal coaction $\delta_{\mathcal A}^n$ determined by the same formula, and the *regular representation* $\Lambda_{\mathcal A}:(C^*({\mathcal A}),\delta_{\mathcal A})\to (C^*_r({\mathcal A}),\delta_{\mathcal A}^n)$ is both a maximalization and a normalization. For $s\in G$, we write ${\raisebox{2pt}{\ensuremath{\chi}}}_s$ for the characteristic function of $\{s\}$, viewed as an element of $B(G)=C^*(G)^*$. If $(A,\delta)$ is a coaction, we write $$\label{def:delta_s} \delta_s=({\operatorname{id}}\otimes{\raisebox{2pt}{\ensuremath{\chi}}}_s)\circ\delta,$$ which is a projection of norm one from $A$ onto the spectral subspace $A_s$. 0.2cm If $A$ is a $C^*$-algebra and $P$ is a discrete semigroup with identity $e$, a *product system over $P$ of $C^*$-correspondences over $A$* consists of a semigroup $X$ equipped with a semigroup homomorphism $d \colon X \to P$ such that: (1) $X_p := d^{-1}(p)$ is a $C^*$-correspondence over $A$ for each $p\in P$; (2) $X_e = {_A}A_A$; (3) the multiplication on $X$ implements isomorphisms $X_p \otimes_A X_q \cong X_{pq}$ for $p,q \in P \setminus \{e\}$; and (4) multiplication implements the right and left actions of $X_e = A$ on each $X_p$. For $p \in P$ we let $\phi_p:A\to{\mathcal L}(X_p)$ be the homomorphism that implements the left action. Given $p, q \in P$ with $p \not= e$ there is a homomorphism $\iota^{pq}_p \colon {\mathcal L}(X_p) \to {\mathcal L}(X_{pq})$ such that $\iota^{pq}_p(S)(xy) = (Sx)y$ for all $x \in X_p$, $y \in X_{q}$ and $S \in {\mathcal L}(X_p)$. Upon identifying ${\mathcal K}(X_e)=A$, we let $\iota^q_e \colon {\mathcal K}(X_e)\to {\mathcal L}(X_{q})$ be given by $\iota^q_e=\phi_q$, see [@SY §2.2]. Recall that for a $C^*$-correspondence $Y$ over $A$, a map $\psi:Y\to B$ and a homomorphism $\pi:A\to B$ into a $C^*$-algebra form a *Toeplitz representation* if $\psi(x\cdot a)=\psi(x)\pi(a)$ and $\pi(\langle x, y\rangle)=\psi(x)^*\psi(y)$ for all $a\in A$, $x,y\in Y$. A map $\psi$ of a product system $X$ into a $C^*$-algebra $B$ is a *Toeplitz representation* if $\psi(xy)=\psi(x)\psi(y)$ for all $x,y\in X$ and $(\psi\vert_{X_p},\psi\vert_{X_e})$ is a Toeplitz representation of the $C^*$-correspondence $X_p$, for all $p\in P$. 0.2cm We recall from [@N] that a quasi-lattice ordered group $(G, P)$ consists of a subsemigroup $P$ of a (discrete) group $G$ such that $P\cap P^{-1}=\{e\}$ and every finite subset of $G$ with a common upper bound in $P$ admits a least common upper bound in $P$, all taken with respect to the left-invariant partial order on $G$ given by $x\leq y$ if $x^{-1}y\in P$. We write $x\vee y<\infty$ to indicate that $x,y$ have a common upper bound in $P$, and then $x\vee y$ denotes their least common upper bound in $P$. If no common upper bound of $x,y$ exists in $P$ we write $x\vee y=\infty$. The semigroup $P$ is *directed* if $x\vee y<\infty$ for all $x,y\in P$. Given a quasi-lattice ordered group $(G,P)$, a product system $X$ over $P$ is called *compactly aligned* if $\iota^{p \vee q}_p(S) \iota^{p \vee q}_q(T) \in {\mathcal K}(X_{p\vee q})$ whenever $S \in {\mathcal K}(X_p)$ and $T \in {\mathcal K}(X_q)$, and $p \vee q < \infty$ cf. [@CLSV] or [@F Definition 5.7] in case each $X_p$ is essential. If $\psi:X\to B$ is a Toeplitz representation, there are homomorphisms $\psi^{(p)}:{\mathcal K}(X_p)\to B$ such that $\psi^{(p)}(\theta_{x,y})=\psi_p(x)\psi_p(y)^*$ for all $p\in P$ and $x,y\in X$, [@P]. When $X$ is compactly aligned, $\psi$ is said to be *Nica covariant* if $\psi^{(p)}(S)\psi^{(q)}(T)$ is $\psi^{(p\vee q)}(\iota^{p \vee q}_p(S) \iota^{p\vee q}_q(T))$ in case $p\vee q<\infty$ and is zero otherwise, see [@F]. Fowler introduced a $C^*$-algebra ${{\mathcal T}_{\rm cov}}(X)$ and showed it is universal for Nica covariant Toeplitz representations of $X$, [@F]. Here we shall use the notation ${\operatorname{\mathcal{NT}}}(X)$ instead of ${{\mathcal T}_{\rm cov}}(X)$ because, as advocated for in [@BanHLR Remark 5.3], the choice of ${{\mathcal T}_{\rm cov}}(X)$ for a $C^*$-algebra generated by a universal representation was unfortunate. Fowler introduced also a Cuntz-Pimsner algebra of $X$ by imposing usual Cuntz-Pimsner covariance in the sense of [@P] in each fibre $X_p$. Given a quasi-lattice ordered group $(G,P)$ and a compactly aligned product system $X$ over $P$ of $C^*$-correspondences over $A$, Sims and Yeend [@SY] introduced a new notion of Cuntz-Pimsner covariance for a Toeplitz representation of $X$. The definition is quite complicated and we will not give it here. It was proved in [@SY Theorem 4.1] that the universal Cuntz-Nica-Pimsner covariant representation $j_X$ of $X$ is injective, meaning that $j_X\vert_{X_e}$ is injective, if $X$ is $\tilde\phi$-injective (see [@CLSV §2.4] for the definition of this concept). The universal $C^*$-algebra for Cuntz-Nica-Pimsner covariant representations of $X$, denoted ${\operatorname{\mathcal{NO}}_{\!X}}$, is then nontrivial. Gauge-invariant uniqueness for ${\operatorname{\mathcal{NT}}}(X)$ and ${\operatorname{\mathcal{NO}}_{\!X}}$ {#NO} =========================================================================================================== Fix a quasi-lattice ordered group $(G,P)$ and a compactly aligned product system $X$ over $P$ of $C^*$-correspondences over $A$. There is a canonical coaction $({\operatorname{\mathcal{NT}}}(X),\delta)$ of $G$, and we let $\mathcal{B}$ be the associated Fell bundle. If $X$ is $\tilde\phi$-injective, there is also a canonical coaction $({\operatorname{\mathcal{NO}}_{\!X}},\nu)$ of $G$, whose associated Fell bundle is denoted by $\mathcal{N}$. It was shown in [@CLSV Remark 4.5] that $C^*(\mathcal{B})\cong {\operatorname{\mathcal{NT}}}(X)$ and $C^*(\mathcal{N})\cong {\operatorname{\mathcal{NO}}_{\!X}}$. Equivalently, both coactions $\delta$ on ${\operatorname{\mathcal{NT}}}(X)$ and $\nu$ on ${\operatorname{\mathcal{NO}}_{\!X}}$ are maximal in the sense of [@EKQ]. The following terminology was introduced in [@CLSV Definition 4.10]: ${\operatorname{\mathcal{NO}}_{\!X}}$ has the *gauge-invariant uniqueness property* provided that a surjective homomorphism $\varphi:{\operatorname{\mathcal{NO}}_{\!X}}\to B$ is injective if and only if: 1. \[it:carries coaction\] there is a coaction $\beta$ of $G$ on $B$ such that $\varphi$ is $\nu-\beta$ equivariant, and 2. \[it:inj on A\] the homomorphism $\varphi\vert_{j_X(A)}$ is injective. The *gauge-invariant uniqueness theorem* for ${\operatorname{\mathcal{NO}}_{\!X}}$ is [@CLSV Corollary 4.11] and gives a number of necessary and sufficient conditions for ${\operatorname{\mathcal{NO}}_{\!X}}$ to have the gauge-invariant uniqueness property. For instance, ${\operatorname{\mathcal{NO}}_{\!X}}$ has the gauge-invariant uniqueness property precisely when the gauge-coaction $\nu$ is normal. Thus the gauge-invariant uniqueness theorem holds for ${\operatorname{\mathcal{NO}}_{\!X}}$ provided that $\nu$ is both maximal and normal. In the next result we recast the gauge-invariant uniqueness property for ${\operatorname{\mathcal{NO}}_{\!X}}$ by asking for a maximal coaction on the target algebra. The apparently short proof follows from the general uniqueness theorems worked out in the context of Fell bundles in the appendix, and illustrates the power of coaction techniques. \[GIUTNO\] Let $(G,P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A surjective homomorphism $\pi:{\operatorname{\mathcal{NO}}_{\!X}}\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ and there is a *maximal* coaction $\beta$ on $B$ such that $\pi$ is $\nu-\beta$ equivariant. Apply Corollary \[cor:giut-max-coact\] to $({\operatorname{\mathcal{NO}}_{\!X}}, \nu)$ and $\pi$. Compared to [@CLSV Corollary 4.12], Theorem \[GIUTNO\] does not require amenability of $G$, and can be applied to arbitrary $\tilde\phi$-injective compactly aligned product systems over $P$ (for which $j_X$ is an injective representation). The drawback is that $\pi$ needs to be injective on the entire fixed-point algebra for $\nu$, and not just on the coefficient algebra. In practice, the injectivity of $\pi$ on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ is likely to be difficult to establish. However, when the compactly aligned product system satisfies one of the two conditions: the left actions on the fibres of $X$ are all injective, or $P$ is directed and $X$ is $\tilde\phi$-injective, then [@CLSV Theorem 3.8] says that $\pi$ is injective on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ precisely when it is injective as a Toeplitz representation, i.e. its restriction to $j_X(A)$ is an injective homomorphism. Suppose that $G$ is a nonabelian finite-type Artin group. Then $G$ and its positive cone $P$ form a quasi-lattice ordered group. By [@CLac1], $P$ is directed and $G$ is not amenable. Then, if $X$ is the product system over $P$ with fibers ${{\mathbb{C}}}$, the algebra ${\operatorname{\mathcal{NO}}_{\!X}}$ is isomorphic to $C^*(G)$ and does not have the gauge-invariant uniqueness property (see [@CLSV Remark 5.4] for details). However, since ${\operatorname{\mathcal{NO}}_{\!X}}^\nu={{\mathbb{C}}}$, Theorem \[GIUTNO\] implies that a surjective homomorphism $\pi:C^*(G)\to B$ is injective if and only if $B$ carries a compatible maximal coaction. Since also $({\operatorname{\mathcal{NT}}}(X), \delta)$ is a maximal coaction, we have a version of [Theorem ]{} for ${\operatorname{\mathcal{NT}}}(X)$. Let $(G,P)$ be a quasi-lattice ordered group and $X$ a compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A surjective homomorphism $\pi:{\operatorname{\mathcal{NT}}}(X)\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NT}}}(X)^\delta$ and there is a maximal coaction $\beta$ on $B$ such that $\pi$ is $\delta-\beta$ equivariant. Apply Corollary \[cor:giut-max-coact\] to $({\operatorname{\mathcal{NT}}}(X),\delta)$ and $\pi$. \[thm:giut-Tcov\] Let $(G,P)$ be a quasi-lattice ordered group and $X$ a compactly aligned product system over $P$ of $C^*$-correspondences over $A$. The coaction $({\operatorname{\mathcal{NT}}}(X), \delta)$ is normal precisely when the following is satisfied: a surjective homomorphism $\pi:{\operatorname{\mathcal{NT}}}(X)\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NT}}}(X)^\delta$ and there is a coaction $\beta$ on $B$ such that $\pi$ is $\delta-\beta$ equivariant. Apply Corollary \[cor:characterise-normal\]. Let $(G,P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. The coaction $({\operatorname{\mathcal{NO}}_{\!X}}, \nu)$ is normal precisely when the following is satisfied: a surjective homomorphism $\pi:{\operatorname{\mathcal{NO}}_{\!X}}\to B$ is injective if and only if $\pi$ is injective on ${\operatorname{\mathcal{NO}}_{\!X}}^\nu$ and there is a coaction $\beta$ on $B$ such that $\pi$ is $\nu-\beta$ equivariant. Apply Corollary \[cor:characterise-normal\]. Next we recall from [@CLSV] that given a quasi-lattice ordered group $(G, P)$ and a compactly aligned product system $X$ over $P$ satisfying one of the following two conditions: the left actions on the fibres of $X$ are all injective, or $P$ is directed and $X$ is $\tilde\phi$-injective, then the $C^*$-algebra ${\operatorname{\mathcal{NO}}_{\!X}}^{r}:=C_r^*(\mathcal{N})$ and the normalization $\nu^n$ of $\nu$ have the co-universal property of [@CLSV Theorem 4.1]. This co-universal property was used to identify various reduced crossed product type $C^*$-algebras in the form ${\operatorname{\mathcal{NO}}_{\!X}}^{r}$ for appropriate $X$, and also to investigate the gauge-invariant uniqueness property in several contexts. Our abstract uniqueness results for Fell bundles allow us to give a characterization of injectivity of homomorphisms $\pi:B\to {\operatorname{\mathcal{NO}}_{\!X}}^{r}$ that is an alternative to [@CLSV Corollary 4.9]. \[GIUT-NOXreduced\] Let $(G, P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. A homomorphism $\pi:B\to {\operatorname{\mathcal{NO}}_{\!X}}^{r}$ is injective if and only if there is a normal coaction $\beta$ of $G$ on $B$ such that $\pi$ is $\beta-\nu^n$ equivariant and $\pi\vert_{B_e}$ is injective. Apply Corollary \[cor:giut-nor-coact\]. To see how this relates to [@CLSV], suppose $X$ is a compactly aligned product system over $P$ such that the left actions on the fibres of $X$ are all injective, or $P$ is directed and $X$ is $\tilde\phi$-injective. Suppose also that $\pi$ arises from the co-universal property of ${\operatorname{\mathcal{NO}}_{\!X}}^{r}$ applied to an injective Nica covariant Toeplitz representation $\psi:X\to B$, where there is a coaction $\beta$ of $G$ on $B$ making $\pi$ a $\beta-\nu^n$ equivariant homomorphism. It is proved in [@CLSV Corollary 4.9] that $\pi$ is injective if and only if $\beta$ is normal and $\psi$ is Cuntz-Nica-Pimsner covariant. In Theorem \[GIUT-NOXreduced\] the last condition is replaced by $\pi\vert_{B_e}$ being injective. Let $(G, P)$ be a quasi-lattice ordered group and $X$ a $\tilde\phi$-injective compactly aligned product system over $P$ of $C^*$-correspondences over $A$. The coaction $({\operatorname{\mathcal{NO}}_{\!X}}^{r}, \nu^n)$ is maximal precisely when the following is satisfied: a homomorphism $\pi:B\to {\operatorname{\mathcal{NO}}_{\!X}}^{r}$ is injective if and only if there is a coaction $\beta$ of $G$ on $B$ such that $\pi$ is $\beta-\nu^n$ equivariant and $\pi\vert_{B_e}$ is injective. Apply Corollary \[cor:characterise-max\]. Inner coactions {#section:inner} =============== In this section we study inner coactions in relation to faithfulness of representations. First we recall some notation. The multiplier algebra $M(C_0(G)\otimes C^*(G))$ is identified with the algebra of continuous bounded functions on $G$ with values in $M(C^*(G))$ equipped with the strict topology. Let $w_G$ be the unitary element of $M(C_0(G)\otimes C^*(G))$ given by the canonical embedding of $G$ in $M(C^*(G))$. Given a coaction $(A,\delta)$ and a $C^*$-algebra $D$, nondegenerate homomorphisms $\mu:C_0(G)\to M(D)$ and $\pi:A\to M(D)$ form a *covariant pair* for $(A,\delta)$ provided that the diagram $$\xymatrix@C+70pt{ A \ar[r]^-\delta \ar[d]_\pi &A\otimes C^*(G) \ar[d]^{\pi\otimes{\operatorname{id}}} \\ M(D) \ar[r]_-{\operatorname{Ad}{\overline}{\mu\otimes{\operatorname{id}}}(w_G)\circ({\operatorname{id}}\otimes 1)} &M(D\otimes C^*(G)) }$$ commutes, or, equivalently, since $G$ is discrete, provided that $$\label{def:cov-pair-coaction} \pi(a_x)\mu({\raisebox{2pt}{\ensuremath{\chi}}}_y)=\mu({\raisebox{2pt}{\ensuremath{\chi}}}_{xy})\pi(a_x)$$ for all $a_x\in A_x$, and all $x,y\in G$ (see e.g., [@EQ Section 2]). By [@QuiggFull Lemma 1.11], any nondegenerate homomorphism $\mu:C_0(G)\to M(A)$ implements an *inner* coaction $\delta^\mu$ on $A$ via $$\delta^\mu(a)=\operatorname{Ad}{\overline}{\mu\otimes{\operatorname{id}}}(w_G)(a\otimes 1).$$ Note that $({\operatorname{id}}_A, \mu)$ forms a covariant pair for every inner coaction $(A,\delta^\mu)$. Every inner coaction is normal, by [@QuiggFull Proposition 2.3] (see also [@BKQ Lemma A.2]). If $(A, G, \delta^\mu)$ is an inner coaction, then $a\in A_e$ if and only if $a$ commutes with $\{\mu({\raisebox{2pt}{\ensuremath{\chi}}}_x): x\in G\}$. Indeed, if $a\in A_e$ then $a$ commutes with every $\mu({\raisebox{2pt}{\ensuremath{\chi}}}_x)$ by . Conversely, if $a$ commutes with every $\mu({\raisebox{2pt}{\ensuremath{\chi}}}_x)$ then $a$ commutes with $\mu(C_0(G))$, hence $a\otimes 1$ commutes with $\mu(C_0(G))\otimes C^*(G)$, and therefore with ${\overline}{\mu\otimes{\operatorname{id}}}(w_G)$, so $a\in A_e$. \[rmk:inner-via-projections\] We note that a necessary and sufficient condition for a coaction $\delta$ on $A$ to be inner is that there is a family $\{p_x: x\in G\}$ of orthogonal projections in $M(A)$ that sum strictly to $1$ in $M(A)$ and satisfy $$\label{cond:et} a_xp_y=p_{xy}a_x$$ for all $a_x\in A_x \text{ and }x, y\in G$. Indeed, if $(A, \delta)$ is an inner coaction, there is a nondegenerate homomorphism $\mu:C_0(G)\to M(A)$ such that ${\operatorname{id}}_A$ and $\mu$ satisfy , which turns into by letting $p_y=\mu({\raisebox{2pt}{\ensuremath{\chi}}}_y)$. Conversely, given a coaction $(A, \delta)$ and a family of projections satisfying , let $\mu:C_0(G)\to M(A)$ be the unique homomorphism satisfying $\mu({\raisebox{2pt}{\ensuremath{\chi}}}_y)=p_y$ for $y\in G$. Then $\mu$ is nondegenerate because $\sum_{y\in G}p_y=1$ strictly in $M(A)$, and $({\operatorname{id}}_A, \mu)$ forms a covariant pair by . Unravelling the definitions, we have $\delta=\delta^\mu$. \[thm:abstract-CKUT\] Let $(A, \delta)$ be an inner coaction. A surjective homomorphism $\varphi:A\to B$ onto a $C^*$-algebra $B$ is injective if and only if $\varphi\vert_{A_e}$ is injective. Since $\delta$ is inner, there is a nondegenerate homomorphism $\mu:C_0(G)\to M(A)$ such that $\delta=\delta^\mu$. Define a nondegenerate homomorphism $\nu:C_0(G)\to M(B)$ by $\nu={\overline}\varphi\circ\mu$. Then $\delta^\nu$ is an inner coaction on $B$, and the computation $$\begin{aligned} \varphi(a_x)\nu({\raisebox{2pt}{\ensuremath{\chi}}}_y)&=\varphi(a_x)\overline{\varphi}(\mu({\raisebox{2pt}{\ensuremath{\chi}}}_y))=\varphi(a_x\mu({\raisebox{2pt}{\ensuremath{\chi}}}_y))\\ &=\varphi(\mu({\raisebox{2pt}{\ensuremath{\chi}}}_{xy})a_x) \text{ by } \eqref{def:cov-pair-coaction}\\ &=\nu({\raisebox{2pt}{\ensuremath{\chi}}}_{xy})\varphi(a_x)\end{aligned}$$ for $a_x\in A_x$ and $x,y\in G$ shows that $\varphi$ is $\delta-\delta^\nu$ equivariant. The theorem therefore follows from Proposition \[GIUTFB\] (2). Suppose that $(A, \delta)$ is a coaction of $G$. An ideal $I$ in $A$ is *$\delta$-invariant* if the restriction of $\delta$ to $I$ gives rise to a coaction of $G$ on $I$. If this is the case, we let $\delta\vert_I$ be the restricted coaction on $I$. \[cor:eiut\] Let $(A, \delta)$ be a coaction of $G$ and $I$ a $\delta$-invariant ideal in $A$ such that the coaction $\delta\vert_I$ of $G$ on $I$ is inner. If $I$ is an essential ideal in $A$, then a homomorphism $\varphi:A\to B$ is injective if and only if $\varphi\vert_{A^\delta}$ is injective. For the non-trivial direction, suppose that $\varphi\vert_{A^\delta}$ is injective. Then $\varphi$ is injective on $I^{\delta\vert_I}=A^\delta\cap I$. Since $\delta\vert_I$ is inner, Theorem \[thm:abstract-CKUT\] implies that $\varphi\vert_I$ is injective. But $I$ is an essential ideal, and so $\varphi$ is injective. $C^*$-algebras of quasi-lattice ordered groups {#subsect:Nica-alg-semigp-cp} ============================================== In this section we recall Nica’s constructions of $C^*$-algebras associated to isometric representations of quasi-lattice ordered groups, we give a quick review of subsequent constructions, and we make connections with coaction theory. Let $(G, P)$ be a quasi-lattice ordered group. A semigroup homomorphism $V$ of $P$ into the isometries on a Hilbert space $H$ such that $V_e=I$ and $V_sV_t=V_{st}$ for all $s,t\in P$ is called an (isometric) representation of $P$. Let $\{\varepsilon_t\}_{t\in P}$ be the canonical orthonormal basis of $l^2(P)$. The *Toeplitz* or *Wiener-Hopf representation* of $P$ on $l^2(P)$ is given by $T_s\varepsilon_t=\varepsilon_{st}$, for $s,t\in P$. The *Toeplitz algebra* (or Wiener-Hopf algebra) ${\mathcal T}(G, P)$ is the $C^*$-subalgebra of $B(l^2(P))$ generated by the image of $T$. Nica noticed that $T_sT_s^*T_tT_t^*=T_{s\vee t}T_{s\vee t}^*$ when $s\vee t<\infty$ and is zero otherwise. Such representations of $P$ are now called Nica covariant, and $C^*(G,P)$ is the universal $C^*$-algebra generated by a Nica covariant representation $v$ of $P$ (see [@N; @LacR1]). By [@N Proposition 3.2], the family $\{T_sT_t^*: s,t \in P\}$ spans a dense subalgebra of ${\mathcal T}(G, P)$. The *diagonal subalgebra* of ${\mathcal T}(G, P)$ is ${\mathcal D}={\operatorname{\overline{span\!}\,\,}}\{T_sT_s^*: s\in P\}$. We next recall some facts from [@LacR1]. Let $(G, P)$ be a quasi-lattice ordered group, and for each $s\in P$ write $1_s$ for the characteristic function of the set $\{t\in P:s\leq t\}$. Then $B_P={\operatorname{\overline{span\!}\,\,}}\{1_s: s\in P\}$ is a commutative $C^*$-subalgebra of $l^\infty(P)$, and $C^*(G,P)$ is the semigroup crossed product $B_P\rtimes P$ arising from translation $t\mapsto (1_s\to 1_{ts})$ on $B_P$, see [@LacR1 Corollary 2.4]. By [@LacR1 §6.1], there is a coaction $\delta$ of $G$ on $C^*(G,P)$ such that $\delta(v_s)=v_s\otimes s$ for all $s\in P$, and $B_P$ is the fixed-point algebra $C^*(G,P)^\delta$. Moreover, [@LacR1 Proposition 2.3] shows that every representation of $C^*(G,P)$ is determined by a Nica covariant representation of $P$. We let ${\lambda_T}$ denote the representation of $C^*(G, P)$ determined by $T$, and note that it carries $1_s$ to $T_sT_s^*$ for all $s\in P$. It follows from [@SY Proposition 5.6] that if $X={{\mathbb{C}}}\times P$ is the trivial product system over $P$ with fibers $X_p={}_{{{\mathbb{C}}}}{{\mathbb{C}}}_{{{\mathbb{C}}}}$ for all $p\in P$, then ${\operatorname{\mathcal{NT}}}(X)\cong C^*(G,P)$. Since $\delta$ is maximal by [@CLSV Remark 4.5], we shall view it as a coaction on $C^*(G, P)=C^*(\mathcal{B})$ (recall that we let $\mathcal B$ denote the associated Fell bundle over $G$) with fixed point algebra equal to $B_P$. Recall from [@EQ] that $C^*_r(\mathcal{B})$ is identified with the normalization $(C^*(\mathcal{B}))^n.$ \[prop:normalising-delta\] The representation $\lambda_T$ is both a maximalization and a normalization from $(C^*(G, P), \delta)$ onto $({\mathcal T}(G, P), \delta^n)$. In particular, ${\mathcal T}(G, P)\cong C^*_r(\mathcal{B})$. Using reduced coactions, it was shown in [@QuiggRa Proposition 6.5] that there is a normal coaction $\eta$ on ${\mathcal T}(G, P)$ such that $\eta(T_sT_t^*)=T_sT_t^*\otimes st^{-1}$ for $s,t \in P$. Then $\lambda_T: (C^*(G, P), \delta)\to ({\mathcal T}(G, P), \eta)$ is equivariant. We noted in the preliminaries that the regular representation $\Lambda_{{\mathcal B}}:(C^*(G, P), \delta)\to (C^*_r(\mathcal{B}), \delta^n)$ is both a maximalization and a normalization. Since $\lambda_T$ is injective on $B_P$ by [@LacR1 Corollary 2.4(1)], Proposition \[GIUTFB\], parts (\[it:delta-max\]) and (\[it:epsilon-normal\]), imply that $\lambda_T$ is also both a maximalization and a normalization. Since all maximalizations are isomorphic, and similarly for normalizations, we therefore have $C^*(G, P)\cong ({\mathcal T}(G, P))^m$ and ${\mathcal T}(G, P)\cong (C^*(G, P))^n$. Nica [@N Definition 4.2] defined $(G, P)$ to be amenable if the representation $\lambda_T$ is an isomorphism. His definition motivated Exel’s definition of amenable Fell bundles in [@E]. Our Proposition \[prop:normalising-delta\] shows that the Fell bundle $\mathcal{B}$ is amenable when $(G, P)$ is amenable in Nica’s sense. Finite exhaustive sets of strictly positive elements {#section:fessub} ==================================================== Throughout this section let $(G,P)$ be a quasi-lattice ordered group. A [FESSPE]{} of $(G,P)$ is a finite subset $F\subset P{\setminus}\{e\}$ such that $FP=P{\setminus}\{e\}$. “[FESSPE]{}” stands for “finite exhaustive set of strictly positive elements”, and the existence of such an $F$ is easily seen to be equivalent to the existence, for each $x\in G$, of a finite set of strict upper bounds $S$ of $x$ (i.e., $x\lneqq y$ for all $y\in S$) that is exhaustive in the sense that every strict upper bound of $x$ has a lower bound in $S$ — namely, take $S=xF$. This condition was introduced in [@N], and was shown in [@N Proposition 6.3] to be equivalent, among others, to the fact that ${\mathcal T}(G,P)$ contains the compact operators ${\mathcal K}(l^2(P))$. As remarked in [@N], all pairs $(G,P)$ with $P$ finitely generated have a [FESSPE]{}. In particular, $({{\mathbb{F}}}_n, {{\mathbb{F}}}_n^+)$ has a [FESSPE]{} for all $n\geq 1$. The pair $({{\mathbb{F}}}_\infty, {{\mathbb{F}}}_\infty^+)$ does not have a [FESSPE]{} since in this case the Toeplitz algebra is isomorphic to ${\mathcal O}_\infty$ and is therefore simple. Another example of a quasi-lattice ordered group not having a [FESSPE]{} is $({{\mathbb{Q}}}_+^*, {{\mathbb{N}}}^\times)$, endowed with the order given by $r\leq s {\ensuremath{\Leftrightarrow}}r \text{ divides }s$. No finite set of non-zero positive integers different from $1$ can contain a lower bound for every element in ${{\mathbb{N}}}^\times\setminus\{1\}$. It is possible for $(G,P)$ to have a [FESSPE]{} but not be finitely generated. For example, consider $G = (R,+)$ and $P={0}\cup[1,\infty)$. Then $(G,P)$ is quasi-lattice ordered and has a [FESSPE]{} (and $P-P=G$), but is not finitely generated. The following result is the essential-inner uniqueness theorem for ${\mathcal T}(G, P)$ when $(G, P)$ has a [FESSPE]{}. \[thm:describe-inner-Nica-alg\] Let $(G, P)$ be a quasi-lattice ordered group and $\delta^{n}$ the canonical normal coaction on ${\mathcal T}(G, P)$. Assume $(G, P)$ has a [FESSPE]{}. Then the following assertions hold. [(a)]{.nodecor} ${\mathcal K}(l^2(P))$ is a $\delta^{n}$-invariant ideal in ${\mathcal T}(G, P)$ and $\delta^{n}\vert_{{\mathcal K}(l^2(P))}$ is an inner coaction. [(b)]{.nodecor} Let $\varphi$ be a homomorphism of ${\mathcal T}(G,P)$ into a $C^*$-algebra $B$. Then the following are equivalent: 1. \[it:phi-inj\] The homomorphism $\varphi$ is injective. 2. \[it:phi-inj-fix\] The homomorphism $\varphi$ is injective on ${\mathcal D}$. 3. \[it:phi-inj-chie\] We have $\varphi(p_e)\not=0$, where $p_e$ is the rank-one projection onto $\varepsilon_e$. To prove this theorem we shall need some preparation. The equivalence of (1) and (4) in the next result is implicit in [@N Proposition 6.3]. We first recall a couple of facts about the Nica spectrum of $(G, P)$. The spectrum of the commutative algebra ${\mathcal D}$ is the space $\Omega$ of all non-empty, hereditary, directed subsets $A$ of $P$, see [@N §6] for definitions and details. Assigning the set $A_\gamma=\{s\in P: \gamma(T_sT_s^*)=1\}$ to a character $\gamma$ of ${\mathcal D}$ gives a homeomorphism of the character space of ${\mathcal D}$ onto $\Omega$. Let $\iota:P\to \Omega$ be the map $t\mapsto [e,t]$ from[@N §6.3, Remark 1], where $[e,t]:=\{s\in P: s\leq t\}$. Since $\lambda_T$ is an isomorphism of $B_P$ onto ${\mathcal D}$, there is a homeomorphism $\widehat{B_P}\to \Omega$ given by $\gamma\to A_\gamma$ for $A_\gamma=\{t\in P: \gamma(1_t)=1\}$, see [@Lac1]. Under this homeomorphism, $[e,t]$ corresponds to the character $\gamma$ of $B_P$ given by $\gamma(1_x)=1_x(t)$ for all $x\in P$. \[lem:essential-ideal\] Let $(G, P)$ be a quasi-lattice ordered group. The following statements are equivalent: 1. \[it:pair-to-ideal1\] $(G, P)$ has a [FESSPE]{}. 2. \[it:pair-to-ideal2\] $c_0(P)$ is contained in $B_P.$ 3. \[it:pair-to-ideal3\] $c_0(P)$ is an essential ideal in $B_P$. 4. \[it:pair-to-ideal4\] $c_0(\iota(P))$ is an essential ideal in ${\mathcal D}$. Let $F$ be a [FESSPE]{} for $(G,P)$, and for $x\in P$ define $1_{\{x\}}\in l^\infty(P)$ by $$1_{\{x\}}(y)=\begin{cases} 1{& \text{if }}y=x\\ 0{& \text{otherwise}}. \end{cases}$$ Then $c_0(P)$ is generated by the projections $1_{\{x\}}$ for $x\in P$. To establish $\Rightarrow$ it suffices to prove that $$\label{eq:Chi-x} \prod_{a\in F}(1_x-1_{xa})=1_{\{x\}}$$ for all $x\in P$. Take $y\in P$, and note that if $x=y$ then $1_{xa}(x)=0$ for all $a\in F$, so the left hand side of evaluated at $y$ is equal to $1_x(x)=1$. If $x^{-1}y\in P\setminus \{e\}$, there is $a\in F$ such that $a\leq x^{-1}y$. Hence $(1_x-1_{xa})(y)=1_x(y)-1_{xa}(y)=0$, and so the product on the left hand side of evaluated at $y$ is zero. The remaining possibility for $y$ is $x{^{-1}}y\notin P$, in which case the left side is obviously zero; this establishes $\Rightarrow$. The implication $\Rightarrow$ is clear, and $\Rightarrow$ follows because the isomorphism $B_P\to {\mathcal D}$ carries the ideal $c_0(P)$ onto $c_0(\iota(P))$. It remains to prove $\Rightarrow$. Since $c_0(\iota(P))$ is essential, $\iota(P)$ is an open and dense subset of $\Omega$. Thus the relative topology on $\iota(P)$ is the original topology on $P$, and so each interval $[e,t]$ is open in $\Omega$. Then the implication $2\Rightarrow 4$ from [@N Proposition 6.3] shows that $(G, P)$ has a [FESSPE]{}. The next result is a sharpening of [@N remark 2.2.3]. \[lem:lub-products\] Let $(G, P)$ be a quasi-lattice ordered group. Assume $a,b,z\in G$ are such that at least one of $a,b$ is in $P$ and at least one of $za, zb$ is in $P$. Then $a\vee b<\infty$ precisely when $z(a\vee b)<\infty$, in which case $z(a\vee b)=za\vee zb$ as elements of $P$. Suppose $w:=za\vee zb\in P$. Then $a\leq z^{-1}w$ and $b\leq z^{-1}w$. Since at least one of $a,b$ is in $P$ we necessarily have $z^{-1}w\in P$. Thus $a\vee b< \infty$. Since the order is left-invariant, $w\leq z(a\vee b)$. Then $z^{-1}w\leq a\vee b$, so necessarily $z^{-1}w=a\vee b$. Now suppose $a\vee b<\infty$. Then by left invariance $za\leq z(a\vee b)$ and $zb\leq z(a\vee b)$. It follows that $z(a\vee b)\in P$. Therefore $za\vee zb\leq z(a\vee b)$, from which equality follows as in the previous paragraph. \[prop:spectral-stuff\] Let $(G, P)$ be a quasi-lattice ordered group having a [FESSPE]{}. The assignment $$\label{eq:et} e_y=\begin{cases} 1_{\{y\}}&\text{ if }y\in P\\ 0&\text{ if }y\in G\setminus P \end{cases}$$ for $y\in G$ defines a family of mutually orthogonal projections in $C^*(G, P)$ such that $$\label{need-delta-inner} c_xe_y=e_{xy}c_x$$ for all $c_x\in C^*(G, P)_x$, and $x, y\in G$. Note that the family $\{e_y\}$ gives rise to a nondegenerate homomorphism $\mu:c_0(G)\to c_0(P)$. Since $C^*(G, P)$ is the closed span of monomials $v_pv_q^*$, we have $$\label{eq:spectral-subsp-full} C^*(G, P)_x=\begin{cases} {\operatorname{\overline{span\!}\,\,}}\{v_pv_q^*: x=pq^{-1}\}&\text{ if }x\in PP^{-1}\\ 0&\text{ otherwise}. \end{cases}$$ It therefore suffices to prove when $c_x$ is of form $v_pv_q^*$ with $x=pq^{-1}\in PP^{-1}.$ [[**Case 1**]{}: $y, xy\in P$.]{} We must show that $v_pv_q^*e_y=e_{xy}v_pv_q^*$. By we have $e_y=\prod_{a\in F}(1_y-1_{ya})$ and $e_{xy}=\prod_{a\in F}(1_{xy}-1_{xya})$. If $y\vee q=\infty$ then Lemma \[lem:lub-products\] implies $xy\vee p=\infty$. Hence Nica covariance of $v$ implies that $v_q^*1_y=v_q^*v_yv_y^*=0$ and $1_{xy}v_p=v_{xy}v_{xy}^*v_p=0$. Since also $ya\vee q=\infty$ for all $a\in F$ (because $y\leq ya$ for all $a\in F$), we likewise have $xya\vee p=\infty$, and therefore $c_x1_{ya}=1_{xya}c_x$. In all, is satisfied. If $y\vee q<\infty$ then $xy\vee p=x(q\vee y)<\infty$ by Lemma \[lem:lub-products\]. We then have $$\begin{aligned} c_x1_y&=v_pv_q^*1_y =v_pv_q^*v_yv_y^*\notag \\ &=v_pv_{q^{-1}(q\vee y)}v^*_{y^{-1}(q\vee y)}v_y^*\notag\\ &=v_{x(q\vee y)}v^*_{q\vee y}\notag\\ &=v_{xy\vee p}v^*_{q\vee y}\label{eq:finiteqy}\\ &=v_{xy}v_{(xy)^{-1}(xy\vee p)}v _{p^{-1}(xy\vee p)}^*v^*_q\notag\\ &=1_{xy}v_pv_q^*=1_{xy}c_x.\label{computations-y}\end{aligned}$$ Let $a\in F$. Two sub-cases arise: $ya\vee q=\infty$, in which case also $xya\vee p=\infty$, and $c_x1_{ya}=1_{xya}c_x=0$ follows as in the previous paragraph. The second sub-case has $ya\vee q<\infty$, which entails $xya\vee p<\infty$, and replacing $y$ with $ya$ in the computations leading to shows that $c_x1_{ya}=1_{xya}c_x$. The equality is thus satisfied in case 1. [[**Case 2**]{}: $y\in P$, $xy\notin P$.]{} We must show $v_pv_q^*e_y=0$. Equivalently, we must show $$\label{eq:cxey-0} \prod_{a\in F}v_pv_q^*(1_y-1_{ya})=0.$$ Again, two sub-cases arise. If $q\vee y=\infty$, then also $q\vee ya=\infty$ for all $a\in F$, and by Nica covariance we see that $v_q^*v_y=0=v_q^*v_{ya}$ for all $a\in F$. Hence follows. In case $q\vee y\in P$, we have $v_pv_q^*1_y= v_{xy\vee p}v^*_{q\vee y}$ by (where the use of Lemma \[lem:lub-products\] is legitimate because $p,q,y\in P$). To establish we claim that there exists $a'\in F$ such that $v_pv_q^*(1_y-1_{ya'})=0$. The assumption $xy\notin P$ implies $q^{-1}y\notin P$, and so $y^{-1}(q\vee y)\in P\setminus \{e\}$. Thus by assumption there is $a'\in F$ with $a'\leq y^{-1}(q\vee y)$. This gives $ya'\vee q\leq q\vee y$, and since the reverse inequality is satisfied because $F\subset P$ we get $q\vee ya'= q\vee y\in P$. Applying Lemma \[lem:lub-products\] yields $xya'\vee p=x(q\vee ya')=x(q\vee y)=xy\vee p$, and invoking equation  where $y$ is replaced by $ya'$ gives $v_pv_q^*1_{ya'}=v_{xya'\vee p}v^*_{q\vee ya'}$. The claim is therefore proved, and case 2 is finished. [[**Case 3**]{}: $y\notin P$, $xy\in P$.]{} We must show that $e_{xy}v_pv_ q^*=0$. Either $xy\vee p=\infty$, in which case $xya\vee p=\infty$ for all $a\in F$, and follows by Nica covariance, or $xy\vee p\in P$. If this last alternative happens, the choice of $y$ implies that $p\nleqq xy$, so $(xy)^{-1}(xy\vee p)\in P\setminus \{e\}$. The [FESSPE]{} $F$ supplies $a'\in F$ with $a'\leq (xy)^{-1}(xy\vee p)$, and similarly to case 2 we get $1_{xy}v_pv_q^*=1_{xya'}v_pv_q^*$, from which again follows. [[**Case 4**]{}: $y\notin P, xy\notin P$.]{} Then both sides of are zero. Since $(G, P)$ has a [FESSPE]{}, [@N Proposition 6.3] gives an inclusion ${\mathcal K}(l^2(P))\subset {\mathcal T}(G, P)$. Clearly ${\mathcal K}(l ^2(P))$ is an ideal in ${\mathcal T}(G, P)$. To prove part (a) we will show that ${\mathcal K}(l ^2(P))$ is $\delta^{n}$-invariant. This will give a coaction $\delta^{n}_{{\mathcal K}(l^2(P))}$ on ${\mathcal K}(l ^2(P))$ obtained as restriction of $\delta^{n}$. We shall then construct mutually orthogonal projections $\{p_x: x\in G\}$ in $B(l^2(P))$ such that $\sum_{x\in G}p_x=I$ in weak-operator topology on $B(l^2(P))$ and $a_xp_y=p_{xy}a_x$ for all $a_x\in {\mathcal K}(l ^2(G, P))_x$ and $x,y\in G$. Remark \[rmk:inner-via-projections\] therefore provides an inner coaction $\delta^\mu$ on ${{\mathcal K}(l^2(P))}$ with $\delta^{n}_{{\mathcal K}(l^2(P))}=\delta^\mu$. Let $F$ be a [FESSPE]{} of $(G,P)$. For every $x\in G$ define a projection in $B(l^2(P))$ by $$p_x=\begin{cases} \lambda_T(1_{\{x\}}){& \text{if }}x\in P\\ 0{& \text{if }}x\in G{\setminus}P. \end{cases}$$ Note that for $x\in P$ we have $$p_x=\prod_{a\in F}(T_xT_x^*-T_{xa}T_{xa}^*).$$ With $\xi\otimes \eta$ denoting the rank-one operator $(\xi\otimes \eta)(\zeta)=\eta\langle \xi, \zeta\rangle$ in $B(l ^2(P))$ we see that $p_e=\varepsilon_e\otimes \varepsilon_e$. So $p_e\in {\mathcal K}(l^2(P))$. Since also $p_e\in {\mathcal D}$, it follows that $\delta^{n}(p_e)=p_e\otimes 1$. For $x,y\in P$, the product $T_xp_eT_y^*$ is the rank-one operator $e_y\otimes e_x$, see also the proof of [@N Proposition 6.3]. Thus ${\mathcal K}(l^2(P))$ is the closed span of monomials $T_xp_eT_y^*$ for $x,y\in P$. But $$\delta^{n}(T_xp_eT_y^*)=\delta^{n}(T_x)\delta^{n}(p_e)\delta^{n}(T_y^*)=(T_xp_eT_y^*)\otimes xy^{-1},$$ showing that $\delta^{n}({\mathcal K}(l^2(P)))\subset {\mathcal K}(l^2(P))\otimes C^*(G)$. Since $$(T_xp_eT_y^*)\otimes z=(T_xp_eT_y^*\otimes xy^{-1})(1\otimes yx^{-1}z),$$ we have $${\operatorname{\overline{span\!}\,\,}}\delta^{n}({\mathcal K}(l ^2(P)))(1\otimes C^*(G))={\mathcal K}(l^2(P))\otimes C^*(G).$$ Thus ${{\mathcal K}(l^2(P))}$ is $\delta^{n}$-invariant, and by restriction $\delta^{n}$ gives a coaction $\delta^{n}\vert_{{\mathcal K}(l^2(P))}$ on ${\mathcal K}(l ^2(P))$. Since $\lambda_T$ is a maximalization by Proposition \[prop:normalising-delta\], it carries $C^*(G, P)_x$ isometrically onto ${\mathcal T}(G, P)_x$ for every $x\in G$. [Lemma ]{} implies that $a_xp_y=p_{xy}a_x$ for every $x\in PP^{-1}$, $a_x\in {\mathcal T}(G, P)_x$ and $y\in G$. In particular, we may take $a_x\in {\mathcal K}(l^2(P))\cap {\mathcal T}(G, P)_x$, which shows that $$\delta^\mu\vert_{{\mathcal K}(l^2(P))_x}=\delta^{n}\vert_{{\mathcal K}(l^2(P))\cap {\mathcal T}(G, P)_x}.$$ Hence $\delta^{n}\vert_{{\mathcal K}(l^2(P))}$ coincides with $\delta^\mu$, and thus is an inner coaction, as claimed in (a). For (b), obviously it suffices to show $\Rightarrow$ and $\Rightarrow$. The implication $\Rightarrow$ follows from (a) and Corollary \[cor:eiut\] because ${\mathcal K}(l^2(P))$ is an essential ideal in $B(l^2(P))$, hence in ${\mathcal T}(G, P)$. For the implication $\Rightarrow$, suppose $\varphi(p_e)\not=0$. For every $x\in P$ we have $T_xp_e=p_xT_x$, and $T_x$ is an isometry, so $\varphi(T_x)\varphi(p_e)\not=0$. Then $\varphi(p_x)\varphi(T_x)\not=0$, and hence $\varphi(p_x)\not=0$. By linearity and density it follows that $\varphi(M_f)\not=0$ for all $f\in c_0(\iota(P))\subset {\mathcal D}$. Since $c_0(\iota(P))$ is an essential ideal in ${\mathcal D}$, it follows that $\varphi$ is injective on ${\mathcal D}$. In the above proof of Theorem 6.3, we appealed to Corollary 4.3 for the implication (2) $\Rightarrow$ (1), and then for (3) $\Rightarrow$ (2) we employed an elementary argument. In fact, however, in this particular case we can prove (3) $\Rightarrow$ (1) directly, as follows. We have mentioned that FESSPE guarantees ${\mathcal K}(l^2(P))\subset {\mathcal T}(G,P)$, and then (3) implies that $\varphi$ is nonzero on the simple, essential ideal ${\mathcal K}(l^2(P))$, and hence is faithful. Nevertheless, we wanted to show how the method involving Corollary 4.3 can be applied, because we feel that it will be useful more generally. A result similar to Theorem \[thm:describe-inner-Nica-alg\] (a) can be proved for $C^*(G, P)$, as follows. \[cor:6.8\] In the notation of Lemma \[prop:spectral-stuff\], the set $$\label{eq:ideal-of-full-algebra} \mathcal{J}:={\operatorname{\overline{span\!}\,\,}}\{v_xe_ev_y^*: x,y\in P\}$$ is a $\delta$-invariant ideal of $C^*(G, P)$, and $\delta\vert_{\mathcal{J}}$ is an inner coaction. Since $C^*(G, P)={\operatorname{\overline{span\!}\,\,}}\{v_sv_t^*: s,t\in P\}$, it suffices to show that $\mathcal{J}$ is a subalgebra of $C^*(G, P)$ and that $v_sv_t^*v_xe_ev_y^*$ and $v_xe_ev_y^*v_sv_t^*$ are in $\mathcal{J}$ for all $s,t,x,y\in P$. By , $v_t^*v_xe_e=v_t^*e_xv_x=0$ unless $t^{-1}x\in P$, in which case $v_sv_t^*v_xe_ev_y^*=v_sv_{t^{-1}x}e_ev_y^*\in \mathcal{J}$. If $y\vee s=\infty$, Nica covariance of $v$ implies that $v_y^*v_s=0$. Otherwise $v_y^*v_s=v_{y^{-1}(y\vee s)}v_{s^{-1}(y\vee s)}^*$, and implies that $e_ev_{y^{-1}(y\vee s)}=0$ unless $y^{-1}(y\vee s)=e$. If $y\vee s=y$ it follows that $v_xe_ev_y^*v_sv_t^*=v_xe_ev_ {t(s^{-1}y)}^*\in \mathcal{J}$. Now clearly $\mathcal{J}$ is closed under taking adjoints, and by the previous computations it follows that $v_xe_ev_y^*v_se_ev_t^*$ is zero unless $s=y$, in which case it equals $v_xe_ev_t^*$, so it lies in $\mathcal{J}$. That $\mathcal{J}$ is $\delta$-invariant follows as in the proof of part (a) of Theorem \[thm:describe-inner-Nica-alg\] because $\delta(v_xe_ev_y^*)=v_xe_ev_y^*\otimes xy^{-1}$ and $(v_xe_ev_y^*)\otimes z=(v_xe_ev_y^*\otimes xy^{-1})(1\otimes yx^{-1}z)$. Hence Lemma \[prop:spectral-stuff\] implies that $\delta\vert_{\mathcal{J}}$ is $\delta^\mu$, and therefore is an inner coaction. We obtain an essential-inner uniqueness theorem for $C^*(G, P)$ if the ideal $\mathcal{J}$ of is essential. If the ideal $\mathcal{J}$ of is essential in $C^*(G, P)$, then $C^*(G, P)$ has the essential-inner uniqueness property of [Corollary ]{}. This holds in particular if $(G, P)$ has the approximation property for positive definite functions in the sense of Nica. If $\mathcal{J}$ is essential, then the conclusion follows immediately from [Corollary ]{} and [Corollary ]{}. For the other part, note that, as remarked in \[Lac99\], if $(G, P)$ has the approximation property of Nica then for every ideal $\mathcal{I}$ of $C^*(G, P)$ we have $\mathcal{I}=\{X\in C^*(G, P): \Phi(X^*X)\in \Phi(\mathcal{I})\}$, where $\Phi$ is the conditional expectation from $C^*(G, P)$ into $B_P$, see [@LacR1 Corollaries 2.4 and 3.3]. Now $\Phi$ is faithful by [@N §4.3 and 4.5]. Therefore, if $\mathcal{I}$ is non-trivial then by faithfulness of $\Phi$ also $\Phi(\mathcal{I})$ is non-trivial as an ideal of $B_P$. By Lemma \[lem:essential-ideal\] there exists $e_x\in c_0(P)$ such that $e_x\in \Phi(\mathcal{I})$. It follows that $e_x\in\mathcal{I}$, so $\mathcal{I}\cap \mathcal{J}$ is non-trivial, and hence $\mathcal{J}$ is essential. The next result is a converse to Theorem \[thm:describe-inner-Nica-alg\]. \[thm:inner-to-FES\] Suppose there is a family $\{q_x: x\in G\}\subset {\mathcal T}(G, P)$ of mutually orthogonal projections such that 1. $q_y\in {\mathcal D}'$ and $T_pq_y=q_{py}T_p$ for all $y\in G$ and $p\in P$, and 2. $\sum_{y\in G}q_y=I$ in the weak operator topology of $B(l^2(P))$. Then $(G, P)$ has a [FESSPE]{}. To prove this theorem we will need a lemma. \[lem:Toe-mult-free\] Let $(G, P)$ be a quasi-lattice ordered group and ${\mathcal D}$ the diagonal subalgebra of ${\mathcal T}(G, P)$. Then the commutant ${\mathcal D}'$ is contained in $l^\infty(P)$. Let $M\in {\mathcal D}'\subset B(l^2(P))$. We claim that there is $g\in l^\infty(P)$ such that $M=M_g$. For each $p\in P$ define $f_p:=M\varepsilon_p$ in $l^2(P)$. Using that $MM_{1_p}=M_{1_p}M$ implies that $f_p=M_{1_p}f_p$, and therefore $f_p$ has support included in $\{t\in P: p\leq t\}$. On the other hand, if $p\leq t$ and $p\neq t$, then the commutation relation $MM_{1_t}=M_{1_t}M$ implies that $M_{1_t}f_p=0$, showing that $f_p$ has support the single point $\{p\}$. Thus there is $g:P\to {{\mathbb{C}}}$ such that $M\varepsilon_p=g(p)\varepsilon_p$ for all $p\in P$. Since $\Vert M\varepsilon_p\Vert_2\leq \Vert M\Vert$ for all $p\in P$, it follows that $\vert g(p)\vert\leq \Vert M\Vert$ for all $p\in P$. This means $g\in l^\infty$. The claim, hence the lemma, are proved. We will show that ${\mathcal T}(G, P)\supseteq {\mathcal K}(l^2(P))$, and then apply [@N Proposition 6.3] to conclude that $(G, P)$ has a [FESSPE]{}. It suffices to show that ${\mathcal T}(G, P)$ contains all rank one projections $\varepsilon_y\otimes \varepsilon_x$ on $l^2(P)$, where $x,y\in P$. For this it suffices to establish that $q_e=p_e$, because then we will have $\varepsilon_y\otimes\varepsilon_x=T_xq_eT_y^*$ as in the proof of Theorem \[thm:describe-inner-Nica-alg\]. By assumption (1), $T_pT_p^*q_y=q_yT_pT_p^*$ for all $p\in P$ and $y\in G$. Thus $q_y\in{\mathcal D}'$ for all $y\in G$, and so $q_y\in l^\infty(P)$ by Lemma \[lem:Toe-mult-free\]. Write $q_y=M_{{\raisebox{2pt}{\ensuremath{\chi}}}_{E(y)}}$ where $\emptyset\neq E(y)\subset P$ for every $y\in G$. Since $\sum_{y\in G}q_y=I$, the family $\{E(y)\}_{y\in G}$ is a mutually disjoint family such that $P=\cup_{y\in P}E(y)$. We claim that $$\label{eq:Et-single} E(y)=\{y\} \text{ for all }y\in P.$$ Towards the claim, we prove first that $pE(y)=E(py)$ for all $p, y\in P$. By assumption (1), $T_pM_{{\raisebox{2pt}{\ensuremath{\chi}}}_{E(y)}}=M_{{\raisebox{2pt}{\ensuremath{\chi}}}_{E(py)}}T_p$ for $p\in P$. Applying both sides to $\varepsilon_u$ gives $$\begin{cases}\varepsilon_{pu}&\text{ if }u\in E(y)\\ 0&\text{ if }u \notin E(y)\end{cases}=\begin{cases} \varepsilon_{pu}&\text{ if }pu\in E(py)\\ 0&\text{ if }pu\notin E(py) \end{cases}$$ when $p,u, y\in P$. Thus it suffices to prove when $y=e$. Let $y\in P$ such that $e\in E(y)$. Then $e\in E(y)=yE(e)\subseteq yP$. This forces $y\in P\cap P^{-1}$, so $y=e$. Hence $e\in E(e)$, which also implies $p\in E(p)$ for all $p\in P$. If $p\in E(e)$, then $p\in E(p)\cap E(e)$. This intersection is non-empty precisely when $p=e$. In other words, we have established $E(e)=\{e\}$, from which and hence the theorem follow. It was asserted in [@N §6.3, Remark 4] that ${\mathcal K}(l^2(P))$ is an induced ideal from ${\mathcal D}$ when $(G, P)$ has a [FESSPE]{}. However, no proof was given of this claim. Here we show that ${\mathcal K}(l^2(P))$ is contained in the ideal of ${\mathcal T}(G, P)$ induced from $c_0(P)$. We conjecture that the two are equal, but we have not been able to prove this. To recall terminology, let $(G, P)$ be a quasi-lattice ordered group. Let $\Phi$ be the conditional expectation from $C^*(G, P)$ onto $B_P$ constructed in [@LacR1] and $\Phi^{n}$ the conditional expectation from ${\mathcal T}(G, P)$ to ${\mathcal D}$ associated to the coaction $\delta^{n}$ of Proposition \[prop:normalising-delta\]. The representation $\lambda_T$ intertwines $\Phi$ and $\Phi^{n}$. Since $\delta^{n}$ is normal, $\Phi^{n}$ is faithful on positive elements. In [@N §6] Nica associates to an invariant ideal $I$ in ${\mathcal D}$ the induced ideal ${\operatorname{Ind}}{I}=\{X\in{\mathcal T}(G, P): \Phi^{n}(X^*X)\in {\mathcal D}\}$ in ${\mathcal T}(G, P)$. Suppose $(G, P)$ has a [FESSPE]{}. Lemma \[lem:essential-ideal\] says that ${\mathcal I}:=c_0(\iota(P))$ is an essential ideal in ${\mathcal D}$. Further, ${\mathcal I}$ is generated by the projections $$p_y=\prod_{a\in F} (T_yT_y^*-T_{ya}T_{ya}^*),$$ for all $y\in P$. We claim that ${\mathcal I}$ is invariant in Nica’s sense. To see this, let $x\in G$ and write it as $x=\sigma(x)\tau(x)^{-1}$ with $\sigma(x)\in P$ the least upper bound of $x$. For $y\in P$, equation  implies that $$T_{\sigma(x)}T_{\tau(x)}^*p_y (T_{\sigma(x)}T_{\tau(x)}^*)^*=p_{xy}T_{\sigma(x)}T_{\tau(x)}^*T_{\tau(x)}T_{\sigma(x)}^*,$$ which is $p_{xy}T_{\sigma(x)}T_{\sigma(x)}^*$, and lies in ${\mathcal I}$ because ${\mathcal I}$ is an ideal in ${\mathcal D}$. Since $p_y$ span ${\mathcal I}$, the ideal ${\mathcal I}$ is indeed invariant. Now the rank-one operator on $l^2(P)$ taking $\varepsilon_y$ to $\varepsilon_x$ is $X=T_xp_eT_y^*$ and $$\Phi^{n}(X^*X)=T_yp_eT_y^*=p_y\in {\mathcal I},$$ so ${\operatorname{Ind}}{\mathcal I}$ contains all rank-one operators in $B(l^2(P))$. Hence ${\mathcal K}(l^2(P))\subset {\operatorname{Ind}}{\mathcal I}$. Gauge-invariant uniqueness for Fell bundles {#appendix} =========================================== Here we present an abstract “gauge-invariant uniqueness” result for Fell bundles over discrete groups. As applications we obtain gauge-invariant uniqueness results for maximal and for normal coactions. \[GIUTFB\] If $\pi:(A,\delta)\to (B,\varepsilon)$ is a surjective morphism of coactions such that $\pi|_{A_e}$ is injective, then $$\pi\times G:A\times_\delta G\to B\times_\varepsilon G$$ is an isomorphism. Consequently: 1. \[it:epsilon-max\] if $\varepsilon$ is maximal, then $\delta$ is maximal and $\pi$ is an isomorphism; 2. \[it:delta-normal\] if $\delta$ is normal, then $\varepsilon$ is normal and $\pi$ is an isomorphism; 3. \[it:delta-max\] if $\delta$ is maximal, then $\pi$ is a maximalization of $(B,\varepsilon)$, and there is a unique morphism $\varphi:(B,\varepsilon)\to (A^n,\delta^n)$ such that the diagram $$\label{A.1(3)} \xymatrix{ (A,\delta) \ar[dr]^\pi \ar[dd]_{q_A^n} \\ &(B,\varepsilon) \ar@{-->}[dl]^\varphi_{!} \\ (A^n,\delta^n) }$$ commutes, and moreover $\varphi$ is a normalization. 4. \[it:epsilon-normal\] if $\varepsilon$ is normal, then $\pi$ is a normalization of $(A,\delta)$, and there is a unique morphism $\varphi:(B^m,\varepsilon^m)\to (A,\delta)$ such that the diagram $$\label{A.1(4)} \xymatrix{ (B^m,\varepsilon^m) \ar@{-->}[dr]^\varphi_{!} \ar[dd]_{q_A^m} \\ &(A,\delta) \ar[dl]^\pi \\ (B,\varepsilon) }$$ commutes, and moreover $\varphi$ is a maximalization. We first show that $$\pi(A_s)=B_s{\quad\text{for all}\quad}s\in G.$$ Indeed, it is easy to check on the generators that $$\varepsilon_s\circ\pi=\pi\circ\delta_s{\quad\text{for all}\quad}s\in G.$$ Then we have $$\begin{aligned} B_s &=\varepsilon_s(B) \\&=\varepsilon_s(\pi(A)) \\&=\pi(\delta_s(A)) \\&=\pi(A_s).\end{aligned}$$ Since $\pi|_{A_e}$ is injective, it follows that for each $s\in G$ the restriction $\pi|_{A_s}$ maps $A_s$ isometrically onto $B_s$, and hence the associated Fell-bundle homomorphism ${\widetilde}\pi:{\mathcal A}\to{\mathcal B}$ is an isomorphism. The normalization $$\pi^n:(A^n,\delta^n)\to (B^n,\varepsilon^n)$$ of $\pi$ is an isomorphism of coactions, because $A^n\cong C^*_r({\mathcal A})$ and $B^n\cong C^*_r({\mathcal B})$. Let $q^n_A:(A,\delta)\to (A^n,\delta^n)$ and $q^n_B:(B,\varepsilon)\to (B^n,\varepsilon^n)$ be the normalizing maps. We have a commuting diagram $$\xymatrix{ (A,\delta) \ar[r]^-{q^n_A} \ar[d]_\pi &(A^n,\delta^n) \ar[d]^{\pi^n} \\ (B,\varepsilon) \ar[r]_-{q^n_B} &(B^n,\varepsilon^n) }$$ of coaction morphisms, hence a commuting diagram $$\xymatrix{ A\times_\delta G \ar[r]^-{q^n_A\times G}_-\cong \ar[d]_{\pi\times G} &A^n\times_{\delta^n} G \ar[d]^{\pi^n\times G}_\cong \\ B\times_\varepsilon G \ar[r]_-{q^n_B\times G}^\cong &B^n\times_{\varepsilon^n} G }$$ of homomorphisms. Thus $\pi\times G$ is an isomorphism. Now (1)–(4) follow from the theory of maximalizations and normalizations: First of all, (1) and (2) follow immediately from [@clda Proposition 3.1]. For (3), [@BKQ Proposition 6.1.11] shows that $\pi$ is a maximalization. Let $q^n_B:(B,\varepsilon)\to (B^n,\varepsilon^n)$ be the normalization of $(B,\varepsilon)$. Then $q^n_B\circ \pi:(A,\delta)\to (B^n,\varepsilon^n)$ also is a normalization, by [@BKQ Proposition 6.1.7]. Since all normalizations of $(A,\delta)$ are isomorphic, there is an isomorphism $\theta$ making the diagram $$\xymatrix@C+30pt{ (A,\delta) \ar[dr]^\pi \ar[dd]_{q_A^n} \\ &(B,\varepsilon) \ar[d]^{q^n_B} \\ (A^n,\delta^n) &(B^n,\varepsilon^n) \ar@{-->}[l]^-\theta_-{\cong} }$$ commute. Put $\varphi=\theta\circ q^n_B:(B,\varepsilon)\to (A^n,\delta^n)$. Then $\varphi$ is a normalization since $q^n_B$ is and $\theta$ is an isomorphism, and the diagram $$\xymatrix@C+30pt{ (A,\delta) \ar[dr]^\pi \ar[dd]_{q_A^n} \\ &(B,\varepsilon) \ar[d]^{q^n_B} \ar[dl]_\varphi \\ (A^n,\delta^n) &(B^n,\varepsilon^n) \ar[l]^-\theta_-{\cong} }$$ commutes. To see that $\varphi$ is the unique morphism making the diagram commute, suppose that $\varphi'$ is another. Since $q_A^n$ is also a maximalization (by [@BKQ Proposition 6.1.15]) it follows from the theory of maximalization that both $\varphi$ and $\varphi'$ have the same maximalization (namely ${\operatorname{id}}_A$), and hence are equal since the maximalization functor is faithful (by [@BKQ Corollary 6.1.19]). \(4) is proved similarly to (3): [@BKQ Proposition 6.1.7] shows that $\pi$ is a normalization, and if $q^m_A:(A^m,\delta^m)\to (A,\delta)$ is a maximalization then $\pi\circ q^m_A$ is also a maximalization, by [@BKQ Proposition 6.1.11], so there is an isomorphism $\theta$ making the diagram $$\xymatrix@C+30pt{ (B^m,\varepsilon^m) \ar[dd]_{q_B^m} \ar@{-->}[r]^-\theta_-{\cong} &(A^m,\delta^m) \ar[d]^{q^m_A} \\ &(A,\delta) \ar[dl]^\pi \\ (B,\varepsilon) }$$ commute. Then $\varphi:=q^m_A\circ\theta$ is a maximalization of $(A,\delta)$ making the diagram $$\xymatrix@C+30pt{ (B^m,\varepsilon^m) \ar[dd]_{q_B^m} \ar[r]^-\theta_-{\cong} \ar[dr]_\varphi &(A^m,\delta^m) \ar[d]^{q^m_A} \\ &(A,\delta) \ar[dl]^\pi \\ (B,\varepsilon) }$$ commute. To prove that $\varphi$ is the unique morphism making the diagram commute, if $\varphi'$ is another then, since $q_B^m$ is also a normalization (by [@BKQ Proposition 6.1.14]) both $\varphi$ and $\varphi'$ have the same normalization (namely ${\operatorname{id}}_B$), and hence are equal since the normalization functor is faithful (by [@BKQ Corollary 6.1.19]). \[cor:giut-max-coact\] Let $(A, \delta)$ be a maximal coaction and $\pi:A\to B$ a surjective homomorphism. Then $\pi$ is injective if and only if $\pi\vert_{A_e}$ is injective and there is a maximal coaction $\varepsilon$ of $G$ on $B$ such that $\pi$ is $\delta-\varepsilon$ equivariant. The forward direction is immediate. Assume now that $\pi\vert_{A_e}$ is injective and there is a maximal coaction $\varepsilon$ of $G$ on $B$ such that $\pi$ is $\delta-\varepsilon$ equivariant. Then $\pi: (A, \delta)\to (B, \varepsilon)$ is a surjective morphism of coactions. Hence $\pi$ is an isomorphism by Proposition \[GIUTFB\], part . The following is parallel to [Corollary ]{}: \[cor:giut-nor-coact\] Let $(B, \varepsilon)$ be a normal coaction and $\pi:A\to B$ a surjective homomorphism. Then $\pi$ is injective if and only if there is a normal coaction $\delta$ of $G$ on $A$ such that $\pi$ is $\delta-\varepsilon$ equivariant and $\pi\vert_{A_e}$ is injective. The forward direction is immediate. Assume now that there is a normal coaction $\delta$ of $G$ on $A$ such that $\pi$ is $\delta-\varepsilon$ equivariant and $\pi\vert_{A_e}$ is injective. Then $\pi: (A, \delta)\to (B, \varepsilon)$ is a surjective morphism of coactions. Hence $\pi$ is an isomorphism by Proposition \[GIUTFB\], part . \[cor:characterise-normal\] Let $(A, \delta)$ be a coaction. The following are equivalent: 1. \[it:delta-normal-cor\] $\delta$ is normal; 2. \[it:general-giut\] A surjective homomorphism $\pi:A\to B$ is injective if and only if $\pi\vert_{A_e}$ is injective and there is a coaction $\varepsilon$ on $B$ such that $\pi$ is $\delta-\varepsilon$ equivariant. Assume . Let $\pi:A\to B$ be an isomorphism. Then trivially $\pi\vert_{A_e}$ is injective and $\pi$ carries $\delta$ to a (normal) coaction on $B$. If on the other hand $\pi:A\to B$ is surjective, $\pi\vert_{A_e}$ is injective, and $B$ carries a coaction $\varepsilon$ such that $\pi$ is $\delta-\varepsilon$ equivariant, then by Proposition \[GIUTFB\], part  $\pi$ is an isomorphism. This proves $\Rightarrow$. Now assume . Since the normalization map $q_A^n:(A, \delta)\to (A^{n}, \delta^{n})$ is equivariant and satisfies $q_A^n\vert_{A_e}$ is injective, by hypothesis $q_A^n$ is injective. Hence it is an isomorphism, so $\delta$ is normal since $\delta^n$ is. The following is parallel to [Corollary ]{}: \[cor:characterise-max\] Let $(B, \varepsilon)$ be a coaction. The following are equivalent: 1. \[it:epsilon-max-cor\] $\varepsilon$ is maximal; 2. \[it:general-giut-2\] A surjective homomorphism $\pi:A\to B$ is injective if and only if there is a coaction $\delta$ on $A$ such that $\pi$ is $\delta-\varepsilon$ equivariant and $\pi\vert_{A_e}$ is injective. Assume . Let $\pi:A\to B$ be an isomorphism. Then trivially $\pi{^{-1}}$ carries $\varepsilon$ to a (maximal) coaction on $A$ and $\pi\vert_{A_e}$ is injective. 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Operator Theory **64** (2010), 349–376. [^1]: This research was supported by the Research Council of Norway and the NordForsk Research Network “Operator Algebra and Dynamics” (grant \#11580). [^2]: and all our coactions will be full

at (current page.center) [{width="\paperwidth" height="\paperheight"}]{}; [\ ]{} ------------------------------------------------------------------------ [ <span style="font-variant:small-caps;">Academic Year 2018 – 2019</span> ]{} [ ]{} Cumrun Vafa proposed in [@rif1] a new unifying model for the principal series of FQHE which predicts non-Abelian statistics of the quasi-holes. The many-body Hamiltonian supporting these topological phases of matter is invariant under four supersymmetries. In the thesis we study the geometrical properties of this Landau-Ginzburg theory. The emerging picture is in agreement with the predictions in [@rif1]. The $4$-SQM Vafa Hamiltonian is shown to capture the topological order of FQHE and the $tt^{*}$ monodromy representation of the braid group factors through a Temperley-Lieb/Hecke algebra with $q = \pm \exp(\pi i/\nu)$. In particular, the quasi-holes have the same non-Abelian braiding properties of the degenerate field $\phi_{1,2}$ in Virasoro minimal models. Part of the thesis is dedicated to minor results about the geometrical properties of the Vafa model for the case of a single electron. In particular, we study a special class of models which reveal a beautiful connection between the physics of quantum Hall effect and the geometry of modular curves. Despite it is not relevant for phenomenological purposes, this class of theories has remarkable properties which enlarge further the rich mathematical structure of FQHE. The thesis is based on the following papers:\ \ R. Bergamin and S. Cecotti, “ FQHE and $tt^{*}$ geometry ”, arXiv:1910.05022 \[hep-th\], 2019.\ \ R. Bergamin, “$tt^{*}$ Geometry of Modular Curves”, arXiv:1803.00489 \[hep-th\],\ published in JHEP 1908 (2019) 007. Acknowledgements {#acknowledgements .unnumbered} ================ Voglio ringraziare Mamma, Papá e Stefano per tutto quel che hanno fatto per me in questi anni di studi. Un supporto non solo morale, ma anche alimentare. Una menzione speciale va infatti alle torte salate e dolci che mi ritrovavo in valigia ogni domenica sera di ritorno a Trieste. Di questo anche i miei coinquilini ringraziano sentitamente. Ringrazio di cuore tutti i coinquilini con cui ho abitato la Geppa in questi quattro anni intensissimi: Sofy, Bruno, Ane, Ale, Mizu, StronzEle, Milly, Segret, Leo, Fra, Carly e Skodella. Li ringrazio per il loro grande affetto e supporto morale, nonché per tutti i bei momenti trascorsi assieme come una vera famiglia. Ringrazio di cuore anche Giovanni, Sara, Luca, Matteo, Giulia, Alessandro, Andrea e Jenny per la loro amicizia incondizionata. Perché anche se trascorro molto tempo lontano da casa, al mio ritorno trovo sempre delle persone su cui contare.Ringrazio anche Fabio e Cristina per la loro grande ospitalitá. Quella di Fabio é la naturale evoluzione dell’ osmiza: se magna, se beve e se ride, ma al posto del vinello scrauso trovi Whisky americano di grande qualitá. Ci tengo molto a ringraziare i miei colleghi ed amici in SISSA per il tempo trascorso assieme e le interessanti discussioni scentifiche. In particolare ringrazio Giulio Ruzza e Matteo Caorsi, che sono sicuramente i ragazzi che ho importunato di piú con le mie domande. Voglio ringraziare il prof. Sergio Cecotti per avermi fatto conoscere l’argomento del fractional quantum Hall effect ed in generale delle fasi topologiche della materia, che trovo uno dei piú interessanti argomenti della fisica teorica di oggi. Lo ringrazio in particolare di avermi reso partecipe di nuove ed interessanti scoperte al riguardo. Ringrazio anche tutti i professori che mi hanno dato tempo e disponibilitá, oltre al personale della caffetteria e la mensa, sempre allegro, gentile e disponibile. Ringrazio infine i ragazzi della palestra Audace per avermi fatto conoscere il crossfit e per la loro grande simpatia, disponibilitá ed entusiasmo durante gli intensi allenamenti. Come recita il famoso detto: “Mens sana in corpore sano”. Introduction and Overview ========================= The theory of topological phases of matter is one of the most interesting subject in theoretical physics. The remarkable discovery of these quantum states showed the limit of the Landau-Ginzburg theory of phase transitions. The topological universality classes cannot be classified with the concepts of symmetry and symmetry breaking: we can have either states in the same topological class but with different symmetries, or states in inequivalent topological classes but with the same symmetries. These quantum phases of matter encode a new kind of order, the topological order, which is not associated to any local order parameter, but rather to global non-local observables [@wen]. From these considerations one can understand the importance of the discovery of quantum Hall effect. A system of interacting electrons moving on a $2d$ surface in a strong magnetic field at low temperature exhibits very surprising properties like the quantization of the Hall conductivity $$\sigma_{xy}=\frac{e^{2}}{2\pi \hslash}\nu.$$ Initially $\nu$ was found to be integer. The quantization of a physical quantity is not new in quantum mechanics, but in this context it acquires a new meaning. The robustness of the conductivity under deformations of the magnetic field within a certain range reveals the topological nature of the Hall states. Subsequently, it was found that $\nu$ can also assume very specific rational values. The most prominent fractions experimentally are $\nu = 1/3$ and $\nu = 2/5$, but many others have been observed. This quantum number has the interpretation of filling fraction of the lowest Landau levels. It is for a rational $\nu$ that the most interesting phenomena of the quantum Hall effect happen. It turns out that the charged excitations of these systems, the quasi-holes, carry a fraction of the charge of the electron, and, more remarkably, they behave like particles of anyonic statistics. From a theoretical point of view what provides the connection between quantum mechanics and geometry is the Berry’s connection. The definition of this mathematical object arises from the Schroedinger equation when we study the adiabatic evolution of quantum systems. The Berry’s holonomy induced by taking a quasi-hole around another one contains the informations about the statistics of these quantum objects. In chapter \[chapter1\] we briefly review the theory of quantum Hall states and the relation between topological order and anyonic particles.\ Despite the discovery of FQHE [@discovery] dates back more than thirty years ago, saying if the quasi-holes are abelian or non-abelian particles still represents a challenging problem for both theorists and experimentalists. From the theoretical point of view, many models have been developed to explain the observed filling fractions. Among these, the Laughlin’s proposal [@ab1] and the idea of hierarchy states of Haldane and Halperin [@ab2; @ab3], as well as Jain’s composite fermion theory [@ab4], predict abelian anyonic statistics for the principal series of FQHE. Also the possibility of non-abelian statistics has been explored by several models for other filling fractions [@nab1; @nab2; @nab3; @nab4]. More recently, C.Vafa proposed in [@rif1] a unifying model of FQHE which leads to new predictions for the statistics of the quasi-holes. He claims that the effective theory of FQH systems with $$\nu=\frac{n}{2n\pm 1}$$ can be realized in the framework of the AGT correspondence [@rif3]. The compactification of the six dimensional $\mathcal{N}=(2,0)$ theory of $A_{1}$ type on a punctured Riemann surface leads to a correspondence between the Nekrason partition functions [@rif24] of certain $SU(2)$ gauge theories and the correlators of the Liouville CFT. In this set up the punctured Riemann surface is identified with the target manifold of FQHE, where the punctures correspond to insertions of quasi-holes, and the Liouville conformal blocks are the wave functions of FQHE. In Appendix \[classS\] we provide a short review about the Gaiotto theory and the relation between topological string amplitudes and Liouville chiral blocks which inspires the Vafa proposal. This construction motivates in addition a microscopic description of FQHE states in terms of a $\mathcal{N}=4$ supersymmetric Hamiltonian. Choosing the plane as Riemann surface, we have a Landau-Ginzburg model with superpotential $$\label{susymod} \mathcal{W}(z)= \sum_{i=1}^{N}\left( \sum_{a=1}^{n}\log (z_{i}-x_{a})-\sum_{k=1}^{M}\log(z_{i}-\zeta_{k}) \right) + \frac{1}{\nu}\sum_{i<j}\log (z_{i}-z_{j}),$$ where $z_{i},i=1,...,N $ are the electron coordinates and $x_{a},\zeta_{\alpha}$ are respectively the positions of quasi-holes and magnetic fluxes. The term $\log(z-x_{a})$ is the two dimensional Coulombic potential which describes the interaction between an electron and a point-like source at $x_{a}$ with charge $1$, while the term $\sum_{i<j}\log (z_{i}-z_{j})$ keeps track of the Coulomb repulsion between electrons. In order to reproduce the large macroscopically uniform magnetic field we have to consider a uniform distribution of the flux sources $\zeta_{k}$ in $\mathbb{C}$.\ In this thesis we want to study the geometrical properties of the model \[susymod\] and verify the predictions of [@rif1]. In chapter \[wedisc\] we analyze the Vafa Hamiltonian in relation to microscopic physics of FQHE. We show that the degeneracy of the lowest Landau level of an electron moving on a generic Riemann surface in a uniform magnetic field $B$ can be mapped to the degeneracy of a supersymmetric system with four supercharges. Considering the complex plane as an example, in the holomorphic gauge a generic state in the lowest Landau level can be written as $$\Psi(z)= f(z)e^{-B \vert z \vert^{2}},$$ where $f(z)$ is an holomorphic function. In the correspondence the holomorphic part of the wave function defines canonically an element of the chiral ring $\mathcal{R}=\mathbb{C}[z]/\partial W$ of a $\mathcal{N}=4$ LG model with superpotential $W$. This space has the structure of a Frobenious algebra and its elements label the vacua of the supersymmetric system [@rif14]. Moreover, we argue that any Hamiltonian describing the motion in a plane of many electrons coupled to a strong magnetic field is described (at the level of topological order) by Vafa’s $\mathcal{N}=4$ Hamiltonian independently of the details of the interactions between the electrons. This result implies that the LG model with superpotential \[susymod\] represents the correct universality class of the fundamental many-electron theory.\ Varying the parameters in $W$ we get a Berry’s connection $D$ on the bundle of vacua which satisfies a set of equations called $tt^{*}$ geometry [@rif10]. One can define in terms of $D$ the $tt^{*}$ Lax connection $$\nabla_{\zeta}= D+\frac{1}{\zeta}C, \hspace{1cm} \overline{\nabla}_{\zeta}= \overline{D}+ \zeta\overline{C},$$ where $C,\overline{C}$ denotes the action of chiral and antichiral operators on the vacua and $\zeta \in \mathbb{C}^{\times}$ is an arbitrary parameter. The $tt^{*}$ equations can be rephrased as flatness conditions for $\nabla_{\zeta},\overline{\nabla}_{\zeta}$. This connection admits flat sections $\Psi_{i}$ which satisfy [@rif23; @rif11] $$\nabla_{\zeta}\Psi_{i}= \overline{\nabla}_{\zeta}\Psi_{i}=0.$$ The advantage of having a $4$-susy Hamiltonian in the same universality class of FQHE is that we can use the tools of $tt^{*}$ geometry to study the statistics of the quasi-holes. According to the Vafa’s program, the topological order of FQHE is captured by the parallel transport of the flat connection $\nabla_{\zeta}$. We can project the coupling constant space of the model on the configuration space of $N$ identical particles on the plane $$Y_{n}=C_{n}/S_{n},$$ where $$C_{n}=\left\lbrace (x_{1},...,x_{n}) \in \mathbb{C}^{n} \vert x_{i} \neq x_{j} \ \mathrm{for} \ i \neq j \right\rbrace$$ is the space of $n$ ordered distinct points and $S_{n}$ is the permutation group of $n$ objects. Hence, the $tt^{*}$ Lax connection restricted to the vacuum bundle $$\mathcal{V}\rightarrow Y_{n}$$ provides a monodromy representation of the braid group of the quasi-holes $B_{n}=\pi_{1}(Y_{n})$. In the chapters \[reviewtt\],\[chap5\],\[chap6\] we discuss all the necessary tools to compute the $tt^{*}$ monodromy representation of the Vafa model. The chapter \[reviewtt\] contains a review of supersymmetric quantum mechanics and basics of $tt^{*}$ geometry that the reader can find in literature. The chapters \[chap5\],\[chap6\] cover more advanced topics in $tt^{*}$ geometry which are relevant for our problem. In chapter \[chap5\] we discuss the concept of statistics in $tt^{*}$ geometry. In particular, since the electrons are fermionic particles, we study in full generality the fermionic sector of a LG model of $N$ identical particles. We also discuss the formulation of $tt^{*}$ geometry in models with a non simply connected target manifold and multivalued superpotential. The Vafa superpotential \[susymod\] belongs to this family of theories. In order to treat properly these models one has to introduce the concept of covering spaces and extend the vacuum space with the so called “$\theta$-sectors”. The common denominator of the chapter is the $tt^{*}$ functoriality, which turns out to be a very powerful tool to generate isomorphisms between quantum theories. In chapter \[chap6\] we study a special class of $tt^{*}$ geometries which we call\ “ very complete”. These models are quite peculiar. In the language of $2d$ $\mathcal{N}=(2,2)$ theories, being very complete means that all the operators in the chiral ring of the UV conformal fixed point are IR relevant or marginally non-dangereous and there are no wall-crossing phenomena. We show that in this case the UV limit of the Berry connection is an $SL(2,\mathbb{C})$ Knizhnik-Zamolodchikov connection [@rif65; @rif67] on $C_{n}$ with global coordinates given by the critical values of the superpotential $w_{i}$. The UV limit consists in rescaling the critical coordinates $w_{i}\rightarrow \beta w_{i}$ and taking $\beta\rightarrow 0$. In this regime the Lax connection and the Berry connection coincide and, if the model is symmetric under permutations of the $w_{i}$, the UV Berry connection provides a unitary representation of the full braid group $B_{n}$.\ In chapter \[chap7\] we study the $tt^{*}$ geometry of the Vafa Hamiltonian. We show that the model is very complete and symmetric. The corresponding UV Berry connection takes the form of a Kohno connection $$\mathcal{D}=d+ \lambda \sum_{i<j}s_{\ell}^{i}s_{\ell}^{j} \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}},$$ acting on the space $V^{n+M}=\bigotimes_{i} V_{i}$ with $V_{i}\simeq \mathbb{C}^{2},i=1,...,n+M$ and $s_{\ell}^{i}, \ell=1,2,3$ is the $su(2)$ generator acting on the $V_{i}\simeq \mathbb{C}^{2}$ factor, namely $$s_{\ell}^{i}= 1 \otimes....\otimes 1 \otimes \frac{1}{2}\sigma_{\ell} \otimes 1 \otimes ....\otimes 1.$$ It is shown in [@rif65] that the monodromy representation of the flat connection above is a Hecke algebra representation of the braid group $B_{n+M}$ which factorizes through the Temperley-Lieb algebra $A_{n+M}(q)$ with $$q= \exp (\pi i \lambda).$$ One can restrict to the monodromy representation of $B_{n}$ with the projection $p:Y_{n+M}\rightarrow Y_{n}$. It turns out that the parameter $\lambda$ is related to the filling fraction $\nu$ by $$q^{2}=e^{2\pi i /\nu},$$ which gives the two possibilities $q=\pm e^{i\pi/\nu}$. It is believed that $3d$ topological Chern-Simons theories are effective description of FQHE states. Since the braiding of conformal blocks in $2d$ WZW models is the same of the Wilson lines in non-Abelian Chern-Simons theories, it is natural to require the UV Berry connection to be a Knizhnik-Zamolodchikov connection for $SU(2)$ current algebra with level $k$ quantized in integral units, namely $$\lambda=\pm \frac{2}{k+2}, \hspace{0.5cm} k \in \mathbb{Z}.$$ This condition leads to the determine the values of the filling fraction which are consistent with $tt^{*}$ geometry. In the case of $q=e^{i\pi/\nu}$ we get $$\nu= \frac{b}{2 b \pm 1}, \ b \in \mathbb{N}, \hspace{1cm} \nu= \frac{b}{2(b \pm 1)}, \ b \ \mathrm{odd},$$ where the first one corresponds to the principal series of FQHE. From point of view of [@rif1], the element $\sigma_{i}^{2}$ of the pure braid group for the principal series has two distinct eigenvalues, in correspondence with the two different fusion channels of the $\phi_{1,2}$ operator in the minimal $(2n, 2n\pm1)$ Virasoro model. The ratio of the two eigenvalues is $$q^{2}=\frac{\exp [ 2\pi i (h_{1,3}-2h_{1,2} ) ] }{\exp [ 2\pi i (h_{1,1}-2h_{1,2})] }=\exp(2\pi i/\nu).$$ As predicted in [@rif1], the quasi-holes have the same non-Abelian braiding properties of the degenerate fields $\phi_{1,2}$ in Virasoro minimal models. On the other hand, the less natural solution $q=-e^{i\pi/\nu}$ gives other two series of filling fractions. These are respectively $$\nu= \frac{m}{m+2}, \hspace{0.5cm} m=k+2 \in \mathbb{N}\geq 2, \hspace{1cm} \nu= \frac{m}{3m-2}, \hspace{0.5cm} m=k+2\geq 2,$$ where the first series contains the values of $\nu$ corresponding to the Moore-Read [@nab2] and Read-Rezayi models [@nab3].\ The reader can find in appendix other minor results about the geometrical properties of the Vafa model for the case of a single electron. In particular, in \[papermodjhep\] we study a special class of models which reveal a beautiful connection between the physics of quantum Hall effect and the geometry of modular curves. The analysis is based on [@rif54]. Despite it is not relevant for phenomenological purposes, this class of theories has remarkable properties which enlarge further the rich mathematical structure of FQHE. Among the main results, the theorems about the cusps counting and classification are recovered in a physical language. From our investigation of this family of models, the algebraic properties of the modular curves emerge in an elegant manner. Topological Order, FQHE and Anyons {#chapter1} ================================== A New Kind of Order -------------------- The fractional quantum Hall effect opened a new chapter in condensed matter physics. For a long time, before the discovery of the quantum Hall states, Landau’s symmetry breaking theory defined the fundamental paradigm of many-body physics. We know that at sufficiently high temperature matter is in form of gas. In this regime the particles are weakly interacting and the motion of a single constituent is not influenced by the other ones. When temperature decreases the particles become more and more correlated and start to develop a regular pattern. In this regime we observe an emergent collective behaviour revealing an internal structure, which we also call order. The concept of order allows to classify different states of matter and is intimately related to the concept of phase transition. One says that two many-body states has the same order if we can change one states into the other with a smooth deformation of the Hamiltonian. If this is not possible we encounter a phase transition. The definition of order organizes the states of matter in equivalence classes which are said universality classes in the Wilsonian language. According to Landau’s theory there is a deep relation between order and symmetry. Different orders are associated with different symmetries, which are described by local parameters, and a phase transition involves the breaking of some symmetry. This picture is able to explain a large class of experimentally observed states of matter and predict the existence of gaplessness excitations in the materials due to symmetry breaking. The existence of the quantum Hall states reveals the limits of Landau’s paradigm and shows the existence of a new type of order, usually called topological order [@wen]. These states of matter are created by confining electrons on a $2d$ interface between two different semiconductors under strong magnetic fields and low temperature. In this regime the electrons are strongly correlated and behave collectively like a quantum liquid. The topological phases are not classified by symmetries and cannot be described by local order parameters. Indeed, all the quantum Hall states have the same symmetries and a phase transition between them does not involve any symmetry breaking. One can define a topological universality class as a family of Hamiltonians with gapped spectrum which can be smoothly deformed to each other without the emergence of gappless excitations. The topological order can be characterized by global, non-local observables, which are robust under any local perturbations that can break symmetries. These are essentially the degeneracy of the ground state, which depends only on the topology of the target space, and the non-local behaviour of the quasi-hole and quasi-particle excitations. The Quantum Hall Effect: General Setting and Phenomenology ---------------------------------------------------------- We recall some basic facts about the phenomenology of quantum Hall effect. More details can be found in [@wen; @tong] and references there. The Hall effect was originally discovered by Edwin Hall in 1879. The physical set-up is very simple to construct: one has to take a bunch of electrons, restrict them to move on a two dimensional plane and turn on an electric field $\vec{E}$ and magnetic field $\vec{B}$ respectively parallel and orthogonal to the plane. The classical Hall effect is simply the consequence of the motion of charged particles in a magnetic field. In a static regime the electric force acting on the charge carriers is balanced by the Lorentz force of the magnetic field. In natural units we have the condition $$q_{e}\vec{E}= q_{e} \vec{v} \times \vec{B},$$ where $q_{e}=-e$ is the charge of the electron and $\vec{v}$ is the velocity of the particles. Denoting with $n$ the density of the electrons, a current $\vec{j}=n\vec{v}$ is made to flow in the normal direction to the electric field. Moreover, the norms $E,j$ are related by $$\label{ohm} E=j \frac{B}{n q_{e} c}=j \frac{h}{q_{e}^{2}} \frac{1}{\nu}$$ where $h,c$ are respectively the Planck constant and the speed of light, while $$\nu= \frac{nhc}{q_{e}B}= \frac{n}{B/\Phi_{0}}=\frac{\mathrm{number \ of \ particles}}{\mathrm{number \ of \ flux \ quanta }}$$ is the filling fraction. In the above equality we introduced $\Phi_{0}=\frac{hc}{q_{e}}$ as the quantum unit of magnetic flux. Recalling the Ohm’s law $$\vec{E}= \rho \vec{j}$$ where $$\rho= \begin{pmatrix} \rho_{xx} & \rho_{xy} \\ -\rho_{xy} & \rho_{yy} \end{pmatrix}$$ is the resistivity tensor, from \[ohm\] we obtain the relations $$\rho_{xx}=0, \hspace{1cm} \rho_{xy}= \frac{h}{q_{e}^{2}}\frac{1}{\nu}.$$ According to the classical theory, the Hall resistance $\rho_{xy}$ is proportional to the magnetic field at fixed electron density. Indeed, experimentally one finds that $\rho_{xy}\propto B$ at weak fields. However, in the early 1980s it was found that for strong magnetic fields ($\sim 10$ T) and very low temperatures ($\sim 1$ K) the Hall resistance develops a plateau structure as in Figure \[plateau\]. In particular, the filling fraction $\nu$ is quantized in this regime and labels the different plateaus. These new states of matter behave like an incompressible fluid of uniform density. Initially $\nu$ was found to be an integer. This phenomenon is called integer quantum Hall effect (IQHE) and was discovered by von Klitzing in 1980 [@vonk]. In the experiment the electrons are confined on an interface between two different semiconductors which play the role of $2d$ plane. The quantization of a physical quantity is rather common at the microscopic level, but the case of the filling fraction is different from the usual quantization phenomena in quantum mechanics. The profile of the resistivity shows that the quantum Hall states are robust under deformations of the magnetic field within a certain range. Moreover, the plateau spectrum turns out to be indipendent from the local, microscopic details of the material. These features reveal the topological nature of these phases of matter. ![Plateau structure[]{data-label="plateau"}](plateau.png) Subsequently, it was found that $\nu$ can also take very specific rational values. This phenomenon is called fractional quantum Hall effect (FQHE) and was discovered by Tsui and Stormer in 1982 [@discovery]. The most prominent fractions experimentally are $\nu = 1/3$ and $ \nu=2/5 $, but many other fractions have been seen. The majority of them has odd denominator and can be recasted in the so called principal series $\nu=\frac{n}{2n\pm 1}$. It is in these states that the most remarkable things happen. The charged excitations of the Hall fluid, the quasi-holes, carry a fraction of the charge of the electron, as if the electron split itself into several pieces. It is not just the charge of the electron that fractionalises: this happens to the statistics of the electron as well. It is known that quantum particles in two spatial dimensions can have statistics which does not correspond to the classical bosonic or fermionic one. After the braiding of two identical particles the wave function of the many-body system picks up a phase $e^{i\theta}$, where the parameter $\theta$ determines the statistics. The statistics of the particles in two dimensions are $$\begin{cases} \theta=0 \hspace{1.8 cm} \mathrm{boson} \\ 0< \theta < \pi \hspace{1cm} \mathrm{anyon}\\ \theta= \pi \hspace{1.8cm} \mathrm{fermion}\end{cases}$$ In more complicated examples even this description breaks down: the resulting objects are called non-Abelian anyons and provide physical realization of non-local entanglement in quantum mechanical systems. In this case the Hilbert space of the system is degenerate and a braiding operation induces a unitary transformation on a generic state. While the fractional charge of quasi-holes has been measured experimentally, a direct detection of their statistics is more challenging. It is confirmed that the quasi-holes of the fractional Hall states do not posses classical statistics, but if they are Abelian or non-Abelian anyons is still an open question. Landau Levels {#eleclandau} ------------- The quantum Hall effect is based at the microscopic level on the dynamics of charged particles in a magnetic field. The Hamiltonian of a quantum particle in a constant, uniform magnetic field has discrete spectrum. The structure of the energy levels, known as Landau levels, allows to describe the plateaus at integer values of the filling fraction [@wen]. In particular we are interested in the lowest Landau level, namely the ground state. We ignore for the moment the Coulomb interaction and assume the electrons to be spin polarized. An electron of mass $m$ in a uniform magnetic field is described by the Hamiltonian $$H=-\frac{1}{2m} (\partial_{i}-iq_{e}A_{i})^{2}, \hspace{1cm} \hslash=c=1,$$ where we choose the symmetric gauge $$(A_{x},A_{y})= \frac{B}{2} (-y,x), \hspace{1cm} B= \partial_{x}A_{y}-\partial_{y}A_{x}.$$ To find the energy levels is convenient to introduce the complex coordinate $z=x+iy$ and the holomorphic and anti-holomorphic derivatives $$\partial_{z}= \frac{1}{2}(\partial_{x}-i\partial_{y}), \hspace{1cm} \partial_{\bar{z}}= \frac{1}{2}(\partial_{x}+i\partial_{y}).$$ One can rewrite the Hamiltonian in complex coordinates as $$H=-\frac{1}{m}\left( D_{z}D_{\bar{z}}+D_{\bar{z}}D_{z}\right)$$ where $D_{z},D_{\bar{z}}$ are the covariant derivatives $$\begin{split} & D_{z}= \partial_{z}-iq_{e}A_{z}, \hspace{1cm} A_{z}= \frac{1}{2}(A_{x}-iA_{y})=\frac{B}{4i}\bar{z}, \\ & D_{\bar{z}}= \partial_{\bar{z}}-iq_{e}A_{\bar{z}}, \hspace{1cm} A_{\bar{z}}= \frac{1}{2}(A_{x}+iA_{y})=-\frac{B}{4i}z. \end{split}$$ One can exploit the commutation relation $\left[D_{z},D_{\bar{z}} \right]=\frac{1}{2}m\omega_{B}$, where $\omega{B}=q_{e}B/m$ is the cyclotron frequency, to simplify the expression of $H$ as $$H=-\frac{2}{m} D_{z}D_{\bar{z}} + \frac{1}{2}\omega_{B}.$$ The constant $E_{0}=\frac{1}{2}\omega_{B}$ is the energy of the first Landau level. A wave function $\Psi(z,\bar{z})$ in this subspace satisfies the condition $$D_{\bar{z}}\Psi(z,\bar{z})=0.$$ Assuming $q_{e}B>0$ we introduce the magnetic length $l_{B}$ defined by $$l_{B}^{2}= \frac{1}{q_{e}B}$$ which represents the characteristic length scale governing the magnetic phenomena in the quantum regime. The Hamiltonian above is conjugated by the relation $$H=e^{-\vert z \vert^{2}/4l_{B}^{2}} \tilde{H} e^{+\vert z \vert^{2}/4l_{B}^{2}}$$ to the operator $$\tilde{H}=-\frac{2}{m} \tilde{D}_{\bar{z}}\tilde{D}_{z} + E_{0}$$ where $$\begin{split} & \tilde{D}_{z}= e^{+\vert z \vert^{2}/4l_{B}^{2}} D_{z} e^{-\vert z \vert^{2}/4l_{B}^{2}} = \partial_{z}-\frac{1}{2l_{B}^{2}}\bar{z}, \\ & \tilde{D}_{\bar{z}}= e^{+\vert z \vert^{2}/4l_{B}^{2}} D_{\bar{z}} e^{-\vert z \vert^{2}/4l_{B}^{2}} = \partial_{\bar{z}}. \end{split}$$ We note that the non unitary transformation $e^{+\vert z \vert^{2}/4l_{B}^{2}}$ makes the anti-holomorphic part of the gauge connection vanishing. Rewriting a generic state in the Hilber space as $$\Psi(z,\bar{z})= \tilde{\Psi}(z,\bar{z}) e^{-\vert z \vert^{2}/4l_{B}^{2}},$$ where $\tilde{\Psi}(z,\bar{z})$ is a smooth function, the condition to be in the lowest Landau level becomes $$\tilde{D}_{\bar{z}}\tilde{\Psi}(z,\bar{z})= \partial_{\bar{z}}\tilde{\Psi}(z,\bar{z})=0,$$ namely $\tilde{\Psi}(z,\bar{z})$ must be an holomorphic function on the complex plane. A natural basis of holomorphic functions is given by the monomials $z^{k}$ with $k\geqslant 0$. Hence, a basis of vacua is $$\Psi_{0;k}= z^{k}e^{-\vert z \vert^{2}/4l_{B}^{2}}.$$ These wave functions have a circular shape and $k$-th state is peaked at the radious $r_{k}=\sqrt{2k}l_{B}$. Noting that the ring of the $k$-th vacuum encloses $m$ flux quanta $\Phi_{0}=2\pi l_{B}^{2}$, we learn that there is one state for every flux quantum. Hence, the number of states in the lowest Landau level is equal to the number of flux quanta $\Phi/\Phi_{0}$, where $\Phi$ is the magnetic flux of $B$. One can obtain the wave functions corresponding to the $n$-th excited Landau levels as follows $$\Psi_{n;k}= D_{z}^{n} \Psi_{0;k},$$ where the energy of the $n$-th state is $E_{n}=(\frac{1}{2}+n)\omega_{B}$. It is clear that all the Landau levels have the same degeneracy.\ In the case of $N$ free electrons in a uniform magnetic field one has to fill the Landau levels with Fermi statistics. It is not difficult to guess that for a quantum Hall state at an integer filling fraction $\nu=n \in \mathbb{N}$ the first $n$ Landau levels are completely filled. The finite gap $\Delta=\omega_{B}$ for the excitations explains why at low temperature we observe these plateau, while the Pauli exclusion principle explains the incompressibility of the Hall fluid. The wave function for the $\nu=1$ state is $$\Psi(z_{1},...,z_{N})= \begin{vmatrix} 1 & 1 & ... \\ z_{1} & z_{2} & ... \\ z_{1}^{2} & z_{2}^{2} & ...\\ . & . & . \end{vmatrix} e^{-\sum_{i}\vert z_{i} \vert^{2}/4l_{B}^{2}}= \prod_{i<j}(z_{i}-z_{j})e^{-\sum_{i}\vert z_{i} \vert^{2}/4l_{B}^{2}}$$ where $N$ must be equal to the number of flux quanta. The above wave function describes a circular droplet with radious $R=2\pi N l_{B}^{2}$ and uniform density. Indeed, each electron occupies an area $\pi r_{k+1}^{2}-\pi r_{k}^{2}=2\pi l_{b}^{2}$ and so the density has the constant value $\rho=1/2\pi l_{B}^{2}$. For magnetic fields $B \sim 1 \ T$ we have $\ell_{B}\sim 10^{-8} m$. In quantum Hall effect experiments the length scale of the sample is of order $R\sim cm $ and so the typical number of electrons is $N\sim 10^{12}$. The Laughlin Wave Function {#laughlin} -------------------------- In order to describe the fractional states one can try to apply the logic that we used in the integer case. Let us consider the states with $ 0<\nu<1$. For such values of the filling fraction we naturally conclude that the lowest Landau level is partially filled by the electrons, whose number must be $N=\nu \mathcal{N}$, where $\mathcal{N}=\Phi/\Phi_{0}$ is the capacity of the lowest Landau level. It is immediate to see that the free electrons description cannot work in the fractional case. Indeed, if we fill a fraction of the lowest Landau level with Fermi statistics, in the thermodynamical limit we get a hugely degenerate ground state. The number of states is $\begin{pmatrix} \mathcal{N} \\ \nu \mathcal{N} \end{pmatrix}$, which is approximately $ \left( \frac{1}{\nu}\right) ^{\nu \mathcal{N}} \left( \frac{1}{1-\nu}\right) ^{(1-\nu) \mathcal{N}}\sim 10^{10^{12}} $ for $\mathcal{N}\sim 10^{12}$. Hence, it is clear that the fractional quantum Hall states are strongly correlated system in which the Coulomb interaction plays a fundamental role. This partially removes the degeneracy of the lowest Landau level in such a way that we can observe plateaus at fractional valules of $\nu$. Solving the Schoredinger equation and find the exact wave function for these systems is hopeless. The first approach to describe these plateaus is due to Laughlin [@ab1]. He proposed a wave function for the $\nu=\frac{1}{m}$ states, with $m$ an odd integer. His guess is based on some physical considerations. First of all, any wave function in the lowest Landau level must have the form $$\Psi(z_{1},...,z_{N})= f(z_{1},...,z_{N})e^{-\sum_{i}\vert z_{i} \vert^{2}/4l_{B}^{2}},$$ where $f(z_{1},...,z_{N})$ must be holomorphic and anti-symmetric under the exchange of any two particle positions because of the Fermi statistics. Moreover, this wave function should describe a fluid of uniform density on a disk of radious $R=\sqrt{2mN}l_{B}$, compatibly with the fact that the number of states in the full Landau level is $\mathcal{N}=mN$. The Laughlin proposal is $$\label{Laugh} \Psi(z_{1},...,z_{N})= \prod_{i<j}(z_{i}-z_{j})^{m}e^{-\sum_{i}\vert z_{i} \vert^{2}/4l_{B}^{2}}.$$ We see that the highest power for each electron coordinate appearing in $f(z_{1},...,z_{N})$ is $m(N-1)$. This means that the maximum radious for each particle is $R\sim \sqrt{2mN}l_{B}$, where we can replace $N-1$ with $N$ in the thermodynamical limit. Correspondingly, the area of the droplet is $A \sim 2\pi m N l_{B}^{2}$ and the density of the Hall fluid is $\rho=\frac{1}{2\pi m l_{B}^{2}}$. The Laughlin wave function captures the relevant physics of the fractional quantum Hall states at the filling fractions $\nu=1/m$, but is not the exact ground state of the many-body Hamiltonian. However, it is possible to construct an Hamiltonian whose ground state is precisely given by the Laughlin state [@tong]. According to the Laughlin proposal, this Hamiltonian should be in the same universality class of the FQHE Hamiltonian for $N=\nu\mathcal{N}$ electrons and support the same topological phase as ground state. This motivates why one can use the above wave function to describe the $\nu=1/m$ states rather than the exact one.\ Following the Laughlin proposal, other models have been developed to explain and describe the other filling fractions, such as the construction of Hierarchy states proposed by Haldane and Halperin [@ab2; @ab3]. We will explain this idea in section \[effective\]. Berry’s Connection ------------------ Topological phases encode the non trivial low energy dynamics of certain gapped systems below the energy scale of the minimal excitation. By definition they cannot have local propagating degrees of freedom and what distinguish them from the trivial phase are their geometrical properties. The mathematical object which puts in relation quantum mechanics with geomery and topology is the Berry’s connection [@berry]. Since it plays a central role in the study of FQHE we briefly recall its definition (see [@tong; @wilczek] for a more detailed review). Let us consider an Hamiltonian $$H(x_{i};\lambda_{j})$$ where $x_{i}$ are the degrees of freedom of the system, as for instance the positions and spins of the particles, and $\lambda_{j}$ is a set of parameters determined by some external apparatus that an observer can vary. These parameters can be thought as coordinates on a space of couplings $\mathcal{M}$ which parametrizes a family of theories. We assume that, apart from some isolated critical points, for every choice of $\lambda_{j}$ the spectrum of the Hamiltonian is gapped and the ground state has degeneracy $N\geqslant 1$. One defines the vacuum bundle $\mathcal{V} \rightarrow \mathcal{M}$ where the fiber $\mathcal{V}_{\lambda}$ at a point $\lambda_{j}$ is the vacuum space of $H(x_{i};\lambda_{j})$. Let us a set a local frame $${{\,| {a;\lambda} \rangle}}, \hspace{1cm} a=1,...,N.$$ The definition of the Berry’s connection arises when we study the adiabatic evolution of the system. The adiabatic theorem states that, if we prepare the quantum system in a certain vacuum ${{\,| {a;\lambda} \rangle}}$ for certain $\lambda_{j}$ and vary slowly the parameters compared to the energy gap of the excitations, at every point $\lambda_{j}^{\prime}$ of the path in the space of couplings the perturbed state ${{\,| {\psi_{a}(\lambda^{\prime})} \rangle}}$ is in the ground state $\mathcal{V}_{\lambda^{\prime}}$. More precisely, the adiabatic ansatz for the evolved state is $$\label{ansatz} {{\,| {\psi_{a}(\lambda^{\prime})} \rangle}}= U_{ab}(\lambda^{\prime}){{\,| {b;\lambda^{\prime}} \rangle}}$$ where $U\in U(N)$ since time evolution in quantum mechanics is unitary. We see from this formula that a vacuum state is parallely transported along a certain in path in the space of couplings. Plugging the adiabatic ansatz in the Schoredinger equation provides the definition of a unitary connection on the vacuum bundle which generates the time evolution. This is the Berry’s connection $$(A_{i})_{ba}=-i {{\langle {a;\lambda} |\,}} \partial_{\lambda_{i}}{{\,| {b;\lambda} \rangle}}.$$ The unitary matrix introduced in \[ansatz\] contains also the dynamical phase $e^{-i\int d\lambda E_{o}(\lambda)}$ which we always have in the time evolution of a quantum state. One can ignore this contribution or simply set $E_{o}(\lambda)=0$ by adding a constant to the Hamiltonian. It is interesting in general to evaluate the transformation of a state after a closed loop in the space space of couplings. The associated unitary matrix is the Berry’s holonomy $$\label{holonomy} U=\mathcal{P} \exp \left( -i \oint A_{i}d\lambda^{i} \right),$$ where $\mathcal{P}$ stands for path ordering. In the case of degenerate ground state the Berry’s connection is non-Abelian and the associated holonomy group is a subgroup of $U(N)$ with $N>1$. If the system has a unique vacuum state ${{\,| {\lambda} \rangle}}$, the Berry’s connection is Abelian and the holonomy \[holonomy\] reduces to the Berry’s phase $$e^{i\gamma}= \exp \left( -i\oint A_{i}(\lambda)d\lambda^{i} \right),$$ with $A_{i}(\lambda)= -i {{\langle {\lambda} |\,}} \partial_{\lambda_{i}}{{\,| {\lambda} \rangle}}$. Anyonic Particles ----------------- Topological phases are beleaved to support anyonic particles. These have not to be thought as fundamental particles existing in nature. Indeed, they are artificially realized as charge excitations of quantum liquids. The fusion and braiding rules of anyons are among the geometrical observables which characterize the topological order. Moreover, as we will discuss in the next section, the existence of anyonic particles is deeply related to the topological degeneracy of the ground state. Given their importance in the phenomenology of topological phases, we spend some words to recall some basic properties about a theory of anyons. We know that in $d=3$ spatial dimensions we can have only fermionic and bosonic statistics. This is related to the topology of the configuration space of $N$ identical particles in three dimensions. Denoting with $r_{i},i=1,...,N$ the positions of the particles in $\mathbb{R}^{3}$, the space is $\left( \mathbb{R}^{3N}\setminus\left\lbrace r_{i}=r_{j} \right\rbrace \right)/ \mathcal{S}_{N} $, where $\mathcal{S}_{N}$ is the permutation group of $N$ objects. The fundamental group of this space is $\mathbb{Z}_{2}$. This means that all the possible ways to exchange two particles are topologically equivalent, while a double exchange is topologically trivial. In quantum mechanics the exchange of particles acts as unitary operator on the wave function $\psi(r_{1},r_{2},...,r_{N})$. Hence, the statistics are in correspondence with unitary irreducible representations of $\mathbb{Z}_{2}$. These are one dimensional and the unique possibilities are given by $$\psi(r_{1},r_{2},...,r_{N})= \pm \psi(r_{2},r_{1},...,r_{N})$$ where $\pm 1$ corresponds respectively to the bosonic and fermionic statistics.\ In two spatial dimensions the classification of statistics is much richer. Let us consider $N$ particles on the plane sitting along a line and order them. The image in space-time of the particle worldlines under an exchange of their order is called a braid. One distinguishes braids by their topological class, which means that two braids are considered the same if we can smoothly change one into the other without crossing of the worldlines. Such braidings form an infinite group called braid group $B_{N}$. This coincide with the fundamental group of the confirguration space of $N$ identical particles on the plane. This is $\left( \mathbb{C}^{N}\setminus\left\lbrace z_{i}=z_{j}\right\rbrace \right) / \mathcal{S}_{N}$, where $z_{i}$ are the coordinates of the particles in $\mathbb{C}$. The braid group of $N$ objects is generated by $N-1$ operators $R_{1},...,R_{N-1}$, where $R_{i}$ exchanges the $i$-th and $(i+1)$-th particles in an anti-clockwise direction. These satisfy the defining relations $$\label{yb1} R_{i}R_{j}=R_{j}R_{i} \hspace{1cm} \vert i-j \vert >2$$ and the Yang-Baxter equations $$\label{yb2} R_{i}R_{i+1}R_{i}= R_{i+1}R_{i}R_{i+1}, \hspace{1cm} i=1,...,N-1.$$ Different statistics in two dimensions are associated with unitary irreducible representations of the braid group. Particles that form a one dimensional representation of $B_{N}$ are called Abelian anyons. In this case the braid matrices act on the wave function as phases $$R_{i}=e^{i\pi \alpha_{i}}.$$ The Yang-Baxter equations for these operators require that $e^{i\pi \alpha_{i}}=e^{i\pi \alpha_{i+1}}$, implying that all identical particles have the same phase. If $\alpha=0,1$ we recover the classical statistics, while for $0<\alpha<1$ we have fractional statistics.\ The most general case is given by non-Abelian anyons. The Hilbert space of these objects is a non-Abelian representation of the braid group and can be characterized in an abstract way with the idea of fusion. One introduce a set of variables $a,b,c....$ which denote the different type of anyons in our theory. These objects satisfy a multiplicative algebra, usually called fusion algebra, which reads $$a\star b= \sum_{c}N_{ab}^{c} c$$ where $N_{ab}^{c}$ are non-negative integers telling how many different ways there are to get the anyon $c$ when we bring together two anyons $a$ and $b$. The fusion algebra is commutative and associative, implying that the order in which we fuse anyons is irrelevant. Each anyonic species identifies an irreducible representations of the braid group. Hence, the Hilbert of two anyons $a,b$ decomposes in irreducible representations of dimension $N_{ab}^{c}$ corresponding to the fusion channels $c$ appearing in $a\star b$. The anyons are said non-Abelian if $N_{ab}^{c}\geq 2$ for some $c$. The vacuum of the theory corresponds to the identity element of the algebra $1$ and satisfies $$a\star 1=a$$ for any $a$. This relation implies that the Hilbert space of a single anyon is always one dimensional. Hence, we see that a single anyonic particle does not have any internal degree of freedom. The information contained in the Hilbert space $H_{ab}$ of two non-Abelian anyons $a,b$ is a property of the pair and cannot be associated to a single particle. We learn that anyonic particles are mutually non local objects and the corresponding Hilbert space is said topological since the stored information is related to global properties of the system, namely how the anyons braid. Charged Excitations of The Laughlin State ----------------------------------------- The Berry’s connection allows to study the topological properties of the quantum Hall states and in particular the physics of the charged exctitations. One can introduce $M$ quasi-holes in the Hall fluid at positions $\zeta_{a}, a=1,...,M$. These are created experimentally by inserting thin solenoids in the fluid and have a three dimensional nature of magnetic fluxes. However, from the $2d$ perspective they behave like point like particles. A Laughlin state $\nu=1/m$ with $M$ quasi-holes is described by the wave function $$\Psi(z_{1},...,z_{N};\zeta_{1},...,\zeta_{M})=\prod_{i,a}(z_{i}-\zeta_{a}) \prod_{i<j}(z_{i}-z_{j})^{m}e^{-\sum_{i}\vert z_{i} \vert^{2}/4l_{B}^{2}}.$$ A remarkable property of the quasi-holes is that they carry a fraction of the electron charge. If the elecron has charge $q_{e}=-e$, these particles have charge $e^{\ast}=e/m$. An euristic justification of this fact is that putting $m$ quasi-holes in the same position is equivalent to fix the position of one electron, making it non-dynamical and creating a hole in the fluid. Hence, if we inject a hole into the quantum Hall fluid, this splits into $m$ independent quasi-particles.\ A more rigorous way to show the fractionalization of the electron charge requires to compute the Berry’s connection associated with the Laughlin state. One can interpret the positions of the quasi-holes $\zeta_{j}$ as coordinates in the space of couplings and denote with ${{\,| {\zeta_{1},...,\zeta_{M}} \rangle}}$ a frame on the vacuum bundle which satisfies ${\ensuremath{\langle z_{1},...,z_{N} | \zeta_{1},...,\zeta_{M} \rangle}}= \Psi(z_{1},...,z_{N};\zeta_{1},...,\zeta_{M})$. The calculation of the Berry’s connection is done in detail in [@tong] in the normalized frame $${{\,| {\psi} \rangle}}=\frac{1}{\sqrt{Z}}{{\,| {\zeta_{1},...,\zeta_{M}} \rangle}}$$ where $Z={\ensuremath{\langle \zeta_{1},...,\zeta_{M} | \zeta_{1},...,\zeta_{M} \rangle}}$. It is assumed in the computation that the quasi-holes are not brought too close to each other. The Abelian connection over the configuration space of $M$ quasi-holes is $$A_{\zeta_{i}}=-\frac{i}{2m}\sum_{j\neq i} \frac{1}{\zeta_{i}-\zeta_{j}} + \frac{i\bar{\zeta_{i}}}{4 m l_{B}^{2}}, \hspace{1cm} A_{\bar{\zeta_{i}}}= \overline{A_{\zeta_{i}}}.$$ From the Berry connection one can exctract several informations about the physics of these particles. If we move a quasi-hole $\zeta$ along a closed path $C$ which do not enclose any other particle the corresponding Berry phase is $$e^{i\gamma}= \exp \left( -i\oint_{C} A_{\zeta}d\zeta + A_{\bar{\zeta}}d\bar{\zeta} \right) = \exp \left( i\frac{e\Phi }{m} \right) ,$$ where $\Phi$ is the magnetic flux enclosed by $C$. In this case the phase $\gamma$ has the interpretation of Aharonov-Bohm phase $\gamma=e^{\ast}\Phi$ picked up by a charged particle in a magnetic field. Hence, we see that a quasi-hole carries fractional charge $e^{\ast}=e/m$.\ Another remarkable property of the quasi-holes of the Laughlin state is that they behave like Abelian anyons. The statistics is encoded in the flat part of the connection. The corresponding Berry phase provides a representation of the braid group of $M$ quasi-holes. Ignoring the Aharonov-Bohm contribution, if we take a quasi-hole in $\zeta_{1}$ along a closed loop $C$ encircling another quasi-hole in $\zeta_{2}$ we get the following phase $$e^{i\pi \alpha}=\exp \left( - \frac{1}{2m}\oint_{C} \frac{d\zeta_{1}}{\zeta_{1}-\zeta_{2}} + \mathrm{h.c} \right) =e^{2\pi i/m}.$$ Mooving a particle around another one is equivalent to exchange them two times. Hence, the phase picked up by the wave function after a single braid is $$\alpha=\frac{1}{m}.$$ In the case of $m=1$, namely the integer plateau at $\nu=1$, the quasi-holes behave like fermionic particles, while in a generic Laughlin state with $m>1$ the quasi-holes have Abelian fractional statistics.\ An important property of fractional quantum Hall states is also the topological degeneracy of the ground state, namely the number of ground states depend on the topology of the manifold. This is one of the main future of topological phases and is deeply related to the existence of anyonic particles. This feature becomes manifest if we put the system on a compact manifold of non trivial topology. Let us put the system on a torus and consider the following process. We create from the vacuum a quasi-particle – quasi-hole pair, where the quasi particle is a charged excitation of the Hall fluid with opposite charge $e^{\ast}=-e/m$. We then separate the particles and take them around one of the two different cycles of the torus before making them annihilate. We denote with $T_{1},T_{2}$ the operators which implement the process for the first and second cycle of the torus. Composing these operations one can see that $T_{1}T_{2}T_{1}^{-1}T_{2}^{-1}$ is equivalent to take one anyon around another. Hence, the torus generators satisfy the algebra $$T_{1}T_{2}=e^{2\pi i/M}T_{2}T_{1}.$$ The Hilbert space must be an irreducible representation of this algebra and has dimension $m$. The generalisation of this argument to an arbitrary genus-g Riemann surface shows that the ground state must have degeneracy $m^{\mathrm{g}}$. This number depends only on the topology and not on the local details of the manifold. One can also construct the analog for the Laughlin states on the torus using the Jacobi theta functions, checking that there are exactly $m$ ground states. We note also that the integer plateau given by $m=1$ corresponds to the trivial topological phase. Effective Theory of Laughlin States ----------------------------------- Different quantum Hall states support inequivalent theories of anyonic particles, with different fractional charge and fractional statistics. This means that, despite these states have the same symmetry, they belong to different quantum phases and have different topological order. An efficient and systematic way to study topological order is to construct an effective low energy theory for the FQH states. This captures the topological properties of FQH liquids and provide a universal characterization of topological order. To construct this theory one follows the hydrodynamical approach [@wen]. Below the scale of the minimal excitation, electrons in a strong magnetic field and low temperature form a quantum liquid which we describe with a vector field $J^{\mu}, \mu=0,1,2$, where $J^{0}$ is the electron charge density and $J^{i},i=1,2$ are the components of the electron current density. A Laughlin state $\nu=1/m$ has charge density $J^{0}=-e/2\pi l_{B}^{2}= e^{2} \nu B/2\pi$ and the charge current induced by an external electrostatic field $E_{i}$ is $J^{i}=\sigma_{xy} \epsilon^{ij} E_{j}$, where $\sigma_{xy}= \frac{\nu e^{2}}{2\pi}$ is the Hall conductance. Hence, the vector field $J_{\mu}$ has the following response to a variation of the electromagnetic field $$\delta J^{\mu}=\sigma_{xy}\epsilon^{\mu\nu\lambda}\partial_{\nu}\delta A_{\lambda}.$$ We want to write an action giving the above equation as equation of motion. The electron density can be parametrized in term of a $U(1)$ gauge field $a_{\mu}$ as $$\label{current} J^{\mu}=\frac{e^{2}}{2\pi}\partial_{\nu} a_{\lambda} \epsilon^{\mu \nu \lambda}.$$ This current satisfies automatically the conservation law $\partial_{\mu}J^{\mu}=0$. The gauge field $a_{\mu}$ is the emergent topological degree of freedom of the FQH liquid. The Lagrangian describing the dynamics of this field and reproducing the above equation of motion is $$\mathcal{L}(a_{\mu}, A_{\nu})=-\frac{e^{2} m}{4\pi}a_{\mu}\partial_{\nu}a_{\lambda} \epsilon^{\mu\nu\lambda} + \frac{e^{2}}{2\pi}A_{\mu}\partial_{\nu}a_{\lambda}\epsilon^{\mu \nu \lambda}.$$ The first piece of the Lagrangian is the well known Abelian Chern-Simons term. As one could expect, the effective theory describing the geometrical properties of FQH states is a topological field theory in $2+1$ dimensions. The identification of the inverse filling fraction $\frac{1}{\nu}=m$ with the Chern-Simons coupling, which is quantized by the requirement of gauge invariance at quantum level, leads to a correspondence between Chern-Simons levels and FQH states.\ To provide a complete description of the FQH liquids one has to include in the Lagrangian the quasi-hole and quasi-particle excitations. This requires to introduce another current $j_{\mu}$ which couples to $a_{\mu}$. The Lagrangian gains a new term $$\Delta \mathcal{L}= a_{\mu}j^{\mu}.$$ The gauge invariance of the action is preserved if the current is conserved: $\partial_{\mu}j^{\mu}=0$. Turning off the electromagnetic field $A_{\mu}$, the equation of motion is $$\frac{e^{2}}{2\pi}f_{\mu \nu}= \frac{1}{m} \epsilon_{\mu\nu\rho} j^{\rho}.$$ Let us place a static quasi-hole in the origin. The current is given by $j^{0}=e\delta^{2}(x), j^{1}=j^{2}=0$. The equation of motion becomes $$\frac{1}{2\pi} f_{12}=\frac{1}{em} \delta^{2}(x).$$ We see that the effect of the Chern-Simons term is to attach a magnetic flux $$\Phi=\frac{2\pi}{em}$$ to a particle of charge $q=e$. Consequently, If we take a quasi-hole around another one we get an Aharonov-Bohm phase $e^{iq \Phi}=e^{2\pi i/m}$, which we interpret as the statistical phase generated by a double exchange of the particles. This result agrees with the fractional statistics that we find for the the Laughlin state $\nu=1/m$. Using the definition of electromagnetic current density in \[current\] we also get the equality $$J^{0}=\frac{e^{2}}{2\pi}f_{12}= \frac{e}{m}\delta^{2}(x),$$ from which we correctly the recover the fractionalization of the quasi-hole charge $e^{\ast}=\frac{e}{m}$.\ Recent reviews about the Chern-Simons approach to FQHE can be found in [@tong; @witten]. Other Filling Fractions {#effective} ----------------------- So far we discussed only FQH liquids at the fractions $\nu=\frac{1}{m}$. As we anticipated in section \[laughlin\] one can describe the other plateaus with the idea of hierarchical states proposed by Haldane and Halperin. One starts with a Laughlin state at $\nu=1/m$ and vary the magnetic field to make the system change the plateau. Given their nature of magnetic fluxes, we can increase or decrease the magnetic field by injecting respecticely quasi-particles or quasi-holes in a $\nu=1/m$ state such that they form themselves a Laughlin condensate. The effective Lagrangian of the model is modified as follows [@tong; @wen]. Let us set $e=1$ for simplicity. One writes the quasi-holes current as $$\label{type} j_{\mu}= \frac{1}{2\pi}\epsilon^{\mu\nu\rho}\partial_{\nu}\tilde{a}_{\rho},$$ where $\tilde{a}_{\rho} $ is another emergent Chern-Simons field describing the quasi-hole condensate. The current couples to $a_{\mu}$ and the new Lagrangian reads $$\label{previousmodel} \mathcal{L}(a_{\mu},\tilde{a}_{\nu})= \frac{1}{2\pi}A_{\mu}\partial_{\nu}a_{\lambda}\epsilon^{\mu \nu \lambda}-\frac{m}{4\pi}a_{\mu}\partial_{\nu}a_{\lambda} \epsilon^{\mu\nu\lambda} + \frac{1}{2\pi}a_{\mu}\partial_{\nu}\tilde{a}_{\lambda}\epsilon^{\mu \nu \lambda} - \frac{\tilde{m}}{4\pi}\tilde{a}_{\mu}\partial_{\nu}\tilde{a}_{\lambda},$$ where $\tilde{m}$ is an integer. To compute the Hall conductivity one has to solve the equations of motion for $\tilde{a}_{\mu}$ and then for $a_{\mu}$. One finds that this theory describes a Hall state with filling fraction $$\nu= \frac{1}{m-\frac{1}{\tilde{m}}}.$$ We can compute the charge and statistics of quasi-holes in this new state. There are two type of excitations for this fluid, the ones which couple to $a_{\mu}$ and the ones which couple to $\tilde{a}_{\mu}$. For the first type we find $$m f_{12}-\tilde{f}_{12}= 2\pi \delta^{2}(x), \hspace{1cm} \tilde{m}\tilde{f}_{12}-f_{12} =0,$$ from which we get $$f_{12}= \frac{2\pi}{m-\frac{1}{\tilde{m}}} \delta^{2}(x).$$ For the second type we have $$m f_{12}-\tilde{f}_{12}=0 \hspace{1cm} \tilde{m}\tilde{f}_{12}-f_{12}= 2\pi \delta^{2}(x)$$ which gives $$f_{12}= \frac{2\pi}{ m\tilde{m}-1}.$$ If we set for instance $m = 3 $ and $\tilde{m} = 2$ we find the state $\nu=2/5$, which is among the most prominent among the observed plateau. The resulting charges of the quasi-holes are $e^{\ast} = 2/5$ and $e^{\ast} = 1/5$, which have been measured experimentally. One can now repeat this construction: the quasi-particles of the new state form a condensate described another gauge field, for which we introduce a new Chern-Simons term and current of type \[type\] which couples to $\tilde{a}_{\mu}$. Iterating this procedure we obtain Abelian quantum Hall states with filling fraction $$\nu=\frac{1}{m-\frac{1}{\tilde{m_{1}} - \frac{1}{\tilde{m_{2}}-....}}}.$$ Using the idea of hierarchy one can write down the most general Abelian quantum Hall state [@wen; @tong]. We consider $N$ emergent gauge fields $a_{\mu}^{i}$, with $i=1,...,N$. The effective theory for these fields is $$\mathcal{L}(a_{\mu}^{i})= \frac{1}{4\pi}K_{ij} \epsilon^{\mu\nu\rho}a_{\mu}^{i}\partial_{\nu}a_{\rho}^{j}+\frac{1}{2\pi}t_{i}\epsilon^{\mu\nu\rho} A_{\mu}\partial_{\nu}a_{\nu}^{i}$$ where the $K$ matrix specifies the Chern-Simons coupling and the $t$ vector tells which linear combination of currents play the role of electron current. So far we only considered the so called single-layer FQH states, where the electrons are confined a single $2d$ interface. This setting is described by the choice $t=(1,0,0,...,0)$, namely we have a single charge carrier. However, one can also make samples with multiple layers of interfaces. The electrons confined on different layers under a strong magnetic filed form a multi-layer FQH state. The topological properties of the model are encoded in $K$ and $t$. The Hall conductance can be computed by integrating out the gauge fields and is given by $$\left( K^{-1}\right)^{ij} t_{i}t_{j},$$ while the charge of the quasi-holes which couple to $a_{\mu}^{i}$ is $$(e^{\star})^{i}= \left( K^{-1}\right)^{ij}t_{j}.$$ The statistics between quasi-holes which couple to $a^{i}$ and $a^{j}$ is $$\alpha^{ij}= \left( K^{-1}\right)^{ij}.$$ Moreover, one can show that the degeneracy of the ground state on a surface of genus-g is $\vert \mathrm{det} K \vert^{g}$. When can recover the single-layer model given in \[previousmodel\] with the choice $$K=\begin{pmatrix} m & -1 \\ -1 & \tilde{m} \end{pmatrix} \hspace{1cm } t=(1,0).$$ It is possible to write down a wave function for the hierarchy states, also in the multi-layer case, which extends the Laughlin proposal to more general filling fractions. The interested reader can refer to [@wen] for the details of the construction.\ An alternative way to describe the hierachy states is given by the composite fermion theory proposed by Jain [@ab4]. The basic proposal is that FQH states for electrons can be thought as integer Hall states for new weakly interacting degrees of freedom. These are the composite fermions, namely electrons bounded to magnetic flux vortices. One can find a discussion about this alternative model in [@tong]. Despite the approach is very different from the hierarchy theory, the composite fermion approach reproduces the same filling fractions. Among these we get in particular the principal series $$\nu=\frac{n}{2n \pm 1},$$ also known as Jain’s series. As the hierarchy model, the composite fermion theory predicts Abelian fractional statistics for the quasi-holes.\ The possibility of non-Abelian statistics has been explored for certain FQH states at higher Landau levels [@nab1; @nab2; @nab3; @nab4]. Among these the most prominent is observed at $\nu=5/2$, which consists of fully filled lowest Landau level for both spin up and spin down electrons, followed by a spin polarized Landau level at half filling. This is also known as Moore-Read state and is believed to support non-Abelian anyons of Ising type [@tong]. One can construct effective low energy theories also for non-Abelian quantum Hall states. These naturally involve emergent non-Abelian Chern-Simons fields associated with certain non-Abelian gauge groups. Theory of Edge States {#theoryofedgestates} --------------------- It has been noticed in [@nab2] a remarkable connection between wave functions of FQH states and correlators of certain $2d$ conformal field theories. Let us consider a $\nu=1/m$ Laughlin state with $N$ electrons at positions $z_{i},i=1,...,N$ and $M$ quasi-holes at positions $\zeta_{a},a=1,..,M$. We consider the $c=1$ theory describing a free scalar field $\phi(z)$ and associate to electrons and quasi-holes respectively the chiral vertex operators $V(z_{i})=\exp\left( i\phi(z_{i})/\sqrt{\nu}\right) $ and $ W(\zeta_{a})=\exp \left(i\sqrt{\nu}\phi(\zeta_{a}) \right) $. The holomorphic part of the Laughlin wave function is captured up to a prefactor depending on the $\zeta_{a})$ by the correlator $$\langle \prod_{i,a} V(z_{i})W(\zeta_{a})\rangle= \prod_{i,a}(z_{i}-\zeta_{a}) \prod_{i<j}(z_{i}-z_{j})^{\frac{1}{\nu}}.$$ The contribution of the magnetic field can be included in this set up as follows. To be compatible with the holomorphicity of the correlator, instead of a uniform $B$ field, one considers a lattice of fundamental units of magnetic flux $\Phi(a)=a^{2}/2\pi l_{B}^{2}ì$ at positions $\zeta_{m,n}(a)=am+ian$, $\ m,n \in \mathbb{Z}$, where $a$ is a real parameter. One has to insert in the correlators the vertex operators $$\label{vertex} \prod_{\Lambda(a)\cap D(R)} \exp\left(-i \Phi(a)\sqrt{\nu}\phi(\zeta_{m,n}(a)) \right),$$ where the lattice $\Lambda(a)= a\mathbb{Z}+ia\mathbb{Z}$ is contained in a disk $D(R)$ of radius $R=\sqrt{2mN}l_{B}$. It is manifest in this language that a point-like charge and a magnetic flux unit have the same nature in the two dimensional physics. Hence, the FQH wave function gains the new term $$\label{term} \prod_{i,\Lambda(a)\cap D(R)} \frac{1}{(z_{i}-\zeta_{m,n}(a))}= \exp \left( -\Phi(a)\sum_{i,\Lambda(a)\cap D(R)} \log (z_{i}-\zeta_{m,n}(a)) \right).$$ We know that a correlator of chiral primaries $\langle \prod_{i}e^{i\alpha_{i}\phi(z_{i})} \rangle$ is non-vanishing only if the neutrality condition $\sum_{i}\alpha_{i}=0$ is satisfied. With the inclusion of the magnetic lattice in the correlator one finds the relation $$N= \nu (\mathcal{N}-M),$$ where $\mathcal{N}=\sum_{\Lambda(a)\cap D(R)}\Phi_{a}= R^{2}/2l_{B}^{2}$ is the number of flux units contained in $D(R)$, namely the capacity of the lowest Landau level. In this framework the magnetic field plays the role of neutralizing background charge for the correlator. We see that the above equality reproduces the correct relation between the number of electrons and the magnetic flux, including also the contribution of the quasi-holes.\ In order to recover the uniform constant $B$ field one should take the continuous limit of the lattice and then omit terms suppressed by $\vert z_{i} \vert /R\ll 1$. The immaginary part of the sum $-\Phi(a)\sum_{\Lambda(a)\cap D(R)} \log (z-\zeta_{m,n}(a))$ oscillates very rapidly for $a\ll 1$ and the average of the oscillations is vanishing in the limit $a\rightarrow 0 $. Instead, the real part coincides with the $2$-dimensional electrostatic potential generated by a discrete set of charges, which tends to a continuous distribution for $a\ll 1$. So, we obtain the limit $$-\Phi(a)\sum_{\Lambda(a)\cap D(R)} \log (z-\zeta_{m,n}(a)) \xrightarrow{ a\rightarrow 0, \ R\rightarrow \infty} -\sum_{i} \vert z \vert ^{2}/4l_{B}^{2},$$ which reproduces the non-holomorphic part of the FQHE wave functions.\ The relation between FQHE and $2d$ CFT is not accidental. The $c=1$ chiral theory describes the edge excitations of the Abelian Chern-Simons theory [@wen; @tong]. Contrary to the bulk topological theory, which is gapped by definition, the edge theory is a conformal chiral liquid with gapless excitations. This bulk to boundary correspondence is one of the main property of topological phases and apply also to non-Abelian quantum Hall states. The physics of the edge modes is a reflection of the topological order in the bulk and provides the most powerful way tool to classify topological universality classes. It turns out that the edge excitations have the same spectrum of the quasi-particle and quasi-hole excitations of the fluid. The role of the different kinds of anyons is now played by the different representations of the conformal algebra that appear in a given conformal field theory. Each of these representations is labelled by a primary operator $O_{i}$. In classifying topological order we consider rational conformal field theories, since they have a finite number of primary operators. The CFT algebra encodes also the concept of fusion. Given two operators $O_{i},O_{j}$, the operator product expansion (OPE) between them can contain other representations labelled by some $O_{k}$. The OPEs are naturally interpreted as the fusion rules satisfied by the anyonic particles. One can write the OPE as $$O_{i}\star O_{j}=\sum_{k}N_{k}^{ij}O_{k}$$ where $N_{ij}^{k}$ are integers.\ The CFT description of FQHE provides also the definition of braiding matrices. In general a CFT contains both chiral and anti-chiral modes. A correlation function of primary operators can be written as $$\langle \prod_{i} O(z_{i},\bar{z}_{i}) \rangle= \sum_{p} \vert \mathcal{F}_{p}(z_{i}) \vert^{2},$$ where $\mathcal{F}_{p}(z_{i})$ are multi-branched holomorphic functions of the $z_{i}$ depending on the set of operators inserted in the correlator. These are also known as conformal blocks and provide a basis of quantum Hall wave functions. As we exchange the positions $z_{i}$ of the quasi-particles the conformal blocks will be analytically continued onto different branches. The result can be written as some linear combination of the original functions, from which one derives the braiding properties of the anyons. A New Proposal -------------- According to the previous discussion, to study the topological properties of FQHE is enough to identify the correct CFT. This encodes all the necessary physical informations to characterize the universality classes of FQH states, such as fusion and braiding rules of the quasi-particles excitations. Moreover, it provides a systematic way to construct the quantum Hall wave functions without any guesswork. Following this approach, it has been recently proposed by Vafa a new unifying model for the principal series of FQHE systems with $\nu=\frac{n}{2n\pm 1}$ [@rif1]. The identification of the CFT describing the Hall states comes from consistency conditions between the above description and the CFT paradigms. The starting point is again the Laughlin state $\nu=1/m$, which according to the previous models is described by the $c=1$ CFT. The wave function interpretation of the correlators implies that we have to integrate them over the electron positions $z_{i}$. However, in the context of a CFT this is allowed only if the vertex operator $\exp(i\phi(z_{i})/\sqrt{\nu})$ has dimension $1$. This can be achieved by modifying the $c=1$ theory with the addition of a background term $Q^{\prime}\mathcal{R}\phi$ to the scalar action, where $\mathcal{R}$ is the Ricci curvature and $Q^{\prime}$ is the background charge. This is given by $$\frac{Q^{\prime}}{\sqrt{2}}= Q= \frac{1}{b} + b= i\left( \sqrt{2\nu}-\frac{1}{\sqrt{2\nu}} \right),$$ where $$b=\frac{-i}{\sqrt{2\nu}}.$$ Hence, it turns out that FQHE systems are described by a Liouville theory with action [@rif4; @rif5] $$S=\int d^{2}z \left[ \frac{1}{8\pi} \partial\phi\bar{\partial} \phi + iQ^{\prime}R\phi \right].$$ The central charge of the theory reads $$c=1+6Q^{2}=1-3\frac{(2\nu-1)^{2}}{\nu}.$$ We relax the restriction $\nu=\frac{1}{m}$ and consider a generic rational filling fraction $\nu=\frac{n}{m}$. We obtain $$c=1-6\frac{(2n-m)^{2}}{2nm}$$ which is the central charge of the $2d$ CFT minimal model $(2n,m)$ [@bel]. Here $m$ needs to be odd in order to have $2n$ and $m$ relatively prime. Moreover, since the edge modes in the FQHE are supposed to have correlations which fall off with the distance (see [@unitarity]) we should restrict to unitary CFT’s [@unitarity]. This puts the further constraint $m=2n\pm 1$, from which we recover the Jain’s series $\nu=\frac{n}{2n\pm 1}$. The operator algebra of the $(2n,m)$ minimal models is realized by the degenerate fields [@rif6]. $$\Phi_{r,s}=\exp\left[ i(r-1)\frac{\phi}{\sqrt{\nu}}+i(s-1)\sqrt{\nu}\phi \right]$$ for $1\leq r < 2n, \ 1\leq s < m $, which satisfy the relations [@minimal] $$\Phi_{r_{1},s_{1}} \times \Phi_{r_{2},s_{2}} = \sum_{ \substack{ k=1+\vert r_{1}-r_{2}\vert , k+r_{1}+r_{2}+1=0 \ \mathrm{mod} \ 2 \\ l=1+\vert r_{1}-r_{2}\vert , l+r_{1}+r_{2}+1=0 \ \mathrm{mod}\ 2 } }^{\substack{ k=r_{1}-r_{2}-1 \\ l=r_{1}-r_{2}-1 }} \Phi_{k,l}$$ In particular, the $\Phi_{1,s}$ are identified with the quasi-holes operators, where $\Phi_{1,2}$ is the minimal excitation, and generate insertions of $(z_{i}-\zeta)^{s-1}$ factors in the wave function. This leads to the main difference between this model and the previous ones, where the quasi-holes of the states $\nu=\frac{n}{2n\pm 1}$ are predicted to be Abelian anyons. On the contrary, in this theory the quasi-holes should have the same fusion rules and non-Abelian braiding properties of the $(2n,2n\pm 1)$ minimal models. The embedding of minimal models in Liouville theory allows also to identify the bulk topological theory, which turns out to be a Chern-Simons theory based on an $SL(2,\mathbb{C})$ gauge group. The Vafa’s Hamiltonian ---------------------- Among the several proposals of Vafa in [@rif1], there is a connection between FQHE and supersymmetric $\mathcal{N}=2$ gauge theories in four dimensions, which arise from compactification of six dimensional $(2,0)$ theories on a punctured Riemann surface $\Sigma$, the so called Gaiotto curve [@rif2]. These are labelled by semisimple Lie algebras in the ADE classification. Picking an ADE group, the rank $r$ of the corresponding Dynkin graph is identified with the number of layers of FQHE. We associate to each node in the graph an integer $ N_{a}, a=1,...,r $ which denotes the number of electrons in the $a$-th layer. The worldvolume of FQHE is identified with $\Sigma\times \mathbb{R}$, where $\mathbb{R}$ is the time direction and the Riemann surface $\Sigma$ is the target space of FQHE. For each node we assign also a meromorphic $(1,0)$ form $W_{a}(z)^{\prime}dz$ on $\Sigma$. The meromorphic functions $W_{a}(z)^{\prime}$ encode the interaction between an electron of coordinate $z$ in the $a$-th layer and the quasi-holes in the quantum Hall fluid. We focus on the case of target space $\mathbb{P}^{1}$, where $W_{a}(z)^{\prime}$ are just rational functions on the complex plane. Finally, we introduce for each node of the graph the set of chiral superfields $z_{a,k_{a}}$, $k_{a}=1,...,N_{a}$, which play the role of electron coordinates for the different layers of FQHE. Hence, with the ingredients defined above, a Dynkin diagram of ADE type identifies a $4$-SQM Landau-Ginzburg theory with superpotential $$\label{potuno} \begin{split} \mathcal{W}(z_{a,k_{a}})= & \sum_{a=1}^{r} \sum_{k_{a}=1}^{N_{a}} W_{a}(z_{a,k_{a}}) + \beta \sum_{a=1}^{r} \ \sum_{1\leq k_{a}<h_{a}\leq N_{a}} \log (z_{a,k_{a}}-z_{a,h_{a}})^{2} \\ & -\beta \sum_{a,b=1}^{r}\sum_{k_{a},h_{a}} I_{a,b} \log(z_{a,k_{a}}-z_{b,h_{a}}), \end{split}$$ where $C_{a,b}=2\delta_{a,b}-I_{a,b}$ is the Cartan matrix of the chosen ADE group and $\beta$ can be a generic complex coupling. We are mostly interested in the case of single layer FQHE. This corresponds to pick the $A_{1} \ (2,0)$ six dimensional theory. In the single layer case we have a unique rational function $W^{\prime}(z)$ containing the interaction between electrons and quasi-holes. It is natural to describe the quasi-holes with simple poles. Indeed, the function $W^{\prime}(z)$ has the physical interpretation of electrostatic field generated by the quasi-holes and the two dimensional Coulomb interaction between point-like particles decays with the inverse of the distance. This is also the most general case, since higher degree poles can be obtained by confluent limit of simple poles. Hence, denoting with $\zeta_{\ell}, \ell=1,...,M $ the positions of the punctures on $\mathbb{P}^{1}$, the holomorphic superpotential corresponding to this setting is $$\label{potdue} \mathcal{W}(z_{i};\zeta_{\ell})= \sum_{i=1}^{N} W(z_{i};\zeta_{\ell}) + \beta\sum_{1\leq i<j \leq N} \log (z_{i}-z_{j})^{2},$$ where $$W(z;\zeta_{\ell})= \sum_{\ell=1}^{M} \alpha_{\ell}\log (z-\zeta_{\ell}).$$ This model can be defined for general residues $\alpha_{\ell}$. However, as we are going to discuss in section \[wedisc\], if we want to identify the punctures at positions $\zeta_{\ell}$ either with the quasi-holes or the magnetic fluxes of FQHE one should set respectively $\alpha_{\ell}=\pm 1$.\ In [@rif1] Vafa makes the following predictions: - The degeneracy of the lowest Landau level can be mapped to the degeneracy of ground states of a supersymmetric system. - The $\mathcal{N}=4$ Hamiltonian prescribed by the above superpotential is in the same universality class of the principal series of FQH states and capture the same topological order. - The minimal quasi-holes of the FQHE states have the same braiding properties of the primary field $\phi_{1,2}$ in minimal models, or equivalently of the spin $1/2$ Wilson line operator in $SU(2)$ Chern-Simons theory. Moreover, the Berry connection of the quantum mechanical model corresponds to the $SU(2)$ Knizhnik-Zamolodchikov connection. This provides a Hecke representation $H_{n}(q)$ of the braid group of $n$ quasi-holes, where the parameter $q=\pm e^{\frac{i\pi}{\nu}}$ is function of the filling fraction $\nu$ labelling the FQH states. - There is a distinguished state among the ground states of the supersymmetric Hamiltonian of Laughlin type. This canonical vacuum corresponds to the configuration in which the electrons are as spread out as possible among the lowest Landau levels. Moreover, it is the most symmetric one with respect to the permutation of the quasi-holes and magnetic fluxes. The main goal of this thesis is to show that these predictions connecting FQHE and $\mathcal{N}=4$ supersymmetry are correct. The superpotentials \[potuno\] and \[potdue\] are motivated by the relation between topological string amplitudes, partition functions of ADE matrix models and chiral blocks of Toda theories [@rif7; @rif8; @rif9]. Focusing on the $A_{N-1}$ case, the topological string amplitudes are the objects which connect the Nekrasov partion functions of $\mathcal{N}=2 \ SU(N)$ gauge theories [@rif24] and the chiral conformal blocks of $A_{N-1}$ Toda theories in the context of AGT correspondence. In particular, one can specialize to the case of minimal models, which admit an embedding in the AGT set up with an appropriate refinement of the Nekrasov partition functions [@rif42; @rif43; @rif44]. In the case of single layer FQHE, the $\mathcal{N}=2$ four dimensional theories arising from the compactification on $\Sigma$ are $SU(2)$ gauge theories with a matter sector which depends on the structure of the punctures [@rif2]. According to the original formulation of AGT correspondence [@rif3], the Nekrasov partition functions of $SU(2)$ gauge theories compute the chiral blocks of the Liouville CFT. In Appendix \[classS\] we provide a short review about the Gaiotto theory and the relation between topological string amplitudes and Liouville chiral blocks which motivate the Vafa Hamiltonian. Once identified the effective theory for the principal series of FQHE, Vafa proposes also a microscopic Hamiltonian that in principle may support the topological phases described by the minimal models as ground states. This Hamiltonian is $\mathcal{N}=4$ supersymmetric and is motivated by string theoretical considerations. The key idea of this program is that with extended supersymmetry we have several techniques at our disposal, such as hyperKahler geometry, isomonodromic deformations, triangle and cluster categories, mirror symmetry, integrable and Hitchin systems, etc..., which we may use to understand the phenomenology of FQHE. These condensed matter systems seem to hide a rich mathematical structure beyond Chern-Simons theory and $2d$ CFT, which involves all the main recent developments in string theory and supersymmetric QFT. Supersymmetry and $tt^{*}$ Geometry {#reviewtt} =================================== Supersymmetric Quantum Mechanics -------------------------------- Before exploring the connections between FQHE and supersymmetry, we need to recall some basic knoledge about supersymmetric quantum mechanics and related tools to study the geometry of vacua. We begin with the case of two supercharges. ### $\mathcal{N}=2$ SQM $\mathcal{N}=2$ supersymmetric quantum mechanics is essentially the Witten’s reformulation of Morse theory [@rif19; @rif22]. The theory has a graded Hilbert space $\mathcal{H}=\mathcal{H}^{+}\oplus \mathcal{H}^{-}$, where $\mathcal{H}^{\pm}$ are spaces of bosonic and fermionic states respectively. These are eigenspaces of the Fermi parity operator $I=(-1)^{F}$ which counts the fermion number $F$ of a state modulo two. This operator is Hermitian and satisfies $I^{2}=1$. By definition of $\mathcal{N}=2$ supersymmetry, the theory has an Hermitian operator $\mathcal{Q}$ (and its adjoint $\mathcal{Q}^{\dagger}$) which maps bosonic states to fermionic states and viceversa. The susy charges $\mathcal{Q},\mathcal{Q}^{\dagger}$ and the Fermi parity satisfy the algebraic relations $$\begin{split} & \left\lbrace \mathcal{Q},I\right\rbrace = \left\lbrace \mathcal{Q}^{\dagger} , I \right\rbrace=0 \\ & \mathcal{Q}^{2}=\mathcal{Q}^{\dagger 2}= \left\lbrace \mathcal{Q},\mathcal{Q}^{\dagger} \right\rbrace =0 \end{split}$$ Since they are symmetry operators of the system, the susy charges commute with the Hamiltonian $H$. In particular, the Hamiltonian is the “Laplacian” $$H= \frac{1}{2}\left( \mathcal{Q}\mathcal{Q}^{\dagger}+ \mathcal{Q}^{\dagger}\mathcal{Q}\right) .$$ The operators $\mathcal{Q},\mathcal{Q}^{\dagger},F,H$ generate the supersymmetry algebra of the $\mathcal{N}=2$ theory.\ Since it is a nilpotent operator, one can define the cohomology of $\mathcal{Q}$ in the Hilbert space. The cohomology classes of $\mathcal{Q}$ are in correspondence with the ground states of the theory. The vacuum wave functions ${{\,| {\Psi} \rangle}}$ are eigenstates of the Hamiltonian with zero energy and coincide with the harmonic representatives of the cohomology classes $$H{{\,| {\Psi} \rangle}}=0 \Leftrightarrow \mathcal{Q}{{\,| {\Psi} \rangle}}= \mathcal{Q}^{\dagger}{{\,| {\Psi} \rangle}}=0.$$ We consider a smooth compact manifold $X$ of dimension $\mathrm{dim}_{\mathbb{R}} X=n$, with a local coordinate system $x_{i},i=1,...,n$. To each coordinate $x_{i}$ we associate a couple of creation-annihilation operator $\psi^{i},\psi_{i}$ which satisfy the Clifford algebra $$\begin{split} & \left\lbrace \psi^{i}, \psi^{j} \right\rbrace= \left\lbrace \psi_{i}, \psi_{j} \right\rbrace = 0 \\ & \left\lbrace \psi^{i}, \psi_{j} \right\rbrace= \delta^{i}_{j}. \end{split}$$ The Hilbert space is a representation of this algebra and a general wave function reads $$\Phi=\sum_{k} \phi(x)_{i_{1},...,i_{k}}\psi^{i_{1}}...\psi^{i_{k}}{{\,| {0} \rangle}},$$ where ${{\,| {0} \rangle}}$ is the Clifford vacuum and the functions $\phi(x)_{i_{1},...,i_{k}}$ are totally antisymmetric in the indices. The commutation relation between creation and annihilation operators can be read as the pairing between forms and vectors $dx^{i}(\partial_{j})= \delta^{i}_{j}$. The identification $\psi^{i}\rightarrow dx^{i}$ allows to rewrite the above wave function as a differential form on $M$. Hence, the space of differential forms $\Omega^{*}(X)$ plays the role of Hilbert space, where the Hermitian scalar products between states coincides with the Hodge inner product of forms $$\label{inner} {\ensuremath{\langle \Phi | \Psi \rangle}}= \int_{X} \Psi \wedge \ast \Phi^{\ast}.$$ In this representation the susy charges act as differential operators on the space of forms, while the degree $k$ of a wave form has the interpretation of Fermi number. Moreover, the Hermitian conjugation of operators is defined with respect to the inner product \[inner\] and has the usual expression of Hodge theory.\ The easiest realization of $\mathcal{N}=2$ supersymmetric quantum mechanics is a sigma model on a Riemannian manifold. The supersymmetric charges $\mathcal{Q}, \mathcal{Q}^{\dagger}$ are given by the exterior derivative $d$ and the adjoint $d^{\dagger}=(-1)^{k} \ast^{-1}d \ast$. The corresponding Hamiltonian is the usual Laplacian on a Riemannian manifold $$H=\frac{1}{2}\Delta=\frac{1}{2}(d^{\dagger}d+dd^{\dagger}).$$ The vacuum of this theory is isomorphic to the de Rham cohomology $H^{*}(X;\mathbb{C})$ and the Witten index coincides the Euler character of $X$ $$I_{W}=\mathrm{Tr}_{\mathcal{H}}(-1)^{F}=\sum_{k=0}^{n} (-1)^{k}b_{k}= \chi (X),$$ where the Betti numbers $b_{k}$ count the number of vacua with degree $k$.\ One can deform the sigma model by introducing a smooth real superpotential $V$ on $X$. The susy algebra generators of this theory are the generalized differential operators $$\begin{split} & d_{V}= d+ dV\wedge, \hspace{1cm} d_{V}^{\dagger}=(-1)^{k} \ast^{-1} (d +dV\wedge)\ast \\ \\ & \frac{1}{2}\Delta_{V}=\frac{1}{2} \left( \Delta + g^{ij}\partial_{i}V\partial_{j}V + \partial_{i}\partial_{j}V [\ast dx^{i}, dx^{j}]\right) , \end{split}$$ where $g_{ij}$ is a Riemannian metric on the target manifold. The susy charge $d_{V}$ is conjugated to the exterior derivative by the relation $d_{V}= e^{-V}d e^{V}$. It follows that the cohomology classes $[\tilde{\Psi}]$ of $d_{V}$ are in one to one correspondence with de Rham classes $[\Psi]$ through the map $$\label{conjugacy} \tilde{\Psi}= e^{-V}\Psi.$$ The isomorphism $\mathcal{H}_{\mathrm{vacuum}}\simeq H^{*}(X;\mathbb{C})$ implies that on a compact manifold the dimension of the ground state depends only on the topology of the manifold and not on the choice of the superpotential.\ One can study the ground states of the theory also with a perturbative approach. We introduce the planck constant $\hslash$ and rescale the superpotential as $V\rightarrow \frac{1}{\hslash}V$. In the semiclassical limit $\hslash\ll 1$ the potential energy $\frac{1}{\hslash^{2}}g^{ij}\partial_{i}V\partial_{j}V$ becomes very large except around the critical points of $V$. For small $\hslash$ the eigenfunctions of the Hamiltonian are concentrated around the classical vacua and can be computed with an asymptotic expansion in terms of local data. This is the essence of Morse theory: we reconstruct the cohomology from the local data of a smooth function around its critical points. We consider the case in which $V$ is a Morse function. This implies that the critical points of $V$ are isolated and non degenerate, namely $\mathrm{det} \ \partial_{i}\partial_{j}V\neq 0$. Since the set of Morse functions is open and dense in the space of functions, these properties are very general. One can associate to each perturbative vacuum a Morse index $I_{M}$ which counts the number of negative eigenvalues of the Hessian of $V$ at the critical point. The Morse index has the physical interpretation of fermion number. Since the Witten index is invariant under smooth deformation of the theory, one can perform its computation also in the classical limit. Denoting with $c_{k}$ the number of classical vacua with Morse index $k$, one has $$I_{W}=\sum_{k=0}^{n}(-1)^{k}c_{k}= \sum_{k=0}^{n}(-1)^{k}b_{k}.$$ The classical computation of vacua is exact at every order in perturbation theory, but is in general not correct at the non-perturbative level. Indeed, instanton corrections can lift some of the classical vacua. Hence, the true quantum vacua are generically less than the classical ones and we have $c_{k}\geq b_{k}$. Since the Witten index must be preserved, bosonic and fermionic vacua can be lifted by instantons only in pairs.\ One can extend $\mathcal{N}=2$ SQM to a non compact Riemannian manifold. In the non compact case the Hilbert space is the space of differential forms whose coefficents are $L^{2}$ functions on $X$. In absence of a superpotential, the vacua are harmonic representatives in cohomology classes of $L^{2}$ forms. These states are exact in the absolute cohomology of $X$ and one can write for a vacuum $\Psi$ the relation $$\Psi=d\Phi.$$ However, $\Phi$ is not square integrable on the target space and so the wave form $\Psi$ defines a non trivial cohomology class in the Hilbert space.\ As before, we can introduce a real superpotential $V$ on the target space which generates a potential term in the Hamiltonian. Differently from the compact case, now the structure of vacua depends on the choice of the superpotential. Morse theory admits an extension for non compact manifolds and one can show that the space of vacua is isomorphic to the relative de Rham cohomology $H^{*}(X,X_{V}; \mathbb{C})$, where $$\label{setv} X_{V}=\lbrace x \in X \big \vert \ V<-\Lambda; \ \Lambda \rightarrow \infty \rbrace$$ is the asymptotic region of $X$ where the superpotential tends to $-\infty$. In particular, the relative de Rham classes in $H^{*}(X,X_{V}; \mathbb{C})$ are mapped to $d_{V}$-cohomology classes in the space of $L^{2}$ forms by the map \[conjugacy\]. We can consider as an example the polynomial superpotential on the real line $$V= a x^{2n}+ \mathrm{lower \ order \ terms}, \hspace{1cm} x \in \mathbb{R}.$$ The structure of the ground state depends on the sign of the coefficient $a$ in fron of the highest power of the polynomial. If $a$ is positive the superpotential is bounded from below and so we have $X_{V}=\varnothing$. The vacuum space is isomorphic to the de Rham cohomology of the real line $$H^{\ast}(\mathbb{R})= \begin{cases} H^{0}(\mathbb{R})= \mathbb{C}\\ H^{1}(\mathbb{R})= 0 \end{cases}.$$ One can easily check that the eigenfunction of the Hamiltonian is the zero form $$\psi= ce^{-V},$$ where $c$ is some normalization constant. If $a$ is negative the superpotential is bounded from above and tends to $-\infty$ as $x\rightarrow \pm \infty$. In this case the vacuum is isomorphic to the cohomology with compact support on $\mathbb{R}$. This is Poincaré dual to the de Rham cohomology and reads $$H^{\ast}_{\mathrm{c}}(\mathbb{R})= \begin{cases} H^{0}_{\mathrm{c}}(\mathbb{R})= 0 \\ H^{1}_{\mathrm{c}}(\mathbb{R})= \mathbb{C} \end{cases}.$$ The corresponding vacuum is the fermionic state $$\psi= c e^{V}dx.$$ ### $\mathcal{N}=4$ SQM {#sqm} Supersymmetric quantum mechanics with four supercharges has the same structure of Hodge theory on a Kahler manifold $X$ [@rif14; @rif19]. For any neighborhood of the manifold one can choose a set of holomorphic coordinates $z_{i},i=1,...,n=\mathrm{dim}_{\mathbb{C}}X$ and a local Kahler potential $K$. This defines also the corresponding Kahler metric $g_{ij}=\partial_{i}\bar{\partial}_{j}K$ and Kahler form $\omega_{K}= i g_{i\bar{j}}dz^{i}d\bar{z}^{j}$. The fermionic partners of the bosonic degrees of freedom $z_{i},\bar{z}_{i}$ are four creation and annihilation operators $\psi^{i},\psi_{i},\bar{\psi}^{i},\bar{\psi}_{i}$ which satisfy the Clifford relations $$\begin{split} & \left\lbrace \psi^{i}, \psi^{j} \right\rbrace= \left\lbrace \psi_{i}, \psi_{j} \right\rbrace = 0 \hspace{1cm } \left\lbrace \bar{\psi}^{i}, \bar{\psi}^{j} \right\rbrace= \left\lbrace \bar{\psi}_{i}, \bar{\psi}_{j} \right\rbrace = 0 \\ & \left\lbrace \psi^{i}, \psi_{j} \right\rbrace= \delta^{i}_{j} \hspace{3cm} \left\lbrace \bar{\psi}^{i}, \bar{\psi}_{j} \right\rbrace= \delta^{i}_{j}. \end{split}$$ Similarly to the case with two supercharges, one can identify the creation and annihilation operators with forms and vectors on the target space. The map $\psi^{i}\rightarrow dz^{i}, \bar{\psi}^{i}\rightarrow d\bar{z}^{i}$ realizes the isomorphism between the space of $L^{2}$ smooth forms on $X$, endowed with the inner product \[inner\], and the Hilbert space $\mathcal{H}$ of a $\mathcal{N}=4$ theory. Compared to the Riemannian case, the richer structure of Kahler geometry leads to a larger supersymmetry algebra. The Hilbert space can be decomposed in irreducible representations of the Lefschetz $SU(2)$ symmetry group, whose algebra is generated by the Lefschetz operators $$L=\omega_{K} \wedge , \hspace{1cm} \Lambda= (-1)^{k}\ast^{-1}L \ast$$ and the shifted Fermi number $\tilde{F}=k-n$, where $k=p+q$ is the degree of a form which decomposes in holomorphic and anti-holomorphic degree $p,q$. These operators satisfy the algebraic relations $$\begin{split} & [\tilde{F}, L ]=2L, \\ & [ \tilde{F}, \Lambda ]=-2L, \\ & [ L, \Lambda ]=\tilde{F}. \end{split}$$ The Lefschetz group has the physical interpretation of $R$-symmetry group of the supersymmetry algebra.\ The easiest realization of $\mathcal{N}=4$ SQM is a sigma model with target space $X$. In this theory the supercharges are given by the Dolbeault operators on the Kahler manifold $$\begin{split} & \mathcal{Q}=\partial, \hspace{1cm} \overline{\mathcal{Q}}= \bar{\partial}, \\ & \mathcal{Q}^{\dagger}= \bar{\delta}, \hspace{1cm} \overline{\mathcal{Q}}^{\dagger}= \delta. \end{split}$$ The susy charges are nilpotent and anticommuting operators $$\begin{split} & \partial^{2}=\bar{\partial}^{2}=0 \hspace{1cm} \partial\bar{\partial} + \partial\bar{\partial}=0 \\ & \delta^{2}=\bar{\delta}^{2}=0 \hspace{1cm} \bar{\delta}\delta + \delta\bar{\delta}=0 \end{split}$$ and with the R-symmetry generators satisfy the Kahler identities $$\label{kahlid} \begin{split} & \left[ \bar{\partial}, L \right]=\left[ \partial, L \right] =0 \hspace{1cm} \left[ \bar{\partial}^{\dagger}, \Lambda \right]=\left[ \partial^{\dagger}, \Lambda \right] =0 \\ & \left[ \bar{\partial}^{\dagger}, L \right]= i\partial \hspace{2.12cm} \left[ \partial^{\dagger}, L \right] =-i\bar{\partial} \\ & H=\bar{\partial}\bar{\partial}^{\dagger}+ \bar{\partial}^{\dagger}\bar{\partial}= \partial\partial^{\dagger}+ \partial^{\dagger} \partial= \frac{1}{2}(dd^{\dagger}+ d^{\dagger}d), \end{split}$$ where $H$ is the Hamiltonian. The four susy charges, the R-symmetry generators and the Hamiltonian provide a complete basis of generators for the supersymmetry algebra.\ If the Kahler manifold is non compact we can add to the theory an holomorphic superpotential $W$. A $\mathcal{N}=4$ theory with a superpotential is called Landau-Ginzburg (LG) theory. In presence of $W$ the supercharges become the generalized Dolbeault operators $$\begin{split} & \partial_{W}= \partial+\bar{dW} \wedge, \hspace{1cm} \bar{\partial}_{W}= \bar{\partial} + dW \wedge, \\ & \bar{\delta}_{W}= \delta_{W}^{\dagger}, \hspace{2.2cm} \delta_{W}= \bar{\partial}_{W}^{\dagger}. \end{split}$$ One can check that these operators satisfy the same algebraic relations of $\partial,\bar{\partial}$. In particular, the Hamiltonian is the generalized Laplacian $$H= \partial_{W}\partial_{W}^{\dagger}+ \partial_{W}^{\dagger} \partial_{W}= \bar{\partial}_{W}\bar{\partial}_{W}^{\dagger}+ \bar{\partial}_{W}^{\dagger}\bar{\partial}_{W}= \frac{1}{2} (d_{V}d_{V}^{\dagger}+ d_{V}^{\dagger}d_{V}),$$ where $d_{V}=\partial_{W}+ \bar{\partial}_{W}$ is the generalized exterior derivative of $V=W+\overline{W}$.\ We are interested in the space of supersymmetric vacua of a $\mathcal{N}=4$ Landau-Ginzburg theory on a non compact space. By the Kahler identities \[kahlid\], we can study the cohomology in the Hilbert space of one of the two Dolbeault derivatives. For convenience, one typically considers the complex of $\overline{\mathcal{Q}}=\bar{\partial}_{W}$. Similarly to Hodge theory, the vacuum wave forms coincide with the harmonic representatives of the $\bar{\partial}_{W}$-cohomology classes. There exists a subclass of manifolds that are very common in the applications in which this problem simplifies considerably, namely Stein manifolds. There are several characterizations of this type of spaces, in particular capturing the property of having “many” holomorphic functions. Denoting with $O(X)$ the ring of holomorphic functions on $X$, the following two conditions are satisfied: - $X$ is holomorphically convex, namely for every compact subset $C\subset X$, the so-called holomorphically convex hull $$\bar{C}=\left\lbrace z \in X \big \vert \vert f(z)\vert \leq \mathrm{sup}_{w\in K} \vert f(w)\vert \ \forall f \in O(X) \right\rbrace$$ is also a compact subset of $X$. - $X$ is holomorphically separable, i.e. $\forall x,y \in X$ with $x\neq y$ there is an holomorphic function $f$ such that $f(x)\neq f(y)$. Examples of Stein manifolds are $\mathbb{C}^{n}$, as well as its holomorphic domains, and all the non compact Riemann surfaces. Among the properties of a Stein space, we have that $X$ is holomorphically spreadable, i.e. for every point $x \in X$ there is a set of $n=\mathrm{dim}_{\mathbb{C}} X$ globally defined holomorphic functions which form a local coordinate system when restricted to a neighborhood of $x$. Another relevant property is that the Dolbeault cohomologies $H^{p,q}_{\bar{\partial}}(X)$ have a simple structure, i.e. $$\label{dolbcom} H^{p,q}_{\bar{\partial}}(X)= \begin{cases} 0, \hspace{2cm} q\geq 1 \\ \Omega^{p}(X), \hspace{1cm} q=0 \end{cases}$$ where $ \Omega^{p}(X)$ is the space of holomorphic $p$-forms on $X$. The cohomologies $H^{p,q}_{\partial}(X)$ have a similar structure and can be obtained by complex conjugation. This result implies that a Stein space admits a globally defined Kahler potential $K$, since the corresponding Kahler form is exact in the Dolbeault cohomology and can be written globally as $\omega=i\partial\bar{\partial}K$. Moreover, a Stein space admits geodesically complete Kahler metrics.\ In order to describe the ground states of a Landau-Ginzburg theory it is relevant to define the BRST cohomology of $\overline{\mathcal{Q}}$ in the space of operators. The supercharge acts on an operator $\phi$ as $$\left[ \overline{Q}, \phi \right],$$ where the brackets denote respectively a commutator or anticommutator if the operator has even or odd fermion number. An operator is closed if the commutator with the susy charge is vanishing, while it is exact if there exist an operator $\varphi$ such that $\phi=\left[ \overline{Q}, \varphi \right]$. The BRST cohomology of $\overline{\mathcal{Q}}$ forms a ring under multiplication of operators, also known as chiral ring [@rif14; @rif19] $$\mathcal{R}=\frac{ \overline{\mathcal{Q}}-\mathrm{closed \ opers}}{ \overline{\mathcal{Q}}-\mathrm{exact \ opers}}.$$ In particular, $\mathcal{R}$ is a unital associative algebra defined over $\mathbb{C}$. Let us consider an operator which multiplies a wave form by a smooth function $\chi$. This operator is closed if $$\left[ \overline{Q}, \chi \right]=\bar{\partial}\chi=0,$$ namely if $\chi$ is holomorphic. Moreover, the operator is exact if there exists an holomorphic section $\upsilon$ of the tangent bundle on $X$ such that $$\left\lbrace \iota_{\upsilon}, \overline{\mathcal{Q}} \right\rbrace = \upsilon^{i}\partial_{i}W,$$ where $\iota_{\upsilon}$ is the contraction of differential forms by the vector field $\upsilon$. It turns out that the chiral ring is isomorphic to the commutative quotient algebra $$\mathcal{R}\simeq O(X)/J_{W},$$ where $J_{W}= \langle \partial_{1}W,...,\partial_{n}W\rangle$ is the Jacobian ideal generated by the partial derivatives of $W$. It is know from singularity theory that if the critical point of $W$ are isolated, the quotient algebra is finite dimensional and localizes around the critical points. Denoting with $p_{i},i=1,...,N$ the set of critical points, we have the isomorphism of complex algebras $$\mathcal{R}\simeq \prod_{i=1}^{N} \mathcal{R}_{i}, \hspace{1cm} \mathcal{R}_{i}= O_{p_{i}}/J_{W} \simeq \mathbb{C}^{\mu_{i}}$$ where $O_{p_{i}}$ are the germs of holomorphic functions at $p_{i}$. The dimension of the local ring $\mu_{i}=\mathrm{dim}_{\mathbb{C}} \mathcal{R}_{i}$ is called Milnor number of the critical point $p_{i}$ and is equal to the degeneracy of $p_{i}$. The dimension of the algebra is computed by the sum of the Milnor numbers $\mathrm{dim}_{\mathbb{C}}\mathcal{R}= \sum_{i}\mu_{i}$. If $W$ is non degenerate at the critical points, the chiral ring is a semisimple algebra and can be written as product of simple factors $\mathcal{R}\simeq\prod_{i=1}^{N} \mathbb{C}_{i} \simeq \mathbb{C}^{N}$. In particular, we have $\mathcal{R}_{i}\simeq \mathbb{C}_{i}$ and $\mu_{i}=1$ for each non degenerate $p_{i}$. One can introduce a basis of orthogonal idempotents $e_{i},i=1,...,N$ in which the chiral ring is completely diagonal. This is defined by the condition $$e_{i}(p_{j})=\delta_{ij}$$ and provides a canonical isomorphism between $\mathcal{R}$ and $\mathbb{C}^{N}$ as complex algebra. This basis can always be constructed if $X$ is a Stein manifold and the set of non degenerate critical points has no accumulation points. An holomorphic operator $\phi$ is identified by this map with its set of critical values $$\phi\rightarrow \begin{pmatrix} \phi(p_{1}) \\ . \\ . \\ . \\ \phi(p_{N}) \end{pmatrix}.$$ One can introduce a non degenerate symmetric pairing between chiral operators $$\langle \phi_{i} \phi_{j}\rangle = \eta_{ij}$$ which is defined by the Grothendieck formula [@rif10] $$\label{grotform} \langle \phi \rangle= \frac{1}{(2\pi i)^{n}} \oint \frac{\phi \ dz_{1}...dz_{n}}{\partial_{1}W...\partial_{n}W},$$ where the integration contour encircles the critical points of $W$. Endowed with the above pairing, the chiral ring is a symmetric Frobenius algebra.\ The structure of the vacuum space $\mathcal{V}$ of $\mathcal{N}=4$ Landau-Ginzburg theory is described by the following theorem [@rif14; @rif15]. Under the condition that $X$ is a Stein manifold, with $\mathrm{dim}_{\mathbb{C}}X=n$ and the critical points of $W$ are isolated - The supersymmetric vacua $\Psi_{i}$ are invariant $n$-forms under the Lefschetz symmetry group, namely $$L\Psi_{i}=\Lambda \Psi_{i}=0$$ - The vacuum wave forms are “essentially independent of the Kahler metric”. Denoting with $\Psi_{0}$ a wave function for a Kahler potential $K_{0}$, given another potential $K$ the corresponding wave function is $$\Psi= \Psi_{0}+ \sum_{k=0}^{[n/2]} a_{k} L^{k}\Lambda^{k} \Psi_{0}$$ for some coefficients $a_{k}$. In particular $\Psi^{n,0}= \Psi_{0}^{n,0}$ and $\Psi^{0,n}= \Psi_{0}^{0,n}$. - The Hamiltonian is compatible with complex conjugation: if $\Psi_{i}$ is a basis of vacuum wave functions, also the Hermitian conjugates $$\label{realstructure} \Psi^{\ast}_{i}= M_{\bar{i}}^{j} \Psi_{j}$$ form a complete basis, where the complex conjugation is defined by the real structure $M$ of the ground state. - The space of vacua is isomorphic to the chiral ring as vector space $$\mathcal{V}\simeq \mathcal{R}.$$ Denoting with $\phi_{i}$ a basis of $\mathcal{R}$, the vacuum wave function $\Psi_{i}$ representing the $\phi_{i}$-class reads $$\Psi_{i}= \phi_{i}dz_{1}\wedge....\wedge dz_{n} + \overline{\mathcal{Q}}-\mathrm{exact \ term}.$$ We make a few comments about this result. The vacuum wave functions can be seen as representatives of the $\bar{\partial}_{W}$-cohomology in the Hilbert space. One can show that if the $\bar{\partial}$-cohomology has the structure \[dolbcom\], which is guaranteed by the fact that $X$ is a Stein manifold, the $\bar{\partial}_{W}$-cohomology is concentrated in the middle cohomology group $H^{n}_{\bar{\partial}_{W}}$, which is isomorphic to the vector space $$H^{n}_{\bar{\partial}_{W}} \simeq \tilde{\mathcal{R}}= \frac{\Omega^{n}}{dW \wedge \Omega^{n-1}},$$ namely the space of holomorphic $n$-forms modulo those that can be written as $dW \wedge \alpha$ for some $n-1$ holomorphic form $\alpha$. The map $\phi(z)\rightarrow \phi(z)dz^{1}....dz^{n}$ makes explicit the isomorphism between $\mathcal{\tilde{R}}$ and the chiral ring $$\label{module} \mathcal{\tilde{R}}\simeq \mathcal{R}$$ as modules over $\mathcal{R}$. Moreover, $H^{n}_{\bar{\partial}_{W}}$ has the same local structure of the chiral ring. The fact that the map $dW \wedge :\Omega^{n-1}(M)\rightarrow \Omega^{n}(M)$ fails to be surjective only at the critical points $p_{i}$ of $W$, which are assumed to be isolated, implies that the cohomology is localized around the classical vacua. Indeed, one can prove that $$\label{localization} \mathcal{\tilde{R}} \simeq \bigoplus_{i} \mathcal{\tilde{R}}_{i}, \hspace{1cm} \mathcal{\tilde{R}}_{i}= \frac{\Omega_{i}^{n}(M)}{dW \wedge \Omega_{i}^{n-1}(M)},$$ where $\Omega_{i}^{n}(M)$ are the germs of holomorphic $n$-forms at $p_{i}$. Moreover, each space $\mathcal{\tilde{R}}_{i}$ is isomorphic to the local ring $$\mathcal{R}_{i}=\frac{\mathcal{O}_{p_{i}}}{I_{W}}$$ as module over $ \mathcal{R}_{i}$. It is proved in [@rif14] that each $\bar{\partial}_{W}$-class has representatives with compact support, hence in the Hilbert space, and viceversa a class in the Hilbert space determines a class in the space of smooth forms $\Lambda^{*}(M)$. Therefore, the cohomology $H_{\bar{\partial}_{W}}(M)$ is equivalent to the cohomology in the Hilbert space and we have the isomorphism of vector spaces $$H_{\bar{\partial}_{W}}(M)\simeq H_{\bar{\partial}_{W}}(\mathcal{H})\simeq \mathcal{R}.$$ The vacuum space has vanishing Lefschetz angular momentum $\tilde{F}$ and forms a trivial representation of the R-symmetry group. In particular, the Witten index $I_{W}=\mathrm{Tr}_{\mathcal{H}}(-1)^{\tilde{F}+n} $ counts the number of vacua up to the sign and is given by the sum of the Milnor numbers $$I_{W}= (-1)^{n}\sum_{i} \mu_{i}.$$ If $W$ is a Morse function, all the critical points are non degenerate and the dimension of the ground state is equal to the number of classical vacua. Indeed, since they have equal fermion number, the classical vacua cannot be lifted by instantons corrections and, differently from the case with two supercharges, the perturbative computation is always exact.\ By the Frobenius structure of the chiral ring, the space of vacua is naturally endowed with a symmetric pairing $$\label{symmpair} \eta_{ij}= \frac{1}{(2\pi i)^{n}}\int \Psi_{i} \wedge \ast \Psi_{j}= \oint \frac{\phi_{i}\phi_{j} \ dz_{1}...dz_{n}}{\partial_{1}W...\partial_{n}W}= \langle \phi_{i}\phi_{j} \rangle,$$ where $\phi_{i},\phi_{j}$ are representative in the chiral ring of the vacua $\Psi_{i},\Psi_{j}$. Viceversa, the scalar product of the Hilbert space allows to define an Hermitian pairing between chiral and anti-chiral operators, also known as $tt^{*}$ metric [@rif10] $$g_{j\bar{i}}={\ensuremath{\langle \overline{\phi}_{i} | \phi_{j} \rangle}}= {\ensuremath{\langle \Psi_{i} | \Psi_{j} \rangle}}= \int \Psi_{j} \wedge \ast \Psi_{j}^{\ast}.$$ Using the identities \[realstructure\], \[symmpair\] one finds that the ground state metric $g$, the real structure $M$ and the symmetric pairing $\eta$ are related by $$g_{j\bar{i}}=M_{\bar{i}}^{k} \int \Psi_{j} \wedge \ast \Psi_{k}= \eta_{jk} M_{\bar{i}}^{k}.$$ Moreover, since the real structure satisfies $MM^{*}=1$, where $M^{*}$ denotes the complex (not Hermitian) conjugate matrix, one can write an identity between $\eta$ and $g$ $$\label{realitycon} \eta^{-1}g(\eta^{-1}g)^{*}=1$$ also known as reality constraint.\ A Landau-Ginzburg theory with four supercharges is in particular $\mathcal{N}=2$ supersymmetric. In the $\mathcal{N}=4$ supersymmetry algebra one can define a family of $\mathcal{N}=2$ subalgebras parametrized by an angle $\theta$ as $$d_{V_{\theta}}= d+( e^{i\theta}\partial W+e^{-i\theta} \overline{\partial W})\wedge, \hspace{1cm} \delta_{h_{\theta}}=d_{h_{\theta}}^{\dagger},$$ where the real superpotential $V_{\theta}$ is the harmonic function $$V_{\theta}= 2Re( e^{i\theta}W).$$ On a Kahler space the complex structure is compatible with the Riemannian structure and one can see the vacuum space also as $L^{2}$-cohomology of $d_{V_{\theta}}$.\ The cohomology classes which label the vacua of the supersymmetric system are defined over the complex numbers. However, it is possible to endow the vacua also with an integral structure. By Morse cobordism, we have an isomorphism between the vacuum space and the relative de Rham cohomology [@rif10] $$\mathcal{V}=H^{*}(X,X_{\mathrm{Re} (e^{i\theta}W)};\mathbb{C}),$$ where the set $X_{\mathrm{Re} (e^{i\theta}W)}\subset X$ is defined according to \[setv\]. The vacuum wave forms $\Psi_{i}$ are conjugated to some closed $n$-forms $\omega_{i}$ in the relative de Rham classes by $$\Psi_{i}= e^{-2\mathrm{Re}( e^{i\theta}W)} \omega_{i}, \hspace{1cm} i=1,...,N.$$ The classes in the dual homology group $H_{n}(X,X_{\mathrm{Re} (e^{i\theta}W)};\mathbb{C})$ are represented by non compact cycles with boundary in the region $X_{\mathrm{Re} (e^{i\theta}W)}$. This space has a natural integral structure given by the homology with integer coefficients $$\mathcal{V}^{\vee}\simeq H_{\ast}(X,X_{\mathrm{Re} (e^{i\theta}W)};\mathbb{Z})\otimes_{\mathbb{Z}} \mathbb{C}.$$ In case of Morse superpotential a canonical integral basis is given by the Lefschetz thimbles $B_{a}(\theta),a=1,...,N$ describing the gradient flow of the superpotential $e^{i\theta} W$ [@rif11; @rif19]. These are special Lagrangian middle-dimensional submanifolds of $X$ which arise from the critical point of $W$. The images of these cycles on the $W$-plane are straight lines stretched in the $e^{i\theta}$ direction starting at the critical values of the superpotential. The relative homology is also called the space of branes, because in $2d$ the corresponding objects have the physical interpretation of half-BPS branes [@rif11]. In particular, the angle $\theta$ specifies which linear combinations of the original $4$ supercharges leave the brane invariant. One can use the above formula and the standard pairing between cycles and forms to introduce the so called $tt^{*}$ brane amplitudes $${\ensuremath{\langle B_{a}(\theta) | \Psi_{i} \rangle}}=\Psi_{a,i}(\theta)= \int_{B_{a}(\theta)} e^{2\mathrm{Re}( e^{i\theta}W)} \Psi_{i},$$ which define a non-degenerate $N \times N$ matrix whose components are not univalued as function of the couplings and the phase $e^{i\theta}$. $tt^{*}$ Geometry ----------------- ### $tt^{*}$ Geometry in SQM Once we solve the Schroedinger equation for the zero energy level, it is interesting to study the evolution of the ground states as we vary the parameters of the theory. We consider a family of Landau-Ginzburg models $W(z_{i},t_{a})$ parametrized by a set of holomorphic couplings $t_{a}$. These are local holomorphic coordinates on a complex manifold $\mathcal{P}$. We assume for this family of theories that the target manifold $X$ is a Stein space and the critical points of $W$ are isolated. One defines a complex vector bundle $\mathcal{H}\rightarrow \mathcal{P}$, also called Hilbert bundle, where the fiber is the Hilbert space $\mathcal{H}$. This is a trivial bundle and can be written globally as $\mathcal{H}\times \mathcal{P} \rightarrow \mathcal{P}$. The vacuum bundle $\mathcal{V}\rightarrow \mathcal{P} $ is the subbundle of the Hilbert bundle whose fiber is the space of vacua $\mathcal{H}_{\mathrm{vac}}$. Differently from the Hilbert bundle, the bundle of vacua is non trivial, since the ground state of the Hamiltonian vary non trivially as we change the couplings of the superpotential. The canonical trivial connection of the Hilbert bundle induces by projection $P:\mathcal{H}\rightarrow \mathcal{H}_{\mathrm{vac}} $ a non trivial connection on the vacuum bundle, which is by definition the Berry connection. The equations which prescribe the curvature of the Berry connection are known as $tt^{*}$ equations [@rif10; @rif14]. We recall that a generic wave form $\Psi$ in the Hilbert space has a unique orthogonal Hodge decomposition $$\Psi=\Psi_{0}+\bar{ \partial}_{W}\alpha + \bar{\partial}_{W}^{\dagger} \beta$$ where $\Psi_{0}$ is $W$-harmonic. The operator $P$ projects the state along the vacuum space, namely $$P:\Psi\rightarrow \Psi_{0}.$$ The vacua $\Psi_{k},k=1,...,N$, satisfy the equations $$\begin{split} & (\bar{\partial}+ dW\wedge) \Psi_{k}=0, \\ & (\bar{\partial}+ dW\wedge)^{\dagger} \Psi_{k}=0. \end{split}$$ We take the derivative of these equations with respect to some coupling $t_{a}$. Denoting $W_{a}=\partial_{t_{a}}W$, we find $$\label{connectiondef1} \begin{split} \partial_{t_{a}} \left[ (\bar{\partial}+ dW\wedge) \Psi_{k} \right] =&\ \bar{\partial}_{W}(\partial_{t_{a}} \Psi_{k})+ d(\partial_{a}W)\wedge \Psi_{k} \\ = & \ \bar{\partial}_{W}(\partial_{t_{a}} \Psi_{k} ) + \partial_{W}(W_{a}\Psi_{k})=0. \end{split}$$ and $$\label{connectiondef2} \partial_{t_{a}} \left[ (\bar{\partial}+ dW\wedge)^{\dagger} \Psi_{k} \right]= \bar{\partial}_{W}^{\dagger}(\partial_{t_{a}}\Psi_{k})=0.$$ We introduce the forms $\lambda_{a,k}$ which satisfy $$\bar{\partial}_{W}\lambda_{a,k}= W_{a}\Psi_{k}-P(W_{a}\Psi_{k}), \hspace{1cm} \bar{\partial}_{W}^{\dagger}\lambda_{a,k}=0.$$ In particular, the first equation is consistent with the fact that $W_{a}\Psi_{k}$ is $\bar{\partial}_{W}$-closed. One can rewrite the equations \[connectiondef1\] and \[connectiondef2\] as $$\begin{split} & \bar{\partial}_{W}\left( \partial_{t_{a}}\Psi_{k}-\partial_{W}\lambda_{a,k}\right) =0, \\ & \bar{\partial}_{W}^{\dagger}\left( \partial_{t_{a}}\Psi_{k}-\partial_{W}\lambda_{a,k}\right) =0. \end{split}$$ So, we see that $\partial_{t_{a}}\Psi_{k}-\partial_{W}\lambda_{a,k}$ is in the ground state and can be written as linear combination of $\Psi_{k}$. The coefficients of the linear combination defines the Berry connection $$\partial_{t_{a}}\Psi_{k}-\partial_{W}\lambda_{a,k}= -(A_{a})_{k}^{l}\Psi_{l}.$$ One can introduce the corresponding covariant derivative $D_{a}$ and rewrite this relation as $$D_{a}\Psi_{k}= \partial_{W}\lambda_{a,k}.$$ We can repeat the same steps also for the other vacuum equations $$\begin{split} & (\partial+ d\overline{W}\wedge) \Psi_{k}=0 ,\\ & (\partial+ d\overline{W}\wedge)^{\dagger} \Psi_{k}=0. \end{split}$$ We find the definition of the antiholomorphic part of the Berry connection $$\overline{D}_{a}\Psi_{k}=\bar{\partial}_{W}\tilde{\lambda}_{a,k}$$ , where $\tilde{\lambda}_{a,k}$ are wave forms satisfying $$\begin{split} \partial_{W}\tilde{\lambda}_{a,k}= \overline{W}_{a}\Psi_{k}-P(\overline{W}_{a}\Psi_{k}), \hspace{1cm} \partial_{W}^{\dagger} \tilde{\lambda}_{a,k}=0. \end{split}$$ The bundle of vacua has a natural structure of holomorphic vector bundle. A basis of holomorphic sections is given by the operators in the chiral ring $\mathcal{R}$, which are holomorphic representatives of the vacua in the $\bar{\partial}_{W}$-cohomology. The holomorphic operators provide also a basis for the tangent space of the couplings manifold. Indeed, the deformations of the superpotential $W_{a}$ are elements of the chiral ring. Denoting with $\phi_{k}$ a basis for $\mathcal{R}$, the chiral ring algebra is encoded in the relation $$\phi_{i}\phi_{j}=C_{ij}^{k}\phi_{k},$$ where the numbers $C_{ij}^{k}$ are the structure constants of the ring. These matrices describe the action of the chiral operators on the vacua. Projecting on the ground state the action of $W_{a}$ on a wave function $\Psi_{k}$ one finds $$P(W_{a}\Psi_{k})= W_{a}\phi_{k} \ dz_{1}\wedge...\wedge dz^{n} + \bar{\partial}_{W}(...)= C_{ak}^{l}\phi_{l} \ dz_{1}\wedge...\wedge dz^{n}+ \bar{\partial}_{W}(...).$$ The curvature of the Berry connection can be computed in terms of the chiral ring coefficients $(C_{a})_{k}^{l}$ with Hodge theoretical arguments. One gets the $tt^{*}$ equations [@rif10] $$\begin{split} & [\overline{D}_{a},D_{b}] =-[ \overline{C}_{a},C_{b}] \\ & [ D_{a},D_{b} ] = [ C_{a},C_{b}]=0 \\ & [ \overline{D}_{a},\overline{D}_{b}] = [ \overline{C}_{a},\overline{C}_{b}]=0 \\ \end{split}$$ In addition, the chiral ring coefficients satisfy $$\overline{D}_{a}C_{b}=D_{a}\overline{C}_{b}=0 \hspace{1cm} D_{a}C_{b}=D_{b}C_{a} \hspace{1cm} \overline{D}_{a}\overline{C}_{b}=\overline{D}_{b}\overline{C}_{a}.$$ The structure of the $tt^{*}$ equations depends uniquely on the holomorphic data of the chiral ring, while there is no dependence on the Kahler metric of the target space. In general, once the topology of the target manifold is fixed, the geometry of the vacuum bundle is completely determined by the superpotential. From the above equations one can see that the Berry connection preserves the holomorphic structure of the bundle and is compatible with the metric. In particular, only the $(1,1)$ component of the curvature is non vanishing. Hence, the Berry connection is the unique Chern connection of the bundle of vacua. In an holomorphic trivialization the coefficients of the connections read $$\label{ciao} A_{ak}^{j}= g_{k\bar{l}}(\partial_{a}g^{-1})^{\bar{l} j}, \hspace{1cm} A_{\bar{a} k}^{j}=0,$$ where $g_{k\bar{l}}= {\ensuremath{\langle \overline{\phi}_{l} | \phi_{k} \rangle}}$ is the ground state metric. One can express $\overline{C}_{a}$ in this basis in terms of $C_{a}$ and $g$ as $$(\overline{C}_{a})^{k}_{l}= g_{k\bar{j}} (C^{\dagger})_{\bar{r}}^{\bar{j}}g^{\bar{r}l}= (gC^{\dagger}g^{-1})^{k}_{l}.$$ Plugging the expression \[ciao\] in the $tt^{*}$ equations one obtains a set of differential equations for $g$ $$\bar{\partial}_{i}(g\partial_{j}g^{-1})=\left[ C_{j},\overline{C}_{i}\right].$$ Solving this equation with the appropriate boundary conditions and the reality contraint \[realitycon\] allows to determine the geometry of the vacuum bundle.\ We specialize the analysis to the class of Landau-Ginzburg models in which the critical points of the superpotential $p_{i},i=1,...,N$ are non degenerate. In this case, as we said in the previous section, the chiral ring is semisimple and factorizes as product of simple factors $\mathcal{R}\simeq \prod_{i=1}^{N}\mathbb{C}_{i}$. Moreover, one can introduce a basis of orthogonal idempotents $e_{i}$ which provide a canonical isomorphism between $\mathcal{R}$ and $\mathbb{C}^{N}$. This is also called ‘point basis ’, since the operators satisfy the property $e_{i}(p_{j})=\delta_{ij}$. In this basis the chiral ring algebra diagonalizes completely $$e_{i}e_{j}=\delta_{ij}e_{j},$$ and a generic operator $\Phi=\sum_{i=1}^{N}\Phi(p_{i})e_{i}$ can be identified with its set of critical values $\Phi(p_{i})$. It is convenient to rescale the generators $e_{i}$ in order to normalize the symmetric pairing $\eta_{ij}$ of the chiral ring to the identity. For a Morse superpotential the Grothendieck residue \[grotform\] becomes $$\label{residue} \eta_{ij}=\langle \phi_{i}\phi_{j}\rangle= \sum_{p_{i}} \phi_{i}\phi_{j}\left( \mathrm{det} \, \partial_{k}\partial_{l}W\right)^{-1}.$$ In the basis of orthogonal idempotent we have $$\eta_{ij}= \left( \mathrm{det} \, \partial_{k}\partial_{l}W(p_{i})\right)^{-1} \delta_{ij}, \hspace{1cm} \langle e_{i}\rangle= \left( \mathrm{det} \, \partial_{k}\partial_{l}W(p_{i})\right) ^{-1}.$$ The symmetric pairing can be normalized to the identity with the rescaling $e_{i}\rightarrow\sqrt{\mathrm{det} \, \partial_{k}\partial_{l}W(p_{i})}e_{i}$. The so redefined basis is called canonical basis and is unique up to the sign. Since the Berry connection is compatible also with the symmetric pairing, in the canonical basis we have $$A_{i}^{t}=-A_{i},$$ namely the Berry connection takes value in the algebra of the orthogonal group.\ The LG theories with a Morse superpotential posses a natural system of coordinates in the coupling constant space. The critical map $w:\mathcal{P}\rightarrow \mathbb{C}^{N}$ given by $t_{a}\rightarrow (w_{1}(t_{a}),...,w_{N}(t_{a}))$ is a local immersion and the set of critical values $w_{i}(t_{a})=W(p_{i};t_{a})$ form a local coordinate system on the Frobenius manifold of all couplings of the theory which contains the physical coupling space $\mathcal{P}$ as a submanifold. These are usually called canonical coordinates [@rif12; @rif32; @rif33]. In the canonical basis, the chiral ring coefficients associated to the variations of $w_{i}$ have the simple structure $$\label{canonicalcoeff} (C_{i})^{k}_{l}= \delta_{il}\delta^{k}_{l}.$$ In these coordinates the $tt^{*}$ equations become universal and inequivalent physical systems are distinguished by different boundary conditions.\ It is known that the vacuum bundle of a $\mathcal{N}=4$ Landau-Ginzburg theory and its two dimensional counterpart have the same $tt^{*}$ geometry [@rif10]. Even if the physical interpretation is different, the formulas are exactly the same and one can use the language of $2d$ $\mathcal{N}=(2,2)$ quantum field theory to study the $tt^{*}$ geometry of the model. The usual strategy to solve the $tt^{*}$ equations requires to rescale the superpotential $W\rightarrow \beta W$ and consider variations of the overall coupling $\beta$. In two dimensions this is equivalent to study the RG flow of the theory [@rif10]. Even though the superpotential is protected by non-renormalization theorems, the F-term picks up a factor due to the rescaling of the superspace coordinates. From $z\rightarrow \beta z$ and $\theta \rightarrow \beta^{-1/2} \theta $ we get $$\int d^{2}\theta d^{2}z \ W \longrightarrow \beta \int d^{2}\theta d^{2}z \ W.$$ Variations in the overall factor $\beta$ generate a flow which has UV limit for $\beta \rightarrow 0$ and IR limit for $\beta \rightarrow \infty$.\ We denote with $\dot{\mathcal{P}} \in \mathcal{P}$ the dense open domain in the space of couplings in which $\mathcal{R}$ is semisimple and $w_{i}\neq w_{j}$ for $i\neq j$. In other words, $\dot{\mathcal{P}}$ is the domain in which $W$ is strictly Morse. A quantity of great interest is the Berry’s connection in the direction of the RG flow [@rif12; @rif20] $$Q=\iota_{\varepsilon}A= \sum_{i}w^{i}A_{i},$$ where $\varepsilon=\sum_{i}w^{i}\partial_{w_{i}}=\frac{1}{2}\beta\partial_{\beta}$ is the Euler vector in $\dot{\mathcal{P}}$. The equations $ D_{i}C_{j}= D_{j}C_{i}$ in the canonical coordinates become $$\left[ A_{i},C_{j}\right] = \left[ A_{j},C_{i}\right].$$ From this equality and the definition of $Q$ we find $$\left[ A_{i}, w^{j}C_{j}\right] = \left[ A_{j}w^{j},C_{i}\right] =\left[ Q,C_{i} \right] .$$ The $k,l$ component of the above expression is $$(A_{i})_{kl}w^{l}-w^{k}(A_{i})_{kl}= Q_{ki}-Q_{li},$$ which allows to write the Berry’s connection in terms of $Q$ only as [@rif32] $$\label{basicformula} (g\partial g^{-1})_{kl}=Q_{kl}\frac{d(w_{k}-w_{l})}{w_{k}-w_{l}},$$ where we used the fact that $Q$ is antisymmetric in the canonical basis. Hence, it is enough to know $Q$ in order to specify completely the solution of the $tt^{*}$ equations.\ In $2d \ (2,2)$ models $Q_{ij}$ is a pseudo-index which plays two roles [@rif20]. First, it captures the half-BPS solitons with boundary conditions given by the $i,j$-th vacuum repsectively at $x\rightarrow \pm \infty$. In the IR limit the $tt^{*}$ solution can be written in the form of soliton expansion, with the boundary condition represented by the soliton spectrum. The IR region in canonical coordinates is identified by large masses $m_{ij}= 2\vert w_{i}-w_{j} \vert$ of the kinks interpolating the $i$-th and $j$-th vacua. In the canonical basis the IR expansion $(\beta\rightarrow \infty)$ of the metric and $Q_{ij}$ are [@rif12] $$\begin{split} & g_{i\bar{j}}\simeq \delta_{ij}-\frac{i}{\pi}\mu_{ij} K_{0}(m_{ij}\beta) \\ & Q_{ij} \simeq -\frac{i}{2\pi} \mu_{ij}m_{ij}\beta K_{1}(m_{ij}\beta) \end{split}$$ where $K_{0},K_{1}$ are the modified Bessel function and $\mu_{ij}$ is the soliton matrix which counts with sign the number of soliton species in the $i,j$ sector. The existence of regular solutions requires $\mu_{ij}$ to be real, although they are integral in the physical case. Moreover, the CPT invariance imposes $\mu_{ji}=-\mu_{ij}$.\ The second role of $Q_{ij}$ is that it captures the $U(1)_{R}$ charges of the Ramond vacua at the UV fixed point. As we said, the UV fixed point is reached when $\beta \rightarrow 0$. If there are no Landau poles along the RG flow and the UV theory exists, the model can flow to a SCFT or an asymptotically free theory. In this limit the operators in the chiral ring are the chiral primary fields of the conformal theory [@rif10; @rif12]. It is known that a generic Landau-Ginzburg theory at the critical point gains the $U(1)_{V}$ R-symmetry, which is broken off-criticality by the superpotential. The associated charge is the scaling dimension of the conformal fields. One can choose a basis of chiral primaries for the chiral ring and order the R-charges $q_{i},i=1,...,N=\mathrm{dim}\mathcal{R}$ in a non-decreasing sequence $$0=q_{1}\leq q_{2}\leq...\leq q_{N}=c/3=q_{\mathrm{max}},$$ where $c$ is the central charge of the CFT. In particular, there is always an operator with $0$ charge in the spectrum which is given by the identity operator. The Berry connection in the direction of the RG flow has the same eigenvalues of the generator of the $U(1)_{R}$ symmetry $\boldsymbol{Q}$ of the $(2,2)$ SCFT [@rif10; @rif12; @rif20]. In this sense, the Berry’s connection associated to the RG flow can be seen as an off-criticality definition of the $U(1)_{R}$ generator. The eigenvalues of $Q$ at the critical point are the Ramond charges of the vacua [@rif10] $$q_{i}^{R}=q_{i}-\hat{c}/2,$$ which are symmetrically distributed between $-\hat{c}/2$ and $\hat{c}/2$. ### The Integral Formulation of $tt^{*}$ Geometry {#integral} We previously introduced D-branes to provide the vacua with an integral structure. One can define a dual bundle whose fiber is the lattice of relative homology cycles which pair with the vacuum wave functions. This bundle is parametrized by a phase $\zeta \in \mathbb{P}^{1}, \vert \zeta \vert=1$ which defines which half of supersymmetry is preserved by the D-branes. The pairing between D-branes and vacua in the Hilbert space allows to define a basis of sections for the vacuum bundle $$\Psi_{i,a}= {\ensuremath{\langle \phi_{i} | B_{a}(\zeta) \rangle}}.$$ We remind that, because of the integral structure of D-branes, these amplitudes are locally constant in the couplings $w_{i}$ and the spectral parameter $\zeta$. We can define a family of flat connections parametrized by $\zeta$ [@rif12] $$\begin{split} & \nabla_{i}^{\zeta}= \partial_{i}+(g\partial_{i}g^{-1})-\zeta C_{i}, \\ \\ & \bar{\nabla}_{i}^{\zeta}= \bar{\partial}_{i}-\zeta^{-1}\overline{C}_{i}, \end{split}$$ anso known as $tt^{*}$ Lax connection. The $tt^{*}$ equations for the Berry connection can be rephrased as the flatness conditions for the above connection, i.e. $$(\nabla^{\zeta})^{2}= (\bar{\nabla}^{\zeta})^{2} = \nabla^{\zeta} \bar{\nabla}^{\zeta} + \bar{\nabla}^{\zeta} \nabla^{\zeta}=0.$$ The D-brane amplitudes are horizontal sections of the Lax connection $$\nabla_{i}^{\zeta}\Psi^{a}(\zeta)=\bar{\nabla}_{i}^{\zeta}\Psi^{a}(\zeta)=0.$$ Differently from the Berry connection, the Lax connection is not compatible with the ground state metric and provides a non unitary representation of $\pi_{1}(\mathcal{P})$. Let $\Psi(w_{i},\zeta)$ be a $N\times N$ matrix whose columns are linearly independent solutions of the above linear equation. Taking the analytic continuation of the solutions along a non trivial loop $\gamma$ in the space of couplings we obtain the monodromy $\rho_{\zeta}(\gamma)$ defined by $$\Psi(w_{i},\zeta)\rightarrow \Psi(w_{i},\zeta)\rho_{\zeta}(\gamma).$$ Since the Lax connection is flat, the matrix $\rho_{\zeta}(\gamma)$ depends only on the homotopy class of $\gamma$. The $tt^{*}$ monodromy representation $\rho_{\zeta}$ in $GL(N,\mathbb{C})$ may be conjugated such that it lays in the arithmetic subgroup $SL(N,\mathbb{Z})$. Since the branes are representatives of integral homology classes, for each $\zeta \in \mathbb{P}^{1}$ they define a local system on $\mathcal{P}$ canonically equipped with a flat connection, the Gauss-Manin one. Dually, the brane amplitudes define a $\mathbb{P}^{1}$- family of flat connections on the vacuum bundle which is naturally identified with the $\mathbb{P}^{1}$-family of $tt^{*}$ Lax connections. Hence, modulo conjugation we have $$\rho_{\zeta}: \pi_{1}(\mathcal{P})\rightarrow SL(N,\mathbb{Z}).$$ Since the entries of the matrix $\rho_{\zeta}(\gamma)$ and its inverse are integers, they are locally indipendent from the couplings and the spectral parameter $\zeta$. This implies that the $tt^{*}$ monodromy representations is indipendent from the point $t \in \mathcal{P}$ and $\zeta$.\ The $tt^{*}$ geometry is a set of isomonodromic equations for the Lax linear system and admits a reformulation as Riemann-Hilbert problem [@rif12; @rif33]. It is known that the solution is captured by the monodromy of the flat sections around the movable singularities. These are poles for the equations and have the $2d$ physical interpretation of UV fixed points in the parameter space. A condition we need to incorporate in this setup is the indipendence of the ground state metric from an overall rotations of $w_{i}$. Indeed, an overall phase can always be absorbed in the fermionic measure of the superspace. We can consider the dependence of $\Psi_{a}$ on the RG scale $\beta \in \mathbb{R}_{\geq 0}$ and a phase $e^{i\theta}$ by redefining the canonical coordinates as $$w_{i}\rightarrow \beta e^{i\theta} w_{i}.$$ After the identification $\zeta= e^{i\theta}$, the Lax equations become $$\label{linearprobl} \begin{split} \zeta\partial_{\zeta} \Psi= \left( \beta\zeta C + Q-\beta \zeta^{-1}\overline{C} \right) \Psi ,\\ \\ \beta\partial_{\beta}\Psi= \left( \beta\zeta C + Q+\beta \zeta^{-1}\overline{C} \right) \Psi. \end{split}$$ The compatibility of these equations automatically implies the indipendence of the $tt^{*}$ solution from the angle $\theta$. Indeed, $Q,C, \overline{C}$ are consistently indipendent from $\zeta$.\ The identification of the overall phase with the spectral parameter allows to extend its domain to whole complex plane. Thus, the equation above has two singular points in $\zeta=0,\infty$ and $\Psi(\zeta,w_{i})$ undergoes monodromy as $w_{i}\rightarrow e^{2\pi i} w_{i}$. In this case there are two Stokes sector which can be chosen to be the upper and lower half $\zeta$-plane. The solutions on these two half planes $\Psi_{+},\Psi_{-}$ overlap on the real line and are related by $$\label{RH} \begin{split} &\Psi_{-}(y)=\Psi_{+}(y)S \\ \\ &\Psi_{-}(-y)=\Psi_{+}(-y)S^{t} \end{split}$$ where $y>0$. The stokes matrix $S$ is given by the phase-ordered product of the Stokes jumps $$\label{piclef} M_{ij}=\delta_{ij}-A_{ij}$$ which are generated when $\zeta$ crosses the BPS central charge $Z_{ij}=2(w_{i}-w_{j})$ of a primitive soliton connecting the $j$-th to the $i$-th vacuum. The matrix $A_{ij}$ has a unique non vanishing coefficient in the $ij$ entry. In the basis of Lefschetz thimbles this is equal to the BPS multiplicity $A_{ij}=\mu_{ij}$ of the $ij$ sector. When $\zeta$ spans the whole upper half plane in the anticlockwise sense we find the definition of the Stokes matrix $$S= \prod_{0< \arg Z_{ij} < \pi} M_{ij},$$ where we are assuming that there are no solitons with central charges aligned with the Stokes axes. This can always be achieved with a rotation of the axes.\ One can solve the Riemann-Hilbert problem \[RH\] imposing the correct boundary condition at the infinity of the $\zeta$-plane $$\label{conditmetric} \lim_{\zeta\rightarrow \infty} \Psi(x)\exp \left[ \beta \left( xC+x^{-1}C^{\dagger} \right) \right] =1.$$ Using this boundary condition and the known identity $$\frac{1}{x-y \mp i\epsilon}= P\frac{1}{x-y} \pm i\pi \delta(x-y)$$ one rewrites the Riemann-Hilbert problem as the integral equation $$\begin{split} \Phi(x)_{ij}= \delta_{ij}& +\frac{1}{2\pi i}\sum_{k} \int_{0}^{\infty} \frac{dy}{y-\zeta+i\epsilon} \Phi(y)_{ik} A_{kj}e^{-\beta(y\delta_{kj}+y^{-1}\bar{\delta}_{kj}} \\ & + \frac{1}{2\pi i}\sum_{k} \int_{-\infty}^{0} \frac{dy}{y-\zeta+i\epsilon} \Phi(y)_{ik} A^{\mathrm{t}}_{kj}e^{-\beta(y\delta_{kj}+y^{-1}\bar{\delta}_{kj}}, \end{split}$$ where $\delta_{kj}=w_{k}-w_{j}$. The Riemann-Hilbert problem has a unique solution given by the piecewise constant function $\Psi = (\Psi_{+}, \Psi _{−})$, where $$\Psi_{+}(x)=\Phi(x)\exp\left[-\beta\left( \zeta C+\zeta^{-1}C^{\dagger} \right) \right] .$$ The solution $\Psi = (\Psi_{+}, \Psi _{−})$ satisfy [@rif12; @rif33] $$\begin{split} & \Psi(x)\Psi^{t}(-x)=1 \\ & \overline\Psi(1/\bar{x})=g^{-1}\Psi(x), \end{split}$$ where the second relation means that in the canonical basis the complex conjugation acts on the vacuum wave functions as the ground state metric. Indeed, in this basis we have $\eta=1$ and so the ground state metric and the real structure coincide. From the above relations and the boundary condition \[conditmetric\] one can extract the ground state metric $$g_{i\bar{j}}=\lim_{\zeta\rightarrow 0}\Phi(x)_{i\bar{j}}.$$ The object which specifies the $tt^{*}$ solution is the monodromy of the D-brane states around the singular point $\zeta=0$, also known as the quantum monodromy [@rif12] $$H= S(S^{t})^{-1}.$$ According to the theory of isomonodromic deformations, the monodromy group of the Lax connection is invariant up to conjugacy under deformations of the parameters and can be computed in the limit that we prefer. The eigenvalues of $H$ are the phases $e^{2\pi i q^{R}_{j}}$ which encode the Ramond charges of the chiral primary fields at the UV conformal point. Given the dependence of the Stokes matrix on the BPS multiplicities, the quantum monodromy puts in relation the spectrum of solitons and chiral primaries, providing a map between IR and UV fixed points. The quantum monodromy is an element of $SL(N,\mathbb{Z})$ and may have non trivial Jordan blocks. It is known that a non trivial Jordan structure is related to logarithmic violations of scaling and reveals that the theory is asymptotocially free in the UV [@rif12]. ### Computing the $tt^{*}$ Monodromy Representation {#uvapproach} Our main problem is to compute the $SL(N,\mathbb{Z})$ monodromy representation of $\pi_{1}(\mathcal{P})$ given by the $tt^{*}$ Lax connection. We are mostly interested in the case in which the chiral ring of the theory is semi-simple. As we mentioned before, since $tt^{*}$ is an isomonodromic problem, we have the freedom to deform continuously the model in the coupling constant space. The monodromy groups we find will be related by an overall conjugation. In particular, the Jordan block structure and the eigenvalues of the monodromy matrices are invariant under finite continuous deformations. However, the useful limits in which the computations really simplify are typically singular limits in the coupling constant space. The monodromy representation that we find in these limits may be related to the original by a singular conjugation matrix. The eigenvalues of the matrices will not be modified, but the Jordan blocks may decompose in smaller ones. Indeed, having a certain spacetrum is a closed condition in the matrix space, while having Jordan blocks of dimension $>1$ is a closed condition. Therefore, the monodromy eigenvalues are typically easy to compute, while knowing the Jordan structure is subtler. However, in many situations we know a priori that the monodromy matrix is semi-simple and so we do not loose any information. For instance, as in the case relevant for the FQHE, when $\pi_{1}(\mathcal{P})$ is a complicated non-Abelian group the Jordan blocks are severely restricted by the group relations and so it is plausible that they can be recovered from the knowledge of the eigenvalues. The limits that are relevant for our purposes are the so-called asymmetric limit and the UV limit (in the $2d$ language). In the first approach we rescale the critical values $$w_{i}\rightarrow \beta w_{i}$$ so that the $tt^{*}$ flat connection becomes $$\nabla_{\zeta}=D + \frac{\beta}{\zeta}C, \hspace{1cm} \overline{\nabla}_{\zeta}= \overline{D}+\zeta \beta \overline{C},$$ while the Berry curvature changes as $$F=-\beta^{2}\left[ C, \overline{C}\right] .$$ The asymmetric limit consists in taking the unphysical limit $\beta\rightarrow 0$ and $\zeta\rightarrow \infty$ with $\beta/\zeta$ fixed. The Berry curvature vanishes in the limit, so the metric connection $A$ is pure gauge. The $tt^*$ linear problem \[linearprobl\] reduces formally to $$\left( \partial_{i}+ A_{i} + \frac{\beta}{\zeta}C_{i} \right) \Psi= \bar{\partial}_{i}\Psi=0.$$ A basis of solutions to this equation is given by the integrals $$\Psi_{i}^{a}= \int_{\Gamma_{a}} \phi_{i}(z)e^{\frac{\beta}{\zeta} W(z;w)}dz^{1}\wedge...\wedge dz^{n},$$ where the cycles $\Gamma_{a}$ are an integral basis of branes and $\phi_{i}(z)$ is a basis of the chiral ring. The homology classes of the branes with given $\zeta$ are locally constant in coupling constant space, but jump at loci where (in the 2d language) there are BPS solitons which preserve the same two supercharges as the branes [@rif12]. The jump in homology at such a locus are given by the Picard-Lefshetz transformations of the forms \[piclef\]. In the canonical basis of branes, namely the Lefshetz thimbles, the Stokes jumps can be written in terms of the soliton multiplicities computed in the IR limit of the $2d$ theory. Taking into account all the jumps in homology one encounters along the path (controlled by the 2d BPS spectrum), one gets the monodromy matrix which is automatically integral of determinant $1$. The full monodromy representation is given by the combinatorics of the PL transformations. This approach is convenient from the practical point of view and the monodromy matrices computed is this limit are manifestly integral. On the other side, the fact that we consider a limit which do not correspond to any unitary quantum system tends to make the physics somewhat obscure. For our present purposes the UV approach seems more natural. As discussed before, this just requires to send $\beta\rightarrow 0$, while the spectral parameter $\zeta$ remain fixed and can be taken in the unitary locus $\vert \zeta \vert=1$. One can see that in this limit the Lax connection reduces to the Berry one $$\nabla_{\zeta} +\overline{\nabla}_{\zeta} \xrightarrow{\beta \rightarrow 0} D + \overline{D}.$$ Hence, the Berry connection becomes flat in this limit. Since the monodromy of the flat connection is independent of $\beta$, the flat UV Berry connection should have the same monodromy modulo the subtlety with the size of the Jordan blocks. This concept can be clarified by comparing the quantum monodromy computed in the UV and IR limit of the theory. The holonomy on the $\zeta$-plane of the UV Berry connection, which corresponds to the Lax connection in the same limit, is conjugated to the unitary matrix $e^{2\pi i \boldsymbol{Q}}$. On the other hand, as we discussed previously, the quantum monodromy is computed in the IR limit in terms of the Stokes matrices as $H=SS^{-t}$. The two matrices have the same unitary spectrum, but for asymptotically free theories the IR monodromy has non trivial Jordan blocks. It is clear in general that the monodromy representation computed in the UV limit is unitary, since the Berry connection is metric, but differently from the homological approach the integral structure of the matrices is not manifest. In order to give an explicit description of the holonomy representation of the UV Berry connection we need additional details about the $tt^{*}$ geometry that we want to study. The Vafa Model and the Microscopic Physics of FQHE {#wedisc} ================================================== The Supersymmetric Structure of Landau Levels --------------------------------------------- In this section we start to explore the connections between supersymmetry and the physics of quantum Hall effect. It turns out that the structure of the lowest Landau level naturally admits a $\mathcal{N}=4$ supersymmetric description. One can describe the Landau levels of a single electron on a Riemann surface $\Sigma$ of arbitrary genus by rephrasing the construction in section \[eleclandau\] in a more geometrical language. We consider a complex line bundle $\mathcal{L}\rightarrow \Sigma $ with first Chern class $c_{1}(\mathcal{L})= \Phi/2\pi$, where $\Phi>0$ is the total magnetic flux through the surface $\Sigma$. The complex structure of the Riemann surface is irrelevant in the discussion and we can choose it according to convenience. A state of the system is represented by a smooth section $\psi: \Sigma\rightarrow \mathcal{L}$ of the line bundle. Every complex line bundle over a one dimensional complex manifold is holomorphic and so one can always find a local holomorphic trivialization. The line bundle is endowed with an hermitian metric $h$ which allows to define an inner product on the Hilbert space $${\ensuremath{\langle \psi_{1} | \psi_{2} \rangle}}= \int_{\Sigma} dzd\bar{z} h \ \psi_{1}^{*}\psi_{2}.$$ The lowest Landau level is defined as the kernel of the Hamiltonian operator $$\tilde{H}=-D_{z}D_{\bar{z}} + E_{0}$$ where $D$ is the connection with respect to the $U(1)$ structure group of the bundle. More precisely, $D$ is the Chern connection associated to $h$. In an holomorphic trivialization one has $$D_{z}=-h\partial_{z}h^{-1}, \hspace{1cm} D_{\bar{z}}=\partial_{\bar{z}}.$$ The magnetic field is encoded in the curvature of the connection, which is a closed and real $(1,1)$ form $$F_{h}=\bar{\partial}\partial \log h,$$ where $\partial, \bar{\partial}$ are the Dolbeault operators of the complex manifold. Choosing natural units, in the case of the complex plane one has $h= e^{-B\vert z \vert^{2}/2}$ and $F_{h}=\frac{B}{2}dzd\bar{z}$. The Riemann surface $\Sigma$ endowed with the curvature of the line bundle, which is positive for $B>0$, is naturally a Kahler manifold. The fact that the configuration space of an electron in a constant magnetic field is a Kahler manifold, in particular a Stein space with a globally defined Kahler potential $K=\log h$, provides a first important hint that quantum Hall effect may be related to $\mathcal{N}=4$ supersymmetry. Isomorphism classes of line bundles over a compact Riemann surface are in correspondence with the divisors of the surface modulo linear equivalence. These form a free abelian group on the points of the surface. We write a divisor $D$ as a finite linear combination of points $$D= \sum_{i} n_{i} p_{i},$$ where $p_{i} \in \Sigma$ and $n_{i}$ are integers. We demand the divisor to be effective, namely with $n_{i}>0$. Since the electrons are particles with spin, in this discussion we have also to endow the line bundle with a spin structure. This is associated with a divisor $S$ on the Riemann surface such that $2S$ is in the class of canonical divisors. The divisor identifying the twisted line bundle is the sum $D+S$ and is unique up to linear equivalence. The vacuum wave functions satisfy $D_{\bar{z}} \phi=0$ and so we have the definition $$\mathcal{H}_{\mathrm{LLL}}= \Gamma (\Sigma, \mathcal{L}(D+S)),$$ where $\mathcal{H}_{\mathrm{LLL}} $ is the lowest Landau level and $\Gamma (\Sigma, \mathcal{L}(D+S))$ is the space of holomorphic sections of the twisted line bundle $\mathcal{L}(D+S)$. The Riemann-Roch theorem for a compact genus-$g$ Riemann surface states that the dimension $\ell(D+S)$ of the vector space $\Gamma (\Sigma, \mathcal{L}(D+S))$ is $$\ell(D+S)= \mathrm{deg}(D+S)-g+1$$ where $\mathrm{deg}(D+S)$ is the degree of the divisor. This result holds as long as $D+S$ has degree at least $2g-1$. For definiteness we put the system in a finite box with periodic boundary conditions. This is equivalent to choose an elliptic curve $E$ as target manifold for the electron coordinate. It is convenient to pick an even spin structure associated to a divisor $S=p_{0}-q$, where $p_{0},q$ are distinct points on the elliptic curve which satisfy $2p_{0}=2q$. The choice of $p_{0}$ does not affect the discussion and we can translate it as we prefer. In the case of a genus-$1$ surface the above relation becomes $$\ell(D+S)= \mathrm{deg}(D+S)= \sum_{i}n_{i}.$$ Since $S$ has vanishing degree, the dimension of the lowest Landau level is entirely determined by $D$. This is consistent with the fact that the spin of the electrons in FQH systems is a frozen degree of freedom and is irrelevant for the physics of the ground state. From this equality we see that the divisor $D$ parametrizes topologically the magnetic flux of the system. In particular, the degree of the divisor is equal to the total magnetic flux $\Phi/2\pi$ which defines the degeneracy of the lowest Landau level. The divisor $D+S$ has a defining meromorphic section $\psi_{0}$ with zeros of order $n_{i}$ at $p_{i}$, a simple zero at $p_{0}$ and a single pole at $q$. The map $\psi\rightarrow \psi/\psi_{0}$ provides a canonical identification between holomorphic sections in $\Gamma (E, \mathcal{L}(D+S))$ and meromorphic functions on the elliptic curve with poles at most given by $D_{\infty}=D+p_{0}$ and vanishing in $q$.\ Taking the limit of infinite volume we recover a system of electrons moving on the complex plane. In this case the spin structure is associated to a divisor $S=-q$, where $q$ is a generic point on the plane. The contribution of $\mathrm{deg} \,S=-1$ cancels the $+1$ factor in the Riemann-Roch formula and we have again $$\ell(D-q)= \mathrm{deg}D= \sum_{i}n_{i}.$$ The states in the lowest Landau level are represented by elements $\psi \in \Gamma(\mathbb{P}^{1},\mathcal{L}(D-q))$, which are canonically associated by the map $\psi\rightarrow \psi/\psi_{0}$ to meromorphic functions on $\mathbb{P}^{1}$ with poles at most given by $D$ and vanishing at $q$.\ In order to get the Hilbert space of the $N$ electrons system we simply need to take the antisymmetric tensor product of the single particle space $$\mathcal{H}_{\Phi}= \bigwedge^{N} \Gamma (\Sigma, \mathcal{L}(D+S)),$$ where the dimension is given by Fermi statistics $\begin{pmatrix} \Phi/2\pi \\ N \end{pmatrix}$.\ We can show that the problem of studying the Landau levels of a charged particle in magnetic field is equivalent to find the vacua of a $\mathcal{N}=4$ supersymmetric Hamiltonian. In particular, the holomorphic structure of Landau levels require to have four supercharges. As we recalled in the previous section, a theory in supersymmetric quantum mechanics with four supercharges is specified by the choice of a Kahler potential $K$ and an holomorphic superpotential $W$. The Kahler potential prescribes an hermitian metric on the target space. Once the topology of the Riemann surface is fixed, the choice of the Kahler metric does not affect the structure of the vacua, which depends only on the superpotential. Hence, we can choose $K=\log h$ as globally defined Kahler potential on the Riemann surface. The number of vacua is given by the Witten index, which is equal to the number of zeros counted with multiplicitly of the $1$-form $dW$. In order to compare this description with the previous one we choose as target space the manifold $\mathcal{K}=E\setminus \mathrm{supp}\ F$, where $E$ is an elliptic curve and $F$ is an effective divisor on $E$. As we discussed above, a divisor identifies up to linear equivalence a line bundle with its set of holomorphic sections. However, for a linear combination of points on the Riemann surface one can associate also a $\mathcal{N}=4$ supersymmetric system. Given an effective divisor $D=\sum_{i}n_{i}p_{i}$, we assign a closed meromorphic $1$-form $dW$ on $E$ with zeros in $p_{i}$ of order $n_{i}$. We choose $W^{\prime}(z)$ such that its polar divisor is given by $F$ and in making a precise dictionary between the two models we can set $p_{0} \in \mathrm{supp}\ F$. The ground states of a $\mathcal{N}=4$ theory are in correspondence with cohomology classes of the susy charge $\overline{\mathcal{Q}}= \bar{\partial} + dW \wedge$ in the space of differential forms, which are labelled by holomorphic operators of the chiral ring $\mathcal{R}$. The meromorphic functions $\psi/\psi_{0}$ defines canonically a basis of wave forms for the supersymmetric vacuum space $\mathcal{V}$ through the map $$\frac{\psi}{\psi_{0}}\rightarrow \frac{\psi}{\psi_{0}} W^{\prime} dz + \overline{\mathcal{Q}}(....).$$ Despite we can write $\frac{\psi}{\psi_{0}} W^{\prime}dz=\overline{\mathcal{Q}}(\frac{\psi}{\psi_{0}} )$, the $1$-forms above define non trivial representatives in the $\overline{\mathcal{Q}}$ cohomology, since the meromorphic functions $\frac{\psi}{\psi_{0}}$ are singular at the zeros of the superpotential and so are not elements of the chiral ring.\ When the electrons move on the complex plane the corresponding $\mathcal{N}=4$ Landau-Ginzburg model is defined by a one-form $dW$ which is a rational differential on the plane with a pole of order $\geq 2$ at $\infty$ and zero-divisor $D=\sum_{i}n_{i}z_{i}$, namely $$dW(z)=\frac{\prod_{i}(z-z_{i})^{n_{i}}}{P(z)}, \hspace{0.5 cm} \mathrm{deg}P(z)\leq \sum_{i}n_{i}=\frac{\Phi}{2\pi}.$$ The prescription about the behaviour of $dW$ at $\infty$ implies that the scalar potential $\vert W^{\prime}\vert^{2}$ in the supersymmetric hamiltonian is non vanishing at infinity for all the complete Kahler metrics on the complex plane. This condition ensures the normalizability of the vacuum wave functions and the existence of an energy gap between the ground state and the first excited level of the Hamiltonian. Moreover, it guarantees the absence of run-away vacua in the $2d$ $\mathcal{N}=(2,2)$ version of the theory. In the correspondence with the Landau description we choose the reference point of the spin structure such that $q \not\in \mathrm{Supp} \ D\cup \left\lbrace \infty \right\rbrace$. As in the case of periodic boundary conditions, the meromorphic functions $\frac{\psi}{\psi_{0}}W^{\prime}$ define a basis of chiral operators labelling the supersymmetric vacua of the model. In conclusion, we have the isomorphisms of vector spaces $$\label{isom} U_{\Phi}: \mathcal{H}_{\mathrm{LLL}} \xrightarrow{\sim} \mathcal{V}, \hspace{0.5 cm} U_{\Phi}:\mathcal{H}_{\Phi} \xrightarrow{\sim} \mathcal{V}_{N}= \bigwedge^{N}\mathcal{V}$$ which confirms the first prediction in the list \[list\]. Comparison of the Hermitian Structures -------------------------------------- The correspondence between the lowest Landau level and the vacuum space of the LG model does not guarantee that the two systems have the same geometry of the vacuum bundle. One can conclude that the vacuum sector of a supersymmetric system and the lowest Landau level are two equivalent description of the same physical system if the above isomorphism is also an isometry between Hilbert spaces. If the norm of the Hilbert space is preserved by the above map, the Berry connection induced on the vacuum bundle and consequently the topological order of the ground state are the same in the two systems.\ We focus on the case of the complex plane as target manifold and consider a system with a single electron. The generalization to the multi-particle case is straightforward. In the Landau description the probability of finding an electron at a position $z$ is given by $$P(z)_{\mathrm{LLL}} = \vert f(z) \vert^{2} e^{-B \vert z \vert^{2}},$$ where $f(z)$ is an holomorphic function and $B>0$. On a macroscopic volume the probability measure satisfies $$\label{plll} \log P(z)_{\mathrm{LLL}} = -B\vert z \vert^{2} + \mathrm{subleading \ as \ } \vert z \vert \rightarrow \infty.$$ An exact identification between the hermitian structures of the Hilbert spaces is a too strong requirement. What we can ask is an equivalence in measure, i.e. measurements of long range observables on small but macroscopic domains $U \subset \mathbb{C}$ give the same answers on the two sides. This is enough for our purposes, since the long range observables are the ones that characterize the quantum topological order. According to the previous discussion, the supersymmetric representation of the Landau levels requires to introduce a meromorphic $1$-form $dW$ on the plane, whose primitive plays the role of superpotential. We consider a differential form $dW(z)$ with $\Phi/2\pi$ zeroes and polar divisor of the type $F=F_{f}+ 2\infty$. This has the form $$\label{modelledby} dW(z)= \left( \mu + \sum_{i=1}^{\Phi/2\pi} \frac{a_{i}}{z-\zeta_{i}} \right) dz, \hspace{0.5 cm} F_{f}=\sum_{i} \zeta_{i}$$ where $\mu, a_{i} \in \mathbb{C}^{\times}$ and $\zeta_{i}$ are all distinct. Since we can always redefine the overall phase of the superpotential, we assume $a_{i} \in \mathbb{R}$ without loss of generality. In order to reproduce a macroscopically uniform magnetic field we should take the residues $a_{i}$ all equal and consider a uniform distribution of the flux sources $\zeta_{i}$ in $\mathbb{C}$. The typical separation of the punctures should be much smaller than the size of the macroscopic domain $U \subset \mathbb{C}$ on which we want to measure observables. In the present context being macroscopic means $$\frac{1}{2\pi}\left( \mathrm{magnetic \ flux \ through \ U}\right) =\int_{U} \frac{B}{2\pi}= \# \left\lbrace \zeta_{i} \in U \right\rbrace \gg 1,$$ namely the domain $U$ contains a large number of fluxes. The supersymmetric wave functions have the form $$\psi(z)_{\mathrm{SUSY}}= \Phi(z)dz + \tilde{\Phi}(z)d\overline{z}$$ and the corresponding probability distribution is $$P(z)_{\mathrm{SUSY}}= \vert \Phi(z) \vert^{2} + \vert \tilde{\Phi}(z) \vert^{2}.$$ We can choose a real basis of wave functions such that $\tilde{\Phi}(z)=\overline{\Phi}(z)$. Then, the Schroedinger equation for the zero energy levels is [@rif14] $$\left( -\frac{\partial^{2}}{\partial z \partial\bar{z}} + \bigg\vert \frac{dW}{dz} \bigg\vert^{2}\right) \frac{\Phi}{W^{\prime}}=0.$$ An asymptotic behaviour of the solution in the macroscopic limit which is consistent with the Schroedinger equation is $$\Phi(z)= e^{\pm 2 \mathrm{Re} W(z) + \mathrm{\ subleading \ as \ } \vert z \vert \rightarrow \infty},$$ where the subleading terms are smooth and bounded functions in the domain $U$ which ensure that the wave function is single-valued and normalizable on the whole complex plane. We notice that the function $$2 \mathrm{Re} \, W= \mu z + \overline{\mu z}+ \sum_{i}a_{i}\log\vert z-\zeta_{i}\vert^{2}$$ is the electrostatic potential of a system of point-like charges of size $a_{i}$ at positions $\zeta_{i}$ superimposed to a constant background electric field $\mu$. When averaged on a macroscopic region $U$, it looks like the potential for a continuous charge distribution with density $\sigma(z)$ such that $$\int_{U}d^{2}z \ \sigma(z)= \sum_{\zeta_{i} \in U}a_{i}$$ for any $U \subset \mathbb{C}$. The conclusion is that for any macroscopic domain $U\subset \mathbb{C}$ we have[^1] $$\begin{split} \mathrm{magnetic \ flux \ through \ U} =& \, \frac{i}{2}\int_{U}\bar{\partial}\partial \log P(z)_{\mathrm{SUSY}} \approx \pm i \int_{U} \bar{\partial} \partial (2 \mathrm{Re} \, W)= \\ \\ & \mp 2\pi\sum_{\zeta_{i} \in U} a_{i}, \end{split}$$ where in the last equality we used the Poisson equation of $2d$ electrostatics. It is clear that the background electric potential does not contribute to the magnetic flux of the system. We see that $\log P(z)_{\mathrm{SUSY}}$ matches the behaviour of $\log P(z)_{\mathrm{LLL}}$ in \[plll\] when averaged on any macroscopic domain $U \subset \mathbb{C}$ iff we set respectively $a_{i}=-1$ or $a_{i}=+1$ for all $\zeta_{i}$. The two choices are related by a change of orientation. We fix the conventions so that the external magnetic field is modelled in the susy side by \[modelledby\] with $a_{i} = -1$ for all $i$. Including Defects and Interactions ---------------------------------- It is very natural to introduce defects on the supersymmetric side. One just need to flip the sign of the residues $a_{i}$ for a bunch of punctures. Now there is a small mismatch between the number of vacua and the effective magnetic field measured by the fall-off of the wave function at infinity: we have two extra vacua per defect. The extra vacua are localized near the position of the corresponding defect in the plane and may be interpreted as internal states of the defect. We identify these defects with the quasi-holes of FQHE.\ We have shown in the previous sections that the low energy physics of a system of charged electrons in a uniform magnetic field is described at large $B$ by a quantum system with Hilbert space $\mathcal{H}_{\Phi}$. Moreover, this system admits two equivalent descriptions which are related by the isomorphism $U_{\Phi}$ in \[isom\]. In particular, the original FQHE Hamiltonian $H_{\mathrm{FQHE}}=U_{\Phi}^{-1}\hat{H}U_{\Phi}$ is mapped to some Hamiltonian $\hat{H}$ which can be seen as a deformation of a $4$-SQM model. The free part of the Hamiltonian si supersymmetric and the corresponding superpotential is given by the sum of $N$ copies of the single particle superpotential describing the interaction between a single electron and the punctures. The interacting part of the Hamiltonian can be splitted in two groups: the interactions which preserve supersymmetry and the ones which are susy-breaking. Including the susy-preserving interaction we get a differential $d\mathcal{W}$ of the form $$d\mathcal{W}= \sum_{i}^{N}\left( dW(z_{i})+ \sum_{a=1}^{h}\frac{dz_{i}}{z-x_{a}}\right) + \sum_{i=1}^{N}U_{i}(z_{1},...,z_{N})dz_{i}.$$ The term $dW(z)$ models the interaction between the electrons and the macroscopic magnetic field. The Vafa proposal for this term is $$dW_{\mathrm{Vafa}}(z)=-\sum_{k=1}^{\Phi/2\pi-h} \frac{dz}{z-\zeta_{k}},$$ where $\zeta_{k}$ form a regular lattice. Working on the plane is convenient to add a constant to $dW_{\mathrm{Vafa}}(z)$ in order to regularize the double pole at infinity. This may be seen as integration constant for the electrostatic Poisson equation satisfied by $2\mathrm{Re} W$ and does not affect the magnetic flux of the system. Hence, one obtains $$dW(z)= dW_{\mathrm{Vafa}}(z)+ \mu dz$$ where $\mu\neq 0$. The meromorphic $1$-form $U_{i}dz_{i}$ describes the Coulomb interaction between an electron and the other ones. As a function of the position $z_{i}$ of the i-electron at fixed $z_{j\neq i}$, this form can have poles only when $z_{i} = z_{j}$. Generically $U_{i}dz_{i}$ has only simple poles and the residues must be entire bounded functions on the plane, namely they are constants. Since $\mathcal{W}$ must be symmetric under permutation of the electron coordinates, the most general interacting part of the superpotential reads $$d\mathcal{W}_{\mathrm{int}}= 2\beta\sum_{1\leq i<j\leq N} \frac{d(z_{i}-z_{j})}{z_{i}-z_{j}},$$ where $\beta$ is some complex coupling. We are mostly interested in FQHE on the plane and in such case $\beta$ can be left a generic coplex parameter. However, if we put the system on a finite box with periodic boundary conditions $\beta$ gets quantized to a rational number. In this case $d\mathcal{W}$ is a meromorphic $1$-form on $E^{N}$, where $E$ is an elliptic curve. If we project the model on the target space of a single electron $z_{i}$ by fixing the positions $z_{j},j\neq i$ of the other ones, we obtain a meromorphic differential on $E$ with poles at $\zeta_{k}$ with residue $-1$, at $x_{a}$ with residue $+1$ and at $z_{j}, j\neq i $ with residue $2\beta$. Since the sum of the residues of an elliptic function must vanish, we have the condition $$0= -\left( \frac{\Phi}{2\pi}-h \right) + h + 2\beta (N-1)\approx (2\beta \nu-1)\frac{\Phi}{2\pi},$$ where in the last equality we have used $N\gg 1$ and the FQHE relation $N=\nu \Phi/2\pi$. Hence, we obtain the quantization condition $$2\beta= 1/\nu \in \mathbb{Q}_{>0},$$ which is the value given in [@rif1]. So, the Vafa model on $E^{N}$ is described by the differential $$d\mathcal{W}=\sum_{i=1}^{N}\left( \sum_{a}U(z_{i},x_{a})-\sum_{k}U(z_{i},\zeta_{k}) + \frac{1}{\nu}\sum_{j\neq i}U(z_{i},z_{j})\right) ,$$ where $$U(z,w)=\frac{\wp^{\prime}(w/2)dz}{\wp(z-w/2)-\wp(w/2)}.$$ Emergence of a Single Vacuum ---------------------------- The isomorphism $U_{\Phi}$ maps the original Landau Hamiltonian to the new Hamiltonian $$\hat{H}=H_{\mathcal{W}}+ H_{\mathrm{su.br}},$$ which provides an equivalent description of FQHE systems. The piece $H_{\mathcal{W}}$ is the supersymmetric Hamiltonian associated to the superpotential $\mathcal{W}$ discussed previously, while $H_{\mathrm{su.br}}$ is the susy-breaking part. For large magnetic fields $H_{\mathcal{W}}$ is of order $O(B)$, while $H_{\mathrm{su.br}}$ is of order $O(1)$ and represents a small perturbation. However, this does not mean and we can neglect it when we study the topological order of FQHE. Indeed, the supersymmetric part alone cannot be in the same universality class of FQHE states. As we are going to discuss in the next section, the vacuum sector with Fermi statistics $\mathcal{V}_{\mathrm{Fer}}$ of the Vafa Hamiltonian has dimension $d=\begin{pmatrix} \Phi/2\pi \\ N \end{pmatrix}$. The vacuum bundle of the theory is an holomorphic vector bundle of rank $d$ which is endowed with the $tt^{*}$ flat connection $\nabla$ which extends holomorphically the Berry connection $D$. In particular, the topological order of $H_{\mathcal{W}}$ is captured by the monodromy representations of $\nabla$. According to the Laughlin argument in the original description of FQHE, the degeneracy of the ground state is lifted by the Coulomb interactions which select a single vacuum. In a similar way, the susy-breaking part of the Hamiltonian should select a unique ground state ${{\,| {\mathrm{vac}} \rangle}}$ in the Hilbert space $H_{\Phi}$ which encodes the topological quantum order of FQHE. The sub-bundle over the coupling constant space with fiber spanned by ${{\,| {\mathrm{vac}} \rangle}}$ is endowed with two canonical sub-bundle connections $\nabla^{\mathrm{vac}},D^{\mathrm{vac}}$ which are induced from $\nabla$ and $D$ respectively. In general the sub-bundle curvature is different from the curvatural of the original one and the monodromy of $\nabla^{\mathrm{vac}}$ is a priori neither well defined nor simply related to the one of $\nabla$. Hence, a priori there is no relation between the topological order captured by $H_{\mathcal{W}}$ and the FQHE one. In order to have such relation the following two conditions must be satisfied: - The monodromy representation of $\nabla$ must be reducible with an invariant sub-bundle of rank $1$. The fiber of this eigenbundle defines a unique preferred vacuum for the $\mathcal{N}=4$ Hamiltonian. - The physical vacuum ${{\,| {\mathrm{vac}} \rangle}}$ is mapped by the isomorphism $U_{\Phi}$ to the preferred vacuum of the susy Hamiltonian. The first question is purely related to the supersymmetric model. It is suggested in $\cite{rif1}$ that such a preferred vacuum exists and should correspond to the identity operator. While this sounds as a natural guess, it is in general not true in $tt^{*}$ geometry that the identity operator spans an invariant subspace of the flat connection. This is an extremely non trivial fact that we have to check by studying the monodromy representation of $\nabla$. The validity of the second point is based on the fact that the preferred vacuum, if it exists, should be the most symmetric one under permutations of the quasi-holes. Then one may argue euristically that [@rif1], indipendently from the details of the interactions between electrons[^2], as long as the susy-breaking part $H_{\mathrm{su.br}}$ respects the permutation symmetry of electrons and quasi-holes, the unique vacuum ${{\,| {\mathrm{vac}} \rangle}}$ will also be the maximally symmetric one. This state should correspond to the preferred vacuum of the susy Hamiltonian.\ The conclusion is that, under our mild assumptions, the quantum order of the FQHE is captured by the 4-susy SQM model proposed in [@rif1]. In the present discussion we are actually arguing more than this. Since our considerations do not depend on the details of the interactions between the electrons, we claim that the supersymmetric model represents the correct universality class of any multi-particle system in a strong uniform magnetic field. Symmetry and Statistics {#chap5} ======================= $tt^{*}$ Functoriality {#functandcovering} ---------------------- Supersymmetric quantum mechanics is functorial with respect to branched coverings [@rif10]. Let us consider a Landau-Ginzburg theory with target space $X_{1}$ and superpotential $W(x;\lambda)$ depending on a set of couplings $\lambda \in \mathcal{P}$. We denote with $f:X_{2}\rightarrow X_{1}$ an holomorphic but not globally invertible map between a covering space $X_{2}$ and $X_{1}$. We assume that the map is indipendent from the point of the coupling constant space. One can use $f$ to pull-back the LG model on $X_{2}$. Denoting with $x_{i}, y_{i}, \ i=1,...,n$ a set of coordinates on $X_{1},X_{2}$ respectively, the superpotential of the covering model can be obtained by making the substitution $x_{i}=f_{i}(y_{j})$, namely $$f^{*}W(x_{i})= W(f_{i}(y_{j}))=W_{f}(y_{j}).$$ For the susy charges we have $$\mathcal{Q}= f^{*}\mathcal{Q}= \partial+d\overline{W}_{f} , \hspace{1cm} \overline{\mathcal{Q}}= f^{*}\overline{\mathcal{Q}}= \bar{\partial}+dW_{f},$$ while the Lefschetz operators can be defined by pulling-back the Kahler form on the cover space. The algebraic relations satisfied by the generators of the supersymmetry algebra are preserved by the pull-back operation. The pull-back of the vacuum wave forms of the original theory are not in general ground states for the covering model. Given a basis of vacuum wave functions $\Psi_{k}, \ k=1,...,N_{1}$, the pulled-back forms $f^{*}\Psi_{k}$ are cohomologous to the true vacua on $X_{2}$. Indeed, the conditions solved by the vacua which are compatible with the pull-back are $$\begin{split} & \bar{\partial}_{f^{*}W} f^{*}\Psi_{k}= f^{*} (\bar{\partial}_{W}\Psi_{k})=0 \\ & \partial_{f^{*}W} f^{*}\Psi_{k}= f^{*} (\partial_{W}\Psi_{k})=0 \\ & f^{*}\omega \wedge f^{*}\psi_{k}= f^{*}( \omega \wedge \Psi_{k})=0, \end{split}$$ where $f^{*}\omega$ is the pulled-back Kahler form. The vacuum equations involving the charges $\mathcal{Q}^{\dagger},\overline{\mathcal{Q}}^{\dagger}$ depend D-term and in general are not preserved by the pull-back. The covering theory has generically more vacua than the original LG model. The number of vacua in the new theory $N_{2}$ is related to $N_{1}$ by $$N_{2}= \mathrm{deg}f \cdot N_{1} + \# \mathrm{zeroes} \ \mathrm{of} \ J_{f},$$ where $\mathrm{deg} f$ is the degree of the cover and $J_{f}$ is the Jacobian of $f$. From the pulled-back expression of the vacuum wave functions $$\begin{split} f^{*}\Psi_{k}= & f^{*}\left( \phi_{k}(z_{i}) dz_{1}\wedge....\wedge dz_{n} +\bar{\partial}_{W}(....)\right) \\ = & \phi_{k}(f_{i}(y_{j}))J_{f} \ dy_{1} \wedge....\wedge dy_{n} + \bar{\partial}_{W_{f}}(....) \end{split}$$ one can read how the chiral ring roperators of the vacua transform by pull-back $$f^{\sharp}(\phi_{k})= f^{*}(\phi_{k})J_{f}.$$ The linear map $f^{\sharp}: \mathcal{R}_{1}\rightarrow \mathcal{R}_{2}$ is an isometry for the topological metric $$\label{cond1} \langle f^{\sharp}(\phi_{k}), f^{\sharp}(\phi_{j}) \rangle_{X_{2}} = \langle \phi_{k} ,\phi_{j} \rangle_{X_{1}}.$$ and is compatible with the $\mathcal{R}_{1}$-module structure $$\label{cond2} f^{\sharp}(\phi_{k}\cdot \phi_{j})= f^{*}(\phi_{k})\cdot f^{\sharp}(\phi_{j}) \in \mathcal{R}_{2}.$$ The chiral ring $\mathcal{R}_{2}$ decomposes as direct sum $$\mathcal{R}_{2}= f^{\sharp}(\mathcal{R}_{1}) \oplus f^{\sharp}(\mathcal{R}_{1})^{\perp},$$ where $(\cdot)^{\perp}$ denotes the orthogonal complement with respect to the $tt^{*}$ metric. The $tt^{*}$ functoriality is the statement that $f^{\sharp}: \mathcal{R}_{1}\rightarrow \mathcal{R}_{2}$ is an isometry also for the $tt^{*}$ metric. In order to show this fact one has to check that the two $tt^{*}$ metrics solve the same equations and satisfy the same boundary conditions. Since the classes in $R_{2}$ of the operators $\partial_{\lambda}W$ belong to the subspace $f^{\sharp}(R_{1})$, the chiral ring coefficients $C_{\lambda}$ are functorial by \[cond2\]. Since also the topological metric $\eta_{ij}$ is functorial by the \[cond1\], we conclude that the $tt^{*}$ equations are preserved by $f^{\sharp}$. The boundary conditions which select the correct solution to the $tt^{*}$ equations are encoded in the 2d BPS soliton multiplicities. The BPS solitons are the connected preimages of straight lines in the $W$-plane ending at critical points [@rif12; @rif19]. Since the map $W_{f}:X_{2}\rightarrow \mathbb{C}$: factorizes through $W:X_{1}\rightarrow \mathbb{C}$, so do the counterimages of straight lines, and therefore the counting of solitons agrees in the two theories.\ $tt^{*}$ functoriality preserves also the integral structure of D-branes. If $B\subset X_{2}$ is a D-brane for the LG model on $X_{2}$, then $f(B)$ is a D-brane for the model on $X_{1}$. In particular, $B$ is a special Lagrangian submanifold of $X_{2}$ and this property is invariant under pull-back and push-forward operations by smooth functions. The functorial property of SQM regards also the integral pairing between branes and vacua $${\ensuremath{\langle f(B) | \Psi \rangle}}_{X_{1}}= {\ensuremath{\langle B | f^{*}\Psi \rangle}}_{X_{2}}.$$ One gives in general the following definition: a $tt^{*}$-duality between two $4$-susy theories is a Frobenius algebra isomorphism between their chiral rings $\mathcal{R}_{2}\rightarrow \mathcal{R}_{1}$ which is an isometry for the $tt^{*}$ metric and so for the brane amplitudes. $tt^{*}$ functoriality produces several interesting $tt^{*}$-dual pairs and for an appropriate choice of the respective D-terms implies the equaivalence of the full quantum theories. Covering Spaces {#form} --------------- ### Abelian Covers The functorial property of SQM is particularly important when we study Landau-Ginzburg models in which the target manifold $X$ is not simply connected and the superpotential differential $dW$ is a closed meromorphic $1$-form. Despite it is not possible to define a primitive $W$ on $X$, the model is still well defined, since the Hamiltonian and the susy charges depend only on the derivatives of the superpotential. However, in these systems one can consider deformations of the theory which are not described by operators in the chiral ring [@rif13]. For instance, the chiral operator associated to the RG flow deformation is precisely the superpotential, which is not an holomorphic function on $X$. In order to write and solve the $tt^{*}$ equations with respect to these variations we need to pull-back the model on the universal cover $\mathcal{K} $ of the target manifold. This can be defined in an abstract way as the space of curves $$\label{defcov} \begin{split} & \hspace{3cm}\mathcal{K}= \left\lbrace p:[0,1]\longrightarrow X, p(0)=p^{*} \in X \right\rbrace / \sim, \\ \\ & \sim: \hspace{2cm} p\sim q= \begin{cases} p(1)=q(1), \\ p \cdot q^{-1} =0 \ \mathrm{in \ the \ fundamental \ group} \ \pi_{1}(X,\mathbb{Z}). \end{cases} \end{split}$$ We can pull-back $dW$ on this space and give a formal definition of superpotential: $$W(p)= \int_{p} dW.$$ The fundamental group $\pi_{1}(X)$ plays the role of Galois group of the cover and is a symmetry group of the model on $\mathcal{K}$. The action of a loop generator $\ell$ of $\pi_{1}(X)$ on $W$ is $$\label{homologygenerator} \ell^{*}W(p)= \int_{\ell \cdot p} dW= W(p) + \int_{\ell} dW.$$ Compatibily with the definition of symmetry, the superpotential is left invariant up to a constant factor. The Galois group of the universal cover is generically non abelian and may not have unitary representations. In the context of quantum mechanics, since the symmetry transformations must be unitary, one considers tipically the abelian universal cover of the target space. We note that the action of the fundamental group is abelianized at the level of superpotential and the model naturally descends on the abelian universal cover. This space is the minimal simply connected cover of the target space on which $W$ is single-valued. The Galois group of the abelian cover is obtained by dividing the fundamental group $\pi_{1}(X)$ by the commutator subgroup $\left[ \pi_{1}(X),\pi_{1}(X)\right] $. This subgroup is normal in $\pi_{1}(X)$ and so the quotient $\pi(X)^{\mathrm{Ab}}=\pi_{1}(X)/\left[ \pi_{1}(X),\pi_{1}(X)\right]$ is still a Galois group. More precisely, it is the first homology group $H_{1}(X;\mathbb{Z})$ of the target space. By replacing the fundamental group with the homology group in \[defcov\], one gets the definition of abelian universal cover $\mathcal{A}$ of $X$. The first homology of the target space is an abelian group with a freely generated part and a torsion part. It is clear by the above formula that the torsion subgroup has a trivial action on the superpotential. Hence, one can consider only the torsion-free part $ H_{1}(X;\mathbb{Z})/\mathrm{tor}\simeq\mathbb{Z}^{b_{1}}$, where $b_{1}$ is the first Betti number.\ The vacuum space of the LG model on $\mathcal{A}$ decomposes in a direct sum of unitary irreducible representations of the homology group: $$\mathcal{V}_{\mathcal{A}}= \bigoplus_{\chi \in \mathrm{Hom}\left( H_{1}(X;\mathbb{Z}),U(1)\right) } \mathcal{V}_{\chi}, \hspace{1cm} \mathrm{dim}\mathcal{V}_{\chi}=d,$$ where $d=\mathrm{dim}\mathcal{V}_{X}$ is the dimension of the vacuum space of the theory on $X$. Identifying $H_{1}(X;\mathbb{Z})$ with $\mathbb{Z}^{b_{1}}$, the characters labelling the unitary representations $\mathcal{V}_{\chi}$ can be written as $$\chi_{\vec{\theta}}: \vec{n}\rightarrow e^{i \vec{n}\cdot\vec{\theta}}$$ and we can call $\vec{\theta}$-vacua the states in the eigenspace $\mathcal{V}_{\chi}=\mathcal{V}_{\vec{\theta}}$. Since it is metric, the orthogonal decomposition of the vacuum space in $\theta$-sectors is preserved by the parallel transport with the Berry connection. However, it is not generically left invariant by the $tt^{*}$ Lax connection. The subrepresentation of the $tt^{*}$ monodromy are associated with subgroups of the Galois group. Let $H\in \pi_{1}(X)^{\mathrm{Ab}}$ a subgroup and let $\mathcal{A}_{H}=\mathcal{A}/H$. We have an Abelian cover $\mathcal{A}_{H}\rightarrow X$ with Galois group $\pi_{1}(X)^{\mathrm{Ab}}/H$. This cover is not simply connected and the fundamental group $\pi_{1}(\mathcal{A}_{H})$ is the kernel of the surjective homomorphism $$\beta: \pi_{1}(X)^{\mathrm{Ab}}\rightarrow \pi_{1}(X)^{\mathrm{Ab}}/H,$$ which is precisely the normal subgroup $\mathrm{ker} \beta= H$. One can consistently formulate the $4$-SQM model on the target space $\mathcal{A}_{H}$. The vacuum space of the LG theory on $\mathcal{A}_{H}$ is identified with the $H$-invariant subspace of the theory on the universal cover $$\mathcal{V}_{H}=\bigoplus_{\chi: \chi\vert_{H}=\mathrm{trivial}} \mathcal{V}_{\chi}.$$ The covering map $\mathcal{A}\rightarrow\mathcal{A}/H$ allows to pull-back on $\mathcal{A}$ also the branes and the $tt^{*}$ brane amplitudes of the $\mathcal{A}/H$-theory. Hence, we know by $tt^{*}$ functoriality that $\mathcal{V}_{H}$ must be preserved by the $tt^{*}$ monodromy. We conclude that for each subgroup $H$ of $\pi_{1}(X)^{\mathrm{Ab}}$ we have a monodromy subrepresentation $\mathrm{Mon}_{H}$ of the Lax connection. Moreover, to a sequence of subgroups $$...\subset H_{k} \subset H_{k-1} \subset ....\subset H_{1} \subset H_{0}= \pi_{1}(X)^{\mathrm{Ab}},$$ there corresponds an inverse sequence of $tt^{*}$ monodromy representations $$\mathrm{Mon}_{H_{0}} \subset \mathrm{Mon}_{H_{1}} \subset ....\subset \mathrm{Mon}_{H_{k-1}} \subset \mathrm{Mon}_{H_{k}} \subset ....$$ where $\mathrm{Mon}_{H_{0}}$ is the monodromy representation for the original model defined on $X$.\ We can choose a subgroup $H$ of finite index in $\pi_{1}(X)^{\mathrm{Ab}}$ in such a way that $\pi_{1}(X)^{\mathrm{Ab}}/H$ is a finite Abelian torsion group. In this case the theory on $\mathcal{A}_{H}$ has finite Witten index $$d_{H}=\left[ \pi(X)^{\mathrm{Ab}} : H \right]\cdot d.$$ A periodic character $\chi$ of $\pi_{1}(X)^{\mathrm{Ab}}$ is identified by theta angles $$\vec{\theta} \in (2\pi \mathbb{Q})^{b_{1}}.$$ Denoting with $J_{\chi}$ the finite cyclic group generated by $\chi$ and $H_{\chi} = \mathrm{ker} \chi \in \pi_{1}(X)^{\mathrm{Ab}}$ the corresponding finite-index normal subgroup, we have $$\pi_{1}(X)^{\mathrm{Ab}}/H_{\chi} \simeq J_{\chi}.$$ In this case we may reduce from the infinite universal cover to a finite cover with Galois group $J_{\chi}$. From the physical viewpoint, torsion characters $\chi$ have the special property that they allow a consisten truncation of the chiral ring $\mathcal{R}_{\mathcal{A}}$ to a finite-dimensional ring $\mathcal{R_{\chi}}$. In this way the $\vec{\theta}$-vacua become normalizable, which is a basic requirement in quantum mechanics, while they are never normalizable for non-torsion $\chi$. ### $tt^{*}$ Equations with $\theta$-Vacua {#cov} On the universal cover we have more vacua than on the target space, but also a larger symmetry group to classify them. We denote with $z_{i},i=1,...,d$ the zeroes of $dW$ on $X$, which we assume to be non degenerate. Choosing a representative $p^{0}_{i}$ of $z_{i}$ on the abelian cover, all the critica points of $W$ can be obtained by composing $p^{0}_{i}$ with the generators $\ell_{1},...,\ell_{b_{1}}$ of the torsion-free part of the homology group $H_{1}(X,\mathbb{Z})/\mathrm{torsion}$. Hence, the points $z_{i}$ label equivalence classes of vacua which are isomorphic to the Galois group of the cover.\ A basis for the chiral ring of the theory on $\mathcal{A}$ can be constructed as follows. Since $X$ is a Stein space we can find a basis of holomorphic one forms $\rho_{k},k=1,...,b_{1} \in \Omega(X)$ dual to $\ell_{k}$ whose classes generates $H^{1}(X,\mathbb{Z})/\mathrm{tor}$. Since the homology has an integer structure, we can choose the one forms $\rho_{k}$ such that $$\int_{\ell_{k}}\rho_{k}= c_{k} \in \mathbb{Z}.$$ Without loss of generality we can normalize the constants $c_{k}$ to $1$. On $\mathcal{A}$ we can find holomorphic functions $h_{k}$ such that $dh_{k}=\rho_{k}$. Let $ \lbrace \phi_{a} \rbrace \in \mathcal{R}_{X}$ be holomorphic functions on $X$ forming a basis for the chiral ring of the model, with $\phi_{0}=1_{X}$ and relations $\phi_{a}\phi_{b}=C_{ab}^{c}\phi_{c}$. Then, let $\varpi:\mathcal{A}\rightarrow X$ be the projection map from the abelian universal cover to the target space. The holomorphic functions on $\mathcal{A}$ $$\Phi_{a}(\vec{\theta})= \varpi^{*}\phi_{a} \cdot e^{i \vec{\theta}\cdot \vec{h}}, \hspace{1cm} \vec{\theta} \in [0,2\pi)^{b_{1}},$$ form a basis for $\mathcal{R}_{\mathcal{A}}$ which is diagonal in the characters of the Galois group. Indeed, the action of a generator of the homology $\ell_{k}$ on the above state is $$\ell_{k}( \varpi^{*}\phi_{a} \cdot e^{i \vec{\theta}\cdot \vec{h}})= \varpi^{*}\phi_{a} \cdot e^{i \vec{\theta}\cdot ( \vec{h} + \int_{\ell_{k}}d\vec{h}) } =e^{i\theta_{k}} \varpi^{*}\phi_{a} \cdot e^{i \vec{\theta}\cdot \vec{h}}$$ The product table of $\mathcal{R}_{\mathcal{A}}$ is $$\Phi_{a}(\vec{\theta})\cdot \Phi_{a}(\vec{\varphi})= C_{ab}^{c}\Phi_{c}(\vec{\theta}+\vec{\varphi}).$$ This equation implies that the Ramond charges of the chiral operators are piece-wise linear functions of the angles. The same applies to the UV Berry connection $A(\vec{\theta})^{\mathrm{UV}}$, which is function of the $U(1)_{R}$ charges. The discontinuous jumps of $A(\vec{\theta})^{UV}$ correspond to gauge transformations. On the contrary, the characters of the monodromy representation are continuous. For generic $\vec{\theta}$ the eigenvalues of the monodromy matrices are distinct, and hence no Jordan blocks are present; at characters where we have jumps typically non-trivial Jordan blocks appear.\ Let $H$ a subgroup of $\pi_{1}(X)^{\mathrm{Ab}}$ and $J$ the subgroup of characters which are trivial on $H$. Then, the chiral ring $\mathcal{R}_{H}$ of the model on $\mathcal{A}_{H}$ is spanned by the chiral operators $$\lbrace\Phi_{a}(\vec{\theta})\rbrace_{\vec{\theta} \in J}.$$ Using the fact that $ \ell_{k}$ are symmetries of the model, one finds that the ground state metric diagonalizes with respect to the angles: $${\ensuremath{\langle \overline{\Phi_{j}(\vec{\theta})} | \Phi_{k}(\vec{\theta^{\prime}}) \rangle}}= \delta(\vec{\theta}-\vec{\theta^{\prime}}) g_{k,\bar{j}}(\vec{\theta}), \hspace{1cm} g_{k,\bar{j}}(\vec{\theta})= \sum_{\vec{r} \in \mathbb{Z}^{b_{1}}} e^{i \vec{\theta}\cdot \vec{r}} g_{k,\bar{j}}.$$ In presence of theta sectors, the complex conjugate operator $g^{*}(\vec{\theta})$ must be intended as the Fourier series of the complex conjugated coefficients, namely $$g^{*}(\vec{\theta})=[g(-\vec{\theta})]^{*}.$$ Hence, using the fact that $g(\vec{\theta})$ must be hermitian, the $tt^{*}$ reality constraint in the canonical basis becomes $$\label{odd} g(-\vec{\theta})^{t}=g(\vec{\theta})^{-1}.$$ We can use the action of the homology generators \[homologygenerator\] to compute the critical values of the superpotential. These reads $$W( (\ell_{1}^{n_{1}}....\ \ell_{b_{1}}^{n_{b_{1}}})\cdot p_{0}^{i})= (\ell_{1}^{n_{1}} ....\ \ell_{b_{1}}^{n_{b_{1}}})^{*}W(p_{0}^{i})= W(p_{0}^{i}) + \sum_{i=1}^{b_{1}} n_{i}\omega_{i},$$ where $\omega_{i}=\int_{\ell_{i}}dW$ are constants in the chiral fields. We choose $\omega_{i}$ as the first $b_{1}$ local coordinates on the coupling constant space $\mathcal{P}$ and denote with $t_{a} \in \mathcal{P}$ the remaining couplings such that $\partial_{t_{a}}W$ are well defined holomorphic functions representing elements of $\mathcal{R}_{X}$. The $tt^{*}$ metric can be thought as function of the variables $$(\omega_{i},\theta_{i}) \in (\mathbb{C}\times S^{1})^{b_{1}}$$ at fixed $t_{a}$. One can consistently define the action of the chiral operator $C_{\omega_{i}}$ associated to the coupling $\omega_{i}$ on the theta vacua. This can be see as a $U(d)$ covariant derivative in the $\theta_{i}$-direction $$\label{togeqn} C_{\omega_{i}}= \overline{D}_{\overline{\vartheta}_{i}}= \frac{\partial}{\partial\theta_{i}}+ M_{\omega_{i}}, \hspace{1cm} \overline{C}_{\overline{\omega}_{i}}=-D_{\vartheta_{i}} =\frac{\partial}{\partial\theta_{i}}- g\partial_{\theta_{i}}g^{-1}-g M^{\dagger}_{\omega_{i}} g^{-1},$$ where $M_{\omega_{i}}$ is a $d\times d$ matrix. At fixed $t_{a}$, the component of the $tt^{*}$ flat connection take the form $$D_{\omega_{i}}+ \frac{1}{\zeta}\overline{D}_{\overline{\vartheta}_{i}}= \boldsymbol{D}^{(\zeta)}_{1,i}, \hspace{1cm} \overline{D}_{\overline{\omega}_{i}}-\zeta D_{\vartheta_{i}}= \boldsymbol{D}^{(\zeta)}_{2,i}$$ We can see the $\vartheta_{i}$ as complex coordinates with real part $\theta_{i}$ and introduce the new complex coordinates $(\eta^{\zeta}_{i},\xi^{\zeta}_{i}), i = 1,...,b_{1}$ $$\eta^{\zeta}_{i} = \omega_{i} -\zeta \overline{\vartheta}_{i}, \hspace{1cm} \xi^{\zeta}_{i} = \overline{\omega}_{i} + \frac{1}{\zeta} \vartheta_{i},$$ which defines a $\mathbb{P}^{1}$ family of complex structures parametrized by the twistor variable $\zeta$ and a flat hyperKhaler geometry with holomorphic symplectic structures $d\xi_{i}^{\zeta}\wedge d\eta_{i}^{\zeta}$. The $tt^{*}$ Lax connection annihilates with spectral parameter $\zeta$ all the holomorphic coordinates $(\eta_{i}^{\zeta},\xi_{i}^{\zeta})$ $$\boldsymbol{D}^{(\zeta)}_{\alpha,i}\eta_{j}^{\zeta}= \boldsymbol{D}^{(\zeta)}_{\alpha,i}\xi_{j}^{\zeta}=0, \hspace{1cm} \alpha=1,2$$ which means that, in the complex structure $\zeta$, it is the $(0,1)$ part of a connection $\boldsymbol{A}$ on the hyperkahler space $(\mathbb{R}^{2}\times T^{2})^{b_{1}}$. The Lax equations $$\boldsymbol{D}^{(\zeta)}_{\alpha,i}\Psi(\zeta)=0, \hspace{1cm}\alpha=1,2$$ say that the brane amplitudes $\Psi(\zeta)$ are holomorphic in complex structure $\zeta$ and independent of $\mathrm{Im} \ \vartheta_{i}$ [@rif13]. The $tt^{*}$ equations then say that the curvature of the connection $\boldsymbol{D}(\zeta)$ on the flat hyperKahler manifold is of type $(1,1)$ in all complex structures, i.e. $\Psi(\zeta)$ is a section of a hyperholomorphic vector bundle [@rif13]. The hyperholomorphic condition, supplemented by the condition on translation invariance in $\mathrm{Im}(\vartheta_{i})$, is equivalent to the higher dimensional generalization of the Bogomolnji monopole equations on $(\mathbb{R}^{2} \times S^{1})^{b_{1}}$.\ The $tt^{*}$ geometry decomposes into an Abelian $U(1)$ monopole and a non-Abelian $SU(d)$ monopole. Restricted to the Abelian part, the $tt^{*}$ equations become linear. Writing $L(\vec{\theta})=-\log(\mathrm{det}g(\vec{\theta}))$, they read $$\begin{split} & \left( \frac{\partial^{2}}{\partial\omega_{i}\partial\overline{\omega_{j}}} + \frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}\right) L(\vec{\theta})=0 \\ \\ & \frac{\partial^{2}}{\partial t_{a}\partial \overline{\omega}_{j}}L(\vec{\theta})= \frac{\partial^{2}}{\partial t_{a} \partial \overline{t}_{b}}L(\vec{\theta})=0. \end{split}$$ The Abelian part of the Berry connection is $$A^{\mathrm{Ab}} = \partial L(\vec{\theta}) = \partial_{\omega_{i}} L(\vec{\theta})d\omega_{i}+ \partial_{t_{a}} L(\vec{\theta}) dt_{a}.$$ The $tt^{*}$ relation $[A_{\omega_{i}}, C_{t_{a}}] = [A_{t_{a}} , C_{\omega_{i}} ]$, together with \[togeqn\], implies $$\partial_{\theta_{i}} A_{t_{a}} =[A_{t_{a}},M_{\omega_{i}}]-[A_{\omega_{i}},C_{t_{a}}].$$ Taking the trace gives $\partial_{\theta_{i}} A^{\mathrm{Ab}}_{t_{a}} = 0$. Since $A^{\mathrm{Ab}}$ is odd in $\vec{\theta}$ by \[odd\], we conclude that the $t_{a}$-components of the $U(1)$ connection vanish.\ For the sake of comparison with the literature on representation of braid groups and the Knizhnik-Zamolodchikov equation [@rif61; @rif62] we state the above result in a different way. We write $q_{i} = e^{i\theta_{i}}$ for $i = 1, . . . , b_{1}$. The Frobenious algebra $\mathcal{R}_{\mathcal{A}}$ is a module over the ring $\mathbb{C}[q^{\pm 1}]$ of Laurent polynomials in $q_{1}, ... , q_{b_{1}}$. The isomorphism with the space of branes $$\mathcal{R}_{\mathcal{A}}\simeq \mathcal{B}_{\mathcal{A}}(\zeta)= H_{*}(\mathcal{A},\mathcal{A}_{\mathrm{Re}(\zeta W)};\mathbb{Z})\otimes_{\mathbb{Z}} \mathbb{C}$$ allows us to restrict the scalars to $\mathbb{Z}$. Thus, we have that $\mathcal{B}(\zeta)_{\mathcal{A}} \simeq \mathcal{R}_{\mathcal{A}} \simeq \mathcal{V}_{\mathcal{A}}$ is a free $\mathbb{Z}[q^{\pm 1}]$-module of rank $d$. Moreover, the $tt^{*}$ Lax connection provides the group homomorphism $$\rho: \pi_{1}(\mathcal{P})\rightarrow GL(d,\mathbb{Z}[q^{\pm 1}]).$$ The Problem of Statistics ------------------------- The concept of statistics is crucial in understanding the physics of FQHE. We want now to discuss this problem in the context of $\mathcal{N}=4$ SQM. We consider for simplicitly the case in which the target space is $\mathbb{C}^{N}$. We suppose to have a superpotential $W(z_{1},...,z_{N})$ which is symmetric under arbitrary permutations of the coordinates $z_{i}\leftrightarrow z_{j}$. In other words, the particles of our theory are indistinguishable. Since it is a symmetry of the model, the Hilbert space and in particular its subspace of vacua $\mathcal{V}$ must decompose in irreducible representations of the permutation group of $N$ object $S_{N}$ $$\mathcal{V}=\bigoplus_{\eta \in \mathrm{irrep}(S_{N})} \mathcal{V}_{\eta}.$$ The relevant representation for physical applications are the trivial representation $\mathcal{V}_{\mathrm{s}}$, which corresponds to symmetric wave functions, and the sign representation $\mathcal{V}_{\mathrm{a}}$, which corresponds to antisymmetric wave functions. The symmetric group acts also on the chiral ring $\mathcal{R}$, whose elements labels the zero energy solutions of the Schroedinger equation. Similarly to the Hilbert space, the chiral ring decomposes linearly as $$\mathcal{R}=\bigoplus_{\eta \in \mathrm{irrep}(S_{N})} \mathcal{R}_{\eta}.$$ The space of symmetric chiral operators $\mathcal{R}_{\mathrm{s}}$ is a ring, while all the other components, as the space of antisymmetric chiral operators $\mathcal{R}_{\mathrm{a}}$, are $\mathcal{R}_{\mathrm{s}}$-modules. One can see that the map between chiral operators $\phi \in \mathcal{R}$ and vacuum wave functions $$\phi \rightarrow \phi \ dz_{1}\wedge....\wedge dz_{N} +\overline{\mathcal{Q}}(....)$$ provides the isomorphism of $\mathcal{R}_{\mathrm{s}}$-modules $$\mathcal{R}_{\eta}\simeq \mathcal{V}_{\mathrm{a\cdot\eta}}.$$ This follows from the antisymmetric property of the fermionic operator $dz_{1}\wedge....\wedge dz_{N}$ in the vacuum wave form. Counter intuition, we claim that Fermi (Bose) statistics corresponds to have symmetric (antisymmetric) wave functions. To justify our definition let us count the number of ground states in an important special case. We consider the class of superpotentials which can be written as sum of single field superpotentials $$\mathcal{W}(z_{1},...,z_{N})=\sum_{i=1}^{N}W(z_{i}),$$ where each superpotential $W(z_{i})$ is defined on a target space $\mathbb{C}$. We denote with $\mathcal{R}_{1}$ the one-particle chiral ring and with $d=\mathrm{dim}\mathcal{R}_{1}$ the dimension of the single particle ground state. The chiral ring of the $N$-particle model is given by the tensor product $$\mathcal{R}= \bigotimes^{N} \mathcal{R}_{1}.$$ The symmetric and antisymmetric subspaces of $\mathcal{R}$ are $$\mathcal{R}_{\mathrm{s}}=\bigodot^{N} \mathcal{R}_{1}, \hspace{1cm} \mathcal{R}_{\mathrm{a}}=\bigwedge^{N} \mathcal{R}_{1},$$ which have dimensions respectively $$\mathrm{dim} \mathcal{R}_{\mathrm{s}}= \begin{pmatrix} N+d-1 \\ N \end{pmatrix}, \hspace{1cm} \mathrm{dim} \mathcal{R}_{\mathrm{a}}= \begin{pmatrix} d \\ N \end{pmatrix},$$ which correspond to Bose and Fermi statistics. The above relations remain true also if we add to the superpotential an arbitrary supersymmetric interactions which do not change the behaviour at infinity in field space, since the dimension of the chiral ring is the Witten index $I_{W}=d$. Using the isomorphism $\mathcal{R}\simeq \mathcal{V}$ we have $$\mathcal{V}_{B}= \bigodot^{N} \mathcal{V}_{1}, \hspace{1cm} \mathcal{V}_{F}= \bigwedge^{N} \mathcal{V}_{1}$$ and the $tt^{*}$ metric, connection and brane amplitudes are induced by the single particle ones.\ Coming back to the general case, one can see that any antisymmetric operator $\phi \in \mathcal{R}_{\mathrm{a}}$ can be written as $$\phi=\hat{\phi} \prod_{i<j}(z_{i}-z_{j}),$$ where $\hat{\phi} \in \mathcal{R}_{\mathrm{s}}$. This provides the definition of chiral ring of the Fermi model $$\mathcal{R}_{F}= \mathcal{R}_{\mathrm{s}}/\mathcal{I}_{\mathrm{Van}},$$ where $\mathcal{I}_{\mathrm{Van}}\subset \mathcal{R}_{\mathrm{s}}$ is the annihilator ideal of the Vandermonde determinant $\Delta(z_{i})=\prod_{i<j}(z_{i}-z_{j})$. We have the linear isomorphism of $\mathcal{R}_{\mathrm{s}}$-modules $$\mathcal{R}_{F}\simeq \mathcal{V}_{\mathrm{s}}=\mathcal{V}_{F}, \hspace{1cm} \mathrm{dim} \mathcal{R}_{F}= \begin{pmatrix} d \\ N \end{pmatrix}.$$ In order to study the fermionic sector of the theory we can use the functoriality of $tt^{*}$ geometry. A generic superpotential $\mathcal{W}(z_{1},...,z_{N})$ which is invariant under permutation of the $z_{i}$ can be rewritten as function of the elementary symmetric polynomials $$e_{k}=\sum_{1\leq i_{1}<....<i_{N}\leq N} z_{i_{1}}z_{i_{2}}....z_{i_{k}}, \hspace{0.5cm} e_{0}=1.$$ Hence, the superpotential $\mathcal{W}:\mathbb{C}^{N}\rightarrow \mathbb{C}$ factorizes through the branched covering map $E: z_{k}\rightarrow e_{k}$ of degree $N!$. One can pull-back the vacuum wave forms for the LG theory with superpotential $\mathcal{W}(e_{i})$ on the cover with coordinates $z_{i}$ $$\begin{split} \Psi(e_{1},...,e_{N})= & \ \hat{\phi}( e_{1},...,e_{N} ) de_{1} \wedge...\wedge de_{N} + \overline{\mathcal{Q}}(....) \\ = & \ \hat{\phi}(e_{1},...,e_{N}) \mathrm{det}\left( \frac{\partial e_{i}}{\partial_{z_{j}}} \right) dz_{1} \wedge...\wedge dz_{N} + \overline{\mathcal{Q}}(....) \\ = & \ \hat{\phi}(e_{1},...,e_{N}) \prod_{i<j} (z_{i}-z_{j})dz_{1}\wedge...\wedge dz_{N} + \overline{\mathcal{Q}}(...), \end{split}$$ which defines non trivial $\overline{\mathcal{Q}}$-cohomology classes for the vacua of the theory with superpotential $\mathcal{W}(z_{i})$. The above pull-back leads to a correspondence between the chiral ring $\mathcal{R}_{e}$ of the $\mathcal{W}(e_{i})$-model and the Fermi chiral ring $\mathcal{R}_{F}$ of the $\mathcal{W}(z_{i})$-theory. We have $$E^{\sharp}: {{\,| {h} \rangle}}_{e}\rightarrow {{\,| {h\delta} \rangle}}_{z}, \hspace{1cm} h \in \mathcal{R}_{e}= \mathbb{C}[e_{1},...,e_{N}]/\left( \partial_{e_{1}}\mathcal{W},...,\partial_{e_{N}}\mathcal{W}\right)$$ which implies the isomorphism $$E^{\sharp}(\mathcal{R}_{e})\simeq \mathcal{R}_{F}.$$ By $tt^{*}$ functoriality, the map $E^{\sharp}$ is also an isometry for the $tt^{*}$ metric and sets an equivalence between the Fermi sector of the original theory and the LG model with superpotential $\mathcal{W}(e_{i})$. $tt^{*}$ Geometry of the $\nu=1$ Phase {#nu1state} --------------------------------------- We want to study the fermionic sector of the $N$ particles LG theory with superpotential $$\label{initialtheory} \mathcal{W}(z_{1},...,z_{N})= \sum_{i=1}^{N}\left( \mu z_{i}+ \sum_{\ell=1}^{N}\alpha_{\ell} \log (z_{i}+\zeta_{\ell}) \right)$$ where we let $\alpha_{\ell}, \ \mu$ to be generic complex couplings and take $\zeta_{\ell}$ all distinct. The coupling constant manifold of the theory at fixed $\alpha_{\ell}, \ \mu$ is given by the space of $N$ distinct ordered points on $\mathbb{C}$ $$C_{N}=\left\lbrace (\zeta_{1},...,\zeta_{N}) \in \mathbb{C}^{N} \vert \zeta_{i}\neq \zeta_{j}, i\neq j \right\rbrace .$$ If we demand the residues $\alpha_{\ell}$ to be equal, the manifold of couplings is naturally projected on $Y_{N}=C_{N}/S_{N}$, namely the space of $N$ identical particles on $\mathbb{C}$. In parituclar, it is clear from the discussion in section \[wedisc\] that if we set the couplings $\alpha_{\ell}=-1$ we obtain the Vafa model for the $\nu=1$ phase of quantum Hall effect. Writing the superpotential as $$\mathcal{W}(z_{1},...,z_{N})= \sum_{i=1}^{N}\mu z_{i} + \sum_{\ell=1}^{N}\alpha_{\ell} \log \left( \prod_{i=1}^{N}(z_{i}+\zeta_{\ell}) \right)$$ we note that the argument of the logarithm can be expanded as $$\prod_{i=1}^{N}(z_{i}+\zeta_{\ell})= \sum_{k=0}^{N}e_{k}\zeta_{\ell}^{N-k}= P(\zeta_{\ell}).$$ The polynomials $P(\zeta_{\ell})$ are linear in the dynamical fields and depend holomorphically on the couplings. Hence, one can do a linear redefinition of variables $$e_{\ell}\rightarrow P(\zeta_{\ell})=P_{\ell}.$$ Rewriting the interaction $\mu \sum_{i} z_{i}$ as linear combination of $P_{\ell}$, the superpotential as function of the new variables reads $$\mathcal{W}(P(\zeta_{1}),...,P(\zeta_{N}))= \sum_{\ell=1}^{N} \left( \mu_{\ell}P_{\ell} + \alpha_{\ell}\log P_{\ell} \right).$$ We note that in the dynamical fields $u_{\ell}=P_{\ell}$ the superpotential of the theory can be written as sum of $N$ copies of the single field superpotential $$\label{singleparticlemodel} W(u)= \mu u + \alpha \log u .$$ From this example we see how much powerful $tt^{*}$ functoriality can be. In the starting theory \[initialtheory\] we have a large number of punctures, equal to the dimension of the lowest Landau level, and an equal number of electrons which are correlated by the Fermi statistics. On the other hand, in the $tt^{*}$ dual we have just a unit of magnetic flux concentrated in a single puncture and the particles are not correlated by the statistics. The $tt^{*}$ functoriality ensures that we can study the Berry connection and its holonomy representations in the $tt^{*}$ dual of the model. Since the Hilbert space of the dual model is simply the tensor producs of single particle Hilbert spaces, it is enough to study the $tt^{*}$ geometry of the LG theory with superpotential \[singleparticlemodel\].\ The equation for the vacua $\partial_{P_{k}}\mathcal{W}=0$ is given by a set of $N$ indipendent equations $$\mu_{\ell} + \frac{\alpha_{\ell}}{P_{\ell}}=0,$$ which has unique solution $$P_{\ell}=-\frac{\alpha_{\ell}}{\mu_{\ell}}.$$ As expected, the dimension of the Fermi chiral ring is $\mathrm{dim}\mathcal{R}_{F}=1$.\ We want to study the $tt^{*}$ geometry of the vacua for this theory. The essential couplings are the positions of the punctures $\zeta_{\ell}$ which are coordinates on the space $C_{N}$. Since we have a unique vacuum, the Berry connection must be Abelian. The chiral operators which describe the deformation of the theory with respect to $\zeta_{\ell}$ are $$\partial_{\zeta_{\ell}}\mathcal{W}= \sum_{i=1}^{N}\frac{\alpha_{\ell}}{z_{i}+\zeta_{\ell}}.$$ The chiral ring coefficients $C_{\ell}$ which represent the action of these operators on the vacua are $1\times 1$ matrices. Hence, all the commutators $\left[ C_{\ell}, C_{m}\right]$ are vanishing and the $tt^{*}$ equations read $$\bar{\partial}_{\ell} (g \partial_{m} g^{-1})=0,$$ where the ground state metric $g$ is just a real function of the couplings. This set of equations is solved by $$g=\vert f(\zeta_{1},...,\zeta_{N})\vert^{2},$$ where $f(\zeta_{\ell})$ is an holomorphic function of the couplings $\zeta_{\ell}$. We also have to impose the reality constraint $$\eta^{-1}g(\eta^{-1}g)^{*}= 1.$$ In the canonical basis the Frobenious pairing is the identity and we have $gg^{*}=1$. Since $g$ is a real function we conclude that $$g=1.$$ So, as expected for the $\nu=1$ phase, the Berry connection on $C_{N}$ is vanishing and its monodromy representation in the UV limit is trivial.\ We note that the target manifold is not simply connected and the theory admits non trivial theta sectors. The solution that we found above corresponds to the sector of the Hilbert space with trivial character. To study the dependence of $tt^{*}$ geometry on the theta angles we have to consider deformations of the theory which are not in the chiral ring. The corresponding couplings are the residues $\alpha_{\ell}$ of $d\mathcal{W}$ at the poles $\zeta_{\ell}$. The associated operators $$\partial_{\alpha_{\ell}}\mathcal{W}=\sum_{i=1}^{N} \log (z_{i}+\zeta_{\ell})$$ are not univalued functions on the target space so are not elements of $\mathcal{R}$. By the functoriality of $tt^{*}$ geometry, the ground state metric of the full theory factorizes as tensor product of metrics of the single field models with superpotential \[singleparticlemodel\]. This superpotential describes in $2d$ a free chiral superfield with a twisted mass. The corresponding $tt^{*}$ geometry has already been studied in [@rif13]. The model is defined on $\mathbb{C}\setminus\lbrace 0 \rbrace$ and the Galois group of the universal cover is generated by the loop which encircles the origin. The universal cover $\mathcal{A}$ is Abelian and is simply the complex plane. An explicit map between the covering space and the target manifold is given by the exponential $u=e^{Y}$. Pulling-back the model on the complex plane one has the superpotential $$W(Y)= e^{Y}+\alpha Y,$$ where the action of the Galois group is given by the shift $T:Y\rightarrow Y + 2\pi i$. We can easily construct the unique theta-vacua of this theory. We denote with ${{\,| {0} \rangle}}$ the vacuum state corresponding to some idempotent element $e_{0}$ of $\mathcal{R}_{\mathcal{A}}$. Then, the vacuum space is spanned by $${{\,| {x} \rangle}}= \sum_{n\in \mathbb{Z}} e^{2\pi i n x} \ T^{n}{{\,| {0} \rangle}},$$ where $x$ is the theta-angle normalized in the interval $[0,1]$. We see that the above state is an eigenstate of $T$ with eigenvalue the character $e^{-2\pi ix}$. Acting with the generator of the Galois group $T$ on the classical vacuum corresponding to ${{\,| {0} \rangle}}$ we obtain all the vacua of the covering model. We can set to $0$ the critical value corresponding to ${{\,| {0} \rangle}} $ by adding a constant to the superpotential. Then, the whole set of critical values is simply $$W_{n}= 2\pi i \alpha n.$$ We want to derive the $tt^{*}$ equation in the parameter $\alpha$ and study the critical limit of the solution. The chiral ring operator $C_{\alpha}$ acts on the theta-vacuum as differential operator in the angle $$C_{\alpha}{{\,| {x} \rangle}}= \sum_{n\in \mathbb{Z}} e^{2\pi i n x}2\pi i n T^{n}{{\,| {0} \rangle}}= \frac{\partial}{\partial x} {{\,| {x} \rangle}}.$$ We define the ground state metric on the vacuum bundle $$g(t,x)= {\ensuremath{\langle \overline{x} | x \rangle}} = e^{L(t ,x)},$$ where $L(t,x)$ is a real function of the angles and the RG scale $t=\vert \alpha\vert$. The $tt^{*}$ metric can be expanded in Fourier series as $$g(x) = \sum_{r \in \mathbb{Z}} e^{2\pi i r x} {\ensuremath{\langle \overline{r} | 0 \rangle}}.$$ In order to solve the $tt^{*}$ equation we have to impose the reality constraint of the metric. In presence of theta sectors the coplex conjugation of the metric must be intended as $$g^{*}(x)=[g(-x)]^{*}.$$ Hence, using the fact that $g$ is real, the $tt^{*}$ reality constraint in the canonical basis becomes $$g(-x)=g^{-1}(x),$$ which implies $$L(-x)=-L(x).$$ Thus, the $tt^{*}$ equation for ground state metric reads $$\left( \partial_{\alpha}\partial_{\bar{\alpha}}+ \frac{\partial^{2}}{\partial x^{2}}\right) L(t,x)=0.$$ which is a $U(1)$ monopole equation on $\mathbb{R}^{2}\times S^{1}$ as expected from the general discussion in \[cov\]. The solution can be expanded in terms of Bessel functions as $$L(t, x)= \sum_{m=1}^{\infty} a_{m}\sin( 2\pi m x )K_{0}(4\pi m t )$$ where the coefficients $a_{m}$ are determined by the boundary conditions. In the limit $t \rightarrow 0$ and for $x \neq 0$ we must have the asymptotics $$L(t, x) \overset{t \rightarrow 0}{\sim} -2 (q(x) - \hat{c}/2 )\log t$$ where $q(x)$ is the SCFT $U(1)$ charge of the chiral primary $e^{xY}$ at the UV fixed point. Since $e^{Y}$ has charge $1$, one has $q(x)= x \in [0,1]$ and $\hat{c}=q_{\mathrm{max}}=1$. Therefore, from the limit $$K_{0}(x) \overset{x \rightarrow 0}{\sim} - \log(x/2)$$ we get the relation $$\frac{1}{2} \sum_{m=1}^{\infty} a_{m} \sin( 2\pi m x)= \left(\frac{\lambda}{2\pi}-\frac{1}{2} \right) .$$ From this equality we learn that the coefficients $a_{m}$ are those of the Fourier expansion of the first periodic Bernoulli polynomial, i.e. $$a_{m}= -\frac{2}{\pi} \frac{1}{m}.$$ Consistently with the discussion in \[cov\], also for non vanishing theta angles the components of the $U(1)$ Berry connection in the $\zeta_{\ell}$-directions are vanishing. In the case of trivial character we recover the constant solution $g=1$. The Vacuum Space of the Interacting Theory {#fermtrunc} ------------------------------------------ ### The Heine-Stieltjes Problem We consider a LG model with $N$ chiral fields and superpotential differential $$\label{vafa} d\mathcal{W}=2\beta \sum_{1\leq i < j \leq N} \frac{d(z_{i}-z_{j})}{z_{i}-z_{j}} + \sum_{i=1}^{N} dW(z_{i}),$$ where $dW(z_{i})$ is a rational differential with $d$ zeros and a pole of order $1\leq \ell \leq d+2$ at $\infty$. In the most general case $dW(z_{i})$ has $p=d+2-\ell$ simple poles at finite points $\left\lbrace y_{1},...,y_{p} \right\rbrace \in \mathbb{C}$ which are all distinct, namely $$dW(z)= \frac{B(z)}{A(z)}dz, \hspace{1cm} A(z)=\prod_{s=1}^{p}(z-y_{s}),$$ for some polynomial $B(z)$ of degree $d$ coprime with $A(z)$. In the case of the Vafa model the residues are $\pm 1$ and $2\beta=1/\nu$. The fundamental degrees of freedom of this theory are not the fields $z_{i}$, but the symmetric polynomials $$e_{k}= \sum_{1\leq i_{1}<....<i_{k}\leq N} z_{i_{1}}....z_{i_{k}}.$$ Indeed the superpotential is manifestly invariant under $S_{N}$ permutations of the $z_{i}$ and can be rewritten in these coordinates. The target manifold of the theory is the configuration space of $N$ identical particles on $\mathbb{C}\setminus\left\lbrace \zeta_{\ell} \right\rbrace $, i.e. $$\mathcal{M}_{N}=\left\lbrace (z_{1},...,z_{N}) \in (\mathbb{C}\setminus\left\lbrace \zeta_{\ell} \right\rbrace )^{N} \vert z_{i}\neq z_{j} \right\rbrace / S_{N}.$$ In order to find the classical vacua of the theory one has to solve the equations $\partial_{e_{k}}\mathcal{W}(e_{j})=0$. This algebraic problem can be mapped to a well known differential problem [@rif23]. One can encode a $S_{N}$-invariant vacuum configuration $\lbrace X_{i}\rbrace^{S_{N}}$ in a polynomial of degree $N$ $$P(z)=\prod_{i=1}^{N}(z-X_{i})=z^{N}+\sum_{k=1}^{N}(-1)^{k} e_{k} \ z^{N-k}.$$ The equation for the configuration $\left\lbrace X_{i}\right\rbrace_{a} $ is equivalent to the differential equation for the polynomial $P_{a}(z)$ $$\label{lamè} 2\beta A(z)P_{a}^{\prime\prime}(z)+B(z)P_{a}^{\prime}(z)= f_{a}(z)P_{a}(z)$$ where $a=1,....,d$ labels the vacua of the theory and $f_{a}(z)$ is a polynomial of degree $d-1$ such that $P_{a}(z)$ solves the equation above. It is clear that counting the number of such solutions is equivalent to count the number of vacua up to permutation of the particles. The general problem of finding the couples $(f_{a}(z),P_{a}(z))$ which solve a second order differential equation of type \[lamè\] is called in mathematical literature the Heine-Stieltjes problem, and the generalized eigenvalues $f_{a}(z)$ are called van Vleck polynomials. If $dW$ is a generic rational differential, with just simple poles in $\mathbb{P}^{1}$, the equation \[lamè\] is a generalized $d$-Lamé equation. The $d$-Lamé equation [@rif56] corresponds to the special case in which we set $\beta = 1$ and $dW = d \log Q(z)$, where $Q(z)$ is a polynomial of degree d. Taking the same differential $dW$ , but choosing $\beta = −1$, the superpotential $\mathcal{W}$ becomes the Yang-Yang functional [@rif55; @rif57] (and its exponential the master function [@rif58; @rif59]) of the $sl(2)$ Gaudin integrable model on $V^{\bigotimes d}$, the Heine-Stieltjes equation is equivalent to the corresponding algebraic Bethe ansatz equations, and the roots of $P(z)$ are the Bethe roots [@rif55; @rif60]. In the case of the Vafa model, solving the Heine-Stieltjes equation is equivalent by construction to solve the equation $$\partial_{z_{i}}\mathcal{W}=\sum_{j\neq i}\frac{2\beta}{z_{i}-z_{j}} + \sum_{a}\frac{1}{z_{i}-x_{a}} - \sum_{\ell}\frac{1}{z_{i}-\zeta_{\ell}}=0$$ where the solutions are counted modulo permutations of $S_{N}$. The above equation is a generalization of the Algebraic Bethe Anzatz equation for the Gaudin model, where the $z_{i}$ are the analog of the Bethe roots. The Gaudin model arises from the semi-classical limit of the solutions to the Knizhnik-Zamolodchikov, and it is natural to expect that the relation remains valid in the present slightly more general context. The most relevant case for us is actually when the poles at $\infty$ in $\mathbb{P}^{1}$ is double. As we alrady observed, this is a very convenient limit. In this case the Heine-Stieltjes equation is a confluent generalized $d$-Lamé equation. The ODE isequivalent to the Bethe ansatz equation for the Gaudin model with an irregular singularity [@rif55]. A well known results from Heine-Stieltjes theory is that for generic couplings the number of solutions $(P(z),f(z))$ is at most $$d= \begin{pmatrix} N+d-1 \\ N \end{pmatrix}.$$ By definition, this is also the Witten index of the LG theory with superpotential $\mathcal{W}$ on the target space $\mathcal{M}_{N}$. ### Fermionic Truncation Following Vafa [@rif1], we wish to interpret the SQM model defined by the superpotential \[vafa\] on $\mathcal{M}_{N}$ as a theory of FQHE where $N$ electrons are coupled to $d$ units of magnetic flux and an external background electric field included in $dW(z)$, while the Vandermonde coupling describes the Coulomb interactions between the electrons. We observe that the multiplicity of vacua given by the Heine-Stieltjes problem corresponds to the Bose statistics and not the Fermi one. It is clear what happens when we take the limit of $\beta\rightarrow 0$: in a classical vacuum configuration for $\mathcal{W}$ the $z_{i}$ are close to the vacua of the single particle model and several of them may take different valures in the vicinity of the same one-field vacuum. Since these values differ by orders $O(\beta)$, in the $\beta\rightarrow 0$ limit the vacuum configurations are simply labelled by a set of positive integers $(N_{1},....,N_{d})$ which denote how many particles we put in the vacua of the single field model $z_{1},...,z_{d}$. A natural guess is that, in order to get the correct FQHE phenomenology, one should consider only the subspace $$\mathcal{V}_{\mathrm{Fer}}\subset \mathcal{V}, \hspace{1cm} \mathrm{dim}\mathcal{V}_{\mathrm{Fer}}=\begin{pmatrix} d \\ N \end{pmatrix}$$ which contains the vacua that survive in the $\beta\rightarrow 0$ limit. In this limit all other vacua $\in \mathcal{V}_{\mathrm{Fer}}^{\perp}$ escape at the infinite end of $\mathcal{M}_{N}$ where two or more $z_{i}$ coincide. The fermionic truncation from $\mathcal{V}$ to $\mathcal{V}_{\mathrm{Fer}}$ is geometrically consistent if and only if it is preserved under parallel transport by the $tt^{*}$ flat connection, i.e. if $\mathcal{V}_{\mathrm{Fer}}$ is a subrepresentation of the monodromy representation. Since the flat $tt^{*}$ connection is the Gauss-Manin connection of the local system on $\mathcal{P}$ provided by the BPS branes (for a fixed $\mathcal{\zeta} \in \mathbb{P}^{1}$), this is equivalent to the condition that the model has $\begin{pmatrix} d \\ N \end{pmatrix}$ preferred branes which remain regular as $\beta\rightarrow 0 $ spanning the dual space of $\mathcal{V}_{\mathrm{Fer}}$.\ The fermionic truncation has already been studied by Gaiotto and Witten in a strictly related context [@rif55]. They show that preferred branes with the required monodromy properties do exist. We review their argument in our notation. We assume that the rational one-form $dW$ has a double pole at infinity of strength $\mu$ and $d$ simple poles in general positions. Then we can write $$B(z)= \mu A(z) + \mathrm{lower \ degree}.$$ Hence, the Heine-Stieltjes equation becomes $$2\beta A(z)P(z)^{\prime \prime}+ \left( \mu A(z)+... \right) P^{\prime} (z)=\mu\tilde{f}(z)P(z),$$ where $ f(z)=\mu\tilde{f}(z)$. The monodromy representation is independent of $\mu$ as long as it is non-zero. Taking $\beta \sim O(1)$ and $\mu$ finite but very large (the reasonable regime for FQHE), the above equation up to a $O(1/\mu)$ correction becomes $$A(z)P^{\prime}(z)=\tilde{f}(z)P(z)$$ which implies that up to $O(1/\mu)$ corrections the zeros of $P(z)$ coincide with the zeros of $A(z)$, namely the positions of the punctures. At large $\mu$ these approximate also the zeros of $B(z)$, i.e. the vacua of the single field model. The fermionic truncation amounts to require that their multiplicities are at most one, namely the polynomials $P(z)$ and $P^{\prime}(z)$ are coprime. In this regime, the product of $N$ one-particle Lefshetz thimbles starting at distinct zeros of $B(z)$ is approximatively a brane for the full interacting model. While the actual brane differs from the product of one-particle ones by some $O(1/\mu)$ correction, they are equivalent in homology and this is sufficient to study the $tt^{*}$ monodromy. Since it is dual to the space of branes of the $N$-particles Fermi model, the fermionic sector $\mathcal{V}_{\mathrm{Fer}} \in \mathcal{V}$ defines by construction a sub-representation of the $tt^{*}$ monodromy. So, differently from the case of non-interacting electrons, the fermionic sector is selected by the $tt^{*}$ solution and not by the statistics. Braid Group Representations in $tt^{*}$ Geometry {#chap6} ================================================ To compute the monodromy representation of the Vafa model in the UV approach we need a more in-depth understanding of $tt^{*}$ geometry. It turns out that for a special class of theories the UV Berry connection is a Kohno connection [@rif62; @rif65]. This appears in the theory of the braid group representation [@rif61; @rif64] as a solution of the Knizhnik-Zamolodchikov equations [@rif63]. In this section we go through the details of this beautiful relation. $tt^{*}$ Monodromy and the Universal Pure Braid Representation -------------------------------------------------------------- Following the strategy discussed in \[uvapproach\] we rescale the critical values $w_{i}\rightarrow \beta w_{i}$. We consider $w_{i} \in \dot{\mathcal{P}}$, where $\dot{\mathcal{P}}$ is an open domain in the space of couplings such that the chiral ring is semi-simple. We note that if $w_{i} \in \dot{\mathcal{P}}$ also $\beta w_{i} \in \dot{\mathcal{P}}$ for all $\beta > 0$, so the limiting point indeed lays in the closure $\overline{\dot{\mathcal{P}}}$ of the semi-simple domain. As we approach the UV fixed point of the RG the $tt^{*}$ equations imply that $\bar{\partial}Q\rightarrow 0$. Since $Q$ is Hermitian, we have also $\partial Q\rightarrow 0$, so that $\lim_{\beta\rightarrow 0}Q$ is a constant matrix. Naively, to get the UV Berry connection we just replace this constant matrix in the the basic formula \[basicformula\]. However, this is not the correct way to define the $\beta\rightarrow 0$ limit. Indeed, the formula \[basicformula\] is derived in the canonical trivialization, which becomes too singular in the UV limit: the chiral ring $\mathcal{R}$ is believed to be regular (even as a Frobenius algebra) in the UV limit but, since the limit ring is no longer semi-simple, its generators are related to the canonical ones by a singular change of basis. A trivialization which is better behaved as $\beta\rightarrow 0$ is the natural one given by the orthogonal idempotents $e_{i}$. Starting from \[basicformula\] and performing the diagonal gauge transformation, we get $$A_{kl}=h_{k}Q_{kl}h_{l}^{-1}\frac{d(w_{k}-w_{l})}{w_{k}-w_{l}}-\delta_{kl}d \log h_{l},$$ where $h_{l}=\sqrt{\langle e_{l} \rangle}$. By the residue formula \[residue\], the norm $\langle e_{l} \rangle$ should be a meromorphic function of the critical coordinates with poles at $w_{l}=w_{i}$ for $l\neq i $. Indeed, the superpotential becomes degenerate when two critical points coincide and, for a strictly Morse superpotential, the approach of two critical values imply the same for the corresponding critical points. In the $\beta\rightarrow 0$ limit the connection $A_{kl}$ becomes locally a meromorphic one form with simple poles at $w_{l}=w_{j}$ for $l\neq j$ and invariant under $w_{j}\rightarrow w_{j}+c$ and $w_{j}\rightarrow \lambda w_{j}$. In addition, its contraction with the Euler vector $\xi=\sum_{k}w_{k}\partial_{w_{k}}$ has no poles. Hence, in the UV limit the Berry connection $\mathcal{D}=D+\bar{\partial}$ can be written locally as $$\label{berkohno} \mathcal{D}= d+ \sum_{1\leq i<j \leq d} B_{ij} \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}},$$ where $B_{ij}$ are holomorphic functions in $w_{j}-w_{k}$ homogeneous of degree $0$. Moreover, they should reproduce the correct quantum monodromy $$\label{corrqu} \exp \left[ \sum_{i<j}B_{ij}\right] = (-1)^{r}\exp \left[ 2\pi i \boldsymbol{\mathcal{Q}}\right] , \hspace{0.5cm} \mathrm{up \ to \ conjugacy},$$ where $\boldsymbol{\mathcal{Q}}$ is the $U(1)_{R}$ generator at the UV fixed point written in the natural basis which makes it symmetric traceless. The prefactor $(-1)^{r}$ is generated by the anomalous term $$\iota_{\xi}(-\delta_{kl}d \log h_{l})= \frac{r}{2}\delta_{kl}\frac{d\beta}{\beta},$$ where the norm $\langle e_{l} \rangle$ scales homogeneously as $\lambda^{-r}, \ r \in \mathbb{Q}$ under the rescaling $w_{i}\rightarrow \lambda w_{i}$. The matrices $B_{ij}$ are restricted to satisfy a set of equations which follow from the flatness condition $$\mathcal{D}^{2}=0,$$ as predicted by the $tt^{*}$ equations in the UV limit. Complete and Very Complete $tt^{*}$ Geometries ---------------------------------------------- We recall the definition of configuration space $C_{d}$ of $d$ ordered distinct points in the plane $$\mathcal{C}_{d}=\left\lbrace (w_{1},...,w_{d}) \in \mathbb{C}^{d} \vert w_{i}\neq w_{j} \ \mathrm{for} \ i\neq j \right\rbrace .$$ The cohomology ring $H^{*}(C_{d}, \mathbb{Z})$ is generated by the $\begin{pmatrix} d \\ 2\end{pmatrix}$ $1$-forms $$\omega_{ij}=\omega_{ji}= \frac{1}{2\pi i } \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}}$$ which satisfy the relations [@rif66] $$\omega_{ij}\wedge\omega_{jk}+ \omega_{jk}\wedge\omega_{ki}+ \omega_{ki}\wedge\omega_{ij}=0.$$ The fundamental group $P_{d}=\pi_{1}(C_{d})$ is called the pure braid group of $d$ strings. The configuration space of $d$ unordered points is the quotient space $$Y_{d}=C_{d}/S_{d}$$ and its fundamental group $B_{d}=\pi_{1}(Y_{d})$ is the Artin braid group of $d$ strings. It is an extension of the symmetric group $S_{d}$ by the pure braid group $$1\rightarrow P_{d} \xrightarrow{\iota} B_{d} \xrightarrow{\beta} S_{d}\rightarrow 1.$$ $B_{d}$ has a presentation with $d-1$ generators $\sigma_{i}$ and the relations $$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}= \sigma_{i+1}\sigma_{i}\sigma_{i+1}, \hspace{1cm} \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i} \ \mathrm{for} \ \vert i-j \vert \geq 2.$$ We define the critical value map $$w: \dot{\mathcal{P}}\rightarrow Y_{d}, \hspace{1cm} t\rightarrow \lbrace w_{1},...,w_{d} \rbrace.$$ For semi-simple chiral rings the critical values provides a set of local coordinates for the space of couplings, implying that the above map is an holomorphic immersion. We say that the $tt^{*}$ geometry is **complete** if, in addition, $w$ is also a submersion. This is equivalent to say that $w$ is a local isomorphism and so a covering map from $\dot{\mathcal{P}}$ to $Y_{d}$. The perturbations of the UV fixed point are generated by the chiral primary operators $\phi \in \mathcal{R}$. Not all these deformations can be included in the superpotential, since for some of them the couplings can be UV relevant and the theory can develop Landau poles. Hence, being complete means that all the chiral primary operators are IR relevant or marginal non-dangerous. In this case the dimension of the manifold of physical couplings is precisely $d$.\ We also say that $tt^{*}$ geometry is **very complete** if the canonical projection $p:C_{d}\rightarrow Y_{d}$ factors through the critical value map $w$. This means that the manifold $C_{d}$ plays the role of cover space for $\dot{\mathcal{P}}$ and denoting with $ s: C_{d}\rightarrow \dot{\mathcal{P}}$ the cover map we have $p=w \circ s$. If $tt^{*}$ geometry is very complete we can pull-back the vacuum bundle $\mathcal{V} \rightarrow\dot{\mathcal{P}}$ to a bundle over $C_{d}$ and consider the $tt^{*}$ geometry on the configuration space $C_{d}$. In a very complete $tt^{*}$ geometry, pulled-back to $C_{d}$, the local expression \[berkohno\] becomes global, since in this case the $w_{i}$ are global coordinates and the partials $\partial_{w_{i}} W$ define a global trivialization of the bundle $\mathcal{R} \rightarrow C_{d}$. In the very complete case the entries of the matrices $B_{ij}$ are holomorphic functions on $C_{d}$, homogenous of degree zero and invariant under overall translation, which satisfy \[corrqu\]. We conclude that the matrices $B_{ij}$ should be constant. In the general case the $B_{ij}$ are only locally constant and can have jumps under certain perturbations of the critical values $w_{k}$. Being very complete means essentially that the theory has no wall-crossing phenomena (see [@rif12]). The $B_{ij}$ are further constrained by the flatness condition $\mathcal{D}^{2} = 0$, which leads to the theory of Kohno connections [@rif65; @rif67]. If a connection $\mathcal{D}=d+A$ of the form \[berkohno\] satisfies $\mathcal{D}^{2}=0$, then the following relations hold $$\begin{split} \left[ B_{ij},B_{ik}+B_{jk}\right] = \left[ B_{ij}+B_{ik},B_{jk}\right] =0, \hspace{1cm} & \mathrm{for} \ i<j<k, \\ \left[ B_{ij},B_{kl}\right] =0, \hspace{5cm} & \mathrm{for \ distinct \ i,j,k,l.} \end{split}$$ The above equations are called the infinitesimal pure braid relations. A connection of the form \[berkohno\] where the constant $d\times d$ matrices $B_{ij}$ satisfy the above relations is called a rank-$d$ Kohno connection. A connection of this type defines a representation of the pure braid group $P_{d}$ of $d$ strings $$\sigma: P_{d}\rightarrow GL(d,\mathbb{C})$$ through the parallel transport on the configuration space $C_{d}$ $$\sigma: \gamma \in C_{d}\rightarrow P \exp (-\int_{\gamma} A) \in GL(d,\mathbb{C}).$$ The family of representations $\sigma$ parametrized by the matrices $B_{ij}$ satisfying the infinitesimal pure braid relations is called the universal monodromy [@rif67]. We conclude that for a very complete $tt^{*}$ geometry the UV Berry connection is the universal monodromy representation of the pure braid group $P_{d}$ of $d$ strings specialized to Kohno matrices $B_{ij}$ computed in terms of the $U(1)_{R}$ spectrum of the UV chiral ring. A very important class of very complete $tt^{*}$ geometries are the symmetric ones. In this case the Kohno matrices satisfy $$(B_{\pi(i)\pi(j)})_{\pi(k)\pi(l)}=(B_{ij})_{kl}, \hspace{1cm} \mathrm{for \ all \ } \pi \in S_{d}$$ In this case the connection $\mathcal{D}$ descends to a flat connection on a suitable bundle $\mathcal{V}\rightarrow Y_{d}$, providing a representation of the full Braid group [@rif62; @rif65; @rif67]. This happens when the critical map $w$ is an isomorphism. Ising Model and SQM ------------------- It is known that $\mathcal{N}=4$ SQM is closely related to the Ising model in $2d$ [@rif32]. The ising model consists in a free massive Majorana field $\Psi(w)$ which satisfies the Euclidean Dirac equation $$(\slashed{\partial}-m)\Psi(w)=0,$$ where $m$ is the mass of the field and $$\slashed{\partial}= \begin{pmatrix} 0 & \partial \\ \bar{\partial} & 0 \end{pmatrix}, \hspace{1cm} \Psi(w)= \begin{pmatrix} \Psi_{+}(w) \\ \Psi_{-}(w) \end{pmatrix}.$$ The space of solutions to the Dirac equation is endowed with an inner product, which coincide with the hermitian norm of the first quantized Hilbert space $$\Vert \Psi \Vert^{2}= \int \left( \vert \Psi{+} \vert^{2} + \vert \Psi_{-} \vert^{2}\right) d^{2}w,$$ where $\Psi_{+},\Psi_{-}$ must be $L^{2}$ functions on the plane. It is possible to rephrase the Dirac equation as cohomological problem. The Majorana field $\Psi(w)$ is not univalued on the $W$-plane, but has complicated branching properties because of the insertion of topological defect operators $O_{k}(w_{k})$ at points $w_{k}$. The OPE $$\Psi(w)_{\pm}O_{k}(w_{k})$$ is singular as $w\rightarrow w_{k}$. One considers two possible defect insertions at $w_{k}$, which we denote by $\sigma_{k},\mu_{k}$ [@rif32]. The OPEs of these operators with the Majorana field are $$\label{OPEs} \begin{split} & \Psi_{+}(u)\sigma_{k}(w)\sim \frac{i}{2} (u-w)^{-1/2}\mu_{k}(w), \hspace{0.5 cm} \Psi_{-}(u)\sigma_{k}(w)\sim -\frac{i}{2} (\bar{u}-\bar{w})^{-1/2}\mu_{k}(w) \\ & \Psi_{+}(u)\mu_{k}(w)\sim \frac{1}{2} (u-w)^{-1/2}\sigma_{k}(w), \hspace{0.5 cm} \Psi_{-}(u)\mu_{k}(w)\sim \frac{1}{2} (\bar{u}-\bar{w})^{-1/2}\sigma_{k}(w) \end{split}$$ up to $O(\vert u-w\vert^{1/2})$ contributions. Although the defect operators $\mu_{k}, \sigma_{k}$ have the same OPE singularities with the fermion field $\Psi(w)$ as the Ising order/disorder operators, they are not in general mere Ising order/disorder operators since globally they have different topological properties. The fermion field $\Psi(w)$ is univalued on a suitable connected cover of the plane punctured at the insertion points. By Riemann existence theorem [@rif68], this can be extended to a cover of the Riemann sphere $$W: \Sigma\rightarrow \mathbb{P}^{1}$$ branched at $\lbrace w_{1},...,w_{d}, \infty \rbrace $. In case of a cover of finite degree $m$, $f$ is specified by its Hurwitz data at the $(d + 1)$ branching points [@rif68]. The Hurwitz data consist of an element $\pi_{k} \in S_{m}$ for each finite branching point $w_{k}$, while $\pi_{\infty} = (\pi_{1}\pi_{2}.... \pi_{d})^{-1}$. The monodromy group of the cover is the subgroup of the permutation group $S_{m}$ generated by the $\pi_{k}$’s. When $m$ is infinite the monodromy group is an infinite group and the geometry is a bit more involved. Hence, the topological defect operator $O_{k}(w_{k})$ inserted at the $k$-th branching point is specified by the choice between $\sigma$-type and $\mu$-type and the monodromy element $\pi_{k} \in S_{m}$.\ One can replace the Majorana field $\Psi(w)$ with the $1$-form on the complex plane $$\psi(w) = \Psi_{+}dw+\Psi_{-}d\overline{w}.$$ The map $\Psi\rightarrow \psi$ is consistent with the structure of the Hilbert space of the Dirac equation. Indeed, the hermitian product of the wave functions is identified with the inner product of forms $$\label{isingprod} \Vert \Psi \Vert^{2} = \int \psi \wedge \ast \psi^{\ast}.$$ It is natural to introduce the operators $\overline{\mathcal{D}},\mathcal{D}$ which act on forms as $$\overline{\mathcal{D}}\psi=\bar{\partial}\psi + dw\wedge \psi, \hspace{1cm} \mathcal{D}\psi=\partial\psi + d\overline{w}\wedge \psi.$$ These operators are nilpotent $$\mathcal{D}^{2}= \overline{\mathcal{D}}^{2}= \overline{\mathcal{D}}\mathcal{D} + \mathcal{D}\overline{\mathcal{D}}=0,$$ and satisfy the Kahler-Hodge identites \[kahlid\] with the Lefschetz operators $L,\Lambda$ on the plane. In particular, we have $$\overline{\mathcal{D}}^{\dagger}=i \left[ \Lambda, \mathcal{D}\right] .$$ It is easy to see that the Dirac equation for $m=1$ can be written as the cohomological relations $$\label{compar} \mathcal{D}\psi= \overline{\mathcal{D}}\psi=0,$$ and using the above Kahler identity we also have $$\overline{\mathcal{D}}^{\dagger}\psi=0.$$ Hence, by standard Hodge theory, the normalizable solutions to the Dirac equation are the harmonic representatives of $\overline{\mathcal{D}}$-cohomology classes in the space of $L^{2}$ forms on the complex plane. One considers the following system of solutions [@rif32] $$\label{isingcorr} \begin{split} \psi_{i}(z;\zeta)= & \langle \Psi_{+}(W(z))\mu_{1}(w_{1})...\sigma_{j}(w_{j})...\mu_{d}(w_{d})\rangle \frac{W^{\prime}(z)dz}{\zeta \tau(w_{j})} + \\ & \langle \Psi_{-}(W(z))\mu_{1}(w_{1})...\sigma_{j}(w_{j})...\mu_{d}(w_{d})\rangle \frac{\overline{W}^{\prime}(\bar{z})\zeta d\bar{z}}{ \tau(w_{j})} \end{split}$$ where $z$ is a local coordinate on the cover $\Sigma$, $\zeta \in \mathbb{P}^{1}$ is an arbitrary parameter such that $\vert \zeta \vert =1$ and the normalization constant is the Sato-Miwa-Jimbo $\tau$-function $$\tau=\langle \mu(w_{1})...\mu(w_{n})\rangle.$$ The wave functions $\psi_{i}(w,\zeta)$ are singular at the branching points $w_{i}$. These singularities can be computed using the OPEs \[OPEs\] and encode the cohomology class of the $j$-th vacuum. We can use the hermitian scalar product \[isingprod\] to define a ground state metric on the Hilbert space spanned by the above Ising one-forms. Studying variations of the metric in the branching points $w_{k}$, one finds that the bundle with fiber spanned by the local system of solutions \[isingcorr\] has the same $tt^{*}$ geometry of the vacuum bundle of a $4$-SQM model with semi-simple $d$-dimensional chiral ring [@rif32]. In this formal correspondence the insertion points $w_{k}$ are identified with the critical coordinates on the coupling constant space of the SQM model. Hence, $4$-SQM and the off-critical Ising model provide two equivalent description of the same geometry.\ The relation between the Ising model on the $W$-plane and $4$-SQM is especially simple when the superpotential depends on a single chiral field $z$. Let us consider a $\mathcal{N}=4$ Landau-Ginzburg model on a Stein space $\Sigma$ and a Morse superpotential $W$. Comparing the expression of $\mathcal{D},\overline{\mathcal{D}}$ with the susy charges $$\mathcal{Q}= \partial + d\overline{W}\wedge, \hspace{1 cm} \overline{\mathcal{Q}}= \partial + dW\wedge,$$ we see that the holomorphic map $$W: \Sigma\rightarrow \mathbb{C}$$ provides the relations $$\mathcal{Q}=W^{*}\mathcal{D},\hspace{1cm} \overline{\mathcal{Q}}=W^{*}\overline{\mathcal{D}}.$$ Namely, one can map the Schroedinger equation for the vacua of the LG model on $\Sigma$ to the Euclidean Dirac equation on the $W$–plane. In this correspondence, the one-form $\psi_{i}(z,\zeta)$ is identified with the Schroedinger wave-function of the $i$-th vacuum corresponding to the idempotent $e_{i} \in \mathcal{R}_{w}$.\ We see that the formulation in critical coordinates is universal. The informations which distinguish different models and make precise the dictionary between SQM and Ising model are encoded in the Hurwitz data. The wave functions are multi-valued on the $W$-plane with branching points just given by $w_{k}$. The Hurwitz data should be chosen such that $\psi_{i}(z,\zeta)$ are univalued in the coordinate $z$ of the target manifold $\Sigma$. This space can be seen as a cover of the $W$-plane with cover map $W : \Sigma \rightarrow \mathbb{C}$ branched at the critical points $w_{k}$. By the Riemann existence theorem, the Hurwitz data specify uniquely up to isomorphism the cover $(W,\Sigma)$ of the punctured plane.\ A similar formula for the vacuum wave functions in the $N$-fields case can be constructed as follows. We consider the inverse image of a point $w$ on the $W$-plane. This has the homotopy type of a bouquet of $N-1$ spheres [@rif12; @rif19]. We set a basis of cycles $S_{\alpha}(w), \alpha=1,...,d$ for the middle dimensional homology of the fiber. A vacuum wave function $\Psi$ is a $N$-form on the target space $X$. Pulling-back $\Psi$ on the $w$-level curve of the superpotential and integrating it over the cycles $S_{\alpha}(w)$, we get a set of $d$ one-forms on the $W$-plane $$\label{singlebutmulti} \psi_{\alpha}(w)= \int_{S_{\alpha}(w)}\Psi.$$ If we transport the homology cycles along a closed loop along in the $W$-plane (punctured at the critical values) we come back with a different (integral) basis of $(N-1)$-cycles $S_{\alpha}^{\prime} (w) = M_{\alpha\beta}S_{\beta}(w)$. The integral matrix $M_{\alpha \beta}$ is described by the Picard-Lefshetz theory [@rif12]. Thus, the \[singlebutmulti\] is best interpreted as a single but multivalued wave-function $\psi(w)$ on the $W$-plane branched at the critical points, whose monodromy representation is determined by the Picard-Lefshetz formula. Let $\Sigma$ be the minimal branched cover of the $W$-plane such that $\psi_{\alpha}(w)$ is single valued. The map $W:X\rightarrow \mathbb{C}$ factorizes through $\Sigma$. Pulling-back the wave forms on $\Sigma$ by the covering map $w:\Sigma\rightarrow \mathbb{C}$, one can replace the original LG model with $N$ chiral fields and Morse superpotential $W$ with an LG model with target space $\Sigma$ and superpotential $w$ given by the factorization of $W$ thorugh $\Sigma$. Hence, we can use the formula for the single field case and write $$\begin{split} \psi_{i}(w;\zeta)= & \langle \Psi_{+}(w)\mu_{1}(w_{1})...\sigma_{i}(w_{i})...\mu_{d}(w_{d})\rangle\frac{dw}{\zeta \tau(w_{j})}+ \\ & \langle\Psi_{-}(w)\mu_{1}(w_{1})...\sigma_{i}(w_{i})...\mu_{d}(w_{d})\rangle \frac{\zeta d\bar{w}}{ \tau(w_{j})}. \end{split}$$ The branes $B_{k}(\zeta)$ of the model on the $W$-plane are straight lines starting at the critical points $w_{k}$ and strethced towards the $\mathrm{Re} \ w/\zeta=-\infty$ direction. Denoting with $\Gamma_{k}(\zeta)$ the support of $B_{k}$ on $\Sigma$, we can define the brane amplitudes $${\ensuremath{\langle \Gamma_{k}(\zeta) | \Psi_{i} \rangle}}= \int_{\Gamma_{k}(\zeta)} e^{\beta w/\zeta + \beta \zeta \bar{w} }\psi_{i}(w;\zeta).$$ In the physical $2d$ (2,2) LG model, the UV limit consists in sending to zero the overall coupling $\beta$. But, as one can see comparing the Dirac equation with the vacuum equation \[compar\], $\beta$ is also the mass $m$ of the Majorana fermion in the context of the QFT on the $W$-plane. Hence, the physical UV limit of the $2d $ LG model coincides with the UV limit of the fermion theory on the $W$-plane. As $\beta\rightarrow 0$ the left and right modes of the fermion $\Psi$ decouple and the multi-valued correlation functions become sums of products of left/right conformal blocks. The statement holds (roughly) for all $tt^{*}$ quantities: in the UV they become some complicate combination of conformal blocks. Then the differential equations they satisfy, the $tt^{*}$ equations, should be related in a simple way to the PDEs for the conformal blocks. From this viewpoint the fact that in the UV limit the Berry connection, coinciding with the $tt^{*}$ Lax one in the $\beta\rightarrow 0$ limit, has the Kohno form, which is typical of the monodromy action on conformal blocks, is not surprising. In connecting the $tt^{*}$ monodromy with the braid representation of conformal blocks, we need to use the precise disctionary between the two. From \[isingcorr\] we see that the wave functions, being normalized, are to be seen as ratios of $n$-point functions in the $W$-plane CFT $$\frac{\langle \Psi_{\pm}(w)\mu_{1}(w_{1})...\sigma_{i}(w_{i})...\mu_{d}(w_{d})\rangle}{\langle \mu_{1}(w_{1})...\mu_{n}(w_{n})\rangle}$$ rather than correlators. Hence, the actual braid representation on the CFT operators is the $tt^{*}$ one twisted by the one defined by the $\tau$-function. Hecke Algebra Representations of the $tt^{*}$ Connection {#hecke} -------------------------------------------------------- We want to determine the matrices $B_{ij}$ for a very complete $tt^{*}$ geometry. $B_{ij}$ is the residue of the pole of the UV Berry connection as $w_{i}\rightarrow w_{j}$. In order to compute $B_{ij}$ it is enough to consider this limit from the point of view of the $2d$ model. Without loss of generality, we may deforme the $D$-terms so that the only light degrees of freedom are the BPS solitons interpolating between vacua $i$ and $j$ of mass $2\vert w_{i}-w_{j} \vert$. We may integrate out all other degrees of freedom and we end up with an effective IR description with just these two susy vacua. The theory that we get may be not UV complete, but this is not an issue, since it is just an effective description valid up to some non-zero energy scale. From the viewpoint of SQM, the $2d$ BPS solitons are BPS instantons. The effect of these BPS instantons is to split the two vacua not in energy as it happens in non-susy QM, but in the charge $q$ of the $U(1)_{R}$ symmetry which emerges in the $w_{i}-w_{j} \rightarrow 0$ limit. In this limit there is also an emergent $\mathbb{Z}_{2}$ symmetry which interchanges the classical vacua. The $U(1)_{R}$ symmetry is broken off-criticality to $\mathbb{Z}_{2}$ and the eigenstates are the symmetric and antisymmetric combination of the classical vacua [@rif10; @rif12]. The corresponding charges must be opposite by PCT symmetry. We renumber the critical values $w_{k}$ such that $w_{i},w_{j}$ are $w_{1},w_{2}$. With a convenient choice of the relative phases of the two states, the upper-left $2 \times 2$ block of the $U(1)_{R}$ generator $\boldsymbol{\mathcal{Q}}$ takes the form $$\boldsymbol{\mathcal{Q}}_{\mathrm{upper-left}}= -\lambda \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}= -\lambda\sigma_{1},$$ for some constant $\lambda$. We have to remember that in the trivialization given by the idempotent basis the UV Berry connection gains the extra piece $\delta_{kl}d\log h_{l}$ which generates an additional term in the above formula proportional to the identity. To be general we consider a shift by a constant $\mu$ $$-\lambda \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix} \rightarrow -\lambda \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix}+ \mu \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$ From the Ising model point of view, at each of the two critical points $w_{1,2}$ we may insert either a $\sigma$-like defect or a $\mu$-like one and different choices correspond to different vacua of the original LG theory. The matrix $\sigma_{1}$ has the effect of flipping the two-vacua system, which corresponds to the exchange $ \sigma\leftrightarrow \mu$ in the correlators. It is therefore convenient to introduce a two-component notation $$\Sigma_{k,\alpha}(w)= \begin{pmatrix} \sigma_{k}(w) \\ \mu_{k}(w) \end{pmatrix}.$$ The UV conformal blocks for the effective theory with two vacua have the form $$\langle \Psi_{\pm}(w) \Sigma_{1,\alpha}(w_{1})\Sigma_{2,\alpha}(w_{2}) \rangle \in V_{1}\otimes V_{2}$$ where $V_{a} \simeq \mathbb{C}^{2},a=1,2$, are two copies of the fundamental representation space of $sl(2, \mathbb{C})$. We note that the amplitudes representing the vacua span only a two dimensional subspace of $V_{1} \otimes V_{2}$ of dimension $2$. The action of $B_{12}$ can be extended on the full four dimensional space of conformal blocks as $$B_{12}=-\lambda\left( \sigma_{+}^{1}\otimes \sigma_{-}^{2} + \sigma_{-}^{1}\otimes \sigma_{+}^{2} \right) -\mu\ \sigma_{3}^{1}\otimes \sigma_{3}^{2},$$ where $\sigma^{\ell}_{a}$ is the Pauli matrix acting on the $a$-th copy $V_{a}$ of $\mathbb{C}^{2}$. We conclude that the UV Berry connection $\mathcal{D}$ of a very complete $tt^{*}$ geometry with $d$ vacua has the general form $$\label{genkohno} \mathcal{D}=d -2\sum_{i<j}\left( \lambda_{ij}s_{\ell}^{i}s_{\ell}^{j}+ \mu_{ij}s_{3}^{i}s_{3}^{j} \right) \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}}$$ and act on the sections of a bundle $\boldsymbol{V}\rightarrow C_{d}$ whose fibers are modelled on the vector space $$V^{\otimes d}= V_{1}\otimes V_{2} \otimes......\otimes V_{d},$$ spanned by the conformal blocks of Ising type. The matrices $s_{\ell}^{a}, \ell=1,2,3$ are the $su(2)$ generators acting on the $V_{a}\simeq \mathbb{C}^{2}$ factor, namely $$s_{\ell}^{a}= 1 \otimes....\otimes 1 \otimes \frac{1}{2}\sigma_{\ell} \otimes 1 \otimes ....\otimes 1.$$ The natural connection on the CFT conformal blocks may differ from $\mathcal{D}$ by a line bundle twisting; Indeed, the $d$-point functions representing the vacua are rescaled by the normalization factor $\tau(w_{j})={\langle \mu_{1}(w_{1})...\mu_{n}(w_{n})\rangle}$. This corresponds to replacing $\mathcal{D} \rightarrow \mathcal{D} + 1 \cdot d \log f$ for some multivalued holomorphic function $f$.\ The actual brane amplitudes live in a rank $d$ sub-bundle $\mathcal{V}$ of the rank $2^{d}$ bundle $\boldsymbol{V}$. The $tt^{*}$ Lax equations requires this sub-bundle to be preserved by parallel transport with the connection $\mathcal{D}$. To see this, consider the total angular momentum $$L_{\ell}= \sum_{a} s_{\ell}^{a}, \hspace{0.5cm} \ell=1,2,3.$$ One can easily check that $L_{3}$ commutes with all the infinitesimal braiding $B_{ij}$ and so the eigenbundles $\mathcal{V}_{m}$ of $L_{3}$ are preserved by the parallel transport of $\mathcal{D}$. In particular, since the correlators representing the vacua has $d-1$ insertions of $\mu$-type and a single insertion of $\sigma$-type, the vacuum bundle corresponds to the $L_{3}$-eigenbundle with eigenvalue $m=1-d/2$. So, we have $$\mathcal{V}=\mathcal{V}_{1-d/2}.$$ The constants $\lambda_{ij},\mu_{ij}$ are restricted by further conditions: - $\mathcal{D}$ is flat acting on $\mathcal{V}$ - Since $\mathcal{D}$ coincide with the Lax connection in the UV limit, the monodromy representation must be arithmetic. - If the very complete $tt^{*}$ geometry is symmetric, the constant $\lambda_{ij},\mu_{ij}$ must be indipendent from $i,j$, i.e. $\lambda_{ij}=\lambda, \mu_{ij}=\mu$. A well known solution to the Knizhnik-Zamolodchikov equations is [@rif65; @rif67] $$\lambda_{ij}=\frac{\lambda}{2}, \hspace{0.5cm} \mu_{ij}=0,$$ which gives $$\label{symmkohno} \mathcal{D}=d+ \lambda\sum_{i<j} \frac{s_{\ell}^{i}s_{\ell}^{j}}{w_{i}-w_{j}}d(w_{i}-w_{j}).$$ This connection is automatically flat for all $\lambda$ when acting on sections of the big bundle $\boldsymbol{V}$ and is also symmetric. On the other hand, it is easy to check that the only symmetric solution to the flatness condition for a connection of the form \[genkohno\] is given by \[symmkohno\]. We conclude that a symmetric very complete $tt^{*}$ geometry has a UV Berry connection of the above form.\ Since $\mathcal{D}$ is flat on the larger bundle $\boldsymbol{V}$, the UV limit of the $tt^{*}$ linear problem, $\mathcal{D}\Psi = 0$ with $\Psi \in \Gamma(C_{d} , \mathcal{V} )$, may be extended to the big bundle $$\mathcal{D}\boldsymbol{\Psi} = 0, \hspace{1cm} \boldsymbol{\Psi} \in \Gamma(C_{d} , \boldsymbol{V} ).$$ in which we recognize the $sl(2)$ Knizhnik-Zamolodchikov equation for the $d$-point functions in the $2d$ WZW model with group $SU(2)$ [@rif63]. In that context $\lambda$ is quantized in discrete values $$\lambda=\frac{2}{k+2}$$ for the $2d$ $SU(2)$ current algebra at level $k \in \mathbb{Z}$. Since the connection \[symmkohno\] is invariant under the symmetric group $S_{d}$, the representation of the pure braid group $P_{d}$ extends to a representation of the full braid group of $d$ strands $B_{d}$. A representation of this type is a Hecke algebra representation [@rif67], which we briefly recall. Given $q \in \mathbb{C}^{\times}$, the Hecke algebra of the symmetric group $S_{d}$, $H_{d}(q)$ is the $\mathbb{C}$-algebra [@rif64] with generators $1,g_{1},...,g_{d-1}$ and relations $$g_{i}g_{i+1}g_{i}=g_{i+1}g_{i}g_{i+1}, \hspace{0.5cm} g_{i}g_{j}=g_{j}g_{i}, \ \vert i-j \vert\ \geq 2, \hspace{0.5cm} (g_{i}+1)(g_{i}.-q)=0.$$ For $q=1$ we just recover the group algebra $\mathbb{C}[S_{d}]$ of the symmetric group. Comparing the relations of the braid group $B_{d}$ with the above ones, we see that the map $\sigma_{i}\rightarrow g_{i}$ provides an algebra homomorphism $$\varpi: \mathbb{C}[B_{d}]\rightarrow H_{d}(q).$$ A linear representation $\rho$ of the braid group $B_{d}$ is called a Hecke algebra representation if it factorizes through $\varpi$. In such a representation the generators $\rho(g_{i})$ have at most two distinct eigenvalues: $−1$ and $q$.\ The Hecke algebra may be rewritten in terms of generators $e_{i}=(q-g_{i})/(1+q)$ satisfying the relations $$\label{first} e_{i}^{2}=e_{i}, \hspace{1cm} e_{i}e_{j}=e_{j}e_{i}, \ \vert j-1 \vert \geq 2,$$ $$\label{second} e_{i}e_{i+1}e_{i}-\beta^{-1}e_{i}=e_{i+1}e_{i}e_{i+1}-\beta^{-1}e_{i+1}$$ with $\beta=2+q+q^{-1}$. The Temperley-Lieb algebra $A_{d}(q)$ [@rif69] is the $\mathbb{C}$-algebra over the generators $1, e_{1},...,e_{d-1}$ satisfying the relation \[first\], while the relation \[second\] is replaced by the stronger condition $$e_{i}e_{i+1}e_{i} - \beta^{-1}e_{i} = 0.$$ A special class of Hecke algebra representations of $B_{d}$ are the ones which factorize through the Temperley-Lieb algebra $A_{d}(q)$. It is shown in [@rif65] that the monodromy representation of the flat connection \[symmkohno\] is a Hecke algebra representation of the braid group $B_{d}$ which factorizes through the Temperley-Lieb algebra $A_{d}(q)$ with $$q= \exp (\pi i \lambda)$$ given by the correspondence $$\sigma_{i}\rightarrow q^{-3/4}(q-(1+q)e_{i}), \hspace{0.5cm} i=1,....,d-1.$$ The eigenvalues of the monodromies $\sigma_{i}$ belong to the set $$\left\lbrace q^{1/4}, -q^{-3/4} \right\rbrace .$$ By arithmeticity of the $tt^{*}$ monodromy representation, in the present context we have to require the eigenvalues of the braiding matrices to be roots of unity. Hence, the Hecke algebra representations which appear in $tt^{*}$ are the ones with $\lambda \in \mathbb{Q}$ and $q$ a root of $1$. Moreover, the identification of $\mathcal{D}_{\mathcal{V}}$ with the UV Berry connection of a very complete $tt^{*}$ geometry entails that its monodromy representation is unitary. As we shall see momentarily, the monodromy representation of the braid group $B_{d}$ defined by restricting the Knizhnik-Zamolodchikov connection to the $tt^{*}$ sub-bundle is isomorphic to the Burau one, which is known to be unitary [@rif61; @rif70; @rif71]. We note further that the Knizhnik-Zamolodchikov connection preserves not only the $tt^{*}$ vacuum bundle $\mathcal{V}$, but also all the eigenbundles $\mathcal{V}_{l,m} \in \mathcal{V}$ of given total angular momentum, i.e. $$\psi \in \mathcal{V}_{l,m} \Leftrightarrow (L^{2}-l(l+1))\psi=(L_{3}-m)\psi=0$$ with $m=l,l-1,l-2,....,-l,$ for $l \in \frac{1}{2}\mathbb{N}$.\ The Braid group representations which factorize through the Temperley-Lieb algebra appear in many areas of mathematical physics. In particular, they describe the braiding properties of the $(p,q)$ Virasoro minimal models [@rif72]. In Virasoro minimal models the primary operators have dimension $$h_{r,s}= \frac{(pr-qr)^{2}-(p-q)^{2}}{4pq}, \hspace{1cm} 1\leq r \leq q-1, \ \ \ \ 1\leq s \leq p-1.$$ The braiding the operator $\phi_{1,2}$ correspond to the Temperley-Lieb algebra with $$q = e^{\pi i \lambda} = e^{2\pi i q/p}$$ The fusion rule for $\phi_{1,2}$ is $$\phi_{1,2}\cdot\phi_{1,2}= 1 +\phi_{1,3}.$$ Transporting a field $\phi_{1,2}$ around another one we get a matrix with eigenvalues $$\left\lbrace e^{2\pi i(h_{1,3}-2h_{1,2})}, e^{2\pi i(h_{1,1}-2h_{1,2})} \right\rbrace = \left\lbrace e^{2\pi i q/2p} , e^{-6\pi i q/2p} \right\rbrace = \left\lbrace q^{1/2} , q^{-3/2} \right\rbrace$$ where one has $$h_{1,1}-2h_{1,2}=1-3\frac{q}{2p}, \hspace{1cm} h_{1,3}-2h_{1,2}=\frac{q}{2p}.$$ . $tt^{*}$ Geometry of the Vafa model {#chap7} =================================== Generalities ------------ In this last section we want to study the $tt^{*}$ geometry of the Vafa model for FQHE. We consider $N$ electrons moving on the plane $\mathbb{C}$. We denote with $z_{i}, i=1,...,N$ and $x_{a}, a=1,...,n$ the positions of electrons and quasi-holes respectively, and with $\zeta_{\alpha},\alpha=1,...,M$ the support of the polar divisor $D_{\infty}$ parametrizing the magnetic flux. We take the $d=M+n$ points $\zeta_{\alpha},x_{a}$ all distinct. Hence, the target space of the SQM Vafa model of FQHE is $$\label{space} X_{d,N}=\left\lbrace (z_{1},...,z_{N}) \in \left( \mathbb{C}\setminus \lbrace x_{a},\zeta_{\alpha}\rbrace \right)^{N} \big\vert z_{i}\neq z_{j}, \ \mathrm{for} \ i\neq j \right\rbrace \big/ S_{N}.$$ In the experimental set-up $N$ is very large, while $N/d = \nu$ and $n$ are fixed. Despite this, we shall keep $N$ arbitrary[^3] as our arguments apply to any $N$. The superpotential of the LG model is $$\label{formby} \mathcal{W}= \beta \sum_{i<j}\log (z_{i}-z_{j})^{2} + \sum_{i}\left( \mu z_{i} + \sum_{a=1}^{n}\log (z_{i}-x_{a}) - \sum_{\alpha=1}^{M} \log(z_{i}-\zeta_{\alpha})\right) ,$$ which can be rewritten in terms of the elementary symmetric polynomials as $$\mathcal{W}= \beta \log \mathrm{discr}(P)+ \mu e_{1} + \sum_{a=1}^{n}\log P(x_{a})- \sum_{\alpha=1}^{M}\log P(\zeta_{\alpha}),$$ where $P(z)= \sum_{k=0}^{N} (-1)^{k}e_{k}z^{N-k} $. As explained in section \[wedisc\], we have introduced the coupling $\mu$ to make the problem better behaved. As long it is non zero, it can be fixed to the value that we prefer, since the $tt^{*}$ monodromy is indipendent from the couplings. We note that the superpotential $\mathcal{W}$ is not univalued in $X$. In this model we have two kinds of couplings: the residues of $d\mathcal{W}$ at its poles and the positions $x_{a}, \zeta_{\alpha}$. The residues of the poles at $x_{a}$ and $\zeta_{\alpha}$ are frozen to the values $±1$ by the arguments in \[wedisc\] and do not play any role in the discussion. The only relevant coupling is $\beta$. Working in a periodic box $\beta$ is frozen to the rational number $1/2\nu$. On the other hand, when the electrons move on the plane the model is well defined for arbitrary values of $\beta$. Since the monodromy representation is independent of $\beta$ we are free to deform it away from its physical value $1/2\nu$ to simplify the analysis. The non-frozen couplings are the $x_{a}$ and the $\zeta_{\alpha}$. These form a set of $d$ distinct points in $\mathbb{C}$ in which are identified modulo permutation the ones with equal charge. The manifold of essential couplings is therefore $$\mathcal{P}=C_{n+M}/S_{n}\times S_{M}.$$ The $\zeta_{\alpha}$ are homogeneously distributed in order to reproduce the macroscopic magnetic field. The details of the distribution are not important for our purposes and we can focus on the projection on $Y_{n}$. The fundamental group $\pi_{1}(\mathcal{P})$ contains $B_{n}$. The UV Berry connection yields a family of unitary arithmetic representations of $\pi_{1}(\mathcal{P})$. Restricting to $B_{n}$ we get a monodromy representation $$\rho(\vec{\theta}):B_{n}\rightarrow GL(\mathcal{V}_{\theta})$$ which is parametrized by the character $\vec{\theta}$ of the Galois group of the universal cover $\pi_{1}(X_{d,N})$. The dimension of the vacuum sector at fixed $\vec{\theta}$ is $d_{d,N}= \begin{pmatrix} N+d-1 \\ N \end{pmatrix}$, which reduces to $d_{d,N}^{f}= \begin{pmatrix} d \\ N \end{pmatrix}$ after the fermionic truncation discussed in \[fermtrunc\].\ The fundamental group of the target manifold $X_{d,N}$ is $$\pi_{1}(X_{d,N})= \mathcal{B}(N,S_{0,d+1}),$$ where $\mathcal{B}(N,S_{g,p}) $ is the braid group of $N$ strings on the surface $S_{g,p}$ of genus $g$ with $p$ punctures. This has the following presentation $$\begin{split} & \mathrm{generators}: \sigma_{1},\sigma_{2},...,\sigma_{N-1},z_{1},z_{2},...,z_{p-1} \\ & \mathrm{relations}: \begin{cases} \sigma_{i}\sigma_{i+1}\sigma_{i}= \sigma_{i+1}\sigma_{i}\sigma_{i+1}, \hspace{2cm} \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i} \ \ \mathrm{for} \vert i-j \vert \geq 2, \\ z_{j}\sigma_{i}=\sigma_{i}z_{j} \ \mathrm{for}\ i\neq 1, \hspace{2.3 cm} \sigma_{1}^{-1}z_{j}\sigma_{1}^{-1}z_{j}= z_{j}\sigma_{1}^{-1}z_{j}\sigma_{1}^{-1}, \\ \sigma_{1}^{-1}z_{j}\sigma_{1}^{-1}z_{l}= z_{l}\sigma_{1}^{-1}z_{j}\sigma_{1}^{-1} \ \ \mathrm{for} \ j<l. \end{cases} \end{split}$$ The $\sigma_{i}$ generates a subgroup of $\mathcal{B}(N,S_{0,d+1})$ which is isomorphic to the Artin Braid group $\mathcal{B}_{N}$. The abelianized Galois group $\mathcal{B}(N,S_{0,d+1})^{\mathrm{Ab}}\simeq \mathbb{Z}^{d+1}$ is the homology group of the target space generated by $\sigma,z_{1},...,z_{d}$. The dual cohomology group $H^{1}(X_{d,N},\mathbb{Z})$ has the generators $$\frac{1}{2\pi i} d \log \mathrm{discr}(P), \hspace{1cm} \frac{1}{2\pi i }d\log P(x_{a}), \hspace{1cm} \frac{1}{2\pi i} d\log P(\zeta_{\alpha}).$$ For each of this generator we have an angle $$\theta, \hspace{1cm} \phi_{a}, \hspace{1cm} \varphi_{\alpha}.$$ It is natural to consider the quasi-holes and the magnetic-flux units to be indistinguishable and take the corresponding angles to be all equal $\phi_{a}=\phi$ and $\varphi_{\alpha} = \varphi$. This is equivalent to take the quotient group of $H_{1}(X_{d,N},\mathbb{Z})$ dual to the subgroup of $H^{1}(X_{d,N},\mathbb{Z})$ generated by the differentials $$\frac{1}{2\pi i} d \log \mathrm{discr}(P), \hspace{1cm} \frac{1}{2\pi i }d\prod_{a} \log P(x_{a}), \hspace{1cm} \frac{1}{2\pi i} d\log \prod_{\alpha} P(\zeta_{\alpha}),$$ and define the LG model on the Abelial cover $\mathcal{A}_{H}=\mathcal{A}/H$, where $H$ is the kernel of the group homomorphism $H_{1}(X_{d,N},\mathbb{Z})\rightarrow \mathrm{Gal}(\mathcal{A}_{H})$. In particular, the Galois group of the cover is $\mathrm{Gal}(\mathcal{A}_{H})\simeq \mathbb{Z}^{3}$ and so we have three angles $\theta,\phi$ and $\varphi$. Setting $q=e^{i\theta},t=e^{i\phi}$,and $y=e^{i \varphi}$ we conclude that in the LG model with indistinguishable defects the BPS branes span a free module of rank $d_{d,N}$ over the ring $\mathbb{Z}[q^{\pm 1}, t^{\pm 1}, y^{\pm 1}]$. The normalizability of the ground states requires to $q, t $ and $y$ to be roots of unity. Moreover, as we are going to see, the physical FQHE requires further truncations. The Physical FQHE ----------------- The FQHE quantum system is a particular version of the 4-susy LG model with superpotential $\mathcal{W}$ given previously. Quasi-holes and magnetic-flux units are indistinguishable, but we have still to fix the characters $\theta, \phi$ and $\varphi$. The roles of the punctures $\zeta_{\alpha}$ is to reproduce the uniform constant magnetic field $B$ through the isomorphism discuss in section \[wedisc\], which holds for the trivial character $\phi=0 \ \mathrm{mod} \ 2\pi$. Since the quasi-holes are just insertion of mangetic fluxes with the opposite sign, it is natural to take also $\varphi=0 \ \mathrm{mod} \ 2\pi$. We remain with just one non-trivial angle $\theta$ associated to the Vandermonde coupling $\beta$. In order to have normalizable states the angle $\theta$ should be a rational multiple of $2\pi$ $$\theta= \pi \left( 1+\frac{a}{b}\right) , \hspace{1cm} a \in \mathbb{Z}, b \in \mathbb{N}, \hspace{0.5cm} \big\vert \mathrm{gcd}(a,b)=1 \hspace{0.5cm} \mathrm{and} \ -b\leq a \leq b.$$ The $tt^{*}$ reality structure relates $\theta$ with $-\theta$ and so $a$ with $-a$. So, if a phase with pair $(a,b)$ exists, it is natural to expect also a dual phase associated to $(-a,b)$. These corresponds to the same representation of the Galois group and we can write $(\pm a,b)$ with $1\leq a \leq b$ to cover both phases at once. The LG model which we obtain with these restrictions is defined on the abelian cover of $X$ associated to $\theta$ $$\mathcal{A}_{\theta}=\left\lbrace p :[0,1] \rightarrow X, p_{0}=p_{*} \right\rbrace / \sim_{b},$$ where $$\sim_{b}= \begin{cases} p(1)=p^{\prime}(1) \\ p^{\prime}=p\cdot \sigma^{2bk} \ \mathrm{in} \ H^{1}(X_{d,N},\mathbb{Z}), \ k \in \mathbb{Z}. \end{cases}$$ The dimension of the generic $\theta$-sector of the ground state $V_{\theta}$ corresponds to the bosonic statistics $d_{d,N}$. To get the correct physical counting of states we need to consider its fermionic truncation, i.e. to keep only the states which are stable as $\beta\rightarrow 0$. Since the monodromy is independent of $\beta$ (as long as it is not zero), and its limit as $\beta\rightarrow 0$ is smooth after the fermionic truncation, we may set $\beta=0$ while keeping track of the non-trivial topology of the target space through the character $\theta$. In the language of [@rif23] this is called $\theta$-limit. According to the discussion in \[cov\], switching a non zero $\theta$ is equivalent to pull-back the operators $\phi_{a}$ of the fermionic chiral ring $\mathcal{R}_{F}$ on $\mathcal{A}_{H}$ and define the basis $$\Phi_{a}(\theta)= \varpi^{*}\phi_{a}e^{i\theta h},$$ where $\varpi: \mathcal{A}_{\theta}\rightarrow X$ and the chiral field $e^{ih}$ is proportional to $$\mathrm{discr}(P)^{\theta/2\pi}= \prod_{i<j}(z_{i}-z_{j})^{\theta/\pi}.$$ It follows that the entries of the $tt^{*}$ monodromy matrices will take values in $\mathbb{Z}[q^{\pm1}]$, with $q=e^{i\theta}$. Keeping into account the Jacobian factor arising from the map $\lbrace z_{i}\rbrace \rightarrow \lbrace e_{k} \rbrace $, the vacuum wave functions in terms of $z_{i}$ contains the factor $$\prod_{i<j}(z_{i}-z_{j})^{1+\theta/\pi}, \hspace{0.5cm} 0\leq \theta \leq 2\pi .$$ The comparison with the Laughlin wave function \[Laugh\] leads to the identification $$\label{fillingab} \frac{1}{\nu}= 1 + \frac{\theta}{\pi}= 2 \pm \frac{a}{b}$$ which gives $1\leq 1/\nu \leq 3$. In particular, the minimal $b$-torsion character, $a=1$ yields the FQHE principal series $$\nu=\frac{b}{2b \pm 1}, \ b \in \mathbb{N}.$$ Although this series is the most natural LG quantum systems of the form \[formby\], it is by no means the only possibility in the present framework. We are going to discuss the allowed values of the filling fraction in \[detlam\]. The above analysis recalls in some sense the idea of Jain’s composite fermion theory [@ab4]. If we consider the character $e^{2\pi i (1+\theta/\pi)} $ as the Aharonov-Bohm phase generated by taking an electron around another one, we can interpret the degrees of freedom of the Vafa model in the $\theta$-limit as free fermions bounded to $\frac{\theta}{\pi}=1/\nu-1$ units of magnetic flux. Similarly to the Jain’s proposal, the picture emerging from the $4$-SQM model is that we can treat the fractional quantum Hall effect as an integer one for composite fermions. Fermi Statistics and Hecke Algebra Representations -------------------------------------------------- The fermionic truncation allows us to effectively put the coupling $\beta$ to zero and consider the LG Fermi model of $N$ chiral fields with superpotential $$\mathcal{W}(e_{i})= \sum_{i=1}^{N} W(z_{i})$$ where the one-field model is $$W(z)= \mu z +\sum _{a=1}^{n}\log(z-x_{a})-\sum_{\alpha=1}^{M}\log(z-\zeta_{\alpha}).$$ Following [@rif55] we can prove that the $tt^{*}$ geometry of the single particle model is very complete. Since the monodromy representation is indipendent from $\mu$, we can take $\mu\gg 1$. The equations for the classical vacua $$\mu= \sum_{\alpha}=\frac{1}{z-\zeta_{\alpha}}-\sum_{a}\frac{1}{z-x_{a}},$$ have $d=n+M$ solutions of the form $z=x_{a}+O(1/\mu)$ or $z=\zeta_{\alpha}+ O(1/\mu)$. Up to an irrelevant rescaling by a factor $\mu^{-1}$, the set of critical values is $$\left\lbrace w_{1},...,w_{n+M}\right\rbrace = \left\lbrace x_{1},...,x_{n},\zeta_{1},...,\zeta_{M}\right\rbrace + O\left( \frac{1}{\mu}\log \mu \right).$$ The cover of the coupling constant space $\mathcal{P}=C_{n+M}/S_{n}\times S_{M}$ is the configuration space of $n+M$ distinct points $C_{n+M}$, which can be seen also as the cover of the space $Y_{n+M}=C_{n+M}/S_{n+M}$ where the critical map $w:\mathcal{P}\rightarrow Y_{n+M}$ take values. By the above equality we see that the canonical projection $p:C_{n+M}\rightarrow Y_{n+M}$ factorizes thorugh the critical map. This shows that the $tt^{*}$ geometry of the one-field model is very complete and so the UV Berry connection is a flat $sl(2)$ Kohno connection $$\mathcal{D}=d+ \lambda \sum_{i<j}s_{\ell}^{i}s_{\ell}^{j} \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}},$$ which provides a monodromy representation of $\pi_{1}(Y_{n+M})=B_{n+M}$ acting on the space $V^{n+M}=\bigotimes_{i} V_{i}$ with $V_{i}\simeq \mathbb{C}^{2},i=1,...,n+M$, restricted to the subspace $\mathcal{V}_{1}$ with total angular momentum $$L_{3}=1-(n+M)/2.$$ Taking the projection $p:Y_{n+M}\rightarrow Y_{n}$ we obtain a monodromy representation of the subgroup $B_{n}$ which factors through a Temperley-Lieb algebra with $q=e^{\pi i \lambda}$.\ The vacuum bundle of the $N$ particle model is $$\mathcal{V}=\bigwedge^{N} \mathcal{V}_{1}\rightarrow \mathcal{P}$$ and the UV Berry connection is the one induced on the $N$-th rank antisymmetric representation by the single field UV Berry connection. It is convenient to introduce the “Grand-canonical ” bundle $$\mathcal{W}= \bigoplus_{N=0}^{d}\mathcal{V}_{N}\rightarrow \mathcal{P}, \hspace{1cm} \mathrm{rank} \ \mathcal{W}=\sum_{N=0}^{d} \begin{pmatrix} d \\ N \end{pmatrix}= 2^{d},$$ where the total number of states is $2^{d}$ since each of the $d=n+M$ one-particle (vacuum) states may be either empty or occupied. In the above direct sum of bundles we included $\mathcal{V}_{0}$ which does not correspond to any LG model, since the number of fields is zero. This can be done without any harm by the isomorphism $\mathcal{V}_{0}\simeq \mathcal{V}_{d}$. The bundle $\mathcal{V}_{d}$ corresponds to the trivial $\nu=1$ phase and should have a trivial monodromy representation. It is easy to see that $$\mathrm{fiber}(\mathcal{W})\simeq V^{\otimes d},$$ where $V^{\otimes d}$ is the fiber of the big bundle $\boldsymbol{V}\rightarrow \mathcal{P}$. According to the discussion in section \[hecke\], one associates to each vacuum of the single particle model a spin degree of freedom $s_{\ell}^{j}$. A spin down (up) corresponds to the insertion in the Ising-like correlators of a $\mu \ (\sigma)$-like operator at the critical value $w_{j}$ of the $j$-th vacuum. We can define a correspondence with the fiber of the Grand bundle as follows: a spin down (up) means that the $j$-th vacuum is empty (occupied). Hence, a vacuum with occupied state $\left\lbrace j_{1},....,j_{N}\right\rbrace $ corresponds linearly to the element of the $N$-particles Fermi chiral ring $$\label{frame} \sum_{\sigma \in S_{N}}\mathrm{sign}( \sigma)e_{j_{1}}(z_{\sigma(1)})e_{j_{2}}(z_{\sigma(2)}).......e_{j_{N}}(z_{\sigma(N)}) \in \left( \mathcal{R}_{N} \right)_{a},$$ where $e_{j}(z) ,j=1,...,d$ are the orthogonal idempotents of the one-particle chiral ring $\mathcal{R}_{1}$. So we conclude that $$\mathcal{W}\simeq \boldsymbol{V},$$ and the linear PDEs satisfied by the brane amplitudes are just the $sl(2)$ Knizhnik-Zamolodchikov equations up to a twist by normalization factors. In this correspondence the angular momentum $L_{3}=\sum_{j=1}^{d}s_{3}^{j}$ is related to the operator number of particles by $$\hat{N}=L_{3} + \frac{d}{2},$$ where $L_{\ell}=\sum_{j=1}^{d}s^{j}_{\ell}$ are the generators of $sl(2)_{\mathrm{diag}}$. The underlying one-particle model, having a very complete symmetric $tt^{*}$ geometry, defines a Kohno connection acting on $V^{\otimes d}$ that we argued has the $sl(2)$ KZ form up to an overall twist, i.e. $$\label{twisting} \mathcal{D}=d + \lambda(\theta)\sum_{i<j}s_{\ell}^{i}s_{\ell}^{j} \frac{d(w_{i}-w_{j})}{w_{i}-w_{j}} + \xi(\theta) \sum_{i<j}d\log(w_{i}-w_{j}).$$ As we shall see, it turns out that the constant $\xi(\theta)$ is related to $\lambda(\theta)$ by consistency conditions. The factor $\lambda(\theta)$ should be some piece-wise linear function of $\theta$ and in the context of physical FQHE is related to the filling fraction labelling the different quantum phases. We are going to determine it in the next section. The eigenbundles $\mathcal{V}_{N}$ are preserved by parallel transport with $\mathcal{D}$, and hence define a monodromy representation of $B_{n}$ which is the one associated to the $N$-field Fermi theory. These representations are generically reducible, since also the eigenbundles of the total angular momentum $L^{2}=\sum_{\ell}L_{\ell}L_{\ell}$ are preserved by parallel transport. We have the monodromy invariant decomposition $$\mathcal{W}=\bigoplus_{l=0, \ l=\frac{1}{2}\mathbb{N}}^{d/2} \ \bigoplus_{m=-l}^{l}\mathcal{V}_{l,m}, \hspace{1cm}\mathcal{V}_{l,m}= \mathrm{ker} \left( L^{2}-l(l+1)\right) \bigcap \mathrm{ker} \left( L_{3}-m\right)$$ and $$\label{decomp} \mathcal{V}_{N}=\bigoplus_{l= \vert N-d/2 \vert }^{d/2} \mathcal{V}_{l,m=N-d/2}.$$ Since the monodromy representation centralizes with respect to $sl(2)_{\mathrm{diag}}$ we have $$\mathcal{V}_{l,m} \simeq \mathcal{V}_{l,m^{\prime}}, \hspace{0.5cm} -l \leq m,m^{\prime} \leq l,$$ from which it follows that the monodromy representations $\mathcal{V}_{N},\mathcal{V}_{d-N}$ are isomorphic. The rank of $\mathcal{V}_{l,m}$ is given by the Catalan triangle $$\mathrm{rank} \ \mathcal{V}_{l,m}= \begin{pmatrix} d \\ \frac{d}{2}-l \end{pmatrix} - \begin{pmatrix} d \\ \frac{d}{2}-l -1 \end{pmatrix} \hspace{0.5cm} \mathrm{for} \ 0\leq l \leq \frac{d}{2}, \ -l\leq m \leq l.$$ In the decomposition \[decomp\] of $\mathcal{V}_{N}$ there is a unique eigenbundle of rank $1$ which corresponds to $\mathcal{V}_{d/2,N-d/2}$, namely the one which maximizes $L^{2}$. This is spanned by the unique monodromy invariant vacuum ${{\,| {d/2,N-d/2} \rangle}}$ of the Fermi model with $N$ electrons. It is tempting to identify this state with the preferred vacuum discussed in \[wedisc\]. For a given number of particles $N$, the determinant of the brane amplitude matrix spans the fiber of a rank $1$ eigenbundle of the UV Berry connection. Hence, we have the isomorphism of bundles $$\mathrm{det}: \bigwedge^{N}\mathcal{V}_{1}\rightarrow \mathcal{V}_{d/2,N-d/2}\simeq \mathcal{V}_{d/2,d/2}.$$ The isomorphism between $\mathcal{V}_{d/2,N-d/2}$ and the vacuum bundle of the trivial phase $\mathcal{V}_{N=d}$ implies that the preferred vacuum should have trivial monodromy. This allows to fix the constant $\xi(\theta)$. The restriction of the Kohno connection to the subbundle $\mathcal{V}_{d/2,d/2}$ is $$\mathcal{D}_{\mathcal{V}_{d/2,d/2}}= d+ \frac{\lambda(\theta)}{4}\sum_{i<j}d\log(w_{i}-w_{j}).$$ In order to have a trivial UV Berry connection we have to demand $$\xi(\theta)=-\frac{\lambda(\theta)}{4}.$$ In other words, the normalized amplitudes $\Psi_{\mathrm{norm}}$ of the Grand-bundle are related to the KZ ones $\Psi$ by $$\Psi_{\mathrm{norm}}= \frac{\Psi}{\Psi_{\mathrm{priv}}}$$ where $\Psi_{\mathrm{priv}}$ is a parallel section of the line-bundle $\mathcal{V}_{d/2,d/2}$. In particular, the normalized monodromy is trivial for the $\nu = 1 $ phase.\ Since it left invariant by the UV Berry connection, the preferred vacuum is the most symmetric one under permutations of the spin degrees of freedom. This state should correspond to some linear combinations of the idempotents $e_{i}$ of the chiral ring $\mathcal{R}_{N}$ such that the electrons are as spread out as possible among the single particle vacua. As discussed at the end of section \[wedisc\], as long as the susy-breaking part of the interacting Hamiltonian preserves the symmetry between the holes and the units of fluxes, it lifts the degeneracy keeping the most symmetric state as the ground state. So it is natural to identify this state with the true FQHE ground state under the isomorphism developed in \[wedisc\]. Determining $\lambda(\theta)$ and the allowed filling fractions {#detlam} --------------------------------------------------------------- A direct computation of $\lambda(\theta)$ is quite hard and subtle. We can fix it by imposing consistency conditions. An intrinsecally defined quantity which is indipendent from the bundle trivialization is $q(\theta)^{2}$, namely the ratio of the two eigenvalues of $\sigma_{i}^{2} \in P_{n+M}$ which correspond to the operation of transporting a quasi-hole around another one and getting back to the original position. We have the relation $$q(\theta)^{2}= e^{2\pi i \lambda(\theta)}.$$ By the functorial property of the Hecke algebra representations, the parameter $\lambda(\theta)$ should be a universal function indipendent from $n$ and $M$. Moreover, since it is related to the Ramond charges of the LG model, it should be a piece-wise linear function of the angle defined mod $1$, i.e. $$\lambda(\theta)= C_{1}+C_{2}\frac{\theta}{\pi} \ \mathrm{mod} \ 1,$$ for some real constants $C_{1},C_{2}$. We can assume $C_{2}>0$ by changing the sign of $\theta$. The periodicity in $\theta$ and the $tt^{*}$ reality structure which relates $\pm \theta$ imply $$q(\theta+2\pi)^{2}=q(\theta)^{2}, \hspace{1cm} q(-\theta)^{2}= q(\theta)^{-2}.$$ These conditions require $2C_{1}=0 \ \mathrm{mod } \ 1$ and $2C_{2}=0 \ \mathrm{mod } \ 1$. Imposing the same condition on the ratio $q(\theta)$ of the two eigenvalues of $\sigma_{i} \in B_{n+M}$ one obtains the stronger conditions $C_{1}=0 \ \mathrm{mod } \ 1$ and $C_{2}=0 \ \mathrm{mod } \ 1$. Demanding $q(\theta)$ to be strictly periodic of $2\pi$ (and not of a fraction $2\pi/C_{2}$), we find the solution $$\lambda(\theta)= \frac{\theta}{\pi} \ \mathrm{mod} \ 1.$$ This identification is quite natural, since the $tt^{*}$ monodromy is defined over the ring $\mathbb{Z}[e^{\pm i \theta}]$, while the KZ monodromy is defined over $\mathbb{Z}[e^{\pm i \pi \lambda(\theta)}]$. Hence, the above solution leads to the identification of the two Laurent polynomial rings. By the relation \[fillingab\] we have $$\label{piùsol} q(\theta)^{2}= e^{2\pi i/\nu}.$$ We assume that the UV Berry connection is a Knizhnik-Zamolodchikov connection for $SU(2)$ current algebra with level $k$ quantized in integral units, namely $$\lambda=\pm \frac{2}{k+2}, \hspace{1 cm} k \in \mathbb{Z}.$$ From the point of view of $tt^{*}$ geometry we do not know any compelling argument to require $k+2 \in \mathbb{Z}$. However, it is believed that $2d$ WZW models describe the modes which live at the edge of a non-Abelian Chern-Simons theory with boundary. Further, it turns out that the braiding of their conformal blocks coincides with the braiding of Wilson lines in the Chern-Simons theory. Hence, if we want our $\mathcal{N}=4$ theory to admit an effective IR description in terms of a $3d$ topological field theory, it is natural to consider a subclass of LG models compatible with the above condition. By the $tt^{*}$ reality structure we have to consider the solutions with both signs. We first study the class of solutions with $C_{1}$ an odd integer. This is the most natural case, since we have $$e^{\pm 2\pi i/(k+2)}= e^{i\pi /\nu}.$$ The equation $$\label{secondsol} \pm \frac{2}{k+2}= 1 +\frac{\theta}{\pi} \ \mathrm{mod} \ 2= \pm \frac{a}{b} \ \mathrm{mod} \ 2$$ has solutions $a=1$ with $k$ even and $a=2$ with $b$ and $k$ odd. In agreement with the prediction in [@rif1], the first case corresponds to the principal series of FQHE $$\nu= \frac{b}{2b \pm 1}, \hspace{0.5cm} b \in \mathbb{N}.$$ From point of view of [@rif1], the element $\sigma_{i}^{2}$ of the pure braid group for the principal series has two distinct eigenvalues, in correspondence with the two different fusion channels of the $\phi_{1,2}$ operator in the minimal $(2n, 2n\pm1)$ Virasoro model. The ratio of the two eigenvalues is $$q^{2}=\frac{\exp [ 2\pi i (h_{1,3}-2h_{1,2} ) ] }{\exp [ 2\pi i (h_{1,1}-2h_{1,2})] }=\exp(2\pi i/\nu),$$ which is in agreement with the $tt^{*}$ result. The model admits also another list of filling fractions with denominatior divisible by $4$, namely $$\nu= \frac{b}{2(b \pm 1)}, \hspace{0.5cm} b \ \mathrm{odd}$$ which correspond to the second solution of \[secondsol\].\ We may consider also the less natural class of solutions in which $C_{1}$ is an even integer. In this case we have $$q(\theta)=-e^{i\pi/\nu}= e^{\pm 2\pi i /(k+2)}$$ Demanding $1\leq 1/\nu \leq 3$ we find two possible series of filling fractins. One corresponds to $$1+ \frac{2}{k+2}= 1 + \frac{\theta}{\pi} \ \mathrm{mod} \ 2= \frac{1}{\nu} \ \mathrm{mod} \ 2$$ which implies $$\nu= \frac{m}{m+2}, \hspace{1cm} m=k+2 \in \mathbb{N}\geq 2.$$ This series contains the values of $\nu$ corresponding to the Moore-Read [@nab2] and Read-Rezayi models [@nab3]. The second possibility is $$3-\frac{2}{k+2}= \frac{1}{\nu} \ \mathrm{mod} \ 2$$ which gives $$\nu= \frac{m}{3m-2}, \hspace{1cm} m=k+2\geq 2.$$ Comparison with the Homological Approach ---------------------------------------- We can compare the $tt^{*}$ monodromy representation computed in the UV limit with the one obtained in the asymmetric limit. In general the two are different unless the UV limit is regular. However, for a very complete $tt^{*}$ geometry, where $\pi_{1}(\mathcal{P})$ = $B_{d}$, the asymmetric limit monodromy yields the so-called homology representation of the braid group, which is essentially equivalent to the monodromy of the Knizhnik-Zamolodchikov connection. Following again [@rif55], we can write an explicit integral representation of solutions to the $sl(2)$ Knizhnik-Zamolodchikov equation $$\label{integralez} \Psi_{\mathrm{KZ}}= f(\zeta_{1},...,\zeta_{d}) \int_{\Gamma} \prod_{i,\alpha}(z_{i}-\zeta_{\alpha})^{-\frac{\lambda}{2}}e^{\tilde{\mu} \sum_{i} z_{i}}\prod_{i<j}(z_{i}-z_{j})^{\lambda} dz_{1} \wedge \cdot \cdot \cdot \wedge dz_{N}$$ where $\tilde{\mu} \in \mathbb{C}^{\times} $ and $$\label{overallfactor} f(\zeta_{\alpha})=\prod_{\alpha <\beta}(\zeta_{\alpha}-\zeta_{\beta})^{\frac{\lambda}{4}}e^{-\frac{\tilde{\mu}}{2}\sum_{\alpha}\zeta_{\alpha}}.$$ The amplitudes \[integralez\] are known to compute the Virasoro conformal blocks [@rif55] $$\langle \phi_{1,2}(\zeta_{1})...\phi_{1,2}(\zeta_{d})V_{\tilde{\mu}}(\infty)\rangle$$ containing insertions of the primary field $\phi_{1,2}$ of the Virasoro algebra and an irregular vertex operator $V_{\tilde{\mu}}$ [@rif29; @rif30; @rif31]. The central charge $c=1+6 Q^{2}, \ Q=b+\frac{1}{b}$ is related to the parameter $\lambda$ of the Kohno connection by $b^{2}= -2/\lambda$. The integration contour $\Gamma$ denotes a basis of reltative cycles which ensure the convergence of the integral. One denotes with $C_{a}, \ a=1,...,d$ a basis of Lefschetz thimbles for the homorphic potential $$W(z;\zeta_{\alpha})= \tilde{\mu} z -\frac{\lambda}{2}\sum_{\alpha=1}^{d}\log(z-\zeta_{\alpha}) ,$$ which has the same functional form of the Vafa superpotential for a single electron and $d$ quasi-holes, but with ‘renormalized’ couplings. Then, $\Gamma$ can be constructed by taking products of cycles $$C_{j_{1},...,j_{N}}= C_{j_{1}} \times C_{j_{2}} \times ..... \times C_{j_{N}},$$ in which $j_{i}\neq j_{k}$ for $i\neq k$. Hence, for a fixed $N$ we have $\begin{pmatrix}d \\ N \end{pmatrix}$ integration cycles. As suggested in [@rif1], it is natural to identify the above integrals with the asymmetric limit of the $tt^{*}$ brane amplitudes given by the pairing between branes and the topological vacuum, i.e. the one corresponding to the identity operator. After the rescaling $w_{j} \rightarrow \beta w_{j}$ we have to send $\beta$ and the spectral parameter $\zeta$ to zero in such a way that the ratio $\beta/\zeta$ remains finite. Hence, we write $$\label{correspasymm} \Psi^{\mathrm{asym}}_{\Gamma}= \int_{\Gamma} e^{\frac{\beta}{\zeta} \mathcal{W}} \prod_{i<j}(z_{i}-z_{j})^{1+\frac{\theta}{\pi}} dz_{1} \wedge \cdot \cdot \cdot \wedge dz_{N},$$ where $$\mathcal{W}= \sum_{i=1}^{N}\left( \mu z_{i} + \sum_{\alpha=1}^{d} \log(z_{i}-\zeta_{\alpha}) \right) .$$ The correspondence requires $\beta/\zeta=-\lambda/2$ and $\tilde{\mu}=-\beta\mu/\zeta$. The identity operator gets corrected by the factor $\prod_{i<j}(z_{i}-z_{j})^{1+\frac{\theta}{\pi} }$ which is generated by the projection on the fermionic sector and pull-back on the abelian cover $\mathcal{A}_{\theta}$. The relative cycles $\Gamma$ provide a basis for the space of branes $B_{N}(\zeta)$ of the $N$ particle Fermi model, which has the structure of a $\mathbb{Z}[e^{\pm i \theta}]$-module of rank $\begin{pmatrix}d \\ N \end{pmatrix}$. The monodromy representation of the Knizhnik-Zamolodchikov connection of the $N$ particle sector coincides with the monodromy representation of the Gauss-Manin connection acting on $B_{N}(\zeta)$.\ We observe that the $tt^{*}$ amplitudes differ from the Virasoro conformal blocks by the overall factor $f(\zeta_{\alpha})$. This is a multi-valued function of the $\zeta_{\alpha}$ and generates the bundle twisting of the $tt^{*}$ connection \[twisting\]. Compared to the UV limit, the asymmetric limit allows an explicit check of the normalization of the $tt^{*}$ flat sections. From \[overallfactor\] we find modulo an exact term $$\mathcal{D}\rightarrow \mathcal{D} + \xi\sum_{\alpha < \beta}d\log(\zeta_{\alpha}-\zeta_{\beta})$$ where $$\xi=-\frac{\lambda}{4}= \frac{1}{2b^{2}}.$$ The comparison between \[integralez\] and \[correspasymm\] leads to the identification $$e^{i\pi \lambda(\theta)}=e^{i\pi/\nu}$$ which is the solution to \[piùsol\] with positive sign. However, there are other normalizations of the brane amplitudes which are compatible with $tt^{*}$ periodicity and reality structure. Hence, one may have in principle also the other solution $$e^{i\pi \lambda(\theta)}=-e^{i\pi/\nu}.$$ The actual normalization of the $tt^{*}$ flat sections is subtle to determine. It is reasonable to expect that not all the filling fractions given in the previous section are physical and, once fixed the sign ambiguity, some of them may be ruled out. The Long-Range Limit of the Vafa Hamiltonian ============================================ We discussed in section \[wedisc\] the equivalence between the Landau description of an electron in a uniform magnetic field and a certain class of $4$-SQM Landau-Ginzburg theories. In the case of a particle moving on the complex plane, the Vafa superpotential is given by $$W= \mu z + \sum_{i} \Phi_{i}\log(z-\zeta_{i}),$$ where $\mu,\Phi_{i} \in \mathbb{C}^{\times}$ and $\zeta_{i}$ are all distinct. In order to reproduce a large macroscopically uniform magnetic field we have to take the residues $\Phi_{i}$ all equal and consider a uniform distribution of the flux sources $\zeta_{i}$ in $\mathbb{C}$. We may proceed by analogy with the CFT description of FQHE and consider a lattice of fundamental units of magnetic flux $\Phi(a)=a^{2}\frac{B}{2\pi}$ at positions $\zeta_{m,n}(a)=am+ian$, $\ m,n \in \mathbb{Z}$, where $a$ is a real parameter. The lattice $\Lambda(a)= a\mathbb{Z}+ia\mathbb{Z}$ is contained in a disk $D(R)$ of radius $R\gg a $ which defines the size of the sample. As we already discussed in section \[theoryofedgestates\], one can recover the uniform constant $B$ field by taking the continuous limit of the lattice and then sending $R$ to infinity. The immaginary part of the logarithmic superpotential oscillates very rapidly for $a\ll 1$ and the average of the oscillations vanishes in the limit $a\rightarrow 0 $. Instead, the real part of $W$ coincides with the $2$-dimensional electrostatic potential of a discrete set of charges and tends in the same limit to the potential of a continuous charge distribution. Setting $\mu=0$, we remain with a real superpotential $$\label{limitsuperpotential} \Phi(a)\sum_{\Lambda(a)\cap D(R)} \log (z-\zeta_{m,n}(a)) \xrightarrow{ a\rightarrow 0, \ R\rightarrow \infty} B \ \vert z \vert ^{2}/4.$$ This convergence is natural from the point of view of electrostatics, but it is non-trivial in the context of SQM. The model is based on $\mathcal{N}=4$ supersymmetry which requires the superpotential to be holomorphic. So, the fact that only the real part of the superpotential survives in the limit seems to be inconsistent with supersymmetry. Moreover, the very different analytical properties of the final limit compared to the finite series suggest that this convergence cannot be intended in the strong topology. In the present section we want to clarify the interpretation of this limit from the point of view of supersymmetry. The function $$W(z;a,R)= \Phi(a)\sum_{\Lambda(a)\cap D(R)} \log (z-\zeta_{m,n}(a))$$ is multivalued and cannot be really considered as the superpotential of the theory. Conversely, the derivative $$\label{derw} \partial_{z}W(z;a,R)= \Phi(a)\sum_{\Lambda(a)\cap D(R)} \frac{1}{z-\zeta_{m,n}(a)}$$ is a well defined meromorphic function on the plane. The dimension of the degenerate vacuum space is given by the number of classical vacua, i.e. the solutions to the equation $\partial_{z} W=0$. These are counted by the Witten index of the theory which grows with the number of lattice points $$I_{W}=\mathrm{Tr}(-1)^{F}= -(\#(\Lambda(a) \cap D(R))-1 ),$$ where the mignus sign keeps into account that the vacua have fermion number $F=1$. In the limit $a\rightarrow 0, R\rightarrow \infty$ the Witten index goes like $\vert I_{W} \vert \simeq \pi R^{2}/a^{2}$, which is an extremely large number of vacua. The meaning of the $a\rightarrow 0$ limit is that we are taking measurements of observables on a scale much larger than the lattice parameter $a$. As physically expected, it turns out that for this specific class of measures the outcomes are correctly reproduced by an effective model with a single vacuum.\ Let us begin with understanding the physics of the system in the continuous limit. It is clear that the wave functions of the theory must vanish at the points of the lattice $\Lambda(a)$ and, in the limit of $a\rightarrow 0$, they oscillate rapidly on a length scale comparable with the lattice parameter $a$. If we consider measures of observables on a scale much larger than $a$, we are not able to detect anymore the fluctuation of the wave functions and see the discrete structure of the set of charges. One can check that long-range measurements do not see the many vacua also by the correlators of topological obervables in the two dimensional $\mathcal{N}=(2,2)$ version of the theory. In a generic $\mathcal{N}=(2,2)$ Landau-Ginzburg model with fundamental chiral fields $X_{i}$ and superpotential $W$, the topological sector is described by operators $\Phi_{k}$ in the chiral ring $\mathcal{R}=\frac{\mathbb{C}\left[ X_{i}\right] }{\partial_{j} W}$, which are in one to one correspondence with the vacua of the model. We recall that, if the critical points $p_{\alpha}$ of $W$ are non degenerate, the chiral ring is semisimple and we have $\mathcal{R}\simeq \mathbb{C}^{\#\mathrm{classical \ vacua}}$. The canonical isomorphism between these $\mathbb{C}$-algebras is provided by a basis of minimal orthogonal idempotents defined by $$\Phi_{\beta}(z_{\alpha})= \delta_{\alpha\beta}.$$ In this basis each operator of the chiral ring is represented by its set of values at the critical points.\ Coming back to our theory, we see that for $a\rightarrow 0$ the classical vacua becomes very close to each other and to the lattice singularities. The holomorphic functions representing $\Phi_{\alpha}$ on the target space must vanish on the critical points of $W$ except $z_{\alpha}$ and so become rapidly oscillating for $a\ll 1$. Hence, it is clear that these operators are irrelevant for long-range measurements and one should consider a basis of generators for the chiral ring which are smooth in this limit. A suited basis is given by the monomials $z^{k},\ k \in \mathbb{N}$, which are defined indipendently from the vacua structure. Let us consider the correlators of these observables in the limit $a\rightarrow 0$. Setting $\Phi_{k}=z^{k},\Phi_{j}=1$ in the Grothendieck formula \[grotform\] for the topological two points function we get $$\langle z^{k} \rangle= \frac{1}{(2\pi i)} \int_{\Gamma} \frac{z^{k}dz}{\Phi(a)\displaystyle\sum_{\Lambda(a)\cap D(R)} \frac{1}{z-\zeta_{m,n}(a)}}.$$ In this computation one has to choose a contour $\Gamma$ which encircles all the critical points of $W$. It is clear that these will be located in the region of the disk containg the holes and so one must take a contour encircling $D(R)$. In the limit of $a\rightarrow 0$ at fixed $R$, the discrete set of charges approaches a continuous distribution on $D(R)$. If the coordinate $z$ is outside the disk, the derivative of the superpotential tends to $$\Phi(a)\sum_{\Lambda(a)\cap D(R)} \frac{1}{z-\zeta_{m,n}(a)}\sim \frac{B R^{2}}{2z} + O(a),$$ in which we recognize the $2$-dimensional electrostatic field generated by a radially symmetric charge distribution. Hence, we find $$\langle z^{k} \rangle \propto \int_{\Gamma} dz z^{k+1} +O(a) \xrightarrow{a\rightarrow 0} 0.$$ The triviality of the topological correlators reflect the fact that measurements at a scale $z$ much larger than $a$ do not detect the many vacua of the theory.\ It is already expected from \[limitsuperpotential\] which should be the effective model which captures the relevant physics of the FQHE. The fact that in the limit we obtain a real superpotential implies a passage from the initial $\mathcal{N}=4$ to a $\mathcal{N}=2$ theory in which the holomorphicity is not required. Now we are able to give a precise physical and mathematical sense to the transition between the two formalisms.\ The considerations above justify the introduction of an effective Hilbert space, different from the previous one, on which we can make long-range measurements of observables without seeing the lattice structure. This is the Schwartz space $S(\mathbb{C})$ of rapidly decreasing functions with bounded derivatives. The concept is that expectation values of observables in this Hilbert space coincide with the expectation values in the original Hilbert space in the long-wave regime. It is clear now that the limit of the superpotential and all the other operators must be intended in the context of the weak topology induced on the space of operators by the Hermitian scalar product of $S(\mathbb{C})$. Hence, the correct limit we have to study regards the matrix element $$\label{matrixelement} \begin{split} & \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \Phi(a)\sum_{\Lambda(a)\cap D(R)} \Big \langle \phi \Big\vert \ \frac{1}{z-\zeta_{m,n}(a)} \Big\vert \psi \Big \rangle =\\ \\ & \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \int _{\mathbb{C}} d^{2}z \ \Phi(a)\sum_{\Lambda(a)\cap D(R)} \frac{\phi^{*}(z)\psi(z)}{z-\zeta_{m,n}(a)}. \end{split}$$ with $\phi,\psi \in S(\mathbb{C})$. The details of the computation are given in \[computation\]. As one can expect we find $$\begin{split} \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \langle \phi \vert \ \partial_{z}W(z;a,R) \vert \psi \rangle= \Big \langle \phi \Big\vert \frac{B}{2}\bar{z} \Big\vert \psi \Big \rangle \end{split}$$ which gives the same limit discussed in the previous section, (actually the derivative) $$\label{weak} \partial_{z}W(z;a,R)\rightarrow \frac{B}{2}\bar{z}.$$ Hovewer, the mathematical structure that we have introduced here gives to the above formula a precise mathematical meaning. It is evident that this convergence must be intended only in the weak sense of quantum measure. A convergence in norm $L^{p}$ is excluded for evident reasons of integrability, as well as the uniform one, given the complitely opposite behavour between the initial function \[derw\] and the final limit. Indeed, the former is a meromorphic function with simple poles at the lattice points and vanishing at infinity, while the latter is a pure antiholomorphic object with no singularities at finite points. This is consistent with the previous considerations: long-distance experiments cannot detect the lattice structure.\ Contrary to $\partial_{z}W$, we cannot define the weak limit of $\vert \partial W \vert ^{2}$ and $\partial ^{2}W$ because the corresponding matrix elements are divergent. Indeed, the weak convergence does not preserve in general products and derivative of operators (otherwise it would be a strong convergence). This fact, together with the lost of holomorphicity of the superpotential, implies that part of the supersymmetry algebra is not preserved by the weak limit (as expected), included the Hamiltonian.\ Adopting the formalism of section \[sqm\] we can write the susy charges as the generalized Dolbeault operators $$\begin{split} & \bar{\partial}_{W}= \bar{\partial} + \partial W \wedge ,\hspace{2cm} \delta_{W}= \bar{\partial}_{W}^{\dagger}, \\ \\ & \partial_{W}= \partial + \overline{\partial W} \wedge \hspace{2cm} \bar{\delta}_{W}= \partial_{W}^{\dagger}. \end{split}$$ The limit \[weak\] is not compatible with the structure of the susy charges because it exchanges holomorphic with antiholomorphic derivatives of $W$. Instead, there is no ambiguity if we consider real combinations of the generators. Only the real part of the superpotential has a limit and so only the $N=2$ subalgebra generated by $$d_{h}= \bar{\partial}_{W} + \partial_{W}= d+ d h \ \wedge, \hspace{2cm} \delta_{h}= \delta_{W}+ \bar{\delta}_{W} =d_{h}^{\dagger},$$ where $ h= \mathrm{Re} \ W $, is compatible with the weak limit. This corresponds to the $\theta=0$ algebra in the family of $\mathcal{N}=2$ subalgebras. Learning from \[weak\] that $$dh \rightarrow \frac{B}{2}(z+\bar{z})$$ we conclude that the initial $\mathcal{N}=4$ theory is equivalent for long-range measurements to a $\mathcal{N}=2 $ one with real superpotential $$\label{effectivesuper} h(z,\bar{z})= \frac{B}{2}\vert z \vert^{2}.$$ Depending on the sign of $B$ we obtain the superpotential of a two dimensional harmonic oscillator or repulsor. Both the models are studied in [@rif19]. According to the convention of the previous section we take $B$ to be positive. One can find the vacua of this theory by exploiting the similarity relation between $d_{h}$ and the exterior derivative. The $d_{h}$-cohomology classes in the Hilbert space are conjugated by $e^{-h}$ to de Rham cohomology classes on the complex plane. Since only $H^{0}(\mathbb{C})$ is non trivial, the theory has a unique ground state described by the $0$-form $$\Phi(z,\bar{z})= e^{ -\frac{B}{2} \vert z \vert^{2} },$$ in which we recognize the typical $B$ field factor which appears in the wave functions of the Landau levels.\ We know that the Witten index of a supersymmetric theory is robust under continuous deformations and, the fact that we started from a model with a huge number of vacua and we found in the end a theory with just one vacuum, may lead to some doubts. However, this paradox does not arise because the $\mathcal{N}=2$ model is not the limit of the initial one in a strict sense. Indeed, the new Hamiltonian is not directly related to the previous one and acts on a different Hilbert space. The meaning of the weak limit is that for long distance measurements we can replace the original $\mathcal{N}=4$ theory with an effective $\mathcal{N}=2$ one with superpotential \[effectivesuper\]. Weak Limit of $\partial W$ {#computation} ========================== Let us take $\phi,\psi$ in the Schwartz space $S(\mathbb{C})$ of smooth functions with bounded derivatives. We want to compute the limit \[matrixelement\] $$\begin{split} & \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \Phi(a)\sum_{\Lambda(a)\cap D(R)} \Big \langle \phi \Big\vert \ \frac{1}{z-\zeta_{m,n}(a)} \Big\vert \psi \Big \rangle =\\ \\ & \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \int _{\mathbb{C}} d^{2}z \ \Phi(a)\sum_{\Lambda(a)\cap D(R)} \frac{\phi^{*}(z)\psi(z)}{z-\zeta_{m,n}(a)}, \end{split}$$ where $\Lambda(a)=a\mathbb{Z}+ia \mathbb{Z} $, $D(R)=\lbrace z \in \mathbb{C}, \vert z \vert \leq R\rbrace$ and $\Phi(a)= \frac{a^{2}B}{2\pi}.$ Let us first take $a\rightarrow 0$ with fixed $R$. Performing the change of variable $z\rightarrow z+\zeta_{m,n}(a)$, we get $$\begin{split} & \lim_{a \rightarrow 0} \ \int _{\mathbb{C}} \frac{d^{2}z}{z} \ \Phi(a)\sum_{\Lambda(a)\cap D(R)} \phi^{*}(z+\zeta_{m,n}(a))\psi(z+\zeta_{m,n}(a))= \\ \\ &\lim_{a \rightarrow 0} \ \Big \langle \frac{1}{z} \Big\vert \Phi(a) \phi^{*}(z+\zeta_{m,n}(a))\psi(z+\zeta_{m,n}(a)) \Big \rangle. \end{split}$$ Given that $\frac{1}{z}$ acts on the space of test functions as a linear continuous functional[^4], one can take the limit under the sign of integration. In the expression $$a^{2}\sum_{\Lambda(a)\cap D(R)} \phi^{*}(z+\zeta_{m,n}(a))\psi(z+\zeta_{m,n}(a))$$ we recognize the Riemann sum of $ \phi^{*}(z+\zeta)\psi(z+\zeta)$ with partition of $D(R)$ given by squares of area $a^{2}$. The smoothness and boundedness of the Schwartz functions allow to write $$\begin{split} \lim_{a \rightarrow 0} a^{2} &\sum_{\Lambda(a)\cap D(R)} \phi^{*}(z+\zeta_{m,n}(a))\psi(z+\zeta_{m,n}(a))= \\ \\ & \int_{D(R)} d^{2}\zeta \ \phi^{*}(z+\zeta)\psi(z+\zeta). \end{split}$$ Then, after another shift of $z$ we remain with $$\label{secondl} \lim_{R\rightarrow \infty} \int _{\mathbb{C}} d^{2}z \ \phi^{*}(z)\psi(z) \ \frac{B}{2\pi} \int_{D(R)} d^{2}\zeta\ \frac{1}{z-\zeta}.$$ Let us focus on the integral in $\zeta,\bar{\zeta}$. Since we are going to take the limit of infinite volume, we expand the integral for $\vert z \vert /R \ll 1$. Introducing the new variable $w=\zeta-z$ and denoting with $R_{z}(\theta)=\vert \zeta_{\partial D}(\theta)-z \vert= \sqrt{\vert z \vert ^{2}+R^{2}-z\overline{\zeta_{\partial D}(\theta)} -\bar{z} \zeta_{\partial D}(\theta)}, $ the distance between $z$ and the point $ \zeta_{\partial D}(\theta) \in \partial D(R)$ with phase $\theta \in [0,2\pi]$, we find $$\begin{split} & \int_{D(R)} d^{2}\zeta\ \frac{1}{z-\zeta}=-\int_{0}^{2\pi} d\theta R_{z}(\theta) e^{-i\theta}= \\ \\ &-\int_{0}^{2\pi} d\theta \left( R-\frac{z}{2R}\overline{\zeta_{\partial D}(\theta)} -\frac{\bar{z}}{2R} \zeta_{\partial D}(\theta)+ O(1/R) \right) e^{-i\theta}. \end{split}$$ Using $ \zeta_{\partial D}(\theta)=R e^{i\theta} + O(1)$ and sending $R$ to infinity one gets $$\int_{D(R)} d^{2}\zeta\ \frac{1}{\zeta-z}=\pi\bar{z} + O(1/R) .$$ Plugging this large $R$ expansion in the integral \[secondl\] one obtains a convergent series of integrals weighted by powers of $1/R$. In the limit of infinite $R$ we keep only the leading term and write finally $$\begin{split} \lim_{R\rightarrow \infty} \ \lim_{a \rightarrow 0} \ \langle \phi \vert \ \partial_{z}W(z;a,R) \vert \psi \rangle= \Big \langle \phi \Big\vert \ \frac{B}{2} \bar{z} \Big\vert \psi \Big \rangle . \end{split}$$ FQHE and Gauge Theories {#classS} ======================= In this section we review some well known facts about class $\mathcal{S}$ theories and the $2d/4d$ correspondence. We also recall the correspondence between matrix models and gauge theories. These are the basic ingredients which motivates the Vafa proposal and allow to formulate a precise dictionary between FQHE and supersymmetric gauge theories in $4d$. $\mathcal{T}_{n,g}$ Gauge Theories ---------------------------------- We recall some known facts about class $\mathcal{S}$ theories, namely four dimensional $\mathcal{N}=2$ gauge theories which arise from the compactification of six dimensional theories. In particular, we focus on the subclass which according to Vafa are related to FQHE. A theory in this class is denoted with $\mathcal{T}_{n+3,g}\left[A_{1} \right] $ and is associated to a Riemann surface $\mathcal{C}_{n+3,g}$ of genus $g$ and $n+3$ equivalent punctures. These models are identified [@rif2] with the twisted compactification of the $A_{1} (2,0)$ six-dimensional SCFT on the $g$-Riemann surface in the presence of $n$ defect operators. In the context of FQHE the surface $\mathcal{C}_{n+3,g}$ has also the interpretation of target space for the electrons, hence we are interested in the case of $g=0$. The first model of the series correspond to the three-punctured sphere. This theory simply contains four free hypermultiplets and the gauge dynamics is absent. Much more interesting are the $\mathcal{T}_{n+3,0}\left[A_{1} \right] $ models with $n>0$ which identify $\mathcal{N}=2 \ SU(2)$ gauge theories with massive deformations. The gauge group contains $n$ $SU(2)$ factors, each of them coupled to $N_{f}=4$ flavours. These theories are therefore conformal and admit a space of exactly marginal gauge couplings. The parameter space of gauge couplings turns out to coincide with the moduli space $\mathcal{M}_{n+3,0}$ of complex structures of the Riemann sphere with $n+3$ punctures $\mathcal{C}_{n+3,0}$. The space of couplings has boundaries where one of the $SU(2)$ gauge groups become weakly coupled. To each puncture corresponds also a flavour subgroup $SU(2)$. So, in total we have an $SU(2)^{n+3}$ flavour group. Each punture is also associated to a mass parameter $m_{a},a=1,...,n+3$ of the corresponding flavour subgroup $SU(2)_{a}$. These theories enjoy $S$-duality, which has the geometric interpretation of fundamental group $\pi_{1}$ of $\mathcal{M}_{n,0}$. The action of S-duality on the flavour symmetry groups is a permutation action and coincides with the permutation of the $n+3$ punctures on the sphere and the corresponding mass parameters. The $S$-duality can rearrange in different ways the matter content of the theory, but in each possible $S$-frame we remain always with $n$ $SU(2)$ gauge groups coupled to four hypermultiplet doublets and $n+3$ $SU(2)$ flavour groups. The various weakly coupled $S$-dual frames of the theory coincide with the different ways a sphere with $n+3$ points can degenerate completely to a set of $n + 1$ three-punctured spheres attached together at $n$ nodes. Labelling with $SU(2)_{i}, i=1,..n$ the gauge groups, in a possible weakly coupled $S$-duality frame we have two hypermultiplet doublets in the fundamental of $SU(2)_{1}$ with masses $m_{1}\pm m_{2}$, $n-1$ hypermultiplets in the bifundamental of $SU(2)_{i}\times SU(2)_{i+1}$ with masses $m_{i+2}$ and other two hypermultiplets in the fundamental of $SU(2)_{n}$ with masses $m_{n+2}\pm m_{n+3} $. The linear gauge quiver describing this frame is\ \[superconfquiver\] (NM) \[rectangle,draw,scale=1.5\] at (0,0) [$2$]{}; (NP) \[rectangle,draw,scale=1.5\] at (13,0) [$2$]{}; (NL)\[circle,draw\] at (2,0) [$SU(2)$]{}; (NR) \[circle,draw\] at (4,0)[$SU(2)$]{}; (NL.east)–(NR.west); (5,0) – (8,0); (NC)\[circle,draw\] at (9,0) [$SU(2)$]{}; (ND) \[circle,draw\] at (11,0)[$SU(2)$]{}; (NC.east)–(ND.west); (0.38,0) – (1.21,0); (11.78,0) – (12.65,0); where the rectangular nodes on the two sides represent the two couples of fundamental hypermultiplets, the circular nodes are the gauge groups and the lines between the gauge nodes are the bifundamental hypermultiplets.\ A canonical example is the $\mathcal{N}=2$ $SU(2)$ gauge theory with $N_{f}=4$ fundamental flavours. The theory has an exact marginal coupling $\tau= \frac{\theta}{\pi} + \frac{4\pi i}{g^{2}}$, since the number of flavours is twice the number of colours. The flavour group $SU(2)^{4}$ is enhanced to $SO(8)$ and the four hypermultiplet doublets transform in the eight dimensional vector representation. In this case the $S$-duality group acts by fractional linear transformation of $SL(2,\mathbb{Z})$ on $\tau$ and by triality on $SO(8)$. Hence, the space of marginal couplings parametrized by $\tau $ is $\mathbb{H}/SL(2,\mathbb{Z})$, which is the complex structure moduli space $\mathcal{M}_{4,0}$ of a sphere with four equivalent punctures. If we quotient the upper half plane only by the subgroup $\Gamma(2)$ we get the moduli space of a sphere with four marked punctures, i.e. the modular curve $\mathbb{H}/\Gamma(2)\simeq \mathbb{P}^{1}\setminus\lbrace 0,1,\infty\rbrace $. A natural parametrization of this space is given by the cross-ratio $q$ of the positions of the four punctures. This can be seen as a coordinates on $\mathbb{P}^{1}\setminus{0,1,\infty}$ and is related to $\tau$ by the $\Gamma(2)$-invariant modular lambda function $\lambda(\tau)=\frac{\theta_{2}^{4}(0,\tau)}{\theta_{3}^{4}(0,\tau)}$ which realizes the isomorphism $\mathbb{H}/\Gamma(2)\simeq \mathbb{P}^{1}\setminus\left\lbrace 0,1,\infty\right\rbrace $. The action of $\Gamma(2)$ does not permute the punctures of the sphere and the moduli space has three cusps at $\tau=0,1,\infty$ corresponding to the three weakly coupled frames of the theory. The full $S$-duality group permutes the punctures among themselves and simultaneously the associated mass parameters, mapping between each other also the three weakly coupled description of the theory which are therefore physically equivalent.\ The compactification of $A_{1}(2,0)$ theories on a puctured Riemann surface provides also the construction of a canonical Seiberg-Witten curve encoding the physics of the Coulomb phase of the theory. This curve is a ramified double cover of $\mathbb{C}_{n+3,g}$ defined by the equation $$y^{2} = \phi_{2}(z),$$ where $(y,z)$ are local coordinates in the cotangent bundle of $C_{n+3,g}$. The SW differential is the canonical one form $d\lambda=ydz$, while $d\lambda^{2}=\phi_{2}(z)dz^{2}$ is the associated Jerkin-Strebel quadratic differential with appropriate poles at the punctures. In the case of the $\mathcal{T}_{n+3,0}$ the quadratic differential on the $n+3$-punctured sphere has double poles at the punctures $\zeta_{a}$ with coefficients given by the square of the corresponding mass parameters.\ More general theories in the $A_{1}$ class are associated to a quadratic differential with poles of higher order [@rif29]. These theories are obtained from superconformal theories by tuning some mass parameter $m_{a}$ to be very large, adjusting at the same time the coupled marginal gauge coupling in the UV so that the running coupling in the IR remains finite. From a six-dimensional perspective, the limiting procedure brings two or more standard punctures together to produce a single puncture with a larger degree of divergence. It is relevant for FQHE the case in which two punctures in the $\mathcal{T}_{n+3,0}$ theory collide to generate an irregular puncture. This appears as a quartic pole in the quadratic differential. In the $n=0$ case one obtains an Argyres-Douglas system of $D_{2}$ type, namely a theory of a free hypermutliplet doublet and $SU(2)^{2}$ flavour group. For $n>0$ we find an $SU(2)^{n}$ gauge theory in which $n-1$ $SU(2)$ factors are coupled to $N_{f}=4$ flavour and one $SU(2)$ gauge group is coupled to three fundamentals. Hence, one of the couplings has negative beta function and a Yang-Mills scale $\Lambda_{ \mathrm{YM}}$ is generated at one loop. This coupling vanishes in the UV limit, while the other ones are marginal. So, the UV limit of the theory is decribed by a $\mathcal{C}_{n+2,0}$ punctured sphere and corresponds to the superconformal $\mathcal{T}_{n+2,0}$ theory. The space of couplings is identified with the complex structure moduli space of a sphere with $n+1$ equivalent punctures and a marked puncture. Each puncture contributes to the flavour group with an $SU(2)$ factor, giving in total a $SU(2)^{n+2}$ flavour group. The $S$-duality group which permute the equivalent punctures in the conformal case now is broken to a subgroup which act only on the regular ones, acting simultaneously on the punctures and the corresponding mass parameters. This residual group plays the role of fundamental group of the space of couplings. The theory admits different weakly coupled description which correspond to the possible ways in which the punctured sphere degenerates to $n$ spheres with three regular punctures, corresponding to $\mathcal{T}_{3,0}$ models, and a sphere with a regular and an irregular puncture which is identified with a $D_{2}$ system. Labelling with $SU(2)_{i}, i=1,..n$ the gauge groups, in a possible weakly coupled $S$-frame we have one hypermultiplet doublet in the fundamental of $SU(2)_{1}$ with masses $m_{0}$, $n-1$ hypermultiplets in the bifundamental of $SU(2)_{i}\times SU(2)_{i+1}$ with masses $m_{i}$ and other two hypermultiplets in the fundamental of $SU(2)_{n}$ with masses $m_{n}\pm m_{n+1}$. The linear gauge quiver describing this frame is\ \[newquiver\] (NM) \[rectangle,draw,scale=1.5\] at (0,0) [$1$]{}; (NP) \[rectangle,draw,scale=1.5\] at (13,0) [$2$]{}; (NL)\[circle,draw\] at (2,0) [$SU(2)$]{}; (NR) \[circle,draw\] at (4,0)[$SU(2)$]{}; (NL.east)–(NR.west); (5,0) – (8,0); (NC)\[circle,draw\] at (9,0) [$SU(2)$]{}; (ND) \[circle,draw\] at (11,0)[$SU(2)$]{}; (NC.east)–(ND.west); (0.38,0) – (1.21,0); (11.78,0) – (12.65,0); Compared to the superconformal quiver \[superconfquiver\], now attached to the the first $SU(2)$ node we have a rectangular node representing a single hypermultiplet doublet, implying that $SU(2)_{1}$ is coupled only to three flavours. A simple example is given by the $\mathcal{N}=2$ $SU(2)$ gauge theory with $N_{f}=3$ fundamental flavours. Since $N_{f}<2N_{c}$ the theory is asymptotically free in the UV. The flavour group $SU(2)^{3}$ is enhanced to $SO(6)$ and the three hypermultiplet doublets transform in the sixth dimensional vector representation. The fundamental group of the complex structure moduli space of a sphere with two equivalent and a marked puncture has a unique generator. This acts by permuting the two regular punctures and corresponds to the residual $S$-duality transformation that we have in all the asymptotically free $SU(2)$ gauge theories with $N_{f}<4$. In the UV limit the theory flows to the $\mathcal{T}_{3,0}$ theory of four free hypermultiplets described by the three punctured sphere. Matrix Models and $2d/4d$ Correspondence {#largen} ----------------------------------------- One of the key ingredients to connect FQHE to the gauge theories introduced above is the relation between matrix models and four dimensional $\mathcal{N}=2$ gauge theories [@rif8; @rif9; @rif23]. This connection arises from string theory as follows. One considers the type IIB superstring on a background $CY \times \mathbb{R}^{4}$, where $CY$ denotes a fixed Calabi-Yau threefold with a non compact holomorphic curve $Y\subset CY$, with $N$ $D3$-branes on a subspace $Y \times \mathbb{R}^{2}$. The theory living on $\mathbb{R}^{4}$ admits $\Omega$-deformation [@rif24] with parameters $\epsilon_{1}$ for a rotation transverse to the brane and $\epsilon_{2}$ along the brane. The complex structure moduli of $CY$ is controlled by a set of parameters $t_{k}$ which play the role of couplings in the four dimensional theory. In this set up one considers a topological string on $CY$ with $N$ branes on $Y$. It has been found [@rif9; @rif25; @rif26] that the Nekrasov partition function $Z(t_{k},\epsilon_{1},\epsilon_{2})$ of the $4d$ gauge theory coincides with the open topological string partition function $Z(t_{k},g_{s})$ with $\epsilon_{2}=-\epsilon_{1}=g_{s}$. This admits the representation of matrix model partition function. In the case of $SU(2)$ gauge theories only a single $N\times N$ matrix $\Phi$ is required and we have $$\label{partition} \begin{split} Z(t_{k},\epsilon_{1},\epsilon_{2})= & \int_{N\times N}d\Phi \exp\left( -\frac{1}{g_{s}}\mathrm{Tr} W(t_{k},\Phi)\right) \\ \\= &\int \prod_{i=1}^{N}dz_{i} \prod_{1\leq i< j \leqslant N} (z_{i}-z_{j})^{\frac{\epsilon_{1}}{\epsilon_{2}}} \exp\left( -\frac{1}{\epsilon_{2}} \sum_{i} W(t_{k};z_{i}) \right) , \end{split}$$ where $W(t_{k};z_{i})$ is the potential governing the dynamics of the matrix eigenvalues $z_{i}$ and the $\beta$-ensemble is specified by the parameter $\beta=\epsilon_{1}/\epsilon_{2}$ to which the Van der Monde determinant $\Delta(x_{i})= \prod_{1\leq i< j \leqslant N} (z_{i}-z_{j})$ is raised. The string partition function can be expanded perturbatively in the string coupling as $$Z(t_{k},g_{s})= \exp \left( \sum_{g \geq 0}g_{s}^{2g-2} \mathcal{F}_{g}(t_{k}) \right) ,$$ where $\mathcal{F}_{g}(t_{k})$ denotes the genus $g$ amplitude of topological strings. In the double-scaling limit $g_{s}\rightarrow 0, N\rightarrow\infty$ with $g_{s}N$ fixed, the string partition function reproduces the prepotential as determined by the Seiberg-Witten solution $$\mathcal{F}_{0}(t_{k},a_{j})= \mathrm{Lim}_{g_{s}\rightarrow 0, N\rightarrow \infty} \ g_{s}^{2}\log Z(t_{k},g_{s})$$ where $a_{j}$ are the electric central charges of the Gauge theory. The geometry of the Seiberg-Witten curve emerge naturally in the large $N$ limit of the matrix model, providing the definition of the Coulomb branch coordinates in terms of the matrix model parameters. The expectation value of operators are computed in this regime with the saddle-point method, which requires to know the critical configurations of the eigenvalues. Taking $N\rightarrow\infty$ and $g_{s}\rightarrow 0$ with the ’t Hooft coupling $g_{s}N$ fixed, the set of eigenvalues becomes continuum and it is distributed according to a density function $\rho(z)=\frac{1}{N}\sum_{i=1}^{N}\delta(z-X_{i})$, where the set $X_{i}$ denotes a vacuum configuration of the eigenvalues. Once exponentiated the Van der Monde determinant in the path integral one obtains the effective action $$\label{mathcalpot} \frac{1}{g_{s}}\mathcal{W}(z_{i})= \sum_{i=1}^{N}\frac{1}{g_{s}}W(z_{i})+\sum_{i<j}\log(z_{i}-z_{j}).$$ A certain critical configuration $X_{i}$ must be a solution of the classical equation of motions $$\label{eqmot} \frac{1}{g_{s}}W^{\prime}(X_{i})+\sum_{j\neq i}\frac{1}{X_{i}-X_{j}}=0.$$ It can be shown that the saddle-point solution is captured in the continuum limit by the so-called spectral curve $$\label{ris} y^{2}-W^{\prime}(z)^{2} - f(z)=0$$ where $f(z)$ is the so-called quantum correction $$\label{spectr} f(z)= 2g_{s}\sum_{i=1}^{N}\frac{W^{\prime}(z)-W^{\prime}(X_{i})}{z-X_{i}}.$$ One can define in terms of $X_{i}$ the function $$\label{seiberwitten} y(z)=g_{s}\sum_{i}\frac{1}{z-X_{i}} + W^{\prime}(z).$$ which has simple poles at $X_{i}$. This function solves the $\ref{ris}$ in the large $N$ limit and encodes the data about the density of eigenvalues. In this regime the domain filled by the eigenvalues consists of $m$ disconnected intervals $A_{j}$ called cuts. One can show that the number of branch cuts is equal to the number of critical points of the single field potential $W(z)$. So, a saddle-point configuration corresponds to a distribution $N = N_{1} + N_{2} + . . .N_{m}$ over the cuts $A_{1},A_{2},...,A_{m}$. The density of eigenvalues on a certain $A_{j}$ is given by the jump of $y(z)$ accross the interval $$\rho(z)=\frac{g_{s}N}{2\pi i}\left(y( z+i0)-y(z-i0) \right).$$ One can compute the filling fractions $a_{j}=g_{s}N_{j}$ of a specific cut by doing a countour integral around the cut $$a_{j}= \frac{1}{2\pi i} \oint_{A_{j}} y(z)dz.$$ In the correspondence with the gauge theory the $1$-form $y(z)dz$ plays the role of Seiberg-Witten differential and the spectral curve arising in the double scaling limit is naturally identified with the Seiberg-Witten curve. The filling fractions $a_{j},j=1,...,m$ parametrizing the vacua of the theory have the natural interpretation of Coulomb parameters and the cycles encircling the branch cuts are the A-cycles of the gauge theory. These are dualized by B-cycles which connects the branch cuts to a cut-off point $\Lambda$. These cycles move eigenvalues from the branch cuts to infinity and generate a variation of the filling fraction $a_{j}$. The open string amplitude allows to identify the ‘free energy’ of the matrix model with the Seiberg-Witten prepotential $\mathcal{F}_{0}(a_{1},...,a_{m})$. This functional can be used to express the B-cycles in terms of their duals by $$\label{magnchem} a_{D}^{j}=\frac{\partial \mathcal{F}_{0}}{\partial a_{j}}= \int_{B_{j}} y(z)dz,$$ hence the magnetic central charge $a_{D}^{j}$ has the interpretation of ‘energy cost’ to bring an eigenvalue from infinity to the $j$-th branch cut.\ The matrix models which correspond to $\mathcal{T}_{n+3,0}[A_{1}]$ are the multi-Penner models with logarithmic superpotential [@rif3; @rif7; @rif8; @rif9] $$W(\Phi)= \sum_{a=1}^{n+2} m_{a}\log \left( \Phi-\zeta\right).$$ The target space of the matrix eigenvalues is the Riemann sphere with $n+3$ punctures at the positions $\zeta_{a}, a=1,...,n+2$ and $\zeta_{n+3}=\infty$. In this case the equation $$W^{\prime}(z)=\sum_{a=1}^{n+2}\frac{1}{z-\zeta_{a}}=0$$ has $n+1$ solutions and so a saddle-point configurations is determined by $n+1$ filling fractions $a_{j}=g_{s}N_{j}$, $j=1,...,n+1$. Using the expression of $W^{\prime}(z)$ one can easily see that $f(z)$ in \[spectr\] takes the form $$f(z)=\sum_{a}^{n+2} \frac{c_{a}}{z-\zeta_{a}}$$ where the coefficients $c_{a}$ are determined by the filling fractions. Moreover, they satisfy the constraint $\sum_{a}c_{a}=0$ which follows from the relation $\sum_{i=1}^{N}W^{\prime}(X_{i})=0$. Using the expressions for $f(z)$ and $W^{\prime}(z)$ in \[ris\] we find for the spectral curve the equation $$\label{quadraticdiff} y^{2}=\frac{P_{2n+2}(z)}{\Delta_{n+2}(z)^{2}},$$ where $$\Delta_{n+2}(z)= \prod_{a=1}^{n+2}(z-\zeta_{a})$$ and $P_{2n+2}(z)$ is a polynomial of degree $2n+2$. While in the denominator we find the positions of the punctures, the $2n+3$ coefficients of $P_{2n+2}(z)$ encode two types of data. There are the $n+2$ coefficients $m_{a}^{2}$ of the double poles of the quadratic differential $y^{2}dz^{2}$ at $z=\zeta_{a}$, to which one should include the residue $m_{0}^{2}$ of the double pole at $\infty$. As we said in the previous section, the residue of a certain double pole is the squared of the mass parameter of the flavour subgroup $SU(2)$ associated to the corresponding puncture. One can read the mass parameters associated to the punctures from the residues of $y(z)$ at the simple poles using the general solution in \[seiberwitten\]. It is immediate to see that the masses satisfy the relation $$\label{masses} \sum_{a=0}^{n+2} m_{a}=\sum_{j=1}^{n+1}a_{j}=-g_{s}N,$$ from which we learn that the t’Hooft coupling corresponds (up to the sign) to the mass parameter of diagonal subgroup of the flavour group. The remaining parameters are combinations of the coefficients $c_{i}$, or equivalently of the filling fractions $a_{j}$. Since the sum $\sum_{j=1}^{n+1}a_{j}$ is already fixed in terms of the masses, we have $n$ indipendent moduli $$S_{j}=a_{j+1}-a_{j}, \hspace{1cm} j=1,...,n,$$ which define a basis of $A$-cycles encicrcling appropriately the $A_{j}$ and $A_{j+1}$ branch cuts. The period $S_{j}$ is the Coulomb parameter associated to the $j$-th $SU(2)$ factor of the gauge group $SU(2)^{n}$.\ An important implication of the double scaling limit is that the number of eigenvalues $N$ becomes naturally a continuum variable and one can consider variations of the total number of fields. This is equivalent to take a many-particle system in the grand canonical ensemble where the number of particles in not fixed. One can use the B-cycles in \[magnchem\] to bring particles from infinity to the branch cuts and vary the free energy of the system. Given that $N$ is not fixed anymore in this regime, the sum of the filling fractions $\sum_{j}a_{j}=g_{s}N$ becomes dynamical and we gain an extra modulus to parametrize the vacua in the Coulomb phase. By the relation \[masses\] we learn that the diagonal combination of the masses is promoted to a vev of a scalar field, implying that taking the continuum limit $N\rightarrow \infty$ has the effect of gauging the $U(1)$ diagonal subgroup of the flavour group.\ The class of matrix models which corresponds to the sphere with $n+1$ regular punctures and an irregular puncture is a generalization of the Penner model involving a linear interaction. The single eigenvalue potential reads $$W(z)= \mu z + \sum_{a=1}^{n+1}\frac{m_{a}}{z-\zeta_{a}}.$$ Using the \[spectr\] one can easily check that $f(z)$ takes again the form $$f(z)=\sum_{a=1}^{n+1}\frac{c_{a}}{z-\zeta_{a}}$$ for some coefficients $c_{a}$ depending on the vacuum configuration $X_{i}$ we are considering. The relation $\sum_{i}W^{\prime}(X_{i})=0$ implies in this case the constraint $\sum_{a=1}^{n+1}c_{a}=2g_{s}N\mu $. The single field superpotential has $n+1$ vacua and so the spectral curve has $n+1$ branch cuts where the particles are distributed. The solution \[seiberwitten\] for $y(z)$ has simple poles at the regular punctures and the corresponding residues are the flavour mass parameters $m_{a}$. At infinity, where the irregular puncture is located, we find a simple pole, whose residue is the mass $m_{0}$, and a double pole with coefficient the coupling $\mu$. As in the previous case the sum of the filling fractions $\sum_{j}a_{j}=g_{s}N$ is fixed in term of the masses according to the relation \[masses\] and the ’t Hooft coupling corresponds to the mass associated to the diagonal subgroup of the flavour group. The meromorphic quadratic differential associated to the spectral curve can be recasted in the form \[quadraticdiff\]. As in the Penner model the denominator contains double poles at the regular punctures $\zeta_{a}$. The polynomial $P_{2n+2}(z)$ of degree $2n+2$ at the numerator is specified by $2n+3$ parameters. Among these we have the $n+1$ masses $m_{a}^{2},a=1,...,n+1$ corresponding to the regular punctures and the mass $m_{0}^{2}$ of the irregular puncture at infinity. The remaining $n+1$ degrees of freedom are the $n$ moduli of the Coulomb branch and the coupling $\mu$. This is identified in the large $N$ correspondence with the Yang-Mills scale $\Lambda_{\mathrm{YM}}$. As we discussed before, in the large $N$ limit the mass parameter $g_{s}N$ is promoted to an electric central charge and the vacua of the theory are parametrized by $n+1$ Coulomb parameters.\ The open string amplitude introduced in \[partition\] specifies a perturbative expansion, but does not give an unambiguous non-perturbative answer. To get the non perturbative contributions to the partition function one has to specify a contour for the matrix integral. There is not a unique choice and one has many non perturbative complitions of the partition function. A natural basis of integration domains is given by the Lefschetz thimbles $D_{\alpha}$ of the potential $\mathcal{W}(z_{1},...,z_{N})/\epsilon_{2}$ in \[mathcalpot\]. These cycles are attached to the critical points of $\mathcal{W}$ and are labelled by the saddle point configurations $\alpha=(a_{1},...,a_{m})$. Moreover, they have boundary where $\mathrm{Re}\left( \mathcal{W}/\epsilon_{2}\right) \rightarrow + \infty$ and ensure the convergence of the integral. So, one can define the non perturbative topological string amplitudes [@rif7; @rif8; @rif9; @rif25; @rif26] $$\label{topstring} \begin{split} Z_{\alpha}(\epsilon_{1},\epsilon_{2}) & =\int_{D_{\alpha}} \prod_{i=1}^{N}dz_{i} \prod_{1\leq i< j \leqslant N} (z_{i}-z_{j})^{\frac{\epsilon_{1}}{\epsilon_{2}}} \exp\left( -\frac{1}{\epsilon_{2}} \sum_{i} W(t_{k};z_{i}) \right) \\ \\ & = \int_{D_{\alpha}} \exp\left( -\frac{1}{\epsilon_{2}} \sum_{i} \mathcal{W}(t_{k};z_{i}) \right), \end{split}$$ where $$\mathcal{W}(t_{k};z_{i})= \sum_{i} W(t_{k};z_{i}) - \epsilon_{1} \sum_{i<j}\log (z_{i}-z_{j}).$$ The matrix model representation of the Nekrasov partition functions allows to connect $\mathcal{N}=2$ gauge theories in $4d$ and two dimensional $\mathcal{N}=(2,2)$ Landau-Ginzburg models with superpotential $\mathcal{W}(t_{k};z_{i})$. The key point is that the open string amplitudes can be identified with the flat sections of the $tt^{*}$ Lax connection in the asymmetric limit [@rif23]. This can be realized by compactifying the theory on a spatial circle of radious $R$ in such a way that the superpotential gets rescaled by $\mathcal{W}\rightarrow R\mathcal{W}$. With a rotation of the fermionic measure of the superspace one can also introduce an overall phase $\zeta$ multiplying the superpotential $\mathcal{W}\rightarrow \mathcal{W}/\zeta, \overline{\mathcal{W}}\rightarrow \zeta\overline{\mathcal{W}} $. Then one has to consider an analytic continuation of the phase $\zeta$ away from the locus $\vert \zeta \vert=1$ and take the limit $R\rightarrow 0, \zeta\rightarrow 0$ with $\epsilon_{2}=\zeta/R$ finite. This is equivalent to a non unitary deformation of the theory in which we set $\overline{\mathcal{W}}=0$ and rescale $W\rightarrow W/\epsilon_{2}$. Among the $tt^{*}$ brane amplitudes, a set of distinguished elements is given by the pairing of the D-brane states ${{\,| {D_{\alpha}} \rangle}}$ with the topological vacuum ${{\,| {0} \rangle}}$ corresponding to the identity operator. We denote such wave functions as $$\psi_{\alpha}= {\ensuremath{\langle 0 | D_{\alpha} \rangle}}.$$ In the asymmetric limit one has an explicit formula for the above overlap which reads $$\lim_{ \mathrm{asym} }\psi_{\alpha}(\epsilon_{1},\epsilon_{2})= Z_{\alpha}(\epsilon_{1},\epsilon_{2}),$$ where one has to redefine the Nekrasov deformation $\epsilon_{2}$ with respect to the open string convention by setting $\epsilon_{2}=\zeta/\tilde{R}$. As one can see from the expression above the Nekrasov parameter $\epsilon_{1}$ plays the role of Van der Monde coupling and has mass dimension $1$. One can naturally extend the $2d/4d$ correspondence also to SQM. The one dimensional $\mathcal{N}=4$ version of the theory has the same structure of vacua and BPS spectrum, which implies that one can study the geometry of the vacuum bundle either in one or two dimensions [@rif10]. The superpotential in SQM is dimensionless and arises from the compactification of the $2d$ theory. The matching scale is given by the radious $R$ of the $tt^{*}$ circle and we have $\mathcal{W}_{\mathrm{SQM}}=-\mathcal{W}_{2d}/\epsilon_{2}$. According to the AGT correspondence, the amplitudes of the Penner matrix models compute the conformal blocks of the Liouville CFT and hence the FQHE wave functions in the Vafa’s model. To match the normalization of the Liouville correlators given in [@rif1] one has to identify $-m_{a}/\epsilon_{2}=1$ and $1/\nu=\epsilon_{1}/\epsilon_{2}$. The AGT correspondence has been generalized in [@rif29] also to class $\mathcal{S}$ theories with irregular punctures in the Gaiotto surface. It turns out that the Nekrasov partition functions of these models reproduce irregular conformal blocks of the Liouville CFT. The presence of higher degree poles in the quadratic differential corresponds to insertions in the Liouville correlators of the so called irregular vertex operators [@rif30; @rif31]. As for the mass parameters, the relevant coupling $\mu_{2d}$ of the $2d$ theory is related to its dimensionless counterpart $\mu_{1d}$ in SQM by $\mu_{1d}=-\mu_{2d}/\epsilon_{2}$. $tt^{*}$ Geometry of the One Electron Model with Two Quasi-Holes ================================================================ We consider a class of $2d$ $\mathcal{N}=(2,2)$ Landau-Ginzburg theory described by the superpotential $$W(z)= \mu z + \log z + \log(z-\rho).$$ The $4$-SQM version of this theory corresponds to the Vafa model for a single electron on the plane with two quasi-holes at distinct positions $0,\rho \in \mathbb{C}$ and a background electrostatic field $\mu \in \mathbb{C}^{\times}$. The above function is multi-valued and cannot be strictly considered as the superpotential of the model. We want to define this theory on the Abelian universal cover of the target space and study the correspoding $tt^{*}$ geometry. As discussed in section \[integral\], the solution to the $tt^{*}$ equations is captured by the Stokes matrices which describe the jumps of the brane amplitudes $\Psi(\zeta)$ on the $\zeta$-plane. The monodromy data and the leading IR behaviour of the $tt^{*}$ metric and connection are determined by the BPS spectrum of the model. The inital step is to find the classical vacua of the theory on the universal cover. The solutions $z_{\pm}$ to the equation $\partial_{z} W=0$ denote two equivalence classes of vacua which are isomorphic to the homology group $H_{1}(\mathbb{C}\setminus(0,\rho);\mathbb{Z})$. A natural basis for this group is given by the two loop generators $\ell_{0},\ell_{\rho}$ encircling the holes positions. The vacuum space of the theory on the Abelian cover $\mathcal{A}$ decomposes in a direct sum of irreducible representations of the homology group: $$\mathcal{V}_{\mathcal{A}} \simeq \mathrm{L}^{2}\left( \mathrm{Hom} \left( \mathbb{Z}^{2}, U(1) \right) \right) \otimes \mathbb{C}^{2},$$ where $ H_{1}(\mathbb{C}\setminus(0,\rho);\mathbb{Z})=\mathbb{Z}^{2}$. We denote the orthogonal idempotents of $\mathcal{R}_{\mathcal{A}}$ corresponding to the classical vacua on the universal cover with $${{\,| {\pm;m,n} \rangle}}= \ell_{0}^{m}\ell_{\rho}^{n} {{\,| {\pm;0,0} \rangle}}$$ where ${{\,| {\pm;0,0} \rangle}}$ corresponds to some representative of $z_{\pm}$ on the universal cover. In terms of these states one can construct eigenstates of the loop generators $$\label{point} \begin{split} {{\,| {\pm;\phi,\varphi} \rangle}}= \sum_{m,n} e^{-i(m\phi + n\varphi)} {{\,| {\pm;m,n} \rangle}} \end{split}$$ where the angles $\phi,\varphi \in [0,2\pi]$ label representations of $H_{1}(\mathbb{C}\setminus(0,\rho),\mathbb{Z})$. By the fact that the homology is an abelian symmetry of the model, the $tt^{*}$ geometry diagonalizes completely with respect to the angles $\phi,\varphi$. In particular, introducing the labels $k,j=\pm$, the ground state metric in the point basis \[point\] can be expanded in Fourier series as $$\label{fourier} \begin{split} & g_{k,\bar{j}}( \phi,\varphi) = \sum_{r,s} e^{i( r\phi + s\varphi)} g_{k,\bar{j}}(r,s), \\ \\ & g_{k,\bar{j}}(r,s)= {\ensuremath{\langle \overline{j;r,s} | k;0,0 \rangle}}. \end{split}$$ From the IR expansion $$\label{expansion} {\ensuremath{\langle \overline{j;r,s} | k;0,0 \rangle}} \sim \delta_{k,j}\delta_{r,0}\delta_{s,0}-\frac{i}{\pi} \mu_{k,0,0;j,r,s} K_{0}(2\vert w_{k,0,0}-w_{j,r,s} \vert)$$ we see that the one-soliton multiplicities $\mu_{k,0,0;j,r,s} $ are weighted by the Fourier phase factors $e^{i(r\phi+s\varphi)}$. The number of solitons saturating the Bogomonlyi bound can be obtained by solving the BPS equation [@rif12] $$\partial_{\sigma}z= \alpha \overline{\partial_{z}W}$$ where $\sigma$ is the spatial variable of the field $z$ and $\alpha= \triangle W/ \vert \triangle W \vert$ identifies the BPS sector we are considering. At spatial infinity $\sigma= \pm \infty$ one has also to impose the corresponding critical points as boundary condition. One can count the number of solutions to the BPS equations by plotting with a program the flow of the vector field $$\label{flow} V= \begin{pmatrix} \mathrm{Re} z \\ \mathrm{Im} z \end{pmatrix}= \begin{pmatrix} \mathrm{Re} \ \bar{\alpha}\partial W \\ -\mathrm{Im} \ \bar{\alpha}\partial W \end{pmatrix} .$$ Let us start with the case of $\alpha=\pm i$. The solitons in this sector projected on the target manifold are respectively anti-clockwise and clockwise loops based at $z_{\pm}$. The spectrum that we find is the same of the models $$\label{IRtheories} \begin{split} & W_{-}(z)= \mu z + 2\log z, \\ \\ & W_{+}(z)= \log z + \log (z-\rho), \end{split}$$ which describe the dynamics of the two vacua at $z_{\pm}$ respectively in the IR limits $\rho\rightarrow 0$ and $\mu\rightarrow 0$. In the case of $\alpha=i$ one finds $$\label{primitive} \begin{split} & \vert \mu_{-,m,n;-,r,s} \vert = \delta_{m,r+1}\delta_{n,s+1}, \\ \\ & \vert \mu_{+,m,n;+,r,s} \vert = \delta_{m,r+1}\delta_{n,s}+\delta_{m,r}\delta_{n,s+1}, \end{split}$$ where the sign of the soliton multiplicities will be fixed by the computation of the Stokes matrices.\ Let us consider now the solitons connecting $z_{-}$ to $z_{+}$. Given that the quantum monodromy is not affected by the choice of the parameters (up to conjugation) we can count the soliton solutions for $\mu=\rho=1$. The soliton equation for these BPS states must be solved with the central charges $$\label{solboundcond} \Delta W^{+-}_{k}= z_{+}-z_{-} +\log \bigg \vert \frac{z_{+}(z_{+}-1)}{z_{-}(z_{-}-1)}\bigg\vert +2\pi i(k+1/2) , \ \ \ \ \ k\in \mathbb{Z}$$ and boundary conditions $x(-\infty)= z_{-}, \ x(+\infty)=z_{+}$, where $z_{\pm}= (1\pm \sqrt{5})/2$. Let us fix a representative $z_{-;0,0}$ of $z_{-}$ on the spectral cover and denote with ${{\,| {-;0,0} \rangle}}$ the corresponding state in the Hilbert space. \(A) at (0,0); (B) at (8.1,0.2); (x) at (6,0); (x) at (10,0); ((0,0) to \[out=-55, in=250\] (8,0); It is convenient to label the representatives of $z_{\pm}$ on the spectral cover with homotopy classes of curves which connect a fixed point $z_{\ast} \in \mathbb{C} \setminus \left\lbrace 0,\rho \right\rbrace $ to $z_{\pm}$. Let us choose $z_{\ast}=z_{-}$. In this way the curves $\gamma^{-}_{r,s}$ corresponding to the points ${{\,| {-;r,s} \rangle}}$ are simply the closed cycles $\ell_{0}^{r}\cdot\ell_{\rho}^{s}$ based at $z_{-}$. Then, we identify the curve $\gamma^{+}_{0,0}$ in figure \[curve\] which connects $z_{-}$ to $z_{+}$ with the state ${{\,| {+;0,0} \rangle}}$. All the other curves $\gamma^{+}_{r,s}$ going from $z_{-}$ to $z_{+}$ which label the representatives ${{\,| {+;r,s} \rangle}}$ of $z_{+}$ are in the same homotopy class of $\gamma^{+}_{0,0}\cdot\ell_{0}^{r}\cdot\ell_{\rho}^{s}$. The solutions to the BPS equation which interpolate $z_{-;00}$ and $z_{+;r,s}$ belong to homotopy classes $\gamma^{+}_{r,s}$ for certain couples of integers $(r,s)$. The multiplicity matrix that we obtain by studying the flow of \[flow\] for $\alpha= \Delta W^{+-}_{k}/\vert \Delta W^{+-}_{k}\vert $ is $$\label{spec} \vert \mu_{-,0,0;+,r,s} \vert = \delta_{r,s}+\delta_{r+1,s}$$ \(A) at (0,0); (B) at (8.1,0.2); (x) at (6,0); (x) at (10,0); (x) at (11,0.5); (x) at (5,-2); (x) at (2,1.7); ((0,0) to \[out=-55, in=250\] (8,0); ((0,0) to \[out=55, in=-250\] (8,0); ((0,0) to \[out=-55, in=250\] (11,0); ((11,0) to \[out=75, in=45\] (8,0); and the central charge of the BPS particle in the $(+,r,s;-,0,0)$ sector turns out to be $$\label{centralcharge} \Delta W^{+-}_{r,s}= z_{+}-z_{-} +\log \bigg \vert \frac{z_{+}(z_{+}-1)}{z_{-}(z_{-}-1)}\bigg\vert +2\pi i(r+s+1/2) .$$ One can easily see that the braiding of the quasi-holes generates a spectral flow of the BPS spectrum. As we exchange the two singularities in the anticlockwise sense the curves $\gamma^{\pm}_{r,s}$ transform respectively as $$\label{rule} \gamma^{-}_{r,s}\rightarrow \gamma^{-}_{s,r}, \hspace{1cm} \gamma^{+}_{r,s}\rightarrow \gamma^{+}_{s,r+1},$$ which implies that the entire BPS spectrum can be generated by the fundamental soliton $\gamma^{+}_{0,0}$ by iterating the braiding of the quasi-holes. At the same time the states ${{\,| {\pm;r,s} \rangle}}$ transform as $${{\,| {-;r,s} \rangle}}\rightarrow {{\,| {-;s,r} \rangle}}, \hspace{1cm} {{\,| {+;r,s} \rangle}}\rightarrow {{\,| {+;s,r+1} \rangle}},$$ which implies $${\ensuremath{\langle \overline{+;r,s} | -;0,0 \rangle}} \rightarrow {\ensuremath{\langle \overline{+;s,r+1} | -;0,0 \rangle}}.$$ On the other hand, from the IR expansion of the metric one gets also $$\begin{split} {\ensuremath{\langle \overline{+;r,s} | -;0,0 \rangle}} & \sim -\frac{i}{\pi}\mu_{-,0,0;+,r,s} \ K_{0}(2\vert w_{k,0,0}-w_{j,r,s} \vert) \\ \\ & \rightarrow -\frac{i}{\pi}\mu_{-,0,0;+,r,s} \ K_{0}(2\vert w_{k,0,0}-w_{j,s,r+1} \vert). \end{split}$$ The compatibility of these two expressions puts the condition on the soliton matrix $$\mu_{-,0,0;+,r,s} = \mu_{-,0,0;+,s,r+1}$$ which implies that the sign of the BPS multiplicity in the $(-,0,0;+,r,s)$ sector is the same for each couple $(r,s)$. This sign has not an intrinsic meaning, since it can always be absorbed by changing the sign of one of the theta vacua in \[point\] without affecting the diagonal elements $g_{k,\bar{k}}( \phi,\varphi) $.\ We are able now to compute the Stokes operators of the system. In the basis of theta vacua ${{\,| {-;\phi,\varphi} \rangle}},{{\,| {+;\phi,\varphi} \rangle}}$ these are $2\times2$ matrices where the entries are functions of the angles $\phi,\varphi$. We choose the real axis of the $\zeta$-plane as Stokes axis. The Stokes matrix can be written as ordered product of three matrices $$S=S_{3}S_{2}S_{1}$$ which we are going to describe. The first contribution $S_{1}$ is generated by the phase-ordered product of the solitons with positive real and immaginary part. For $\mu=\rho=1$ we have $\mathrm{Re} \ W(z_{-})> \mathrm{Re} \ W(z_{+})$. So, these BPS states have central charges $-\Delta W^{+-}_{r,s} $ and connect the points $z_{+;r,s}$ to $z_{-;0,0}$ with $r<0,s\leq 0 $. Each of them generates a Stokes jump $$\label{stokesgenerated} M_{r,s}= \begin{pmatrix} 1 & \mu_{-,0,0;+,r,s} \ e^{i(r\phi+s\varphi)} \\ 0 & 1 \end{pmatrix}$$ where the phase factor comes from the Fourier expansion of the metric and we take the soliton multiplicities $\mu_{-,0,0;+,r,s}=-( \delta_{r,s}+\delta_{r+1,s})$ with the mignus sign. Taking the product of these factors for $r<0,s\leq 0 $ we get $$\begin{split} S_{1}=& \prod_{r<0,s\leq 0}M_{r,s}= \begin{pmatrix} 1 & -\sum_{r<0,s\leq 0} e^{i(r\phi+s\varphi)} (\delta_{r,s}+\delta_{r+1,s})\\ 0 & 1 \end{pmatrix} \\ \\ =&\begin{pmatrix} 1 & \frac{1+e^{i\varphi}}{1-e^{i(\phi+\varphi)}}\\ 0 & 1 \end{pmatrix}. \end{split}$$ When $\zeta$ crosses the positive immaginary axis the Stokes matrix picks up the contribution $S_{2}$ of the closed solitons bases at $z_{\pm}$. This matrix has a diagonal structure and the entries $S_{\pm}$ correspond to the Stokes jumps of the abelian models \[IRtheories\]. In presence of aligned critical values there are subtleties in the computation of the Stokes factors. Let us suppose to have $N$ vacua and a set of critical values $w_{k},k=1,....,m\leq N$ on a straight line in the W-plane with one physical soliton (and the time-reversed) connecting only the consecutive ones. As discussed in [@rif37], one finds in the leading IR behavour of the ground state metric terms looking like there were solitons in the $k,k+n$ sector with multiplicity $\pm 1/n$. This could seem strange, since the number of BPS particles should be an integer. However, what is really required to be integral is the Stokes matrix $S\in SL(N,\mathbb{Z})$ and not the soliton matrix $\mu_{ij}$. In this case one has to define a matrix $F$ belonging to the Lie algebra $sl(N,\mathbb{Q})$ that encodes also the BPS particles with rational multiplicities. In order to have an integral Stokes operator we have to exponentiate the algebra element $$\label{stokes} S=e^{F}.$$ Coming back to our model, one has to deal with an infinite number of vacua on the spectral cover with pure immaginary critical values. The correct Stokes jumps as function of the angles can be extracted by the asymmetric limit of the $tt^{*}$ brane amplitudes for the Landau-Ginzburg models $W_{\pm}(z)$ in \[IRtheories\]. The computation in [@rif23] provides the following functions $$\label{magnfun} \begin{split} & F_{-}(\phi,\varphi)= \log\left( 1-e^{-i(\phi+\varphi)}\right) \\ \\ & F_{+}(\phi,\varphi)= \log\left( 1-e^{-i\phi}\right) + \log\left( 1-e^{-i\varphi}\right) - \log\left( 1-e^{-i(\phi+\varphi)}\right) \end{split}$$ from which one can obtain the Stokes jump $$S_{2}= \begin{pmatrix} e^{F_{-}} & 0 \\ 0 & e^{F_{+}} \end{pmatrix}.$$ The matrix $S_{3}$ includes the contribution of the solitons with negative real part and positive immaginary part. These BPS particles connect the points $z_{-,r,s}$ with $r,s\leq 0$ to $z_{+,0,0}$ and have central charges $\Delta W^{+-}_{-r,-s}$. The corresponding multiplicities are $\mu_{+;0,0;-,r,s}=- \mu_{-;r,s;+,0,0}=-\mu_{-;0,0;+,-r,-s}.$ Each of these particle generates a contribution to the Stokes matrix which reads $$M_{r,s}= \begin{pmatrix} 1 & 0 \\ -\mu_{-,0,0;+,-r,-s} \ e^{i(r\phi+s\varphi)} & 1 \end{pmatrix}.$$ Multiplying the Stokes jumps of these solitons for $r,s\leq 0$ one finds $$\begin{split} S_{3}=& \prod_{r,s\leq 0}M_{r,s}= \begin{pmatrix} 1 & 0 \\ \sum_{r,s\leq 0} e^{i(r\phi+s\varphi)} (\delta_{-r,-s}+\delta_{-r+1,-s}) & 1 \end{pmatrix} \\ \\ =&\begin{pmatrix} 1 & 0 \\ \frac{1+e^{-i\varphi}}{1-e^{-i(\phi+\varphi)}} & 1 \end{pmatrix}. \end{split}$$ Putting all the pieces together we get $$S(\phi,\varphi)= \begin{pmatrix} e^{F_{-}(\phi,\varphi)} & \frac{1+e^{i\varphi}}{1-e^{i(\phi+\varphi)}}e^{F_{-}(\phi,\varphi)} \\ \frac{1+e^{-i\varphi}}{1-e^{-i(\phi+\varphi)}}e^{F_{-}(\phi,\varphi)} & e^{F_{+}(\phi,\varphi)} + \frac{2(1+\cos\varphi)}{(1-e^{i(\phi+\varphi)})(1-e^{-i(\phi+\varphi)})}e^{F_{-}(\phi,\varphi)} \end{pmatrix}.$$ The Stokes matrix encoding the contribution of the BPS states with central charges in the lower part of the $\zeta$-plane corresponds to the operator $S^{-t}(\phi,\varphi)$. This has not to be intended as the inverse transpose in the $2\times2$ matrix sense. The correct definition is $$S^{-t}(\phi,\varphi)=\left[ S(-\phi,-\varphi)\right] ^{-t},$$ which requires also to change the sign of the angles.\ In the present case we obtain $$S^{-t}(\phi,\varphi)= \begin{pmatrix} e^{-F_{-}(-\phi,-\varphi)} + \frac{2(1+\cos\varphi)e^{-F_{+}(-\phi,-\varphi)}}{(1-e^{i(\phi+\varphi)})(1-e^{-i(\phi+\varphi)})} & -\frac{(1+e^{i\varphi})e^{-F_{+}(-\phi,-\varphi)} }{1-e^{i(\phi+\varphi)}}\\ -\frac{(1+e^{-i\varphi})e^{-F_{+}(-\phi,-\varphi)}}{1-e^{-i(\phi+\varphi)}} & e^{-F_{+}(-\phi,-\varphi)} \end{pmatrix}.$$ So, the quantum monodromy $H=S^{-t}S$ of this system is $$\begin{split} H(\phi,\varphi)= & \begin{pmatrix} e^{F_{-}(\phi,\varphi)-F_{-}(-\phi,-\varphi)} & \frac{1+e^{i\varphi}}{1-e^{i(\phi+\varphi)}}\left( e^{F_{-}(\phi,\varphi)-F_{-}(-\phi,-\varphi)} -e^{F_{+}(\phi,\varphi)-F_{+}(-\phi,-\varphi)}\right) \\ 0 & e^{F_{+}(\phi,\varphi)-F_{+}(-\phi,-\varphi)} \end{pmatrix} \\ \\ = & \begin{pmatrix} -e^{-i(\phi+\varphi)} & -(1+e^{i\varphi})e^{-i(\phi + \varphi)} \\ 0 & -1 \end{pmatrix} \end{split}$$ where in the second equality we have used the expression for $F_{\pm}$ in \[magnfun\]. We see that despite the Stokes matrices are singular for some values of the angles, the quantum monodromy has smooth coefficients. This is equivalent to the statement that the phases of the quantum monodromy $e^{2\pi i q^{R}_{\pm}(\phi,\varphi)}$ are smooth in the angles, while the Ramond charges $q^{R}_{\pm}(\phi,\varphi)$ are allowed to have integral jumps. The Ramond charges of the theta-vacua reads $$q^{R}_{-}=-B_{1}((\phi+\varphi)/2\pi), \hspace{1cm} q^{R}_{+}= B_{1}((\phi+\varphi)/2\pi) -B_{1}(\phi/2\pi) - B_{1}(\varphi/2\pi),$$ where $B_{1}(x)=x-1/2$ is the first periodic Bernoulli polynomial of variable $x \in [0,1]$. $tt^{*}$ Geometry of Modular Curves {#papermodjhep} =================================== Quantum Hall Effect and Modular Curves -------------------------------------- In this section we are going to study a special class of models which reveal a beautiful connection between the physics of quantum Hall effect and the geometry of modular curves [@rif54]. Despite it is not relevant for phenomenological purposes, this class of theories has remarkable properties which enlarge further the rich mathematical structure of FQHE. We recall that the prototype of one-particle supersymmetric model which is relevant for the FQHE physics is given by the $\mathcal{N}=4$ Landau-Ginzburg theory with superpotential $$\label{class} W(z)= \sum_{\zeta \in L} e(\zeta)\log(z-\zeta),$$ where $L$ is a discrete set of $\mathbb{C} $ and $e(\zeta)$ are real numbers. The variable $z$ is interpreted as the electron coordinate and the term $e(\zeta)\log(z-\zeta)$ is the two dimensional coulombic potential which describes the interaction between the electron and an external charge. In the standard setting of the FQH systems the source of electrostatic interaction is taken to be $L= \Lambda \cup S$, with $\Lambda$ a lattice and $S$ a set of positions of quasi-holes. The effect of the lattice is to reproduce the constant macroscopic magnetic field with $e(\lambda)=1$ units of magnetic flux at a point $\lambda \in \Lambda$. At this level the expression of the superpotential is just symbolic, since the sum is taken over an infinite set of points and the function is multi-valued. Therefore it requires a more precise definition according to the class of models that one is considering.\ For a generic choice of the parameters defining $W(z)$, the classical vacua are isolated, and the elements in the chiral ring are identified with their set of values at the critical points. However, finding the classical vacua and studying the $tt^{*}$ geometry of these models is rather complicated, unless one arranges the set of quasi-holes in some special configuration to have an enhancement of symmetry. In this way we can construct degenerate models of FQHE which are not realistic for phenomenology, but at least analitically treatable. We want to consider a particular class of theories of this type which have an abelian subgroup of symmetry acting transitively on the set of vacua. This is the most convenient limit, since in this case the Berry’s connection can be completely diagonalized in a basis of eigenstates of such symmetry and the $tt^{*}$ equations can be derived. It turns out that these models are parametrized modulo isomorphisms by the family of Riemann surfaces $Y_{1}(N)= \mathbb{H}/\Gamma_{1}(N)$, labelled by an integer $N\geq 2$, also known as modular curves for the congruence subgroup $\Gamma_{1}(N)$ of the modular group $SL(2,\mathbb{Z})$ [@rif45]. Each point of the curve of level $N$ identifies a theory where $\partial_{z}W(z)$ is an ellitpic function with a $\mathbb{Z}_{N}$ symmetry generated by a torsion point of the elliptic curve $\mathbb{C}/\Lambda$. For each connected component of the space of models one can define its spectral cover as the complex manifold whose points identify a model and a vacuum [@rif13]. These are the modular curves $Y(N)= \mathbb{H}/\Gamma(N)$ for the principal congruence subgroup $ \Gamma(N)$. More precisley, $Y(N)$ is a cover of $Y_{1}(N)$ of degree $N$, which is the number of vacua in the fundamental cell of the torus. The $tt^{*}$ equations simplify considerably on the spectral curve and can be normalized in the form of $\hat{A}_{N-1}$ Toda equations [@rif10]. These appear in all the models with a $\mathbb{Z}_{N}$ symmetry group which is transitive on the vacua. The modular curves are manifolds with cusps, which represent physically the RG flow fixed points of the theory. These are in correspondence with the equivalence classes of rationals with respect to the congruence subgroups. An outstandig fact is that for a given $N$ all the $\hat{A}_{Q-1}$ models with $Q \vert N$ are embedded in this class of theories as critical limits, providing the regularity conditions for the solutions to the equations. An exception is the case of $N=4$, where only $\hat{A}_{3}$ models appear. The beauty of the modular curves is that they possess various geometrical structures. For instance, in the modular curves $Y(N)$ of level $N=3,4,5 $ the cusps are located at the vertices of platonic solids inscribed in the Riemann sphere.\ The theory of modular curves is also strictly related to number theory. These surfaces can be seen as projective algebraic curves defined over the real cyclotomic extension of the rationals $\mathbb{Q}(\zeta_{N}+\zeta_{-N})$, with $\zeta_{N}=e^{2\pi i/N}$. At the level of superpotential, the action of the Galois group is reproduced by a third congruence subgroup which enters in this theory, i.e. $\Gamma_{0}(N)$. An interesting implication is that the solutions of the $N$-Toda equations are related by the action of the Galois group, since the UV cusps described by the $\hat{A}_{N-1}$ models are all in the same orbit of $\Gamma_{0}(N)$.\ Another interesting phenomenon appearing in this class of theories is that, despite the covariance of the $tt^{*}$ equations, neither the superpotential nor the ground state metric, and therefore the Berry’s connection, are invariant under the action of $\Gamma(N)$. This apparent contraddiction finds a consistent explanation in the context of the abelian universal cover of the model.\ \ The rest of the chapter is organized as follows: In section \[sec2\] we classify up to isomorphisms all the models of the type \[class\] with an abelian subgroup of symmetry acting transitively on the vacua. The underlying structure of the modular curves and the relation between geometry and number theory arise naturally in the derivation. In section \[sec3\] we provide an explicit description of these models. The target manifold is not simply connected and one needs to pull-back the model on the universal cover in order to define the Hilbert space and write the $tt^{*}$ equations. On this space the symmetry group contains also the generators of loops around the poles in the fundamental cell. The symmetry algebra is non-abelian on the universal cover and the abelian physics of the punctured plane can be recovered at the level of quantum states by considering trivial representations of the loop generators. We show that this can be done consistently with the $tt^{*}$ equations in \[trunc\]. In section \[sec4\] we study the modular properties of these systems. First we consider the transformation of $\partial_{z}W(z)$ under the congruence subgroups and then of the superpotential. In particular we focus on the critical value of one the vacua which we use to write the $tt^{*}$ equations in the Toda form. This can be defined as holomorphic function only on the upper half plane and shifts by a costant under a transformation of $\Gamma(N)$. We connect this phenomenon to the geometry of the modular curves in the simple cases of genus $0$, i.e. with $2\leq N \leq 5$. The details of the computation of the constant for a generic $N$ are instead given in \[modulartrnasofrmationmass\]. The modular transformations have also the effect of changing the basis of the symmetry generators and act on the states by modifying the representation of the symmetry group. We study this action and how the components of the ground state metric are transformed. In \[final\] we provide a classification of the cusps. In particular we study the behavour of the superpotentials around these points and distinguish between UV and IR critical regions. Finally, we discuss the boundary conditions of the solutions and how they are related by the Galois group. Classification of the Models {#sec2} ---------------------------- ### Derivation Our first aim is to classify (up to isomorphisms) all the models in the class of FQHE theories \[class\] with an abelian subgroup of symmetry acting transitively on the vacua. Since the punctured plane is not a simply connected space, one cannot define for these models a superpotential on the target manifold. So, we have to start the classification from the derivative $$\partial_{z} W(z)= \sum_{\zeta \in \Lambda} \frac{1}{z-\zeta} + \sum_{s \in S} \frac{e(s)}{z-s},$$ where $$\begin{split} & \Lambda= 2 \pi \mathbb{Z} \oplus 2\pi \tau \mathbb{Z}, \ \tau \in \mathbb{H}, \\ \\ & e: S \longrightarrow \mathbb{C}. \end{split}$$ We stress again that the expression above is just formal and represents a meromorphic function with a simple pole at each point of $L=\Lambda \cup S$. Moreover, in this classification we allow the charges of the quasi-holes to be complex. This is mathematically consistent, since the superpotential is a complex function.\ The action of the abelian group is transitive on the vacua and, in the case of a non trivial kernel, can always be made faithful. The transitivity implies that the set of vacua is a copy of the abelian group. In particular, given that the zeroes of $\partial_{z} W(z)$ cannot have accumulation points, it must be finitely generated. The abelian subgroups of $\mathbb{C}$ satisfying this property are lattices. Since the group acts freely also on the set of poles and the principal divisor of $\partial_{z} W(z)$ has degree $0$[^5], it is immediate to conclude that also $L$ is a lattice, as well as (the faithful representation of) the abelian subgroup of symmetries of our model.\ If we want $L$ to be at least a pseudosymmetry for $\partial_{z} W(z)$, the function $e(s)$ must be extended to a multiplicative periodic character: $$\begin{split} \partial _{z}W(z+\zeta)=e(\zeta) \partial_{z} W(z),\ \ \ \ \ \ \ \ \ \ e(\zeta+\lambda)= e(\zeta), \end{split}$$ for each $\zeta \in L$, $\lambda \in \Lambda$. By definition of homomorphism, the kernel of $e$ must be a subgroup of $L$. Up to a redefinition of the initial set of holes, this is represented by the sublattice $\Lambda$, which is a symmetry for $\partial_{z} W(z)$ in a strict sense.\ Up to this point, we have a model for each lattice $L$ and a character $e$ which is periodic of a sublattice $\Lambda \subset L$.[^6]\ Given the periodicity of $e$, we can equivalently restrict the analysis to primitive characters, i.e. with trivial kernel: $$e: L/ \Lambda \longrightarrow \mathbb{C},$$ where $ L/ \Lambda \simeq \mathbb{Z}_{N_{1}} \oplus \mathbb{Z}_{N_{2}}$ for two positive integers $N_{1}, N_{2}$ such that $N_{1} \mid N_{2}$. The fact that $e$ is primitive implies the isomorphism $ \mathbb{Z}_{N_{1}} \oplus \mathbb{Z}_{N_{2}} \simeq \mathbb{Z}_{N_{1}N_{2}}$. But, according to the chinese remainder theorem, this can be true only if the two integers are coprime. The consistency between the two conditions on the integers $N_{1},N_{2}$ requires that $N_{1}=1$ and $N_{2}=N\geq 2$, with $$L/ \Lambda \simeq \mathbb{Z}_{N}, \ \ \ \ \ \ N \geq 2.$$ If we set $N=1$ we obtain the trivial case in which there are no quasi-holes.\ In conclusion, the models are classified by couples $( E_{\Lambda},Q)$, where - $E_{\Lambda}$ is the elliptic curve $ \mathbb{C}/ \Lambda$, - $ Q \in E[N]= \left\lbrace P \in \mathbb{C}/ \Lambda \mid NP \in \Lambda \right\rbrace $ such that $e(Q)= e^{\frac{2\pi i}{N}}$. The set $E[N]$ is called the N-torsion subgroup of the additive torus group $\mathbb{C}/ \Lambda$.\ Once $\Lambda$ and the level $N$ are fixed, the choice of the torsion point specifies an embedding of the cyclic subgroup $L/ \Lambda \simeq \mathbb{Z}_{N}$ in the elliptic curve. In particular, the torsion point must be of order $N$, i.e. such that $NQ \in \Lambda$ but $nQ \not \in \Lambda$ for $ 1<n<N$. ### Modular Curves {#modular} Since we are classifying models up to strict equivalence, we have to identify those which are related by an isomorphism. It is known that there is a bijection between the set of equivalence classes of elliptic curves endowed with a $N$-torsion point and the space $$Y_{1}(N)= \mathbb{H}/\Gamma_{1}(N),$$ where $\Gamma_{1}(N)$ is a subgroup of $SL(2,\mathbb{Z})$ defined by the congruence condition $$\Gamma_{1}(N)= \left\lbrace \gamma \in SL(2,\mathbb{Z}) : \gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix}= \begin{pmatrix} 1 & * \\ 0 & 1 \end{pmatrix} \mathrm{mod} \ N \right\rbrace .$$ A complete proof of this result can be found in [@rif45]. It is worth to recall that a matrix $\gamma \in SL(2,\mathbb{Z})$ acts on the upper half plane by the usual fractional linear transformation $$\tau^{\prime}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\tau= \frac{a\tau+b}{c\tau+d},$$ and induces on the points of the elliptic curve the isogeny $z + \Lambda_{\tau} \longrightarrow m z + \Lambda_{\tau^{\prime}} $, for some $m \in \mathbb{C}$ such that $m\Lambda_{\tau}=\Lambda_{\tau^{\prime}}$. These maps are the only bijections which preserve the group structure of the elliptic curve. With these definitions it is immediate to show that the enhanced elliptic curve $(E_{\Lambda_{\tau^{\prime}}},Q)$ is isogenous to $\left( E_{\Lambda_{\tau}}, 2\pi/N +\Lambda_{\tau}\right)$, where $\tau^{\prime}=\gamma(\tau)$ for some $\gamma \in SL(2,\mathbb{Z})$, and that transformations of $\Gamma_{1}(N)$ are the only ones which preserve the choice of the torsion point. So, we can define the moduli space for $\Gamma_{1}(N)$ as $$S_{1}(N)= \lbrace \left( E_{\Lambda_{\tau}}, 2\pi/N +\Lambda_{\tau}\right) , \tau \in \mathbb{H} \rbrace / \sim$$ where $\tau \sim \tau^{\prime}$ if and only if $\Gamma_{1}(N)\tau=\Gamma_{1}(N)\tau^{\prime}$, and state the bijection $$S_{1}(N) \overset{\sim}{\longrightarrow} Y_{1}(N).$$ The space $Y_{1}(N)$ is topologically a complex manifold with cusps and can be compactified. One first has to extend the action of $SL(2,\mathbb{Z})$ to the rational projective line $\mathbb{P}^{1}(\mathbb{Q})= \mathbb{Q} \cup \lbrace \infty \rbrace$. Given $\begin{pmatrix} p & q \\ r & t \end{pmatrix} \in SL(2,\mathbb{Z})$, we have $$\begin{split} & \mathbb{H} \longrightarrow \mathbb{\overline{H}}= \mathbb{H} \cup \mathbb{P}^{1}(\mathbb{Q}), \\ \\ \begin{pmatrix} p & q \\ r & t \end{pmatrix} \frac{a}{c}=& \frac{pa+qc}{ra+tc}, \hspace{2cm } \begin{pmatrix} p & q \\ r & t \end{pmatrix} \infty = \frac{p}{r} \end{split}$$ with $a,c \in \mathbb{Z}$ such that $\mathrm{gcd}(a,c)=1$.\ The set of cusps is given by $C_{\Gamma_{1}(N)}=\mathbb{P}^{1}(\mathbb{Q})/ \Gamma_{1}(N)$ and is finite. By adding these points to $Y_{1}(N)$ one obtains $$Y_{1}(N)\longrightarrow X_{1}(N)= \overline{\mathbb{H}}/\Gamma_{1}(N).$$ The space $X_{1}(N)$ is called modular curve for $\Gamma_{1}(N)$ and can be shown to have the structure of a compact Riemann surface. Since the two dimensional lattice becomes degenerate when $\tau$ approaches the rational projective line, the cusps cannot be strictly considered as members of this class of theories, but rather as critical limits[^7].\ So, we learn that the whole space of models can be written as union of connected components labelled by the integer $N$: $$\mathcal{A}= \bigcup _{N \geq 2} X_{1}(N),$$ where $X_{1}(N)$ parametrizes the subclass of theories of level $N$. It follows from the derivation that each point on $\mathcal{A}$ identifies a model up to isomorphisms.\ The next step is to classify the vacua for this family of theories. Let us consider the modular curve of level $N$. The derivative of the superpotential is an elliptic function on the elliptic curve $\mathbb{C}/ \Lambda_{\tau}$ which has a simple pole at each point $kQ + \Lambda_{\tau},\ k=0,...,N-1$ and a simple zero at each point $P+kQ+ \Lambda_{\tau}, \ k=0,...,N-1$ for some $P \in C/ \Lambda_{\tau}$ such that $\partial_{z} W(P)=0$. Given that the principal divisor is vanishing by Abel’s theorem, we get $$\mathrm{div}(\partial_{z} W(z))= \sum_{0\leq k < N} ([P]+k[Q])-\sum_{0\leq k < N} (k[Q])= N[P]=0.$$ So we deduce that also $P$ must be a torsion point and, once known, we can construct the whole set of vacua with the action of the symmetry generators. In order to find $P$, one can define the Weil pairing of order $N$ between torsion points: $$\begin{split} & e_{N}: E[N] \ \mathrm{x} \ E[N]\longrightarrow \mu_{N}, \ \ \ \ \ \ \ \ \ \ \mu_{N}=\lbrace z \in \mathbb{C}\mid z^{N}=1\rbrace, \\ \\ \ \ \ \ & \hspace{3cm}e_{N}(P,Q)= \frac{F_{P}(z+Q)}{F_{P}(z)}, \end{split}$$ where $F_{P}(z)$ is any elliptic function with simple poles at $kQ+\Lambda_{\tau}$ and simple zeroes at $P+kQ+\Lambda_{\tau}$. Since $F_{P}(z)$ is unique up to multiplicative constants, there is no ambiguity in the definition. The Weil pairing is a sort of inner product on $E[N]$ and can be shown to be alternating, bilinear and non degenerate [@rif2; @rif3]. These properties imply that, once the torsion point of the poles $Q$ is picked, a torsion point of the vacua $P$ is coupled by the condition $e_{N}(P,Q)= e^{\frac{2\pi i}{N}}$. Moreover, another point $P^{\prime}$ which has the same pairing with $Q$ must be of the form $P^{\prime}=P+kQ$ for some integer $k$. We can also give an explicit expression for $e_{N}$ [@rif2]. Chosen $\left( 2\pi/N + \Lambda_{\tau}, 2\pi \tau /N+ \Lambda_{\tau}\right) $ as basis of generators for the torsion group, the formula for the Weil pairing of $P,Q \in E[N]$ is $$e_{N}(P,Q)= e^{2\pi i \ \mathrm{ det} \gamma /N},$$ where $$\begin{bmatrix} P \\ Q \end{bmatrix}= \gamma \begin{bmatrix} 2\pi/N + \Lambda_{\tau} \\ 2\pi \tau/N + \Lambda_{\tau} \end{bmatrix}, \ \ \ \ \ \mathrm{for} \ \gamma \in M_{2}(\mathbb{Z}).$$ This definition is actually independent of how the basis of the torsion group is chosen. One can see that, if $P$ and $Q$ generate $E[N]$, the matrix $\gamma$ is invertible and $e_{N}(P,Q)$ is a primitive complex $N$-th root of unity. Moreover, such expression is preserved under isogenies.\ Now we can classify models and vacua with triplets $(E_{\Lambda_{\tau}}, P, Q)$ which denote elliptic curves with associated two torsion data. If we set $Q=2\pi/N +\Lambda_{\tau}$ with a modular transformation, a torsion point satisfying the Weil condition is $P=-2\pi \tau/N+\Lambda_{\tau}$. It is straightforward to see that the congruence subgroup of $SL(2,\mathbb{Z})$ which preserves both the torsion points is: $$\Gamma(N)= \left\lbrace \gamma \in SL(2,\mathbb{Z}) : \gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \mathrm{mod} \ N\right\rbrace,$$ also called principal congruence subgroup. Similarly to the case of $S_{1}(N)$, one shows that the moduli space for the enhanced elliptic curves of principal type is $$S(N)= \lbrace \left( E_{\Lambda_{\tau}}, -2\pi\tau/N+\Lambda_{\tau}, 2\pi/N +\Lambda_{\tau}\right) , \tau \in \mathbb{H} \rbrace / \sim$$ where $\tau \sim \tau^{\prime}$ if and only $\Gamma(N)\tau=\Gamma(N)\tau^{\prime}$. This set of equivalence classes is therefore isomorphic to the complex manifold $$Y(N)= \mathbb{H}/ \Gamma(N).$$ As for $Y_{1}(N)$, we can compactify such space by adding the set of cusps $C_{\Gamma(N)}= \mathbb{P}^{1}(\mathbb{Q})/\Gamma(N)$ and obtain the compact Riemann surface $$Y(N)\longrightarrow X(N)= \overline{\mathbb{H}}/\Gamma(N),$$ which is known as the N-modular curve of principal type.\ This curve represents the spectral cover of the space of models of level $N$ [@rif12]. Given that $\Gamma(N)$ is a normal subgroup of $\Gamma_{1}(N)$ with coset group $\Gamma_{1}(N)/\Gamma(N)\simeq \mathbb{Z}_{N}$, the degree of the cover is $[\Gamma_{1}(N):\Gamma(N)]=N$, which is exactly the number of vacua up to periodic identification.\ A series of results which allows to count the cusps and describe the set $C_{\Gamma(N)}$ can be found in [@rif2]. In particular, denoting with $\begin{bmatrix} a \\ c \end{bmatrix}, \begin{bmatrix} a^{\prime} \\ c^{\prime} \end{bmatrix}$ two vectors of $\mathbb{Z}^{2}$, one can show that $$\label{width1} \begin{bmatrix} a^{\prime} \\ c^{\prime} \end{bmatrix}= \gamma \begin{bmatrix} a \\ c \end{bmatrix}, \ \mathrm{for} \ \gamma \in \Gamma(N) \Longleftrightarrow \begin{bmatrix} a^{\prime} \\ c^{\prime} \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix} \ \mathrm{mod} \ N.$$ From this result, letting $s=a/c$ and $s^{\prime}= a^{\prime}/c^{\prime}$ two elements of $\mathbb{Q}\cup\left\lbrace \infty\right\rbrace $ such that $\mathrm{gcd}(a,c)= \mathrm{gcd}(a^{\prime},c^{\prime})=1$, it follows the equivalence relation $$\label{width2} \Gamma(N)s^{\prime}= \Gamma(N)s \Longleftrightarrow \begin{bmatrix} a^{\prime} \\ c^{\prime} \end{bmatrix} = \pm \begin{bmatrix} a \\ c \end{bmatrix} \ \mathrm{mod} \ N,$$ where the sign ‘$-$’ keeps into account the projective action of the modular group on $\mathbb{P}^{1}(\mathbb{Q})$. Once collected the rationals in $\Gamma(N)$-classes according to this theorem, one can go futher and find also the cusps of $\Gamma_{1}(N)$. It is enough for this purpose to identify the cusps of $\Gamma(N)$ which belong to the same orbit of $T:\tau\rightarrow \tau +1$, i.e. the generator of $\Gamma_{1}(N)/\Gamma(N)$. Indeed, these points represent vacua of the same theory and are projected on the same cusp of $X_{1}(N)$.\ From a geometrical point of view the cusps are the points for which the covering map $X(N)\rightarrow X_{1}(N)$ degenerates and the dimension of the orbit of $T$ is generically less than $N$. In physical terms, these represent the fixed points of the RG flow in the space of couplings. Being more precise, the cusps of $\Gamma(N)$ which are stabilized by $T$ are expected to be UV fixed points of the RG flow. Indeed, in this limits the zeroes of $\partial_{z}W(z)$ tend to a unique vacuum of order $N$ and the BPS states of the theory become consequently massless, implying that we are approaching a conformal field theory [@rif12]. On the other hand, in the IR limits the vacua become infinitely separated from each other and decouple at the leading order. If they are not simple zeroes of $\partial_{z}W(z)$, i.e. the corresponding orbit of $T$ has dimension between $1$ and $N$, these cusps represent again conformal fixed points on the spectral cover. Instead, if the vacuum is a simple zero and the orbit of the cusp under $T$ has dimension exactly $N$, we are dealing with a free massive field theory. We will provide a more precise description of the physics around the cusps in section $5$. ### The Role of Number Theory Another congruence subgroup of $SL(2,\mathbb{Z})$ which plays an important role in this classification is $$\Gamma_{0}(N)= \left\lbrace \gamma \in SL(2,\mathbb{Z}): \gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} * & * \\ 0 & * \end{pmatrix} \mathrm{mod} \ N \right\rbrace.$$ The three congruence subgroups that we have defined satisfy the chain of inclusions $\Gamma(N)\subset \Gamma_{1}(N) \subset \Gamma_{0}(N) \subset SL(2,\mathbb{Z})$. Moreover, $\Gamma_{1}(N)$ is a normal subgroup of $\Gamma_{0}(N)$. The action of $\Gamma_{0}(N)$ on the space of models is more clear when we choose $Q= \frac{2\pi}{N}$. It is evident that $\gamma \in \Gamma_{0}(N)$ does not leave the model invariant and maps the torsion point into an inequivalent one $\gamma^{*}Q= 2\pi \frac{(c\tau+d)}{N} \sim \frac{2\pi d}{N}$, for an integer $d$ coprime with $N$. The effect on the model corresponds to modify the character by the formula $\sigma_{a}(e(Q))= e(Q)^{a}=e^{\frac{2\pi i a}{N}}$, where $a$ is the inverse of $d$ in $\mathbb{Z}_{N}$. Therefore, it is manifest that $\Gamma_{0}(N)$ reproduces the action of the Galois group of the cyclotomic extension $[\mathbb{Q}(\zeta_{N}):\mathbb{Q}]$. This number field is obtained by adjoining a primitive $N$-th root of unity $\zeta_{N}$ to the rational numbers. Such remarkable connection reveals the algebraic nature of the modular curves. The above formula suggests to define a character $e_{l}(Q)=e^{\frac{2\pi i l}{N}}$ depending on an integer $l$ coprime with $N$, which we call co-level of the modular curve. It is clear that, for a fixed lattice $\Lambda_{\tau}$, a different choice of the co-level corresponds to pick a different point on the space of models. With this definition, a transformation of $\Gamma_{0}(N)$ can be seen as a permutation of the co-levels.\ We can be more precise about these statements by keeping into account that $\Gamma_{0}(N)$ contains the matrix $-I$. We know that a point on $X_{1}(N)$ parametrizes an elliptic curve and a specific embedding of $\mathbb{Z}_{N}$. However, a cyclic group has always two generators which are one the inverse of the other. It is clear that $-I$ acts on the model as a parity transformation, since it does not change the point on $X_{1}(N)$ but inverts the sign of the torsion point. This means that the co-levels $l$ and $N-l$ are actually two descriptions of the same model. So, except for the trivial case of $N=2$, since $-I$ is in $ \Gamma_{0}(N)$ but not in $\Gamma_{1}(N)$ the degree of the cover $\mathbb{H}/ \Gamma_{0}(N)\rightarrow \mathbb{H}/ \Gamma_{1}(N)$ is $\left[ \Gamma_{0}(N):\Gamma_{1}(N) \right]/2 = \phi(N)/2$, where $\phi(N)$ is the Euler totient function which count the elements of $\left\lbrace 0,...,N-1\right\rbrace$ coprime with $N$. As one can expect, this is also the degree of the number field defining this class of theories, which is actually the real cyclotomic extension $\mathbb{Q}(\zeta_{N}+\zeta_{-N})$. The cusps of $\Gamma_{1}(N)$ fall in equivalence classes of the Galois group described by the set $C_{\Gamma_{0}(N)}= \mathbb{P}^{1}(\mathbb{Q})/\Gamma_{0}(N)$. We will discuss the orbits of $\Gamma_{0}(N)$ in the set of critical theories in section $5$. Geometry of the Models {#sec3} ---------------------- ### The Model on the Target Manifold Now we want to translate our abstract classification into an explicit description of these models. A fundamental property of the elliptic functions is that they are uniquely specified (up to multiplicative constants) by the positions and orders of their zeroes and poles. Let us focus on the modular curve of level $N$ and co-level $l$ and choose $(P,Q)=(-\frac{2\pi l \tau}{N}+\Lambda_{\tau},\frac{2\pi }{N}+\Lambda_{\tau})$ as torsion points. For convenience we invert the lattices of poles and vacua with respect to the previous derivation. This can be done with the translation $z \rightarrow z- \frac{2\pi l \tau}{N}$. The derivative of the superpotential for this class of theories is $$\partial_{z} W^{(N,l)}(z;\tau)= \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \left[ \zeta \left( z-\frac{2\pi}{N}(l\tau + k);\tau\right) +\frac{2 \eta_{1}k}{N}\right],$$ where $ z \in \mathcal{S}= \mathbb{C} \setminus \left\lbrace \frac{2\pi}{N}(l\tau+k)+\Lambda_{\tau}, \ k=0,....,N-1; \ \Lambda_{\tau}= 2\pi\mathbb{Z} \oplus 2\pi \tau \mathbb{Z}\right\rbrace $.\ The Weierstrass zeta function $\zeta(z;\tau)$ is defined with the conventions $\eta_{1}=\zeta(\pi;\tau)$, $ \eta_{2}=\zeta(\pi \tau;\tau)$ [@rif47]. This function is meromorphic on $\mathcal{S}$ with simple poles and simple zeroes respectively in $\frac{2\pi(l\tau+k)}{N}+\Lambda_{\tau}$ and $\frac{2\pi k}{N}+\Lambda_{\tau}$, $k=0,...,N-1$. By definition of elliptic function, the sum of the residue inside the fundamental cell is vanishing. From a physical point of view this means that the flux of the magnetic field is cancelled by that of the quasi-holes charges.\ The algebra of the abelian symmetry group of this model is generated by three operators $\sigma,A,B$, defined by the actions $$\label{transform} \begin{split} & \sigma: z\longrightarrow z+ 2\pi/N ;\hspace{2cm} \partial_{z} W^{(N,l)}\left( z+2\pi/N;\tau\right) = e^{\frac{2\pi i l}{N}}\partial _{z}W^{(N,l)}\left( z;\tau\right), \\ \\ & A: z\longrightarrow z+2\pi; \hspace{2.4cm} \partial_{z} W^{(N,l)}\left( z+2\pi;\tau\right) = \partial_{z} W^{(N,l)}\left( z;\tau\right), \\ \\ & B: z\longrightarrow z +2\pi \tau ; \hspace{2.1cm} \partial_{z} W^{(N,l)}\left( z+2\pi \tau;\tau\right) = \partial_{z} W^{(N,l)}\left( z;\tau\right) , \end{split}$$ with the additional relation $ \sigma^{N}= A$. These transformations follows from the quasi-periodicity properties of the Weierstrass function. The double periodicity of $\partial _{z}W^{(N,l)}\left( z;\tau\right) $ allows to identity points wich differ by an element of $\Lambda_{\tau}$, so that the model is naturally projected on the torus $\mathcal{K}=\mathcal{S}/\Lambda_{\tau} $. With this identification we can work with just $N$ critical points, denoted by the equivalence classes $\left[ \frac{2\pi k}{N}\right], k=0,...,N-1 $. From the property $\zeta(-z;\tau)=-\zeta(z;\tau)$, we find that the superpotentials of co-levels $l$ and $N-l$ are actually related by a parity transformation $$\label{colevel} \partial_{z} W^{(N,N-l)}(-z;\tau)=-\partial_{z} W^{(N,l)}(z;\tau)$$ and therefore, as pointed out in the previous section, they describe the same model with inverse torsion points generating the $\mathbb{Z}_{N}$ symmetry.\ We can also write for this class of theories a symbolic ‘primitive’ $$\label{symb} W^{(N,l)}(z;\tau)= ``\sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log \Theta \begin{bmatrix} \frac{1}{2}-\frac{l}{N} \\ \frac{1}{2}-\frac{k}{N}\end{bmatrix}\left( \frac{z}{2\pi};\tau \right) ",$$ where $\Theta\begin{bmatrix} \alpha \\ \beta \end{bmatrix}(z;\tau)$ is the theta function of character $(\alpha,\beta) \in \mathbb{R}$ and quasi-periods $1,\tau$. This function is multi-valued and cannot be really considered as superpotential of the model. Indeed, the target manifold is not simply connected and we cannot find a true primitive on this space. ### Abelian Universal Cover In order to define a superpotential, as well as the Hilbert space of the theory, we need to pull-back the model on the abelian universal cover of $\mathcal{S}$. We remind the definition of universal cover as space of curves $$\label{def} \begin{split} & \hspace{3cm}\mathcal{H}= \left\lbrace p:[0,1]\longrightarrow \mathcal{S}, p(0)=p^{*} \in \mathcal{S} \right\rbrace / \sim, \\ \\ & \sim: \hspace{2cm} p\sim q= \begin{cases} p(1)=q(1), \\ p \cdot q^{-1} =0 \ \mathrm{in \ the \ homology \ group} \ H_{1}(\mathcal{S},\mathbb{Z}). \end{cases} \end{split}$$ By the pull-back of $ \partial_{z} W^{(N,l)}(z;\tau)$ on this space we can define the superpotential of the theory: $$W^{(N,l)}(p;\tau)= \int_{p} \partial_{z} W^{(N,l)}(z;\tau) dz .$$ Contrary to \[symb\], this is a well defined function which assigns a single value to each equivalence class of paths in $\mathcal{H}$.\ In the covering model we have many more vacua than before, but at the same time a larger symmetry group to classify them. An evident subgroup is the deck group of the cover, i.e. the homology. Let us fix a point $z \in \mathcal{S}$. We choose as local basis of $H_{1}(\mathcal{S};\mathbb{Z})$ the one given by the curves $\gamma_{a},\gamma_{b}$, obtained by applying respectively the lattice vectors $a=2\pi,b=2\pi \tau$ to the base point $z$, and the anticlockwise loops $\ell_{0},...,\ell_{N-2}$ encircling the first $N-1$ poles contained in the cell with sides $\gamma_{a},\gamma_{b}$ (Figure \[homology\]). The action on the superpotential of the corresponding operators is $$\label{integrals} \begin{split} & A^{*}W^{(N,l)}(p;\tau)= \int_{\gamma_{a}\cdot p} \partial_{z} W^{(N,l)}(z;\tau)dz = W^{(N,l)}(p;\tau) + \int_{\gamma_{a}} \partial_{z} W^{(N,l)}(z;\tau)dz, \\ \\ & B^{*}W^{(N,l)}(p;\tau)= \int_{\gamma_{b}\cdot p} \partial_{z} W^{(N,l)}(z;\tau)dz = W^{(N,l)}(p;\tau) + \int_{\gamma_{b}} \partial_{z} W^{(N,l)}(z;\tau)dz, \\ \\ & L_{k}^{*}W^{(N,l)}(p;\tau)= \int_{\ell _{k}\cdot p} \partial_{z} W^{(N,l)}(z;\tau)dz = W^{(N,l)}(p;\tau) + \oint_{\ell_{k}} \partial_{z} W^{(N,l)}(z;\tau)dz. \end{split}$$ Compatibly with the definition of symmetry, the superpotential shifts by a constant under the action of the homology generators.\ \ \(A) at (7,0); (B) at (1,1.5); (Z) at (0,0); (x) at (1.5,1); (x) at (3,1); (x) at (6.8,1); (x) at (11,1); (x) at (13,1); (x) at (4.87,1); (x) at (8.88,1); (L) at (2,2); (L) at (4,2); (L) at (7.5,1.7); (L) at (11.5,1.47); (0,0)–(13,0); (13,0)–(15,3); (0,0)–(2,3); (2,3)–(15,3); ((0,0) to \[out=35, in=180\] (2,2); (0,0) to \[out=0, in=0\] (2,2); ((0,0) to \[out=2, in=180\] (4,2); (0,0) to \[out=0, in=0\] (4,2); ((0,0) to \[out=0, in=180\] (7.5,1.7); (0,0) to \[out=0, in=0\] (7.5,1.7); ((0,0) to \[out=0, in=180\] (11.5,1.47); (0,0) to \[out=0, in=0\] (11.5,1.47); In order to define the action of $\sigma$ on $\mathcal{H}$ we need to specify a path $\gamma_{\sigma}$ connecting $z$ to $z+\frac{2\pi}{N}$. Let us fix the point $p^{*}$ in the definition \[def\]. We choose a path $\gamma^{*}$ connecting $p^{*}$ to $p^{*}+\frac{2\pi}{N}$. If the point is not orizontally aligned with the poles we can take for instance a straight line. Now, given a curve $p$ in $\mathcal{H}$, we define $\gamma_{\sigma}$ such that $\gamma_{\sigma} \cdot p \ $= $ \sigma (p) \cdot \gamma^{*}$ in $H_{1}(\mathcal{S};\mathbb{Z})$, where $\sigma(p)$ denotes the curve $p$ shifted by $\frac{2\pi}{N}$ (Figure \[sigma\]). Choosing the integration constant so that $W^{(N,l)}(\gamma^{*};\tau)=0$, we get the transformation of the superpotential $$\label{exact} \begin{split} \sigma^{*}W^{(N,l)}(p;\tau) = & \int_{\gamma_{\sigma}\cdot p} \partial_{z} W^{(N,l)}(z;\tau)dz= \int_{\sigma (p)\cdot \gamma_{*}} \partial_{z} W^{(N,l)}(z;\tau)dz= \\ \\ & e^{\frac{2\pi i l}{N}} W^{(N,l)}(p;\tau) + W^{(N,l)}(\gamma^{*};\tau)= e^{\frac{2\pi i l}{N}} W^{(N,l)}(p;\tau) . \end{split}$$ Inequivalent definitions of $\gamma_{\sigma}$ differ by compositions with the loops $\ell_{k} $ and generate a different constant $W^{(N,l)}(\gamma^{*};\tau)$ in the transformation. Indipendently from the choice, by composing $N$ times $\gamma_{\sigma}$ we obtain an curve in $H_{1}(\mathcal{S};\mathbb{Z})$ homologous to $\gamma_{a}$, consistently with the operatorial relation $\sigma^{N}=A$.\ \ \(A) at (2.5,0); (B) at (4.5,2); (P) at (0,0); (A) at (6,1.3); (A) at (0.7,1.3); (A) at (5,0); (A) at (2,2); (A) at (7,2); (0,0)–(5,0); ((0,0) to \[out=60, in=200\] (2,2); (2,2) to \[out=-10, in=190\] (7,2); ((5,0) to \[out=60, in=200\] (7,2); The generators $\sigma, A,B,(L_{k})_{k=0,...,N-2}$ represent a complete basis for the symmetry group of the covering model. It is clear from Figure \[algebr\] that the algebra is non-abelian on the space of curves. Indeed, the generator $\sigma$ do not commute with $B$ and $L_{k}$ according to the algebraic relations $$\label{identity} \sigma B= L_{0} B \sigma, \hspace{2cm} \sigma L_{k}= L_{k+1} \sigma, \ \ \ k=0,...,N-2,$$ where we have $L_{N-1}= \prod\limits_{k=0}^{N-2}L_{k}^{-1}$. We see that the obstruction to abelianity on the universal cover is represented by the generators of loops. In the faithful representation on the plane these act trivially and $A,B,\sigma$ commute. This is not inconsistent, since the physics must be abelian on the target manifold but not necessarily on the covering model. The abelianity which we have asked in the classification can be recoverd at the quantum level by projecting the Hilbert space on the trivial representation of $L_{k}$. We will discuss this point in the next section. There are not problems instead in the commutation between $\sigma$ and $A$ and clearly the subgroup given by the homology is abelian. \(A) at (1.2,1); (B) at (4.5,0.9); (A) at (9.5,1); (B) at (10.5,1); (X) at (3.4,1); (X) at (10,0.8); (X) at (5,-3.2); (X) at (8,-3.2); (s) at (2.6,0.8); (X) at (10,0); (s) at (1,0); (s) at (6.5,-4); (s) at (5.5,-2.5); (s) at (8.5,-2.5); (s) at (10.3,2); (0,0)–(3,0); (0,0)–(3,2); (3,0)–(6,2); (5,-4)–(8,-4); ((3,2) to \[out=-100, in=170\] (2.8,0.7); ((2.8,0.7) to \[out=00, in=200\] (6,2); ((10,2) to \[out=180, in=180\] (10,0); ((10,0) to \[out=30, in=00\] (10,2); ((5,-4) to \[out=150, in=180\] (5,-2); ((5,-4) to \[out=30, in=00\] (5,-2); ((8,-4) to \[out=150, in=180\] (8,-2); ((8,-4) to \[out=30, in=00\] (8,-2); Now we can classify the vacua of the model defined on $\mathcal{H}$. The vacua on the target manifold are represented by the points $z_{k}= \frac{2\pi k}{N}, k=0,...,N-1,$ up to periodic identification of $\Lambda_{\tau}$. For these points there is an ambiguity in the definition of the cycle $\gamma_{b}$, since there is a pole along the side of the fundamental cell. We choose conventionally the homology path which bypasses the pole on the left. Now that the homology based on the critical points is well defined, we fix $p^{*}$ away from the set of vacua and pick a curve $p_{0}$ connecting $p^{*}$ to $z_{0}$ as representative of $z_{0}$ in $\mathcal{H}$. All the other representatives in the corresponding fiber can be obtained by composing $p_{0}$ with the cycles of $H_{1}(\mathcal{S},\mathbb{Z})$ based on $z_{0}$. Moreover, we can map fibers over different points to each other with the action of $\sigma$. Therefore, the vacua on $\mathcal{H}$ are labelled by the curves $$\label{vacua} p_{k;m,n,n_{0},...,n_{N-2}}= \gamma_{a}^{m} \cdot \gamma_{b}^{n} \cdot \prod_{r=0}^{N-2} \ell_{r}^{n_{r}} \cdot \gamma_{\sigma}^{k} \cdot p_{0}$$ where $k=0,...,N-1,$ and $m,n,n_{0},...,n_{N-2} \ \in \mathbb{Z}$. By definition of abelian universal cover, each fiber is a copy of the homology group.\ We can also provide the corresponding critical values by computing the integrals in \[integrals\]. Let us take again the curve $p_{0}$. By the relation $A=\sigma^{N}$ and the residue formula we have $$\begin{split} & A^{*}W^{(N,l)}(p_{0};\tau)= W^{(N,l)}(p_{0};\tau),\\ \\ & L_{k}^{*}W^{(N,l)}(p_{0};\tau)= W^{(N,l)}(p_{0};\tau) + 2\pi i \ e^{\frac{2\pi i l k}{N}}. \end{split}$$ Moreover, from the first algebraic identity in \[identity\], we easily get $$B^{*}W^{(N,l)}(p_{0};\tau)= W^{(N,l)}(p_{0};\tau) + \frac{2 \pi i}{e^{\frac{2\pi i l}{N}}-1}.$$ Finally, by acting with the generators $\sigma, A,B,(L_{k})_{k=0,...,N-2}$ on $W^{(N,l)}(p_{0};\tau)$, we obtain the whole set of critical values $$\label{critical} \begin{split} W^{(N,l)} & (p_{k;m,n,n_{0},...,n_{N-2}};\tau) = A^{* m} B^{* n} \prod_{0\leq r \leq N-2} L_{r}^{* n_{r}} \sigma^{* k} \ W^{(N,l)}(p_{0};\tau)= \\ \\ & e^{ \frac{2 \pi i l k}{N}} W^{(N,l)}(p_{0};\tau)+ n \frac{2 \pi i}{e^{\frac{2\pi i l}{N}}-1} + \sum_{r=0}^{N-2} n_{r} \ 2\pi i \ e^{\frac{2\pi i l r}{N}} . \end{split}$$ $tt^{*}$ Equations {#trunc} ------------------ ### General Case Our next aim is to derive the $tt^{*}$ equations for this class of models. We first introduce the point basis $\Phi_{k;m,n,n_{0},...,n_{N-2}}$ defined by $$\label{rep} \Phi_{k;m,n,n_{0},...,n_{N-2}} ( p_{k^{\prime};m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}};\tau) =\delta_{k,k^{\prime}}\delta_{m,m^{\prime}} \delta_{n,n^{\prime}} \prod_{r=0}^{N-2} \delta_{n_{r},n_{r}^{\prime}}.$$ The isomorphism between $\mathcal{R}$ and $\mathbb{C}^{\# \mathrm{ classical \ vacua}}$ is realized by the identification $$\Phi_{k;m,n,n_{0},...,n_{N-2}}\longrightarrow {{\,| {p_{k;m,n,n_{0},...,n_{N-2}}} \rangle}}= A^{m} B^{ n} \prod_{r=0}^{N-2} L_{r}^{ n_{r}} \sigma^{ k}{{\,| {p_{0}} \rangle}},$$ where the operatorial action of the symmetry algebra is naturally transported on the Hilbert space. The vacuum space of the theory on $\mathcal{H}$ decomposes in a direct sum of irreducible representations of the homology group: $$\mathcal{V}_{\mathcal{H}} \simeq \mathrm{L}^{2}\left( \mathrm{Hom} \left( \mathbb{Z}^{N+1}, U(1) \right) \right) \otimes \mathbb{C}^{N},$$ provided that $N+1$ is the rank of $H_{1}(\mathcal{S},\mathbb{Z})$ and $N$ the number of vacua on $\mathcal{S}$. A point basis of ‘theta-vacua’ for $\mathcal{V}_{\mathcal{H}}$ is $$\label{pointb} \begin{split} {{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}=& \ e^{\frac{-i \alpha k}{N}} \sum_{m,n,n_{r} \in \mathbb{Z}} e^{-i( m\alpha + n\beta+\sum_{r=0}^{N-2}n_{r}\lambda_{r} )} \\ \\ & A^{m} B^{n} \prod_{r=0}^{N-2} L_{r}^{ n_{r}} \sigma^{ k}{{\,| {p_{0}} \rangle}}, \end{split}$$ where the angles $\alpha, \beta, \lambda_{0},...,\lambda_{N-2} \in [0,2\pi]$ label representations of $H_{1}(\mathcal{S},\mathbb{Z})$. Since the homology is abelian, the angles are defined simultaneously in this common basis of eigenstates: $$\begin{split} & A{{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}=e^{i\alpha} {{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}, \\ \\ & B{{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}=e^{i\beta} {{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}, \\ \\ & L_{r}{{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}=e^{i\lambda_{r}} {{\,| {k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2}} \rangle}}. \end{split}$$ Using the symmetries $A,B,L_{r}$, one finds that in the point basis the ground state metric diagonalizes with respect to the angles: $$\begin{split} & {\ensuremath{\langle \overline{j;\alpha^{\prime}, \beta^{\prime}, \lambda_{0}^{\prime},...,\lambda_{N-2}^{\prime}} | k;\alpha, \beta, \lambda_{0},...,\lambda_{N-2} \rangle}}= \\ \\ & \delta(\alpha-\alpha^{\prime})\delta(\beta-\beta^{\prime}) \prod_{r=0}^{N-2} \delta( \lambda_{r}-\lambda_{r}^{\prime}) \ g_{k,\bar{j}}( \alpha, \beta, \lambda_{0},...,\lambda_{N-2}), \end{split}$$ where the matrix coefficients $g_{k,\bar{j}}( \alpha, \beta, \lambda_{0},...,\lambda_{N-2})$ have the Fourier expansion: $$\begin{split} & g_{k,\bar{j}}( \alpha, \beta, \lambda_{0},...,\lambda_{N-2}) = e^{\frac{i}{N}\alpha (j-k)}\sum_{r,s,t_{q} \in \mathbb{Z}} e^{i( r\alpha + s\beta+\sum_{q=0}^{N-2}t_{q}\lambda_{q} )} g_{k,\bar{j}}(r,s,t_{0},...,t_{N-2}), \\ \\ & g_{k,\bar{j}}(r,s,t_{0},...,t_{N-2})= {\ensuremath{\langle \overline{p_{j;m,n,n_{0},...,n_{N-2}}} | p_{k;m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}} \rangle}} \vert_{m-m^{\prime}=r, n-n^{\prime}=s, n_{q}-n_{q}^{\prime}=t_{q}} . \end{split}$$ The metric cannot be diagonalized also as $N \times N$ matrix, since $\sigma$ do not commute with $B$ and $L_{r}$. The pull-back of the Landau-Ginzburg model on the universal cover is a necessary operation to write the $tt^{*}$ equations, but the price to pay is to have more vacua and a larger space of solutions. We are not interested in the most general solution to the $tt^{*}$ equations, but in the subclass reproducing the abelian physics of the punctured plane. Moreover, the importance of an abelian symmetry group with a transitive action on the vacua is that we can completely diagonalize the ground state metric. Without this property, even writing the $tt^{*}$ equations becomes an hopeless problem. Therefore, we have to truncate the $tt^{*}$ equations in such a way that the solutions are compatible with the abelian representation of the symmetry group. We cannot just set the loop angles to be $0$, since these are differential variables which enter in the equations. So, we have to check that the ansatz of a solution with constant vanishing $\lambda_{r}$ is consistent with all the equations. From the equality $$\frac{2 \pi i}{e^{\frac{2\pi i l}{N}}-1}= \frac{2\pi i}{N} \sum_{k=0}^{N-1} k e^{\frac{2\pi i l k}{N}}= \frac{2\pi i}{N} \sum_{k=0}^{N-2} (k-N+1) e^{\frac{2\pi i l k}{N}},$$ we learn that the combination $\tilde{B}= B^{N}\prod\limits_{k=0}^{N-2} L_{k}^{N-k-1}$ leaves invariant the superpotential. Moreover, one can see that it commutes with $\sigma$ $$\begin{split} &\sigma \tilde{B}= \sigma B^{N}\prod_{k=0}^{N-2} L_{k}^{N-k-1}= B^{N}L_{0}^{N}\prod_{k=0}^{N-3} L_{k+1}^{N-k-1} \prod_{k=0}^{N-2} L_{k}^{-1} \sigma = \\ & B L_{0}^{N-1} \prod_{k=0}^{N-3} L_{k+1}^{N-(k+1)-1}\sigma = B^{N}\prod\limits_{k=0}^{N-2} L_{k}^{N-k-1}\sigma = \tilde{B}\sigma \end{split}$$ where we have used the commutation relations in \[identity\]. Thus, if we choose $A,\tilde{B},L_{r}$ as generators of the homology, we can find a basis in which the eigenvalues of $\sigma$ are simultaneously defined with the two angles $\alpha,\tilde{\beta}= N\beta + \sum_{k=0}^{N-2}(N-k-1)\lambda_{k}$ corresponding to $A,\tilde{B}$.\ The couplings we can vary in these models are the lattice parameter $\tau$ and the usual overall scale multiplying the superpotential. To avoid clash of notations, we denote with $\mu$ this coupling. The corresponding chiral ring coefficients in the point basis of the covering model read $$\begin{split} \left( C_{\tau}\right)_{ k;m,n,n_{0},...,n_{N-2}}^{k^{\prime};m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}}= & \ \mu \partial_{\tau} W^{(N,l)}\Phi_{k^{\prime};m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}} (p_{k;m,n,n_{0},...,n_{N-2}};\tau) \\ = & \ \delta_{k,k^{\prime}}\delta_{m,m^{\prime}} \delta_{n,n^{\prime}} \prod_{r=0}^{N-2} \delta_{n_{r},n_{r}^{\prime}} \ \mu \partial_{\tau} W^{(N,l)}(p_{0};\tau)e^{\frac{2\pi i l k}{N}}, \\ \\ \left( C_{\mu}\right)_{ k;m,n,n_{0},...,n_{N-2}}^{k^{\prime};m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}}= & \ W^{(N,l)}\Phi_{k^{\prime};m^{\prime},n^{\prime},n_{0}^{\prime},...,n_{N-2}^{\prime}} (p_{k;m,n,n_{0},...,n_{N-2}};\tau) \\ = & \ \delta_{k,k^{\prime}}\delta_{m,m^{\prime}} \delta_{n,n^{\prime}} \prod_{r=0}^{N-2} \delta_{n_{r},n_{r}^{\prime}} \\ & \mathrm{x} \ \left( W^{(N,l)}(p_{0};\tau)e^{\frac{2\pi i l k}{N}} +\sum_{r=0}^{N-2} n_{r} \ 2\pi i \ e^{\frac{2\pi i l r}{N}} \right), \end{split}$$ where we have replaced $B$ with $\tilde{B}$ in the computation of the critical values in \[critical\].\ The action of $C_{\tau}$ and $C_{\mu}$ is naturally projected on the theta-sectors of $\mathcal{V}_{\mathcal{H}}$: $$\begin{split} C_{\tau}^{\mathrm{point \ basis}}(\alpha,\tilde{\beta} , \lambda_{0},...,\lambda_{N-2})_{j,k}=& \ \mu \partial_{\tau} W^{(N,l)}(p_{0};\tau) e^{\frac{2\pi i l k}{N}} \delta_{j,k}, \\ \\ C_{\mu}^{\mathrm{point \ basis}}(\alpha, \tilde{\beta}, \lambda_{0},...,\lambda_{N-2})_{j,k}=& \left( W^{(N,l)}(p_{0};\tau)e^{\frac{2\pi i l k}{N}}- 2\pi \sum_{r=0}^{N-2}e^{\frac{2\pi i l r}{N}} \frac{\partial}{\partial \lambda_{r}} \right) \delta_{j,k}. \end{split}$$ The ground state metric satisfy the following set of equations $$\label{set} \begin{split} & \partial_{\bar{\tau}}( g \partial_{\tau} g^{-1})= \left[C_{\tau} ,g C_{\tau}^{\dagger}g^{-1}\right], \\ \\ & \partial_{\bar{\mu}}( g \partial_{\mu} g^{-1})= \left[C_{\mu} ,g C_{\mu}^{\dagger}g^{-1}\right], \\ \\ & \partial_{\bar{\mu}}( g \partial_{\tau} g^{-1})= \left[C_{\tau} ,g C_{\mu}^{\dagger}g^{-1}\right], \end{split}$$ as well as the complex conjugates. If we demand that $\lambda_{0}=...=\lambda_{N-2}=0$ and $\frac{\partial g}{\partial \lambda_{r}}=0$ for each $r$, the derivatives $\frac{\partial}{\partial \lambda_{r}}$ do not contribute to the $tt^{*}$ equations and the F-term deformations can be truncated at the non trivial part $$\begin{split} C_{\tau}^{\mathrm{point \ basis}}(\alpha, \tilde{\beta})_{j,k}=& \mu \partial_{\tau} W^{(N,l)}(p_{0};\tau) e^{\frac{2\pi i l k}{N}} \delta_{j,k}, \\ \\ C_{\mu}^{\mathrm{point \ basis}}(\alpha, \tilde{\beta})_{j,k}=& W^{(N,l)}(p_{0};\tau)e^{\frac{2\pi i l k}{N}} \delta_{j,k}, \end{split}$$ with $\tilde{\beta}=N\beta$. It is clear that with this ansatz the equations given above can be written universally as a unique equation with respect to the critical value $w=W^{(N,l)}(p_{0};\tau)$: $$\label{canonical} \partial_{\bar{w}}( g \partial_{w} g^{-1})= \left[C_{w} , g C_{w}^{\dagger}g^{-1}\right],$$ where $$C_{w}^{\mathrm{point \ basis}}(\alpha, \tilde{\beta})_{j,k}= e^{\frac{2\pi i l k}{N}} \delta_{j,k}.$$ The equations in the parameters $\tau$ and $\mu$ can be obtained from this one by specifying the variation of $w$ and $\bar{w}$. If the equation in the canonical variable is solved with the boundary conditions given by the cusps, the solution automatically satisfies all the equations in \[set\]. Hence, it is convenient to pull-back the equation on the spectral cover of the model where one can use $w= W^{(N,l)}(p_{0};\tau)$ as unique complex coordinate to parametrize models and vacua. By consistency, the cusps should reproduce the correct regularity conditions for the truncated equations. We are going to discuss the boundary conditions for the $tt^{*}$ equations in section \[final\] and we will see that this is the case.\ Now that we have recovered the abelian representation of the symmetry group, we can construct a common basis of eigenstates for $\sigma,A,B$ in which the ground state metric diagonalizes completely. Given the action of $\sigma$ on the point basis $$\sigma{{\,| {k;\alpha, \beta} \rangle}}=e^{\frac{i \alpha }{N}} {{\,| {k+1;\alpha, \beta} \rangle}},$$ we can define a set of $\sigma$-eigenstates as $$\label{eigen} \begin{split} & {{\,| {j;\alpha, \beta} \rangle}}= \sum_{k=0}^{N-1} e^{- \frac{2 \pi i l k j}{N}} {{\,| {k;\alpha, \beta} \rangle}}, \\ \\ & \sigma {{\,| {j;\alpha, \beta} \rangle}}= e^{\frac{ i}{N} (\alpha + 2\pi l j)} {{\,| {j;\alpha, \beta} \rangle}}. \end{split}$$ With this normalization we have the periodicity $${{\,| {k+N;\alpha, \beta} \rangle}}={{\,| {k;\alpha, \beta} \rangle}}$$ and we note that the given definitions are consistent with the relation $\sigma^{N}=A$.\ In the $\sigma$-basis the metric becomes $$g_{k,\bar{j}}( \alpha, \beta) = \delta_{k,j} \ e^{\varphi_{k}( \alpha, \beta)},$$ where the $\varphi_{k}( \alpha, \beta), k=0,..,N-1,$ are real functions of the angles. It is straightforward to derive the $tt^{*}$ equations satisfied by these functions. Using $$C_{w}^{\sigma-\mathrm{basis}}(\alpha, \beta)= \begin{pmatrix} 0 & 1 & 0 & . . . & 0 & 0 \\ 0 & 0 & 1 &. . . & 0 & 0 \\ . & . & . & & . & . \\ . & . & . & & . & . \\ 0 & 0 & 0 & . . . & 0 & 1 \\ 1 & 0 & 0 & . . . & 0 & 0 \\ \end{pmatrix},$$ one finds the well known $\hat{A}_{N-1}$ Toda equations: $$\label{toda} \begin{split} & \partial_{\bar{w}}\partial_{w} \varphi_{0} + e^{\varphi_{1}-\varphi_{0}}-e^{\varphi_{0}-\varphi_{N-1}}=0 ,\\ \\ & \partial_{\bar{w}}\partial_{w} \varphi_{i} + e^{\varphi_{i+1}-\varphi_{i}}-e^{\varphi_{i}-\varphi_{i-1}}=0, \hspace{2cm} i=1,...,N-2 \\ \\ & \partial_{\bar{w}}\partial_{w} \varphi_{N-1} + e^{\varphi_{0}-\varphi_{N-1}}-e^{\varphi_{N-1}-\varphi_{N-2}}=0, \end{split}$$ where the dependence on the angles and $w$ is understood. These equations appear tipically in the models with a $\mathbb{Z}_{N}$ symmetry group acting transitively on the vacua. We are going to discuss the solution to these equations in section \[final\]. In particular, we will see that the RG fixed points reproduce the appropriate boundary conditions for the Toda equations, providing a consistency check for the abelian truncation.\ We can also provide an expression for the symmetric pairing of the topological theory. The operators in the chiral ring corresponding to the basis in \[pointb\] with $\lambda_{r}=0$ and \[eigen\] are respectively $$\begin{split} & \chi_{k}(\alpha, \beta)= e^{\frac{-i \alpha k}{N}} \sum_{m,n \in \mathbb{Z}} e^{-i( m\alpha + n\beta)} \Phi_{k;m,n}, \\ \\ & \Psi_{j}(\alpha, \beta)= \sum_{k=0}^{N-1} e^{- \frac{2 \pi i l k j}{N}} \chi_{k}(\alpha, \beta). \end{split}$$ The topological metric can be computed with the formula \[residue\] for one chiral superfield. In the point basis the expression is $$\begin{split} \mathrm{Res}_{W} \left[ \chi_{k}(\alpha, \beta), \chi_{m}(\alpha^{\prime}, \beta^{\prime}) \right] = \delta(\alpha-\alpha^{\prime})\delta(\beta-\beta^{\prime}) \eta_{k,m}^{\mathrm{point \ basis}}(\alpha, \beta) \end{split}$$ where $$\eta_{k,m}^{\mathrm{point \ basis}}(\alpha, \beta) = \left( \partial^{2} _{z} W^{(N,l)}(0;\tau)\right) ^{-1} e^{-\frac{ i k}{N} (2\alpha + 2\pi l )}\delta_{k,m} .$$ In the basis of $\sigma$-eigenstates one gets $$\begin{split} & \mathrm{Res}_{W} \left[ \Psi_{n}(\alpha, \beta), \Psi_{m}(\alpha^{\prime}, \beta^{\prime}) \right] = \delta(\alpha-\alpha^{\prime})\delta(\beta-\beta^{\prime})\ \eta_{n,m}^{\sigma-\mathrm{basis}}(\alpha, \beta), \end{split}$$ where $$\eta_{n,m}^{\sigma-\mathrm{basis}}(\alpha, \beta)=\left( \partial^{2} _{z} W^{(N,l)}(0;\tau)\right) ^{-1} \sum_{k=0}^{ N-1} e^{-\frac{ i k}{N} (2\pi l (m+n+1) + 2\alpha )} .$$ We can choose a basis and use one of these expressions to impose the $tt^{*}$ reality constraint, where the complex conjugates $g^{*}, \eta^{*}$ in terms of the matrices $g(\alpha,\beta)$, $\eta(\alpha,\beta)$ are respectively $$\begin{split} & g^{*}(\alpha, \beta)=\left[ g(-\alpha,- \beta) \right] ^{*}, \\ & \eta^{*}(\alpha, \beta)=\left[ \eta(-\alpha, -\beta) \right]^{*}. \end{split}$$ This condition gives $g(-\alpha, -\beta)$ in terms of $g(\alpha, \beta)$, but an explicit computation is hard in general. In the case of a trivial representation of the homology, the reality constraint simplifies, becoming the same of the $A_{N}$ models [@rif10]. ### Peculiarities of the N=2 Level We briefly discuss some peculiar aspects of the class of models of level $2$. The derivative of the superpotential is $$\partial_{z} W(z;\tau)= \zeta ( z-\pi \tau;\tau)-\zeta(z-\pi \tau -\pi ;\tau) -\eta_{1},$$ where $ z \in \mathcal{S}= \mathbb{C} \setminus \left\lbrace \pi \tau, \pi \tau +\pi +\Lambda_{\tau}; \ \Lambda_{\tau}= 2\pi\mathbb{Z} \oplus 2\pi \tau \mathbb{Z}\right\rbrace $. The simple poles and simple zeroes are located respectively in $\pi \tau, \pi \tau +\pi +\Lambda_{\tau}$ and $0, \pi+\Lambda_{\tau}$. In this case the Galois group acts trivially and we have only one co-level.\ In addition to the generators $\sigma, A, B,L$ already discussed, the symmetry group of the models of level $2$ contains also the parity transformation $$\iota : z\longrightarrow -z; \hspace{2cm} \partial_{z} W(-z;\tau)=- \partial_{z} W(z;\tau),$$ which follows from \[colevel\] for $N=2$ and $l=1$.\ This operator satisfies the following commutation relations with the generators of the abelian group: $$\begin{split} & \iota A= A^{-1} \iota, \\ & \iota B= B^{-1} \iota, \\ & \iota L= L \iota, \\ & \iota \sigma= \sigma^{-1} \iota . \end{split}$$ The action of $ \iota$, as well as its algebraic relations, can be extended to the abelian universal cover. There are inequivalent choices which differ by compositions with the loop generator $L$. Indipendently from the definition, the set of curves with $p(1)=0$ is left invariant by the action of $\iota$.\ The presence of an extra symmetry, as well as the possibility of working with $2 \times 2$ matrices, guarantees some semplifications in the derivation of the $tt^{*}$ equations. A point basis of theta-vacua for this family of theories is $$\begin{split} & {{\,| {0;\alpha, \beta, \lambda} \rangle}}= \sum_{m,n,j \in \mathbb{Z}} e^{-i( m\alpha + n\beta+j\lambda)} A^{m} B^{ n} L^{j} {{\,| {p_{0}} \rangle}}, \\ & {{\,| {1;\alpha, \beta, \lambda} \rangle}}= \sum_{m,n,j \in \mathbb{Z}} e^{-i( m\alpha + n\beta+j\lambda)} A^{m} B^{ n} L^{j} \sigma{{\,| {p_{0}} \rangle}}. \end{split}$$ Setting $\lambda=0$, the operators $\sigma$ and $\iota$ act on these states as: $$\begin{split} & \sigma {{\,| {0;\alpha, \beta} \rangle}}= {{\,| {1;\alpha, \beta} \rangle}}, \hspace{2cm} \sigma {{\,| {1;\alpha, \beta} \rangle}} = e^{i \alpha} {{\,| {1;\alpha, \beta} \rangle}}, \\ & \iota {{\,| {0;\alpha, \beta} \rangle}}= {{\,| {0;-\alpha, -\beta} \rangle}}, \hspace{1.5cm} \iota {{\,| {1;\alpha, \beta} \rangle}} = e^{-i \alpha} {{\,| {1;-\alpha, -\beta} \rangle}}. \end{split}$$ The ground state metric in this basis is represented by the $2$ x $2$ hermitian matrix $$g(\alpha, \beta)= \begin{pmatrix} g_{0\bar{0}}(\alpha, \beta) & g_{0\bar{1}}(\alpha, \beta) \\ g_{1\bar{0}}(\alpha, \beta) & g_{1\bar{1}}(\alpha, \beta) \end{pmatrix}.$$ The symmetries $\sigma$ and $\iota $ imply respectively $$g_{0\bar{0}}(\alpha, \beta)= g_{1\bar{1}}(\alpha, \beta), \hspace{2cm} g_{0\bar{1}}(\alpha, \beta)= e^{i\alpha} g_{1\bar{0}}(\alpha, \beta),$$ and $$g_{0\bar{0}}(\alpha, \beta)= g_{0\bar{0}}(-\alpha, -\beta), \hspace{2cm} g_{1\bar{0}}(\alpha, \beta)= e^{-i\alpha} g_{1\bar{0}}(-\alpha, -\beta).$$ Therefore the metric can be written as $$g(\alpha, \beta)= \begin{pmatrix} A(\alpha, \beta) & e^{\frac{i\alpha}{2}}B(\alpha, \beta) \\ e^{\frac{-i\alpha}{2}}B(\alpha, \beta) & A(\alpha, \beta) \end{pmatrix},$$ where $A(-\alpha, -\beta)=A(\alpha, \beta)$ and $B(-\alpha, -\beta)=B(\alpha, \beta)$. The transpose and complex conjugate of $g$ read $$\begin{split} & g^{\mathrm{T}}(\alpha, \beta)=\left[ g(-\alpha,- \beta) \right] ^{\mathrm{T}}, \\ & g^{*}(\alpha, \beta)=\left[ g(-\alpha, -\beta) \right]^{*}, \end{split}$$ and therefore $$g^{\mathrm{\dagger}}(\alpha, \beta)= \left[ g(-\alpha,- \beta) \right] ^{\mathrm{\dagger}}.$$ The hermiticity of $g$ implies that $A(\alpha, \beta)$ and $B(\alpha, \beta)$ are real functions, while the positivity requires that $A(\alpha, \beta) > 0$. The matrix $C_{w}$ and the topological metric in this basis are $$C_{w}(\alpha, \beta)= \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \hspace{2cm } \eta(\alpha, \beta)= \left( \partial^{2} _{z} W(0;\tau)\right) ^{-1} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ We see that, by the parity properties of $A(\alpha, \beta)$ and $B(\alpha, \beta)$ implied by $ \iota$, the reality constraint \[realityconstraint\] reduces to $$\vert \partial^{2} _{z} W(0;\tau) \vert ^{2} ( A(\alpha, \beta)^{2}-B(\alpha, \beta)^{2} )=1.$$ As a conquence, the metric can be parametrized in term of a single function of the angles $$\begin{split} & A(\alpha, \beta)= \frac{1}{\vert \partial^{2} _{z} W(0;\tau) \vert} \cosh \left[ L(\alpha, \beta)\right], \\ & B(\alpha, \beta)= \frac{1}{\vert \partial^{2} _{z} W(0;\tau) \vert} \sinh \left[ L(\alpha, \beta)\right] . \end{split}$$ Finally, by plugging the given expressions of $g(\alpha, \beta)$ and $C_{w}$ in \[canonical\] we find the Sinh-Gordon equation $$\partial_{\bar{w}}\partial_{w} L(\alpha, \beta) = 2 \sinh \left[ 2 L(\alpha, \beta)\right].$$ Modular Properties of the Models {#sec4} -------------------------------- ### Modular Transformations of $ \partial_{z} W^{(N,l)}(z;\tau)$ The underlying structure of this class of models is the theory of the modular curves. In particular, as discussed in \[modular\], the space of models and its spectral cover are described respectively by the modular curves for $\Gamma_{1}(N)$ and $\Gamma(N)$. Now that we have provided an explicit description of these systems, we want to investigate the modular properties of the superpotential, as well as the ground state metric, with respect to the relevant congruence subgroups of $SL(2,\mathbb{Z})$. Let us first consider $\partial_{z} W^{(N,l)}(z;\tau)$. Given a modular transformation $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_{1}(N)$, the derivative of the superpotential transforms as $$\partial_{z} W^{(N,l)}\left( \frac{z}{c\tau + d};\frac{a\tau +b}{c\tau + d}\right) =$$ $$\begin{split} & (c\tau+d) \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \biggl[ \zeta \left( z- \frac{2\pi}{N}((a\tau+b)l + k (c\tau + d)) ;\tau \right) + 2\frac{d \eta_{1}+c \eta_{2}}{N} k \biggr] = \\ \\ &(c\tau+d) \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \left[ \zeta \left( z-\frac{2\pi}{N}bl- \frac{2\pi}{N}(l\tau+k) ;\tau \right) +\frac{2\eta_{1}k}{N} \right]= \\ \\ & (c\tau+d) \partial_{z} W^{(N,l)}\left( z-\frac{2\pi}{N}bl ;\tau\right), \end{split}$$ where we have used the fact that $a,d=1$ mod $N$, $c=0$ mod $N$ and the modular properties of the Weierstrass function: $$\label{weierstrass} \begin{split} & \zeta \left(\frac{z}{c\tau + d} ; \frac{a\tau +b}{c\tau + d} \right)= (c\tau+d)\zeta(z;\tau), \\ \\ & \eta_{1}\left( \frac{a\tau +b}{c\tau + d}\right) = (c\tau+d) (d \eta_{1}+c \eta_{2}). \end{split}$$ We see that $\Gamma_{1}(N)$ preserves only one of the torsion point. The transformation $\frac{2\pi\tau}{N} \rightarrow \frac{2\pi(a\tau+b)}{N}$ results in a shift of poles and vacua by $\frac{2\pi}{N}bl$. If we set $b=0$ mod $N$, i.e. $\gamma \in \Gamma(N)$, also the order of vacua is preserved modulo periodicity.\ On the other hand, a matrix $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_{0}(N)$ acts on $\partial_{z} W^{(N,l)}(z;\tau)$ as $$\label{dw} \begin{split} & \partial_{z} W^{(N,l)}\left( \frac{z}{c\tau + d};\frac{a\tau +b}{c\tau + d}\right) = \\ \\ & (c\tau+d) \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \left[ \zeta \left( z- \frac{2\pi}{N}((a\tau+b)l + k (c\tau + d)) ;\tau \right) +2\frac{d \eta_{1}+c \eta_{2}}{N} k \right]= \\ \\ & (c\tau+d) \sum_{k=0}^{N-1} e^{\frac{2\pi i al k}{N}} \left[ \zeta \left( z-\frac{2\pi}{N}bl- \frac{2\pi}{N}(al\tau+k) ;\tau \right) +\frac{2\eta_{1}k}{N} \right]= \\ \\ & (c\tau+d) \partial_{z} W^{(N,al)}\left( z-\frac{2\pi}{N}bl ;\tau\right), \end{split}$$ where in the third line we used the fact that $a=d^{-1}$ mod $N$. As discussed in section \[modular\], $ \Gamma_{0}(N)$ changes the co-level with the map $l \rightarrow al$, acting as the Galois group of the real cyclotomic extension. ### Modular Transformations of the Physical Mass {#modphysmass} It turns out that the superpotential is not invariant under a transformation of $\Gamma(N)$, but it shifts by a constant. On one side, it is not wrong to say that the model is left invariant, since a constant is irrilevant in determining the physics of a system. But, at the same time, the ‘physical mass’ $W^{(N,l)}(p_{0};\tau)$ is the parameter that we have used to write the $tt^{*}$ equations. This means that $\Gamma(N)$ changes the coordinate of the model on the spectral cover, with a consequent transformation of the ground state metric. An analogous constant shift is induced on the superpotential also by $\Gamma_{1}(N)$ and $\Gamma_{0}(N)$, in addition to the effects discussed in the previous paragraph. Before giving an interpretation of such phenomena and solving the apparent contraddiction, we want to compute these constants.\ The fact that $W^{(N,l)}(p_{0};\tau)$ transforms non trivially under $\Gamma(N)$ is connected to the geometry of the modular curves. These spaces are not simply connected and a critical value can be defined only on the universal cover, i.e. the upper half plane, where $\Gamma(N)$ plays the role of deck group. This is also strictly related to the fact that the superpotential is defined on the universal cover of the target space, where we have more than $N$ vacua. On the other hand, variations of $W^{(N,l)}(p_{0};\tau)$ with respect to some coordinate on the modular curve must be modular functions of $\Gamma(N)$. Let us give some examples. Among the curves $X(N)$, the cases with $N\leq 5$ are the only ones with genus $0$ and therefore the easiest to describe [@rif51]. Genus $0$ means in particular that the modular function field has a unique generator, also called Hauptmodul. This function is unique up to Mobius transformations and modular with respect to $\Gamma(N)$. Moreover, it is the projective coordinate which realizes the isomorphism between $X(N)$ and the Riemann sphere punctured with the cusps. Denoting with $x$ the coordinate on the sphere, we can define the one-form $dW(x)=C(x)dx$, where $C(x)=\partial_{x}W(p_{0};x)$ is the chiral ring coefficient which describes variations of the critical value with respect to the Hauptmodul. Except the case of $N=5$, for these curves we have only the co-level $1$ (and the inverse), so we suspend the notation $(N,l)$. From the theory of projective algebraic curves, $C(x)$ must be a rational function with coefficients in the cyclotomic field. Moreover, the poles must be located in the free IR cusps, where the physical mass is expected to diverge. Thus, $C(x)$ can be fixed up to a multiplicative constant by requiring the correct transformation properties under $\Gamma_{1}(N)$ and $\Gamma_{0}(N)$. In particular, from the transformations of $\partial_{z} W^{(N,l)}(z;\tau)$ under $\Gamma_{1}(N)$, we learn that $T: \tau \rightarrow \tau +1$ acts on the one form as an automorphism of the sphere with the formula $$T^{*}dW(x)= dW(p_{0};T^{*}x)=dW(\sigma^{-1}(p_{0});x)=e^{-\frac{2\pi i }{N}} dW(x),$$ where $l=1$ is understood.\ Starting from $X(2)$, we have $3$ cusps which can be viewed as the $3$ vertices of the equatorial triangle of a double triangular pyramid inscribed in $\mathbb{P}^{1}$. A set of representatives for the cusps is given by $$C_{\Gamma(2)}=\left\lbrace 0,1,\infty \right\rbrace$$ where $\infty$, being a fixed point of $\tau \rightarrow \tau +1$, is a UV cusp and $0,1$ identify two decoupled vacua. Moreover, the quotient group $PSL(2,\mathbb{Z})/\Gamma(2)$ is isomorphic to $\mathbb{S}_{3}$, which is the symmetry group of a triangle. We can use $x(\tau)=1-\lambda(\tau)$ as coordinate on the sphere, where $\lambda(\tau): \mathbb{H}/\Gamma(2) \overset{\sim}{\longrightarrow} \mathbb{P}^{1}/\lbrace 0,1, \infty \rbrace$ is the modular lambda function defined by $$\lambda(\tau)= \frac{\theta_{2}^{4}(0;\tau)}{\theta_{3}^{4}(0;\tau)}.$$ The $3$ cusps fall in the point $x(\infty)=1, x(0)=0,x(1)= \infty$ and $T$ acts on $x$ by the transformation of $PSL(2,\mathbb{C})$ $$T^{*}x =\frac{1}{x}.$$ From what we have said $C(x)$ must have a simple pole in $x=0$ and $x=\infty$. Asking that $T^{*}dW(x)= -dW(x)$, we find $$dW(x)= \alpha \frac{dx}{x}$$ where $\alpha$ is a complex constant. We see that there is a unique generator of $\Gamma(2)$ which acts non trivially on the critical value. This generates loops around the cusp in $0$ (as well as in $\infty$) and produces the constant shift of the superpotential.\ In the case of $N=3$ the cusps are located at the $4$ vertices of a regular tetrahedron inscribed in the Riemann sphere. It is not a coincidence that $PSL(2,\mathbb{Z})/\Gamma(3)$ is isomorphic to the symmetry group of this solid figure, i.e. $\mathbb{A}_{4}$. The set of cusps is $$C_{\Gamma(3)}=\left\lbrace 0,1/2,1,\infty\right\rbrace .$$ Also in this case $\infty$ is the only UV cusp, while the other rationals are in the same orbit of $T$ and identify free theories. The integers coprime with $3$ are $1$ and $2 =-1$ mod $3$ which describe the same theory. The generator of the function field is $$J_{3}(\tau)= \frac{1}{i\sqrt{27}}q_{\tau}^{-\frac{1}{3}} \left( \prod_{n=1}^{\infty} \frac{1-q_{\tau}^{n/3}}{1-q_{\tau}^{3n}} \right)^{3} = \frac{1}{i\sqrt{27}} \left( \frac{\eta(\tau/3)}{\eta(3\tau)}\right)^{3}$$ where $q_{\tau}=e^{2\pi i \tau}$ and $$\eta(\tau)= q_{\tau}^{1/24} \prod _{n=1}^{\infty} (1-q_{\tau}^{n})$$ is the Dedekind eta function. The generator of $\Gamma_{1}(3)/ \Gamma(3)$ acts on the Hauptmodul as $$T^{*} J_{3}=\zeta_{3} + \zeta_{3}^{2}J_{3}$$ where we use the notation $ \zeta_{k}= e^{2\pi i/k}$. Denoting $x=J_{3}$, the one-form with values in the chiral ring is $$dW(x)= \alpha\sum_{k=0}^{2} \frac{\zeta_{3}^{-k} dx}{x-x_{k}}$$ where $x_{0}=J_{3}(0)=0,x_{1}=J_{3}(1)=\zeta_{3}$ and $x_{2}=J_{3}(1/2)=1+\zeta_{3}$, with $x_{k+1}=T^{*}x_{k}$. The cusp at $\tau=\infty$ is instead sent to $x=\infty$ on the sphere. It is straightforward to check that $dW(x)$ satisfies $T^{*}dW(x)= \zeta_{3}^{-1}dW(x)$.\ Another spherical version of a platonic solid appears for $N=4$. In this curve the cusps are the $6$ vertices of a regular octrahedron in $\mathbb{P}^{1}$ with symmetry group $PSL(2,\mathbb{Z})/\Gamma(4)\sim \mathbb{S}_{4}$. Also in this case the co-levels are just $\pm 1$. The critical points are given by $$C_{\Gamma(4)}=\left\lbrace 0,1/3,1/2,2/3,1,\infty\right\rbrace .$$ In this curve we have two UV cusps, i.e. $\infty$ and $1/2$, which are both fixed points of $T$. The remaining ones represent instead free IR critical points. The Hauptmodul of level $4$ is $$J_{4}(\tau)=\zeta_{8}^{3}\sqrt{8}q_{\tau}^{1/4} \prod_{n=1}^{\infty} \frac{(1-q_{\tau}^{4n})^{2}(1-q_{\tau}^{n/2})}{(1-q_{\tau}^{n/4})^{2}(1-q_{\tau}^{2n})},$$ which transforms under $T$ as $$T^{*}J_{4}= \frac{\zeta_{4}J_{4}}{1-J_{4}}.$$ The fixed points of this map are $0$ and $1-\zeta_{4}$, which correspond respectively to $\tau= \infty, \frac{1}{2}$. The $4$ IR cusps are all in the same orbit of $T$ and fall in the points $J_{4}(0)=\infty, J_{4}(1)=-\zeta_{4}, J_{4}(2/3)=1/(1+\zeta_{4}), J_{4}(1/3)=1$. These must be simple poles for $dW(x)$, with $x=J_{4}$, which reads $$dW(x)= \alpha \left( \frac{1}{x-1} -\frac{1}{x+\zeta_{4}}+ \frac{\zeta_{4}}{x-1/(1+\zeta_{4})}\right)dx,$$ and satisfies $T^{*}dW(x)=\zeta_{4}^{-1}dW(x)$.\ $X(5)$ is the last case of genus $0$. This modular curve has $12$ cusps which identify the vertices of a regular icosahedron. The quotient $PSL(2,\mathbb{Z})/\Gamma(5)$ acts on the Riemann sphere as $\mathbb{A}_{5}$, the symmetry group of this platonic solid. Among the $12$ cusps $$C_{\Gamma(5)}=\left\lbrace 0,2/9,1/4,2/7,1/3,2/5,1/2,5/8,2/3,3/4,1,\infty \right\rbrace$$ the UV fixed points are $2/5$ and $\infty$, while $\left\lbrace 0,2/9,1/4,3/4,1\right\rbrace $ and $\left\lbrace 2/7,1/3,1/2,5/8,2/3 \right\rbrace $ represent the $5$ decoupled vacua in two inequivalent IR limits. This curve has two inequivalent co-levels, i.e. $l=1,2$ and the Galois group acts on the set of cusps by exchanging the two UV limits and the two groups of IR theories. As in the previous cases, we can use as projective coordinate on $\mathbb{P}^{1}(\mathbb{C})$ the Hauptmodul $$J_{5}(\tau)= \zeta_{5} q_{\tau}^{-1/5} \prod_{n=1}^{\infty} \frac{(1-q_{\tau}^{5n-2})(1-q_{\tau}^{5n-3}) }{(1-q_{\tau}^{5n-4})(1-q_{\tau}^{5n-1}) }$$ which transforms under $T$ as $$T^{*}J_{5}= \zeta_{5}^{-1}J_{5}.$$ The UV cusps $2/5,\infty$ are sent by $J_{5}$ respectively to $x=0,\infty$, which are the fixed points of $T$. We know from \[dw\] that a transformation $\gamma \in \Gamma_{0}(N)$ with $b=0$ mod $N$ acts on the cyclotomic units through the Galois group and permutes the residue of the poles in $dW(x)$. It follows that if two IR cusps are in the same orbit of such $\gamma$, their residue must be related by the correspondent Galois transformation. It is the case for instance of $0$ and $5/8$ if we set $b=5$ and $c=8$. Imposing also the correct transformation property under $T$ for $l=1$, one can repeat the procedure and fix all the coefficients up to an overall constant. Setting $x_{1;k}=\zeta_{5}^{k}J_{5}(0)$ and $x_{2;k}=\zeta_{5}^{k}J_{5}(5/8)$, with $J_{5}(0)=1+\zeta_{5}+\zeta_{5}^{2}$ and $J_{5}(5/8)=(\zeta_{5}^{2}-\zeta_{5}^{4})/(1+\zeta_{5}-\zeta_{5}^{3}-\zeta_{5}^{4})$, we find $$dW(x)= \alpha \sum_{k=0}^{4} \left( \frac{1}{x-x_{1;k}} + \frac{1}{x-x_{2;k}}\right)\zeta_{5}^{k} dx.$$ where the residue of $J_{5}(0)$ and $J_{5}(5/8)$ are both normalized to $1$.\ By summarizing, the generators of loops around the IR cusps on the sphere correspond to the subset of generators of $\Gamma(N)$ which act non trivially on the critical value. Once the normalization is fixed, the constant generated by modular transformations can be computed with the residue formula. Moreover, the free IR cusps are all in the same orbit of $\Gamma_{0}(N)$, which acts by multiplication on the coefficients of $dW(x)$. This implies in particular that all the poles have the same order, which must be $1$ from the non trivial monodromy of $W(p_{0};x)$, and their residue are related by Galois transformations.\ A similar procedure could be carried on in principle also for modular curves of higher genus, but it is more complicated. It is instead much more convenient to find a general expression of the critical value as function on the upper half plane and study its modular properties. Let us take the multi-valued function in \[symb\]. Since the constants generated by the modular transformations are indipendent from the point, we can set $z=0$. The expression of $W^{(N,l)}(0;\tau)$ reads $$W^{(N,l)}(0;\tau)= \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log \left[ \Theta \begin{bmatrix} \frac{1}{2}-\frac{l}{N} \\ \frac{1}{2}-\frac{k}{N}\end{bmatrix}\left( 0;\tau \right) e^{- 2 \pi i \left( \frac{l}{N}-\frac{1}{2}\right)\left( \frac{k}{N}-\frac{1}{2}\right) }\right] .$$ where the phase $e^{- 2 \pi i \left( \frac{l}{N}-\frac{1}{2}\right)\left( \frac{k}{N}-\frac{1}{2}\right)}$ is a convenient normalization constant. This function remains ill defined as long as we do not specify the determination of the logarithm. This is equivalent to choose, for a fixed $\tau$, a representative of $z=0$ and determine its critical value. We first set the notations $$\begin{split} & q_{\tau}= e^{2\pi i \tau}, \hspace{2cm} q_{z}= e^{2\pi i z}, \\ \\ & \hspace{1.3 cm} z= u_{1} \tau+ u_{2}, \end{split}$$ with $ u_{1},u_{2} \in \mathbb{Z}/ N $. Then, we recall the definition of Siegel functions: $$\label{siegel} g_{u_{1},u_{2}}(\tau)=- q_{\tau}^{B_{2}(u_{1})/2} e^{2\pi i u_{2}(u_{1}-1)/2}(1-q_{z}) \prod _{n=1}^{\infty} (1-q_{\tau}^{n}q_{z})(1-q_{\tau}^{n}/q_{z}),$$ where $B_{2}(x)= x^{2}-x+\frac{1}{6}$ is the second Bernoulli polynomial. Because of their modular properties, these objects are a sort of ‘building blocks’ for the modular functions of level $N$. In particular, all the Hauptmoduls defined previously can be expressed in terms of $ g_{u_{1},u_{2}}(\tau)$ and the Dedekind eta function [@rif48; @rif49; @rif50]. The theta functions $ \Theta \begin{bmatrix} \frac{1}{2}-u_{1} \\ \frac{1}{2}-u_{2}\end{bmatrix}\left( 0;\tau \right)$ have the $q$-product representation $$\begin{split} \Theta \begin{bmatrix} \frac{1}{2}-u_{1} \\ \frac{1}{2}-u_{2}\end{bmatrix}\left( 0;\tau \right)= & - q_{\tau}^{B_{2}(u_{1})/2}q_{\tau}^{1/24} e^{2\pi i (u_{1}-1/2)(u_{2}-1/2)} (1-q_{z}) \\ & \mathrm{x} \prod _{n=1}^{\infty}(1-q_{\tau}^{n}) (1-q_{\tau}^{n}q_{z})(1-q_{\tau}^{n}/q_{z}) \end{split}$$ and can be written in terms of $g_{u_{1},u_{2}}(\tau) $ and $\eta (\tau) $ as $$\Theta \begin{bmatrix} \frac{1}{2}-u_{1} \\ \frac{1}{2}-u_{2}\end{bmatrix}(0;\tau)= i g_{u_{1},u_{2}}(\tau) \eta (\tau) e^{ 2 \pi i u_{1}\left( u_{2}-1\right) /2}.$$ Since the Siegel and Dedekind functions have neither zeroes nor poles, there is a single-valued branch of $\log \Theta \begin{bmatrix} \frac{1}{2}-u_{1} \\ \frac{1}{2}-u_{2}\end{bmatrix}\left( 0;\tau \right)$ on the upper half plane. Therefore, the critical value can be consistently defined as holomorphic function of $\tau \in \mathbb{H}$. Provided the above relations, we can rewrite $W^{(N,l)}(0;\tau)$ as $$\label{mass} W^{(N,l)}(0;\tau)= \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log E_{\frac{l}{N},\frac{k}{N}}(\tau),$$ where $E_{\frac{l}{N},\frac{k}{N}}(\tau)$ is the Siegel function of characters $ u_{1}=\frac{l}{N}, u_{2}=\frac{k}{N}$ without the root of unity $e^{2\pi i u_{2}(u_{1}-1)/2} $. Under an integer shift of the characters, these functions satisfy [@rif48] $$\label{shift} E_{u_{1}+1,u_{2}}(\tau)=-e^{-2\pi i u_{2}} E_{u_{1},u_{2}}(\tau), \hspace{2cm} E_{u_{1},u_{2}+1}(\tau)=E_{u_{1},u_{2}}(\tau).$$ Moreover, being the Siegel functions up to a multiplicative constant, they have good modular properties. For $\gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2,\mathbb{Z})$, they transform with a phase: $$\label{prop} \begin{split} & E_{u_{1},u_{2}}(\tau+b)= e^{\pi i b B_{2}(u_{1})} E_{u_{1},u_{2}+b u_{1}}(\tau) , \ \ \ \ \mathrm{for } \ c = 0 , \\ \\ & E_{u_{1},u_{2}}(\gamma (\tau))= \varepsilon (a,b,c,d) e^{\pi i \delta} E_{u_{1}^{\prime},u_{2}^{\prime}}(\tau) , \ \ \ \ \mathrm{for } \ c\neq 0, \end{split}$$ where $$\begin{split} \varepsilon(a,b,c,d)= & \begin{cases} e^{i\pi (bd(1-c^{2})+c(a+d-3))/6}, \ \ \ \ \mathrm{if} \ c \ \mathrm{is \ odd}, \\ -ie^{i\pi (ac(1-d^{2})+d(b-c+3))/6}, \ \ \ \ \mathrm{if} \ d \ \mathrm{is \ odd}, \end{cases} \\ \\ \delta= & u_{1}^{2}ab + 2 u_{1}u_{2}bc + u_{2}^{2}cd - u_{1}b-u_{2}(d-1), \end{split}$$ and $$\label{character} u_{1}^{\prime}= a u_{1}+c u_{2}, \hspace{1cm} u_{2}^{\prime}= b u_{1}+d u_{2}.$$ In order to compute the constants generated by the modular transformations, we need to evaluate the difference $$\label{diff} \chi_{u_{1},u_{2}}(\gamma)=\log E_{u_{1},u_{2}}(\gamma (\tau))-\log E_{u_{1}^{\prime},u_{2}^{\prime}}(\tau),$$ for $\gamma \in SL(2,\mathbb{Z})$. Here we assume the characters of the Siegel functions to be normalized such that $0 < u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime} < 1$. This can always be achieved by the property \[shift\]. The computation for the case of $\Gamma(N)$ has already been done in [@rif49]. In \[modulartrnasofrmationmass\] we follow closely that derivation, adapting it to the general case. For $c=0$ the transformations belong to the coset group $\Gamma_{1}(N)/\Gamma(N) \simeq \mathbb{Z}_{N}$ and we obtain $$\chi_{u_{1},u_{2}}(\gamma)= 2\pi i \frac{1}{2}B_{2}(u_{1}).$$ It is clear that in this case we cannot appreciate a modular shift of the critical value. Indeed, these transformations simply translate the vacua: $$\label{trivial} W^{(N,l)}(0;\tau+b)=e^{-\frac{2\pi i b l^{2}}{N}} W^{(N,l)}(0;\tau).$$ On the other hand, for $c \neq 0$, we get the formula $$\label{formula} \begin{split} \chi_{u_{1},u_{2}}(\gamma)= & \ 2\pi i \frac{1}{2} \left( B_{2}(u_{1}) \frac{a}{c} + B_{2}(u_{1}^{\prime}) \frac{d}{c} -\frac{2}{c} B_{1}(u_{1}^{\prime}) B_{1}( \langle d u_{1}^{\prime}- u_{2}^{\prime} c \rangle ) \right) \\ \\ & -\frac{2 \pi i}{c} \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x \neq 0} } [ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c}, \end{split}$$ where $B_{1}(x)=x-1/2$ is the first Bernoulli polynomial, $\langle x \rangle $ represents the fractional part of $x$ and the symbol $ [ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c}$ denotes $$[ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c}= \frac{e^{2\pi i x \bigl( \frac{ \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle-du_{1}^{\prime} }{c}+u_{2}^{\prime} \bigr)} }{(1-e^{-2\pi i x d/c})(1-e^{2\pi i x/c})}.$$ This result turns out to be indipendent from the branch of the logarithm and in particular from $\tau$. This is consistent with the fact that the modular shift of the critical value is indipendent from the vacuum that we choose. From the general formula we can reduce to the cases of the congruence subgroups. Let us consider $\gamma \in \Gamma_{0}(N)$. Using the fact that $c=0$ mod $N$ and $ad=1$ mod $N$, we have to plug in the above expression: $$\begin{split} u_{1}^{\prime}= \langle a u_{1} \rangle ,& \hspace{2cm} u_{2}^{\prime}= \langle d u_{2}+b u_{1} \rangle, \\ \\ & \langle d u_{1}^{\prime}-u_{2}^{\prime}c \rangle= u_{1}. \end{split}$$ If $\gamma \in \Gamma_{1}(N)$, these becomes $$\begin{split} u_{1}^{\prime}= u_{1},& \hspace{2cm} u_{2}^{\prime}= \langle u_{2}+ b u_{1} \rangle, \\ \\ & \langle d u_{1}^{\prime}-u_{2}^{\prime}c \rangle= u_{1}. \end{split}$$ The case of $\gamma \in \Gamma(N)$ follows from this by requiring further $b=0$ mod $N$.\ Setting $u_{1}= \frac{l}{N}$, $u_{2}= \frac{k}{N}$ and summing over $k$ with the residue $e^{\frac{2\pi i l k}{N}}$, we find the modular transformations of the physical mass. In sequence $$\label{transf} \begin{split} & \gamma \in \Gamma_{0}(N): \\ & W^{(N,l)}(0;\gamma(\tau))= e^{-\frac{2\pi i al^{2}b}{N}} W^{(N,al)}(0;\tau) + \Delta W^{(N,l)}_{\Gamma_{0}(N)}(\gamma), \\ \\ & \gamma \in \Gamma_{1}(N): \\ & W^{(N,l)}(0;\gamma(\tau))= e^{-\frac{2\pi i l^{2}b}{N}} W^{(N,l)}(0;\tau) + \Delta W^{(N,l)}_{\Gamma_{1}(N)}(\gamma), \\ \\ & \gamma \in \Gamma(N): \\ & W^{(N,l)}(0;\gamma(\tau))= W^{(N,l)}(0;\tau) + \Delta W^{(N,l)}_{\Gamma(N)}(\gamma), \end{split}$$ with $$\label{delta} \begin{split} & \Delta W^{(N,l)}_{\Gamma_{0}(N)}(\gamma)= -\frac{2 \pi i N}{c} \ e^{-\frac{2\pi i al^{2}b}{N}}\sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x= -al \ \mathrm{mod} \ N}} \frac{e^{2\pi i x \bigl( \frac{l/N-d \langle al/N \rangle }{c} \bigr)} }{(1-e^{-2\pi i x d/c})(1-e^{2\pi i x/c})}, \\ \\ & \Delta W^{(N,l)}_{\Gamma_{1}(N)}(\gamma)= -\frac{2 \pi i N}{c} \ e^{-\frac{2\pi i l^{2}b}{N}}\sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x= -l \ \mathrm{mod} \ N}} \frac{e^{2\pi i x \bigl( \frac{l}{N}\frac{1-d}{c} \bigr) }}{(1-e^{-2\pi i x d/c})(1-e^{2\pi i x/c})}, \end{split}$$ $$\Delta W^{(N,l)}_{\Gamma(N)}(\gamma)= -\frac{2 \pi i N}{c} \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x= -l \ \mathrm{mod} \ N}} \frac{e^{2\pi i x \bigl( \frac{l}{N}\frac{1-d}{c} \bigr) }}{(1-e^{-2\pi i x d/c})(1-e^{2\pi i x/c})},$$ where the constraints on $x$ follow from the summation over $k$. These formulas are coherent with the transformations of $\partial_{z}W^{(N,l)}(z;\tau)$ that we found in the previous paragraph.\ We can check in the simple case of $N=2$ that the formulas above give the results obained with the geometrical approach. The physical mass has the expression $$W(0;\tau)= \log \frac{\Theta_{4}(0;\tau)}{\Theta_{3}(0;\tau)} = \frac{1}{4}\log(1- \lambda(\tau)),$$ where $1-\lambda(\tau)= \left( \Theta_{4}(0;\tau)/ \Theta_{3}(0;\tau) \right) ^{4}$ is the Hauptmodul of level $2$ that we have defined previously. Although $1-\lambda(\tau)$ is invariant under transformation of $\Gamma(2)$, the logarithm is not. We can use the expression for $\Delta W_{\Gamma(2)}$ to derive the modular transformations of $\log(1-\lambda(\tau))$. $\Gamma(2)$ is freely generated by the matrices $$T_{1}= \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \hspace{2cm} T_{2}= \begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix} .$$ Using respectively the \[trivial\] and the \[transf\], \[delta\] with $N=2, l=1$, one finds $$\begin{split} & \log(1-\lambda(\tau+2))= \log(1-\lambda(\tau)), \\ \\ & \log\left( 1-\lambda\left( \frac{\tau}{1-2\tau}\right)\right) = \log(1-\lambda(\tau))+2\pi i. \end{split}$$ It is clear that $T_{2}$ is the generator of anticlockwise loops around the IR cusp in $\tau=0$. Indeed, the constant is the same we obtain with the residue formula. ### Modular Transformations of the Ground State Metric The modular shift of the superpotential seems to contraddict the statement that the model is invariant under transformations of $\Gamma(N)$. But, if we assume the perspective of the universal cover, there is no contraddiction at all. Indeed, the physical mass parametrizes not only models, but also vacua. The modular transformation simply changes the initial choice of the vacuum $p_{0}$ with another one of the same fiber in the universal cover. Therefore, the new coordinate on the spectral curve describes the same model, but a different vacuum.\ The $tt^{*}$ equations for these class of theories are manifestly covariant under transformations of the congruence subgroups. In particular, the covariance under $\Gamma_{1}(N)$ implies that the equation naturally descends on the space of models. However, as a consequence of the modular shift, the ground state metric is not left invariant. Indeed, matrices of $\Gamma(N)$ and $\Gamma_{1}(N)$ change the basis of lattice generators and consequently the representation of the homology group. Besides this effect, a transformation of $\Gamma_{0}(N)$ changes also the torsion point of the $\mathbb{Z}_{N}$ symmetry, resulting in a permutation of the metric components. Let us consider this more general case. From the transformation of $\partial_{z} W^{(N,l)}(z;\tau)$ under $\Gamma_{0}(N)$ one can read how the generators of the symmetry group are modified. We note that the new function is still periodic of $2\pi \tau$. Therefore, we can consider again $B$ as a generator of $H_{1}(\mathcal{S};\mathbb{Z}) $ in the new model. The operators $L_{k}$ are left invariant by the transformation as well, since the corresponding homology cycles have the same definition in the model of co-level $al$. Differently, $\sigma$ and $A$ change in relation to the transformation of the torsion point of the vacua. For a $\gamma \in \Gamma_{0}(N)$, we can write $$\begin{split} & \gamma^{*}B=B, \\ & \gamma^{*}L_{k}= L_{k} \\ & \frac{2\pi}{N} \longrightarrow \frac{2\pi}{N} (c\tau + d ) \Longrightarrow \begin{cases} \gamma^{*}\sigma= \sigma^{d} B^{\frac{c}{N}}= \tilde{\sigma} B^{\frac{c}{N}} \\ \gamma^{*}A = A^{d} B^{c}= \tilde{A} B^{c}, \end{cases} \end{split}$$ where we denote with $\tilde{\sigma}=\sigma^{d}$ the operator associated to the torsion point of co-level $al$, and with $\tilde{A}=A^{d}$ the homology cycle satisfying $\tilde{A}=\tilde{\sigma}^{N}$. The fact that the generator of loops are not involved in the modular transformations is consistence with the truncation of the $tt^{*}$ equation that we discussed in section $3$.\ We have also to include the shift $z\rightarrow z -\frac{2\pi}{N}bl$, which implies the vacuum transformation $$p_{0} \longrightarrow \sigma^{-bl}(p_{0}) \Longrightarrow {{\,| {p_{0}} \rangle}} \mapsto \sigma^{-bl}{{\,| {p_{0}} \rangle}} .$$ Let us study what these transformations mean at the level of vacuum states. We consider again trivial representations of $L_{k}$. The action of $\gamma$ on the point basis is $$\gamma {{\,| {k;\alpha, \beta} \rangle}}=\ e^{\frac{-i \alpha k}{N}} \sum_{m,n \in \mathbb{Z}} e^{-i( m\alpha + n\beta )} (A^{d} B^{c})^{m} B^{n} (\sigma^{d} B^{\frac{c}{N}})^{ k} \sigma^{-bl}{{\,| {p_{0}} \rangle}}$$ $$\begin{split} & = \ e^{-i\frac{kd}{N}\left( \frac{\alpha-\beta c }{d}\right) } \sum_{m,n\in \mathbb{Z}} e^{-i \left( m \left( \frac{\alpha-\beta c}{d}\right) + n\beta \right)} A^{m} B^{ n} \sigma^{kd-bl}{{\,| {p_{0}} \rangle}} \\ \\ & = \ e^{-i\frac{bl}{N} \left( \frac{\alpha-\beta c}{d} \right) } {{\,| {kd-bl; (\alpha-\beta c)/d, \beta} \rangle}}. \end{split}$$ We note that, in the case of $ b=0$ mod $N$ and $d=1$ mod $N$, the fiber index $k$ is left invariant. This follows from the fact that a transformation of $\Gamma(N)$ preserves the torsion point up to a shift of a lattice vector, which moves the base point $p_{0}$ without changing the fiber. The $\sigma$-eigenstates transform consequently as $$\gamma {{\,| {j;\alpha, \beta} \rangle}}=\ e^{-i\frac{bl}{N} \left( \frac{\alpha-\beta c}{d} \right) } \sum_{k=0}^{N-1} e^{- \frac{2 \pi i l k j}{N}} {{\,| {kd-bl;(\alpha-\beta c)/d , \beta} \rangle}}$$ $$\begin{split} = & \ e^{-i\frac{bl}{N} \left( 2\pi a l j + \frac{\alpha-\beta c}{d}\right)} \sum_{k=0}^{N-1} e^{- \frac{2 \pi i al k j}{N}} {{\,| {k;(\alpha-\beta c)/d , \beta} \rangle}} \\ \\ = & \ e^{-i\frac{bl}{N} \left( 2\pi a l j + \frac{\alpha-\beta c}{d}\right)} {{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}. \end{split}$$ The action of the symmetry group generators $\tilde{\sigma},\tilde{A},B$ on the transformed states is given by $$\begin{split} & \tilde{\sigma}{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}= e^{\frac{2\pi i l j}{N}} e^{\frac{i}{N}(\alpha-\beta c)}{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}, \\ \\ & \tilde{A}{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}= e^{i(\alpha-\beta c)}{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}, \\ \\ & B{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}= e^{i\beta}{{\,| {aj;(\alpha-\beta c)/d, \beta} \rangle}}. \end{split}$$ From the eigenvalues of the new basis, we learn that the ground state metric transforms in the following way: $$\label{metrictransformation} \gamma^{*}\varphi_{j}(w;\alpha,\beta)= \varphi_{j}(\gamma^{*}w;\alpha,\beta)= \varphi_{aj}(w;\alpha-\beta c,\beta).$$ As anticipated, we see that a transformation of $\Gamma(N)$ changes the representation of the homology, resulting in the character shift $\alpha\rightarrow \alpha-\beta c$. In the case of $\Gamma_{0}(N)$, since the operators $\tilde{\sigma}$ and $\sigma$ are inequivalent and have a different set of eigenstates, we appreciate also the exchange of the metric components along the diagonal. This effect takes place specifically for $N>2$, where we have a non trivial co-level structure. We note further that transformations of the coset $\Gamma_{1}(N)/\Gamma(N)$, i.e. with $c=0$, leave the metric completely invariant. Physics of the Cusps {#final} -------------------- ### Classification of the Cusps Now that we have discussed the modular properties of the solution, we want to describe its behaviour around the boundary regions of the domain. These are represented by the cusps of the modular curve $\mathbb{H}/\Gamma(N)$, i.e. the equivalence classes of $\Gamma(N)$ in $\mathbb{Q} \cup \lbrace\infty \rbrace$. First of all, we have to understand which type of model each cusp corresponds to. Let us begin with the cusp at $\tau= i\infty$. It is convenient to come back to the initial lattices of poles and vacua with the shift $z\rightarrow z + \frac{2\pi l \tau}{N}$. Moreover, we rewrite the derivative of the superpotential in terms of $\Theta_{1}(z;\tau)$ as $$\label{new} \partial_{z} W^{(N,l)}(z;\tau)= \ \frac{1}{2} \sum_{k=0}^{N-1} e^{\frac{2\pi i lk}{N}} \frac{\Theta_{1}^{\prime}\left( \frac{1}{2}\left( z-\frac{2\pi k}{N}\right) ;\tau\right) }{\Theta_{1}\left(\frac{1}{2}\left( z-\frac{2\pi k}{N}\right) ;\tau\right)}.$$ Using the relation [@rif47] $$\frac{\Theta_{1}^{\prime}(z,\tau)}{\Theta_{1}(z ;\tau)}= \cot z +4 \sum_{n=1}^{\infty} \frac{q_{\tau}^{n}}{1-q_{\tau}^{n}}\sin 2nz,$$ with $q_{\tau}=e^{2\pi i \tau} $, the expression above becomes $$\begin{split} \partial_{z} W^{(N,l)}(z;\tau)= & \frac{1}{2} \sum_{k=0}^{N-1} e^{\frac{2\pi i lk}{N}} \biggl[ \cot\left(\frac{1}{2}\left( z-\frac{2\pi k}{N}\right) \right) + \\ & 4 \sum_{n=1}^{\infty} \frac{q_{\tau}^{n}}{1-q_{\tau}^{n}}\sin \left( 2n\left( \frac{1}{2}\left( z-\frac{2\pi k}{N}\right) \right) \right) \biggr]. \end{split}$$ The theta function in these expressions is normalized with quasi-periods $\pi,\pi\tau$. Let us denote with $S$ the infinite sum in $n$. Manipulating the expression, we get $$\begin{split} S= & \ 2 \sum_{k=0}^{N-1} e^{\frac{2\pi i lk}{N}} \sum_{n=1}^{\infty} \frac{q_{\tau}^{n}}{1-q_{\tau}^{n}}\sin \left( n \left( z-\frac{2\pi k}{N}\right)\right) \\ \\ = & -i \sum_{n=1}^{\infty} \frac{q_{\tau}^{n}}{1-q_{\tau}^{n}} \left( e^{inz}\sum_{k=0}^{N-1} e^{\frac{2\pi i k}{N}(l-n)} - e^{-inz}\sum_{k=0}^{N-1} e^{\frac{2\pi i lk}{N}(l+n)} \right) . \end{split}$$ The two sums over $k$ are not $0$ if and only if $n$ satisfies respectively $n=l$ mod $N$ and $n=-l$ mod $N$. Therefore, the derivative of the superpotential becomes $$\begin{split} \partial_{z} W^{(N,l)}(z;\tau) = & \ \frac{1}{2} \sum_{k=0}^{N-1} e^{\frac{2\pi i lk}{N}} \cot \left( \frac{1}{2}\left( z-\frac{2\pi k}{N}\right) \right) \\ \\ & -iN \left( \sum_{\overset{n=1}{n=l \ \mathrm{mod} \ N}}^{\infty} \frac{q_{\tau}^{n}}{1-q_{\tau}^{n}} e^{inz}- \sum_{\overset{n=1}{n=-l \ \mathrm{mod} \ N}}^{\infty}\frac{q_{\tau}^{n}}{1-q_{\tau}^{n}} e^{-inz} \right). \end{split}$$ Taking the limit $\tau\rightarrow i\infty$, the series in $q_{\tau}$ are truncated at the leading order. Moreover, since the vacua $-\frac{2\pi l \tau}{N} + \frac{2\pi k}{N} + \Lambda_{\tau}$ escape to infinity for large $\tau$, we have also to take $z \rightarrow i\infty$. Thus, we obtain $$\partial_{z}W^{(N,l)}(z;\tau) \xrightarrow{ \tau\rightarrow i\infty} -iN \left( q_{\tau}^{l } \ e^{ilz} - q_{\tau}^{(N-l)} \ e^{-i(N-l)z} \right).$$ Integrating this expression, we find $$\label{limit} W^{(N,l)}(z;\tau) \xrightarrow{ \tau\rightarrow i\infty} -N\left( q_{\tau}^{l }\ \frac{e^{ilz}}{l} + q_{\tau}^{(N-l)} \ \frac{e^{-i(N-l)z}}{N-l} \right) ,$$ wich we recognize as the superpotential of a $\hat{A}_{N-1}$ model of co-level $l$. In particular, the case of $N=2$ corresponds to the Sinh-Gordon model $$\label{sinhgordon} W(z;\tau) \sim q_{\tau} \cos z .$$ Now let us consider the rationals. We associate to a cusp $\frac{a}{c}$ with $\mathrm{gcd}(a,c)=1$ a modular transformation $\gamma_{\frac{a}{c}}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ which sends $i\infty$ to $\frac{a}{c}$. In this definition $b,d$ are integers such that $\gamma_{\frac{a}{c}} \in SL(2,\mathbb{Z})$ and clearly the case of $c=0$ corresponds to take again $\tau=i\infty$. One can study the behaviour of the model around $ \tau=\frac{a}{c}$ by acting on $ \partial_{z} W^{(N,l)}(z;\tau)$ with $\gamma_{\frac{a}{c}}$ and then taking the limit $\tau \rightarrow i\infty $. Using the modular properties \[weierstrass\] of the zeta function, we have $$\begin{split} & \partial_{z} W^{(N,l)}\left( \frac{z}{c\tau + d};\frac{a\tau + b}{c\tau + d}\right) = \\ \\ & (c\tau+d) \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \left[ \zeta \left( z- \frac{2\pi}{N} k (c\tau + d) ;\tau \right) +2\frac{d \eta_{1}+c \eta_{2}}{N} k \right], \end{split}$$ where the torsion point of the poles is now $\frac{2\pi}{N}(c\tau + d)$. Let us introduce the integers $Q= \mathrm{gcd} (c,N)$, with $1 \leq Q \leq \mathrm{min} \left\lbrace c,N \right\rbrace$, $j=\frac{N}{Q}$ and $r=\frac{c}{Q}$. Clearly we have $\mathrm{gcd} (r,j)=1$. By these definitions, we can split the sum over $k$ by writing $k=m+jp$, with $m=0,...,j-1$ and $p=0,...,Q-1$. Let us consider for the moment the cusps with divisor $Q>1$. One obtains $$\begin{split} & \partial_{z} W^{(N,l)}\left( \frac{z}{c\tau + d};\frac{a\tau + b}{c\tau + d}\right) = \\ \\ & (c\tau +d)\sum_{m=0}^{j-1} e^{\frac{2\pi i l m}{N}} \sum_{p=0}^{Q-1} e^{\frac{2\pi i l p}{Q}} \biggl[ \zeta \left( z- \left( \frac{2\pi r}{j} \tau +\frac{2\pi d}{N} \right)m -\frac{2\pi}{Q} d p;\tau \right) \\ \\ & + 2 \left( \frac{r\eta_{2} }{j} + \frac{d\eta_{1} }{N}\right)m + \frac{2 d \eta_{1}}{Q}p \biggr]. \end{split}$$ As $\tau$ becomes very large, for $m\neq 0$ the torsion point approaches $\frac{2\pi r}{j}\tau$.\ Moreover, by the formulas [@rif53] $$2\eta_{1}= \frac{G_{2}(\tau)}{2\pi}, \hspace{2 cm} 2\eta_{2}= \frac{\tau G_{2}(\tau)-2\pi i }{2\pi},$$ where $G_{2}(\tau)$ is the Eisenstein series $$G_{2}(\tau)= \sum_{c ,d \in \mathbb{Z}\setminus \left\lbrace 0 \right\rbrace } \frac{1}{(c\tau +d)^{2}},$$ and the asymptotic behaviour $$G_{2}(\tau)\xrightarrow{ \tau\rightarrow i\infty} 2\zeta(2),$$ where $\zeta(z)$ is the Riemann zeta function, one has $$\frac{\eta_{2}}{\eta_{1}} \xrightarrow{ \tau\rightarrow i\infty} \tau.$$ Thus, we get the limit $$\begin{split} & \partial_{z} W^{(N,l)}\left( \frac{z}{c\tau + d};\frac{a\tau + b}{c\tau + d}\right) \xrightarrow{ \tau\rightarrow i\infty} (c\tau +d) \sum_{p=0}^{Q-1} e^{\frac{2\pi i l p}{Q}} \biggl[ \zeta \left( z -\frac{2\pi}{Q} d p;\tau \right)+ \frac{2d\eta_{1}}{Q} p \biggr]_{\tau \rightarrow i\infty} + \\ \\ & (c\tau +d)\sum_{m=1}^{j-1} e^{\frac{2\pi i l m}{N}} \sum_{p=0}^{Q-1} e^{\frac{2\pi i l p}{Q}} \biggl[ \zeta \left( z- \frac{2\pi r}{j} \tau m ;\tau \right) + \frac{2 r\eta_{2}}{j}m \biggr]_{\tau \rightarrow i\infty} = \\ \\ & (c\tau +d) \sum_{p=0}^{Q-1} e^{\frac{2\pi i l p}{Q}} \biggl[ \zeta \left( z -\frac{2\pi}{Q} d p;\tau \right)+ \frac{2d\eta_{1}}{Q} p \biggr]_{\tau \rightarrow i\infty} = \\ \\ & (c\tau +d) \sum_{p=0}^{Q-1} e^{\frac{2\pi i al p}{Q}} \biggl[ \zeta \left( z -\frac{2\pi}{Q} p;\tau \right)+ \frac{2\eta_{1}}{Q}p \biggr]_{\tau \rightarrow i\infty}, \end{split}$$ where in the last line we have used the fact that $ad=1$ mod $Q$. So, we learn from this expression and the limit \[limit\] that the cusp $\frac{a}{c}$ with divisor $Q=\mathrm{gcd} (c,N)$ correspond to an $\hat{A}_{Q-1}$ model of co-level $al$: $$\label{ahat} W^{(N,l)}(z;\tau) \xrightarrow{ \tau\rightarrow \frac{a}{c}} -Q\left( q_{\tau}^{al }\ \frac{e^{ialz}}{al} + q_{\tau}^{(Q-al)} \ \frac{e^{-i(Q-al)z}}{Q-al} \right),$$ which is a theory with $Q$ vacua up to periodicity $z \sim z+2\pi$.\ In the case of $Q=1$, the poles $\frac{2\pi k}{N}(c\tau +d) + \Lambda_{\tau}$ are all pushed to infinity when $\tau$ becomes large except for $k=0$. Thus, using the \[new\] we can write symbolically $$W \sim `` \log \Theta_{1}(z/2;\tau\rightarrow i\infty) ",$$ and using the asymptotics $$\Theta_{1}(z;\tau) \xrightarrow{ \tau\rightarrow i\infty} 2 q_{\tau}^{\frac{1}{8}}\sin z,$$ one finds that these cusps are decribed by the multi-valued superpotential $$\label{free} W(z)=`` \log \sin \left( z/2 \right) ".$$ This model represents the free version of our class of theories, with a $1$ dimensional lattice of poles and one of vacua.\ We know that the cusps of $X_{1}(N)$ are ramification points of the cover $X(N)\rightarrow X_{1}(N)$. Denoting with $\Gamma_{\frac{a}{c}}$ the stability group of the cusp $a/c$ in $\Gamma_{1}(N)$, one can write the equality [@rif45] $$\label{cond} \gamma_{\frac{a}{c}}^{-1}\Gamma_{\frac{a}{c}}\gamma_{\frac{a}{c}}= \left\langle \pm \begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix} \right\rangle,$$ which is satisfied with one of the two signs. This relation means that the generator of $\Gamma_{\frac{a}{c}}$ is conjugated to $\pm \begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix}$ in $ \gamma_{\frac{a}{c}}^{-1} \Gamma_{1}(N) \gamma_{\frac{a}{c}} $.\ The number $h$ is called width of the cusp and represents the minimal integer such that $a/c +h \sim a/c$ in $\Gamma(N)$. The absolute value can be equal or less than $N$ and count the number of degenerate vacua of the model labelled by the cusp $\left[ \frac{a}{c} \right]$ of $\Gamma_{1}(N)$. The cusps which satisfy the relation with the plus sign are called regular. From the theorems \[width1\], \[width2\], the stability condition can be written as: $$\begin{bmatrix} a+ch \\ c \end{bmatrix}= \begin{bmatrix} a \\ c \end{bmatrix} \ \mathrm{mod} \ N.$$ It is clear that the minimal integer $h$ satisfying this condition is $h=j=\frac{N}{Q}$. Thus, for the cusps with divisor $Q$, the $N$ vacua split in $j$ decoupled theories which appear on $X(N)$ as $Q$-degenerate points. These are described by the $\hat{A}_{Q-1}$ models \[ahat\] for $1<Q\leq N$, or by the free theories \[free\] if $Q=1$. In particular, the cusps with $Q=N$, or equivalently $c=0$ mod $N$, are the UV limits, since the vacua tend to a unique point on the spectral curve.\ The exceptions to this picture are represented by the so called irregular cusps, i.e. those which satisfy the \[cond\] with the mignus sign. In this case the stabilizer of the cusp belongs to $-\Gamma_{1}(N)$ and we have $$\begin{bmatrix} a+ch \\ c \end{bmatrix}= -\begin{bmatrix} a \\ c \end{bmatrix} \ \mathrm{mod} \ N.$$ If we exclude the trivial case of $N=2$ where $1\sim-1$ and require $a$ and $c$ to be coprime, we find that the stability condition is satisfied only by cusp $1/2$ for the curve of level $4$. This is known to be the unique irregular cusp for $\Gamma_{1}(N)$. Despite we have $Q=N/Q=2$, the width of the cusp is $h=1$, and the corresponding theory is actually a $\hat{A}_{3}$ model with $4$ vacua. The superpotential is the Sinh-Gordon one in \[sinhgordon\] as for the cusps with divisor $2$, but we have to impose the identification $z \sim z+ 4\pi$.\ Now we want to determine the positions of the cusps on the W-plane. Using the expression \[mass\] for the critical value, we have $$\begin{split} W^{(N,l)}(0;\tau\rightarrow \frac{a}{c})=& \ W^{(N,l)}(0;\gamma_{\frac{a}{c}}(\tau\rightarrow i\infty))= \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log E_{\frac{l}{N},\frac{k}{N}}(\gamma_{\frac{a}{c}}(\tau\rightarrow i\infty)) \\ \\= & \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log E_{\big<\frac{al+ck}{N}\big>,\big<\frac{dk+bl}{N}\big>}(\tau\rightarrow i\infty) + \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \chi_{\frac{l}{N};\frac{k}{N}}(\gamma_{\frac{a}{c}}), \end{split}$$ where $ \chi_{\frac{l}{N};\frac{k}{N}}(\gamma_{\frac{a}{c}})$ is given by the formula in \[formula\].\ Let us consider the limit of the first piece. The leading order of $E_{u_{1},u_{2}}(\tau)$ for $\tau \rightarrow i\infty$ is $$\mathrm{ord}_{i \infty} E_{u_{1},u_{2}}(\tau)= \frac{1}{2} B_{2}\left( \langle u_{1} \rangle \right) .$$ Therefore, we find $$\begin{split} W^{(N,l)}(0;\tau\rightarrow \frac{a}{c})= & \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log E_{\big<\frac{al+ck}{N}\big>,\big<\frac{dk+bl}{N}\big>}(\tau) \xrightarrow{ \tau\rightarrow i\infty} \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \log q_{\tau}^ {\frac{1}{2} B_{2}\left( \big<\frac{al+ck}{N} \bigr> \right) } \\ \\ =& \log q_{\tau}^{K^{(N,l)}_{\frac{a}{c}}}, \end{split}$$ where $${K^{(N,l)}_{\frac{a}{c}}}= \frac{1}{2}\sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} B_{2}\left( \bigg<\frac{al+ck}{N}\bigg> \right). \hspace{2cm}$$ Let us develop this expression. Using the Fourier expansion of the second Bernoulli periodic polynomial $$B_{2}(\langle x \rangle)= -\frac{2!}{(2\pi i)^{2}} \sum_{\overset{m=-\infty}{m \neq 0}}^{\infty} \frac{e^{2\pi i m x}}{m^{2}},$$ we get $$\begin{split} {K^{(N,l)}_{\frac{a}{c}}}= & -\frac{1}{(2\pi i)^{2}} \sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \sum_{\overset{m=-\infty}{m \neq 0}}^{\infty} \frac{e^{2\pi i m \left( \frac{al+ck}{N}\right) }}{m^{2}} \\ \\ = & -\frac{1}{(2\pi i)^{2}} \sum_{\overset{m=-\infty}{m \neq 0}}^{\infty} \frac{e^{2\pi i m \frac{al}{N}}}{m^{2}} \sum_{k=0}^{N-1} e^{\frac{2\pi i k}{N}(l+mc)}. \end{split}$$ The sum $\sum_{k=0}^{N-1} e^{\frac{2\pi i k}{N}(l+mc)} $ is not vanishing if and only if $l+mc=0$ mod $N$, which admits solution only when $c$ is coprime with $N$. Thus, we have $${K^{(N,l)}_{\frac{a}{c}}}= \begin{cases} -\frac{N}{(2\pi i)^{2}}^{-\frac{2\pi i a l^{2}r}{N}} \sum\limits_{\overset{m=-\infty}{m= -lr \ \mathrm{mod} \ N}}^{\infty} \frac{1}{m^{2}} \neq 0, \ \ \ \mathrm{if} \ \mathrm{gcd}(c,N)=1, \\ \\ 0, \ \ \ \mathrm{otherwise}, \end{cases}$$ where $r= c^{-1}$ mod $N$. So, we learn that if $\mathrm{gcd}(c,N)=1$ the cusp order is not vanishing and therefore the critical value is divergent. Coherently with our analysis, these are the IR fixed points described by free theories. In this limits all the vacua decouple and the solitons connecting them become infinitely massive. On the contrary, the cusps with $1< Q \leq N $ have a finite coordinate on the W-plane: $$W^{(N,l)} \left( 0;\frac{a}{c}\right)=\sum_{k=0}^{N-1} e^{\frac{2\pi i l k}{N}} \chi_{\frac{l}{N};\frac{k}{N}}(\gamma_{\frac{a}{c}}).$$ In the case of $c=0$ mod $N$ this becomes $$W^{(N,l)} \left( 0;\frac{a}{c}\right)^{\mathrm{UV}}= \Delta W^{(N,l)}_{\Gamma_{0}(N)}(\gamma_{\frac{a}{c}}).$$ In particular, the cusp $\tau= i \infty$ is located at the origin and provides the boundary condition for the critical limit $\mu\rightarrow 0$.\ It is clear from the last formula that the UV cusps are all in the same orbit of the Galois group of the modular curve. Counting the equivalence classes of $\Gamma_{1}(N)$, these models are labelled by the co-levels $\pm l$ and their number is equal to $\phi(N)/2$. From this point of view, choosing the co-level is equivalent to pick which UV cusp to put in the origin. Also the free IR cusps are all in the same orbit of the Galois group, since we can map $\tau=0$ to a generic rational $a/c$ such that $\mathrm{gcd}(c,N)=1$ with a matrix of $\Gamma_{0}(N)$. Instead, the other IR cusps with divisor $1< Q < N$ can split in different equivalence classes of $\Gamma_{0}(N)$. In general the Galois group maps a cusp $a/c$ with $\mathrm{gcd}(c,N)=Q$ and $\mathrm{gcd}(a,Q)=1$ to another cusp $\frac{a^{\prime}}{c^{\prime}}$ with $\mathrm{gcd}(c^{\prime},N)=Q$ and $\mathrm{gcd}(a^{\prime},Q)=1$. A computation in [@rif45] shows that for a given divisor $Q$ we have $\phi(\mathrm{gcd}(Q,N/Q))$ cusps of $\Gamma_{0}(N)$. ### Boundary Conditions In this last section we provide the boundary conditions for the $tt^{*}$ equations and describe the solution around the cusps. Approaching a critical point, the solution has to match the asymptotic behavour required by the physics of the corresponding cusp. The deformations in the space of couplings near these points regard the overall parameter $\mu$ rescaling the superpotential. Near the UV fixed point the ground state metric can be diagonalized in a basis of vacua with definite Ramond charge: $$g_{i\bar{i}} \xrightarrow{ \mu \rightarrow 0} (\mu\bar{\mu})^{-q_{i}^{R}-n/2 },$$ where $n$ is the complex dimension of the target manifold. In the present case we have $n=1$. So, the solution to the $tt^{*}$ equations near the critical point is given in terms of the Ramond charges of the vacua, or equivalently the charges of the vector R-symmetry. These are given by the scaling dimensions of the chiral primary operators of the CFT.\ Let us start with the cusps corresponding to $\hat{A}_{N-1}$ models. These are Landau-Ginzburg theories with superpotential [@rif12] $$W^{(N,l)}(z;t)= \mu \left( \frac{e^{-lz}}{l} + \frac{e^{(N-l)z}}{N-l}\right) ,$$ and $\mathrm{gcd}(l,N)=1$. These are integrable models with a $\mathbb{Z}_{N}$ symmetry generated by $\sigma: z\rightarrow z + \frac{2\pi i}{N}$ and $N$ vacua, provided the periodic identification $z \sim z+2\pi i$. The symmetry acts transitively on the critical points, which are given by the condition $e^{Nz}=1$. In the UV limit these theories tend to $\sigma$-models over abelian orbifolds of $\mathbb{CP}^{1}$. These are known to be asymptotically free theories with central charge $\hat{c}=1$ [@rif12]. The $tt^{*}$ equations in canonical form are the Toda equations in \[toda\] with vanishing $\beta$. Indeed, in this limit the unique non trivial homology operator is $A=\sigma^{N}$ and the $U(1)$ charges defining the solution can depend only on $\alpha$. Since $U(1)$ is broken by the superpotential to $\mathbb{Z}_{N}$, it is clear that the generators of the two symmetry groups have a common basis of eigenstates. Let us first consider the case of $\alpha=0$. A basis of $U(1)$ eigenstates in the chiral ring can be generated with the operators $e^{-z}, e^{z}$. Since the superpotential must have R-charge $1$, these have respectively charge $\frac{1}{l}$ and $\frac{1}{N-l}$. Given that $e^{-lz}= e^{(N-l)z}$ in the chiral ring from the vacua condition, we find that the set of eigenstates split in two ‘towers’ $$\begin{split} & e^{-z} \hspace{1cm} e^{-2z} \hspace{1cm}\cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} e^{-(l-1)z} \\ & e^{z} \hspace{1.2cm} e^{2z} \hspace{1.25cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} e^{(N-l-1)z} \end{split}$$ with $U(1)_{V}$ charges $$\begin{split} & \hspace{0.45cm} \frac{1}{l} \hspace{1.8cm} \frac{2}{l}\hspace{1.5cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1.5cm} \frac{l-1}{l} \\ \\ & \frac{1}{N-l} \hspace{1cm} \frac{2}{N-l}\hspace{1.13cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \cdot \hspace{1cm} \frac{N-l-1}{N-l}. \end{split}$$ Approximately, we can say that the theory splits in two, with a set of operators dominant on the other one according to how we take the limit. We complete the basis by adding the identity $I$ and $e^{-lz}$, which have respectively charge $0$ and $1$. Near the critical point these two operators correspond to a unique marginal degree of freedom which gets a logarithmic correction to the scaling [@rif12]. We point out that in this language the Galois group acts directly on the $U(1)$ charges with the map $l\rightarrow al$ and puts in relation the solutions of the different $\hat{A}_{N-1}$ models. To see that the set of charges is invariant under this map we have to use the chiral ring condition $e^{Nz}=1$. The operatorial equality $e^{-kz}=e^{(N-k)z}$ for a generic $k \in \mathbb{Z}$ implies the equivalence $\frac{k}{l} \sim \frac{N-k}{N-l}$ at the level of corresponding charges. In general, the integer $k$ and the co-level $l$ are periodic of $N$ in the chiral ring. So, one can recast all the charges above as $q_{k}=\frac{k}{l}$, $k=0,...,N-1$ and write the action of the Galois group as $q_{k}\rightarrow\frac{k}{al}$. The relation $\frac{k}{l}=\frac{ak}{al}\sim\frac{k^{\prime}}{al}$, with $k^{\prime}= ak$ mod $N$, shows that the set of charges is left invariant by this map.\ We can include the dependence from the angle $\alpha$ by multiplying the basis above by $e^{\frac{\alpha}{2\pi}z}$. In this way the operators have the correct eigenvalues under $\mathbb{Z}_{N}$ when $\alpha \neq 0$. The $U(1)$ charges as functions of the angle are $$\begin{split} & \frac{1}{l}\left( k-\frac{\alpha}{2\pi}\right) , \ \ \ k=1,...,l-1, \\ \\ & \frac{1}{N-l}\left( k+\frac{\alpha}{2\pi}\right), \ \ \ k=1,...,N-l-1, \\ \\ & \frac{\alpha}{2\pi(N-l)}, \hspace{1cm} 1-\frac{\alpha}{2\pi l} . \end{split}$$ It is clear from \[metrictransformation\] that for $\beta=0$ a transformation of $\Gamma_{0}(N)$ does not change the dependence on $\alpha$ of the metric components. This can be seen at the level of charges by the fact that the map $l\rightarrow al$ is compensated by the rescaling of the angle $\alpha\rightarrow\alpha/d$.\ We note further that, since $\beta$ is vanishing, the UV cusps turn out to be fixed points of $\Gamma(N)$. This is consistent with the fact that $A$ is the unique generator of the homology in this regime.\ The irregular cusp $1/2$ of the modular curve of level $4$ is described by the superpotential $$W(z)= \mu \left( e^{2z} + e^{-2z} \right)$$ with the identification $z \sim z+2\pi i$. This theory has a $Z_{2}$ symmetry generated by $\sigma: z\rightarrow z+\frac{i\pi}{2}$, but $4$ vacua determined by the condition $e^{2z}=e^{-2z}$. This model belongs to $\hat{A}_{3}$ family and is asymptotically a $\sigma$-model on the $\mathbb{CP}^{1}/\mathbb{Z}_{2} $ orbifold. The $tt^{*}$ equations are the Toda ones with $N=4$ and a basis of $U(1)$ eigenstates is given by $$e^{\frac{\alpha}{2\pi}z} \hspace{1cm} e^{\left( 1+\frac{\alpha}{2\pi} \right) z} \hspace{1cm} e^{-\left( 1-\frac{\alpha}{2\pi}\right) z} \hspace{1cm} e^{-\left( 2-\frac{\alpha}{2\pi} \right) z}$$ with charges respectively $$\frac{\alpha}{4\pi} \hspace{1cm} \frac{1}{2}\left( 1+\frac{\alpha}{2\pi}\right) \hspace{1cm} \frac{1}{2}\left( 1-\frac{\alpha}{2\pi} \right) \hspace{1cm} \frac{1}{2}\left( 2-\frac{\alpha}{2\pi} \right) .$$ We conclude by saying that the solution of the $tt^{*}$ equation is singular in the UV cusps: $$\varphi_{i}(t;\alpha) \xrightarrow{t\rightarrow 0} -2\left( q_{i}(\alpha) -\frac{1}{2}\right) \log t.$$ A solution in terms of regular trascendents can be given only on the upper half plane, which is a simply connected space.\ The discussion for the $\hat{A}_{Q-1}$ models for $1< Q < N$ is pretty much the same of the previous paragraph. So, we focus on the free massive theories corresponding to the case of $Q=1$. These IR cusps are Landau-Ginzburg models described by the derivative $$\partial_{z}W(z;\tau)= \mu \cot\left( \frac{z}{2} \right).$$ This function is periodic of $2\pi$ and has simple poles and simple zeroes respectively in $ 2k\pi $ and $\pi + 2k\pi$, $\k \in \mathbb{Z}$. Moreover, it is odd with respect to the parity transformation $\iota: z\rightarrow -z$. Since the target space is not simply connected we need to pull-back the model on the abelian universal cover. A natural basis for the homology is given by the cycles $B,B^{\prime}$ in figure \[algebr\]. From the residue formula and the parity properties of $\partial_{z}W(z)$ one gets the transformations of the superpotential $$\begin{split} & B^{*}W(p)= W(p) - 2\pi i \mu, \\ & B^{\prime *}W(p)= W(p) + 2\pi i \mu. \end{split}$$ Proceeding as in \[point\] we can construct the unique theta-vacua of this theory: $${{\,| {\phi, \psi} \rangle}}= \sum_{n,m \in \mathbb{Z}} e^{-i(m\phi + n\psi)} B^{m}B^{\prime n} {{\,| {0} \rangle}},$$ where we denote with ${{\,| {0} \rangle}}$ some vacuum state of the covering model. Setting to $0$ the corresponding critical value, the whole set is simply $$W_{n,m}= 2\pi i \mu (n-m).$$ We want to derive the $tt^{*}$ equation in the parameter $\mu$. The chiral ring operator $C_{\mu}(\phi, \psi)$ acts on the theta-vacuum as differential operator in the angles $$C_{\mu}{{\,| {\phi, \psi} \rangle}}= \sum_{n,m \in \mathbb{Z}} e^{-i(m\phi + n\psi)}2\pi i(n-m) B^{m}B^{\prime n} {{\,| {0} \rangle}}= 2\pi \left( \frac{\partial}{\partial \phi}-\frac{\partial}{\partial \psi}\right) {{\,| {\phi, \psi} \rangle}}.$$ We define the ground state metric $$g(t,\phi, \psi)= {\ensuremath{\langle \overline{\phi, \psi} | \phi, \psi \rangle}} = e^{L(t ,\phi,\psi)},$$ where $L(t,\phi,\psi)$ is a real function of the angles and the RG scale $t=\vert \mu \vert$. We can normalize the state so that the topological metric is $1$. Thus, the reality constraint implies $$\label{reality} \begin{split} & L(-\phi,-\psi)=-L(\phi,\psi). \end{split}$$ Moreover, by the commutation relations $$\begin{split} & \iota B=B^{\prime -1}\iota, \\ \\ & \iota B^{\prime}=B^{-1} \iota, \end{split}$$ we have also $$\label{parity} L(-\psi,-\phi)=L(\phi,\psi).$$ The $tt^{*}$ equation for $g(t,\phi, \psi)$ reads $$\left( \partial_{\mu}\partial_{\bar{\mu}}+ 4\pi^{2}\left( \frac{\partial}{\partial \phi}-\frac{\partial}{\partial \psi}\right) ^{2}\right) L(t,\phi,\psi)=0.$$ We recognize in this expression the equation of a $U(1)$ Bogomolnyi monopole on $\mathbb{R}\times T^{2}$. Abelian $tt^{*}$ monopoles have been studied in [@rif13; @rif23]. The solution can be expanded in Bessel-MacDonald functions as $$L(t,\phi,\psi)= \sum_{ m_{1},m_{2} \in \mathbb{Z} \setminus \left\lbrace 0 \right\rbrace } A(m_{1},m_{2}) K_{0}\left( 4\pi t \vert m_{1}+m_{2} \vert \right) \exp \left( i \left( m_{1}\phi-m_{2}\psi \right) \right),$$ where the coefficients $A(m_{1},m_{2})$ can be determined by imposing appropriate boundary conditions. One can easily see that the $tt^{*}$ reality constraint \[reality\] implies $$A(-m_{1},-m_{2})=- A(m_{1},m_{2}), \hspace{1cm} A(m_{1},m_{2}) \in i \mathbb{R},$$ while the parity condition \[parity\] requires $$A(m_{2},m_{1})=A(m_{1},m_{2}).$$ Combining these two conditions one gets the further constraint $$L(t, \psi, \phi)=-L(t,\phi,\psi).$$ According to the discussion in section \[trunc\], in order to have the abelianity of the solution one should consider trivial representations of the loop generator. If we demand the loop angle to vanish, namely $\phi=\psi$, we simply find the trivial solution $$g(t,\phi, \phi)=1.$$ Conclusions ----------- In this chapter we have shown how the $tt^{*}$ geometry of the modular curves is rich of interesting phenomena and outstanding connections between geometry, number theory and physics. These Riemann surfaces parametrize a family of supersymmetric FQHE models in which the usual setting degenerates in a doubly periodic physics on the complex plane. In the subclass of theories of level $N$, the elliptic functions playing the role of superpotentials have $N$ vacua and $N$ poles in the fundamental cell, with the corresponding residues which add up to zero by definition. The cancellation of the total flux between the magnetic field and the quasi-holes guarantees the enhancement of symmetry that makes possible to face analitically these models. In particular, the presence of an abelian symmetry group with a transitive action on the vacua allows to diagonalize the ground state metric, as well as to find the necessary topological data to write the $tt^{*}$ equations. This requires to pull-back the model on the abelian universal cover of the target manifold, where we have seen that the physics is non-abelian. On this space the symmetry group is enlarged with the generators of loops around the poles, which are responsible for the non trivial commutation relations between the generators of the algebra. Hovewer, the abelianity that we have required in the classification can be recovered at the quantum level. In particular, the ansatz of a solution with vanishing loop angles is consistent with all the $tt^{*}$ equations, which can be recasted as Toda equations in the canonical coordinates.\ Studying the modular properties of these models, we have underlined that the non trivial modular transformations of the superpotential are a natural consequence of the geometry of the modular curves. A critical value as coordinate on the spectral cover can be defined only on the upper half plane, since the F-term variations are rational functions in projective coordinates on the modular curves. This has been studied in the easiest cases of the platonic solids inscribed in the Riemann sphere, but for surfaces of higher genus it is more convenient to parametrize the critical value in terms of the fundamental units of the modular function field. The congruence subgroups have a not trivial effect also on the components of the ground state metric, since they change the representation of the abelian symmetry group.\ The known results and theorems about the cusps counting and classification have been recovered in a physical language when we have classified the critical limits of this family of theories. One of the main point is that the width of a cusp allows to determine the UV or IR nature of the corresponding RG fixed point.\ Our investigation has also revealed the algebraic properties of the modular curves. As we pointed out, the most remarkable connection with number theory is that the Galois group of the real cyclotomic extensions acts on the regularity conditions of the $\hat{A}_{N-1}$ Toda equations. This follows from the fact that the $\hat{A}_{N-1}$ models play the role of UV critical limits and belong to the same orbit of the Galois group. Modular Transformations of $\log E_{u_{1},u_{2}}(\tau)$ {#modulartrnasofrmationmass} ======================================================= In section \[modphysmass\] we have setted the notations $$\begin{split} & q_{\tau}= e^{2\pi i \tau}, \hspace{2cm} q_{z}= e^{2\pi i z}, \\ \\ & \hspace{1.3 cm} z= u_{1} \tau+ u_{2}, \end{split}$$ with $ u_{1},u_{2} \in \mathbb{Z}/ N $, and defined the modular units $$\label{expan} E_{u_{1},u_{2}}(\tau)= q_{\tau}^{B_{2}(u_{1})/2} (1-q_{z}) \prod _{n=1}^{\infty} (1-q_{\tau}^{n}q_{z})(1-q_{\tau}^{n}/q_{z}),$$ which are the Siegel functions up to the root of unity $e^{2\pi i u_{2}(u_{1}-1)/2}$. These objects satisfy [@rif48] $$E_{u_{1}+1,u_{2}}(\tau)=-e^{-2\pi i u_{2}} E_{u_{1},u_{2}}(\tau), \hspace{2cm} E_{u_{1},u_{2}+1}(\tau)=E_{u_{1},u_{2}}(\tau),$$ and transform under $\gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2,\mathbb{Z})$ as $$\label{prop1} \begin{split} & E_{u_{1},u_{2}}(\tau+b)= e^{\pi i b B_{2}(u_{1})} E_{u_{1},u_{2}+b u_{1}}(\tau) , \ \ \ \ \mathrm{for } \ c = 0 , \\ \\ & E_{u_{1},u_{2}}(\gamma (\tau))= \varepsilon (a,b,c,d) e^{\pi i \delta} E_{u_{1}^{\prime},u_{2}^{\prime}}(\tau) , \ \ \ \ \mathrm{for } \ c\neq 0, \end{split}$$ where $$\varepsilon(a,b,c,d)= \begin{cases} e^{i\pi (bd(1-c^{2})+c(a+d-3))/6}, \ \ \ \ \mathrm{if} \ c \ \mathrm{is \ odd}, \\ -ie^{i\pi (ac(1-d^{2})+d(b-c+3))/6}, \ \ \ \ \mathrm{if} \ d \ \mathrm{is \ odd}, \end{cases}$$ $$\delta= u_{1}^{2}ab + 2 u_{1}u_{2}bc + u_{2}^{2}cd - u_{1}b-u_{2}(d-1),$$ and $$u_{1}^{\prime}= a u_{1}+c u_{2}, \hspace{1cm} u_{2}^{\prime}= b u_{1}+d u_{2}.$$ With these definitions, we want to compute the difference $$\chi_{u_{1},u_{2}}(\gamma)=\log E_{u_{1},u_{2}}(\gamma (\tau))-\log E_{u_{1}^{\prime},u_{2}^{\prime}}(\tau),$$ for $\gamma \in SL(2,\mathbb{Z})$ and generic characters $u_{1},u_{2} \in \mathbb{Z}/N$. From \[prop1\] we know that there is a power of $E_{u_{1},u_{2}}(\gamma (\tau))/E_{u_{1}^{\prime},u_{2}^{\prime}}(\tau)$ which is equal to one. This number is $12 N$ for $\Gamma(N)$ and $12 N^{2}$ for $\Gamma_{1}(N),\Gamma_{0}(N)$ and the whole $SL(2,\mathbb{Z})$. Therefore, the difference $\chi_{u_{1},u_{2}}(\gamma)$ must be equal to $2\pi i$ times a rational number. Given that the upper half plane is simply connected, this number is independent of $\tau$. Moreover, since $\log E_{u_{1},u_{2}}(\tau) $ is single-valued on the upper half plane, it is also indipendent from the branch of the logarithm. A natural choice, suggested by the $q$-expansion of the Siegel functions, is the principal branch on $\mathbb{C}$ with the negative real axis deleted. From now on we will use this determination. Because $E_{u_{1},u_{2}}(\tau)$ changes by a phase under an integer shift of the characters, we can assume without loss of generality the canonical normalization $0 < u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime} < 1$.\ Let us first consider the case with $c=0$. These transformations belong to the coset group $\Gamma_{1}(N)/\Gamma(N) \simeq \mathbb{Z}_{N}$ and are generated by $\gamma (\tau)= \tau+1$. Using the expansion of the Siegel function in \[expan\] we easily obtain $$\chi_{u_{1},u_{2}}(\gamma)= 2\pi i \frac{1}{2}B_{2}(u_{1}).$$ From now on we assume $c\neq 0$ and write $\gamma(\tau)= \frac{a\tau +b}{c\tau +d}= \frac{a}{c}-\frac{1}{c^{2}\tau + cd}$.\ Using again the \[siegel\] we have $$\log E_{u_{1},u_{2}}(\tau)= 2\pi i B_{2}(u_{1}) \tau + \log(1-q_{z}) + \sum_{n=1}^{\infty} \left( \log(1-q_{\tau}^{n}q_{z})+ \log(1-q_{\tau}^{n}/q_{z})\right).$$ With $\tau$ in the upper half plane and the characters canonically normalized, the conditions of absolute convergence for the standard series of the principal logarithm are satisfied. Therefore, using series expansions like $$\log(1-q_{z})=-\sum_{m=1}^{\infty} \frac{q_{z}^{m}}{m}$$ for the logarithms in the expression, we obtain $$\log E_{u_{1},u_{2}}(\tau)= 2\pi i \frac{1}{2}B_{2}(u_{1})\tau - \mathcal{Q}(z;\tau),$$ where $$\mathcal{Q}(z;\tau)= \sum_{m=1}^{\infty} \frac{1}{m} \frac{q_{z}^{m}+ (q_{\tau}/q_{z})^{m}}{1-q_{\tau}^{m}}.$$ Then, let us put $$\tau= -\frac{d}{c}+i y, \ \mathrm{with} \ y>0, \hspace{2cm} \gamma(\tau)= \frac{a}{c} + \frac{i}{c^{2}y}.$$ Since it is indipendent of $\tau$, we can calculate $ \chi_{u_{1},u_{2}}(\gamma)$ in the limit $y\rightarrow 0$, i.e. $\tau \rightarrow -\frac{d}{c}$ and $\gamma(\tau) \rightarrow i \infty$, by applying the Abel limit formula. Setting $$z_{\gamma}= u_{1} \gamma(\tau)+u_{2}, \hspace{2cm} z^{\prime}= u_{1}^{\prime}\tau + u_{2}^{\prime},$$ and keeping only the immaginary parts, since $ \chi_{u_{1},u_{2}}(\gamma)$ is pure immaginary, we have to evaluate the expression $$\begin{split} \chi_{u_{1},u_{2}}(\gamma)=&\ 2\pi i \frac{1}{2} \left( B_{2}(u_{1}) \frac{a}{c} + B_{2}(u_{1}^{\prime}) \frac{d}{c} \right) \\ \\ &- \lim_{\tau\rightarrow -\frac{d}{c}}\left( \mathrm{Im} \mathcal{Q}(z_{\gamma};\gamma({\tau}))- \mathrm{Im} \mathcal{Q}(z^{\prime};\tau)\right) . \end{split}$$ Let us start with $\mathrm{Im} \mathcal{Q}(z_{\gamma};\gamma({\tau}))$. As $\gamma(\tau) \rightarrow i \infty$, $q_{z_{\gamma}}$ and $q_{\gamma(\tau)}/q_{z_{\gamma}}$ approach $0$, therefore $$\lim_{\tau\rightarrow -\frac{d}{c}} \ \mathrm{Im} \mathcal{Q}(z_{\gamma};\gamma({\tau}))=0.$$ Now it is the turn of $\mathcal{Q}(z^{\prime};\tau)$. We can decompose it in two pieces : $$\begin{split} \lim_{\tau\rightarrow -\frac{d}{c}}\ \mathrm{Im} \mathcal{Q}(z^{\prime};\tau)= & \ \lim_{\tau\rightarrow -\frac{d}{c}} \ \mathrm{Im} \sum_{c \ \nmid \ m} \frac{1}{m} \mathcal{Q}_{m}(z^{\prime};\tau) \ + \ \lim_{\tau\rightarrow -\frac{d}{c}} \ \mathrm{Im} \sum_{c \mid m} \frac{1}{m}\mathcal{Q}_{m}(z^{\prime};\tau) \\ = & \ L^{\prime} + L^{\prime \prime} , \end{split}$$ where $$\mathcal{Q}_{m}(z^{\prime};\tau)=\frac{q_{z^{\prime}}^{m}+ (q_{\tau}/q_{z^{\prime}})^{m}}{1-q_{\tau}^{m}}.$$ The symbols $L^{\prime}$ and $L^{\prime \prime}$ denote the sum respectively for $c \nmid m$ and $c \mid m $. We introduce $$r= e^{-2\pi y}, \hspace{1.5cm} M=N\vert c \vert, \hspace{1.5cm} \zeta= e^{-2\pi i d/c}, \hspace{1.5cm} \lambda= e^{2\pi i\left( -\frac{d}{c}u_{1}^{\prime} + u_{2}^{\prime} \right)}.$$ It is shown in [@rif49; @rif50] that the partial sums of these series are uniformly bounded. Therefore, we are allowed to take the limit under the sign of summation. Let us consider first $L^{\prime \prime}$. Using the notation above and taking the immaginary part, we have $$L^{\prime \prime}= \lim_{r\rightarrow 1} \sum_{ c \mid m} \frac{r^{u_{1}^{\prime}m}-r^{(1-u_{1}^{\prime})m}}{1-r^{m}} \frac{1}{2m}(\lambda^{m}-\lambda^{-m}).$$ Taking the limit under the summation sign, one gets $$\begin{split} L^{ \prime \prime }= & \sum_{ c \mid m} (1-2u_{1}^{\prime}) \frac{\lambda^{m}-\lambda^{-m}}{2m}= \sum_{m=1}^{\infty}(1-2 u_{1}^{\prime}) \frac{1}{2\vert c \vert m}( \lambda^{\vert c \vert m}-\lambda^{-\vert c \vert m}) \\ \\ = & \ (1-2u_{1}^{\prime})\frac{1}{2 \vert c \vert}\sum_{m=1}^{\infty}\frac{1}{m}\left( e^{2\pi i (-d \varepsilon(c)u_{1}^{\prime}+ u_{2}^{\prime} \vert c \vert )m} - e^{-2\pi i (-d \varepsilon(c)u_{1}^{\prime}+ u_{2}^{\prime} \vert c \vert )m} \right), \end{split}$$ where $ \varepsilon (c)= \vert c \vert / c $. If $t$ is real and not integer, it holds the Fourier expansion $$\sum_{m=1}^{\infty} \frac{1}{m}(e^{2\pi i m t}- e^{-2\pi i m t})= -2\pi i B_{1}\left( \langle t \rangle \right) ,$$ where $ B_{1}(x)=x-\frac{1}{2}$ is the first Bernoulli polynomial. Thus $$L^{ \prime \prime }= -2\pi i (1-2u_{1}^{\prime})\frac{1}{2 \vert c \vert} B_{1}\left( \langle -d \varepsilon(c)u_{1}^{\prime}+ u_{2}^{\prime} \vert c \vert \rangle\right) = -\frac{2\pi i}{c} B_{1}\left( u_{1}^{\prime}\right) B_{1}\left( \langle d u_{1}^{\prime}- u_{2}^{\prime} c \rangle \right) .$$ Now we turn to the last piece $L^{\prime}$. Taking the limit under the summation sign, we obtain $$L^{\prime}=\lim_{\tau\rightarrow -\frac{d}{c}} \ \mathrm{Im} \sum_{c \ \nmid \ m} \frac{1}{m}\mathcal{Q}_{m}(z^{\prime};\tau) = \sum_{c \ \nmid \ m} \frac{1}{m} \varphi (m),$$ where $$\varphi(m)= \mathcal{Q}_{m}(z^{\prime};\tau)\vert_{\tau= -d/c} = \frac{\lambda^{m}+ (\zeta/\lambda)^{m}}{1-\zeta^{m}}.$$ Since $ \varphi(-m)=-\varphi(m)= \overline{\varphi(m)}$, we note that $\varphi(m)$ is pure immaginary and an odd function of $m$ mod $M=N \vert c \vert$. Now, for each class $x \in \mathbb{Z}/M\mathbb{Z}$ and $2x \not \in M\mathbb{Z}$, we define $$f(x)= \sum_{m=1}^{\infty} \frac{a(m,x)}{m}$$ where $$a(m,x)= \begin{cases} 0 \hspace{0.8cm} \mathrm{if} \ m \neq \pm x \ \mathrm{mod} \ M \\ 1 \hspace{0.8cm} \mathrm{if} \ m = x \ \mathrm{mod} \ M \\ -1 \hspace{0.5cm} \mathrm{if} \ m =- x \ \mathrm{mod} \ M. \end{cases}$$ Then, $L^{\prime}$ can be rewritten as $$L^{\prime}= \frac{1}{2}\sum_{\overset{x \in \mathbb{Z}/M\mathbb{Z},}{ \overset{\ x \neq 0 \ \mathrm{mod} \ c\mathbb{Z},}{ \ 2x \not \in M\mathbb{Z}} }}\varphi(x) f(x).$$ In [@rif49] is shown that $$f(x)= \frac{-i \pi}{M}\left[ \frac{1}{1-e^{2\pi i x/M}} - \frac{1}{1-e^{-2\pi i x/M}} \right].$$ Let $\omega= e^{2\pi i /N \vert c \vert}$. Using this expression $L^{\prime}$ becomes $$\begin{split} L^{\prime}= & \frac{-\pi i}{2M}\sum_{c \ \nmid \ x}\frac{\lambda^{x}+(\zeta/\lambda)^{x}}{1-\zeta^{x}}\left[ \frac{1}{1-\omega^{x}}-\frac{1}{1-\omega^{-x}} \right] \\ \\ = & \frac{-\pi i}{2M}\sum_{c \ \nmid \ x} \biggl[ \frac{\lambda^{x}}{(1-\zeta^{x})(1-\omega^{x})} + \frac{(\zeta / \lambda)^{x}}{(1-\zeta^{x})(1-\omega^{x})} \\ \\ & - \frac{\lambda^{x}}{(1-\zeta^{x})(1-\omega^{-x})} - \frac{(\zeta / \lambda)^{x}}{(1-\zeta^{x})(1-\omega^{-x})} \biggr]. \end{split}$$ Changing $x$ to $-x$ in the last two terms, we find $$L^{\prime}= -\frac{\pi i}{M} \left[ \sum_{c \ \nmid \ x} \frac{\lambda^{x}}{(1-\zeta^{x})(1-\omega^{x})} + \sum_{c \ \nmid \ x} \frac{(\zeta / \lambda)^{x}}{(1-\zeta^{x})(1-\omega^{x})} \right] .$$ This expression can be further simplified. We decompose the sum by introducing the variable $$\begin{split} & x= y + k \vert c \vert, \\ \\ 0 < y < \vert c \vert , & \hspace{2cm} 0 \leqslant k \leqslant N-1. \end{split}$$ Let us denote with $S$ the partial sum in the variable $k$ of the first term in $L^{\prime}$. One gets $$\begin{split} S= & \frac{\lambda^{y}}{1-\zeta^{y}}\sum_{k=0}^{N-1} \frac{\lambda^{k \vert c \vert}}{1-\omega^{y+k \vert c \vert}} \\ \\ = & -\frac{1}{M}\frac{\lambda^{y}}{1-\zeta^{y}} \sum_{r=0}^{N-1}r\omega^{ry} \sum_{k=0}^{N-1} (\lambda \omega^{r})^{k \vert c \vert}. \end{split}$$ The sum on the right is $0$ unless $ (\lambda \omega^{r})^{\vert c \vert}=1$. Using the definitions of $\lambda$ and $\omega$ in terms of $u_{1}^{\prime}, u_{2}^{\prime}, d,c$, we see that $(\lambda \omega^{r})^{\vert c \vert}=1$ if and only if $$r= Nd u_{1}^{\prime}-Ncu_{2}^{\prime} \ \ \mathrm{mod} \ N.$$ Letting consequently $r=N \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle + sN$ with $ 0 \leq s \leq \vert c \vert -1$, we have $$\begin{split} S & = -\frac{1}{\vert c \vert} \frac{\lambda^{y}}{1-\zeta^{y}} \sum_{\overset{0 \leq r \leq N-1,}{ r= N u_{1}^{\prime}d-Ncu_{2}^{\prime} \ \mathrm{mod} \ N}} r\omega^{ry} \\ \\ & = -\frac{1}{\vert c \vert} \frac{\lambda^{y}}{1-\zeta^{y}} \sum_{s=0}^{\vert c \vert -1}(N \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle + sN)e^{2\pi i \frac{y}{Nc} (N \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle + sN)} \\ \\ & = -\frac{1}{\vert c \vert} \frac{\lambda^{y}}{1-\zeta^{y}}\ e^{2\pi i \frac{y}{c} \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle} \sum_{s=0}^{\vert c \vert -1} s N e^{2\pi i ys/c} \end{split}$$ $$\begin{split} & = N \frac{\lambda^{y}}{1-\zeta^{y}} e^{2\pi i \frac{y}{c} \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle} \frac{1}{1-e^{2\pi i y/c}} \\ \\ & = N \frac{e^{2\pi i y \bigl( \frac{ \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle-du_{1}^{\prime} }{c}+u_{2}^{\prime} \bigr)} }{(1-e^{-2\pi i y d/c})(1-e^{2\pi i y/c})}. \end{split}$$ In order to write the final result in a more compact way, we introduce the symbol $$[ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c}= \frac{e^{2\pi i x \bigl( \frac{ \langle d u_{1}^{\prime}-cu_{2}^{\prime}\rangle-du_{1}^{\prime} }{c}+u_{2}^{\prime} \bigr)} }{(1-e^{-2\pi i x d/c})(1-e^{2\pi i x/c})}.$$ Noting that the second sum in $L^{\prime} $ can be obtained from the first one with the substitution $ u_{1}^{\prime}\rightarrow 1- u_{1}^{\prime}, u_{2}^{\prime} \rightarrow -u_{2}^{\prime}$, we get $$L^{\prime}= -\frac{\pi i}{c} \left[ \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ x \neq 0 } } [ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c} + \sum_{\overset{ x \in \mathbb{Z}/c\mathbb{Z},}{ x \neq 0} } [ x, 1-u_{1}^{\prime}, -u_{2}^{\prime} ]_{d,c} \right] .$$ From the property $[ -x,1- u_{1}^{\prime},- u_{2}^{\prime} ]_{d,c}=[ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c} $, one obtains further $$L^{\prime}= -\frac{2 \pi i}{c} \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ x \neq 0 } } [ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c} .$$ Putting all the pieces together, we finally have $$\begin{split} \chi_{u_{1},u_{2}}(\gamma)= & \ 2\pi i \frac{1}{2} \left( B_{2}(u_{1}) \frac{a}{c} + B_{2}(u_{1}^{\prime}) \frac{d}{c} -\frac{2}{c} B_{1}(u_{1}^{\prime}) B_{1}( \langle d u_{1}^{\prime}- u_{2}^{\prime} c \rangle ) \right) \\ \\ & -\frac{2 \pi i}{c} \sum_{\overset{x \in \mathbb{Z}/c\mathbb{Z},}{ \ x \neq 0 }} [ x, u_{1}^{\prime}, u_{2}^{\prime} ]_{d,c}. \end{split}$$ [100]{} K. von Klitzing, G. 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This ensures that the dynamics of the system at large $\vert z \vert$ is governed by the magnetic field $B$, so that on any macroscopic domain the probability of finding the electrons at positions $z_{i}$ satisfies $\log P(z_{i})\sim -B\sum_{i}\vert z_{i} \vert^{2}$. The interaction term may diverge when two particles become closer. This is not an issue, since the hyperplanes $z_{i}=z_{j}$ are not part of the configuration space of $N$ identical particles and the wave functions of the Hilbert space are vanishing there. [^3]: Because of Fermi statistics we should demand at least $N\leq m+M$. [^4]: It is straighforward to see that $1/z$ is a tempered distribution of function type. [^5]: This is true for a compact Riemann surface, as it turns out to be the target manifold. [^6]: We are going to show that $e$ is non trivial. [^7]: It will be clear later that the cusps can be interpreted as RG flow fixed points.

--- abstract: 'We explore the contribution of intracluster stars (ICS) to the chemical enrichment history of the intracluster medium (ICM). In contrast to scenarios in which all the metals originate in cluster galaxies and are then transported into the ICM, intracluster stars enrich the ICM [*in situ*]{}, thereby contributing 100% of their supernovae ejecta directly into the ICM. Modeling the ICS as an ancient, single burst stellar population with a normal initial mass function, we generate ICM iron abundances in the range of the observed values of several tenths solar. Large observational and theoretical uncertainties preclude us from concluding that the intracluster stars are the primary contributor of metals to the ICM in general. However, for the two clusters in our sample, and one from the literature, for which all of the required observational constraints exist, we are able to reproduce between half and all of the ICM iron with this simple model. Due to the ubiquity of intracluster stars in clusters and their direct connection with the ICM, we conclude that all models of the chemical enrichment history and energy budget of the ICM should account for the impact of the ICS.' author: - 'Dennis Zaritsky, Anthony H. Gonzalez, and Ann I. Zabludoff' title: Intracluster Stars and the Chemical Enrichment of the Intracluster Medium --- Introduction {#sec:intro} ============ The chemical enrichment history of the intracluster medium (ICM) is arguably the least understood aspect of galaxy cluster evolution. Particularly puzzling are the relatively high Fe abundance and moderately large $[\alpha/$Fe$]$ [@renzini93; @mush96; @tamura]. The former has been interpreted as evidence for large gas outflows from galaxies [@renzini93; @renzini97], while the latter, in combination with constraints from cluster galaxies, is interpreted as evidence for a non-standard initial mass function [@mush96; @portinari although see [@pipino] for a contrasting result]. Both of these inferences have a wide ranging impact on a variety of astronomical topics. Here we present evidence that the first argument should be significantly revised, and that the second argument should be revisited. Our proposal is straightforward. The intracluster stars (ICS) that we have found without exception in a sample of 24 clusters [@paper1 hereafter Paper I] must chemically pollute the ICM via supernovae. Such intergalactic SNe have already been observed [@galyam]. We evaluate the degree to which the ICS plays a role with the simplest modeling we can envision. As the vast literature on this topic demonstrates, there are many variables and unknowns in this type of model. We will adopt [*plausible*]{}, often typical, values of the relevant variables and demonstrate that the intracluster stars make a significant, and perhaps dominant, contribution to the enrichment history of the ICM. This study is not intended to be a complete or comprehensive treatment of the ICS enrichment of the ICM, but it illustrates the likely importance of the ICS in the chemical abundance budget of galaxy clusters. Observational Constraints on the Intracluster Components {#sec:data} ======================================================== In Paper I we present observations of 24 clusters that span a range of velocity dispersions and Bautz-Morgan types [@bautzmorgan1970]. The sample consists of nearby clusters ($0.03<z<0.13$) that contain a dominant brightest cluster galaxy (BCG) with a major axis position angle that lies within 45 degrees of the east-west axis (our drift scan direction). We present details of these unique data and our reduction procedure in Paper I. From modeling the two dimensional optical surface brightness distribution, we conclude in Paper I that the BCG light can be divided into two components. The inner component is similar to normal giant ellipticals in extent, luminosity, and scaling relations. The outer component is closely related in scale and ellipticity to the distribution of galaxies in the cluster. We associate the inner component with the BCG and the outer component with the ICS. In Paper I, we determine the luminosity of the outer component, which we will use here as a measure of the intracluster stellar content. Standard measurements of BCG light underestimate the contribution of this low surface brightness outer component (for example, assuming a single $r^{1/4}$ surface brightness profile to evaluate the total magnitude leads to a 50% underestimate of the total luminosity of the BCG plus ICS; Paper I). For constraints on the properties of the ICM, we have searched the literature for X-ray gas masses [@reiprich] and chemical abundances [@white]. There are only two clusters in our sample for which such data exist (Table \[tab:obs\]). We have adopted the gas mass inside of $r_{500}$, the radius at which the mean mass density exceeds 500 times the cosmological value, as the measure of the gas mass. We measure the ICS luminosity directly only within 300 kpc, but our model fit provides an estimate of the total luminosity. The mismatched apertures limit the precision with which comparisons can be made, although even if the apertures are matched, the comparison is complicated by flows of material across the aperture during the cluster’s evolution. The gas masses have been rescaled to our adopted cosmology ($H_0 = 70$ km/sec/Mpc, $\Omega_m = 0.3$ and $\Omega_\Lambda = 0.7$), and the observed abundances have been rescaled due to the difference between the older value of the photospheric solar abundance by mass [0.0026, @anders89], and the more recent one that we adopt here [0.0017, @grevesse]. This conversion increases the canonical Fe cluster abundance from 0.3 to 0.46 Fe$_\odot$. Models and Results {#sec:models} ================== This type of study usually relies on analytic models of the enrichment history [@renzini93; @bm; @maoz; @portinari], but we choose to model the chemical evolution with the publicly available modeling package, PEGASE.2 [@pegase], which provides estimates of the Type I and II supernovae rates as a function of age for a user-selected star formation history and initial mass function. We use these models because they also provide luminosities and colors of the resulting stellar population, which we need to scale our predictions for the observed ICS population. To mitigate ambiguity in our results, we adopt a very simple star formation history (a single burst of star formation that peaks at $z > 10$ and is finished by $z = 4$), an initial chemical abundance by mass fraction of 0.004 for the ISM (although adopting values of 0.0001 and 0.08 does not significantly affect the results), a standard IMF [@scalo], and all the default PEGASE parameters (such as a binary fraction of 0.5). Changing the star formation history so that the peak star formation rate is at $z =$ 3.5 and the star formation does not cease entirely until $z \sim 2$ decreases the predicted abundances by $\sim 10\%$. Varying the adopted initial chemical abundance only affects aspects of the stellar evolutionary modeling, such as the predicted stellar colors. We always assume that the ICM abundance is zero until we add the contribution from the ICS. \[fig:sne\] By integrating the rate of SNe predicted from the PEGASE models (Figure \[fig:sne\]) and adopting Fe yields for SNe, we obtain the total Fe mass ejected over time. For the yields we assume that a Type I ejects 0.7$M_\odot$ of Fe per supernova and that a Type II ejects 0.05$M_\odot$ of Fe per supernova. This yield for a Type I is the canonical value adopted by various other studies [see @maoz; @portinari], while that for Type II’s is the mean value [@elmhamdi] over the observed range, 0.0016 to 0.26 $M_\odot$ [@hamuy]. Because Type I’s dominate the enrichment in our models, the results are fairly robust to decreasing the Type II yield from our adopted value. Increasing the Type II yield leads to more pronounced changes. However, increasing the Type II yield obviously produces a larger total Fe mass and therefore is not a conservative approach. This discussion sidesteps the difficulties others have encountered in reproducing $[\alpha$/Fe$]$ or specific element ratios [@renzini93; @mush96]. However, any solution that has been proposed previously, such as non-standard initial mass functions, will also work with our ICS scenario, with the added flexibility that now one does not need to fit both the ICM and galaxy abundance patterns with the same model. Therefore, although we do not pursue the \[$\alpha$/Fe\] question here, we conclude that it is no more difficult a problem than in the standard models. Future models should be able to take advantage of the wealth of information becoming available on the spatial gradients of these element ratios [@tamura]. We present two different calculations that provide quantitative estimates of the importance of the ICS. First, we model clusters as a general population, adopting typical ratios of such quantities as gas mass to stellar mass. Second, we model the only three clusters for which all of the required quantities are measured. In both approaches, we combine the PEGASE supernovae rates with the yields described above to calculate the Fe mass that 1 M$_\odot$ of initial material produces after 14 Gyr (this 1 M$_\odot$ of material results in 0.59 M$_\odot$ of stellar mass, including remnants, at the current time). In the first type of calculation, we utilize the ratios of gas mass to galaxy mass in stars ($M_{GAS}/M_{GAL}$) and of ICS to total cluster luminosity ($L_{ICS}/L_{TOTAL}$), and assume that that the ICS and galactic stars have the same M/L ratio, to calculate the ratio of ICM gas mass to ICS mass. We then use this ratio, and the calculated Fe abundance per solar mass of ICS, to calculate an ICM Fe abundance. Values for $M_{GAS}/M_{GAL}$ vary between $\sim 2$ to 10 [@arnaud; @fukugita], although a combination of recent optical work with the Sloan Digital Sky Survey [@bell] and a large ROSAT sample [@ettori] favors the upper end of that range [@mushotzky04]. In the limited observational work to date, values for $L_{ICS}/L_{TOTAL}$ range from 0.1 [@feldmeier2004] to 0.2 [@paper0 adjusted for the division into two components of Abell 1651 from Paper I]. If, as models predict [@sl; @murante], the stars in the ICS are on average older than those in the galaxies, then our assumption of equal M/L for the ICS and galactic stars leads to an underestimate of the relative amount of mass in the ICS. This is therefore a conservative assumption in our determination of the produced Fe. Instead of adopting specific values for the various ratios, we provide results over the range of plausible values (Figure \[fig:ratios\]). Figure \[fig:ratios\] demonstrates that for the ICS to chemically enrich the ICM to cluster-like values requires both a fairly high $L_{ICS}/L_{TOTAL}$ and a low $M_{GAS}/M_{GAL}$ relative to the ranges of values explored. However, to reach a level where the ICS contributes significantly to the ICM ($\sim 0.1 - 0.2$ Fe$_\odot$) requires values of the two scaling ratios that lie well within the observational ranges [@arnaud; @paper0; @feldmeier2004]. Furthermore, because M/L is likely to be larger for the ICS than for the galaxies (see above), our scaling may underestimate the contribution from the ICS by up to a factor of two. \[fig:ratios\] The second type of calculation provides a more direct comparison in the few clusters for which the relevant data are available. For two of our clusters, we find corresponding $M_{GAS}$ and metallicity measurements in the literature (Table \[tab:obs\]). We scale the Fe mass, as calculated from the PEGASE models and our adopted yields, using the M/L from those models and the total luminosity of the outer component (Paper I). The ICM without the contribution of the ICS is assumed to have zero metallicity. The calculated metallicity resulting from the ICS contribution is presented in Table \[tab:obs\] for comparison. $M_{ICS}$ represents the stellar mass, including remnants, in the intracluster volume. [lrrrrrrrr]{} Abell 1651 & 0.0853 & 6.5 &1.73&$-$24.95 & ...& 0.9&0.44&0.22\ Abell 1914$^a$&0.1712&12.9&2.46&...&$-$24.94&1.8&0.37&0.22\ Abell 3112 & 0.0759 & 4.5 &1.53&$-$25.74 & ...&1.9&0.68&0.66\ \[tab:obs\] It is unfortunate that only two of our 24 clusters have published values of $M_{GAS}$ and Fe abundance. However, for those two clusters, we find that the ICS can account for a large fraction of the metals in the ICM, and we reproduce the relative rankings of metallicity for the two clusters. There are numerous adjustments that can be made due to uncertainties in the luminosity of the ICS, the region over which $M_{GAS}$ and the ICS are measured, the adopted yields, the modeling of the SNe rates, the star formation history, initial mass function, and so forth. In Table \[tab:obs\] we also include results for a cluster observed by [@feldmeier2004]. Despite some differences in their definition and measurement of the ICS, our model again reproduces a sizable fraction of the ICM iron abundance in this cluster. The purpose of this exercise is primarily to demonstrate that for plausible values of the various parameters, the ICS can contribute a significant amount of iron to the ICM. A similar conclusion has been reached using scaling relationships and infrared photometry by [@lm]. A complete model of cluster enrichment must include the contribution from all stars, whether currently in the ICS or in galaxies, and must account for the metals both in the ICM and in galaxies. Our approach has ignored 1) the contribution of metals to the ICM from stars currently in galaxies, because we are interested in determining the contribution from the newly measured ICS component, and 2) the metals currently locked in galaxies, again because we are interested in the connection between the ICS and ICM. In effect, we have treated the current cluster galaxies as closed box systems and set them aside. This approach produces a conservative estimate of the metal enrichment of the ICM. It is important to discuss one possible refinement of our chemical enrichment model. We make no direct assumption about the origin of the ICS, which, in simulations, arises from the tidal disruption of cluster galaxies early in the cluster’s history [@sl; @murante; @willman2004]. If these simulations are correct, then the metals in the ICM are some combination of the metals already present in those first cluster galaxies and the subsequent evolution of the stars stripped from those galaxies. So far in this paper, we have discussed only the latter issue. We now consider the former issue by asking what fraction of the metals in those ICS progenitor galaxies enters the ICM. If the parent galaxies of the ICS were completely disrupted, then the answer is 100% and our model implicitly includes this possibility. If some of the parent galaxies survive to the current time, then our calculation overestimates the ICS metal contribution to the ICM by whatever amount of metals is locked in these galaxies. Without knowing when the ICS formed and its detailed star formation history, we cannot estimate the magnitude of this effect, but it may be significant. For example, for a star formation model that peaks sharply at $z \sim 3$, an age comparable to the mean age of the ICS in the simulation by [@sl], about 25% of the Fe forms prior to the mean age of the ICS stars. If the stars are stripped from these galaxies shortly thereafter, the galaxies might retain as much as 25% of all the metals that we attribute to the ICS. Nevertheless, even if our calculation overestimates the ICS metal contribution to the ICM by as much as 25%, the ICS still contributes between 35% and 75% of the metals in the three clusters that we model in detail (Table 1). Conclusions {#sec:discussion} =========== We calculate that the chemical enrichment due to intracluster stars (ICS), a ubiquitous population in galaxy clusters (Paper I), is a significant component of the chemical enrichment history of the intracluster medium (ICM). Furthermore, our model suggests that for at least some clusters the ICS could contribute the bulk of the metals in the ICM. Unlike models of ICM enrichment in which the metals originated in the cluster members we see today, the simplest version of our model has no uncertainty related to the fraction of the processed material from the ICS that pollutes the ICM. In fact, once the relevant measurements (ICM mass, ICM metallicity, and ICS mass) are made sufficiently precise, our argument could be turned around to place limits on the metal outflow from cluster galaxies. The ICS, and the associated SNe, should also be an important factor in the energy budget of the ICM (see [@bm] for such calculations relating to the SNe within the cluster galaxies and [@domainko] for the effects of ICS SNe on large-scale cluster gas dynamics). Lastly, Table \[tab:obs\] suggests that the factor of $\sim$ 2 scatter in abundances among clusters [@baum; @horner] is related to the balance between the mass of the ICM gas and that of the ICS. We speculate that the trend seen in Fe abundance versus temperature [@horner01; @buote] may reflect an interplay between the ICS fraction, which appears to decrease in less massive clusters [@feldmeier2004], and the gas fraction, which varies in a complex manner as a function of temperature [@sanderson]. This paper is not a detailed exploration of the parameter space currently accessible to models of ICM enrichment by the ICS, but rather a demonstration case for the importance of including the ICS in any accounting of the chemical enrichment history of clusters. As such, our conclusions are based on simple models that adopt typical values of such unknown quantities as the initial mass function and stellar yields. Future work should aim to obtain the requisite observational constraints to test these models in detail across a larger sample of clusters. We thank Richard Mushotzky for comments on a preliminary draft and Wayne Baumgartner for providing data prior to publication. We also thank Yen-Ting Lin and Joe Mohr for communicating their results prior to publication. DZ acknowledges support from a David and Lucile Packard Foundation Fellowship and NASA grant NAG5-13583. AIZ is supported by NSF grant AST-0206083 and NASA LTSA grant NAG5-11108. AHG is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-0407085. Anders, E., & Grevesse, N. 1989, GeCoA, 53, 197 Arnaud, M., Rothenflug, R., Boulade, O., Vigroux, L., & Vangioni-Flam, E. 1992, , 254, 49 Baumgartner, W.H., Loewenstein, M., Horner, D.J., & Mushotzky, R.F. 2004, , in press. 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--- abstract: 'Recent research shows that Hawking radiation (HR) is also possible around the trapping horizon of a wormhole. In this article, we show that the HR of gravitino (spin-$3/2$) particles from the traversable Lorentzian wormholes (TLWH) reveals a negative Hawking temperature (HT). We first introduce the TLWH in the past outer trapping horizon geometry (POTHG). Next, we derive the Rarita-Schwinger equations (RSEs) for that geometry. Then, using both the Hamilton-Jacobi (HJ) ansätz and the WKB approximation in the quantum tunneling method, we obtain the probabilities of the emission/absorption modes. Finally, we derive the tunneling rate of the emitted gravitino particles, and succeed to read the HT of the TLWH.' author: - 'I. Sakalli [[email protected] ]{}' - 'A. Ovgun [[email protected] ]{}' title: Gravitinos Tunneling From Traversable Lorentzian Wormholes --- Introduction ============ An interesting phenomenon that corresponds to spontaneous emissions (as if a black body radiation) from a black hole (BH) is the HR. It is a semi-classical outcome of the quantum field theory [@Haw; @Haw1]. HR dramatically changed our way of looking to the BHs; they are not absolutely black and cold objects, rather they emit energy with a characteristic temperature: HT. Event horizon, where is an irreversible point (in classical manner) for any object including photons is the test-bed of the gedanken experiment for the HR. The studies concerning this phenomenon have been carrying on by using different methods. In particular, the quantum tunneling [@PW] of particles with different spins from the various BHs have gained momentum in the recent years (the reader may be referred to [@Vaz; @Jing; @kerner1; @kerner2; @kerner3; @Mann; @yang; @sharifa; @Kruglova; @ran2; @sharif; @ran; @ChenZhou; @ali2; @ali1; @ali3; @sucu; @gohar1; @huan; @singh; @MDehghani; @hale] and references cited therein). Recently, it has been shown that HR of the bosons with spin-0 (scalar particles) and spin-1 (vector particles) from the TLWH [@wh], which is a bridge or tunnel between different regions of the spacetime is possible by using the POTHG [@Diaz; @Diaz1; @aliwh]. Wormhole has been extensively studied in different areas [@garattini; @kuhfittig1; @rahaman1; @rahaman2; @rahaman3; @halilsoy]. However, HT of the TLWH appears to be negative because of the phantom energy (exotic matter: the sum of the pressure and energy density is negative) that supports the broadness of the wormhole throat [@wh]. In addition, it is a well-known fact that the virtual particle-antiparticle pairs are created near the horizon. In a BH spacetime the real particles with positive energy and temperature are emitted towards spatial infinity [@wald]. However, in the POTHG which is analog to the white hole geometry, the antiparticles come out from the horizon [@Helou2]. In other words, our analysis predicts that the energy spectrum of the antiparticles leads to a negative temperature for the TLWH. For the subject of the white hole radiation, the reader may refer to [@peltola]. As it is shown by Caldwell et al [@Cald], the dark matter (DM) [@dm] could have a phantom energy. In this regard, the phantom energy can keep apart every bound object until the Cosmos eventuates in the Big-Rip [@BRP]. On the other hand, DM does not emit, reflect or absorb light, making it not just dark but entirely transparent. But if the DM particles strolling around a BH or  a wormhole can produce gamma-rays would give a possibility to study the radiation of this mysterious matter [@gamma1; @dm1]. DM has many candidates, and gravitino (spin-$3/2$) is one of them [@Kawas; @David]. Gamma-ray decay of the gravitino DM has been very recently studied in [@dm1]. So, HR of the gravitinos from the BHs and/or wormholes could make an impact on the production of the DM. Behaviors of the gravitino’s wave function are governed by the RSEs [@Mann; @rsch1]. So, our main motivation in this paper is to investigate the HR of the gravitino tunneling from the TLWH geometry. Using the RSEs and HJ method, we aim to regain the standard HT of the TLWH. The structure of this paper is as follows. In Sec. II, we introduce the 3+1 dimensional TLWH [@Diaz1] and analyzes the RSEs for the gravitino particles in the POTHG of the TLWH [@hawyard; @hayward1; @hayward2; @MisnerSharp; @Topology]. We show that the RSEs are separable when a suitable HJ änsatz is employed. Then the radial equation can be reduced to a coefficient matrix equation that makes us possible to compute the probabilities of the emission/absorption of the gravitinos. Finally, we calculate the tunneling rate of the radiated gravitinos, and retrieve the standard HT of the TLWH. We summarize and discuss our results in Sec. III. Quantum Tunneling of Gravitinos From 3+1 Dimensional TLWH ========================================================= For the wave equation of the gravitino (spin-3/2) particles, we start with the massless (the mass has no remarkable effect in the computation of the quantum tunneling [@Mann]) RSEs [@rsch1; @ferm; @ferm1; @ferm2]: $$\begin{gathered} i\gamma^{\nu}\left( D_{\nu}\right) \Psi_{\mu}=0,\label{1}\\ \gamma^{\mu}\Psi_{\mu}=0,\label{2nn}$$ where $\Psi_{\mu}\equiv\Psi_{\mu a}$ is a vector-valued spinor and the $\gamma^{\mu}$ matrices satisfy $\left\{ \gamma^{\mu},\gamma^{\nu}\right\} =2g^{\mu\nu}$. The first equation is the Dirac equation applied to every vector index of $\Psi$, while the second is a set of additional constraints to ensure that no ghost state propagates; that is, to ensure that $\Psi$ represents only spin-$3/2$ fermions, with no spin-$1/2$ mixed states [@Mann; @ferm]. The covariant derivative obeys $$D_{\mu}=\partial_{\mu}+\frac{i}{2}\Gamma_{\mu}^{\alpha\beta}J_{\alpha\beta}, \label{3}$$ where $$\begin{aligned} \Gamma_{\mu}^{\alpha\beta} & =g^{\beta\gamma}\Gamma_{\mu\gamma}^{\alpha },\nonumber\\ J_{\alpha\beta} & =\frac{i}{4}\left[ \gamma^{\alpha},\gamma^{\beta}\right] ,\nonumber\\ \{\gamma^{\alpha},\gamma^{\beta}\} & =2g^{\alpha\beta}\times I. \label{4}$$ The metric of TLWH in the generalized retarded Eddington-Finkelstein coordinates (REFCs), which is the POTHG, is given by [@Diaz1] $$ds^{2}=-Fdu^{2}-2dudr+r^{2}\left( d\theta^{2}+\sin^{2}\theta d\varphi ^{2}\right) , \label{5}$$ where $F=$ $1-2M/r$. Misner-Sharp energy is represented by $M=\frac{1}{2}r(1-\partial^{a}r\partial_{a}r)$ which becomes $M=\frac{1}{2}r_{h}$ on the trapping horizon ($r_{h}$) [@MisnerSharp]. Marginal surfaces having $F(r_{h})=0$ are the past marginal surfaces in the REFCs [@Diaz]. For solving the RSEs, we use the following Dirac $\gamma$-matrices:$$\begin{aligned} \gamma^{u} & =\frac{1}{\sqrt{F}}\left( \begin{array} [c]{cc}-i & -\sigma^{3}\\ -\sigma^{3} & i \end{array} \right) ,\,\,\ \gamma^{r}=\sqrt{F}\left( \begin{array} [c]{cc}0 & \sigma^{3}\\ \sigma^{3} & 0 \end{array} \right) ,\nonumber\\ \gamma^{\theta} & =\frac{1}{r}\left( \begin{array} [c]{cc}0 & \sigma^{1}\\ \sigma^{1} & 0 \end{array} \right) ,\,\,\,\ \gamma^{\phi}=\frac{1}{r\text{sin}\theta}\left( \begin{array} [c]{cc}0 & \sigma^{2}\\ \sigma^{2} & 0 \end{array} \right) , \label{6}$$ where the Pauli matrices are given by $$\sigma^{1}=\left( \begin{array} [c]{cc}0 & 1\\ 1 & 0 \end{array} \right) ,\sigma^{2}=\left( \begin{array} [c]{cc}0 & -i\\ i & 0 \end{array} \right) ,\sigma^{3}=\left( \begin{array} [c]{cc}1 & 0\\ 0 & -1 \end{array} \right) . \label{7}$$ Gravitino wave function ($\psi$) has two spin states \[spin up (i.e. positive $r-$direction) and spin down (i.e. negative $r-$direction)\]: $$\psi_{\nu\uparrow}=\left( a_{\nu},0,c_{\nu},0\right) e^{\frac{i}{\hbar }S_{\uparrow}(u,r,\theta,\phi)}, \label{8}$$ $$\psi_{\nu\downarrow}=\left( 0,b_{\nu},0,d_{\nu}\right) e^{\frac{i}{\hbar }S\downarrow(u,r,\theta,\phi)}, \label{9}$$ where $S(u,r,\theta,\phi)$ denotes the gravitino action which is going to be expanded in powers of $\hbar$, and $ a_{\nu},b_{\nu},c_{\nu},d_{\nu } $ are the arbitrary constants. Here we shall only consider the spin up case, since the spin down case is fully analogous with it. The action for the spin-up states can be chosen as follows $$\begin{aligned} S_{\uparrow}(u,r,\theta,\phi) & =S_{\uparrow0}(u,r,\theta,\phi)+\hbar S_{\uparrow1}(u,r,\theta,\phi)\nonumber\\ & +\hbar^{2}S_{\uparrow2}(u,r,\theta,\phi)+.... \label{10}$$ Therefore, the corresponding RSEs become $$\begin{aligned} \frac{1}{\sqrt{F}}\left[ \left( ia_{0}-c_{0}\right) (\partial _{u}S_{\uparrow0})\right] +\sqrt{F}(-c_{0}\partial_{r}S_{\uparrow0}) & =0,\label{11}\\ \frac{1}{r\text{sin}\theta}(-ic_{0}\partial_{\phi}S_{\uparrow0})+\frac{1}{r}(-c_{0}\partial_{\theta}S_{\uparrow0}) & =0,\label{12}\\ \frac{1}{\sqrt{F}}\left[ \left( -a_{0}-ic_{0}\right) (\partial _{u}S_{\uparrow0})\right] +\sqrt{F}(-a_{0}\partial_{r}S_{\uparrow0}) & =0,\label{13}\\ \frac{1}{r\text{sin}\theta}(-ia_{0}\partial_{\phi}S_{\uparrow0})+\frac{1}{r}(-a_{0}\partial_{\theta}S_{\uparrow0}) & =0, \label{14}$$ with the constraints equations: $$\begin{aligned} \frac{-a_{0}+c_{0}}{\sqrt{F}}+\sqrt{F}c_{1}+\frac{d_{2}}{r}-\frac{id_{3}}{r\text{sin}\theta} & =0\label{15}\\ \frac{-d_{0}}{\sqrt{F}}-\sqrt{F}d_{1}+\frac{c_{2}}{r}+\frac{ic_{3}}{r\text{sin}\theta} & =0\label{16}\\ \frac{a_{0}}{\sqrt{F}}+\sqrt{F}a_{1}+\frac{b_{2}}{r}-\frac{ib_{3}}{r\text{sin}\theta} & =0\label{17}\\ \frac{-b_{0}+id_{0}}{\sqrt{F}}-\sqrt{F}b_{1}+\frac{a_{2}}{r}+\frac{ia_{3}}{r\text{sin}\theta} & =0 \label{18n}$$ Equations (15-18) are not important here. Because these equations give an independent wave solution, so that they have no effect on the action [@Mann]. Afterwards, the separation of variables method is applied to the action $S_{\uparrow0}(u,r,\theta,\phi)$: $$S_{\uparrow0}=Eu-W(r)-j_{\theta}\theta-j_{\phi}\phi+K, \label{19n}$$ where $E$ and $\left( j_{\theta},j_{\phi}\right) $ are energy and angular constants, respectively. However, $K$ is an arbitrary complex constant. Thus, Eqs. (11-14) reduce to $$\begin{gathered} -\frac{ia_{0}}{\sqrt{F}}E+\frac{c_{0}}{\sqrt{F}}E-c_{0}\sqrt{F}W^{\prime }=0,\label{20}\\ \frac{-c_{0}}{r}\left( j_{\theta}+\frac{i}{\text{sin}\theta}j_{\phi}\right) =0,\label{21}\\ \frac{a_{0}}{\sqrt{F}}E+\frac{ic_{0}}{\sqrt{F}}E-a_{0}\sqrt{F}W^{\prime }=0,\label{22}\\ \frac{-a_{0}}{r}\left( j_{\theta}+\frac{i}{\text{sin}\theta}j_{\phi}\right) =0. \label{23n}$$ Equations (21) and (23) are about the solutions of $\left( j_{\theta},j_{\phi}\right) $, and they do not have contribution to the tunneling rate. For this reason, we simply ignore them. Namely, the master equations for the tunneling rate are Eqs. (20) and (22). To analyze them, we first consider the case of $a_{0}=ic_{0}$ [@yian]. Using Eqs. (20) and (22), we now have a solution for $W(r)$ as $$W_{1}=\int\frac{2E}{F}dr. \label{24n}$$ The integrand has a simple pole at $r=r_{h}$. Choosing the contour as a half loop going around this pole from left to right and integrating, one obtains $$W_{1}=\frac{i2\pi E}{F^{\prime}(r_{h})}=\frac{i\pi E}{\kappa|_{H}}. \label{25n}$$ where $\kappa|_{H}=\left. \partial_{r}F/2\right\vert _{r=r_{H}}$ is the surface gravity at the horizon. On the other hand, if one sets $a_{0}=-ic_{0}$, this time Eqs. (20) and (22) admit the following solution for $W(r):$ $$W_{2}=0. \label{26n}$$ Hence, we can derive the ingoing/outgoing imaginary action solutions as $$\operatorname{Im}S_{1}=\operatorname{Im}W_{1}+\operatorname{Im}K, \label{27n}$$ $$\operatorname{Im}S_{2}=\operatorname{Im}W_{2}+\operatorname{Im}K=\operatorname{Im}K. \label{28}$$ We can now set $S_{1}$ for the action of absorbed (ingoing) gravitinos. We can tune their probability: $$\Gamma_{in}=\exp(-2\operatorname{Im}S_{1}), \label{29n}$$ to %100 by letting $$K=-\frac{i\pi E}{\kappa|_{H}}. \label{30n}$$ Consequently, the probability of the emitted (outgoing) gravitinos becomes $$\Gamma_{out}=\exp(-2\operatorname{Im}S_{2})=\exp(\frac{2\pi E}{\kappa|_{H}}). \label{31n}$$ Recalling the definition of the tunneling rate: $$\Gamma=\frac{\Gamma_{out}}{\Gamma_{in}}=\exp(\frac{2\pi E}{\kappa|_{H}}), \label{32n}$$ which is also equivalent to the Boltzmann factor: $\Gamma=\exp(-E/T)$, we read the HT of the TLWH as follows $$T_{H}=-\frac{\kappa|_{H}}{2\pi}, \label{33n}$$ which is a negative temperature. This result implies that if the trapping horizon remains in the past outer region, the wormhole throat would have a negative temperature [@Diaz1]. The phantom energy, which is the special case of the exotic matter could be the reason of that negative temperature [@Diaz; @Diaz1; @aliwh; @Diaz2; @Saridakis; @de1; @Helou]. On the other hand, when $K=0$ in the action (19), it is possible to obtain the positive temperature: $T_{H}=+\frac{\kappa |_{H}}{2\pi }$. Although, the latter remark contradicts with the previous results [@Diaz1; @aliwh] (and whence, one may easily get rid of the case of $K=0$), however Hong and Kim [@hong] showed that possibility of negative/positive temperature of the wormhole depends on the exotic matter distribution. Conclusion ========== In this work, we have studied the HR of the gravitino particles from the TLWH in 3+1 dimensions. TLWH has been introduced in the POTHG. We have analyzed the RSEs in the background of the TLWH with the help of HJ method. The probabilities of the emitted/absorbed gravitino particles from the trapped horizon of the TLWH have been computed. After comparing the obtained tunneling rate with the Boltzmann factor, we have recovered the standard HT of the TLWH, which is a negative temperature. This is the special condition in which the high-energy states are more occupied than lower-energy states [@sci1]. Another possibility of the negative temperature may originate from the exotic matter distribution of the wormhole [@hong]. Meanwhile, very recently it has been claimed by [@Helou2] that HR does not occur in the POTHG. In fact, the latter debatable remark is based on the study of [@ellis] stating that cosmic matter flux may turn the HR off. On the other hand, Hayward show that switching off the radiation causes the wormhole to collapse to a Schwarzschild BH [@hayward3]. In summary, gravitinos can tunnel through wormhole \[simply this can be thought as a wormhole with one entrance (BH) and one exit (white hole)\]. In such a case, gravitinos tunnel from the BH with positive temperature, while they tunnel through the white hole with negative temperature. Thus our calculations are based on the exit of the wormhole, just as the white hole case. 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SHEP 04-31\ CERN-PH/TH-2005-042\ \ [Daniel F. Litim]{}\ [*School of Physics and Astronomy\ University of Southampton, Southampton SO17 1BJ, U.K.*]{}\ [*Theory Division, CERN, CH-1211 Geneva 23.*]{}\ [**Abstract**]{}\ [ Several functional renormalisation group (RG) equations including Polchinski flows and Exact RG flows are compared from a conceptual point of view and in given truncations. Similarities and differences are highlighted with special emphasis on stability properties. The main observations are worked out at the example of $O(N)$ symmetric scalar field theories where the flows, universal critical exponents and scaling potentials are compared within a derivative expansion. To leading order, it is established that Polchinski flows and ERG flows – despite their inequivalent derivative expansions – have [*identical*]{} universal content, if the ERG flow is amended by an adequate optimisation. The results are also evaluated in the light of stability and minimum sensitivity considerations. Extensions to higher order and further implications are emphasized.]{} [**1. Introduction**]{}\ It is an experimental fact that physical correlation lengths are diverging in the vicinity of a critical point like a second order phase transition. The absence of dimensionful length scales implies scale invariance of physical correlation functions. Then, properties of physical systems close to a scaling regime or a critical point are characterised by universal exponents and scaling laws [@Zinn-Justin:1989mi]. In quantum field theory, functional methods like the renormalisation group are important tools in the study of strongly coupled systems and critical phenomena. The Wilsonian renormalisation group is based on the successive integrating-out of momentum degrees of freedom from a path integral representation of the theory, thereby interpolating between a given classical theory and the full quantum effective action [@Polchinski; @continuum; @Ellwanger:1994mw; @Morris:1994qb] (for reviews, see [@reviews]). The strength of these methods is their flexibility when it comes to approximations, in particular for theories with strong couplings or large correlation lengths. Furthermore, powerful optimisation criteria are available to increase the reliability within given truncations [@Litim:2000ci; @Litim:2001up; @Litim:2001fd]. Wilsonian flows play an important role in the study of universal scaling phenomena in gauge theories and gravity [@Pawlowski:2003hq; @gauge; @gravity]. In this Letter, we compare different implementations of a Wilsonian cut-off, the Exact Renormalisation Group based on an infrared momentum cut-off for the full effective action $\Gamma$, and the Polchinski renormalisation group based on an ultraviolet momentum cut-off for the action $S$. Both approaches correspond to exact flows, meaning that the endpoint of the fully integrated flow is given by the physical theory. This equivalence, in general, is lost in given truncations due to the qualitative differences in these approaches. We analyse the structural differences both in the full flows and within a derivative expansion. We also establish the remarkable result that the universal information encoded within the Polchinski renormalisation group to leading order in a derivative expansion is equivalent to the Exact Renormalisation Group, if the latter is amended by an adequate optimisation. Extensions to higher order and further implications of this result are equally discussed.\ [**2. Generalities**]{}\ Wilsonian flows are based on integrating-out momentum modes from a path integral representation of quantum field theory. A Wilsonian flow connects a short-distance effective action - typically the classical action - with the full physical theory. Given that the main physical information is contained in the integrated flow, the key properties of the different implementations proposed in the literature deserve a more detailed study. The Exact Renormalisation Group is based on a cutoff term $\Delta S_k=\s012\int\, \phi\, R\,\phi$, added to the Schwinger functional. The operator $R(q)$ introduces a momentum cutoff at momentum scale $q^2\approx k^2$. This induces a $k$-dependence on the level of the effective action. In its modern form, the flow for an effective action $\Gamma_k$ for bosonic fields $\phi$ is given by the simple one-loop expression [@continuum; @Ellwanger:1994mw; @Morris:1994qb] \[ERG\] \_t\_k\[\] = \_t R Here, $t\equiv\ln k$ is the logarithmic scale parameter, and the trace denotes a loop integration and a sum over fields and indices, and $\Gamma_k^{(2)}[\phi](p,q)\equiv \delta^2\Gamma_k/\delta\phi(p)\delta\phi(q)$. The regulator $R$ can be chosen freely, though within a few restrictions which ensure that the flow is well-defined, thereby interpolating between an initial action in the ultraviolet (UV) and the full quantum effective action in the infrared (IR) [@reviews]. In momentum space, the flow receives its main contributions for momenta in the vicinity of $q^2\approx k^2$, because large momentum modes are suppressed by $\partial_t R$, and small momentum modes are suppressed because $R$ is an IR cutoff. The flow depends on fields and couplings only through the full propagator. More generally, any exact RG flow for $\Gamma_k$ with a one-loop structure always depends linearly on the full propagator, and hence on the inverse of $\Gamma^{(2)}_k$ [@Litim:2002xm]. The linear dependence on the full propagator implies that an exact flow for $\Gamma_k$ is small whenever the full propagator is quantitatively small in the momentum regime where $\partial_t R$ is peaked, $e.g.$ for large fields or strong coupling. Hence, the Wilsonian flow is essentially local both in field- and momentum-space. Owing to this structure, the flow is amiable to truncations local in the fields (vertex functions) in the range where is non-trivial. An integration of the flow in given truncations requires a good control over the full propagator. The stability of – in the regions of momentum- and field-space where it is non-trivial – is controlled by the cutoff propagator, and hence by the momentum cutoff $R$ [@Litim:2000ci]. Therefore, the convergence of truncated flows can be increased by appropriate choices of $R$. Efficient optimisation criteria based on the flow are available [@Litim:2000ci; @Litim:2001up; @Litim:2001fd]. A different version of an exact renormalisation group has been introduced by Polchinski [@Polchinski], and is based on an ultraviolet regulator $K(q^2/\Lambda^2)$ for propagators in the path integral, where $\Lambda$ denotes the ultraviolet scale parameter. The Polchinski flow equation for the Wilsonian action $S_\Lambda[\phi]$ for a scalar field theory is given by [@Ball:1995ji] \[PRG\] \_t S\_= 12 \_t P\_(q) , where $S^{(1)}[\phi](q)\equiv \delta S[\phi]/\delta\phi(q)$, $P_\Lambda(q)=K(q^2/\Lambda^2)/q^2$ and $t\equiv \ln \Lambda$. The cutoff function $K$ is chosen such that high momentum modes are suppressed, $K(q^2/\Lambda^2\to\infty)\to 0$. The scale dependence introduced via $K$ induces the scale dependence of $S_\Lambda[\phi]$ on $\Lambda$. By construction, the flow equation is exact in the same sense as the ERG flow . The Polchinski flow depends on the fields via functional derivatives of $S$ – linearly through the operators $S^{(2)}$ and $P^{-1}_\Lambda \,\phi\,S^{(1)}$, and non-linearly through $S^{(1)}\, S^{(1)}$. Each of these terms can grow large for large fields and/or large couplings, even in the momentum regime where $\partial_t P_\Lambda$ is non-vanishing. In general, they do not cancel amongst each other. Hence, the right-hand side of remains non-trivial in large parts of field space. This makes it more difficult, within given truncations, to identify stable flows or regions in field space where the flow remains small. On the other hand, the polynomial non-linearities in are simpler than those in as they do not involve an inverse of $S^{(n)}$, also leading to a smaller number of different terms in the flow for a $n$-point function $\partial_t S^{(n)}$ for sufficiently large $n$. These aspects have been considered as a major benefit of the Polchinski flow, $e.g.$ [@Ball:1995ji]. Below, we shall see that it is precisely the non-linear dependence of on the inverse of $\Gamma_k^{(2)}$ which implies stability of the flow, whereas the polynomial non-linearities in are responsible for less stable solutions. Flow equations based on a proper-time regularisation have received considerable interest recently [@Liao:1994fp; @Bohr:2000gp; @Litim:2001hk; @Litim:2002hj]. They derive from a proper-time regularisation of the effective action valid to one-loop order [@Liao:1994fp]. Any proper-time flow at momentum scale $k$ can be represented in the basis of \[PTRG\] \_t\_k\[\] = ( )\^m, or linear combinations thereof, where the parameter $m\ge 1$ characterises the momentum cutoff [@Litim:2001hk; @Litim:2002xm]. Path integral derivations of proper-time flows have been worked out in [@Litim:2001hk; @Litim:2002xm; @Litim:2002hj]. In their simplest form, they make use of background fields, where plain momenta $q^2$ in the cutoff is replaced by $\Gamma_k^{(2)}[\phi]$ evaluated at some background field [@Litim:2002xm]. Therefore, an exact proper-time flow contains additional flow terms proportional to $\partial_t \Gamma^{(2)}_k$ on its right-hand side [@Litim:2002hj]. Implicit to this approach is that differences between fluctuation and background field are neglected. The proportionality of to (powers of) the full propagator is responsible for an increased suppression of the flow at large fields or couplings with increasing $m$. In this respect, proper-time flows are similar to ERG flows and, consequently, allow for analogous optimisations [@Litim:2001up; @Litim:2001dt]. Given the established link between exact proper time flows and ERG flows, it suffices, in the remainder, to elaborate on the differences between and .\ [**3. Derivative expansion**]{}\ Both and are exact flows. In consequence, the physical content of the fully integrated flows should be identical. This equivalence, in general, cannot be maintained within specific approximations, unavoidable as soon as either method is applied to a non-trivial physical problem. In order to highlight the similarities and differences between and , we study both flows for an $O(N)$ symmetric real scalar field $\phi^a$, $a=1\cdots N$, in $d$ dimensions within a derivative expansion [@Golner:1986], which is the most commonly used expansion scheme for critical phenomena (see also [@Pawlowski:2003hq; @Blaizot:2004]). A priori, little is known about its convergence as there is no small expansion parameter associated to it [@Litim:2001dt]. The ERG flow for $\Gamma_k$, and the Polchinski flow for $S_\Lambda$ are linked by a Legendre transform and additional momentum-dependent field rescalings. This implies that derivative expansions for the ERG and the Polchinski RG are inequivalent. To leading order in the derivative expansion, an Ansatz for the effective action $\Gamma_k$ contains a standard kinetic term and the effective potential $U_k$, \[AnsatzGamma\] \_k=d\^dx (U\_k(|) + 12 \_\^a\_\_a ) , where $\bar\rho=\s012\phi^a\phi_a$. Introducing dimensionless variables $\rho=\bar\rho k^{2-d}$, $U(\bar\rho)=u(\rho)\,k^d$, and using , the flow equation for the effective potential is \[ERG-Flow\] \_t u+du-(d-2) u’ =(N-1) (\_1)+ (\_2) with $\omega_1=u'$ and $\omega_2=u'+2\rho u''$. The function $\ell(\omega)$ encodes the non-trivial flow, and reads \[Id\] () =v\_d\^\_0dy y\^[d/2]{} with $y\equiv q^2/k^2$, $r(y)=R(q^2)/q^2$, $\partial_t r(y) = -2 y r'(y)$, and $v^{-1}_d=2^{d+1}\pi^{d/2}\Gamma(d/2)$. The flow is a second order non-linear partial differential equation. All non-trivial information regarding the renormalisation flow and the regularisation scheme are encoded in the function . The momentum integration is peaked and regularised for large momenta due to the cutoff term $\partial_t r(y)$, and for small momenta due to $r(y)$ in the numerator. All terms on the left-hand side of are cutoff independent, and display the intrinsic scaling of the variables that have been chosen for the parametrisation of the flow. By making use of rescaling in the fields and in the effective potential, the numerical factor $\sim v_d$ can be removed. Rescalings of the fields and the infrared scale parameter cannot remove the explicit cutoff dependence in . The $R$-dependence of the flow can be characterised by appropriate moments of $R$ [@Litim:2000ci]. In [@Litim:2000ci; @Litim:2001up; @Litim:2001fd; @Litim:2001dt], ideas have been put forward to increase the stability and physical content of truncated RG flows by choosing ‘optimised’ regulators $R$. The main observation is that the infrared cutoff $R$, in addition to regularising the flow, also controls its convergence and stability properties. This fact entails that specific cutoffs lead to improved results already at a fixed order in a systematic expansion. An optimisation then corresponds to identifying the RG flows with best stability properties. As an example, consider the flow in an expansion in vertex functions about vanishing field. The truncated propagator is $G_k(q^2)=(q^2+R)^{-1}$. Its contribution to the flow is largest in the momentum range where $G_k(q^2)$ is maximal. As a function of momenta, the cutoff propagator in units of $k$ achieves the maximum $C^{-1}(R)=\max_{q^2/k^2}\ \left[G_k(q^2) \, k^2\right]$. Consequently, the flow displays an increased stability if the maximal propagator contributions $\sim C^{-1}(R)$ remain as small as possible, $e.g.$ for regulators $R$ for which the ’gap parameter’ $C(R)$ becomes maximal, \[Opt\] \_R C(R). This condition requires an appropriate normalisation for the cutoff. Hence, states that the gap $C(R)$ should be maximal with respect to the cutoff function $R$. To leading order, the gap criterion, and its solution, is independent of the specific theory studied. Note that is a very mild condition: it only fixes one parameter in $R$ out of infinitely many. This implies that the subspace of optimised cutoffs is still infinite dimensional. Optimised flows have a number of interesting properties: their radius of convergence for amplitude expansions is increased [@Litim:2000ci], they factorise thermal and quantum fluctuations in the flow [@Litim:2001up], they entail a minimum sensitivity condition [@Litim:2001fd], they improve the derivative expansion [@Litim:2001dt], they lead to a fast decoupling of heavy modes, and they lead to an improved approach to convexity for theories with spontaneous symmetry breaking. An important optimised cutoff is given by [@Litim:2001up] \[Ropt\] R\_[opt]{}(q\^2) = (k\^2-q\^2)(k\^2-q\^2). In momentum space, the cutoff is distinguished because it solves in the entire domain $q^2<k^2$, and not only at the minimum of the inverse cutoff propagator. Hence, implements the gap criterion in a global manner. In the space of all optimised cutoff propagators, the cutoff corresponds to the convex hull of optimised inverse cutoff propagators (cf. Fig. 1 in [@Litim:2001up]), reflecting the extremal property of . In more physical terms, the cutoff leaves the propagation of large momentum modes $q^2>k^2$ unchanged, $q^2+R\approx q^2$. In turn, the propagation of infrared modes with $q^2<k^2$ is cut off leading to an effective mass term $q^2+R\approx k^2$. When expressed in terms of , the flow becomes \[ERG-FlowOpt\] \_t u +d u -(d-2) u’ = +. The numerical factor $4v_d/d$ has been absorbed into the potential and the fields by an appropriate rescaling. Now we turn to the Polchinski flow. We use an Ansatz for $S_\Lambda[\phi]$ analogous to . For comparison with , we introduce the dimensionless effective potential $u(\rho)=U_\Lambda/\Lambda^d$ and the dimensionless field variable $\rho=\s012 \phi^a\phi_a \Lambda^{2-d}$ Then, for $N\neq 0$, the Polchinski flow reads [@Ball:1995ji; @Comellas:1997tf; @Comellas:1998ep] \[PRG-Flow\] \_t u -d u +(d-2) u’ = u’+02N u” -2(u’)\^2 . We also have performed a finite renormalisation of the fields and the potential. The crucial observation at this point is that the flow equation is cutoff independent. The main differences and similarities between the ERG flow , and the Polchinski flow are summarised as follows:\ $(i)$ [*Non-linearities—*]{}The essential nonlinearities in the flows and are very different. For the Polchinski flow, they reduce to a quadratic term $\sim \rho\cdot (u')^2$. For the ERG flow, the non-linearities appear solely in a denominator, a direct consequence of the structural form of . When expanded in powers of the fields, the flow contains all and arbitrarily high powers of $u'$, $2 \rho u''$ and products thereof.\ $(ii)$ [*Stability—*]{}The structure of the non-linearities influence the stability properties of the flows. In the strong coupling domain or at large fields, the non-trivial part of the ERG flow (the right-hand side of ) is small, effectively suppressed by powers of the propagator. The non-linearities of the Polchinski flow (the right-hand side of ) are unbounded for large fields and couplings.\ $(iii)$ [*Cutoff independence—*]{}The terms on the right-hand sides of and originate from the non-trivial flow of the potential and contain the essential non-linearities. Their structure constitutes the main qualitative difference between the two flows. The ERG flow depends on infinitely many moments of the regulator $R$. This can be seen explicitly by expanding the flow in powers of $\omega_1$ and $\omega_2$. On a scaling solution, the Polchinski flow is fully scheme independent. This is in marked contrast to the ERG flow, where the regulator dependence cannot be removed by rescalings of the fields. A weak scheme dependence may persist even after an integration of the truncated flow.\ 0.0006 (1000,815) (320,400)[ ${\nu_{\rm opt} = 0.649562\cdots}$]{} (320,350)[ ${\nu_{\rm phys} \approx 0.63}$]{} (870,440)[ ${\nu_{\rm min}}$]{} (870,480)[ ${\nu_{\rm sharp}}$]{} (870,670)[ ${\nu_{\rm max}}$]{} (440,230)[ $R$]{} (40,780)[ $\nu$]{} .04 -1.8cm [**4. Universality**]{}\ Next, we switch to $d=3$ dimensions and compare the Polchinski and ERG flows on a quantitative level in the vicinity of a scaling solution, the Wilson-Fisher fixed point. The numerical values for universal critical exponents from Polchinski flows, due to their scheme independence to leading order in a derivative expansion, can be seen as a benchmark test for any other approach in the same approximation. For ERG flows, this check is non-trivial since depends on the cutoff. Universal critical exponents $\nu$ and subleading corrections-to-scaling exponents have been deduced in the literature from the non-trivial scaling solution $\partial_t u'_*=0$ as eigenvalues $\lambda$ of small perturbations $\delta u'$ around the fixed point, $\partial_t(u'_*+\delta u')=\partial_t\ \delta u'=\lambda\ \delta u'$ ($\nu=-1/\lambda_0$, where $\lambda_0$ is the single negative eigenvalue). The flow $\partial_t\ \delta u'$ and the eigenperturbations $\delta u'$ depend on the scaling solution $u'_*$. We begin with the ERG flow . The maximal ranges of attainable values for the critical exponents $\nu(R)$, for all $N$, has been discussed in [@Litim:2002cf]. The main result is depicted in Fig. 1 for $N=1$, which contains $\sim 10^3$ data points for $\nu(R)$, based on qualitatively different classes of cutoff functions including exponential, compact and algebraic ones, combinations thereof, discontinuous cutoffs and cutoffs with sliding scales. The main result represented by Fig. 1 is that the range of values for $\nu(R)$ is bounded both from above and from below. The numerical value at the upper bound \[max\] \_[max]{}=\_[R]{} (R), corresponds to the large-$N$ limit $\nu =1$. The upper boundary is achieved for a Callan-Symanzik type flow with mass-like regulator $R\to k^2$. In this limit, the corresponding flow ceases to be a Wilsonian flow in the strict sense because the momentum integration in is no longer cut-off in the ultraviolet limit. In the light of the optimisation, these flows have poor convergence and stability properties, and do not represent solutions to . This behaviour is equally reflected in the increasingly poor numerical convergence of solutions to for cutoffs with $\nu$ in the vicinity of . Conversely, in the vicinity of the lower boundary \[nu-Opt\] \_[min]{}=\_[R]{} (R), the flows have good convergence and stability behaviour. Results $\nu(R)$ from generic optimised flows are typically less than 1% away from the lower boundary . Hence, the numerical value $\nu_{\rm min}$ and all regularisations leading to it, are distinguished. Most interestingly, the minimum is achieved for the cutoff , \[min\] \_[opt]{}\_[min]{}. Therefore, the value has maximal reliability in the present truncation and is taken as the physical prediction to this order. Fig. 1 can also be interpreted in the light of the principle of minimum sensitivity (PMS) [@Stevenson:1981vj]. We emphasize that the latter is only applicable because $\nu(R)$ is globally bounded. Then, the PMS condition corresponds to the choice of cutoffs $R_{\rm PMS}$, for which $\delta \nu(R)/\delta R$ vanishes. For one-parameter families of cutoffs $R(b)$ parametrised by $b$, the PMS condition often has several solutions for $b_{\rm PMS}$ and $\nu_{\rm PMS}$, $e.g.$ [@Liao:1999sh; @Litim:2001fd]. Requiring that $\nu(R)$ is ‘globally’ extremal with respect to the regularisation identifies both boundaries and as solutions of a global minimum sensitivity condition. Here, ‘global’ refers to the fact that the extrema in Fig. 1 are achieved within the entire space of regulators $R$, and not just ‘locally’ for some $n$-parameter subclasses thereof (see [@Litim:2002cf]). Hence, the PMS condition by itself, neither locally nor globally, is sufficient to provide a unique physical prediction for $\nu$. In turn, the optimisation condition singles out a unique prediction: locally, for one-parameter families of cutoffs, it leads to values for $\nu$ in the vicinity of [@Litim:2001dt]; globally, the value is singled out straightaway. Hence, the optimisation which has let to the choice is equivalent to a global extremisation of $\nu(R)$. This provides an explicit example for the more general result of [@Litim:2001fd], which states that the optimisation entails a minimum sensitivity condition, while the converse, in general, is not true. ---------------- --------------------------------- ------------------------------------ $\quad N\quad$ $\quad{\rm Polchinski}\quad$ $\quad\quad R_{\rm opt}\quad\quad$ \[.5ex\] $0$ ${}\quad$ ${}\quad$ — $\ 0.592083^{a}\ {}$ $1$ ${}\quad$ $\ 0.64956^{c,e} {}$ $\ 0.649562^{a}\ {}$ $2$ ${}\quad$ $\ 0.7082^{d}\ \ {}$ $\ 0.708211^{a}\ {}$ $3$ ${}\quad$ $\ 0.7611^{d}\ {}$ $\ 0.761123^{a}\ {}$ $4$ ${}\quad$ $\ 0.8043^{d}\ {}$ $\ 0.804348^{a}\ {}$ \[-1ex\] ---------------- --------------------------------- ------------------------------------ : Critical exponent $\nu$ (see text). We continue with a comparison of all critical exponents that have been published to date within both the optimised ERG flow and the Polchinski flow . These are: the critical exponent $\nu$ (Table I), the smallest correction-to-scaling exponent $\omega$ (Table II) and the asymmetric corrections-to-scaling exponent $\omega_5$ (Table III). Based on the optimised ERG flow , the exponents $\nu$ and $\omega$ have been given for all $N$ in [@Litim:2002cf] with up to six significant figures $(a)$. The exponent $\omega_5$ has been computed in [@Litim:2003kf] for $N=1$ $(b)$. Cutoff independence of the Polchinski flow implies unique results for universal eigenvalues. The critical exponents $\nu$ and $\omega$ have been computed with up to four significant digits in [@Ball:1995ji] for $N=1$ $(c)$, and in [@Comellas:1997tf] for $N=1\cdots 4$ $(d)$. Results up to five digits for the exponents $\nu, \omega$ and $\omega_5$ have recently been stated in [@Bervillier:2004mf] for $N=1$ $(e)$. Except for $N=\infty$, there are no published results based on the Polchinski flow for $N>4$. However, it has recently been indicated [@Bervillier:2005za] that the results from Polchinski flow and optimised ERG flow also agree for $N>4$. In the large $N$ limit, the spread of $\nu(R)$ with $R$ is absent, and the results for critical exponents becomes unique, $\nu(R)=1$, and $\omega_n(R)=2n-1$, $n=1,2,\cdots$ in agreement with the corresponding limit of the Polchinski flow [@D'Attanasio:1997ej]. Hence, it is most remarkable that all universal critical exponents computed either from the Polchinski flow or from the optimised ERG flow, agree to all significant figures, for the leading and subleading critical exponents, and for different universality classes! This high degree of coincidence leads to the important conjecture that the universal content of the partial differential equations and is equivalent [^1].\ ---------------- --------------------------------- ------------------------------------ $\quad N\quad$ $\quad {\rm Polchinski}\quad$ $\quad\quad R_{\rm opt}\quad\quad$ \[.5ex\] $0$ ${}\quad$ ${}\quad$ — $\ 0.65788^{a}\ \ \ {}$ $1$ ${}\quad$ $\ 0.65574^{c,e}\ {}$ $\ 0.655746^{a}\ {}$ $2$ ${}\quad$ $\ 0.6712^{d}\ {}$ $\ 0.671221^{a}\ {}$ $3$ ${}\quad$ $\ 0.6998^{d}\ {}$ $\ 0.699837^{a}\ {}$ $4$ ${}\quad$ $\ 0.7338^{d}\ {}$ $\ 0.733753^{a}\ {}$ ---------------- --------------------------------- ------------------------------------ : Subleading correction-to-scaling exponent $\omega$. [**5. Stability**]{}\ Next, we analyse the locality and stability structure of flows and show that the non-universal properties of RG flows in the vicinity of a scaling solution are vastly different. Consider the fixed point solution itself, which is non-universal and not measurable in any experiment. Using , the nontrivial scaling solution with $u'_\star \neq$ const. and $\partial_t u'_\star=0$ of the Polchinski RG obeys the differential equation \[FlowPRGupScal\] 2 u\_’ - (d-2)u\_” =2(u\_’)\^2 - 02N u\_”’ -(1+02N-4u\_’)u\_”. For large fields, the scaling potential behaves as \[solutionPRGExpansionLarge\] u\_() +[subleading]{}. An analytical solution for $N=\infty$ has been given in [@Kubyshin:2001gz]. On the level of the RG flow, this behaviour stems from a cancellation between the canonical scaling of the potential and its non-linear renormalisation. From , we conclude that the non-trivial quantum contributions to the Polchinski flow diverges like \[PRG-largefield\] \_t u -d u +(d-2) u’ +[subleading]{} for large fields $\rho$ close to a scaling solution. Within the optimised ERG, the non-trivial scaling solution, using , obeys \[FlowPotential3\] 2 u’\_- (d-2)u”\_= -(N-1) - . Analytical solutions for the limit $N=\infty$ have been given in [@Litim:1995ex]. For $N\neq\infty$, and in the vicinity of $\rho=0$, the scaling solution can be obtained analytically as a Taylor expansion in the field [@Litim:2002cf]. In the limit of large fields $\rho\gg 1$, we find \[solutionExpansionLarge\] u\_()\^[1+]{} +[subleading]{} for arbitrary regulator, where $\alpha=2/(d-2)$ is positive for $d>2$. Here, the large-field behaviour is solely due to the canonical scaling dimension of the fields. From , and for large fields $\rho \gg 1$, we conclude that the non-linear part of the ERG flow for the potential behaves as \[ERG-largefield\] \_t u + du -(d-2) u’ \^[-]{}+[subleading]{}. Hence, the right-hand side of is suppressed for all $d>2$. ---------------- ------------------------------- ------------------------------------ $\quad N\quad$ $\quad {\rm Polchinski}\quad$ $\quad\quad R_{\rm opt}\quad\quad$ \[.5ex\] $1$ $\ 1.8867^{e}\ {}$ $\ 1.8867^{b}\ {}$ ---------------- ------------------------------- ------------------------------------ : Asymmetric correction-to-scaling exponent $\omega_5$. More generally, the result holds for generic ERG flows where the right-hand side of decays $\propto 1/\omega$ for large $\omega$. The power-law behaviour is altered for cutoffs which effectively introduce non-localities due to their momentum structure, $e.g.$ the mass-like cutoff (no large momentum decay), the sharp cutoff, or cutoffs with an algebraic large-momentum decay like the quartic cutoff $R\sim k^4/q^2$, [*e.g.*]{} [@Morris:1997xj]. They lead, respectively, to with a large-field behaviour $\propto \rho$, $\propto \ln\rho$ and, in three dimensions, $\propto \rho^{-3/2}$. The different power law exponents $\alpha(R)$ as a function of the cutoff are displayed in Fig 2 for three dimensions. The minimum \_[min]{}=\_R (R) is achieved for Callan-Symanzik type flows and the Polchinski flow, $\alpha_{\rm min}=-1<0$. A negative $\alpha$ also indicates that an additional renormalisation of the flow is necessary due to an insufficiency in the integrating-out of momentum modes. This is well-known for Callan-Symanzik type flows [@reviews]. We stress, however, that the set of flows with negative $\alpha$ is of measure zero; generic ERG flows have positive $\alpha$. The sharp cutoff marks the boundary between ERG flows with insignificant ($\alpha>0$) and significant ($\alpha<0$) contributions for large fields, and hence the boundary between flows which are essentially local, respectively non-local, in the fields. The maximum \_[max]{}=\_R (R) is achieved for generic ERG flows including optimised ones, $\alpha_{\rm max}=2/(d-2)>0$. Note that the few ’non-local’ ERG flows with $\alpha$ in the range $[\alpha_{\rm min},\alpha_{\rm sharp}]$ lead to critical exponents $\nu$ in Fig. 1 in the range $[\nu_{\rm max},\nu_{\rm sharp}]$, whereas all ’local’ ERG flows with $\alpha$ within $[\alpha_{\rm sharp},\alpha_{\rm max}]$ – the overwhelming majority of all ERG flows – lead to values within the narrow window $[\nu_{\rm sharp},\nu_{\rm min}]$ [@Litim:2002cf]. We conclude that flows with underlying non-localities have the tendency to deviate strongly from the physical theory and display a strong cutoff dependence, while local flows display only a weak cutoff dependence, thereby remaining close to the physical theory. In the Ising universality class, the exponent $\nu$ from non-local (local) flows deviates between 10-50% (3-10%) from the physical value. This quantifies the link between the locality structure of the flow, its stability, and its vicinity to the physical theory. Summarising, unlike the universal parts the non-universal scaling solutions derived from ERG or Polchinski flows are vastly different. This result also extends to the non-universal eigenperturbations at criticality. The non-trivial quantum corrections to the flow at large fields are strongly suppressed for generic ERG fows , while they remain large in the Polchinski case . This is a direct consequence of the structural differences in the basic flows and . Despite of having the same universal content, in the light of Fig. 2 the optimised flow and the Polchinski flow have maximally distinct locality structures. Note that good locality properties of flows are at the root for stable numerical integrations, and the quantitative smallness of quantum corrections in large domains of field space improves the convergence of RG flows. Based on the result that ERG flows with increasing non-localities show an increasingly strong cutoff dependence, we expect that Polchinski flows display a similar behaviour as soon as the leading-order degeneracy with respect to the cutoff is lifted by higher order operators in extended truncation.\ 0.0006 (1000,815) (900,400)[$\alpha_{\rm min}$]{} (900,500)[$\alpha_{\rm sharp}$]{} (900,710)[$\alpha_{\rm max}$]{} (450,230)[ $R$]{} (40,800)[$\alpha$]{} .04 -1.8cm [**6. Extensions**]{}\ A global analysis of critical exponents in the full space of cutoffs has only been performed to leading order in the derivative expansion, which is the most important order as higher order effects should be suppressed proportional to the anomalous dimension $\eta$ of the order of a few percent. Still, it is useful to briefly review the results achieved so far beyond leading order in the light of the preceeding discussion. Within Polchinski flows, the leading order scheme independence is lost as soon as higher order derivative operators are taken into account [@Ball:1995ji]. To order ${\cal O}(\partial^{n})$, the flow depends explicitely on $n$ scheme-dependent parameters which cannot be removed by further rescalings. To order ${\cal O}(\partial^{2})$, a part of the cutoff dependence has been probed for different projections on the anomalous dimension, [*i.e.*]{} [@Ball:1995ji; @Comellas:1998ep; @Bervillier:2004mf; @Bervillier:2005za]. All published results for $\nu$ and $\eta$ to order ${\cal O}(\partial^{2})$ have in common that the spurious dependence on remaining unphysical cutoff parameters is monotonous, without displaying local extrema. The range of numerical results includes the physical values. Unfortunately, none of the truncations admits a minimum sensitivity condition, and further conditions have to be invoked to remove the scheme dependence. From a structural point of view, the comparatively strong cutoff dependence beyond leading order is not unexpected and fully in line with the stability considerations detailed in the preceeding section. It remains to be seen whether the next order in the expansion has a stabilising effect on the series [@Bervillier:2005za]. Within the ERG, parts of the cutoff space have been probed quantitatively to order ${\cal O}(\partial^2)$ [@Tetradis:1993ts; @VonGersdorff:2000kp; @Canet:2002gs] and to order ${\cal O}(\partial^4)$ [@Canet:2003qd]. Two observations have to be made in the present context: first of all, the critical exponents $\nu$ and $\eta$ remain bounded, similar to Fig. 1, in the parameter range considered. Furthermore, they attain local extrema as functions of the cutoff indicating the existence of a global boundary equivalent to those displayed in Fig. 1. The boundedness of $\nu(R)$ within a given order of the derivative expansion is an important ingredient in the convergence of the series. Secondly, the set of stable flows, as identified through the optimisation, remains stable to higher orders: typically, flows with regulators $R$ such that $\nu(R)$ to leading order is in the vicinity of $\nu_{\rm opt}$ remain in the vicinity of the local extrema of $\nu(R)$ even beyond leading order. This confirms the validity of the underlying optimisation. For sufficiently stable flows, higher order corrections remain quantitatively small, thus increasing the convergence of the derivative expansion [@Litim:2001dt]. Finally, we point out that the qualitative differences beyond leading order are also reflected in the explicit cutoff dependence of either flow. ERG flows depend only on ’global’ properties of the regulator $R$ through specific momentum integrals $a_n(R)$ of the form $$a_n(R)\sim \int_0^\infty dy \frac{-y^{d/2+1-n}\, r'(y)}{[1+r(y)]^n}$$ and similar [@Litim:2000ci]. This follows from expanding the flow in powers of $u'$ and $u'+2\rho u''$. The coefficients $a_n(R)$ receive their main contributions for momenta $q^2\approx k^2$. Therefore, small changes in the momentum behaviour of the cutoff $R\to R+\delta R$ induce small changes in all coefficients $a_n(R)$. Furthermore, many different cutoff functions $R$ can lead to equivalent sets of coefficients $a_n$. Hence, the precise small- or large-momentum structure of $R$ is at best of subleading relevance to the flow. In turn, Polchinski flows in a derivative expansion depend both on ’global’ and on ’local’ characteristics of the regulator $K(q^2/\Lambda^2)$, and in particular on its derivatives at vanishing momenta. For example, the Polchinski flow for the wave function renormalisation depends on the ratio $B(K)= K''(0)/K'(0)^2$, and the anomalous dimension at criticality $\eta$ is even proportional to $B$ [@Ball:1995ji] (see also [@Comellas:1998ep; @Bervillier:2004mf; @Bervillier:2005za]). Small modifications in the cutoff $K\to K+\delta K$ at small momenta can induce large changes in $B$ including its sign, and, therefore, induce comparatively large alterations in the flow and the physical observables. These structural differences can be seen as a further indication for the increased stability of ERG flows as opposed to Polchinski flows.\ [**7. Discussion and conclusions**]{}\ We compared several functional renormalisation group equations based on Wilsonian cutoffs. The main structural differences between Polchinski flows and ERG flows are due to the non-linearities of their right-hand sides. In the literature, it has sometimes been argued that the simple non-linearities of the Polchinski flow as opposed to those of ERG flows are a benefit to the formalism. Here, we arrived at the opposite conclusion: the non-linearities in ERG flows involve the inverse of $\Gamma^{(2)}_k$, and, therefore, guarantee that the flow remains small in large regions of field and momentum space. This structure implies that ERG flows are amiable to systematic expansions ($e.g.$ in vertex functions) and allow for a straightforward optimisation, since truncational variations in the flow are suppressed in large parts of field space. On the other side, the non-linearities of the Polchinski flow appear to be algebraically simpler. The price to pay is that the flow remains non-trivial in a larger domain of field space, including the region of large fields. These differences in the locality and stability behaviour favour the flows in particular for numerical implementations. The structural differences have been made explicit within a derivative expansion. To leading order, critical exponents from the Polchinski flow are scheme independent and, therefore, serve as a benchmark test for functional RG flows in corresponding approximations. We have established the remarkable result that the optimised ERG flow and the Polchinski flow have identical universal eigenvalues, for all $O(N)$ symmetric scalar theories, for the leading and subleading critical exponents, and for the asymmetric corrections-to-scaling exponent! This equivalence is non-trivial in that the corresponding flows and differ substantially, both in their structure and in their non-universal scaling solutions. We conjecture that this result extends to all universal observables to leading order in the derivative expansion. This equivalence, however, does not persist in an obvious manner beyond the leading order, where universal observables from Polchinski flows depend strongly on remaining unphysical parameters. This is in marked contrast to the results from ERG flows which remain bounded, similar to the leading order. The comparatively large cutoff sensitivity and the nontriviality of the Polchinski flow for large fields – a consequence of the non-linearities in – require a better conceptual understanding before definite conclusions can be drawn concerning its convergence properties. For ERG flows, on the other hand, a coherent picture has emerged. In given truncations, an appropriate optimisation leads to an increased stability of the flow. The cutoff dependence of physical observables is, thereby, largely reduced to a small range in the vicinity of the physical theory. This pattern is established quantitatively within a derivative expansion, both to leading order and beyond. 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--- abstract: '[The calculation of autocorrelation functions represents a routinely used tool to characterise quantum states of light. In this paper, we evaluate the $g^{(2)}$ function for detected photons in the case of mesoscopic multi-mode twin-beam states in order to fully investigate their statistical properties starting from measurable quantities. Moreover, we show that the second-order autocorrelation function is also useful to estimate the spurious effects affecting the employed Silicon-photomultiplier detectors.]{}' author: - Giovanni Chesi - Alessia Allevi - Maria Bondani title: 'Autocorrelation functions: a useful tool for both state and detector characterisation' --- Introduction ============ The autocorrelation functions, formally introduced by Glauber in 1963 [@glauber63], represents one of the standard tools used to characterise quantum states of light, such as to discriminate between bunched and antibunched light [@loudon]. Correlation functions are usually evaluated at the single-photon level: The light state under examination is divided at a balanced beam splitter and the two outputs are detected by means of two avalanche diodes [@ave10; @elsaesser; @somaschi]. We have recently demonstrated that the same scheme can also be adopted to characterise mesoscopic optical states, namely pulsed states containing sizeable numbers of photons in each pulse [@antibunching; @QMETRO17]. Instead of single-photon detectors, photon-number-resolving (PNR) ones are needed in such a case [@JMO; @harder16]. For instance, in our work we have employed hybrid photodetectors to prove a behaviour of sub-Poissonian states analogous to antibunching [@antibunching].\ In the original Glauber’s definition, the autocorrelation functions are expressed in terms of normal ordered operators. Nevertheless, in practical situations, analogous definitions expressed in terms of measurable quantities could be desirable [@pra12]. Indeed, the link between the autocorrelation functions for photons and those for, e.g., detected photons can be easily found provided the model for the detection process is known.\ Moreover, the calculation of these function can also be useful to extract information about the detectors used to reveal the light under study.\ In this paper, we emphasize the versatility of the $g^{(2)}$ autocorrelation functions for both the above-mentioned purposes. On the one hand, we show that from the evaluation of autocorrelations we can extract some information about the features of the employed detectors. On the other hand, we prove that correlation functions represent a useful criterion for the characterisation of quantum correlations as well as of nonclassical states. Characterisation of Silicon photomultipliers {#sec1} ============================================ As already stated in the Introduction, our measurements have been performed in the mesoscopic intensity regime. In such a case, the $g^{(2)}$ function can be easily evaluated by direct detection of the state under examination. As to the detectors, we decided to employ a commercial class of PNR detectors, namely Silicon photomultipliers (SiPMs). These detectors consist of avalanche diodes, called cells, arranged in a matrix of pixels connected to a common output. Every diode is reverse-biased and operates in Geiger-M$\ddot{\rm u}$ller regime [@akindinov; @bondarenko; @saveliev; @piemonte; @renker]. During the last fifteen years, SiPMs have been employed in many scientific applications, ranging from particle physics experiments to positron emission tomography and biomedical research. Thanks to their structure, SiPMs are endowed with a good photon-number-resolving capability, which makes them appealing for Quantum Optics, to detect mesoscopic quantum states of light [@ramilli]. However, this possibility has been till now prevented by the occurrence of stochastic spurious events, such as dark counts and optical cross-talk [@afek; @kala12], as well as by a limited quantum efficiency. In the following, we consider the new generation of SiPMs produced by Hamamatsu, in which the cross-talk probability has been lowered and the quantum efficiency increased [@hama].\ According to the model presented in Refs. [@ramilli; @chesi], the response of a SiPM detector can be seen as the convolution of different terms, corresponding to the different detector features.\ First of all, we assume that the detection process is described by a Bernoullian distribution $B_{m, n}(\eta)$. This means that the distribution of detected photons, $P_{\rm el}(m)$, is linked to that of photons, $P_{\rm ph}(n)$, through $$\begin{aligned} {\rm P_{el}}(m) &=& \sum_{n=m}^\infty B_{m,n}(\eta) {\rm P_{ph}}(n) \nonumber\\ &=& \sum_{n=m}^\infty {{n}\choose{m}} \eta^m (1- \eta)^{n-m} {\rm P_{ph}}(n),\end{aligned}$$ where $\eta$ is the detection efficiency, $n$ is the number of incident photons, and $m$ that of detected photons. Dark counts are spurious avalanches triggered by thermally generated charge carriers. Since they are independent and uncorrelated events, their statistics is Poissonian $$\label{Pdc} {\rm P_{dc}}(m) = \frac{(\langle m \rangle_{\rm dc})^m}{m!}\exp{(-\langle m \rangle_{\rm dc})},$$ where $\langle m \rangle_{\rm dc}$ is the mean value of dark counts.\ Second, we consider optical cross-talk events [@gola] that arise when the electrons accelerated during the avalanche process produce brehmsstralung radiation that may trigger avalanches in a neighbouring cell. Hereafter, we assume the cross-talk probability distribution to be [@ramilli] $$C_{k,l}(\epsilon) = {{l}\choose{k-l}} \epsilon^{k-l}(1- \epsilon)^{2l-k},$$ where $\epsilon$ is the probability that the avalanche from a cell triggers one neighbour cell, $l$ is the number of photo-triggered avalanches and of dark counts, and $k$ is the resulting number of avalanches including cross talk.\ Finally, we assume that the amplification (both internal and external) of the detector is described by a multiplicative factor, $\gamma$, so that the single-shot output of the detection chain is $x_{\rm out} = \gamma k$. The distribution of the detector output is given by the convolution of all the previous terms: $$\label{Pout} {\rm P}(x_{\rm out}) = \gamma \sum_{m=0}^k C_{k,m}(\epsilon) \sum_{j=0}^m {\rm P_{dc}}(j) {\rm P_{el}}(m-j).$$ By using the two moments of the distribution in Eq. (\[Pout\]), it is possible to define the $g^{(2)}$ function for the SiPM output as $$g^{(2)}( x_{\rm out}) \equiv \frac{\langle x_{\rm out}^2 \rangle}{\langle x_{\rm out} \rangle^2} = \frac{\langle (\gamma k)^2 \rangle}{\langle \gamma k \rangle^2} \equiv g^{(2)}(k) = \frac{\sigma^2(k)}{\langle k \rangle^2} + 1, \label{g2out}$$ where $$\begin{aligned} \langle k \rangle &=& (1 + \epsilon) (\langle m \rangle + \langle m\rangle_{\rm dc})\\ \sigma^2(k) &=& (1+\epsilon)^2 \left(\sigma^2(m)+\langle m\rangle_{\rm dc}\right) + \epsilon(1+\epsilon)(\langle m \rangle + \langle m\rangle_{\rm dc})\nonumber\end{aligned}$$ are the mean value and the variance of $k$, respectively. Note that $g^{(2)}(k)$ can be also linked to the expression of the autocorrelation function for photons, $$g^{(2)}(n) = \frac{\langle :n^2: \rangle}{\langle n \rangle^2}.$$ In fact, it can be demonstrated that Eq. (\[g2out\]) can be re-written as $$g^{(2)}(k) = 1 + (g^{(2)}(n) -1)\left( 1- \frac{(1+ \epsilon)\langle m \rangle_{\rm dc}}{\langle k \rangle} \right)^2 + \frac{1}{\langle k \rangle} \frac{1+3\epsilon}{1+\epsilon}. \label{g2new}$$ In the Introduction we claimed that the evaluation of the autocorrelation function expressed in terms of measurable quantities can help the determination of the detector features, such as the mean value of dark counts, $\langle m \rangle_{\rm dc}$, and the cross-talk probability, $\epsilon$. Moreover, the expression in Eq. (\[g2new\]) also contains information about the light through the term $g^{2}(n)$. For instance, in the case of multi-mode thermal light with $\mu$ modes equally populated $g^{2}(n) = 1 + 1/\mu$.\ To experimentally prove these statements, we generated a multimode twin-beam (TWB) state by means of parametric down conversion (see Fig. \[setup\]). The pump field was the fourth harmonic (at 262 nm, 3.5-ps pulse duration) of a Nd:YLF laser regeneratively amplified at 500 Hz, whose pulses were sent to a $\beta$-barium-borate nonlinear crystal (BBO2, cut angle = 46.7 deg, 6-mm long). ![Sketch of the experimental setup. See the text for details. []{data-label="setup"}](setup1.pdf){width="0.7\columnwidth"} Two twin portions of TWB states were spatially and spectrally filtered by means of irises and interference filters centered at 523 nm, respectively. The two light components were then delivered to two SiPMs through two multi-mode fibers (600-$\mu$m core diameter). We used two commercial SiPMs (MPPC S13360-1350CS) produced by Hamamatsu [@hama1]. Such detectors are endowed with a moderate rate of dark count at room temperature ($\sim$ 140 kHz) and a low cross-talk probability ($\sim 2 \%$). Each detector output was amplified and integrated by means of two synchronous boxcar-gated integrators (SR250, Stanford Research Systems). In particular, we adopted a small value of gate width (10 ns) in order to keep the effects of dark count and delayed cross talk as small as possible. As shown in Fig. \[signal\], the chosen value corresponds to the width of the signal peak. ![Typical single-shot detector signal.[]{data-label="signal"}](tipicosegnale){width="0.7\columnwidth"} The experimental measurements were performed at different values of pump energy, which was modified through a half-wave plate (HWP) followed by a polarizing cube beam splitter (PBS). At each energy value, $10^5$ single-shot acquisitions were performed.\ As declared above, the generation of our TWB state is intrinsically multi-mode [@paleari; @pra12]. This means that each arm of TWB is described by a multi-mode thermal statistics. By imposing this condition, Eq. (\[g2new\]) reads as $$\label{g2mthXC} g^{(2)}(k) = 1 + \frac{1}{\mu} \left(1- \frac{(1+\epsilon) \langle m \rangle_{\rm dc}}{\langle k \rangle} \right)^2 + \frac{1}{\langle k \rangle} \frac{1+3 \epsilon}{1+ \epsilon}.$$ Note that, at variance with the autocorrelation function for photons, the maximum value of Eq. (\[g2mthXC\]) can be larger than 2. ![Autocorrelation function at different mean numbers of $k$ measured in each TWB arm. The two panels correspond to the two arms. Black dots + error bars: experimenta data; magenta circles: theoretical fitting curve according to Eq. (\[g2mthXC\]); blue line: classical boundary. The values of the mean error [@meanerror] with respect to red circles are 0.003 in the upper panel and 0.002 in the lower panel.[]{data-label="g2single"}](g2ch1 "fig:"){width="0.7\columnwidth"} ![Autocorrelation function at different mean numbers of $k$ measured in each TWB arm. The two panels correspond to the two arms. Black dots + error bars: experimenta data; magenta circles: theoretical fitting curve according to Eq. (\[g2mthXC\]); blue line: classical boundary. The values of the mean error [@meanerror] with respect to red circles are 0.003 in the upper panel and 0.002 in the lower panel.[]{data-label="g2single"}](g2ch2 "fig:"){width="0.7\columnwidth"} In the two panels of Fig. \[g2single\], we plot $g^{(2)}(k)$ as a function of the mean value of $k$ for each TWB arm. The experimental data are shown as black dots + error bars, whereas the theoretical fitting curve (magenta circles) was calculated according to Eq. (\[g2mthXC\]), in which we assumed $\mu = 1000$ (reasonable value for a TWB in our experimental conditions [@SiPMOL]) and left $\epsilon$ and $\langle m \rangle_{\rm dc}$ as free parameters. In particular, we got: $\epsilon = 0.008$, and $\langle m \rangle_{\rm dc} = 0.001$ in the first arm (upper panel) and $\epsilon = 0.007$, and $\langle m \rangle_{\rm dc} = 0.001$ in the second arm (lower panel). The absolute values of $\epsilon$ and $\langle m \rangle_{\rm dc}$ obtained from the fit are quite small. However, in order to quantify their relevance for the calculation of the second-order autocorrelation function, we evaluated the relative variation of $g^{(2)}(k)$ by expanding the function up to the first order of $\alpha$ $$\label{deriv} \frac{\Delta g^{(2)}(k)}{g^{(2)}(k)} = \left. \frac{1}{g^{(2)}(k)|}\right|_{\alpha = 0} \left. \frac{\partial g^{(2)}(k)}{\partial \alpha}\right|_{\alpha = 0} \Delta \alpha \equiv \beta \Delta \alpha,$$ where $\alpha$ is either $\epsilon$ or $\langle m \rangle_{\rm dc}$, both evaluated in 0. We note that, for any choice of the parameters, $\beta$ in Eq. (\[deriv\]) is always smaller than 1. Thus, since the quantities $\Delta \alpha$ vary in very narrow ranges, the variation of $g^{(2)}(k)$ is negligible. For this reason, in the following Section we do not consider the contribution of cross talk and dark count in the calculation of the $g^{(2)}$ function connected to other variables. Characterisation of nonclassical states of light {#sec2} ================================================ The TWB states generated as described in Sect. \[sec1\] are optical states endowed with nonclassical correlations. This nature can be proved by means of suitable nonclassicality criteria [@klyshko96; @short83; @vogel; @arimondo]. Among them, the most used is the noise reduction factor, $R$, which is defined as the ratio between the variance of the photon-number difference between the two twin arms and the shot noise level, namely $$\label{eq:NRF} R = \frac{\sigma^2(m_1 - m_2)}{\langle m_1 + m_2 \rangle}$$ Values of $R$ lower than 1 testify nonclassicality [@pra07]. To further characterise the TWB states, we evaluated the autocorrelation function for the photon-number difference, that is $g_{diff}^{(2)}(m) = \langle (m_1 - m_2)^2 \rangle / \langle m_1 - m_2 \rangle^2$. We note that this function can be expressed in terms of the noise reduction factor $R$ as $$\label{g2diffteo} g_{diff}^{(2)}(m) = 1+\frac{R~\langle m_1 + m_2 \rangle}{\langle m_1 - m_2 \rangle^2}.$$ In Fig. \[g2diff\] we plot the measured values of $g_{diff}^{(2)}(m)$ together with the expectation in Eq. (\[g2diffteo\]), in which we used the expression of $R$ for multi-mode thermal TWB states, namely $$\label{NRFeta} R = 1 - \frac{2 \sqrt{\eta_1 \eta_2}\sqrt{\langle m_1 \rangle \langle m_2 \rangle}}{\langle m_1 \rangle + \langle m_2 \rangle} + \frac{(\langle m_1 \rangle - \langle m_2 \rangle)^2}{\mu (\langle m_1 \rangle + \langle m_2 \rangle)},$$ where $\eta_1$ and $\eta_2$ are the quantum efficiencies in the two arms, calculated in the experimental values of $\langle m_j \rangle$, $\eta_j$ and $\mu$, with $j=1,2$ [@lamperti]. The experimental data shown in the figure are well superimposed to theory. Note that, since the values of $R$ are within the range (0.87-0.9), the values of $g_{diff}^{(2)}(m)$ are in general quite large, no matter the absolute values of $\langle m_1 \rangle$ and $\langle m_2 \rangle$. This fact emphasizes that the term $\langle m_1 - m_2 \rangle^2$ is quite small, thus proving a good balancing between the light detected in the two arms.\ While the evaluation of $g_{diff}^{(2)}(m)$ is useful to investigate the balancing between the numbers of photons detected in the two arms, the calculation of the analogous expression for photons, $i.e.$ $g_{diff}^{(2)}(n)$ can give information about the quantum nature of photon-number correlations. ![Measured $g_{diff}^{(2)}(m)$ as a function of the mean number of photons detected in each TWB arm. Black dots + error bars: experimental data; magenta circles: theoretical expectation according to Eq. (\[g2diffteo\]).[]{data-label="g2diff"}](g2diff){width="0.7\columnwidth"} By definition $$\label{g2nodiff} g_{diff}^{(2)}(n) = \frac{\langle:(n_1 - n_2)^2:\rangle}{\langle n_1 -n_2 \rangle^2} = \frac{\langle(n_1 - n_2)^2\rangle}{\langle n_1 -n_2 \rangle^2} - \frac{\langle n_1 + n_2\rangle}{\langle n_1 -n_2 \rangle^2}.$$ Assuming that the detection efficiency in the two arms of TWB is the same, $\eta_1 = \eta_2 = \eta$, it is possible to demonstrate that the relation between $g_{diff}^{(2)}(n)$ and $g_{diff}^{(2)}(m)$ reads as follows: $$\label{g2difflink} g_{diff}^{(2)}(n) = g_{diff}^{(2)}(m) - \frac{\langle m_1 + m_2\rangle}{\langle m_1 -m_2 \rangle^2}.$$ By using Eq. (\[g2diffteo\]) in Eq. (\[g2difflink\]), the autocorrelation function $g_{diff}^{(2)}(n)$ can be directly connected to the noise reduction factor $$\label{g2diffcrit} g_{diff}^{(2)}(n) -1 = (R-1)\frac{\langle m_1 + m_2\rangle}{\langle m_1 -m_2 \rangle^2}.$$ According to Eq. (\[g2diffcrit\]), the negativity of the quantity $[g_{diff}^{(2)}(n)-1]$ can be used as nonclassicality criterion as an alternative to $R<1$. In Fig. \[g2ndiff\] we show the values of $[g_{diff}^{(2)}(n) -1]$ calculated from the experimental data as black dots + error bars. In the same figure we also plot the theoretical expectation according to Eq. (\[g2diffcrit\]), in which $R$ was evaluated according to Eq. (\[NRFeta\]). The negativity of the plotted quantity proves the quantum nature of photon-number correlations.\ Since all the measured TWB states are nonclassically correlated, the employed SiPMs can be used to perform multi-photon conditioning operations in order to produce sub-Poissonian states of light [@laurat03; @laurat04; @ourj06; @Iskhakov16]. Indeed, we have already demonstrated that, when a certain number of photons is selected in one TWB arm, the distribution of photons in the other arm is narrower than a Poissonian distribution [@lamperti]. This fact can be quantified either by calculating the Fano factor, $F(m) = \sigma^2_m/\langle m \rangle$, of the obtained photon-number statistics or evaluating the $g^{(2)}$ function for photons. ![Measured $[g_{diff}^{(2)}(n)-1]$ as a function of the mean number of photons detected in each TWB arm. Black dots + error bars: experimental data; magenta circles: theoretical expectation according to Eq. (\[g2diffcrit\]).[]{data-label="g2ndiff"}](g2ndiff){width="0.7\columnwidth"} In fact, it is possible to demonstrate [@antibunching] that $$\label{eq:g2cond} g^{(2)}(n) - 1 = \frac{F(m)-1}{\langle m \rangle}.$$ In the case of conditional states obtained by multi-mode thermal TWB states, the expression of $F(m)$ is not trivial, but it is analytic $$\begin{aligned} \label{eq:Fanocond} \hspace{-0.1cm}F(m) \hspace{-0.3cm}&=& \hspace{-0.3cm}(1- \eta)+ (1- \eta) \cdot\\ &\cdot & \hspace{-0.3cm} \frac{\langle m \rangle(m_{\rm cond}+\mu)(\langle m \rangle+\eta \mu)}{(\langle m \rangle+ \mu) [(m_{\rm cond}+\mu)(\langle m \rangle+\eta \mu)-\eta \mu (\langle m \rangle+\mu)]} ,\nonumber\end{aligned}$$ where $\eta$ is the overall detection efficiency, $\langle m \rangle$ the mean value of the unconditioned state, and $m_{\rm cond}$ the conditioning value, that is the value of photons measured in one arm according to which the values of the other arm are selected.\ In Fig. \[g2cond\] we plot the experimental values of $[g^{(2)}(n) -1]$ (black dots + error bars) as a function of different conditioning values. Even if the negativity of the plotted quantity is not so large, it is sufficient to prove that the produced states are nonclassical. In the same figure, we also show the theoretical expectations (magenta circles) according to Eq. (\[eq:g2cond\]), in which Eq.(\[eq:Fanocond\]) was used. As in the case of the $g^{(2)}$ function for the photon-number difference, also for the Fano factor we used the experimental values of $\eta$, $\mu$ and $\langle m \rangle$. In particular, for the data in the figure $\langle m \rangle = 2.64$. We notice that the data are well superimposed to theory. ![Measured values of $[g^{(2)}(n)-1]$ as a function of the conditioning value. Black dots + error bars: experimental data; magenta circles: theoretical expectation according to Eq. (\[eq:g2cond\]).[]{data-label="g2cond"}](g2condmis3){width="0.7\columnwidth"} Conclusions =========== In conclusions, we have shown the twofold usefulness of the $g^{(2)}$ autocorrelation function written in terms of measurable quantities. On the one hand, we have proved that its evaluation can be used to characterise the main features of the employed detector. In particular, the spurious stochastic features that affect SiPMs can be easily determined from the expression of the second-order autocorrelation function. Moreover, for the specific choice of operational detector parameters and of the integration gate width used to acquire the detector output we have proved that such effects are substantially negligible. On the other hand, we have used the $g^{(2)}$ function to fully investigate the statistical properties and the correlated nature of mesoscopic multi-mode thermal TWB states. 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--- abstract: 'The density operators obtained by taking partial traces do not represent *proper mixtures* of the subsystems of a compound physical system, but *improper mixtures*, since the coefficients in the convex sums expressing them never bear the ignorance interpretation. As a consequence, assigning states to these subsystems is problematical in standard quantum mechanics (*subentity problem*). Basing on the proposal provided in the SR interpretation of quantum mechanics, where improper mixtures are considered as true nonpure states conceptually distinct from proper mixtures, we show here that proper and improper mixtures can be represented by different density operators in the quaternionic formulation of quantum mechanics, hence they can be distinguished also from a mathematical viewpoint. A simple example related to the quantum theory of measurement is provided.' author: - | Fabio Masillo,[^1] Giuseppe Scolarici,[^2] Sandro Sozzo[^3]\ Dipartimento di Fisica and Sezione INFN\ Università del Salento, via Arnesano, 73100 Lecce, Italy --- Introduction ============ In the complex formulation of quantum mechanics (CQM) a physical system $\Omega$ is associated with a separable complex Hilbert space ${\mathscr H}^{{\mathbb C}}$ and the states of $\Omega$ are represented by density operators on ${\mathscr H}^{{\mathbb C}}$, which reduce to one-dimensional (orthogonal) projection operators in the case of *pure* states. Every density operator $\rho$ representing a *mixed* state, or *proper mixture*, $S_M$ of $\Omega$ can be expressed in many ways as a convex combination of pure states, and a decomposition $\rho=\sum_{i} p_{i}| \psi_{i} \rangle\langle \psi_{i}|$ exists in which every coefficient $p_i$ denotes the probability that $\Omega$ be in the state $S_i$ represented by the projection operator $| \psi_{i} \rangle\langle \psi_{i}|$. A proper mixture can be produced by performing a (nonselective) measurement, thus obtaining the wave function collapse which is a nonlinear process in quantum mechanics. The probability $p_i$ expresses our ignorance about the real state of $\Omega$, hence also about the result of a measurement testing whether the property $E_i$ of $\Omega$ represented by $|\psi_{i}\rangle\langle\psi_{i}|$ is possessed by $\Omega$. Yet, if $\langle\psi_i|\psi_j \rangle=\delta_{ij}$, every $E_i$ is *objective* in $S_M$, in the sense that it can be considered as either possessed or not possessed by $\Omega$ independently of any measurement. Let now $\Omega$ be a compound system, made up of two subsystems $\Omega_1$ and $\Omega_2$, prepared in a pure entangled state $S_P$ represented by the projection operator $| \psi \rangle\langle \psi|$. Let $|\psi\rangle=\sum_{i}\sqrt{p_i}|\phi_{i}(1)\rangle |\chi_{i}(2)\rangle$$, 0 < p_i < 1$, be the biorthogonal decomposition of $|\psi\rangle$. If one considers $\Omega_1$ only, the physical information provided by CQM on it can be attained by taking the partial trace of $|\psi\rangle\langle\psi|$ with respect to $\Omega_2$, thus getting $\rho_1= Tr_{2} |\psi \rangle\langle \psi|=\sum_{i} p_{i} |\phi_{i}(1)\rangle\langle\phi_{i}(1)|$. The density operator $\rho_1$ is formally similar to $\rho$. Yet, a coefficient $p_i$ in it denotes the probability of actualizing the property $E_i(1)$ of $\Omega$ represented by the projection operator $|\phi_{i}(1)\rangle\langle\phi_{i}(1)| \otimes I_2$ whenever a measurement occurs, but it cannot denote the probability that $\Omega_1$ actually be in the state $S_{i}(1)$ represented by $|\phi_{i}(1)\rangle\langle\phi_{i}(1)|$. Indeed, $E_i(1)$ should then be objective, as the property $E_i$ considered above, while it is *nonobjective* in $S_P$ according to the standard interpretation of quantum mechanics (that is, one cannot consider $E(1)$ as either possessed or not possessed by $\Omega$ in the state $S_P$ if a measurement is not performed).[^4] Basing on this conclusion, one can show that no decomposition of $\rho_1$ bears the above *ignorance interpretation*. Hence, some authors say that $\rho_1$ represents an *improper mixture*, distinguishing it from a proper mixture as $\rho$ (see, *e.g.*, [@hk69]–[@blm91]). As a consequence, the density operators obtained by taking partial traces generally neither represent pure nor mixed states of the component subsystems in standard quantum mechanics, so that these subsystems can never be considered as independent entities, which raises the so-called *subentity problem* [@a99a]–[@a99b; @a00]. A proposal of solution of the subentity problem has recently been forwarded by one of the authors [@gs06] in the framework of the *Semantic Realism* (or *SR*) interpretation of quantum mechanics [@gs96a]–[@ga02]. The SR interpretation has been worked out to show that the mathematical apparatus of quantum mechanics is compatible with objectivity of physical properties, which avoids the objectification problem together with a number of quantum paradoxes. In this perspective, improper mixtures are considered as new states of the physical system and they cannot be distinguished in CQM from proper mixtures. Yet, the unitary evolution of proper mixtures is different from the evolution of improper mixtures, generally nonunitary, which suggests, according to this interpretation, a more suitable mathematical representation of these physically different entities. In 1936, by using lattice theoretic arguments, Birkhoff and von Neumann [@bn36] concluded that the set of states of a quantum system can be represented by a vector space over the real $\mathbb{R}$, complex $\mathbb{C}$, or quaternionic $\mathbb{Q}$, fields. While the real number formulation of quantum mechanics is essentially equivalent to CQM [@st60], the research on quaternionic quantum mechanics (QQM) began much later with a series of papers by Finkelstein *et al.* in the sixties [@fjss60], and pursued up to now. A systematic study of QQM is given in [@a95]. The possibility of a generalization of quantum mechanics based on quaternion fields instead of complex fields is still controversial. Yet, the rich structures emerging from such a generalization proved to be useful in the description of entanglement, dynamical maps and decoherence phenomena in quantum physics [@Kossa1]–[@as]. Furthermore, we will see in this paper that these structures provide rigorous tools to solve the subentity problem. Let us briefly sketch the content of the next sections. After briefly resuming the qualitative solution of the subentity problem that raises from the SR interpretation of quantum mechanics in Sec. 2, we will discuss in Sec. 3 how mixtures are represented in QQM. In particular, we will show that proper and improper mixtures can be represented by different density operators in QQM, and that this mathematical representation is compatible with their different time evolutions in CQM. Finally, we will apply these results in Sec. 4, and consider a specific example regarding the measurement process. The subentity problem in the SR interpretation of quantum mechanics \[sr\] ========================================================================== The SR interpretation of quantum mechanics [@gs96a]–[@ga02] has been worked out in order to show that both CQM and QQM can be embodied into a more general framework in which objectivity of physical properties holds and quantum probabilities are reinterpreted as conditional instead of absolute. The SR interpretation avoids the objectification problem and other quantum paradoxes. Moreover, it provides a nonstandard solution of the subentity problem [@gs06], which will be summarized in the following. \(i) A physical system $\Omega$ is associated with a set $\mathcal S$ of *states* and a set $\mathcal E$ of *physical properties*. Each state is operationally defined as a class of statistically equivalent *preparing devices*. Each property is operationally defined as a class of statistically equivalent *ideal registering devices*. \(ii) We assume that all properties in $\mathcal E$ are *objective*, *i.e.*, for every $E \in \mathcal E$, the outcome of a registering device $r \in E$, when applied to an individual sample (or, *physical object*) $x$ of $\Omega$, does not depend on the measurement procedure.[^5] \(iii) It follows from (ii) that the probability of finding a given result when performing a measurement on a physical object $x$ can be interpreted as expressing our ignorance about the properties possessed by $x$ (in this sense one can say that it is *epistemic*) in the SR interpretation, whatever the state of the physical object may be. The distinction between pure and nonpure states may still be introduced basing on the different values of the probabilities of the properties in these states, but not on different interpretations (epistemic or not) of the probabilities themselves. In particular, one can accept the standard representation of states by means of density operators, and characterize pure states as the states whose representing density operators reduce to projection operators. \(iv) Since every state is operationally defined as an equivalence class of preparing devices, if one considers a state $S$ represented by the density operator $\sum_{i} p_i |\psi_i\rangle\langle\psi_i|$, an ensemble of physical objects in the state $S$ can be realized by a *mixed* preparing device, *i.e.*, a device that mixes physical objects prepared by devices belonging to the states $S_1$, $S_2$, …represented by the projection operators $|\psi_1\rangle\langle\psi_1|$, $|\psi_2\rangle\langle\psi_2|$, …, respectively. In this case a coefficient $p_i$ cannot only be interpreted as in (iii), but also as the probability that a given physical object in the state $S$ actually be in the state $S_i$. Nevertheless, there is no evident physical reason, according to the SR interpretation, for assuming that $S$ contains only mixed preparing devices. \(v) It follows from (iv) that in the case of compound physical systems the density operators obtained by taking partial traces can be accepted as representing states in which also preparations occur that are not mixed in the sense specified in (iv). \(vi) The probabilistic definition of states in (i) groups together, in the case of nonpure states, mixed with nonmixed preparing devices, that therefore cannot be distinguished by means of measurements. This opens the way to a possible solution of the problem of explaining how both unitary and nonunitary evolutions may occur for the same density operator, since it suggests distinguishing mixed from nonmixed preparing devices by introducing a new equivalence relation on the set of all preparing devices, strictly contained in the physical equivalence relation defined in (i). Thus, every state $S$ would be associated with a family of *hidden states*, which would be equivalent with respect to measurements but could have different behaviours with respect to time evolution. Mixtures in quaternionic quantum mechanics ========================================== We recall in the first part of this section some basic notations, properties and results of QQM (for an exhaustive discussion of quaternionic matrices, see, *e.g.*, [@Zhang]) in order to provide in the second part different mathematical representations for proper and improper mixtures. A physical system $\Omega$ is associated in QQM with a quaternionic $n$-dimensional right Hilbert space ${\mathscr H}^{{\mathbb Q}}$ [@a95] (for the sake of simplicity, we will limit ourselves to consider finite-dimensional quaternionic Hilbert spaces; this will allow us to denote operators and the associated matrices by the same symbols). The states of $\Omega$ are represented by positive hermitian operators on ${\mathscr H}^{{\mathbb Q}}$ with unit trace (as in CQM). More precisely, a pure state $S_P$ of $\Omega$ is represented by a density operator $\rho_{\psi}=|\psi \rangle \langle \psi |$ (where $|\psi\rangle$ is a unit vector of ${\mathscr H}^{{\mathbb Q}}$) with rank one, while a mixed state $S_M$ of $\Omega$ is represented by a density operator $\rho$ with rank greater than one. The observables of $\Omega$ are represented by hermitian operators on ${\mathscr H}^{{\mathbb Q}}$. Moreover, the expectation value of an observable $\mathcal A$, represented by the quaternionic hermitian operator $A$, in the pure state $S$, represented by the unit vector $|\psi\rangle$, is given by [@a95] $$\label{expec} \langle A \rangle_{\psi} = \langle \psi |A|\psi \rangle =\textrm{Re} \textrm{Tr} (A|\psi \rangle \langle \psi |)=\textrm{Re} \textrm{Tr} (A\rho_{\psi}).$$ Expanding $A=A_{\alpha}+jA_{\beta}$ and $\rho=\rho_{\alpha}+j\rho_{\beta}$ (where $j$ denotes one of the quaternion imaginary units) in terms of the complex matrices $A_{\alpha}$, $A_{\beta}$, $\rho_{\alpha}$ and $\rho_{\beta}$, it follows that the expectation value $\langle A\rangle_{\psi}$ may depend on $A_{\beta}$ or $\rho_{\beta}$ only if both $A_{\beta}$ and $\rho_{\beta}$ are different from zero. Indeed, $$\label{esplexpc} \langle A\rangle_{\rho}=\textrm{Re}\textrm{Tr}(A\rho)=\textrm{Re}\textrm{Tr}(A_{\alpha}\rho_{\alpha}-A_{\beta}^{\ast}\rho_{\beta}),$$ where $\ast $ denotes complex conjugation. Thus, the expectation value of an observable $\mathcal A$, represented by the hermitian operator $A$, in the state $S$, represented by the density matrix $\rho$, depends on the quaternionic parts of $A$ and $\rho $ only if both the observable and the state are represented by genuine quaternionic matrices. However, if an observable $\mathcal A$ is represented by a pure complex hermitian matrix, its expectation value in the state $S$ does not depend on the quaternionic part $j\rho _{\beta }$ of the density matrix $\rho =\rho _{\alpha}+j\rho_{\beta}$ representing $S$. Moreover, the expectation value predicted in CQM in the state represented by the density matrix $\rho_{\alpha}$ coincides with the expectation value in the state represented by $\rho$ predicted by QQM, since $$\textrm{Tr}( A \rho_{\alpha})=\textrm{Re} \textrm{Tr}(A \rho_{\alpha})=\textrm{Re}\textrm{Tr}(A\rho).$$ This simple observation is relevant in our approach since it enables us to merge CQM in the (more general) framework of QQM without modifying any theoretical prediction (as long as complex observables are taken into account), thus eluding or postponing any comparison between these formulations. Let us now denote by $M(\mathbb{Q})$ and $M(\mathbb{C})$ the space of $n \times m$ quaternionic and complex matrices, respectively, and let $M=M_{\alpha}+j M_{\beta} \in M(\mathbb{Q})$. We define the *complex projection* $$P: M(\mathbb{Q}) \rightarrow M(\mathbb{C})$$ by the relation $$\label{complexProjection} P(M)=\frac{1}{2}[M-i M i]=M_{\alpha}.$$ When we consider time-dependent quaternionic unitary dynamics, $$\rho (t)=U(t)\rho (0)U^{\dagger }(t), \label{unitev}$$ where $$U(t)=(U_{\alpha }+jU_{\beta })(t)=T_{o}\,e^{-\int_{0}^{t}duH(u)} \label{timeordev}$$ and $T_{o}$ denotes the time ordering operator, the differential equation associated with the time evolution for $\rho $ reads $$\frac{d}{dt}\rho (t)=-[H(t),\rho (t)], \label{eveqtimdep}$$ where $H(t)=H_{\alpha }+jH_{\beta }=-\left( \frac{d}{dt}U(t)\right) U^{\dagger }(t)$. Finally, Eqs. (\[unitev\]) and (\[eveqtimdep\]) reduce to $$\rho _{\alpha }(t)=U_{\alpha }\rho _{\alpha }(0)U_{\alpha }^{\dagger }+U_{\beta }^{\ast }\rho _{\alpha }^{\ast }(0)U_{\beta }^{T}+U_{\alpha }\rho _{\beta }^{\ast }(0)U_{\beta }^{T}-U_{\beta }^{\ast }\rho _{\beta }(0)U_{\alpha }^{\dagger } \label{finitecomppro}$$ and $$\frac{d}{dt}\rho _{\alpha }=-[H_{\alpha },\rho _{\alpha }]+H_{\beta }^{\ast }\rho _{\beta }-\rho _{\beta }^{\ast }H_{\beta }, \label{gencompro}$$ respectively, for the complex projection of the density matrix [@Asor]. Now, we focus our attention on the complex projection $\rho _{\alpha }$ of a quaternionic density matrix $\rho =\rho _{\alpha }+j\rho _{\beta }$. First of all, it follows from the hermiticity of $\rho$ and $\rho_{\alpha}$ that $$\textrm{Tr}\rho_{\alpha}=\textrm{Re}\textrm{Tr} \rho_{\alpha}=\textrm{Re}\textrm{Tr}\rho=\textrm{Tr}\rho,$$ *i.e.*, the complex projection of any quaternionic density matrix is trace preserving. Moreover, we recall that [@Asor]: **Proposition 1.** *The complex projection $\rho_{\alpha }$ of any quaternionic density matrix $\rho=\rho_{\alpha}+j\rho_{\beta}$ is a complex density matrix*. The following statement provides instead informations about the rank of the complex projection $\rho _{\alpha }$ of any quaternionic density matrix $\rho =\rho_{\alpha}+j\rho_{\beta}$ [@as]. **Proposition 2.** *Let $\rho=\rho_{\alpha}+j\rho_{\beta}$ be a $n$-dimensional quaternionic density matrix, and let $\mathrm{rank}\,\rho =m$. Then, $m$ $\leq \mathrm{rank}\,\rho_{\alpha}\leq 2m$.* Conversely [@gallipoli], **Proposition 3.** *Let $\rho_{\alpha}$ be a $n$-dimensional complex density matrix with $\mathrm{rank}\,\rho _{\alpha }=m>1$, and let $[x]$ denote the integer part of $x$. Then, for any $m^{\prime}$ with $\left[ \frac{m+1}{2}\right] $ $\leq m^{\prime }\leq m$ there exists a (skew-symmetric) complex matrix $\rho _{\beta }$ such that $\rho =\rho _{\alpha}+j\rho _{\beta }$ is a density matrix with $\mathrm{rank}\,\rho=m^{\prime }$.* As a consequence of the above two propositions, we can conclude that: **Proposition 4.** *Any complex density matrix $\rho _{\alpha }$ can be obtained as the complex projection of a quaternionic pure density matrix $\rho =\rho _{\alpha }+j\rho _{\beta }$ if and only if $\mathrm{rank} \rho _{\alpha }=2$*. Let us now discuss how the above results can be used to represent proper and improper mixtures by different density matrices in QQM. To this aim, we observe that every complex density matrix $\rho_\alpha$ can be associated with a set $[\rho_\alpha]$ of quaternionic density matrices as follows: $$\rho_\alpha\longrightarrow[\rho_\alpha]=\{\rho=\rho_\alpha+j\rho_\beta\},$$ where the $\rho_{\beta}$s must be chosen in a such way that $\rho$ is still a density matrix (see Propositions 1–4). The above mapping can be inverted by means of the complex projection in Eq. (\[complexProjection\]), and it introduces an equivalence relation $\approx$ on the set of density matrices defined, for every $\rho, \rho\prime$, as $\rho \approx \rho\prime$ iff $P(\rho)=P(\rho\prime)$. Hence, the set of quaternionic density matrices is partitioned into equivalence classes, and each class contains one and only one complex density matrix, which thus represents the class. Each equivalence class $[\rho_{\alpha}]$ can further be partitioned by distinguishing the quaternionic density matrices $\rho=\rho_{\alpha}+j\rho_{\beta}$, $\rho_{\beta} \ne 0$, from the complex matrix $\rho=\rho_{\alpha}$. It is important to observe that a complex unitary dynamics, *i.e.*, a dynamics given by a complex unitary matrix $U_{\alpha}$, preserves both the first and the second partition. Indeed, we get from Eqs. (\[unitev\]) and (\[finitecomppro\]) that $$\rho(t)=\rho_\alpha(t)+j \rho_\beta(t)=U_{\alpha}\rho_\alpha(0)U_{\alpha}^{\dagger}+j U_{\alpha}^{*}\rho_\beta(0)U_{\alpha}^{\dagger},$$ $$\rho_\alpha(t)=U_{\alpha}\rho_\alpha(0)U_{\alpha}^{\dagger}.$$ On the contrary, a quaternionic unitary dynamics in general neither preserves the first nor the second partition (see again Eqs. (\[unitev\]) and (\[finitecomppro\])). As stated in the previous sections, the distinction between proper and improper mixtures is strictly connected in CQM with the existence of entangled states and the partial trace procedure. In particular, we notice that the latter operation can give rise to the following situations for the subsystems. \(i) A separable state of the compound system generally produces proper mixtures of the component subsystems. \(ii) An entangled state of the compound system generally produces improper mixtures of the component subsystems. Clearly, if the unitary evolution of the compound system is factorizable in CQM, it transforms entangled states into entangled states, and separable states into separable states. Then we demand that, if a mathematical distinction between proper and improper mixtures exists in QQM, then the unitary subdynamics associated in CQM with each subsystem should be such that proper mixtures are transformed into proper mixtures and improper mixtures are transformed into improper mixtures. Therefore, we are led to represent proper mixtures by the quaternionic density matrices for which $P(\rho)=\rho$, and improper mixtures by the quaternionic density matrices for which $P(\rho)\neq\rho$. Of course, a complex unitary dynamics does not modify this distinction. In fact, if $\rho_\beta(0)=0$, then $\rho_\beta(t)=U_{\alpha}^{*}\rho_\beta(0)U_{\alpha}^{\dagger}=0$. We stress once again that the density matrices $\rho$ representing proper and improper mixtures produce the same expectation values on complex observables. We finally observe that the above partition is not preserved in CQM by nonfactorizable dynamics of the compound system. In fact, in this case the subsystems can exchange entanglement each other during their evolution, which is thus nonunitary. We remind that nonunitary dynamics in CQM can in many cases be described in terms of the complex projection of a quaternionic unitary evolution [@compent; @Asor], [@as]. The measurement process: an illustrative example ================================================ Our main aim in this section is to illustrate an example in which QQM represents proper and improper mixtures with different density operators, thus allowing one to distinguish them not only at a conceptual but also at a mathematical level. The proposed example is meaningful since it concerns with the description of the measurement process as a dynamical process and is highly problematical in standard quantum mechanics. We will consider a very simple and schematic model to describe the interaction between the measured system and the measuring apparatus that occurs in a measurement. More precisely, let $\Omega_m$ be a microscopic physical system, for instance, a spin-$\frac{1}{2}$ quantum particle, associated with the complex Hilbert space ${\mathscr H}_{m}^{\mathbb C}={\mathbb C}^{2}$, and let $\mathcal A$ be the observable “spin of $\Omega_m$ along the direction $\vec{n}$”, represented in CQM by the hermitian operator $A=\frac{1}{2}\hbar \vec{\sigma}\cdot \vec{n}$, where $\vec{\sigma}$ are the Pauli matrices. Let $s_1$ and $s_2$ be the pure states of $\Omega_m$ corresponding to the eigenvectors $| +_{\vec{n}} \rangle$ and $| -_{\vec{n}} \rangle$ of $A$, respectively. Let us schematize the apparatus that performs an ideal measurement of $\mathcal A$ on $\Omega_m$ by means of a macroscopic physical system $\Omega_M$ that can be described in CQM by the Hilbert space ${\mathscr H}_{M}^{\mathbb C}={\mathbb C}^{2}$. Suppose that $\Omega_M$ is initially in the macroscopic state $S_0$ represented by the unit vector $|0 \rangle$ corresponding to the value 0 on the reading scale of the apparatus, and that $\Omega_M$ possesses further macroscopic states $S_1$ and $S_2$ represented by the unit vectors $|u_{\vec{n}} \rangle$ and $| d_{\vec{n}} \rangle$, respectively (corresponding to the values $up$ and $down$, respectively, on the same scale), and let $\{|u_{\vec{n}} \rangle,|d_{\vec{n}} \rangle \}$ be an orthonormal basis on ${\mathscr H}_{M}^{\mathbb C}$. Furthermore, assume that there exists a one-to-one correspondence between the states $s_1$ and $s_2$ of $\Omega_{m}$ and the states $S_1$ and $S_2$ of $\Omega_{M}$. Let $s_0$ be the initial state of $\Omega_m$ represented by the unit vector $|\varphi_0\rangle=c_+ |+_{\vec{n}} \rangle + c_- |-_{\vec{n}} \rangle$. Finally, assume that the interaction between $\Omega_m$ and $\Omega_M$ is represented by a complex unitary operator $U(0,t)$. This interaction can be described as follows. $$\label{unitcomp} \begin{CD} | \Psi(0)\rangle=|\varphi_0\rangle |0\rangle =[c_+ |+_{\vec{n}} \rangle + c_- |-_{\vec{n}} \rangle] |0\rangle @>>U(0,t)> |\Psi (t)\rangle= c_+ |+_{\vec{n}} \rangle |u_{\vec{n}} \rangle+c_- |-_{\vec{n}} \rangle |d_{\vec{n}} \rangle, \end{CD}$$ where $|\Psi(t)\rangle$ is expressed by a biorthogonal decomposition, hence it represents a pure entangled state of the compound system $\Omega_m+\Omega_M$. The physical information on $\Omega_m$ can be obtained in CQM by taking the partial trace of the density operator $\rho_{\alpha}(t)=|\Psi(t)\rangle\langle\Psi(t)|$ with respect to the physical system $\Omega_M$. Then, we get $$\label{partialtrace} \rho_{\alpha}^{m}(t)=Tr_{M} \rho_{\alpha}(t)=\langle u_{\vec{n}} | \rho_{\alpha}(t) | u_{\vec{n}} \rangle+\langle d_{\vec{n}} | \rho_{\alpha}(t) | d_{\vec{n}} \rangle=|c_+|^{2} | +_{\vec{n}}\rangle\langle +_{\vec{n}}|+|c_-|^{2} | -_{\vec{n}}\rangle\langle -_{\vec{n}}| .$$ This density operator represents an improper mixture $M_{S}^{I}$ of $\Omega_m$ (see Sec. 1). If we now consider the physical system $\Omega_m$ separately, and apply the projection postulate of CQM in the case of a nonselective measurement of the observable $\mathcal A$ we instead obtain, via Lüders’ rule, $$\label{projpost} \rho_{\alpha}^{m}=|+_{\vec{n}}\rangle\langle +_{\vec{n}}|\varphi_0\rangle\langle\varphi_0|+_{\vec{n}}\rangle\langle +_{\vec{n}}|+|-_{\vec{n}}\rangle\langle -_{\vec{n}}|\varphi_0\rangle\langle\varphi_0|-_{\vec{n}}\rangle\langle -_{\vec{n}}|=\rho_{\alpha}^{m}(t).$$ This density operator formally coincides with the one obtained in Eq. (\[partialtrace\]) but it represents a proper mixture $M_{S}^{P}$ of $\Omega_m$ in this case (see again Sec. 1). It is well known that the main problem of the standard quantum theory of measurement is reconciling the two above descriptions, in particular, recovering the objectification, that is, the occurrence of definite outcomes for the macroscopic apparatus (*objectification problem*). Any attempt at providing a consistent description of the measuring process within the quantum formalism and its standard interpretation leads to the so-called *von Neumann’s chain* and to the ensuing quantum paradoxes (in particular, *Schrödinger’s cat* and *Wigner’s friend* paradoxes). We have observed in Sec. 2 that the above problem disappears in the SR interpretation of quantum mechanics because of objectivity of the properties of both $\Omega_m$ and $\Omega_M$. However, $S_{M}^{P}$ and $S_{M}^{I}$ cannot be distinguished in CQM also if the SR interpretation is adopted, since they are represented by the same complex density operator. Yet, we can use the arguments in Sec. 3 to accomplish this task in QQM. According to the partition introduced in Sec. 3, the improper mixture $S_{M}^{I}$ can be represented in QQM by the density operator $\rho^{m}=\rho_{\alpha}^{m}+j \rho_{\beta}^{m}$, where $\rho_{\beta}^{m} \ne 0$, while the proper mixture $S_{M}^{P}$ can be represented by the density operator ${\rho}'^{m}=\rho_{\alpha}^{m}$. It must be noted that $S_{M}^{I}$ can also be purified[^6] (see Proposition 4) by choosing $$\rho^{m}=|c_{+}|^{2}|+_{\vec{n}}\rangle\langle +_{\vec{n}}|+|c_{-}|^{2}|-_{\vec{n}}\rangle\langle -_{\vec{n}}|+|-_{\vec{n}}\rangle j c_{+}^{*}c_{-}^{*} \langle +_{\vec{n}}|-|+_{\vec{n}}\rangle j c_{+}^{*}c_{-}^{*} \langle -_{\vec{n}}|,$$ which is the projection operator on the 1–dimensional subspace generated by the unit vector $|+_{\vec{n}}\rangle c_{+} +|-_{\vec{n}}\rangle c_{-}j$. One realizes at once that $\rho^{m}$ and ${\rho}'^{m}$ produce the same expectation values on complex observables, whereas their expectation values are different if purely quaternionic observables are taken into account. The above result is relevant from our viewpoint since it shows that the mixtures $S_{M}^{I}$ and $S_{M}^{P}$ can be experimentally distinguished, at least in principle, in QQM. [99]{} K. E. Hellwig and K. Kraus, *Comm. Math. Phys.* **11**, 214 (1969). B. 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Phys.* **30**, 1539 (2000). C. Garola, *Found. Phys.* **32**, 1597 (2002). G. Birkhoff and J. von Neumann, *Ann. Math.* **37**, 823 (1936). E. C. G. Stueckelberg, *Helv. Phys. Acta* **33**, 727 (1960); **34**, 621 (1961); **35** 637 (1962). D. Finkelstein, J. M. Jauch, S. Sciminovich and D. Speiser, *J. Math. Phys.* **3**, 207 (1962); **4**, 136 (1963); **4**, 788 (1963). S. L. Adler, *Quaternionic Quantum Mechanics and Quantum Fields* (Oxford UP, New York, 1995) and references therein. A. Kossakowski, *Rep. Math. Phys.* **46**, 393 (2000). G. Scolarici and L. Solombrino “Complex entanglement and quaternionic separability” in *The Foundations of Quantum Mechanics: Historical Analysis and Open Questions-Cesena 2004*, C. Garola, A. Rossi and S. Sozzo eds. (World Scientific, Singapore, 2006). M. Asorey and G. Scolarici, *J. Phys. A* **39**, 9727 (2006). M. Asorey, G. Scolarici and L. Solombrino, *Theor. Math. Phys.* **151** 733 (2007). M. Asorey, G. Scolarici and L. Solombrino, *Phys. Rev. A* **76** 12111 (2007). F. Zhang, *Lin. Alg. Appl.* **251**, 21 (1997). [^1]: E-mail address: [email protected] [^2]: E-mail address: [email protected] [^3]: E-mail address: [email protected] [^4]: Nonobjectivity is commonly believed to be an intrinsic and uneliminable feature of standard quantum mechanics because of some mathematical results, as the *Bell-Kochen-Specker theorem* [@b66; @ks67] and the *Bell theorem* [@b64]. Yet, it is the deep root of most problems that afflict the standard interpretation and raises a lot of paradoxes and conceptual difficulties (in particular, the *objectification problem* in the quantum theory of measurement, see, *e.g.*, [@blm91; @bs96]). \[nogo\] [^5]: Objectivity of properties implies that the SR interpretation clashes with the standard interpretation, which asserts instead nonobjectivity of properties on the basis of empirical (*e.g.*, the double-slit experiment) or theoretical (*e.g.*, the no-go theorems mentioned in footnote \[nogo\]) arguments. Hence, the SR interpretation was worked out together with an accurate analysis of those arguments, which singled out some weaknesses in each of them. In particular, theoretical arguments in favor of nonobjectivity turn out to be based on implicit assumptions that, when made explicit, are rather doubtful. Indeed, these assumptions subtend an epistemological perspective that assumes the validity of empirical quantum laws also in physical situations in which quantum mechanics itself states that, in principle, they cannot be checked [@gs96b]–[@ga00]. If this perspective is criticized, nonobjectivity appears as an interpretative choice, not a logical consequence of the theory, and alternative objective interpretations (as the SR) become possible. [^6]: The possibility of considering improper mixtures as new pure states has already been taken into account by Aerts in a series of papers [@a99a]–[@a00] where, however, no mention to the field of the Hilbert space has been made.

--- abstract: 'The exclusive photoproduction of the $J/\psi$ state is investigated in peripheral AA collisions for the energies available at the LHC, $\sqrt{s}=2.76$ TeV and $\sqrt{s}=5.02$ TeV. In order to evaluate the robustness of the light-cone color dipole formalism, previously tested in the ultraperipheral regime, the rapidity distribution and the nuclear modification factor ($R_{AA}$) were calculated for three centrality classes: 30%-50%, 50%-70% and 70%-90%. In the peripheral regime, three scenarios were considered. In the scenario 1, a similar formalism adopted in the UPC regime is used; in the scenario 2, one considers that only the spectators in the target are the ones that interact coherently with the photon; in the scenario 3, the photonuclear cross section is modified using the same geometrical constraints applyed in the scenario 2. The results obtained from the three scenarios were compared with the ALICE measurements (only $J/\psi$ at the moment), showing a better agreement in the more complete approach (scenario 3), mainly in the more central regions (30%-50% and 50%-70%) where the incertainty is smaller.' author: - 'M. B. GAY DUCATI[^1] AND S. MARTINS[^2]' title: 'Exclusive Photoproduction $J/\psi$ in Peripheral Pb-Pb' --- Introduction ============ The ALICE collaboration measured the $J/\psi$ hadroproduction in peripheral collisions Pb-Pb, with $\sqrt{s}=2.76$ TeV, revealing an excess in the production of the meson in very small transverse momentum ($p_T<0.3$ GeV/c) in the range rapidity $2.5<y<4.0$ [@PRL116-222301]. This excess was quantified by the nuclear modification factor, $R_{AA}$, which reaches the values 7 (2) in the centrality classes 70%-90% (50%-70%) within the rapidity range $2.5<y<4.0$. Considering the $J/\psi$ excess could be generated from exclusive photoproduction, the nuclear photoproduction of the $J/\psi$ was calculated in peripheral collisions from the factorization adopted in the ultraperipheral collisions (UPC), with cross section separated in two factors: the equivalent photon flux, $N\left(\omega, b\right)$, and the photonuclear cross section, $\sigma_{\gamma A\rightarrow VA}$. In our approach, the peripheral regime is analysed considering three scenarios: in the scenario 1 is applyed a photon flux with b-dependence and a more realistic electromagnetic form factor is used; in the scenario 2 a geometrical cut is applyed in the photon flux ensuring that only the spectators in the target will interact coherently with the photon; in scenario 3, for completeness, the restriction adopted in the scenario 2 is extended on the photonuclear cross section. Using these three scenarios, the rapidity distribution and the nuclear modification factor, $R_{AA}$, were estimated for 30%-50%, 50%-70% and 70%-90% centrality classes. To calculate the $R_{AA}$, it was adopted the expression developed in [@PLB734-314], $$\begin{aligned} \scalebox{0.7}{$R_{AA}^{hJ/\psi}=\dfrac{N_{AA}^{J/\psi}}{BR_{J/\psi\rightarrow l^+l^-}\cdot N_{events}\cdot\left(A\times\varepsilon\right)_{AA}^{J/\psi}\cdot\left\langle T_{AA}\right\rangle \cdot\sigma_{pp}^{hJ/\psi}}$}\label{eq:raa},\end{aligned}$$ where the measured number of $J/\psi$ ($N_{AA}^{J/\psi}$) is corrected for acceptance and efficiency $\left(\mathcal{A}\times\varepsilon\right)_{AA}^{J/\psi}\sim 11.31\%$ and branching ratio $\text{BR}_{J/\psi\rightarrow \mu^+\mu^-}=5.96$%. Then, the result is normalized to the equivalent number of MB events ($N_{events}\simeq 10.6\times 10^7$), defined in [@PLB734-314], average nuclear overlap function $\left(\langle T_{AA}\rangle\right)$, calculated from [@PRC88-044909], and proton-proton inclusive $J/\psi$ production cross section ($\sigma_{pp}^{hJ/\psi}\sim 0.0514$ $\mu$b), calculated from a parametrization detailed in [@PRL116-222301]. It is not possible to measure the individual contribution of each mechanisms of production (hadroproduction and photoproduction) in the kinematical regime of the interest. Therefore, the number of the $J/\psi$ events showed in (\[eq:raa\]) was separated in two terms $$\begin{aligned} N_{AA}^{J/\psi}=\underset{\text{hadro}}{\underbrace{N_{AA}^{hJ/\psi}}}+\underset{\text{photo}}{\underbrace{N_{AA}^{\gamma J/\psi}}}.\end{aligned}$$ Considering the central data and the hypothesis shown in the ALICE paper [@PRL116-222301], it is possible to write the number of the $J/\psi$ events as a function of the average rapidity distribution as [@1804.09836] $$\begin{aligned} N_{AA}^{J/\psi}=\begin{cases} 1.96\times 10^6\,\,d\sigma^{\gamma}_{J/\psi}/dy, & 30\%-50\%\\ 1.34\times 10^6\,\,d\sigma^{\gamma}_{J/\psi}/dy, & 50\%-70\%\\ 0.96\times 10^6\,\,d\sigma^{\gamma}_{J/\psi}/dy, & 70\%-90\% \end{cases}\end{aligned}$$ where $d\sigma^{\gamma}_{J/\psi}/dy$, evaluated in the $2.5<y<4.0$ rapidity range. Theoretical Framework {#ipdpf} ===================== In the ultrarelativistic limit, the differential cross section in the rapidity $y$ and impact parameter $b$, is given by [@PRC93-044912] $$\begin{aligned} \dfrac{d^3\sigma_{AA\rightarrow AAV}}{d^2bdy} = \omega N(\omega,b)\sigma_{\gamma A\rightarrow VA}+\left(y\rightarrow -y\right)\label{pri}.\end{aligned}$$ where $N(\omega,b)$ is a photon flux with b-dependence, $\sigma_{\gamma A\rightarrow VA}$ is the photonuclear cross section, which characterizes the photon-target interaction $\gamma A\rightarrow VA$, $\omega=\frac{1}{2}M_V\text{exp}(y)$ is the photon energy and $M_V$ is the meson mass. In the peripheral collisions ($b<2R_A$), the use of the different eletromagnetic form factor, $F(k^2)$, becomes relevant and, therefore, the following generic formula was adopted for the photon flux[@PPNP39-503] $$\begin{aligned} N\left(\omega,b\right)=\dfrac{Z^{2}\alpha_{QED}}{\pi^{2}\omega}\left|\int_{0}^{\infty}dk_{\perp}k_{\perp}^{2}\dfrac{F\left(k^2\right)}{k^2}J_{1}\left(bk_{\perp}\right)\right|^{2}\label{eq:xx},\end{aligned}$$ where $Z$ is the nuclear charge, $\gamma=\sqrt{s_{NN}}/(2m_{\textrm{proton}})$ is the Lorentz factor, $k_{\perp}$ is the transverse momentum of the photon and $k^{2}=\left(\omega/\gamma\right)^{2}+k_{\perp}^{2}$. A good approximation for the lead nucleus form factor is shown in [@PRC14-1977], in which the Woods-Saxon distribution is rewritten as a hard sphere, with radius $R_A$, convoluted with an Yukawa potential with range $a = 7$ fm. The Fourier transform of this convolution result in $$\begin{aligned} \scalebox{0.7}{$F(k)=\dfrac{4\pi\rho_{0}}{Ak^{3}}\left[\textrm{sin }\left(kR_{A}\right)-kR_{A}\textrm{cos }\left(kR_{A}\right)\right]\left[\dfrac{1}{1+a^{2}k^{2}}\right]$},\label{eq:wsy}\end{aligned}$$ where $A$ is the mass number, $a=0.7$ fm and $\rho_{0}=0.1385\textrm{ fm}^{-3}$. The second factor in (\[pri\]), $\sigma_{\gamma A\rightarrow VA}$, represents the photon-nuclei interaction and can be described in the light cone colour dipole formalism, which includes the partonic saturation phenomenon and the nuclear shadowing effects [@EPJC8-115]. The formalism has already been explored in the last works [@PRD94-094023] in pp, pA and AA collisions. In the last case, the coherent photonuclear cross section of a vector meson $V$ can be factorized as $$\begin{aligned} \sigma_{(\gamma A\rightarrow VA)} = \left.\frac{d\sigma}{dt}\right|_{t=0}\int_{t_{min}}^{\infty}|F(t)|^2dt\label{sdt}\end{aligned}$$ where the forward scattering amplitude, $d\sigma/dt|_{t=0}=|\text{Im }A(x,t=0)|^2/16\pi$, carries the dynamical information of the process and the form factor, $F(t)$, represents the dependence on the spatial characteristics of the target. On the Eq. (\[sdt\]) are also included the parameter $\beta=\text{tan }(\pi\lambda_{eff}/2)$, which restores the real contribution of the scattering amplitude, and $R_g^2(\lambda_{eff})=(2^{2\lambda_{eff}+3}/\sqrt{\pi})[\Gamma(\lambda_{eff}+5/2)/\Gamma(\lambda_{eff}+4)]$, which corresponds to the ratio of off-forward to forward gluon distribution (skewedness effect) relevant for heavy mesons. The parameter $\lambda_{eff}$ can be estimated from relation $\lambda_{eff}\equiv\partial\text{ln}[\text{Im }A(x,t=0)]/[\partial\text{ln}(1/x)]$. Thus, the Eq. (\[sdt\]) is rewritten as $$\begin{aligned} \sigma_{(\gamma A\rightarrow VA)} &=&\frac{|\textrm{Im }A(x,t=0)|^2}{16\pi}\left(1+\beta^2(\lambda_{eff})\right)R^2_g(\lambda_{eff})\nonumber \\ &\times&\int_{t_{min}}^{\infty}|F(t)|^2dt,\label{seg}\end{aligned}$$ where $x=(M^2_V+Q^2)/(Q^2+2\omega\sqrt{s})$ with $Q\sim 0$ for nucleus and $t_{min}=(m_V^2/2\omega\gamma)^2$. In the color dipole formalism, the photon-nuclei forward scattering amplitude is factorized in the overlap between the photon and the vector meson wave functions, and in the dipole-nuclei cross section as $$\begin{aligned} \text{Im }A(x,t=0)=\int d^2r\int\frac{dz}{4\pi}\left(\psi^*_V\psi_{\gamma}\right)_T\sigma^{\text{nucleus}}_{\text{dip}}(x,r).\end{aligned}$$ where $\left(\psi^*_V\psi_{\gamma}\right)_T$ is described with more detail in [@PRD74-074016], and $\sigma^{\text{nucleus}}_{\text{dip}}(x,r)$ is obtained using the Glauber-Gribov picture [@JETP29-483; @JETP30-709], as proposed in [@EPJC26-35] $$\begin{aligned} \scalebox{0.9}{$\sigma_{\textrm{dip}}^{\textrm{nucleus}}(x,r)=2\int d^2b'\left\{1-\textrm{exp}\left[-\frac{1}{2}T_A(b')\sigma_{\textrm{dip}}^{\textrm{proton}}(x,r)\right]\right\}$}\label{nuc}.\end{aligned}$$ In the Eq. (\[nuc\]), $T_{A}(b)$ is the nuclear overlap function and $\sigma_{dip}$ is the dipole-nucleon cross section, which was calculated in this work using the GBW and CGC dipole models. These two models shown a good agreement with the data in the ultraperipheral regime [@PRD94-094023]. The application of the equations (\[eq:xx\]) and (\[seg\]) inside of (\[pri\]) constitutes what we named the **scenario 1**, which produces the first results presented in Table \[dsigdy2\]. The Effective Photon Flux ========================= In order to refine our calculations, an effective photon flux was built as a function of usual photon flux (eq. \[eq:xx\]) following a similar procedure showed in [@PRC93-044912] where considered two hypothesis: (1) only the photons that reach the geometrical region of the nuclear target will be considered and (2) the photons that reach the overlap region will be neglected. Then, the new photon flux can be expressed as [@PRD96-056014] $$\begin{aligned} \scalebox{0.9}{$N^{eff}(\omega,b)=\int N^{usual}(\omega,b_1)\frac{\theta(b_1-R_A)\theta(R_A-b_2)}{A_{eff}(b)}d^2b_2$}\label{Neff}\end{aligned}$$ where the effective interaction area, $A_{eff}(b)$, is given by $$\begin{aligned} \scalebox{0.9}{$A_{eff}(b)=R_A^2\left[\pi-2\textrm{cos}^{-1}\left(\frac{b}{2R_A}\right)\right]+\frac{b}{2}\sqrt{4R_A^2-b^2}$},\end{aligned}$$ In the Figure \[fig:fig3\], the usual photon flux is compared with the effective photon flux for energy of the photon $\omega = 0.01$ GeV and $\omega = 1$ GeV (the photon flux is dominated by photons with energy $\lesssim 0.2$ GeV). As expected, both models become similar as $b\rightarrow\infty$. In the range $4\text{ fm}\lesssim b\lesssim 11\text{ fm}$, the usual photon flux is bigger than the effective photon flux, mainly on the threshold $b\sim R_A\sim 7$. Lastly, the usual photon flux diverges considerably from the effective photon flux as $b\rightarrow 0$. Considering the effective photon flux (Eq. (\[Neff\])) and the photonuclear cross section (\[seg\]), the rapidity distribution for $J/\psi$ photonuclear production was calculated in Pb-Pb collisions at $\sqrt{s} = 2.76$ TeV (Fig. \[psi1s\]) and $\sqrt{s} = 5.02$ TeV (Fig. \[psi1s5020\])for $y<|4.0|$. For each centrality class there is some difference in the $|y|\gtrsim 1.0$ range but, in general, the curves showed a similar behavior in relation to dipole models used. Moreover, analysing the different centrality classes, it was observed an increase of $\sim$ 12% from 70%-90% to 50%-70% and of $\sim$ 13.7% from 50%-70% to 30%-50%, for both dipole models, at $y=0$. Similarly, at $\sqrt{s}=5.02$ TeV, it was observed an increase of the $\sim$ 12% from 70%-90% to 50%-70% and $\sim$ 13.3% from 50%-70% to 30%-50% at $y=0$. Therefore, in the central region, the relative variation between the different centrality classes is not sensitive to the increase of the energy. The ratio $\frac{d\sigma^{5.02}}{dy}/\frac{d\sigma^{2.76}}{dy}$ was also investigated. An increase of approximately 30% in the central rapidity region $|y|<1.5$ was observed for the three centrality classes analyzed. However, this ratio is of the 60% for the same rapidity region in the UPC. Therefore, in this formalism the effective photon flux seems less sensitive with the variation of the energy in comparison with the usual photon flux. The Effective Photonuclear Cross Section ======================================== To ensure more consistence in the application of the effective photon flux, the photonuclear cross section also needs to be limited in accordance with the geometrical constraints adopted in the construction of the effective photon flux, in order to consider only the interaction of the photon with the non-overlap region. This is achieved by applying the $\Theta\left(b_1-R_A\right)$ function into Eq. (\[nuc\]), which produces $$\begin{aligned} \scalebox{1.0}{$\sigma_{\textrm{dip}}^{\textrm{nucleus}}(x,r)$}&\scalebox{0.7}{$=$}&\scalebox{0.7}{$2\int d^2b_2\Theta(b_1-R_A)$}\nonumber \\ & \scalebox{0.7}{$\times$} &\scalebox{0.7}{$\left\{1-\textrm{exp}\left[-\frac{1}{2}T_A(b_2)\sigma_{\textrm{dip}}^{\textrm{proton}}(x,r)\right]\right\}$}\label{sigeff}\end{aligned}$$ where, $b_1^2=b^2+b_2^2+2bb_2\textrm{cos}(\alpha)$. Considering the effective photon flux and photonuclear cross section, the rapidity distribution was calculated and its results for the three centrality classes (scenario 3) are shown in the Table (\[dsigdy2\]). Main Results {#result} ============ The results of the average rapidity distribution for the three scenarios described in the text are shown in the Table (\[dsigdy2\]), where were taken into account the GBW and CGC dipole models. It is observed that in the simplest approach (scenario 1), there is a good agreement with the ALICE data in the more peripheral region where the b-dependence is more weak. For the more central regions, the scenario 2 and scenario 3 are more suitable although both overestimate the central value of the ALICE in the 70%-90% region. In particular, the scenario 3 one has the largest production cross section in the 50-70% centrality class. This is due the dipole-nuclear cross section, which is b-dependent through the $T_A(b)$ function, to be more strongly suppressed in more central collisions. Consequently, the 30%-50% centrality class is more affected than the 50%-70% region. In addition to rapidity distribution, the excess of the $J/\psi$ also was quantified by nuclear modification factor, experimentally defined by Eq. (\[eq:raa\]), and calculated from the results presented in the Table (\[dsigdy2\]). Adopting the CGC model, which shows slightly better results than GBW model, the $R_{AA}$ was calculated for the three scenarios investigated and its results are compared with the ALICE data, Fig. \[raa\]. Similarly to rapidity distribution, the scenario 1 show better agreement in the more peripheral region while the scenarios 2 and 3 are more suitable for more central collisions where the b-dependence is more relevant. More details about each scenario can be found in [@1804.09836]. Summary ======= In this work, the estimates for the rapidity distribution and nuclear modification factor were presented for the $J/\Psi$ production in the centrality classes 30%-50%, 50%-70% and 70%-90%. The ALICE measurements were compared with our estimates, obtained from three different approaches. In the simplest approach (scenario 1), better aggrement was obtained with the data only in the more peripheral region, where there is a considerable uncertainty. For the more consistent approach (scenario 3), the result overestimate in the more peripheral region, however, it agrees better with the data in more central region, where the color dipole formalism is more intensely tested. Although it is not yet possible to confirm that the exclusive photoproduction is fully responsible for the $J/\psi$ excess observed in ALICE, there are indications that it produces a considerable part of the effect. We would like to thank Dr.Ionut Arsene for usefull discussions. This work was partially financed by the Brazilian funding agency CNPq. [99]{} ALICE Collaboration, J. Adam et al., Phys. Rev. Lett. 116, 222301, (2016). ALICE Collaboration, B. Abelev et al., Phys. Lett. B 734, 314, (2014). ALICE Collaboration, B. Abelev et al., Phys. Rev. C88, 044909, (2013). M. B. Gay Ducati and S. Martins, arXiv: 1804.09836 \[hep-ph\]. To appear in PRD. M. K. Gawenda and A. Szczurek, Phys. Rev. C93, 044912, (2016). F. Krauss, M. Greiner and G. Soff, Prog. Part. Nucl. Phys. 39, 503, (1997). K. T. R. Davies and J. R. Nix, Phys. Rev. C 14, 1977, (1976) A. L. Ayala Filho, M. B. Gay Ducati and E. M. Levin, Eur. Phys. J. C8, 115, (1999). M. B. Gay Ducati, F. Kopp, M.V.T. Machado and S. Martins, Phys. Rev. D94, 094023, (2016). H. Kowalski, L. Motyka and G. Watt, Phys. Rev. D74, 074016, (2006). V. N. Gribov, Sov. Phys. JETP 29, 483, (1969). V. N. Gribov, Sov. Phys. JETP 30, 709, (1970). N. Armesto, Eur. Phys. J. C 26, 35, (2002). M. B. Gay Ducati and S. Martins, Phys. Rev. D96, 056014, (2017). [^1]: [email protected] [^2]: [email protected]

--- abstract: 'We discuss actions of free groups on the circle with “ping-pong” dynamics; these are dynamics determined by a finite amount of combinatorial data, analogous to Schottky domains or Markov partitions. Using this, we show that the free group $F_n$ admits an isolated circular order if and only if $n$ is even, in stark contrast with the case for linear orders. This answers a question from [@MR]. Inspired by work in [@markov-partition-vfree], we also exhibit examples of “exotic” isolated points in the space of all circular orders on $F_2$. Analogous results are obtained for linear orders on the groups $F_n\times \mathbb Z$.[^1]' author: - 'Dominique Malicet [^2]' - 'Kathryn Mann [^3]' - 'Cristóbal Rivas [^4]' - 'Michele Triestino [^5]' title: 'Ping-pong configurations and circular orders on free groups' --- Introduction ============ Let $G$ be a group. A (left-invariant) *linear order*, often called a *left order* on $G$ is a total order invariant under left multiplication. Left-invariance directly implies that the order is determined by the set of elements greater than the identity, called the *positive cone*. It is often far from obvious whether a given order can be determined by only finitely many inequalities, or whether a given group admits such a finitely-determined order. This latter question turns out to be quite natural from an algebraic perspective, and can be traced back to Arora and McCleary [@Arora-McCleary] for the special case of free groups. McCleary answered this question for free groups shortly afterwards, showing that $F_n$ has no finitely determined orders [@McCleary2]. The question of finite determination gained a topological interpretation following Sikora’s definition of the *space of linear orders* on $G$ in [@sikora]. This space, denoted $\operatorname{\mathsf{LO}}(G)$, is the set of all linear orders on $G$ endowed with the topology generated by open sets $$U_{(\preceq, X)} :=\{ {\preceq'} \mid {x\preceq' y } \text{ iff } {x \preceq y} \text{ for all } x, y \in X\}$$ as $X$ ranges over all finite sets of $G$. Finitely determined linear orders on $G$ are precisely the *isolated points* of $\operatorname{\mathsf{LO}}(G)$; going forward, we will refer to these as *isolated orders*. This correspondence between isolated points and finitely determined orders is perhaps the simplest instance of the general theme that topological properties of $\operatorname{\mathsf{LO}}(G)$ should reflect algebraic properties of $G$. Presently, several families of groups are known to either admit or fail to admit isolated orders, with proofs that use both purely algebraic and dynamical methods. Some examples of groups that do *not* admit isolated orders include free abelian groups [@sikora], free groups [@McCleary2; @NavasFourier], free products of arbitrary linearly orderable groups [@RivasLO], and some amalgamated free products such as fundamental groups of orientable closed surfaces [@ABR]. Large families of groups which do have isolated orders include braid groups [@DD; @DehornoyBook], groups of the form $\langle x,y\mid x^n=y^m\rangle$ ($n, m\in {\mathbb{Z}}$) [@ItoAlg; @NavasAlg], and groups with triangular presentations [@dehornoy14]. (In fact, all of these latter examples have orders for which the positive cone is finitely generated as a semi-group, a strictly stronger condition.) As a consequence of our work here, we give a family of groups where, interestingly, *both* behaviors occur. \[LO\_thm\] Let $F_n$ denote the free group of $n$ generators. The group $F_n\times {\mathbb{Z}}$ has isolated linear orders if and only if $n$ is even. This result appears to give the first examples of any group $G$ with a finite index subgroup $H$ (in this case $F_n \times {\mathbb{Z}}\subset F_m \times {\mathbb{Z}}$, for $n$ odd and $m$ even) such that $\operatorname{\mathsf{LO}}(G)$ and $\operatorname{\mathsf{LO}}(H)$ are both infinite, but only $G$ contains isolated points. Theorem \[LO\_thm\] also has an interesting consequence regarding the *space of marked groups*. As shown in [@champ-guira Prop. 2.13], the set of left-orderable groups is a closed subset of the space of marked groups on $n$ generators. However, Theorem \[LO\_thm\] implies that this is not the case either for the subset of groups admitting isolated linear orders (or its complement): one may take a sequence of markings of $F_2 \times {\mathbb{Z}}$ so as to approach $F_3\times {\mathbb{Z}}$, and similarly, a sequence of markings on $F_3\times{\mathbb{Z}}$ can be chosen to approach $F_4\times {\mathbb{Z}}$ (see [@champ-guira §2.4]). Thus, Theorem \[LO\_thm\] immediately gives the following. \[cor:marked\] In the space of finitely generated marked groups, having an isolated linear order is neither a closed, nor an open property. The main tool for Theorem \[LO\_thm\], and main focus of this work, is the study of *circular orders* on $F_n$ and the dynamics of their corresponding actions of $F_n$ on ${\mathbb{S}^1}$. It is well known that, for countable $G$, admitting a linear order is equivalent to acting faithfully by orientation-preserving homeomorphisms on the line. In the same vein, a *circular order* on $G$ is an algebraic condition which, for countable groups, is equivalent to acting faithfully by orientation-preserving homeomorphisms on ${\mathbb{S}^1}$. We recall the definition and basic properties in Section \[CO\_sec\]. Any action of $G$ on ${\mathbb{S}^1}$ lifts to an action of a central extension of $G$ by ${\mathbb{Z}}$ on the line, giving us a way to pass between circular and linear orders on these groups, and giving us many dynamical tools for their study. Analogous to $\operatorname{\mathsf{LO}}(G)$, one can define a *space of circular orders* $\operatorname{\mathsf{CO}}(G)$. In [@MR], the second and third authors showed that a circular order on $F_n$ is isolated if and only if the corresponding action on the circle has what they called *ping-pong dynamics*. They gave examples of isolated circular orders on free groups of even rank, but the odd rank case was left as an open problem. Here we answer this question in the negative: \[CO\_thm\] $F_n$ admits an isolated circular order if and only if $n$ is even. Similarly to Corollary \[cor:marked\], one can also prove that the set of groups admitting isolated circular orders is neither closed nor open in the space of marked groups. We prove Theorem \[CO\_thm\] by developing a combinatorial tool for the study of actions on ${\mathbb{S}^1}$ with ping-pong dynamics (similar to actions admitting Markov partitions), inspired by the work in [@DKN2014] and [@markov-partition-vfree]. We expect these to have applications beyond the study of linear and circular orders; one such statement is given in Theorem \[t:gen\_ping-pong\]. The notion of ping-pong dynamics is defined and motivated in the next section. Sections \[CO\_sec\] and \[LO\_sec\] give the application to the study of circular and linear orders, respectively, and the proofs of Theorems \[LO\_thm\] and \[CO\_thm\]. Ping-pong actions and configurations {#s:PingPong} ==================================== \[pp\_action\_def\] Let $G=F_n$ be the free group of rank $n$, freely generated by $S=\{ a_1,\ldots,a_n\}$. A *ping-pong action* of $(G, S)$ on ${\mathbb{S}^1}$ is a representation $\rho:G\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ such that there exist pairwise disjoint open sets $D(a)\subset {\mathbb{S}^1}$, $a\in S\cup S^{-1}$, each of which has finitely many connected components, and such that $\rho(a) \left ({\mathbb{S}^1}\setminus D(a^{-1})\right ) \subset \overline{D(a)}$. We further assume that if $I$ and $J$ are any connected components of $D(a)$, then $\bar{I} \cap \bar{J} = \emptyset$. We call the sets $D(a)$ the *ping-pong domains* for $\rho$. A similar definition is given in [@MR], with the additional requirement that ping-pong domains be closed. The above, more general definition is more natural for our purposes, although we will later introduce Convention \[Closures\] to reconcile the two. The reader may notice that, for a given ping-pong action $\rho$ of $(G, S)$, there can be many choices of sets $D(a)$ satisfying the property in Definition \[pp\_action\_def\]. For instance, if $\rho$ is a ping-pong action such that $\bigcup_{a \in S \cup S^{-1}} \overline{D(a)} \neq {\mathbb{S}^1}$, then one may choose an arbitrary open set $I$ disjoint from $\bigcup_{a \in S \cup S^{-1}} \overline{D(a)}$ and replace $D(a_1)$ with $D(a_1) \cup I$, leaving the other domains unchanged. These new domains still satisfy $\rho(a) \left ({\mathbb{S}^1}\setminus D(a^{-1})\right ) \subset \overline{D(a)}$. Later we will adopt a convention to avoid this kind of ambiguity. #### Motivation: why ping-pong actions? The classical ping-pong lemma implies that ping-pong actions are always faithful, and a little more work shows that the action is determined up to semi-conjugacy by a finite amount of combinatorial data coming from the cyclic ordering and the images of the connected components of the sets $D(a)$ (see Definition \[pp\_config\_def\] and Lemma \[config\_determines\] below, or [@BP Thm. 4.7]). In particular, one can think of ping-pong actions as the family of “simplest possible” faithful actions of $F_n$ on ${\mathbb{S}^1}$, and it is very easy to produce a diverse array of examples. Perhaps the best-known examples are the actions of discrete, free subgroups of ${\mathsf{PSL}}(2,{\mathbb{R}})$ on ${\mathbb{R}}{\mathsf{P}}^1$. For these actions, one can choose domains $D(a)$ with a single connected component. Figure \[PP-0\_fig\] shows an example of the dynamics of such an action of $F_2 = \langle a, b \rangle$. $$\includegraphics[scale=1]{PingPong-0.pdf}$$ Despite their simplicity, ping-pong actions are quite useful. For instance, in [@markov-partition-vfree] ping-pong actions were used to construct the first known examples of discrete groups of real-analytic circle diffeomorphisms acting minimally, but not conjugate to a subgroup of a finite central extension of ${\mathsf{PSL}}(2,{\mathbb{R}})$. This was a by-product of a series of papers concerning longstanding open conjectures of Hector, Ghys and Sullivan on the relationship between minimality and ergodicity of a codimension-one foliation (see for instance [@DKN2009; @FK2012; @DKN2014]). In general, it is quite tractable to study the dynamic and ergodic properties of a ping-pong action (or a Markov system), and this program has been carried out by many authors [@CC1; @CC2; @IM; @Matsumoto; @mane]. Basic properties ---------------- \[reduced\_lem\] Given a ping-pong action of $(G, S)$, there exists a choice of ping-pong domains $D(a)$ such that $\rho(a)\left ({\mathbb{S}^1}\setminus \overline{D(a^{-1})}\right ) = D(a)$ holds for all $a \in S \cup S^{-1}$. Let $\rho$ be a ping-pong action with sets $D(a)$ given. We will modify these domains to satisfy the requirements of the lemma. For each generator $a \in S$ (recall this is the free, *not* the symmetric, generating set), we shrink the domain $D(a)$, setting $D'(a) := \rho(a)\left ({\mathbb{S}^1}\setminus \overline{D(a^{-1})}\right ).$ Applying $a^{-1}$ to both sides of the above expression gives $\rho(a^{-1})(D'(a)) = {\mathbb{S}^1}\setminus \overline{D(a^{-1})}$. Moreover, since the connected components of $D(a)$ have disjoint closures, and the same holds for $D(a^{-1})$, hence also for $D'(a)$, we also have $\rho(a^{-1})\left (\overline{D'(a)}\right ) = {\mathbb{S}^1}\setminus D(a^{-1})$; or equivalently $\rho(a^{-1})\left ({\mathbb{S}^1}\setminus \overline{D'(a)}\right ) = D(a^{-1})$. This is what we needed to show. \[reduced\_con\] From now on, we assume all choices of domains $D(a)$ for every ping-pong action are as in Lemma \[reduced\_lem\]. In particular, this means that, for each $a \in S$, the sets of connected components $\pi_0(D(a))$ and $\pi_0(D(a^{-1}))$ have the same cardinality, and $\rho(a)$ induces a bijection between the connected components of ${\mathbb{S}^1}\setminus \overline{D(a^{-1})}$ and connected components of $D(a)$. \[pp\_config\_def\] Let $\rho$ be a ping-pong action of $(G, S)$. The *ping-pong configuration* of $\rho$ is the data consisting of 1. the cyclic order of the connected components of $\bigcup_{a\in S\cup S^{-1}}D(a)$ in ${\mathbb{S}^1}$, and 2. for each $a\in S\cup S^{-1}$, the assignment of connected components $$\lambda_a:\pi_0\left (\bigcup_{b\in S\cup S^{-1}\setminus \{a^{-1}\}}D(b)\right )\to \pi_0\left (D(a)\right )$$ induced by the action. Note that [*not*]{} every abstract assignment $\lambda_a$ as in the definition above can be realized by an action $F_2 \to {\mathsf{Homeo}}_+({\mathbb{S}^1})$. The following construction gives one way to produce some large families of examples. \[easy\_ex\] For $a \in S$, let $X_a$ and $Y_a \subset {\mathbb{S}^1}$ be disjoint sets each of cardinality $k(a)$, for some integer $k(a) \geq 1$, such that every two points of $X_a$ are separated by exactly one point of $Y_a$. Choose these so that all the sets $X_a \cup Y_a$ are pairwise disjoint as $a$ ranges over $S$. Let $D(a)$ and $D(a^{-1})$ be neighborhoods of $X_a$ and $Y_a$, respectively, chosen small enough so that all these sets remain pairwise disjoint. Now one can easily construct a piecewise linear homeomorphism (or even a smooth diffeomorphism) $\rho(a)$ with $X_a$ as its set of attracting periodic points, and $Y_a$ as the set of repelling periodic points such that $\rho(a)\left ({\mathbb{S}^1}\setminus D(a^{-1})\right ) = D(a)$. The assignments $\lambda_a$ are now dictated by the period of $\rho(a)$ and the cyclic order of the sets $X_a$ and $Y_a$. While the reader should keep the construction above in mind as a source of examples, we will show in Example \[ex:strange\] that not every ping-pong configuration can be obtained in this manner. However, the regularity (PL or smooth) in the construction is attainable in general. The following construction gives one possibility for a PL realization that will be useful later in the text. We leave the modifications for the smooth case as an easy exercise. \[GoodRealization\] Given a ping-pong action $\rho_0$ of $(G,S)$ with domains $\{D_0(a)\}_{a\in S\cup S^{-1}}$ following Convention \[reduced\_con\], one can find another ping-pong action $\rho$ of $(G,S)$ with domains $\{D(a)\}_{a\in S\cup S^{-1}}$such that: 1. the action $\rho$ is piecewise linear, 2. there exists $\mu>1$ such that for any $a\in S\cup S^{- 1}$, one has $\rho(a)'\vert_{D(a^{-1})}\ge \mu$, and 3. the actions $\rho_0$ and $\rho$ have the same ping-pong configuration. Let $\rho_0$ be as in the statement of the Lemma. For each $a \in S \cup S^{-1}$, replace the original domains by smaller domains $D(a) \subset D_0(a)$, chosen small enough so that the largest connected component of $D(a)$ is at most half the length of the smallest connected component of ${\mathbb{S}^1}\setminus D(a^{-1})$. We require also that $D(a)$ has exactly one connected component in each connected component of $D_0(a)$. Now define $\rho(a)$ as a piecewise linear homeomorphism that maps connected components of ${\mathbb{S}^1}\setminus D(a^{-1})$ onto connected components of $D(a)$ linearly following the assignment $\lambda_a$. The next definition and proposition give a further means of encoding the combinatorial data of a ping-pong action. This will be used later in the proof of Theorem \[CO\_thm\]. \[d:graph\] Let $\rho$ be a ping-pong action of $(G, S)$ with domains $D(a)$. For each $a\in S$, we define an oriented bipartite graph $\mathsf{\Gamma}_a$ with vertex set equal to $\pi_0(D(a)) \cup \pi_0(D(a^{-1}))$, and edges defined as follows: - For $I^+\in \pi_0(D(a))$, let $J^+$ denote the connected component of ${\mathbb{S}^1}\setminus D(a)$ adjacent to $I^+$ on the right. Put an oriented edge from $I^+$ to an interval $I^- \in \pi_0(D(a^{-1}))$ if and only if $\rho(a^{-1})(J^+)= \overline{I^-}$. - Similarly, for $I^- \in \pi_0(D(a))$, with $J^-$ the adjacent interval of ${\mathbb{S}^1}\setminus D(a^{-1})$ on the right, put an oriented edge from $I^-\in \pi_0(D(a^{-1}))$ to $I^+\in \pi_0(D(a))$ if and only if $\rho(a)(J^-)= \overline{I^+}$. \[p:graph\] Let $\rho:G\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ be a ping-pong action of a free group $(G,S)$. Then, for each generator $a\in S$, there exists $k(a) \in {\mathbb{N}}$ such that the graph $\mathsf\Gamma_a$ is an oriented $2k(a)$-cycle. First, the construction of the graph ensures that it is bipartite, and that each vertex has at most one outgoing edge. As a consequence of Convention \[reduced\_con\], for any $s\in S\cup S^{-1}$ and connected component $J$ of ${\mathbb{S}^1}\setminus D(s^{-1})$, there exists $I\in \pi_0(D(s))$ such that $\rho(s)(J)= \overline{I}$, so each vertex does indeed have an outgoing edge. Moreover, if $J' \neq J$ is a different connected component, then $\rho(s)(J) \cap \rho(s)(J') = \emptyset$, so each vertex $I$ has a unique incoming edge. This shows that $\mathsf \Gamma_a$ is a union of disjoint cycles, and it remains only to prove that the graph is connected. To show connectivity, let $I^{-}$ be a connected component of $D(a^{-1})$ and consider the connected component $J^+$ of ${\mathbb{S}^1}\setminus D(a)$ such that $\rho(a^{-1})(J^+)=\overline{I^-}$. Let $I^+_1$ and $I^+_2$ be the connected components of $D(a)$ (possibly the same) which are adjacent to $J^+$ on either side. By definition of the graph $\mathsf{\Gamma}_a$, the intervals $I^+_1,I^-,I^+_2$ are consecutive vertices in the same cycle of the graph. And vice versa: if three intervals $I^+_1,I^-,I^+_2$ are consecutive vertices, then $J^+:=\rho(a)(\overline{I^-})$ is the connected component of ${\mathbb{S}^1}\setminus D(a)$ adjacent to both $I^+_1$ and $I^+_2$. This proves that if $I^+_1$ and $I^+_2$ are consecutive connected components of $D(a)$ in ${\mathbb{S}^1}$, then they belong to the same cycle in $\mathsf\Gamma_a$. Hence we easily deduce that *all* connected components of $D(a)$ are in the same cycle in $\mathsf\Gamma_a$. The same also holds for the components of $D(a^{-1})$, and the graph is connected. Left-invariant circular orders {#CO_sec} ============================== We begin by quickly recalling standard definitions and properties. A reader familiar with circular orders may skip to Section \[dynam\_sec\]. \[dfn:CO\] Let $G$ be a group. A *left-invariant circular order* is a function $c:G\times G\times G\to \{0,\pm 1\}$ such that 1. $c$ is *homogeneous*: $c(\gamma g_0,\gamma g_1,\gamma g_2)=c(g_0,g_1,g_2)$ for any $\gamma,g_0,g_1,g_2\in G$; 2. $c$ is a *$2$-cocycle* on $G$: $$c(g_1,g_2,g_3)-c(g_0,g_2,g_3)+c(g_0,g_1,g_3)-c(g_0,g_1,g_2)=0\quad\text{for any }g_0,g_1,g_2,g_3\in G;$$ 3. \[cond:3\] $c$ is *non-degenerate*: $c(g_0,g_1,g_2)=0$ if and only if $g_i=g_j$ for some $i\neq j$. The *space* of all left-invariant circular orders on $G$, denoted $\operatorname{\mathsf{CO}}(G)$, is the set of all such functions, endowed with the subset topology from $\{0,\pm 1\}^{G\times G\times G}$ (with the natural product topology). Although spaces of left-invariant linear orders have been well-studied, there are very few cases where the topology of $\operatorname{\mathsf{CO}}(G)$ is completely understood. Other than a few sporadic examples, the only complete description of spaces of circular orders known to the authors comes from [@CMR], which gives a classification of all groups such that $\operatorname{\mathsf{CO}}(G)$ is finite, and also a proof that $\operatorname{\mathsf{CO}}(A)$ is homeomorphic to a Cantor set for any Abelian group $A$. Given that left-orders on free groups are well understood, a natural next case of circular orders to study is $\operatorname{\mathsf{CO}}(F_n)$. Our main tool for this purpose is the following classical relationship between circular orders and actions on ${\mathbb{S}^1}$ (see [@Calegari; @MR]). \[p:realization\] Given a left-invariant circular order $c$ on a countable group $G$, there is an action $\rho_c:G\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ such that $c(g_0,g_1,g_2) = {\mathsf{ord}}\left (\rho_c(g_0)(x), \rho_c(g_1)(x), \rho_c(g_2)(x)\right )$ for some $x \in {\mathbb{S}^1}$, where ${\mathsf{ord}}$ denotes cyclic orientation. Moreover, there is a canonical procedure for producing $\rho_c$ which gives a well-defined conjugacy class of action. This conjugacy class is called the *dynamical realization of $c$ with basepoint $x$*. A description of this procedure is given in [@MR], modeled on the analogous linear case (see e.g. [@Ghys]). Note that modifying a dynamical realization by *blowing up* the orbit of some point $y \notin \rho(G)(x)$ may result in a non-conjugate action that still satisfies the property $c(g_0,g_1,g_2) = {\mathsf{ord}}\left (\rho_c(g_0)(x), \rho_c(g_1)(x), \rho_c(g_2)(x)\right )$. However, this non-conjugate action cannot be obtained through the canonical procedure. \[r:induce\] The converse to the above proposition is also true: if $G$ is a countable subgroup of ${\mathsf{Homeo}}_+({\mathbb{S}^1})$, then $G$ admits a circular order. A proof is given in [@Calegari Thm. 2.2.14][^6]. As a special case, if $\rho:G \to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ is such that some point $x$ has trivial stabilizer, then we may define an *induced order* on $G$ by $$c(g_1, g_2, g_3):={\mathsf{ord}}(\rho(g_1)(x),\rho(g_2)(x),\rho(g_3)(x)).$$ While one cannot expect in general to find a point with trivial stabilizer, this does hold for ping-pong actions by the following lemma. \[config\_determines\] Suppose that $\rho$ is a ping-pong action of $(G, S)$ with domains $D(a)$. If $x_0 \in {\mathbb{S}^1}\setminus \bigcup_{a \in S \cup S^{-1}} D(a)$, then the orbit of $x_0$ is free and its cyclic order is completely determined by the cyclic order of the elements of $\left \{ \pi_0 \left( \bigcup_{a\in S\cup S^{-1}}D(a) \right), \{x_0\} \right \}$ and the assignments $\lambda_a$. The proof is obtained by a careful reading of the standard proof of the classical ping-pong lemma. Details are given in [@MR Lemma 4.2]. Isolated circular orders on free groups {#dynam_sec} --------------------------------------- In this section we will use ping-pong actions to prove Theorem \[CO\_thm\] from the introduction. As this builds on the framework of [@MR], we start by introducing two results obtained there. Let $G$ be any group, and $\rho: G \to {\mathsf{Homeo}}_+({\mathbb{S}^1})$. Recall that, if $\rho(G)$ does not have a finite orbit, then there is a unique closed, $\rho(G)$-invariant set contained in the closure of every orbit, called the *minimal set* of $\rho(G)$. We denote this set by $\Lambda(\rho)$. If $\Lambda(\rho) = {\mathbb{S}^1}$, the action is called *minimal*. Otherwise, $\Lambda(\rho)$ is homeomorphic to a Cantor set and $\rho$ permutes the connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$. While, for many examples of actions, the permutation will have many disjoint cycles, the next lemma states that this is not the case for dynamical realizations. \[transitive\_lem\] Let $\rho:G\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ be a dynamical realization of a circular order $c$. Suppose that $\rho$ has a minimal invariant Cantor set $\Lambda(\rho)$. Then $\rho$ acts transitively on the set of connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$. Since a ping-pong action of a free group of rank at least 2 cannot have finite orbits, invariance of the minimal set immediately implies that $\Lambda(\rho) \subset \bigcup_{a \in S \cup S^{-1}} \overline{D(a)}$. If additionally, for each $s \neq t \in S \cup S^{-1}$, one has $\overline{D(s)} \cap \overline{D(t)} = \emptyset$, then invariance of $\Lambda(\rho)$ and the definition of ping-pong implies that in fact $\Lambda(\rho) \subset \bigcup_{a \in S \cup S^{-1}} {D(a)}$. Going forward, it will be convenient to have the this stronger condition, which is given by following lemma. Let $\rho_0$ be a ping-pong action of $(G, S)$ with domains $D_0(a)$ for $a \in S \cup S^{-1}$. Then there exists an action $\rho$ with the same ping-pong configuration as $\rho_0$ and with domains $D(a)$ satisfying $\overline{D(s)} \cap \overline{D(t)} = \emptyset$ whenever $s \neq t$. Let $\rho_0$ be a ping-pong action. There are only finitely many points $x$ contained in sets of the form $\overline{D_0(s)} \cap \overline{D_0(t)}$ for $t \neq s \in S \cup S^{-1}$. For each such point $x$, blow up its orbit, replacing each point $y\in \rho_0(G)(x)$ with an interval $I_{y}$; if lengths of the $I_{y}$ are chosen so their sum converges, then we obtain a new circle, say $ \hat{\mathbb S}^1$, with a natural continuous, degree one map $h: \hat{\mathbb S}^1 \to {\mathbb{S}^1}$ given by collapsing each $I_{y}$ to the point $y$. For each $s\in S\cup S^{-1}$, let $D_1(s) \subset \hat{\mathbb S}^1$ be the preimage of $D_0(S)$ under $h$. Since $x$ has trivial stabilizer and $G$ is free, we may extend the action of $G$ to this new circle by allowing $a \in S$ to act as any orientation-preserving map from $I_y$ to $I_{a(y)}$. We now show that we may choose maps in such a way as to achieve a ping-pong action with the desired properties. For each inserted interval $I = [p,q]$ that is adjacent to a set of the form $D_1(s)$ on the left and $D_1(t)$ on the right (where $s, t \in S \cup S^{-1}$), fix points $p_s < p_t$ in the interior of $I$ and extend $D_1(s)$ into $I$ to include $[p, p_s)$ and $D_1(t)$ to include $(p_t, q]$. Having done this on each such interval, let $D(s)$ denote the new extended domains, and note that these have disjoint closures. Now for $a \in S$, define the action of $a$ on such an interval $I = I_y$ as follows. If $\rho_0(a)(y) \in D_0(a)$, the restriction of $\rho$ to $I$ may be any orientation-preserving homomorphism between $I_y$ and $I_{a(y)}$. Otherwise, $I_y$ is adjacent to $D_1(a^{-1})$ either on the right or the left, and we define $\rho(a)$ on $I_y$ to map the chosen point $p_{a^{-1}} \in I_y$ to the point $p_a$ in $I_{a(y)}$. This ensures that $\rho(a)(S^1 \setminus \overline{D(a^{-1})}) = D(a)$, so that these are indeed ping-pong domains for the action. Finally, note that by construction, the ping-pong configuration has not changed. \[Closures\] In a ping-pong action of $(G, S)$, we assume from now on that the domains $D(s)$ satisfy $\overline{D(s)}\cap \overline{D(t)}=\emptyset$ whenever $s \neq t\in S\cup S^{-1}$. It follows easily from invariance of $\Lambda(\rho)$ and the definition of ping-pong that, for actions as in Convention \[Closures\], we have the inclusion $\Lambda(\rho)\subseteq \bigcup_{a \in S \cup S^{-1}} {D(a)}$. The following theorem from [@MR] relates circular orders and ping-pong actions. \[alternate\_thm\] Let $G=F_n$ be a free group. A circular order $c\in \operatorname{\mathsf{CO}}(G)$ is isolated if and only if its dynamical realization $\rho_c:G\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ is a ping-pong action satisfying Convention \[Closures\]. With these tools, we proceed to the main goal of this section. The case where $n$ is even is covered in [@MR]. As explained there, the representation of $G$ into ${\mathsf{PSL}}(2,{\mathbb{R}})$ coming from a hyperbolic structure on a genus $n/2$ surface with one boundary component gives an isolated circular order. (In fact, by taking lifts to cyclic covers, one can obtain infinitely many isolated circular orders in distinct equivalence classes under the action of ${\mathsf{Aut}}(F_n)$ on $\operatorname{\mathsf{CO}}(G)$.) To show that $F_n$ does not admit an isolated circular order when $n$ is odd, we need more work. We begin with some generalities, applicable to free groups of any rank (even or odd). Suppose that $\rho: F_n\to {\mathsf{Homeo}}_+({\mathbb{S}^1})$ is a dynamical realization of an isolated circular order, and fix a free generating set $S = \{a_1, \ldots, a_n\}$ for $F_n$. By Theorem \[alternate\_thm\] and Lemma \[transitive\_lem\], $\rho$ is a ping-pong action with domains satisfying Convention \[Closures\] and the connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$ form a unique orbit. Let $c_0,\ldots, c_r$ be the (finitely many) connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$ that are not contained in any domain $D(s)$. Suppose that $c_i$ has endpoints in $D(s)$ and $D(t)$, for some $s \neq t$. Then, for any generator $u\notin\{s^{-1},t^{-1}\}$, we have that $\rho(u)(c_i)\in D(u)$. In addition, we have that $\rho(s^{-1})(c_i)$ and $\rho(t^{-1})(c_i)$ belong to $\{c_0,\ldots,c_r\}$: indeed, the intersection $c_i\cap D(s)$ is nonempty; its image by $\rho(s^{-1})$ is contained in ${\mathbb{S}^1}\setminus (\Lambda\cup D(s^{-1}))$ and is adjacent to $D(s^{-1})$ because of Convention \[reduced\_con\]; moreover, it has to intersect some $c_j$ because of Convention \[Closures\]. Then we must have $\rho(s^{-1})(c_i)=c_j$. The same holds for $t^{-1}$. This implies that $c_i$ and $c_j$ are in the same orbit if and only if they are equivalent under the equivalence relation $\sim$ on $\{c_0,\ldots,c_r\}$ generated by $$c_i \sim c_j \quad\text{if there exists } t \in S \cup S^{-1} \text{ such that } c_i = \rho(t)(c_j) \text{ and } c_i \cap D(t) \neq \emptyset$$ We will now argue that the number of equivalence classes under this relation can be 1 only if $n$ is even. This is done by using the combinatorial data of the graphs from Definition \[d:graph\] to build a surface with boundary using the disc, and then making an Euler characteristic argument. For each generator $a\in S$, let $k(a)$ be the integer given by Proposition \[p:graph\]. Let $P_a$ be a $4k(a)$-gon (topologically a disc) with cyclically ordered vertices $v_1, v_2, \ldots, v_{4k(a)}$. Choose a connected component $I = [x_1,y_1]$ of $\overline{D(a)}$ and glue the oriented edge $v_1v_2$ to $I$ so as to agree with the orientation of $I \subset {\mathbb{S}^1}$. Then glue the edge $v_3 v_4$ to the connected component of $\overline{D(a^{-1})}$ containing $\rho(a^{-1})(x_1)$, according to the orientation in ${\mathbb{S}^1}$. Let $y_2$ denote the other endpoint of this connected component, and glue $v_5v_6$ to the connected component of $\overline{D(a)}$ containing $\rho(a)(y_2)$. Iterate this process until all edges $v_{2j-1}v_{2j}$ have been glued to ${\mathbb{S}^1}= \partial {\mathbb{D}}$. Our convention to follow the orientation of ${\mathbb{S}^1}$ implies that the resulting surface with boundary is orientable. Note that the remaining (unglued) edges of $P_a$ correspond exactly to the edges of the graph $\mathsf{\Gamma}_a$ from Definition \[d:graph\]; precisely, collapsing each connected component of $D(a)$ and of $D(a^{-1})$ to a point representing a vertex recovers the cycle $\mathsf{\Gamma}_a$. Now repeat this procedure for each generator in $S$, to obtain an orientable surface with boundary, which we will denote by $\Sigma$. A cartoon of the result of this procedure for the ping-pong action of Example \[ex:strange\] is shown in Figure \[fig;surface\_exotic\], and may be helpful to the reader. We claim that the number of boundary components of the surface $\Sigma$ is exactly the number of equivalence classes of the relation $\sim$. To see this, we proceed as follows. By construction, the connected components of $\partial \Sigma \cap \{c_0, \ldots, c_r\}$ are exactly the intervals $c_i$. If some interval $c_i$ has endpoints in $D(s)$ and $D(t)$, then $\partial \Sigma \cap c_i$ is joined to $\rho(t^{-1}) c_i \cap \partial \Sigma$ and $\rho(s^{-1})c_i \cap \partial \Sigma$ by edges of $P_s$ and $P_t$ respectively. Thus, $c_i \sim c_j$ implies that $c_i$ and $c_j$ lie in the same boundary component of $\Sigma$, and the intersection of that boundary component with $\{c_0, \ldots, c_r\}$ defines an equivalence class. This proves the claim. We now compute the Euler characteristic of $\Sigma$ and conclude the proof. Proposition \[p:graph\] implies that the gluing of $P_a$ described in our procedure adds one face and $2k(a)$ edges to the existing surface. Therefore after all the polygons $P_a$ (as $a$ ranges over elements of $S$) have been glued, the surface $\Sigma$ obtained has $\chi(\Sigma)\equiv n+1$ mod 2. Since $\Sigma$ is orientable, $\chi(\Sigma)$ agrees mod 2 with the number of boundary components of $\Sigma$, which, by our claim proved above, agrees with the number of equivalence classes of $c_i$. As discussed above, if $\rho$ is the dynamical realization of an isolated order then this number is equal to 1, hence $n+1 \equiv 1$ mod 2, and $n$ must be even. The proof above can be improved to give a statement about general ping-pong actions: \[t:gen\_ping-pong\] Let $G=F_n$ be the free group of rank $n$ with free generating set $S$. Consider a ping-pong action $\rho$ of $(G,S)$ satisfying Conventions \[reduced\_con\] and \[Closures\]. Let $\Lambda(\rho)$ be the minimal invariant Cantor set for the action. Then the number of orbits of connected components of the complement ${\mathbb{S}^1}\setminus \Lambda(\rho)$ is congruent to $n+1$ mod 2. As in the previous proof, let $c_0, c_1,\ldots,c_r$ be the connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$ that are not contained in any domain $D(s)$, and recall that $G$ permutes the connected components of ${\mathbb{S}^1}\setminus \Lambda(\rho)$. We claim that each cycle of this permutation contains at least one of the $c_i$. Given this claim, we may construct an orientable surface $\Sigma$ as in the proof of Theorem \[CO\_thm\], whose boundary components count the number of cycles. Computing Euler characteristic as above shows that the number of cycles is congruent to $n+1$ mod 2. We now prove the claim. Suppose that $I$ is a connected component of ${\mathbb{S}^1}\setminus \Lambda(\rho)$ contained in some $D(s)$. By Lemma \[GoodRealization\], we can take $\rho(s)$ to be piecewise linear, and such that each $\rho(s^{-1})$ expands $D(s)$ uniformly, increasing the length of each connected component by a factor of some $\mu > 1$, independent of $s$. Iteratively, assuming that $\rho(s_k s_{k-1} \cdots s_1) (I) \subset D(s_{k+1}^{-1})$, then the length of $\rho(s_{k+1} s_k\cdots s_1) (I)$ is at least $\mu^{k+1}\operatorname{\mathrm{length}}(I)$. This process cannot continue indefinitely, so some image of $I$ is not contained in a ping-pong domain. Exotic examples --------------- To indicate some of the potential difficulty of the problem of classifying all isolated orders on $F_n$, we give an example of a ping-pong configuration for $F_2$ that, even after applying an automorphism of $F_2$, cannot arise from the construction in Example \[easy\_ex\]. $$\includegraphics[scale=1]{Surface-2.pdf} \qquad \includegraphics[scale=1]{Surface-3.pdf}$$ $$\includegraphics[scale=.9]{Strange-0.pdf} \quad \includegraphics[scale=.9]{Strange-1.pdf}$$ \[ex:strange\] Let $F_2 = \langle a, b\rangle$ and consider a ping-pong action where $\rho(b)$ is as defined by the graph in Figure \[fig:strange\], and $\rho(a)$ is a hyperbolic element of ${\mathsf{SL}}(2,{\mathbb{R}})$ chosen so that the connected components of the domains for the ping-pong action are in cyclic order as follows: $$D(b^{-1}), D(a^{-1}), D(b^{-1}), D(a), D(b), D(a^{-1}), D(b), D(a)$$ (we are abusing notation slightly here, using each appearance of $D(s)$ to stand for a connected component of $D(s)$). See Figure \[fig;surface\_exotic\] (left) for an illustration of the domains and the surface $\Sigma$ constructed in the proof of Theorem \[CO\_thm\]. Since $\rho(b)$ has two hyperbolic fixed points, and $\rho(a)$ has four, this example is not realized by a ping-pong action in ${\mathsf{PSL}}(2,{\mathbb{R}})$, nor in any finite extension of it. In fact, the ping-pong configuration for $\rho(b)$ alone is atypical, in the sense that it is not the classical ping-pong configuration for a hyperbolic element in ${\mathsf{PSL}}(2,{\mathbb{R}})$ – $\rho(b)$ has a “slow” contraction on the left half of the circle, as two iterations of $\rho(b)$ are needed in order to bring the external gaps of $D(b)\cup D(b^{-1})$ into the component of $D(b)$ with the attracting fixed point. However, the surface $\Sigma$ from this construction has one boundary component, as shown in Figure \[fig;surface\_exotic\] (right), so it corresponds to an isolated circular order in $\operatorname{\mathsf{CO}}(F_2)$. Observe that one can create several examples of this kind, by choosing $\rho(b)$ to have two hyperbolic fixed points, but with an arbitrarily slow contraction (i.e. with $N$ connected components for $D(b)$, $N\in{\mathbb{N}}$ arbitrary) and then choosing $\rho(a)$ to be a $N$-fold lift of a hyperbolic element in ${\mathsf{PSL}}(2,{\mathbb{R}})$. Left-invariant linear orders on $F_n\times {\mathbb{Z}}$ {#LO_sec} ======================================================== The purpose of this section is to prove Theorem \[LO\_thm\], stating that $F_n\times {\mathbb{Z}}$ admits an isolated linear order if and only if $n$ is even. Preliminaries on linear orders {#s:LOpre} ------------------------------ Before proceeding to the proof of Theorem \[LO\_thm\], we recall some standard tools. As for circular orders, linear orders on countable groups have a *dynamical realization* (see for instance [@GOD Prop. 1.1.8]). One quick way of seeing this given what we have already described, is by thinking of a linear order as a special case of a circular order. Indeed, given a linear order $\preceq$ on a group $G$, one defines the cocycle $c_\preceq$ by setting, for distinct $g_1,g_2,g_3\in G$ $$c_\preceq(g_1, g_2, g_3) = \mathsf{sign}(\sigma),$$ where $\sigma$ is the permutation of the indices such that $g_{\sigma(1)} \prec g_{\sigma(2)} \prec g_{\sigma(3)}$. Thus, the construction of the dynamical realization sketched in the proof of Proposition \[p:realization\] may be performed also for a linear order. The result is an action on the circle with a single one global fixed point, which one can view as an action on the line with no global fixed point. Conversely, a faithful action on the real line $\rho:G\to{\mathsf{Homeo}}_+({\mathbb{R}})$ can be viewed as a faithful action on the circle with a single fixed point, and the circular orders produced as in Remark \[r:induce\] will be linear orders on $G$. Next, we recall the notion of convex subgroups, their dynamical interpretation, and their relationship to isolated orders. A subgroup $C$ in a linearly-ordered group $(G,\preceq)$ is *convex* if for any two elements $h,k\in C$, and for any $g\in G$, the condition $h\preceq g \preceq k$ implies $g\in C$. \[r:convex\] Let $G$ be a countable left-ordered group and consider a dynamical realization $\rho$ of $(G,\preceq)$ with basepoint $x$ such that $C$ is a convex subgroup. Let $I$ be the interval bounded by $\inf_{h\in C}\rho(h)(x)$ and $\sup_{h\in C} \rho(h)(x)$. Then $I$ has the following property: $$\label{interval} \text{for any $g\in G$, either $\rho(g)(I) = I$, or $\rho(g)(I) \cap I = \emptyset$.}$$ Moreover, the stabilizer of $I$ is precisely $C$. Conversely, given a faithful action on the real line $\rho:G\to{\mathsf{Homeo}}_+({\mathbb{R}})$, if an interval $I$ has the property , then the stabilizer $C={\mathsf{Stab}_{G}(I)}$ is convex in any induced order with basepoint $x\in I$. It is easy to see that the family of convex subgroups of a linearly ordered group $(G,\preceq)$ forms a *chain*: if $C_1,C_2$ are two convex subgroups of $(G,\preceq)$, then either $C_1\subset C_2$ or $C_2\subset C_1$. Moreover, for any convex subgroup $C\subset G$, the group $G$ acts on the ordered coset space $(G/C,\preceq_C)$ by order-preserving transformation. (The induced order on the coset space is given by $fC<_C gC$ if and only if $fc< gc'$ for every $c,c'\in C$, which makes sense because $C$ is convex.) In particular, this implies that if $C$ is convex in $(G,\preceq)$, then *any* linear order $\preceq_C$ on $C$ may be extended to a (new) order $\preceq'$ on $G$ by declaring $$id\preceq' g \Leftrightarrow \begin{cases} C\preceq_C gC & \text{if }g\notin C, \\ id\preceq' g& \text{if }g\in C. \end{cases}$$ Elaborating on this, one can show the following lemma (see [@GOD Prop. 3.2.53] or [@MatsumotoLO Thm. 2] for details). \[l:chain\] If $(G,\preceq)$ has an infinite chain of convex subgroups, then $\preceq$ is non-isolated in $\operatorname{\mathsf{LO}}(G)$. Let us also introduce a dynamical property that implies that an order is non-isolated. Recall that two representations $\rho_1$ and $\rho_2: G \to {\mathsf{Homeo}}_+({\mathbb{R}})$ are *semi-conjugate* if there is a proper, non-decreasing map $f: {\mathbb{R}}\to {\mathbb{R}}$ such that $f \circ \rho_1 = \rho_2 \circ f$. Let $G$ be a discrete group. Let $\mathsf{Rep}(G,{\mathsf{Homeo}}_+({\mathbb{R}}))$ denote the space of representations (homomorphisms) $G \to {\mathsf{Homeo}}_+({\mathbb{R}})$, endowed with the compact-open topology; let $\mathsf{Rep}_\#(G,{\mathsf{Homeo}}_+({\mathbb{R}}))$ be the subspace of representations with no global fixed points. A representation $\rho \in \mathsf{Rep}_\#(G,{\mathsf{Homeo}}_+({\mathbb{R}}))$ is said to be *flexible* if every open neighborhood of $\rho$ in $ \mathsf{Rep}_\#(G,{\mathsf{Homeo}}_+({\mathbb{R}}))$ contains a representation that is not semi-conjugate to $\rho$. The following lemma is implicit in work of Navas [@NavasFourier] as well as in [@RivasLO]. An explicit proof can be found in [@ABR Prop. 2.8]. \[r:Flexible\] Let $G$ be a discrete, countable group and let $\rho_0$ be the dynamical realization of an order $\preceq$ with basepoint $x\in {\mathbb{R}}$. If $\rho_0$ is flexible, then $\preceq$ is non-isolated in $\operatorname{\mathsf{LO}}(G)$. Though not needed for our work here, we note that a precise characterization of isolated circular and linear orders in terms of a strong form of rigidity (i.e. strong non-flexibility) of their dynamical realizations is given in [@MR]. As mentioned in the introduction, in order to prove Theorem \[LO\_thm\] we use the relationship between circular orders on groups and linear orders on their central extensions by ${\mathbb{Z}}$. For this purpose, we need the notion of [*cofinal elements*]{}. An element $h$ in a linearly-ordered group $(G, \preceq)$ is called *cofinal* if $$\label{eq:cofinal} \text{for all $g\in G$, there exist $m,n\in{\mathbb{Z}}$ such that $h^m \preceq g \preceq h^n$.}$$ \[r:cofinal\] Cofinal elements also have a characterization in terms of the dynamical realization: If $\rho$ is a dynamical realization of $\preceq$ with basepoint $x$, then $h \in G$ is cofinal if and only if $\rho(h)$ has no fixed point. Indeed, if $h$ is not cofinal, then the point $\inf\{\rho(g)(x)\mid h^n\preceq g\text{ for every }n\in {\mathbb{Z}}\}$ is fixed by $\rho(h)$. Conversely, if $h$ satisfies , then the orbit of $x$ under $\rho(h)$ is clearly unbounded on both sides. Given a group $G$ with a circular order $c$, there is a natural procedure to *lift $c$ to* a linear order $\preceq_c$ on a central extension of $G$ by ${\mathbb{Z}}$ such that any generator of the central ${\mathbb{Z}}$ subgroup is cofinal for $\preceq_c$ [@zheleva; @MR]. The following statement appears as Proposition 5.4 in [@MR]. \[LO\_central\] Assume that $G$ is finitely generated and $c$ is an isolated circular order on $G$. If $\preceq_c$ is the lift of $c$ to a central extension $\widehat{G}$ of $G$ by ${\mathbb{Z}}$, then the induced linear order $\preceq_c$ is isolated in $\operatorname{\mathsf{LO}}(\widehat{G})$. Finally, we recall the definition of crossings. Let $G$ be a group acting on a totally ordered space $(\Omega,\leq)$. The action has *crossings* if there exist $f,g\in G$ and $u,v,w\in \Omega$ such that: 1. $u < w < v$. 2. $g^n u < v$ and $f^n v > u$ for every $n \in {\mathbb{N}}$, and 3. there exist $M, N$ in ${\mathbb{N}}$ such that $f^N v < w < g^M u$. In this case, we say that $f$ and $g$ are *crossed*. If $f$ and $g$ are crossed, then the graph of $\rho(f)$ and $\rho(g)$ in the dynamical realization is locally given by the picture in Figure \[fig:crossing\]. $$\includegraphics[scale=.8]{Cross-0.pdf}$$ Our application of the notion of crossings will be through the following lemma. \[l:crossings\] Let $C$ be a convex subgroup of $(G,\preceq)$ and suppose that the (natural) action of $G$ on $(G/C,\preceq_C)$ has no crossings. Then there exists a homomorphism $\tau:G\to {\mathbb{R}}$ with $C$ in its kernel. Moreover, if $C$ is the maximal convex subgroup of $(G,\preceq)$, then $C$ agrees with the kernel of $\tau$. Isolated linear orders on $F_n\times {\mathbb{Z}}$ {#sec_isoFnZ} -------------------------------------------------- We now turn to our main goal of describing isolated linear orders on $F_n\times {\mathbb{Z}}$ and proving Theorem \[LO\_thm\]. We begin by reducing the proof to the statement of Proposition \[prop flexible\] below. Since every central extension of $F_n$ by ${\mathbb{Z}}$ splits, Proposition \[LO\_central\] tell us that $F_{2n}\times{\mathbb{Z}}$ admits isolated linear orders – more precisely, any lift of an isolated order on $F_{2n}$ to $F_{2n}\times{\mathbb{Z}}$ will be isolated. Furthermore, if $$0\to{\mathbb{Z}}\to \hat{G} \overset{\pi}\rightarrow G\to 1$$ is a central extension of $G$ by ${\mathbb{Z}}$ then any linear order $\preceq$ on $\hat{G}$ in which ${\mathbb{Z}}$ is cofinal gives a canonical circular order on $G$ as follows. Let $z$ be the generator of ${\mathbb{Z}}$ such that $z \succ id$. Since $z$ is cofinal, for each $g \in G$, there exists a unique representative $\hat{g} \in \pi^{-1}(g)$ such that $id \preceq \hat{g} \prec z$. Given distinct elements $g_1, g_2, g_3 \in G$, let $\sigma$ be the permutation such that $$id \preceq \hat{g}_{\sigma(1)} \prec \hat{g}_{\sigma(2)} \prec \hat{g}_{\sigma(3)} \prec z.$$ Define $\pi^*(\preceq)(g_1, g_2, g_3) := \mathsf{sign}(\sigma)$. One checks that this is a well defined circular order on $G$. In the proof of Proposition 5.4 of [@MR], it is shown that $\pi^*$ is continuous and is locally injective when $G$ is finitely generated, which implies that an isolated linear order of $F_{2n+1}\times {\mathbb{Z}}$ with cofinal center induces an isolated circular order of $F_{2n+1}$ by this procedure. Since $F_{2n+1}$ has no isolated circular orders, by Theorem \[CO\_thm\], to finish the proof of Theorem \[LO\_thm\] it is enough to show the following: \[prop flexible\] Let $F$ be a free group, and $\preceq$ a linear order on $G=F\times {\mathbb{Z}}$ in which the central factor is not cofinal. Then $\preceq$ is non-isolated. As a warm-up, as well as tool to be used in the proof, we start with a short proof of a special case. \[l:infinite\_rank\] Let $F$ be a free group of infinite rank and $G=F\times {\mathbb{Z}}$. Then no order in $\operatorname{\mathsf{LO}}(G)$ is isolated. Let $f_1,f_2,\ldots$ be a set of free generators of the free factor $F$ and $g$ the generator of the central factor ${\mathbb{Z}}$. Let $\preceq$ be any order on $G$ and $\rho_0$ a dynamical realization with basepoint $x$. For any fixed $n\in {\mathbb{N}}$, we can define a representation $\rho_n:G\to {\mathsf{Homeo}}_+({\mathbb{R}})$ by setting $$\rho_n(g)=\rho_0(g),\quad \rho_n(f_k)=\begin{cases} \rho_0(f_k) &\text{if }k\neq n,\\ \rho_0(f_n)^{-1}&\text{if }k=n. \end{cases}$$ It is easy to see that the orbit of $x$ is free for all the actions $\rho_n$, and that no two distinct representations $\rho_n$ and $\rho_m$ are semi-conjugate one to another. Thus, they determine distinct orders $\preceq_n$; and these orders converge to $\preceq$ in $\operatorname{\mathsf{LO}}(G)$ as $n\to\infty$. We have already eliminated the case where $F$ has infinite rank. If $F$ has rank one, then $F \times {\mathbb{Z}}$ is abelian, and so admits no isolated orders (see [@sikora]). So from now on we assume that the rank of $F$ is finite and at least $2$. Looking for a contradiction, suppose that $\preceq$ is a linear order on $G = F\times {\mathbb{Z}}$ which is isolated, and in which the center is not cofinal. Let $\rho$ be its dynamical realization, and let $z$ be a generator of the central ${\mathbb{Z}}$ subgroup. By Remark \[r:cofinal\], $\rho(z)$ acts with fixed points. Moreover since ${\mathbb{Z}}$ is central, the set of fixed points of $\rho(z)$ is $\rho(G)$-invariant. Since $\rho(G)$ has no global fixed point, this implies that $\rho(z)$ has fixed points in every neighborhood of $+\infty$ and of $-\infty$. We now find a convex subgroup in which $z$ is cofinal. Let $I$ denote the connected component of ${\mathbb{R}}\setminus {\mathsf{Fix}}(\rho(z))$ that contains the basepoint $x_0$ of $\rho$, so in particular $I$ is a bounded interval. Let $C={\mathsf{Stab}_{G}(I)}$. $C$ is a convex subgroup of $(G,\preceq)$ and $z$ is cofinal in $(C,\preceq)$. If $h,k\in C$ and $g\in G$ satisfy $\rho(h)(x_0)<\rho(g)(x_0)<\rho(k)(x_0)$, then for any $n$ we also have $\rho(z^nh)(x_0)<\rho(z^ng)(x_0)<\rho(z^nk)(x_0)$. Since $z$ is central, this implies $$\label{eq:convex} \rho(hz^n)(x_0)<\rho(gz^n)(x_0)<\rho(kz^n)(x_0).$$ Up to replacing $z$ with $z^{-1}$, without loss of generality we may assume that $\rho(z)(x_0)>x_0$. Thus, as $n\to\infty$, the sequence of points $\rho(z^{n})(x_0)$ converges to the rightmost point of $I$, which is fixed by both $\rho(h)$ and $\rho(k)$. We deduce from that $\rho(g)$ also fixes this point. Similarly, considering the limit in as $n\to-\infty$ shows that the leftmost point of $I$ is fixed by $\rho(g)$. Hence $g\in C$, which shows $C$ is convex. Finally, by Remark \[r:cofinal\], the fact that $\rho(z)$ has no fixed points in $I$ implies that $z$ is cofinal in $(C,\preceq)$. Since $(G, \preceq)$ is isolated and $C$ is convex, the restriction of $\preceq$ to $C$ is also an isolated order on $C$. Additionally, the fact that $\preceq$ is isolated implies, by Lemma \[l:chain\], that the chain of convex subgroups of $G$ is *finite*. Let $G'$ denote the smallest convex subgroup properly containing $C$. Since ${\mathbb{Z}}\subset C$, we have that $G'$ is also a direct product of ${\mathbb{Z}}$ and a free group (a subgroup of $F$), and again, our assumptions on $\preceq$ imply that (the restriction of) $\preceq$ is an isolated left-order on $G'$ in which ${\mathbb{Z}}$ is not cofinal. Thus, we may work from now on with $G'$ instead of $G$. Equivalently – and, for notational convenience, this is how we will proceed – we may assume that $C$ is the *maximal* convex subgroup of $G$. For our next claim observe that this maximal convex subgroup $C$ also admits a decomposition of the form $F^*\times{\mathbb{Z}}$, where $F^*$ is a subgroup of $F$. \[cl:F\*\] $F^*$ is a non trivial free group of even rank. Since the restriction of $\preceq$ to $C=F^*\times {\mathbb{Z}}$ is isolated, as before, Lemma \[l:infinite\_rank\] implies that $F^*$ cannot have infinite rank. If $F^*$ were trivial, then the action of $G$ would be semi-conjugate to an action of $F$, thus making very easy to perturb the action of $F\times {\mathbb{Z}}$ and thus the order $\preceq$ (recall that free groups have no isolated orders [@McCleary2]). Thus, $F^*$ is a nontrivial free group of finite rank, and as $z$ is cofinal in the ordering in $C$, its rank must be even (c.f. the remarks at the beginning of Section \[sec\_isoFnZ\]). \[InfiniteIndex\] $C$ has infinite index in $G$. If $C$ had finite index, then the $G$-orbit of the interval $I$ would be bounded. This would imply that the dynamical realization has a global fixed point, which is absurd. Since every nontrivial normal, infinite index subgroup of $F$ has infinite rank, we conclude from Claims \[cl:F\*\] and \[InfiniteIndex\] that $F^*$ (and thus $C$) is not a normal subgroup of $G$. Lemma \[l:crossings\] thus implies that the action of $G$ on $(G/C,\preceq_C)$ has crossings, as otherwise $C$ would be normal. In particular, if we collapse $I$ and its $G$-orbit, we obtain a semi-conjugate action $\bar \rho: G\to {\mathsf{Homeo}}_+({\mathbb{R}})$ which is minimal and has crossings. Using this observation, we now prove the following claim. \[cl:flex\] For any compact set $K \subset {\mathbb{R}}$, there exists $\rho'$ agreeing with $\rho$ on $K$, but not semi-conjugate to $\rho$. Fix a compact set $K$. We will modify the action of $F$ outside $K$ to produce an action of $G$ that is not semi-conjugate to $ \rho$. Suppose as an initial case that there is a primitive element (i.e. a generator in some free generating set) $a$ of $F$ such that $\rho(a)$ has a fixed point $p \notin K$. Without loss of generality, assume $p$ is to the right of $K$, the other case is completely analogous. Since ${\mathsf{Fix}}(\rho(a))$ is $\rho(z)$-invariant, and $\rho(z^n)(p)$ is bounded and accumulates at a fixed point of $\rho(z)$, we may also assume without loss of generality that we have chosen $p$ to be a common fixed point of $\rho(z)$ and $\rho(a)$. We now define $a_+$ and $a_- \in {\mathsf{Homeo}}_+({\mathbb{R}})$ which commute with $\rho(z)$, and have the property that $a_+(x) \geq x$ for all $x \geq p$, and $a_- \circ \rho(a)(x) \leq x$ for all $x \geq p$. For this, let $J$ be any connected component of $({\mathbb{R}}\setminus {\mathsf{Fix}}(\rho(z))) \cap [p, \infty)$. Suppose first that $J$ contains a point of ${\mathsf{Fix}}(\rho(a))$. The fact that $a$ and $z$ commute means that the endpoints of $J$ are preserved by $\rho(a)$. Then define the restriction of $a_+$ to $J$ to agree with $\rho(z)$ if $\rho(z)(x) > x$ on $J$, or with $\rho(z^{-1})$ if $\rho(z)(x) < x$ on $J$. If $J$ contains no point of ${\mathsf{Fix}}(\rho(a))$, then $J \subseteq J'$ where $J'$ is a connected component of ${\mathbb{R}}\setminus {\mathsf{Fix}}(\rho(a)) \cap [p, \infty)$, and we may define $a_+$ to agree with $\rho(a)$ or $\rho(a^{-1})$ there, so as to satisfy $a_+(x)>x$ for $x\in J$. Lastly, set $a_+(x)=x$ for any $x\in {\mathsf{Fix}}(\rho(z))\cap [p,\infty)$. The definition of $a_-$ is analogous. Let $\rho_\pm$ be the actions obtained by replacing the action of $\rho(a)$ by that of $a_\pm$ on $[p, \infty)$ and leaving the other generators unchanged. Since $a_+$ and $a_-$ commute with $\rho(z)$, this defines representations of $G$, and clearly $\rho_+$ and $\rho_-$ are not semi-conjugate. We are left to deal with the case where no primitive element of $F$ has a fixed point outside $K$. In this case, we will perturb the action $\rho$ to obtain a primitive element with a fixed point outside $K$, and hence a non-semi-conjugate action. To do this, we use the fact that the semi-conjugate action $\bar \rho$ has crossings and is minimal. Minimality implies that crossings can be found outside any compact set and, thus, for any compact $K\subset {\mathbb{R}}$ there is $g\in G$ such that ${\mathbb{R}}\setminus {\mathsf{Fix}}(\rho(g))$ has a component outside (and on the right of) $K$. Let $J=(j_0,j_1)$ denote one of those components. Notice that some primitive element $a \in F$ has the property that $\rho(a)(j_0)\in J$, but $\rho(a)(j_1)\notin J$. For if this was not the case, $J$ would satisfy property and as observed in Remark \[r:convex\], $G$ would then have a convex subgroup properly containing a conjugate of $C$. Since $C$ was assumed maximal, this is impossible. Fix a primitive element $a$ with the property above, and let $\bar g$ be the homeomorphism defined as the identity outside $J$ and agreeing with $\rho(g)$ on $J$. Define $\rho^{\bar g}$ by $$\rho^{\bar g}(a)= \bar{g}\rho(a), \text{ and } \rho^{\bar g}(b)=\rho(b) \text{ for any other generator of $F$, and } \rho^{\bar g}(z) = \rho(z).$$ Since $\bar g$ commutes with $\rho(z)=\rho^{\bar g}(z)$, the new action $\rho^{\bar g}$ is a representation of $G$. Moreover, by changing $\bar g$ by some power if necessary, we have that $\rho^{\bar g}(a)$ has a fixed point in $J$. This ends the proof of Claim \[cl:flex\]. To finish the proof of Proposition \[prop flexible\] (and thus that of Theorem \[LO\_thm\]), we note that the *flexibility* of $\rho$ from Claim \[cl:flex\] together with the statement of Lemma \[r:Flexible\] implies that the order is non-isolated, giving the desired contradiction. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank Yago Antolín for suggesting Corollary \[cor:marked\]. K.M. was partially supported by NSF grant DMS-1606254. C.R. was partially supported by FONDECYT 1150691. M.T. was partially supported by PEPS – Jeunes Chercheur-e-s – 2017 (CNRS), Projet “Jeunes Géométres” of F. Labourie (financed by the Louis D. Foundation) and the Réseau France-Brésil en Mathématiques. [^1]: **MSC:** Primary 20F60, 57M60. Secondary 20E05, 37C85, 37E05, 37E10, 57M60. [^2]: LAMA, Université Paris-Est Marne-la-Vallée, CNRS UMR 8050, 5 bd. Descartes, 77454 Champs sur Marne, France. [^3]: Dept. Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912. [^4]: Dpto. Matemática y C.C. Universidad de Santiago de Chile, Alameda 3363, Estación Central, Santiago, Chile. [^5]: IMB, Université Bourgogne Franche-Comté, CNRS UMR 5584, 9 av. Alain Savary, 21000 Dijon, France. [^6]: In [@BS Prop. 2.4] the authors propose an alternative way of inducing an ordering of $G$, different from that in [@Calegari]. However their method is incorrect, as the following example shows: suppose to have three distinct homeomorphisms $f,g,h$, with $f$ coinciding with $g$ on one half circle and with $h$ on the other half. Then for any point $x\in {\mathbb{S}^1}$, there are always two equal points in the triple $(f(x),g(x),h(x))$.

--- author: - 'Luis Salas, and Irene Cruz-González' bibliography: - 'cosmologia.bib' title: An alternative approach to the Finger of God in large scale structures --- Introduction ============ A spherical object observed at a distance in its longitudinal ($||$) and transversal ($\bot$) dimensions, should provide a test of different cosmological models, as first proposed by @ap79. The Alcock-Paczy[ń]{}ski parameter, hereafter $AP$, basically the ratio of $||$ to $\bot$ dimensions, takes a value of one at redshift zero, and increases with $z$ with a strong dependence on the value of the cosmological parameters that make up the Hubble function, introducing a cosmological distortion to the large scale structure observations. This apparently simple comparison is, however, greatly complicated by several factors. First, real-space measurements are not directly attainable and one has to rely on redshift-space. Then, if the proposed object consists of a cluster of quasars or galaxies, or a statistical ensemble of such, proper motions of its constituents, either derived from gravitational collapse or virialized conditions, distort redshift-space measurements causing a degeneracy problem [e.g., @hamilton98]. On cosmological scales, clusters of galaxies or quasars are among the most simple geometric structures that one may conceive. Even if single clusters may have non-spherical or filamentary structures, those should be randomly oriented. As we probe more distant clusters, observations become biased towards brighter and widely separated members, and the numbers become statistically insignificant. A superposition of many such clusters may reduce the problem while retaining spherical symmetry. The two-point correlation function (2PCF), and its Fourier transform, the power spectrum, have been fundamental tools in these studies for more than 40 years [e.g., @peebles80]. Overdense clusters or associations separate from the Hubble flow due to their own gravity, which results in peculiar velocities of its members that distort redshift-space observations. When gravitational fields are small, velocities are well described by linear theory of gravitational collapse [@peebles80]. In the study of these clusters, the 2PCF was initially conceived as a single entity $\xi$ that could be evaluated in either real ($r-$) or redshift ($s-$) space [@peebles80]. @davis_peebles_83 even mentioned that when observing the local universe, if the peculiar velocities were small by comparison, $s-$space would directly reproduce $r-$space and one would have $\xi(\bm r) = \xi (\bm s)$. That should be the case for distant objects, although one should be careful not to mix up the notions of distant from each other and distant from the observer. In the case of the CfA Redshift Survey [e.g., @1983ApJS...52...89H], as described in @davis_peebles_83 peculiar velocities were significant, and the authors chose to go from real $\xi(r)$ to observable $\xi(s)$ by means of a convolution with a pair-wise velocity distribution, tailored to approach the Hubble flow at large distances, known as the [*streaming model*]{}. The convolution integral would at the same time convert $r-$space to $s-$space coordinates. However, the same function $\xi$ would be obtained as a result of the convolution of $\xi$ with a function of velocity, which constitutes an inconsistency. Later on, @kaiser87, hereafter K87, showed that gravitationally induced peculiar velocities by gravitational collapse of overdense structures in the linear regime, produces a power spectrum $P^{(s)}$ for $s-$space different from the one $P^{(r)}$ for $r-$space, that is two different functions for the power spectrum. Both are, however, functions of the $r-$space Fourier frequency [$\bm k$]{}. Then, while $P^{(r)}({\bm k})$ is a spherically symmetric function, $P^{(s)}({\bm k})$ shows an elongation along the line of sight (LoS) direction. Later on, @hamilton92 translated these results to configuration space obtaining the 2PCF in its two flavors: $\xi^{(r)}({\bm r})$ and $\xi^{(s)}({\bm r})$. Again $\xi^{(r)}({\bm r})$ is symmetric and the possibility of a power-law $r^{-\gamma}$ is considered, as had been historically accepted [e.g., @peebles80 who favored $\gamma = 1.8$ ]. Also in perfect agreement with K87, $\xi^{(s)}({\bm r})$ shows a squashing along the LoS direction. @hamilton98 presents in great detail the assumptions that led to his results. He starts by defining selection functions $n^{(r)}(r)$ and $n^{(s)}(s)$ for $r-$space and $s-$space and by numerical conservation obtains a complicated high order expression (his eq. 4.28) for the density contrast $\delta^{(s)}$. From that one can obtain the 2PCF, but a series of approximations are needed (the linear case) first to reduce the right hand side of the equation and end up in his eq. 4.30 for $\delta^{(s)}({\bm s})$. Then, he performs one extra assumption, $\delta^{(s)}({\bm r}) = \delta^{(s)}({\bm s})$, which is not justified by the linear approximation. This changes the left hand side of the equation directly to $\delta^{(s)}({\bm r})$. It may be argued that this approximation is valid in the distant case mentioned above. Consequently, one could easily write $\xi^{(s)}({\bm s})$ in place of $\xi^{(s)}({\bm r})$, shifting between one form and the other as needed. That is an imperative because observable 2PCF are inevitably obtained in $s-$space. Since then many authors have tried the Kaiser linear approximation facing this dilemma and have performed similar approximations. In the description of 2PCF in redshift-space, due to the multipole expansion of the inverse Lagrangian operator derived from the corresponding power spectrum in Fourier space [@hamilton92], there appears a dependence with $\mu$, the cosine of the angle between the ${\bm r}$ (real space) vector and the LoS: $ ~ \mu{(\bm r)}\,=\,r_{||}/|{\bm r}| $. However, it has been a common practice to approximate $\mu$ from redshift-space coordinates as either $ \mu{(\bm s)} = s_{||}/|{\bm s}|$ or $\mu{(\bm {cs})} = c_{||} s_{||}/\sqrt{ c_{\bot}^2 s_{\bot}^2 + c_{||}^2 s_{||}^2 }$ [e.g., @ms96; @Nakamura_1998_ApJ_494_13; @lc14]. Yet in some other cases the approximation $r_{||} = s_{||}$ is specifically made [e.g.,, @2006MNRAS.368...85T] calling it the “distant observer" approximation. But as mentioned above this is really meant to mean a wide separation approximation and does not apply in the small scale regime. Furthermore, the “distant observer" name is also used for the plane-parallel case [e.g., @percival2008], adding to confusion. In some other cases the substitution $r_{||} = s_{||}$ is just performed with no further comment [e.g., @hawkins2003]. Another facet of the same problem has been to expand the redshift-space correlation function as a series of harmonics of that same $\mu{(\bm s)}$, rather than the actual $\mu({\bm r})$ derived in linear theory [e.g., @guo2015; @chuang2012; @marulli2017]. While this is certainly a valid approach, the conclusions of linear theory, like the existence of only monopole, quadrupole and hexadecapole terms in the Legendre polynomial expansion, are not really applicable to the $\mu{(\bm s)}$ case. All these forms of the approximation are really one and the same, and to avoid further confusion (like the term “distant observer") we decided to call it the $\mu(\bm s)$ approximation. When observational data is used to construct the 2PCF $\xi^{(s)}(\bm s)$, it is generally true that simple linear theory predictions are not kept. On one hand, the predicted compression along the viewing direction is observed, but as one approaches the LoS axis the observed structure is mostly dominated by an elongated feature [e.g., @hamaus2015], usually called Finger of God [@huchra88], hereafter FoG. Prominent examples of FoG were found in the Coma Cluster by @deLapparent86 and in the Perseus cluster by @Wegner93. The FoG feature is also commonly observed in the 2PCF of statistical aggregates [e.g., @hawkins2003], making it a common feature in large scale structure. Many studies have been conducted to explain this discrepancy. In general non-linear processes are invoked. Sometimes the non-linearities are assigned to virial relaxation in the inner regions of clusters, while others explore the non-linear terms in the approximation in the derivation of the K87 result. In these categories, we mention a small sample of representative literature. Kinematic relaxation, like the virialized motion of cluster members in the inner regions [@kaiser87; @hamaus2015], are explored by introducing a distribution of pair-wise peculiar velocities for cluster components. There are at least two ways of doing so: First, the [*streaming model*]{} where a velocity distribution $f(V)$ is convolved with $\xi^{(r)}(\bm r)$ to obtain $\xi^{(s)}(\bm s)$, without using the K87 result, similar to @davis_peebles_83 but differentiating $\xi^{(s)}$ from $\xi^{(r)}$. More recent work on distribution functions take great care on this issue [@2011JCAP...11..039S; @2012JCAP...11..014O; @2012JCAP...02..010O] by directly obtaining the power spectra in redshift-space as a function of the $s-$space wave-number. Unfortunately, the expression that results for the power spectra is rather complicated, even when it is conveniently expressed as a series in mass weighted velocity moments. However, it is possible to obtain FoG structures in $\xi^{(s)}(\bm s)$ maps by the convolution with simple velocity distributions, at the same time that a mapping from $r-$ to $s-$space takes place [e.g., @2004PhRvD..70h3007S]. Paradoxically, it is not that easy to obtain the traditional peanut-shape structure that is generally recognized as the K87 limit in $\xi^{(s)}(\bm s)$, unless the limit $s \sim r$ is once again invoked. Second, in the phenomenological [*dispersion model*]{} [c.f., @2004PhRvD..70h3007S; @2006MNRAS.368...85T] a linear K87 spectrum is multiplied in Fourier space by a velocity distribution. This can be seen as a convolution in configuration space, as in @hawkins2003, but the procedure has the disadvantage that it obtains the same function $\xi^{(s)}$ as the result of the convolution of $\xi^{(s)}$ and $f(V)$. It has to be noted, however, that very good fits to the observed data are obtained by this procedure. The same is true for the fits to numerical simulation results at mid spatial frequencies obtained by similar procedures in e.g., @marulli2017. In the streaming model, the velocity distribution function can also be obtained from the interaction of galaxies with dark mater halos , via the halo occupation distribution formalism. Apart from kinematics, non-linear terms also arise in the expansion of the mass conservation or continuity equation in $r-$ and $s-$spaces to obtain the power spectrum or the 2PCF [e.g., @2008PhRvD..77f3530M; @2010PhRvD..82f3522T; @2016JCAP...08..050Z]. Preserving only first order terms yields the K87 result. However, a full treatment of all the terms is possible with the use of perturbative methods. There are diverse techniques: standard, Lagrangian, re-normalized, resumed Lagrangian [for a comparison see @percival2008; @2011MNRAS.417.1913R]. The latter authors however, conclude that the failure of these methods to fit the $l$=2 and 4 terms in the expansion $\xi^{(s)}_l(r)$ on quasi-linear scales of 30 to 80 $h^{-1}$ Mpc, must be due to inaccuracies in the mapping between $r$- and $s$-spaces. So, they favor again the [*streaming model*]{}. Clearly, there is still substantial debate on this subject. In most of these works the necessity to translate their results to observable 2PCFs, $\xi^{(s)}(\bm s)$, is not really addressed. Most authors prefer to display their results in Fourier space as $s$-space power spectrum $P^{(s)}(\bm k_r)$ [e.g., @2008PhRvD..77f3530M; @2012JCAP...11..014O], but with $k_r$ in $r$-space; or display its moments $P^{(s)}_l(\bm k_r)$ [e.g., @2010PhRvD..82f3522T; @2016JCAP...08..050Z]; or power spectra with $k_s$ in $s-$space $P^{(s)}(\bm k_s , \mu_s)$ [e.g., @2012JCAP...02..010O]. Other authors display the correlation function in $r$-space, either as $\xi^{(r)}(\bm r)$ [e.g., @2008PhRvD..77f3530M] or $\xi^{(s)}(\bm r)$ [e.g., @2007MNRAS.374..477T; @2011MNRAS.417.1913R; @2012JCAP...11..014O], or its moments $\xi^{(s)}_l(s)$ [e.g., @2010PhRvD..82f3522T] for $l$=2. Few works try to display directly the 2PCFs $\xi^{(s)}(\bm s)$ [e.g., @ms96; @Nakamura_1998_ApJ_494_13; @2006MNRAS.368...85T; @lc14], but as already mentioned above, usually perform the $\mu(\bm s)$ approximation that amounts to really obtaining $\xi^{(s)}(\bm r)$ instead. To further complicate matters, redshift-space distortions are often treated separately from the cosmological distortions. Both are not easily discernible because both produce stretching or squashing in the LoS direction [@hamilton98; @hamaus2015]. This degeneracy could in principle be resolved because the cosmological and peculiar velocity signals evolve differently with redshift, but in practice the uncertain [evolution of bias (the dimensionless growth rate for visible matter, see eq. \[eq:dpmn26\])]{} complicates the problem [@Ballinger96]. Furthermore, @kaiser87 and @hamilton92 [@hamilton98] do not consider cosmological distortions in their analysis of peculiar motions. Since the earlier works, the inclusion of cosmological distortions has been attempted by several authors [e.g., @ms96; @hamaus2015]. In this paper, we show that a structure quite similar to FoG can be obtained in $\xi^{(s)}(\bm s)$ directly in the linear theory limit of K87. That is, without invoking virial relaxation or the [*streaming model*]{}, nor the non-linearities studied in perturbation theory, but just by avoiding the $\mu(\bm s)$ approximation, in any of its forms ($\mu=s_{||}/|\bm s| $, “distant observer" or $r_{||} = s_{||}$ ), the FoG structure is recovered. This will be accomplished by solving for the function $\bm r ( \bm s)$ with the aid of the projected correlation function of both 2PCFs : $\xi^{(s)}(\bm s)$ and $\xi^{(r)}(\bm r)$. We will stay on the academic power-law approximation $\xi^{(r)}(\bm r) \sim r^{-\gamma}$ in order to be able to show a closed form for the result, and to prove the main point of this paper, i.e. that the FoG feature is derived in the simplest case. [We start with a detailed definition of $r-$ and $s-$space, noting that ]{} frequently $s-$space is expressed in distance units as is $r-$space. But in doing so, one multiplies by a scale factor that invariably introduces a cosmological parameter in the definition; and as a result the named $s-$space is no longer purely observational. [Later on the factor is solved by introducing a fiducial cosmology and solving for the real values.]{} An example can be seen in the analysis made by @padwhite2008 in Fourier space and @xu2013 in configuration space. The latter recognize the need of introducing a two-step transformation, one isotropic dilation and one warping transformation, to transform from real fiducial to real space. However, the real fiducial space is actually redshift-space, and this identification is missing in these works. Therefore, we argue (c.f., Section \[sec:sec2\]) that it is convenient to define the observable-redshift-space ${\bm \sigma}$ (${\sigma}$-space) given by the simple redshift differences and subtended angles that are truly observable, and that do not depend on any choice of cosmological parameters. Multiplying by a units function (scale factor) produces the physical redshift-space ($s-$space): ${\bm s} = K({\bm \Omega},z) ~ {\bm \sigma}$, that is isomorphic to the observable ${ \sigma}$-space, but has actual distance units that are dependent on a particular cosmological set of parameters [$\Omega$]{} and the redshift $z$. The $K({\bm \Omega},z)$ function is chosen so that the physical redshift-space is related to real space ${\bm r}$ by a unitary Jacobian independent of redshift. So that no additional scaling is needed, and the only remaining difference will be precisely in shape. That is why ${\bm \sigma}$ and ${\bm s}$ are more alike, and thus can both be named redshift-space; ${\bm \sigma}$ is the observable redshift-space while ${\bm s}$ is the physical redshift-space. Then, the transformation to real-space necessarily goes through redshift distortions. Furthermore, when we introduce peculiar non-relativistic velocities in this scheme, we will show that it is possible to keep the same relation between observable and physical redshift-spaces, ${\bm s}$ and ${\bm \sigma}$, and that the @kaiser87 effect is recovered independently of redshift (see Section \[sec:sec3\]). That is, now redshift-space will also show an additional gravitational distortion with respect to real-space. To solve for the relation between real-space and redshift-space, we will rely on projected correlations. Projections of the 2PCF in the plane of the sky have been widely used to avoid the complications of dealing with unknown components in redshift-space [e.g., @davis_peebles_83]. This has the advantage that in the case of a symmetric 2PCF in real-space, the 3-D structure can be inferred from the projection. We will show in Section \[sec:sec4\] that since the projections of the 2PCF in real-space and in redshift-space are bound to give the same profile, a relationship can be obtained for the real-space coordinate $r_{||}$ as a function of the corresponding one in redshift-space $s_{||}$. From this, we solve for $\mu{(\bm r)}$ in real-space, and show that a different view of the redshift-space 2PCF emerges. The main result is that the redshift-space 2PCF presents a distortion in the LoS direction which looks similar to the ubiquitous FoG. This is due to a strong anisotropy that arises purely from linear theory and produces a change in scale as one moves into the on-axis LoS direction. As we move out of the LoS, a structure somewhat more squashed than the traditional result by the $\mu(\bm s)$ approximation is obtained. As this effect has been missed before (to the best of our knowledge), we provide a detailed derivation in Sections \[sec:sec2\] to \[sec:sec4\], and show examples of the derived 2PCFs in redshift-space (Section \[sec:sec5\]). Finally, in Section \[sec:sec6\] we summarize our main conclusions. redshift-space {#sec:sec2} ============== Consider the Friedmann-Lemaître-Robertson-Walker metric [e.g., @harrison93] written in units of distance and time as follows: $$\label{eq: dpmn1} {ds}^2=c^2{dt}^2 - d r^2=c^2{dt}^2 - a(t)^2\left({d\chi }^2+S_k(\chi )^2({d\theta }^2+\sin ^2(\theta ){d\varphi }^2)\right) ,$$ with   $S_k=(~sin ~, ~ {Identity}~,~ sinh~ )$ for $k=(1,0,-1)$. Then the co-moving [ present-time ]{} length of an object $d r^0$ that is observed longitudinally is related to a variation in the observed redshift $d z$ by $$\label{eq:dpmn2} d r_{||}^0 = \frac{c d z} {H(z) },$$ where $H(z)$ is the Hubble function and the $^0$ superindex is used to define the present time $t_0$. Similarly, an object with a transversal co-moving dimension ${d r}_{\bot}^0$ subtends an angle $d \theta$ given by the angular co-moving distance [e.g., @hogg99] as $$\label{eq:dpmn3} \frac {{d r}_{\bot}^0}{{d \theta}}=a_0S_k\left(\frac c{a_0}\int _0^z\frac{{dz}'}{H(z')}\right),$$ where $a_0$ is the present day scaling parameter of the metric. Observationally one measures redshift differences $dz$ and subtended angles $d\theta$. We then define the observable redshift-space adimensional quantities ($d\sigma_{||}~, d\sigma_{\bot}$) as $$\label{eq:dpmn4} d\sigma_{||} = dz$$ and $$\label{eq:dpmn5} d\sigma_{\bot} = z d\theta.$$ The physical redshift-space sizes $ds_{||}$ and $ ds_{\bot}$ can then be defined in terms of ${\bm \sigma}$ as $$\label{eq:dpmn6} ds_{||} = K({\bm \Omega},z) d\sigma_{||}$$ and $$\label{eq:dpmn7} ds_{\bot}= K({\bm \Omega},z) d\sigma_{\bot},$$ where $K({\bm \Omega},z)$ has distance units and depends on the cosmology, represented here symbolically by the ${\bm \Omega}$ terms. The relation between real-space and physical redshift-space is then obtained from eqs. to , that is: $$\label{eq:dpmn8} dr^0_{||} = c_{||} ds_{||}$$ and $$\label{eq:dpmn9} dr^0_{\bot} = c_{\bot} ds_{\bot},$$ with $$\label{eq:dpmn10} c_{||}=\frac{ c } {K({\bm \Omega},z) ~ H(z) }$$ and $$\label{eq:dpmn11} c_{\bot} = \frac{ a_0 }{z ~ K({\bm \Omega},z) } ~ S_k\left(\frac c{a_0}\int _0^z\frac{{dz}'}{H(z')}\right).$$ It is clear then that the @ap79 function $AP(z)$, that tests redshift distortions of a particular cosmology, can be written as $$\label{eq:dpmn12} AP(z) = \frac{c_{\bot} (z)} {c_{||}(z)} =\frac{a_0} {c} ~\frac{H(z)} z ~ S_k\left(\frac c{a_0}\int _0^z\frac{{dz}'}{H(z')}\right).$$ Furthermore, from the transformation of physical redshift-space with coordinates $(ds_{\bot},ds_{\bot},ds_{||})$ into real-space $(dr_{\bot}^0,dr_{\bot}^0,dr_{||}^0)$ we get a Jacobian $$\label{eq:dpmn13} \left| {\frac { d^3 \bm s} { d^3 \bm r} } \right| = \frac{1}{c_{||}(z)} ~ \frac{1}{c_{\bot}^{2}(z)} .$$ In order for this transformation to preserve scale we need a unitary Jacobian. This condition can be achieved simply by the following condition: $$\label{eq:dpmn14} K({\bm \Omega},z)= \frac {c}{H(z)} ~ AP(z)^{2/3},$$ as can be seen from eqs. to . Here the dependence on the cosmology is made explicit through the Hubble function. Note that the resulting scale factor $K({\bm \Omega},z) $ approaches the Hubble radius $a_H = c/H_0$ as $z \rightarrow 0$ and decreases approximately as $1/(1+z)$ thereafter. Also note that for redshift $z > 0$, the physical scale that transforms all dimensions of redshift-space, contracts isotropically. Also we remark that $c_{||} $ and $ c_{\bot}$ are of order unity as $z \rightarrow 0$, and satisfy $c_{\bot}/c_{||} = AP(z)$ for all $z$. In fact we have [see also @xu2013]: $$\label{eq:dpmn15} c_{||}(z) = AP(z)^{-2/3},$$ and $$\label{eq:dpmn16} c_{\bot}(z) = AP(z)^{1/3}.$$ Peculiar velocities modify the observed redshift, and therefore alter the relation between real-space and redshift-space giving rise to kinematic distortions. Suppose the near-end of an object is at rest at redshift $z$, while the far-end is moving with peculiar non-relativistic velocity $\vec{\rm v }$. Then it will appear Doppler shifted to an observer at rest at the far-end position, causing eq.  to get the form [see also @ms96; @hamaus2015]: $$\label{eq:dpmn17} c dz = H(z)~ dr_{||}^0 + (1+z) ~ (\vec{\rm v} \cdot \hat{r}),$$ where $\hat{r}$ points in the direction of the far-end, at an angle $d\theta$ from the near-end. Since $ \vec{\rm v} \cdot \hat{r} = {\rm v}_{||} + {\rm v}_{\bot} ~d \theta$, then for small angular separations ($d\theta << 1$) the perpendicular component of the peculiar velocity may be trivialized. Therefore eq.  gets modified to $$\label{eq:dpmn18} dr_{||}^0 = c_{||} ~ ( ds_{||} - ds_{\rm v} ).$$ where $ds_{\rm v}$ (in physical redshift-space) is given by $$\label{eq:dpmn19} ds_{\rm v} = K({\bm \Omega},z) ~d\sigma_{\rm v}$$ and $d\sigma_v$ (in observable redshift-space) is given by $$\label{eq:dpmn20} d\sigma_{\rm v} = (1+z) ~ \frac{{\rm v}_{||}}c.$$ And through the similarity of eqs. and with eqs. and , we note that the concepts of observable redshift-space and physical redshift-space can be extended to include peculiar motions as well. Two point correlation function {#sec:sec3} ============================== Let ${\bm r}$ be real-space Euclidean co-moving coordinates in the close vicinity of a point at redshift $z$, what was defined as $dr$ in the previous section. Then for azimuthal symmetry around the line of sight (aligned to the third axis) we have ${\bm r} $ = $ ( dr_{\bot}^0, dr_{\bot}^0,dr_{||}^0)$. Let ${\bm s}$ denote physical redshift-space coordinates around the same point (in the same tangent subspace), with the third axis along the line of sight. Then, from eqs. and , the Jacobian is $$\label{eq:dpmn21} \left| {\frac { d^3 \bm s} { d^3 \bm r} } \right| = \frac{1}{c_{||}(z)~ c_{\bot}^{2}(z)} ~ \left( 1 + {\frac {(1+z)} {H(z)}} ~~ {\frac {\partial {\rm v}_{||} } {\partial r_{||}}} \right) = 1 + {\frac {(1+z)} {H(z)}} ~~ {\frac{\partial {\rm v}_{||} } {\partial r_{||}}},$$ where we have used the unitary condition on eq. to eliminate the $c_{||}(z) ~ c_{\bot}^{2}(z)$ term. In going from ${\bm r}$ to ${\bm s}$ space, the density change can be related to the change in volume $V$, and the Jacobian by the equation $$\label{eq:dpmn22} \left( {\frac {d\rho} \rho}\right)_{\bm s-r} = - {\frac{dV} V} = 1 - \left| { \frac { d^3 \bm s} { d^3 \bm r} } \right|.$$ This can also be expressed in terms of the contrast density ratios in $\bm s$ and $\bm r$ spaces defined such that $$\label{eq:dpmn23} \left( {\frac {d\rho} \rho}\right)_{\bm s-r} = \delta^{(s)} - \delta^{(r)},$$ where $\delta^{(s)}$ and $\delta^{(r)}$ are two distinct scalar functions of position in either space. This particular definition of $\delta^{(s)}$ requires knowledge of the real-space selection function [@hamilton98], which makes it rarely a first choice. However, the procedure given below allows us precisely to solve for the function ${\bm r}({\bm s})$. In linear theory, @peebles80 shows in eqs. 14.2 and 14.8 that an overdensity of mass $\delta(\bm{r})$ creates a peculiar velocity field similar to the acceleration field produced by a mass distribution. As such, it can be derived from a potential function whose Laplacian is the overdensity itself [e.g., @marion2004] times a constant which is time (or redshift) dependent. That is $$\label{eq:dpmn24} {\bf {\rm v}}({\bm r} ) = -{\frac{ H(z)~f(z)}{ (1+z)}} ~{\bm \nabla} \nabla^{-2} \delta^{(r)}_m({\bm r}),$$ where $\nabla$ is the gradient and $\nabla^{-2}$ is inverse Laplacian, and $$\label{eq:dpmn25} f(z) = {\frac{a(z)}{D(z)}}{ \frac{dD } {da}}.$$ Here $D(z)$ is the growth factor, the temporal component of density. Note that in @peebles80 coordinates are given in the expanding background model ${\bm x}$ which relate to present time real-space coordinates by ${\bm r } = a_0 {\bm x}$; this brings about the $(1+z)$ factor to eq. . The $m$ subscript to $\delta$ emphasizes that all mass is responsible for the velocity field, while $\delta$ without the subscript refers to visible mass in the form of galaxies or quasars. To account for the difference, it is customary to introduce a bias factor $b(z)$ and define the dimensionless growth rate for visible matter $$\label{eq:dpmn26} \beta(z) = {\frac {f(z)}{b(z)}}.$$ Then from eqs. to we get: $$\label{eq:dpmn27} \delta^{(s)}({\bm r}) = \left(1+ \beta(z) ~ \partial_{||}^2 ~ \nabla^{-2} \right)\delta^{(r)}({\bm r}),$$ where $\partial_{||}$ denotes $\partial / \partial r_{||}$ in real space. Note that if we had not required a unity Jacobian (c.f., eq. \[eq:dpmn13\]), then eqs.  and would not had canceled out the $1 - c_{||}(z)^{-1}c_{\bot}(z)^{-2}$ term. We note that this term is not small when $K(\Omega,z) $ is a constant, and will vary by one order of magnitude as $z \rightarrow 1$, and up to three orders of magnitude as $z \rightarrow 10$. So the transformation between observable and physical redshift-spaces cannot be neglected [contrary to @ms96 assumption]. The square modulus of the Fourier transform of eq. gives an expression for the power spectrum, or the Fourier transform of the autocorrelation function (2PCF) $\xi$, which generalizes @kaiser87 results for any redshift $z$ $$\label{eq:dpmn28} \widetilde{\xi^{(s)}}({\bm k}) = \left(1+ \beta(z) ~ \mu_k^2 \right)^2 \widetilde{\xi^{(r)}}({\bm k}),$$ where $\mu_k = k_{r3}/|{\bm k_r} |$ is the cosine of the angle between the $k_{r3}$ component and the wave number vector ${\bm k_r}$ in real-space; and it arises by the Fourier transform property of changing differentials into products. Note that wave number vectors in real-space also differ from their counterparts in redshift-space by the unknown velocity field in eq. . Fourier transforming back into coordinate space gives @hamilton92 result: $$\label{eq:dpmn29} \xi^{(s)}({\bm r}) = \left(1+ \beta(z) ~ \partial_{||}^2 ~\nabla^{-2} \right)^2\xi^{(r)}({\bm r}) .$$ Note that this equation is written in a way that all terms in the right hand side are real-space coordinates ${\bm r}$ dependent, as is the case for the derivatives and inverse Laplacian. Recalling that the solution of the Laplace equation in spherical coordinates consists of spherical harmonics in the angular coordinates and a power series in the radial part, one can write for the case of azimuthal symmetry $$\label{eq:dpmn30} \xi^{(s)}({\bm r}) = \sum_{l=0} \xi_l(r) ~ P_l(\mu{(\bm r)})$$ where $P_l(\mu{(\bm r)})$ are the Legendre polynomial, $$\label{eq:dpmn31} {\mu{(\bm r)}} = {\frac {r_{||}} {|{\bm r}|}},$$ explicitly defined for real-space coordinates, and the harmonics are given by the coefficients $\xi_l(r)$ that can be obtained from eq. through orthogonality properties as $$\label{eq:dpmn32} \xi_l(r) = {\frac{(2l+1)} 2} \int_{-1}^1 P_l(\mu{(\bm r)}) ~\xi^{(s)}({\bm r}) ~d\mu{(\bm r)}.$$ Substituting eq.  in for the case of spherical symmetry in real-space ($\xi^{(r)}({\bm r}) = \xi^{(r)}(r) $), one gets by direct evaluation the classical result given by @hamilton92, see also @hawkins2003. That result consists of only three terms, monopole, quadrupole and hexadecapole ($ l = 0, 2, 4$), all the others evaluate to zero. It is important to note that this is not true when the expansion of eq.  has been done in $\mu{(\bm s)}$ as is assumed by several authors [e.g., @guo2015; @chuang2012; @marulli2017]. When the 2PCF could be approximated by a power-law, $ \xi^{(r)}(r) = (r/r_0)^{-\gamma}$, the solution for eq.  can be written as $$\label{eq:dpmn33} \xi^{(s)}({\bm r}) = g(\gamma,\beta,\mu{(\bm r)}) ~ \xi^{(r)}(r).$$ where $g(\gamma,\beta,\mu{(\bm r)})$ has been written in several equivalent forms [@hamilton92; @ms96; @hawkins2003]. One of these is the following $$\label{eq:dpmn34} g(\gamma,\beta,\mu{(\bm r)}) = 1 + 2 {\frac {1-\gamma \, \mu(\bm r)^2} {3-\gamma}} ~ \beta(z) +{ \frac { \gamma (\gamma+2) \, \mu(\bm r)^4 - 6 \gamma \, \mu(\bm r)^2 + 3} { (3-\gamma) (5-\gamma) }} ~ \beta(z)^2.$$ This function takes values greater than 1 for the equatorial region ($\mu{(\bm r)} \rightarrow 0$), and less than 1 for the polar axis ($\mu{(\bm r)} \rightarrow 1$). Alternatively, it has been mentioned that the quadrupolar term in the multipole expansion dominates the hexadecapole. As a result of either argument the 2PCF [ $\xi^{(\bm s)} (\bm r) $ ]{} seems squashed [ with a peanut shape when displayed in $r$-space ]{}, in agreement with common knowledge. However, we will show below that the stretching of redshift scale along the LoS will counteract this apparent squashing producing a structure similar to a FoG. In order to stay within the linear regime, we ensure not to reach the turnaround velocity by keeping $g(\gamma,\beta,\mu{(\bm r)})$ positive in the polar region. In that case $\beta$ is limited from 0 to an upper limit which is a function of $\gamma$, and equals 2/3 when $\gamma = 1.8$. The $\beta = 0$ case gives the no gravity one in which $\xi^{(s)}({\bm r}) = \xi^{(r)}({\bm r})$. We now remark that $ \mu{(\bm r)} = r_{||}/|{\bm r}| $ (see eq. \[eq:dpmn31\]). But in some works [ [e.g., @ms96; @2006MNRAS.368...85T; @lc14]]{} it has been approximated as $\mu{(\bm s)} = s_{||}/|{\bm s}|$ or as $ \mu{(\bm{cs})} = c_{||} s_{||}/\sqrt{ c_{\bot}^2 s_{\bot}^2 + c_{||}^2 s_{||}^2 }, $ [ or even as $ r_{||} = s_{||}$]{}. We [ have referred ]{} to this as the $\mu{(\bm s)}$ approximation. In principle, given that $\mu$ is a scalar function, either form should be acceptable as long as the ${\bm s}$ and ${\bm r}$ vectors refer to the same point. However, we remark that $r_{||}$ differs from $c_{||} s_{||}$ (see eq. \[eq:dpmn18\]), and that it is usually unknown, since in order to obtain it from $s_{||}$, the infall velocity field must be known. So these approximations should be carefully used. The result in our eq.  has been derived for $r$-space, profiting on the difference between $r$- and $s$- spaces. Plotting this function directly in $r$-space as the independent variable, produces a squashed structure for $\xi^{(s)}(\bm r)$. However, one wants to display the correlation function in $s$-space to compare with observations, not in $r$-space. In order to do so, some authors perform the $\mu(\bm s)$ approximation while others may plainly substitute $s$ for $r$ all the way in eq.  and write $\xi^{(s)}( \bm s) = g(\gamma, \beta, \mu( \bm s)) ~ \xi^{(r)}(s) $ to be able to display $\xi^{(s)} $ in $s$-space. This is certainly wrong because $\bm s$ and $\bm r$ are not just independent names for position, and there exists a relation $\bm s(\bm r)$ between them that is not linear. Specifically, the parallel component is $s_{||} \sim r_{||} + v _{||} $ (eq. \[eq:dpmn18\]), with $v_{||}$ also a (yet unknown) function of position $\bm r$. In the case of small disturbances we expect small velocities (below turnover) that result in a bi-univocal map $\bm s(\bm r)$ and its inverse. So, if we want to display the resulting $\xi^{(s)}$ in $\bm s$ space, one must proceed first to evaluate $\bm r= \bm r(\bm s)$ and then $\xi^{(s)}(\bm r)$ via eq. , or in short $\xi^{(s)}(\bm r (\bm s ) ) = g(\gamma, \beta, \mu( \bm r(\bm s ))) ~ \xi^{(r)}(r(\bm s )) $. We can therefore informally define $\xi^{(s)} (\bm s ) \equiv \xi^{(s)}(\bm r (\bm s ) ) $ and we claim that this is the correct way to evaluate the two-point correlation function on a grid in $s$-space. On the other hand, [ if the $\mu{(\bm s)}$]{} approximation is used one then obtains structures that are squashed in the LoS direction, and with a characteristic peanut-shaped geometry close to the polar axis [see for example @hawkins2003]. One concludes that this geometry fails to reproduce the structure known as “Finger of God" (FoG). The consequence is that other processes are called upon to account for it, such as random motions arising in the virialized inner regions of clusters. We next show below that by avoiding this approximation, it allows us to obtain a geometrical structure quite similar to the FoG feature. Projected correlation function {#sec:sec4} ============================== In order to avoid the complications that redshift-space distortions introduced in the correlation function, such as those produced by gravitationally induced motions or virialized conditions, the projected correlation function $w_{\bot}(r_{\bot})$ is frequently preferred in the analysis. This approach was first suggested in the analysis of CfA data by @davis_peebles_83, who mention that at small redshift separations, peculiar velocities may cause $\xi(s)$ to differ from $\xi(r)$. To avoid this effect, they integrate $\xi(r)$ along the redshift difference to obtain the projected function $w_{\bot}(r_{\bot})$ on the plane of the sky. Then, from it, they recuperate $\xi(r)$ inverting the problem by solving Abel’s integral equation [@bt87] numerically. See also @pisiani2014 for other possibilities. In the case where $\xi(r)$ is a power-law, $w_{\bot}(r_{\bot})$ will be one as well, and the relation between them is analytical [e.g., @krumpe10]. We will show that the projected correlation function can be used to obtain the $r_{||}(s_{||})$ function that allows one to calculate $\mu{(\bm r)}$. We start by noting that the projection on the plane of the sky may be performed either by using the $\xi^{(s)}$ function or its real space counterpart $\xi^{(r)}$. Then we define the projected correlation functions as $$\label{eq:dpmn35} w^{(s)}_{\bot} (s_{\bot},s^*_{||}) = \int_0^{s^*_{||}}{ \xi^{(s)}({ \bm r} (s_{\bot},s_{||} ) ) ~ ds_{||} },$$ and $$\label{eq:dpmn36} w^{(r)}_{\bot} (r_{\bot},r^*_{||}) = \int_0^{r^*_{||}} { \xi^{(r)}(r_{\bot},r_{||}) ~ dr_{||} },$$ where $\xi^{(s)}( \bm r (s_{\bot},s_{||} ) $, given by eq. , may be understood as $\xi^{(s)}( \bm s ) $ as mentioned above. The integral limits should go to infinity to get the total projected functions. However, one can project the correlation function up to a particular real space distance $r^*_{||}$. Furthermore, if we assume that there exists a biunivocal function $s_{||}(r_{||})$, then we can find the corresponding $s^*_{||} = s_{||}(r^*_{||})$. [Boundary conditions are thus well defined [e.g., @Nock2010]. On the one hand slices in r-space (eq. \[eq:dpmn36\]) do not depend on the observers perspective, while on the other (eq. \[eq:dpmn35\]) the limit of the integral (boundary condition) becomes a function that is precisely going to be evaluated.]{} Carrying on, due to number conservation the projections in redshift- and real-space, multiplied by the corresponding area elements that complete the volume where the number of pairs are counted, must be equal. Which leads to $$\label{eq:dpmn37} w^{(s)}_{\bot} (s_{\bot},s^*_{||}) ~ ds_{\bot}^2 =w^{(r)}_{\bot}(r_{\bot},r^*_{||}) ~ dr_{\bot}^2,$$ for all values of $r_{\bot}$ (or its corresponding $s_{\bot}$, see eq. \[eq:dpmn9\]). Inverting the $s_{||}(r_{||})$ map and using eqs.  to , together with and we obtain $$\label{eq:dpmn38} \int_0^{s^*_{{||} }} { g(\gamma, \beta, \mu{(\bm r)})~ \xi^{(r)}(r_{\bot},r_{||}) ~ ds_{||} } = c_{\bot}^2 ~\int_0^{r_{||}(s^*_{{||} })} { \xi^{(r)}(r_{\bot},r_{||}) ~ dr_{||}}.$$ Then, changing variables to $r_{||}$ in the left ($ds_{||} = {\frac {ds_{||} } { dr_{||} } } dr_{||} $), and noting that the equality holds for all values of $s^*_{{||} }$, the integral signs can be omitted. Furthermore, using eqs.  and the equation simplifies to $$\label{eq:dpmn39} c_{||} ~ ds_{||} = {\frac { dr_{||} } { g(\gamma, \beta, \mu (\bm r (r_{\bot},r_{||} ))) } } ,$$ where the dependence $ \mu (\bm r (r_{\bot},r_{||} )) = r_{||} / \sqrt{r_{\bot}^2+r_{||}^2}$ has been emphasized for clarity. Equation completes the metric transformation between redshift- and real-spaces. As a consistency test, we note that in the limit of no gravitational disturbance ($\beta = 0 $) we have $g(\gamma,\beta,\mu{(\bm r)}) = 1$ and eq.  is recovered. Resulting redshift-space and real-space relation {#sec:sec5} ================================================ We integrate eq.  numerically using eq. , to obtain the $s_{||}(r_{||})$ function shown in Figure \[fig:rdes\], for different values of $r_{\bot}/r_e$ indicated for each curve in the figure, where $r_e$ is an arbitrary scaling parameter, $\gamma = 1.8$, $\beta = 0.4$, and $c_{||} =1$. Note that the relation is not linear. If we compare to the identity line ($s_{||} = r_{||} $) shown as a dashed line, we note that sometimes the curves of constant $r_{\bot}$ lie above or below the identity line, or even cross it. So, it can be noted that for on-axis separations (where $r_{\bot} = 0$), the spatial scale in redshift space is stretched, i.e. $s_{||} > r_{||} $, effectively opposing the squashing effect obtained by the rough $\mu{(\bm s)}$ approximation. On the other hand, for $r_{\bot} \rightarrow 1 $ a squashed structure is seen (even more so that the one obtained by the $\mu{(\bm s)}$ approximation) that ultimately converges to the limit $ s_{||} \rightarrow r_{||} $ as we approach the plane of the sky ($ r_{||} = 0$). These geometrical distortions can be better appreciated by their effect on the 2PCF presented in Fig. \[fig:figfinga\]. Here we start from a grid in $s-$space, and transform to r-space using the integral relation (eq. \[eq:dpmn39\]) for the parallel component and eq.  for the perpendicular one. From there, we calculate $\mu{(\bm r)}$ (eq. \[eq:dpmn31\]), $g(\gamma, \beta, \mu{(\bm r)}) $ (eq. \[eq:dpmn34\]), assuming that $\xi^{(r)}({\bm r}) = (r/r_0)^{-\gamma}$; and finally, $\xi^{(s)}({\bm s})$ [ (i.e. $\xi^{(s)}(\bm r ({\bm s}))$ ) ]{} from eq. . The cosmological distortion is governed by the $c_{||}$ and $c_{\bot}$ parameters that depend on the Alcock-Paczyński function $AP$ (see eq. \[eq:dpmn12\]). Its value depends on the cosmological parameters ${\bm \Omega} = (\Omega_m,\Omega_k,\Omega_\Lambda)$ and increases with the redshift $z$ [see figure 1 in @ap79]. Figure \[fig:figfinga\](a) shows the case that corresponds to the parameters used for Figure \[fig:rdes\]: $\gamma = 1.8$, $\beta = 0.4$ and $AP =1$, where the geometrical distortions produced are evident, an elongation in the polar direction and a squashing in the equatorial direction. As can be noted the polar elongation resembles the structure known as FoG. In the other three figures, \[fig:figfinga\](b), \[fig:figfinga\](c) and \[fig:figfinga\](d), we explore the effect of cosmological and gravitational alterations. Figure \[fig:figfinga\](b) shows that the effect of increasing $AP$ is a geometrical distortion that concentrates the structure towards the polar axis direction for $AP=2$ that corresponds to $\Lambda$CDM cosmology at $z=2.6$. In Figure \[fig:figfinga\](c) we explore the effect of changing the dimensionless growth-rate for visible mater $\beta$. This gravitational effect is to enhance the FoG structure as its value increases (recall that its limit value is 2/3). On the other hand, if $\beta$ decreases the structure becomes rounder and the FoG faints accordingly as is shown in Fig. \[fig:figfinga\](d). By comparing figures \[fig:figfinga\](b) and \[fig:figfinga\](c) relative to panel \[fig:figfinga\](a), we note that the same enhanced strength of the FoG feature is obtained in the small scale regions, but the large scale structure is quite different. This is because in the first case the distortion is cosmological while on the second it is gravitational. Although it has not been the purpose of this paper, we may consider different values of the power-law index $\gamma$ and obtain figures similar to those shown in Fig. \[fig:figfinga\]. In some cases they might even resemble some of the cases depicted here. It turns out that lower values may accommodate rounder 2PCFs at mid scales, while a steeper $\gamma$ may also concentrate the structure towards the LoS. Note however, that it is easy to discern those cases by a simple projection on the plane of the sky, as depicted through section \[sec:sec4\]. This is because that projection will erase redshift distortions, both gravitational ($\beta$) and cosmological ($AP$) while preserving the radial structure $\gamma$. As we have indicated, a rounder 2PCF at mid spatial scales is favoured by some works that use the $\mu(s)$ approximation. As can be seen in Fig. \[fig:figfinga\](b), rounder figures can be obtained with lower values of $\beta$. We have estimated that a $\beta = 0.25 $ produces a 2PCF which is equally squashed to that obtained by the $\mu(s)$ approximation for the case $\beta = 0.4 $ for most points in the s-space plane, those with $s_{\bot} > s_{||}$. An increase in the $AP$ parameter may also contribute to alleviate the situation. Another possibility, that was not intended to be covered here, is the case of a more realistic 2PCF $\xi^{(r)}({ r} )$ as the ones inferred from baryon acoustic oscillations (BAOs) [e.g. @slosar2013] or those obtained by the CAMB code [@seljak_zaldarriaga_96]. In order to apply the results of this paper to such cases, one could try breaking the inferred $\xi^{(r)}({ r})$ profile in a series of power-laws and then apply eq.  to each section. If this is not possible, then we would have to give up eqs.  and as a way of simplifying $\xi^{(s)}({\bm r})$. However, the projections in the plane of the sky, i.e. eqs.  and are still valid, and instead of using eq.  to simplify, we would have to go back to the expansion of $\xi^{(s)}({\bm r})$ in multipoles eq. . In that case one would end up with the following equation: $$\label{eq:dpmn40} c_{||} ~ ds_{||} = {\frac { \xi^{(r)}(r) } { \sum_{l=0,2,4} \xi_l(r) ~ P_l(\mu{(\bf r)}) } } dr_{||}$$ instead of eq. . And we would have to find a way to estimate the multipole moments $\xi_l(r)$. Another possibility is to leave $\xi^{(s)}(r)$ in the denominator. Considering these possibilities seems like an interesting task for future works, but it is beyond the scope of this paper. We conclude that a whole range of possibilities in shape and strength of the FoG structure and the squashing of the equatorial zone can be obtained by tuning the parameters $\gamma$, $\beta$, and $AP$. This may provide a path towards solving the usual degeneracy problem between cosmological and gravitational distortions, that can still be seen at a level of 10% in 1$\sigma$ correlated variations in recent work [e.g. @Satpathy2017]. Conclusions {#sec:sec6} =========== We emphasize the importance of distinguishing three spaces in cluster and large scale structure studies: the observable redshift-space ${\bm \sigma}$, the physical redshift-space ${\bm s}$, and the real-space ${\bm r}$. The transformation between ${\bm \sigma}$ and ${\bm s}$ is isotropic dilation that introduces a scale factor dependent on the cosmology. On the other hand, the transformation between ${\bm s}$ and ${\bm r}$ goes through a unitary Jacobian independent of redshift, and only distorts the space by factors related to the Alcock-Paczyński $AP$ function (c.f., eqs. \[eq:dpmn15\] and \[eq:dpmn16\]). Furthermore, when we introduce peculiar non-relativistic velocities in this scheme, we demonstrate that the same relation between observable and physical redshift-spaces ${\bm s} = K({\bm \Omega},z) ~ {\bm \sigma}$ is kept. In the analysis of the 2PCF in the physical redshift-space ${\bm s}$, we recover the @kaiser87 effect independent of redshift in Fourier space, and @hamilton92 results in configuration space. We remark, that there appears a dependence with $\mu$ in real-space ($ \mu{(\bm r)} = r_{||}/|{\bm r}|$), and that it has been a common practice to approximate it from redshift-space coordinates as either $ \mu{(\bm s)} = s_{||}/|{\bm s}|$ or $\mu{(\bm {cs})} = c_{||} s_{||}/\sqrt{ c_{\bot}^2 s_{\bot}^2 + c_{||}^2 s_{||}^2 }$ or $r_{||} = s_{||}$, sometimes called the“distant observer approximation", or simply to substitute $\bm s$ for $\bm r$ in the equations. To avoid further confusion we have called this the $\mu{(\bm s)}$ approximation in any of its forms. We argued that this wrong assumption produces either a squashed or a peanut-shaped geometry close to the LoS axis, for the 2PCF in redshift-space. Since $r_{||}$ is usually unknown, we proposed a method to derive it from $s_{||}$ using number conservation in the projected correlation function in both real- and redshift-spaces. This led to a closed form eq. for the case where the real 2PCF can be approximated by a power-law. From this, we solved for $\mu{(\bm r)}$ in real-space, and showed that a different view of the redshift-space 2PCF emerges. The main result is that the redshift-space 2PCF presents a distortion in the LoS direction which looks quite similar to the ubiquitous FoG. This is due to a strong anisotropy that arises purely from linear theory and produces a stretching of the scale as one moves into the on-axis LoS direction. Moving away from the LoS the structures appear somewhat more squashed than those obtained by the $\mu(s)$ approximation for equivalent values of $\beta$. The implications of this remains an open question. The development presented here produces structures that qualitatively reproduce the observed features of the 2PCF of galaxies and quasars large scale structure. A squashing distortion in the equatorial region is attributed to a mixture of cosmological and gravitational effects. And the FoG feature that is usually attributed to other causes, is instead ascribed to the same gravitational effects derived from linear theory. We conclude that a whole range of possibilities in shape and strength of the FoG structure, and the squashing of the equatorial zone, can be obtained by tuning the parameters [$\gamma$]{}, [$\beta$]{}, and [ $AP$]{}. This provides a path towards solving the usual degeneracy problem between cosmological and gravitational distortions. In a future paper (Salas & Cruz-González in preparation) we apply these results to the galaxies and quasar data obtained by current large scale surveys. Acknowledgements. I.C.G. acknowledges support from DGAPA-UNAM (Mexico) grant IN113417. ![ $s_{||}/r_e $ vs. $ r_{||}/r_e$ for $r_{\bot}/r_e$ from 0 to 1 as indicated in the figure, for $\gamma = 1.8$, $\beta = 0.4$, and $c_{||} =1$. For any value of the scaling parameter $r_e$. The dashed line indicates the identity $s_{||}= r_{||} $ for reference.[]{data-label="fig:rdes"}](figure43_1e.eps){width="\columnwidth"} ![ Redshift-space Two-Point-Correlation-Function (2PCF) $\xi^{(s)}(s_{\bot},s_{||})$ in logarithmically spaced contours at $e$ intervals for any value of the scaling parameter $r_e$. The parameter values are: a) $\gamma = 1.8$, $\beta = 0.4$ and $AP =1$; b) $\gamma = 1.8$, $\beta = 0.4$ and $AP =2$; c) $\gamma = 1.8$, $\beta = 0.5$ and $AP =1$; d) $\gamma = 1.8$, $\beta = 0.2$ and $AP =1$; []{data-label="fig:figfinga"}](figure43-2e.eps){width="\columnwidth"}

--- author: - 'M. Radovich' - 'I. Formicola' - 'M. Meneghetti' - 'I. Bartalucci' - 'H. Bourdin' - 'P. Mazzotta' - 'L. Moscardini' - 'S. Ettori' - 'M. Arnaud' - 'G. W. Pratt' - 'N. Aghanim' - 'H. Dahle' - 'M. Douspis' - 'E. Pointecouteau' - 'A. Grado' bibliography: - 'P100.bib' date: 'received; accepted' title: 'A weak lensing analysis of the PLCK G100.2-30.4 cluster. [^1] ' --- Introduction {#sec:intro} ============ Clusters of galaxies are a crucial probe for a variety of key cosmological issues. Among their many applications, knowledge of the mass function and its redshift evolution allows constraints to be put on the matter power spectrum and limits to be set on several cosmological parameters. The construction of the cluster mass function generally involves the following steps: 1) clusters should be identified in large surveys; 2) mass proxies should be used to estimate their masses; and 3) the mass function should be constructed by counting clusters in given mass and redshift bins. This process relies on the fundamental assumption that the scaling relations between the mass and its proxies are well-calibrated and unaffected by selection biases (or that the biases have been accounted for). It has been successfully implemented both using X-rays surveys as the ROSAT All-Sky Survey, where proxies like the gas mass, the temperature, or the X-ray luminosity have been used to estimate the masses [e.g., @2009ApJ...692.1060V; @2010MNRAS.406.1759M], and in optical photometric optical surveys (e.g. the SDSS), where the masses are usually derived from the cluster richness [e.g., @2009ApJ...703..601R]. More recently, searches for galaxy clusters have been started via the detection of the Sunyaev-Zel’dovich (SZ) effect on the photons of the Cosmic-Microwave-Background. The newly-established millimetric observatories, both ground- (South Pole Telescope and Atacama Cosmology Telescope) and space-based (Planck), have provided catalogues of hundreds to thousands of clusters. The first cluster detections from the Planck all-sky survey were released in the Planck Early SZ sample [@planck2011-5.1a]. A crucial part of the validation process involved use of snapshot XMM-Newton observations; in the first two campaigns, 21 out of 25 newly identified cluster candidates were confirmed to be real [@planck2011-5.1b]. ![Composite $gri$ image of the inner $400\arcsec \times 400\arcsec$ region of the PLCKG100 field. []{data-label="fig:field"}](Plots/field.png){width="9.5"} ![Zoom of the inner $100\arcsec \times 100\arcsec$ region of the PLCKG100 field. Labels mark the 10 brightest galaxies. Several strong lensing gravitational arcs are visible. []{data-label="fig:map"}](Plots/map.png){width="9.5"} The XMM-Newton snapshot observations appeared to reveal a class of massive clusters with low X-ray luminosity and disturbed morphology, which may indicate a population of massive clusters that are under-represented in X-ray surveys. Since the cluster masses in SZ surveys are estimated from the integrated Compton $Y$ parameter via scaling relations calibrated with optical or X-ray observations, this may have an effect on the estimate of the mass function, and thus our understanding of cluster growth and evolution. It is therefore of fundamental importance to study these systems in more detail. Lensing is a powerful method to recover the surface density field of clusters, thus allowing to trace the spatial distribution of dark matter. Simulations have shown that provided wide-field observations of excellent quality are available, lensing allows an accurate measurement of the mass profile of clusters , and hence the radial mass/luminosity (M/L) profile [@Medez10], and its dependence on early- / late-type galaxies in the cluster. Combined with X-ray observations, lensing can help to characterize the dynamical state of the clusters, and probe deviations from hydrostatic equilibrium which may bias the X-ray masses [@2008MNRAS.384.1567M; @2010ApJ...711.1033Z]. In this paper we describe the first weak lensing analysis of the cluster PLCK G100.2-30.4 (hereafter PLCKG100), based on deep SUBARU SuprimeCam images. PLCKG100 was detected via the SZ effect in the first 10 months of the Planck survey and confirmed by XMM-Newton X–ray observations [@planck2011-5.1b]. The cluster redshift was estimated by optical $griz$ observations performed with the 0.80-m IAC80 telescope, and through X-ray spectroscopic observations of the Fe K line, giving $z_{\rm phot} = 0.38\pm0.04$ and $z_{\rm Fe} = 0.31\pm0.04$ respectively [@planck2011-5.1b]. The cluster mass estimate, derived from XMM-Newton X-ray observations via iteration about the $M_{500}$ - $Y_X$ relation of , is $M_{\rm 500} = (5.60\pm0.22) \times 10^{14}$ $M_\odot$, and the \[0.1-2.4\] keV X-ray luminosity was $L_{\rm 500} = (3.36 \pm 0.08) \times 10^{44}$ erg s$^{-1}$. Here we use deep (1hr) $r$–band and shallower ($\sim$ 30 min) $giz$ images obtained with SuprimeCam at the Subaru telescope, to derive the mass of the cluster by weak lensing. The paper is organized as follows. A summary of the observations and data reduction strategy is presented in Sect. \[sec:data\]. Techniques adopted for star/galaxy separation are described in Sect. \[sec:sgclass\]. A comparison with cluster properties derived by optical photometry is given in Sect. \[sec:properties\]. The weak lensing analysis is discussed in Sect. \[sec:wl\]. We attempt to refine the hydrostatic mass estimate from XMM-Newton data in Sect. \[sec:xray\]. Summary and conclusions are given in Sect.\[sec:summary\]. A standard concordance cosmology was adopted: $\Omega_\Lambda = 0.7$, $\Omega_M = 0.3$, $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$, giving a scale of 4.7 kpc/arcsec at the redshift of PLCK G100.2-30.4 ($z=0.36$, as derived in Sect. \[sec:properties\]). Date Band Number Total time Average seeing (arcsec) mag($\sigma=10$) mag($\sigma=5$) ------------ ------ -------- ------------ ------------------------- ------------------ ----------------- 23/07/2012 $g$ 10 1800s 0.7 24.9 25.6 23/07/2012 $r$ 20 3600s 0.5 25.5 26.5 23/07/2012 $i$ 10 1800s 0.6 24.7 25.2 23/07/2012 $z$ 8 1440s 0.6 23.8 24.7 Observations and data reduction {#sec:data} =============================== PLCKG100 (Fig. \[fig:field\], Fig. \[fig:map\]) was observed in the second part of the night on July 23, 2012 in the $griz$ bands, with the wide–field Suprime-Cam camera [@suprimecam] mounted at the 8m Subaru telescope. The camera is composed of 10 CCDs, each with $2048\times4096$ pixels and pixel scale $0\farcs207$, allowing coverage of a field of $34\arcmin \times 27 \arcmin$. A sequence of 180s exposures was obtained following the dithering pattern recommended for SuprimeCam. The deepest (1hr) image was taken in the $r$-band, to be used for the lensing analysis, with optimal seeing conditions ($\sim 0\farcs5$). Images in $g$ and $i$ were obtained with a total exposure time of 1800s as planned, while due to technical reasons only 8 out of the 10 planned exposures could be done in $z$, for a total exposure of 1440s. Details of the observations are reported in Tab.\[tab:obssum\]. The prereduction (overscan correction, bias, flat-fielding and masking of bad columns and autoguider) was based on the SDFRED2 [@2002AJ....123...66Y; @2004ApJ...611..660O] pipeline developed for Suprime–Cam. The tool [AstromC]{} [@Romano10] was then used to compute the astrometric solution describing the deformations in the field of view of individual exposures. For each filter, the astrometric solution was computed taking the NOMAD catalog as reference, and at the same time minimizing the differences in the position of the same sources measured in different exposures: this allowed us to obtain an internal accuracy of $\sim 0.01$ arcsec, better than the accuracy of the NOMAD catalog (rms $\sim$ 0.2 arcsec). Images were then resampled and finally coadded using the [Swarp]{} software[^2] [@SWarp]. Photometric calibration was done by observing a field selected from the Stripe 82 area, and using the SDSS Stripe 82 Standard Star Catalog [@2007AJ....134..973I] to derive photometric zero points and SuprimeCam vs. SDSS color terms. Limiting magnitudes for point–like sources at signal to noise levels of SNR=5,10 were computed as in [@virmosu]: simulated images with the same depth and background rms as each science image were produced using [SkyMaker]{}[^3] software, and stars added at random positions; errors on magnitudes were computed from the difference between input and measured magnitudes, and used to derive the median magnitude for each value of SNR. In addition to the stacked image obtained as described above, we also produced a set of stacks where the PSF was homogenized to the same gaussian shape for all filters. This was done as follows: 1. For each exposure, all CCD images were first regridded and combined in one image using [SWarp]{}, adopting the same center, pixel scale and size as the coadded $r$–band stack derived above. 2. For each of them, stars were identified and the PSF fitted as a function of pixel coordinates using the [PSFex]{} software[^4] [@PSFex], as described later in Sect. \[sec:sgclass\]. 3. The [PSFMATCH]{} task in [IRAF]{} was used to compute the kernel that transforms that PSF into a Gaussian with the target FWHM, chosen to be 0.9 arcsec, the worst seeing value measured in all exposures and bands: since the PSF changes across the field, we divided each image in sections of size $100\times100$ pixels, and computed the kernel in each of them. 4. For each filter, all convolved exposures were finally summed together. These PSF-homogenized stacks were used for the photometric measurements only, not for shape measurements, for which we used the unconvolved $r$–band image. A multi-band photometric catalog was extracted by running [SExtractor]{}[^5] [@SExtractor] in dual mode, taking the $r$–band as the detection image, and measuring the photometry on the convolved stacks. The Galactic dust extinction in the field covered by Suprime-Cam was computed using the Schlegel maps [@schlegel], obtaining an average value and rms: $E_{\rm B-V} = 0.12 \pm 0.01$. For each galaxy, observed magnitudes were corrected in each filter by adopting the dust extinction computed at its position. Star-galaxy classification {#sec:sgclass} ========================== The selection of non-saturated stars and galaxies was undertaken in the magnitude (MAG\_AUTO) vs. size ($\delta$ = MU\_MAX-MAG\_AUTO) space [@Huang11]. MU\_MAX is the peak surface brightness above background, which is constant for saturated stars. As in @Huang11, we rejected those sources with $\delta$ lower than for stars as spurious detections. As a test of both the photometric accuracies and of the star-galaxy classification, we compared the $g-r$ and $i-z$ observed colors of stars with those derived at the Galactic plane position of PLCKG100 by the [TRILEGAL]{} code . [TRILEGAL]{} allows the simulation of broad-band photometry of stars in any Galaxy field. The effect of the Galaxy dust extinction on stellar colors was also included in the models, starting from the average and rms $E(B-V)$ values derived from the Schlegel maps, and assigning to each simulated star a random value within this distribution. The transmission curves obtained by the combination of SUBARU/SuprimeCam CCD, telescope, atmosphere and filter contributions were adopted. Model and observed stellar colors are displayed in Fig.\[fig:stcols\]. The zero point offsets required to optimize the overlap of observed and model colors are: +0.038 mag ($g$), -0.030 mag ($r$), -0.023 mag ($i$), -0.016 mag ($z$). Stars with SNR$>$ 100 were extracted and used in the [PSFex]{} software, designed to model the PSF variations in astronomical images. The code allows to model by a polynomial the pixel-by-pixel PSF intensity in a stamp of a given size as a function of the position in the field of view. Starting from this model, we can build a PSF stamp at each galaxy position, and derive the terms needed for the PSF correction of ellipticities, as described in Sect. \[sec:ksbex\]. Different combinations of polynomial orders and stamp sizes were tried; based on the analysis of the residuals of the correction of the anisotropic PSF component (see later in Sect. \[sec:ksbex\]), we selected a polynomial of order 6 and stamps with size of 20x20 pixels ($\sim 5 \times {\rm FWHM}$). ![Stellar colors ([*top:*]{} $i-z$ vs. $g-r$; [*bottom:*]{} $r-i$ vs. $g-r$): measured data (red dots) are compared to the values obtained by TRILEGAL (green dots).[]{data-label="fig:stcols"}](Plots/PLCKG100_st.png){width="9.5"} Cluster photometric properties {#sec:properties} ============================== Figure \[fig:map\] displays the composite $gri$ image of the cluster field: the brightest 10 galaxies within 1 arcmin from the X–ray peak [@planck2011-5.1b] are marked. Two galaxies of similar brightness ($r \sim $ 18.20 mag) are present in this region. We select as the center of the cluster the galaxy that lies closest to the X–ray peak, and identify it as the Brightest Cluster Galaxy (BCG). Several strong–lensing gravitational arcs are also visible, the brightest of which lies at $\sim$ 40 arcsec from the BCG. Another lies at $\sim$ 60 arcsec from the BCG. ![$r$ vs. $g-r$ color plot of galaxies in the field. Galaxies classified as likely cluster members from their colors are plotted as green dots; the red dot is the BCG galaxy. The best fitting red sequence line is overplotted.[]{data-label="fig:colmag"}](Plots/PLCKG100_colmag.png){width="9.5"} Red sequence ------------ We selected the galaxies located in the inner 5 arcmin around the BCG that were classified as cluster members according to their colors, as described later in Sect. \[sec:sample\]. The red sequence, $g-r = a + b r$, was fitted by a biweight algorithm, giving $a=2.6$, $b=-0.05$ (Fig. \[fig:colmag\]). Early-type galaxies were selected as those lying within $2\sigma$ from the best fitting line. Photometric redshifts were computed using the [Zebra]{} code [@Feldmann06]: the cluster redshift, derived from the average photo-z of $r<22$ mag cluster members, is $z = 0.36 \pm 0.08$. For comparison, we obtain $z=0.41 \pm 0.10$ from the Sloan Digital Sky Survey (SDSS) photometric redshifts (DR9). Richness -------- $N_{200}$, the richness of the cluster within $r_{200}$, was derived both from the selection of red-sequence cluster galaxies, and via making a statistical background subtraction. We used the $r_{200}$ derived from the weak lensing analysis detailed below in Sect. \[sec:wl\]: $r_{200} \sim 6$ arcmin. Absolute $r-$band magnitudes and k-corrections were computed for all galaxies with the cluster redshift, $z=0.36$, and to a magnitude limit of $M_{\rm abs}$ = -20 mag. The richness derived from selection of red sequence galaxies produces $N_{200} = 150$. The richness estimate based on statistical background subtraction was done following @AA.433.415P: the density of background and foreground galaxies was then computed in each of 20 sectors in an annulus with thickness 0.1 deg, starting at $r_{200}$+0.1 deg. Sectors with too few galaxies were rejected, and the average density was finally computed, giving the number of background/foreground galaxies in the $r_{200}$ area of the cluster. In this way, we derived a richness $N_{200} \sim 135$. The richness can be translated to a mass estimate using e.g. Eq. 16 in @2010MNRAS.404.1922A: $\log M_{200} = (0.96 \pm 0.15) (\log N_{200}-1.5) +14.36\pm0.04$. From these two consistent values of the richness, a mass $\log M_{200} \sim (15.0 \pm 0.1) $ $M_\odot$ is derived. Weak lensing analysis {#sec:wl} ===================== Shape measurement {#sec:ksbex} ------------------ The derivation of the shear, and the correction of the effect of the PSF on galaxy shapes, is based on the Kaiser Squires and Broadhurst approach [@kaiser95 KSB hereafter]. In the last decade there has been a large effort to develop new algorithms to improve the accuracy in shape measurement [see @2012MNRAS.423.3163K for a comprehensive review], which is crucial in particular to derive cosmological parameters from weak gravitational lensing (cosmic shear). However, KSB is still widely used in the case of the analysis of massive structures such as galaxy clusters [see e.g. @Okabe10; @2012MNRAS.427.1298H; @2014ApJ...795..163U; @2014MNRAS.439...48A]. Our implementation follows that described in @Huang11, with some differences that are outlined below. The KSB algorithm is based on the measurement of the following quantities for each galaxy, derived from the source brightness moments: the source ellipticity vector[^6] $e_{\rm obs}$, and the [*smear polarizability*]{} ($P^{\rm sm}$) and [*shear polarizability*]{} ($P^{\rm sh}$) tensors. The source ellipticity corrected for the anisotropic component of the PSF, $e_{\rm aniso}$ , is first derived as: $$e_{\rm aniso} = e_{\rm obs} - P^{\rm sm} p,$$ $$p = e^*_{\rm obs}/P^{\rm sm*},$$ where starred terms indicate that they are derived from measurement of stars. This is then related to the unknown intrinsic ellipticity $e$ of the source and to the reduced shear, $g=\gamma/(1-\kappa)$, by: $$e_{\rm aniso} = e +P^{\gamma} g, \label{eq:ellipticity}$$ where the effect of the (isotropic) seeing is corrected by the term $P^{\gamma}$, dubbed by @kaiser97 as the *pre–seeing shear polarizability*: $$P^{\gamma} = P^{\rm sh} - P^{\rm sm} \frac{P^{\rm sh*}}{P^{\rm sm*}} \equiv P^{\rm sh} - P^{\rm sm} q.$$ The tangential and cross components of the reduced shear, $g_t$ and $g_x$ are defined as: $$\label{eq_1.0_ila} g_t=- g_1\cos 2\varphi - g_2\sin 2\varphi;\\ g_x=- g_1\sin 2\varphi + g_2\cos 2\varphi,$$ $\varphi$ being the position angle of the galaxy with respect to the assumed cluster center. The source brightness moments are weighted by a window function to suppress the outer parts of the galaxies which are noise dominated. As in @Huang11, the window function size was set to the value that maximizes the ellipticity signal to noise ratio ${\rm SNe}$, defined as: $${\rm SNe}(\theta) = \frac{\int I(\theta) W(|\theta|)d^2\theta}{\sigma_{\rm sky}\sqrt{\int W^2(|\theta|)d^2\theta}}.$$ Sources with ${\rm SNe}$ $<$ 5 were rejected from the analysis. We further excluded those galaxies with $P_\gamma<0.1$, for which the PSF correction of the ellipticity is less reliable. The final output of the pipeline is the vector $e_{\rm iso} =e_{\rm aniso}/P^\gamma$. Provided that the average intrinsic ellipticity vanishes, $\left\langle e\right\rangle = 0$, the average reduced shear is $\left\langle g\right\rangle = \left\langle e_{\rm iso}\right\rangle$: it is therefore common to set $g = e_{\rm iso}$. As first pointed out by @Hoekstra98, it is important to adopt the same size of the window function to compute the galaxy ellipticities and the stellar correction terms used for the PSF correction. In @Huang11 we computed the stellar terms in bins with $\theta$ varying between the minimum and maximum values allowed for the size of the galaxies (2 and 10 pixels respectively), with a step of 0.5 pixels. For each bin, they were then fitted by a polynomial, as a function of pixel coordinates, so that they could be derived at the position of each galaxy. This approach presents some disadvantages: (i) it is necessary to make a polynomial fit for each PSF correction term and for each window function size; (ii) large window sizes increase the amount of noise in the measurement of moments, even if we are using bright stars, which may contribute to different biases as a function of galaxy size. A more recent approach consists of modelling the PSF [@2012MNRAS.419.2356B], enabling us to reproduce the PSF at each galaxy’s position: this is now feasible with accuracies [@2013ApJS..205...12K] $\sigma(e) < 10^{-3}$. To this end, here we used the <span style="font-variant:small-caps;">PSFex</span> [^7] software, which allows modelling of the PSF as a linear combination of basis vectors and fits their coefficients with a polynomial as a function of the pixel position. For each galaxy position, PSF correction terms are then derived using exactly the same window function adopted for that galaxy. A similar approach was adopted in @Gruen13, who showed that using <span style="font-variant:small-caps;">PSFex</span> to model the PSF provides the accuracy required for weak lensing analysis of galaxy clusters. The accuracy of the PSF fitting was checked by deriving the anisotropy corrected ellipticities for stars, and verifying the absence of systematic effects in the residuals across the field of view (see Fig. \[fig:psfmap\]). Using a polynomial of order 6 as described above, we obtained $\langle e1_ {\rm aniso} \rangle = (-6 \pm 2) \times 10^{-4}$, $\langle e2_ {\rm aniso}\rangle = (2 \pm 1) \times 10^{-4}$. Fig. \[fig:psfanisot\] shows the tangential component of the average stellar ellipticity before and after the anisotropic correction ($< 6 \times 10^{-4}$), at different distances from the cluster centre. All of the steps outlined above are implemented in the <span style="font-variant:small-caps;">lensR</span> pipeline introduced in @Huang11. For each galaxy, we adopted the same weight definition as in @Huang11: $$w=\frac{1}{\Delta {e_0}^2 +\Delta e_{\rm iso}^2 }, \\ \label{eq:well}$$ where $\Delta {e_0} \sim 0.3$ is the typical intrinsic rms of galaxy ellipticities. The uncertainty on the corrected ellipticities, $\Delta e_{\rm iso}$, was computed following a numeric approach. For each star used in the PSF fitting, we compute the difference between the correction terms measured on the real stars and on the PSF model derived by PSFex, and fit them as a function of the position. This allows estimation of an uncertainty on the stellar correction terms at each galaxy’s position. $N$ values of $e_{\rm iso}$ were then computed (e.g. $N$=100), randomly drawing both the observed source ellipticity and the stellar correction terms from a Gaussian distribution built from their errors. The error on $e_{\rm iso}$ used to compute the weight was taken to be the rms of these values. ![PSF anisotropy correction: the first three panels show the ellipticity pattern (measured, derived from the PSFex model and corrected ($e_{\rm aniso}$), X and Y are in pixels). The scale is displayed by the arrows in the middle ($e=0.1$). In the last panel, blue and red dots are the values of $e_{\rm aniso}$ derived on real stars and on the PSFex model respectively; green dots are the values after the correction. []{data-label="fig:psfmap"}](Plots/W-S-R+_coadd.png){width="9.5"} ![Average tangential PSF ellipticity at different distances from the cluster centre, before (black triangles) and after (red circles) the anisotropy correction. The shaded area is plotted at $e_{\rm aniso}^T = \pm 5 \times 10^{-4}$.[]{data-label="fig:psfanisot"}](Plots/anisotan.png){width="9.5"} The background sample {#sec:sample} --------------------- The dilution bias introduced by unlensed galaxies in the sample used for the shape measurement can adversely affect the cluster mass estimate. To minimise the contamination of this sample by unlensed sources, we followed the approach described in @Formicola13. Using the galaxy catalog with photometry in $griz$ bands, we identified background, cluster members and foreground populations in a color-color diagram derived by choosing the appropriate combination of colors to be investigated. Our selection in the color-color (CC) space was based on the simultaneous analysis of galaxy colors and of the shear signal of the galaxies selected as background sources. We used the photometric data from the Cosmic Evolution Survey [COSMOS; @ilbert] to train the color selection. The accurate photometric redshifts of the sample allow a careful characterization of the distribution of the galaxies belonging to different redshift ranges in CC space. At the same time, we tuned the color cuts to maximize the amplitude of the tangential shear component (Eq. \[eq\_1.0\_ila\]) measured from the background sources. Starting from the $i-z$ vs $g-r$ CC plot of the COSMOS galaxies, we defined criteria based on color and magnitude cuts allowing to identify each galaxy population in the PLCKG100 images. To this end, we computed the distance $d$ of each galaxy in the $i-z$ vs $g-r$ diagram from the line represented in Fig. \[col\_plck\] (upper panel). This line (derived as described in [@Formicola13]) separates the over-dense region mainly populated by sources at low redshift, from the over-dense region mainly populated by sources at redshift higher than that of the cluster. The background population was selected according to the following criteria: - galaxies with: $24.5 < r < 26.5$ mag; - or: $d < 0$, $21.8 < r < 24.5$ mag, $i-z > 0.1$; - or: $d \ge 0$, $21.5 < r < 24.5$ mag, $i-z <- 0.3$. The foreground and the cluster member population were identified according to these criteria: - foreground galaxies: $d \ge 0$, $21.0 < r < 24.5$ mag, $-0.1<i-z <0.1$; - cluster member galaxies: $d \ge 0$, $r < 23.5$ mag, $0.65<g-r <2.0$, $0.1<i-z<0.5$. ![Selection in CC space. Upper panel: CC diagram for PLCK G100 galaxies. Cluster members are displayed with orange dots and foreground and background galaxies with green and magenta dots, respectively. Unclassified sources are plotted with black dots. Bottom panel: radial number density profiles of galaxies. The background density profile (magenta squares) shows a decrease in the central region; overplotted is the magnification effect expected for a NFW profile. []{data-label="col_plck"}](Plots/imm2_p.pdf "fig:") ![Selection in CC space. Upper panel: CC diagram for PLCK G100 galaxies. Cluster members are displayed with orange dots and foreground and background galaxies with green and magenta dots, respectively. Unclassified sources are plotted with black dots. Bottom panel: radial number density profiles of galaxies. The background density profile (magenta squares) shows a decrease in the central region; overplotted is the magnification effect expected for a NFW profile. []{data-label="col_plck"}](Plots/magnification_p100.pdf "fig:"){width="\columnwidth"} The total sample of background galaxies is represented with magenta dots in Fig. \[col\_plck\] (upper panel), while foreground and cluster galaxies are marked with green and orange dots, respectively. Radial number density profiles for each identified population are shown in Fig. \[col\_plck\] (bottom panel). The radial number density profile for cluster member sources (orange triangles) indicates a clear identification of the cluster member population as shown by the high clustering at small radii. As expected, the foreground population (green diamonds) appears uniformly distributed. The background sample (magenta squares) shows a drop in number counts towards the cluster centre, which is expected due to the so–called magnification bias [@broad_2008]. Also displayed in Fig. \[col\_plck\] is the depletion in counts expected for a NFW profile with the parameters derived by the weak lensing analysis in Sect. \[sec:weaklens\], showing good consistency with the observations. Using the photo-z derived by @ilbert, we extracted the redshift distribution of the COSMOS galaxies with color and magnitude cuts describing the background population, needed for the evaluation of the critical density in the weak lensing analysis, as described in Sect. \[sec:weaklens\]. n $M_{\rm vir}$ $c_{\rm vir}$ $r_{\rm vir}$ $M_{\rm 2500}$ $r_{\rm2500}$ $M_{\rm 500}$ $r_{\rm 500}$ $M_{\rm 200}$ ---- ------------------------- ------------------------ ------------------------------- ------------------------ ---------------------------- ------------------------ ----------------------------- ------------------------- 1s $11.72_{-1.55}^{+2.35}$ $5.98_{-1.45}^{+1.27}$ $2120.70_{-97.70}^{+133.39}$ $3.31_{-0.43}^{+0.48}$ $518.27_{-23.35}^{+23.74}$ $7.39_{-0.78}^{+1.50}$ $1158.17_{-42.43}^{+73.45}$ $10.25_{-1.26}^{+1.95}$ 1b $15.31_{-2.00}^{+2.43}$ $3.77_{-0.04}^{+0.04}$ $2318.68_{-105.61}^{+116.58}$ $2.85_{-0.34}^{+0.41}$ $493.32_{-20.61}^{+22.64}$ $8.43_{-1.07}^{+1.30}$ $1210.04_{-53.61}^{+59.09}$ $12.91_{-1.67}^{+2.03}$ 1c $15.46_{-1.74}^{+2.85}$ $3.65_{-0.02}^{+0.03}$ $2326.07_{-90.41}^{+134.79}$ $2.78_{-0.33}^{+0.54}$ $489.09_{-20.03}^{+29.93}$ $8.41_{-0.96}^{+1.58}$ $1209.20_{-47.83}^{+71.36}$ $13.00_{-1.46}^{+2.41}$ 1d $15.04_{-2.59}^{+2.47}$ $3.77_{-0.02}^{+0.02}$ $2304.89_{-140.47}^{+119.99}$ $2.80_{-0.47}^{+0.44}$ $490.08_{-28.90}^{+24.63}$ $8.27_{-1.41}^{+1.34}$ $1202.59_{-72.51}^{+61.89}$ $12.68_{-2.17}^{+2.08}$ 1e $14.47_{-2.12}^{+2.66}$ $4.28_{-0.04}^{+0.04}$ $2275.06_{-116.91}^{+131.45}$ $3.07_{-0.43}^{+0.53}$ $505.21_{-24.70}^{+27.66}$ $8.30_{-1.20}^{+1.50}$ $1204.05_{-60.88}^{+68.39}$ $12.34_{-1.80}^{+2.25}$ 2a $11.21_{-1.94}^{+1.98}$ $6.15_{-1.25}^{+1.91}$ $2089.44_{-128.58}^{+116.37}$ $3.24_{-0.40}^{+0.32}$ $514.43_{-22.41}^{+16.38}$ $7.12_{-1.05}^{+1.00}$ $1143.87_{-59.25}^{+51.11}$ $9.83_{-1.63}^{+1.62}$ 2b $14.18_{-1.35}^{+2.99}$ $3.80_{-0.06}^{+0.03}$ $2259.80_{-73.98}^{+149.04}$ $2.66_{-0.23}^{+0.51}$ $481.84_{-14.46}^{+28.99}$ $7.82_{-0.72}^{+1.60}$ $1180.16_{-37.58}^{+75.59}$ $11.96_{-1.13}^{+2.51}$ 2c $15.23_{-2.49}^{+2.61}$ $3.65_{-0.03}^{+0.03}$ $2314.16_{-133.77}^{+125.39}$ $2.74_{-0.47}^{+0.50}$ $486.45_{-29.61}^{+27.83}$ $8.28_{-1.38}^{+1.45}$ $1202.89_{-70.75}^{+66.38}$ $12.80_{-2.10}^{+2.21}$ 2d $15.00_{-2.00}^{+2.38}$ $3.77_{-0.02}^{+0.02}$ $2302.54_{-106.99}^{+115.82}$ $2.79_{-0.36}^{+0.43}$ $489.60_{-22.01}^{+23.78}$ $8.25_{-1.09}^{+1.29}$ $1201.38_{-55.22}^{+59.74}$ $12.64_{-1.68}^{+2.00}$ 2e $14.11_{-2.07}^{+1.45}$ $4.29_{-0.02}^{+0.04}$ $2256.27_{-115.92}^{+74.78}$ $2.99_{-0.42}^{+0.29}$ $501.23_{-24.48}^{+15.77}$ $8.10_{-1.17}^{+0.82}$ $1194.27_{-60.37}^{+38.93}$ $12.04_{-1.75}^{+1.23}$ ![$r-$band luminosity-weighted map in the PLCKG100 field. The overplotted contour lines are the weak lensing *S-map* showing the mass distribution derived by weak lensing, at levels $\sigma=$ 3, 4.5, 6, 8.5. The cross marks the position of the BCG assumed as the centre of the halo.[]{data-label="fig:lumden"}](Plots/PLCKG100_ldist.png){width="9.5"} ![Shear profile detected for PLCKG100 (tangential and cross components, where the latter is expected to be null), and for comparison the values derived adopting the best-fit models. We indicate with mnfw the model obtained using the @2008MNRAS.390L..64D relation and with nfw the model obtained keeping both $M_{vir}$ and $c_{vir}$ as free parameters.[]{data-label="fig:shprof"}](Plots/PLCKG100_shear_cs_ila.png){width="8"} ![Wavelet reconstructed X-ray image in the $[0.5-2.5]$ keV energy band, obtained from the combination of the three XMM EPIC camera images. The contours show the weak lensing *S-map* (see Fig.\[fig:lumden\]), at levels $\sigma=3.5,5,7,9$ levels. The circle is the cluster centroid derived by the XMM analysis. The cross and square mark the position of the two brightest cluster galaxies ($1$ and $2$ respectively in the text).[]{data-label="fig:xmm"}](Plots/PLCKG100_xmm2.png){width="8.5"} Simulations {#sec:simulations} ----------- The ellipticities derived by most of the available shape measurement algorithms are affected [see @step1 and references therein] by a multiplicative ($m$) and an additive ($c$) bias defined as: $e_{\rm obs} -e_{\rm true} = m\ e_{\rm true} + c$. The value of $m$ depends on the PSF type, as well as on the brightness and size of the galaxies, and in the case of KSB it is typically around 10%-20% [@step1]. To investigate this, we produced simulations using the GalSim[^8] software [@2014arXiv1407.7676R], that was specifically designed to simulate sheared galaxies within the [*GREAT*]{} effort [see @great10 and references therein]. GalSim offers the possibility to use both analytic models for the galaxies, and/or a sample of 260000 real galaxies from Hubble Space Telescope COSMOS data. In addition, it can take as input the model produced by PSFex, and hence can produce an image with the same PSF shape and spatial variations as the real image. We built our simulations so that ellipticities were measured on individual stamps of sheared galaxies, in order to avoid the effect of blending from neighbour sources. To evaluate the bias, we first produced images with constant shear values, $g = 0.01, 0.05, 0.1, 0.15, 0.2$, for each ellipticity component. We randomly drew 10000 galaxies from the full sample, and repeated the simulation 10 times. For each shear value, we then verified that the expected input value could be recovered and evaluated the bias as a function of signal to noise ratio, as well as at different positions in the image to check for systematics related to a poor PSF correction. From these tests, we derived a multiplicative bias $m=10\%$, and no significant additive bias; we were not able to detect a significant dependence on the signal to noise ratio. Secondly, we used GalSim to simulate the shear signal produced by a cluster having a NFW profile with redshift, virial mass ($M \sim 10^{15}$ h$^{-1}$ $M_\odot$) and concentration parameter ($c_{\rm 200} =4$) similar to what we find here for PLCK-G100. We made ten such simulations and fitted the results in the same way as in Sect. \[sec:weaklens\], where the only free parameter was the mass and the concentration parameter was set to the input value of the simulation. The mass obtained applying the multiplicative bias was $M_{\rm 200} = (1.0\pm 0.2) \times 10^{15}$ h$^{-1} M_\odot$. [cc]{} {width="8"} & {width="8"}\ {width="8"} & {width="8"}\ Mass estimate {#sec:weaklens} ------------- For the weak lensing analysis we used the sources in the catalog satisfying the following criteria: (i) classified as extended (Sect. \[sec:sgclass\]); (ii) with $P_\gamma > 0.1$, SNRe&gt;10; e1iso, e2iso $<$ 1 ; (iii) outside regions masked due to bad columns, haloes or spikes nearby bright, saturated stars. We further considered only those sources classified as background galaxies in Sect. \[sec:sample\]. This produced a catalog of 17000 sources in a 0.227 sq. deg area, corresponding to a density of 21 gals arcmin$^{-2}$. We first extracted (Fig. \[fig:lumden\]) the S-map [see @Huang11 and references therein], showing the spatial distribution of the weak lensing signal. The peak is consistent with the position of the galaxy that we identified as the BCG of the cluster in Sect. \[sec:properties\]. We therefore took this position as the center of the cluster, and used it to compute the cross and tangential components of the ellipticities whose profiles are displayed in Fig. \[fig:shprof\]. Computing the mass from the shear requires an estimate of the critical surface density: $$\Sigma_{\rm crit} = c^2 (4\pi G D_l\beta)^{-1},$$ where $\beta=D_{ls}/D_s$; $D_{ls}$, $D_s$, and $D_l$ are the angular distances between lens and source, observer and source, and observer and lens, respectively. Since we have no spectroscopic training set, we are not able to judge the reliability of our photometric redshifts for the lensed background galaxies. We therefore adopted the approximation of a constant $\beta$ for all galaxies. This introduces an overestimate of the average shear by a factor $1 +\left(\left\langle\beta^2\right\rangle/\left\langle\beta\right\rangle^2 -1\right) \kappa$ $\sim 1 + 0.2 \kappa$ at $z = 0.36$, as discussed in @apJ...532...88h. The correction of this bias is included in our mass estimate. As described in Sect. \[sec:sample\], the value of $\left\langle \beta \right\rangle $ was computed using the photometric redshifts from the COSMOS sample, applying the color and magnitude cuts adopted for the selection of background galaxies. We find $\left\langle \beta \right\rangle= 0.54$, equivalent to assuming that all galaxies lie at the same redshift $z_s = 1.04$. The shear profile was first fitted assuming a NFW halo and using the analytical expression for the shear given by @Wright00: a Maximum Likelihood fitting approach was adopted [@Radovich08; @Huang11]. In order to see how the mass estimate is affected by the assumption on the cluster centre and on the model fitting parameters, we performed different tests: ([*a*]{}) selecting both of the two brightest galaxies as the cluster centre ($n=1,2$), and ([*b*]{}) trying both a 2-parameter fit (virial mass $M_{\rm vir}$ and concentration $c_{\rm vir}$), and by assuming a relation between $M_{\rm vir}$, $c_{\rm vir}$ and the cluster redshift $z_{\rm cl}$. In the latter case, we first used the relation in @2008MNRAS.390L..64D: $$c_{\rm vir} = A \left(\frac{M}{M_\star}\right)^B (1+z_{\rm cl})^C, \label{eq:cvirmass}$$ with $M_\star= 2. \times 10^{12}/h$ $M_{\odot}$, $A = 7.85$, $B=-0.081$, $C=-0.71$. Then, we adopted the $c-M-z$ relations discussed in @2014ApJ...797...34M [see their Table 2]. These were based on numerical simulations of halos at different redshifts and halo relaxation states, undertaken for the analysis of the CLASH cluster sample. We used their 2D relations derived for NFW models for all halos, relaxed halos and super-relaxed halos. The results are given in Table \[tab:nfw\]. The differences in the fitted parameters are within the uncertainties, with a slightly higher mass derived for $n=1$. A model-independent estimate of the mass was then derived using the aperture densitometry method [see @Huang11 and references therein]. This method provides an estimate of the projected 2D mass within a radius $\theta_m$, $M_{2D}(\le \theta_m)$. As discussed e.g., in @Okabe10, before comparing the aperture (2D) and the NFW (3D) mass we first need to evaluate the correction factor $f(\theta_m) = M_{2D}(\le \theta_m)/M_{3D}(\le \theta_m) > 1$. To this end, we used again the analytical expression given by @Wright00 to compute $f(\theta_m)$ for an NFW halo with the best–fit values of $c_{\rm vir}$, $M_{\rm vir}$ found above (first row in Tab. \[tab:nfw\]). We obtained $f=1.39$ at $r_{500}$ and $f=1.31$ at $r_{\rm vir}$. The deprojected mass values are: $M^{\rm ap}_{500} = (6\pm2) \times10^{14}$ $M_\odot$ and $M^{\rm ap}_{\rm vir} = (1.0\pm0.4)\times 10^{15}$ $M_\odot$. The mass derived by weak lensing can be affected by the presence of large-scale structures along the line of sight [@Hoekstra03; @Hoekstra11]. According to Figs. 6 and 7 in @Hoekstra03, this uncertainty is of the order of $\sim 2 \times 10^{14} h^{-1}$ $M_\odot$ for a cluster with $M=10^{15} M_\odot$ at $z \sim 0.4$, $\theta_{\rm max} = 15$ arcmin, which is comparable to the uncertainties derived in the fitting listed in Table \[tab:nfw\]. ![Radial mass profile derived from the X-ray data, where the hydrostatic mass profile and uncertainties are displayed by the solid line and the grey area. Error bars represent the 1 $\sigma$ uncertainties described in the text. Also displayed are the mass profiles derived from the WL analysis (dashed line: NFW fit; points with error bars: aperture densitometry after deprojection). The inset shows $M_{2500}$, $M_{1000}$, $M_{500}$ and $M_{\rm vir}$ derived from our XMM analysis (blue squares); the Planck analysis (green square); WL NFW, first row in Tab. \[tab:nfw\] (red triangles); WL, aperture densitometry (black circles): a small offset in radius is applied here for display purposes. []{data-label="fig:radmass"}](Plots/mass_xwl.png){width="8.5"} X-ray analysis {#sec:xray} ============== PLCK G100.230.4 has been observed for $\SI{13.7}{\kilo\second}$ with the XMM-Newton (obs-ID $0656202301$) European Photon Imaging Cameras (EPIC), during the follow-up campaign of Planck newly detected clusters. This dataset was re-processed using the XMM-Newton Science Analysis System (SAS) version $14.0.0$ in order to select the relevant calibration files. To remove any flare contamination, i.e. observation periods with an unusual count rate, we applied a 3$\sigma$ clipping and a temporal wavelet filter to the histogram of the photon arrival times, with a bin size of $\SI{50}{\second}$, in the energy band $[1-5]$ and $[10-12] \si{\kilo\electronvolt}$. After the flare removal procedure we find a useful exposure time of $\SI{13.7}{\kilo\second}$ and $\SI{5.2}{\kilo\second}$ for the two EPIC-MOS and the EPIC-PN datasets, respectively. To prepare the following imaging and spectroscopy analyses, we rebinned the flare-cleaned list of photon events in sky coordinates and energy, and associated both a 3D effective exposure and a background noise model to this data cube (see details in @mass_profiles). The background noise was modelled after subtraction of all detected point-like sources using two components: the instrumental noise caused by high energy particles interacting with the detectors, and several astrophysical components, namely the Cosmic X-ray background (CXB) (e.g. @giacconi) plus a diffuse emission of Galactic origin (e.g. @snowden_gala). For the instrumental component we used an analytical model inspired by the works of [@kuntz_particle] and [@xmm_model_lec_mol], which holds for each EPIC camera a continuum plus several fluorescence lines (see details in @bourdin_a521). As proposed by [@kuntz_gala], the Galactic foreground was modelled using two absorbed APEC (@apec_code) spectra, with temperatures of $\SI{0.14}{\kilo\electronvolt}$ and $\SI{0.248}{\kilo\electronvolt}$, respectively. The unresolved CXB spectrum follows a power law with $\Gamma = 1.42$ (@lumb_power). The particle background was normalised in order to match the count rates of each camera outside the target region and in the hard energy band of $[10-12] \si{\kilo\electronvolt}$. The normalization of each astrophysical component was determined using a joint spectral fit outside any cluster emission, in the same region where the particle background was normalised. From the cleaned data we produced an image in the $[0.5-2.5]\si{\kilo\electronvolt}$, which is background subtracted and corrected for any spatial variation of the effective area. From this image we determined the X-ray peak at ($RA=350.56228$, $DEC=28.52041$). Fig. \[fig:xmm\] shows a wavelet denoising of this image. As we can see from this Figure, the X-ray peak (circle) lies within 5$\arcsec$ of the galaxy that we assumed to be the BCG (cross), whereas the second brightest galaxy is 18$\arcsec$ away. It is worth noting that there is also a good agreement between the direction of the elongation of the cluster, going from NE to the SW, as seen by the X-ray image and the isocontours from the lensing analysis. From the photon image we then extracted a point-source excised surface brightness profile, centered on the cluster X-ray peak. In Fig.  \[fig:xprops\], top left panel, we show this surface brightness profile, which we found to reach the background level at $R=\SI{1450}{\kilo\parsec}$. The temperature profile, shown in Fig. \[fig:xprops\] bottom left panel, was extracted in 4 bins from the same circular region as the surface brightness The temperature was measured in each bin by fitting the spectrum in the energy band $[0.3-11]\si{\kilo\electronvolt}$ using an absorbed APEC thermal model with three parameters, namely the normalization, temperature and metallicity. The thermal models were convolved by the instrument response, as described in detail in [@mass_profiles]. The atomic hydrogen column density was fixed to its Galactic value $N_{H}=6.24 \times 10^{20} \si{\centi\meter}^{-2}$ (@lockman_1990). To measure the X-ray mass we followed the “forward” procedure described in and [@rasia12]. In short, analytic profiles of the gas density and temperature [@vikh_prof] were projected using the spectroscopic like temperature [@mazzotta_wt] and fitted to the observed surface brightness and projected temperature. The 3D information were then folded into the hydrostatic mass equation (e.g. @sarazin_1988 or @ettori_2013). The confidence envelopes of the 3D profiles in Fig. \[fig:radmass\] were computed following a Monte-Carlo procedure. More specifically, we generated several realisations of both a Poisson distributed surface brightness profile and a Gaussian distributed temperature profile, that hold the mean values and variances of the derived data quantities. For each of these realisations we fitted a new set of density and 3D temperature profiles, and stored these models when they obeyed the physical prior that the underlying hydrostatic mass profile is increasing and positive. From $N=500$ remaining hydrostatic density, 3D temperature and mass models, we drew the confidence envelopes that encompass $68\%$ of the realisations with minimal distance to their relative best fit. The envelope of the mass profile allowed us to derive a cluster mass of $M_{2500}=3.2^{+0.8}_{-0.9} \times 10^{14}M_{\odot}$, $M_{1000}=4.4^{+0.8}_{-0.7} \times 10^{14}M_{\odot}$ and $M_{500}=5.4^{+1.0}_{-0.9} \times 10^{14}M_{\odot}$, which corresponds to $R_{2500}=541^{+49}_{-59}\si{\kilo\parsec}$, $R_{1000}=816^{+49}_{-46}\si{\kilo\parsec}$ and $R_{500}=1093^{+65}_{-62}\si{\kilo\parsec}$, respectively. The latter mass value is in excellent agreement with the mass derived from iteration about the $M_{500}$ - $Y_X$ relation, $M_{\rm 500} = (5.60\pm0.22) \times 10^{14}$ $M_\odot$ [@planck2011-5.1b]. Figure \[fig:radmass\] also shows a comparison of the X–ray total mass profile with those obtained from WL aperture mass densitometry and WL NFW model fitting (obtained with two free parameters). The inset shows the various mass estimates at density contrasts of 2500, 1000, and 500. The agreement between the different mass measurements is remarkably good at all density contrasts. @planck2013-p15 parameterised the difference between their mass estimate and the true mass by the factor (1-$b$), where $b$ contains all possible sources of bias, such as neglect of the non-thermal pressure support in the hydrostatic assumption, multi-temperature structure in the ICM, instrumental calibration uncertainties, etc. Assuming a weak lensing reconstructed mass of $M_{\rm 500, WL} \sim (7\pm1) \times 10^{14}$ $M_\odot$ and comparing to the mass estimate obtained by @planck2011-5.1b, we obtain (1-$b$) = 0.8 $\pm$ 0.1 from the present observations. @planck2013-p15 obtained a bias factor of (1-$b$) = 0.8 from their calibration of $b$ using numerical simulations, and used this value, with a flat prior of $[0.7, 1.0]$, as the baseline for their cosmological analysis. @planck2015-p24 have updated this analysis to include a calibration of $(1-b)$ using weak lensing mass measurements, taking into account sources of scatter such as triaxiality, the presence of subtructures, and systematic errors in the weak lensing measurement itself. Assuming the weak lensing mass to be the true mass, their analysis yields $(1-b)_{\rm WtG} = 0.69 \pm 0.07$ and $(1-b)_{\rm CCCP} = 0.78 \pm 0.09$ for the samples of @vdlinden14 [Weighing the Giants] and @hoe15 [Canadian Cluster Cosmology Project], respectively. Under the same assumption, the value we obtain from PLCKG100 is compatible within $1\sigma$ with the bias factor derived from these updated measurements. Summary and conclusions {#sec:summary} ======================= We presented here the first weak lensing analysis of the PLCK G100.2-30.4 cluster. The analysis confirmed that this is a massive cluster ($M_{\rm vir} > 10^{15}$ $M_\odot$), in agreement with that expected from the richness estimate. At a density contrast of 500 with respect to the critical density of the Universe at the cluster redshift, the weak lensing mass from aperture densitometry is $M^{\rm ap}_{\rm500} = (6\pm2) \times 10^{14}$ $M_\odot$, after deprojection. An NFW fit in the radial range 1-15 arcmin, shown by the dotted line in Fig. \[fig:radmass\], yields a mass $M_{\rm 500} \sim (7\pm1) \times 10^{14}$ $M_\odot$. We checked the robustness of the weak lensing mass estimates by repeating the analysis with different choices of the cluster centre and of the NFW model fitting: in particular, we compared the results obtained by leaving both $c_{\rm vir}$ and $M_{\rm vir}$ as free parameters, and using different $(c, M, z)$ relations. These yield results consistent with those given above. Comparing the mass derived by weak lensing with the value obtained by the reanalysis of XMM data, we obtain a bias factor of (1-$b$) = 0.8 $\pm$ 0.1. This is compatible within $1 \sigma$ with the bias factor obtained by @planck2015-p24 from their calibration of (1-$b$) using newly-available weak lensing samples from @vdlinden14 and @hoe15. The present results illustrate the importance of high-quality data and the need for larger samples in such cluster mass studies. Acknowledgements ================ We thank the referee for the helpful comments that improved the paper. We acknowledge financial contributions from: contract ASI/INAF I/023/12/0 (LM, MR, MM); PRIN INAF 2009 “Weighing galaxy clusters with strong and weak lensing” (MR, MM); PRIN MIUR 2010-2011 “The dark Universe and the cosmic evolution of baryons: from current surveys to Euclid” (LM, MR,MM); PRIN INAF 2012 “The Universe in the box: multiscale simulations of cosmic structure” (LM); contracts ASI-INAF I/009/10/0 and PRIN-INAF 2012 “A unique dataset to address the most compelling open questions about X-Ray Galaxy Clusters” (SE). MR also acknowledges Prof. P. Benvenuti (Department of Physics and Astronomy, University of Padova), for having made available his computing facilities. [^1]: Based on data collected at Subaru Telescope (University of Tokyo). [^2]: http://www.astromatic.net/software/swarp [^3]: http://www.astromatic.net/software/skymaker [^4]: http://www.astromatic.net/software/psfex [^5]: http://www.astromatic.net/software/sextractor [^6]: In the following, the vector components along the $X$ and $Y$ axes are denoted as e.g. $e1$ and $e2$, respectively. [^7]: http://www.astromatic.net/software/psfex [^8]: https://github.com/GalSim-developers/

--- author: - Tobias Gehring - Vitus Händchen - Jörg Duhme - Fabian Furrer - Torsten Franz - Christoph Pacher - 'Reinhard F. Werner' - Roman Schnabel title: 'Implementation of Quantum Key Distribution with Composable Security Against Coherent Attacks using Einstein-Podolsky-Rosen Entanglement' --- **Secret communication over public channels is one of the central pillars of a modern information society. Using quantum key distribution (QKD) [@Gisin2002; @Scarani2009] this is achieved without relying on the hardness of mathematical problems which might be compromised by improved algorithms or by future quantum computers [@Shor1997]. State-of-the-art QKD requires composable security against coherent attacks for a finite number of samples. Here, we present the first implementation of QKD satisfying this requirement and additionally achieving security which is independent of any possible flaws in the implementation of the receiver. By distributing strongly Einstein-Podolsky-Rosen entangled continuous variable (CV) [@Cerf2001; @Weedbrook2012] light in a table-top arrangement, we generated secret keys using a highly efficient error reconciliation algorithm. Since CV encoding is compatible with conventional optical communication technology, we consider our work to be a major promotion for commercialized QKD providing composable security against the most general channel attacks.** Using a QKD system the communicating parties employ a cryptographic protocol that cannot be broken, neither by todays nor by future technology. The security of the key distributed by such a system is guaranteed on the basis of quantum theory by a mathematical proof, which has to consider the most sophisticated (quantum) attacks on the quantum channel, so-called ‘coherent attacks’. Furthermore, security has to be established in a ‘composable’ fashion, which means that if the distributed key is used in another secure protocol (like one-time-pad encryption), it remains secure in the composition of the two protocols. To make a security proof applicable to actual implementations, it also has to include all effects due to the finite number of distributed quantum states. Additionally, the security proof has to model the source and the detectors correctly. Since a completely correct model of a detector is difficult to achieve, the proof can be made ‘measurement-device independent’ instead, i.e. the proof does not include any detector implementation details at all. Thus, security cannot be compromised by any ‘side-channels’ of the detector’s implementation, including those which may only be discovered in the future. To match the properties of actual implementations and the assumptions made in security proofs, intense theoretical as well as experimental research and development is ongoing. Here, we report the first QKD implementation that generates a (finite) key which is secure against coherent attacks and whose security is composable as well as guaranteed independent of possible side channels of the receiver’s detector. Theoretically, the security of our protocol has been established in Ref. . Following this work we use measurement variables with continuous spectra (continuous variables) and strongly Einstein-Podolsky-Rosen entangled light beams whose actual entanglement strength is a crucial parameter for achieving a positive key rate. Our work was possible only after the security proof of Ref.  was found. Other proofs did also find composable security against coherent attacks [@Renner2009; @Leverrier2013] but only for an unrealistically large number of distributed quantum states. The security of previous experimental continuous-variable implementations, however, were restricted to the class of ‘collective attacks’. While this class of attacks already allows an eavesdropper to possess a quantum memory, all quantum states are attacked identically using a collective Gaussian operation. Although collective attacks are in the limit of an infinite number of distributed quantum states as strong as coherent attacks, it is currently not known whether this holds for a realistic finite key length protocol. For collective attacks a transmission distance of $80$km was achieved with a finite number of distributed quantum states using Gaussian modulated coherent states[@Leverrier2010; @Jouguet2013]. In the discrete-variable regime, first security proofs providing composable security against coherent attacks were given in Refs. . However, all of them have in common that they require perfect single-photon sources, which are experimentally impractical. The recent experimental demonstration reported in Ref.  targeted the QKD protocol considered in Ref.  but it used weak coherent pulses and, thus, an imperfect single-photon source. To compensate for imperfect single-photon sources, the decoy state method has been developed [@Wang2005; @Lo2005]. This method was very recently proven composable secure against coherent attacks [@Lim2014] and was implemented in Ref. , at around the same time as our CV QKD implementation. To also avoid possible side-channel attacks in future discrete-variable QKD implementations, for instance on the employed single photon detectors [@Makarov11], the finite-key measurement-device-independent protocol very recently proven composable secure in Ref.  could be implemented in the future. ![**Einstein-Podolsky-Rosen entanglement source for CV QKD.** (a) The source consists of two continuous-wave squeezed vacuum beams, generated by type I parametric down-conversion at $1550$nm (red), which are superimposed at a balanced beam splitter with a relative phase of $\tfrac{\pi}{2}$. Yellow beam: $775$nm pump field, DBS: Dichroic beam splitter, PS: Phase shifter. (b)-(e) Correlations between Alice’s and Bob’s data, measured by balanced homodyne detection in either the amplitude ($X$) or phase ($P$) quadrature. The data is normalized to the noise standard deviation of a vacuum state. Blue: Einstein-Podolsky-Rosen entangled state used for QKD. Black: Reference measurement of zero-point fluctuations of the ground state (vacuum).[]{data-label="fig:quadrature entanglement"}](Fig1-EPR-Source.pdf){width="8.5cm"} Our implemented protocol uses two continuous-wave optical light fields which were produced by a source at one of the communicating parties (Alice) and whose amplitude and phase quadrature amplitude modulations were mutually entangled [@Furusawa1998]. Using Einstein-Podolsky-Rosen entanglement as a resource makes our protocol a CV equivalent of the BBM92 protocol for discrete variables [@Bennett1992]. The schematic of the experimental setup is illustrated in Fig. \[fig:quadrature entanglement\](a). Two squeezed-light sources [@Eberle2010; @Mehmet2011], each composed of a nonlinear PPKTP crystal and a coupling mirror, were pumped with a bright pump field at $775$nm (yellow) to produce two squeezed vacuum states at the telecommunication wavelength of $1550$nm (red). The two squeezed vacua, both exhibiting a high squeezing of more than $10$dB, were superimposed at a balanced beam splitter with a relative phase of $\pi/2$, thus generating Einstein-Podolsky-Rosen entanglement [@Furusawa1998]. One of the outputs of the beam splitter was kept by Alice, while the other was sent to her communication partner (Bob). The technical details of the source, including the locking scheme, were characterized in Ref. . Figures \[fig:quadrature entanglement\](b)-(e) show the distribution of measurement outcomes obtained by the two parties measuring either the amplitude ($X$) or phase ($P$) quadrature of their respective light field with balanced homodyne detection. Each measurement outcome is truly random since it stems from parametrically amplified zero-point fluctuations. When both parties simultaneously measure either $X$ or $P$ the strong correlations between their outcomes are clearly visible (Fig. \[fig:quadrature entanglement\] (b) and (e)). If the two parties measure different quadratures instead, the measurement outcomes are uncorrelated (Fig. \[fig:quadrature entanglement\](c) and (d)). The strength of the correlations of Alice’s and Bob’s measurement for the same quadratures, which is related to the initial squeezing strength, is a central parameter in our QKD protocol and enters the key length computation directly in the form of an [*average distance*]{} $d_\text{pe}$, introduced below. The precise steps of the QKD protocol are as follows: [@Furrer2012] #### Preliminaries Alice and Bob use a pre-shared key to authenticate the classical communication channel for post processing [@Stinson1994; @Gemmell1994]. Furthermore, Alice and Bob negotiate all parameters needed during the protocol run and Alice performs a shot-noise calibration measurement by blocking the signal beam input of her homodyne detector. #### Measurement Phase Alice prepares an entangled state using her Einstein-Podolsky-Rosen source and sends one of the outputs to Bob along with a local oscillator beam. Both Alice and Bob choose, randomly and independently from each other, a quadrature $X$ or $P$, which they simultaneously measure by homodyne detection of their light fields. The outcome of this measurement is called a sample. This step is repeated until $2N$ samples have been obtained. #### Sifting Alice and Bob announce their measurement bases and discard all samples measured in different quadratures. #### Discretization The continuous spectrum of the measurement outcomes is discretized by the analog-to-digital converter (ADC) used to record the measurement. During the discretization step Alice and Bob map the fine grained discretization of their remaining samples caused by the ADC to a coarser one consisting of consecutive $2^d$ bins. In the interval $[-\alpha,\alpha ]$ a binning with equal length is used, which is complemented by two bins $(-\infty,-\alpha)$ and $(\alpha,\infty)$. The parameter $\alpha$ is used to include the finite range of the homodyne detectors into the security proof. #### Channel Parameter Estimation The secret key length is calculated using the average distance between Alice’s and Bob’s samples. To estimate it, the two parties randomly choose a common subset of length $k$ from the sifted and discretized data, $X_A^\text{pe}$ and $X_B^\text{pe}$, respectively, which they communicate over the public classical channel. Using these, they calculate $$\label{eqn:pe test} d_\text{pe}(X_A^\text{pe}, X_B^\text{pe}) = \frac{1}{k}\sum_{\mu=1}^k {\lvert(X_A^\text{pe})_\mu - (X_B^\text{pe})_\mu\rvert}\ ,$$ and abort if it exceeds a threshold agreed on in the preliminaries step. #### Error Reconciliation Bob corrects the errors in his data to match Alice’s using the hybrid error reconciliation algorithm described below. Afterwards, Alice and Bob confirm that the reconciliation was successful. #### Calculation of Secret Key Length Using the results from the channel parameter estimation and considering the number of published bits during error reconciliation, Alice and Bob calculate the secret key length $\ell$ according to Ref. . If the secret key length is negative, they abort the protocol. #### Privacy Amplification Alice and Bob apply a hash function which is randomly chosen from a two-universal family [@Carter1979], to their corrected strings to produce the secret key of length $\ell$. The key generated by the above protocol is proven to be $\epsilon$-secure against coherent attacks in Ref. , where $\epsilon$ is the so-called composable security parameter. The security proof makes no assumptions on the attacks and only some on our implementation. First of all it assumes that Alice’s measured quadrature angles are precisely adjusted to $X$ and $P$, an assumption that can be overcome as the secure key length proves to be robust against small deviations [@Furrer2012]. Furthermore, the proof requires that Alice’s station is inaccessible to the eavesdropper. This allows Alice to trust her source and to determine the probability for measuring a quadrature amplitude value exceeding the parameter $\alpha$. There are no assumptions on Bob’s measurement device (one-sided device independent) such that even attacks on his local oscillator are fully covered and a shot-noise calibration of his homodyne detector is not necessary. ![**Implementation of Alice’s and Bob’s QKD receivers**. Both parties used balanced homodyne detection (BHD) to measure their part of the quadrature entangled state. The measured quadrature angle was controlled by a computer via a fast fiber-coupled electro-optical modulator (EOM). To make sure that Alice and Bob switched between the same orthogonal quadratures, a phase shifter (PS) was employed to compensate slow phase drifts (see Methods). Optical losses of the transmission channel to Bob were modelled by a variable attenuator consisting of a half-wave plate ($\lambda/2$) and a polarizing beam splitter (PBS). The measurement rate was $100$kHz. PD: Photo Diode.[]{data-label="fig:detectors"}](Fig2-Detectors.pdf){width="8.5cm"} Important for a high key rate is an error reconciliation protocol which has an efficiency close to the Shannon limit. Since in our CV QKD protocol the discretized sample values are non-binary and follow a Gaussian distribution, error reconciliation codes with high efficiency and low error rate are more difficult to achieve than for discrete-variable protocols with uniformly distributed binary outcomes [@Lodewyck2007]. To solve the problem, we designed a two-phase error reconciliation protocol which can exploit the non-uniform distribution efficiently. First the $d_1$ least significant bits of each sample are sent to Bob. Since these bits are only very weakly correlated this step works with an efficiency very close to the Shannon limit. In a second step Alice and Bob use a non-binary low density parity check (LDPC) code over the Galois field $\text{GF}(2^{d_2})$ to correct the $d_2=d-d_1$ most significant bits. $d_1$, $d_2$, as well as the LDPC code were optimized for the different channel conditions and the actually employed code was determined using the $k$ revealed samples from the channel parameter estimation. More details are given in the Methods. ![**Secure key rates achieved by our CV QKD system.** Common parameters: $\alpha=61.6$, $d = 12$, $\epsilon = 2\times 10^{-10}$. (a) Effect of the finite number of distributed quantum states on the secret key rate. The graph shows experimental results (blue points) obtained without the variable attenuator in Bob’s arm. The theoretical model (solid line) is included for comparison and was calculated by reconstructing the covariance matrix for $10^8$ samples. (b) Experimentally obtained secure key rate versus optical attenuation in the transmission line to Bob’s detector for $2 \times 10^8$ measured samples (blue points). The error bars are due to the accuracy of the measurement of the optical attenuation. The theoretical model (solid line) was calculated by reconstructing the covariance matrix of the state corresponding to no attenuation ($0$dB) and using a reconciliation efficiency of $\beta = 94.3\,\%$.[]{data-label="fig:results"}](Fig3a-keyrate_vs_N.pdf "fig:"){width="8.2cm"} ![**Secure key rates achieved by our CV QKD system.** Common parameters: $\alpha=61.6$, $d = 12$, $\epsilon = 2\times 10^{-10}$. (a) Effect of the finite number of distributed quantum states on the secret key rate. The graph shows experimental results (blue points) obtained without the variable attenuator in Bob’s arm. The theoretical model (solid line) is included for comparison and was calculated by reconstructing the covariance matrix for $10^8$ samples. (b) Experimentally obtained secure key rate versus optical attenuation in the transmission line to Bob’s detector for $2 \times 10^8$ measured samples (blue points). The error bars are due to the accuracy of the measurement of the optical attenuation. The theoretical model (solid line) was calculated by reconstructing the covariance matrix of the state corresponding to no attenuation ($0$dB) and using a reconciliation efficiency of $\beta = 94.3\,\%$.[]{data-label="fig:results"}](Fig3b-keyrate_vs_loss.pdf "fig:"){width="8.2cm"} Figure \[fig:results\] shows the experimental results. First we removed the variable attenuator in the transmission line to Bob and executed the protocol for different sample sizes to show the effect of the finite sample size on the secure key rate (Fig. \[fig:results\](a), blue points). For each sample size the number of samples $k$ used for channel parameter estimation was optimized before each run of the QKD protocol to yield maximum key length. The hybrid error reconciliation had a total efficiency of $\beta = 94.6\,\%$ without a single frame error. While we achieved a positive secret key rate with already $5 \times 10^6$ samples, the secret key rate of $0.485$bit/sample achieved for $2 \times 10^8$ samples is close to saturation. The theoretical model, which is the solid line in the figure, is shown for comparison. With the variable attenuator in place, we varied the optical loss of the channel to Bob between $0\,\%$ and $16\,\%$ (see Figure \[fig:results\](b)), which is equivalent to a fiber length of up to $2.7$km when standard telecommunication fibers with an attenuation of $0.2$dB/km are used and a coupling efficiency of $95$$\%$ is taken into account. By measuring a total of $2\times 10^8$ samples we were still able to achieve a secret key rate of about $0.1$bit/sample at an equivalent fiber length of $2.7$km ($\approx 0.76$dB channel loss). This value, as well as the secret key sizes at the other attenuation values, were achieved by having a very high overall error reconciliation efficiency between $\beta = 94.3\,\%$ and $95.5\,\%$, again without a single frame error. The theoretical model shown in the figure reveals that even an optical transmission loss of almost $1.2$dB between Alice and Bob should be possible. This corresponds to an equivalent distance of about $4.8$km, which is already enough to implement CV QKD links with composable security against coherent attacks between parties in, for instance, a city’s central business district. In conclusion, we have for the first time successfully implemented QKD with composable security against coherent attacks in a one-sided measurement-device independent manner. Along with the exploitation of strong Einstein-Podolsky-Rosen entanglement and a new highly efficient error reconciliation algorithm, the innovation of fast controlled random switching between the two measured quadrature angles made the implementation possible. Although attacks on the source of the quantum states are not covered by the security proof [@Furrer2012], these side-channel attacks can be avoided to arbitrarily low probabilities by adding optical isolators. While in our setup Alice and Bob were located on the same optical table, they could in principle be separated and connected by a standard telecommunication fiber (see Methods). Estimations show that our implementation is limited to about $4.8$km. Longer distances will be possible by using optical fibres with less loss, or by using reverse reconciliation [@Furrer2014]. #### **Acknowledgements** This research was supported by the Deutsche Forschungsgemeinschaft (projects SCH 77/5-1 and WE 1240/20-1), the Centre for Quantum Engineering and Space-Time Research (QUEST) and the Vienna Science and Technology Fund (WWTF) (project ICT10-067). TG and VH thank the IMPRS on Gravitational Wave Astronomy for support. FF acknowledges support from Japan Society for the Promotion of Science (JSPS) by KAKENHI grant No. 24-02793. CP would like to thank Gottfried Lechner for very helpful conversations. RFW acknowledges support from the European network SIQS. Methods ======= Experimental Details -------------------- The measurement rate of our implementation was $100$kHz. For each measurement, both Alice and Bob had to choose randomly between the $X$ and $P$ quadrature. The necessary relative phase shifts of $\pi/2$ of the local oscillator with respect to the signal beam were applied to the local oscillator beam by a high-bandwidth fiber-coupled electro-optical phase modulator driven by a digital pattern generator PCI-Express card. Since not only the orthogonality of the measurements is important but also that Alice and Bob measure the same set of quadratures, we compensated slow phase drifts by a phase shifter made of a piezo attached mirror. The error signal for this locking loop was derived by employing an $82$MHz single sideband from the entanglement generation [@Eberle2013] which was detected by the homodyne detector. By lowpass filtering the demodulated homodyne signal at $10$kHz with a sufficiently high order, the high frequency phase changes from the fiber-coupled phase modulator were averaged over. To make the average independent of the chosen sequence of quadratures we used the following scheme. For a choice of the $X$ quadrature, the phase modulator was first set to a phase of $\pi/2$ during the first half of the $10$$\mu$s interval, and then to $0$. For the $P$ quadrature, the phase was first set to $0$ and then to $\pi/2$. Thus, this scheme made sure that the phase did not stay in one quadrature for longer than $10$$\mu$s even in the case where one party chose by chance to measure only one quadrature for a while. The measurement was performed synchronously by Alice and Bob in the second half of the interval after $3$$\mu$s settling time. The data acquisition was triggered by the pattern generator and performed by a two channel PCI-Express card at a rate of $256$MHz. The $200$ acquired samples per channel were digitally mixed down at $8$MHz, lowpass filtered by a $200$-tap finite impulse response filter with a cut-off frequency of $200$kHz and down-sampled to one sample. After the total number of samples were recorded the classical post processing of the QKD protocol was performed. Alice and Bob both employed a local oscillator with a power of $10$mW, yielding a dark noise clearance of about $18$dB. The efficiency of both homodyne detectors was $98$$\%$ (quantum efficiency of the photo diodes $99$$\%$, homodyne visibility $99.5$$\%$). The pump powers for the two squeezed-light sources were $140$mW and $170$mW, respectively. The optical attenuation of the variable attenuator used in Fig. \[fig:results\](b) was measured by determining the strength of the $35.5$MHz phase modulation used to lock one of the squeezed-light sources [@Eberle2013] with Bob’s homodyne detector. The error bars in the figure are due to the accuracy of this measurement. The security of the protocol relies substantially on the use of true random numbers which are needed by Alice and Bob to choose between the $X$ and $P$ quadrature, and to determine a random hash function during privacy amplification. We implemented a quantum random number generator following a scheme of Ref.  which is based on vacuum state measurements performed by a balanced homodyne detector. For this purpose we implemented another balanced homodyne detector with blocked signal port using an independent $6$mW $1550$nm beam from a fiber-laser as local oscillator. The output of the homodyne detector circuit was anti-alias filtered by a $50$MHz fourth-order Butterworth filter and sampled with a sampling frequency of $256$MHz by a data acquisition card. The data was subsequently mixed down digitally at $8$MHz, lowpass filtered with a $200$-tap finite-impulse-response filter with a cut-off frequency of $5$MHz and down-sampled to $2$MHz. The generation of the random numbers from the data stream followed the procedure in Ref. . While in our implementation both parties were located on the same optical table and the quantum states including the local oscillator for Bob’s homodyne detection were transmitted through free space, a separation is in principle possible by using standard telecommunication fibers. To send both the entangled state and the local oscillator to Bob, they could be, for instance, time multiplexed. Using a dedicated fiber for both beams would also be possible. To achieve synchronization between the two parties, a modulated $1310$nm beam could be employed which could be send along with the local oscillator by wavelength division multiplexing. Classical Post Processing ------------------------- The main post-processing is performed with the AIT QKD software. For the current protocol the following algorithms are combined: (i) the binning of the synchronized outcomes, (ii) the estimation algorithm for CV QKD, (iii) the reconciliation algorithm for CV QKD, (iv) the confirmation algorithm, and (v) the privacy amplification algorithm. All classical messages during the protocol are authenticated with a message authentication code using a pre shared secret key to select a random function from a set of (almost strongly two-universal) polynomial hash functions. \(i) First, Bob’s samples in the $P$ quadrature are multiplied by $-1$ to account for the anti-correlation. Alice and Bob then discretize their sifted samples into $2^d$ bins of equal size $ \delta $ in the interval $[-\alpha, \alpha]$. The remaining outcomes associated to the intervals $(-\infty,-\alpha)$ and $(\alpha,\infty)$ are joined to $(-\alpha,-\alpha+\delta)$ and $(\alpha-\delta,\infty)$, respectively. The $2^d$ bins are identified with the key generation alphabet $ {\chi_\text{kg}}= \{0,1\}^d $ and each bin (symbol) has a unique binary representation of $d$ bits. Alice and Bob obtain the binned sifted samples $X^{\text{sift}}_A \in {\chi_\text{kg}}^{N}$ and $X^{\text{sift}}_B\in {\chi_\text{kg}}^{N}$, respectively. Throughout the experiment we have used a key generation alphabet of size $\vert \chi_\text{kg} \vert =2^{12}$. \(ii) In the estimation module for CV QKD the average distance between Alice’s and Bob’s binned symbols is estimated. Alice chooses a random index set ${\mathcal{E}}\subset \{1,2,\dots,N\}$ of size $\vert {\mathcal{E}}\vert = k$ for estimation and communicates ${\mathcal{E}}$ together with the corresponding binned symbols $X_A^\text{pe}:=X^{\text{sift}}_A({\mathcal{E}})$ to Bob. Bob determines his corresponding binned raw key symbols $X_B^\text{pe}:=X^{\text{sift}}_B({\mathcal{E}})$, calculates the mean difference $d_\text{pe}$ between $X_A^\text{pe}$ and $X_B^\text{pe}$ (see Eq. (\[eqn:pe test\])), and checks that $d_\text{pe} \leq d_\text{pe}^0$. Here, $d_\text{pe}^0$ has been determined before the run of the protocol by a theoretical estimation given the characterization of the source, the fiber loss and excess noise. If the test passes they continue with the protocol and both parties remove the $k$ estimation samples from their sifted samples to form their raw keys $X_A:=X^{\text{sift}}_A\setminus X_A^\text{pe} \in {\chi_\text{kg}}^{N-k}$ and $X_B:=X^{\text{sift}}_B\setminus X_B^\text{pe} \in {\chi_\text{kg}}^{N-k}$. \(iii) The reconciliation module for CV QKD implements the hybrid reconciliation protocol. As the security analysis assumes direct reconciliation, Bob has to correct his raw key $X_B$ to match with Alice’s $X_A$ to generate a common raw key $X$. The hybrid reconciliation used to correct Bob’s noisy raw key operates directly on the key generation alphabet $ {\chi_\text{kg}}$. In preparation for the hybrid reconciliation, two additional alphabets ${\hat{\chi}}$ and ${\check{\chi}}$ are introduced such, that $ \chi_\text{kg} = {\hat{\chi}} \times {\check{\chi}} $. Hence, each symbol $x \in {\chi_\text{kg}}$ has a unique decomposition $x=({\hat{x}},{\check{x}})$ with $ {\hat{x}} \in {\hat{\chi}} $ and ${\check{x}} \in {\check{\chi}}$. We take for ${\hat{x}}$ the $d_2$ most significant bits of the binary representation of $x$, and for ${\check{x}}$ the remaining $d_1=d-d_2$ least significant bits of the binary representation of $x$. We thus decompose the raw keys as $X=({\hat{X}},{\check{X}})$, where ${\hat{X}}$ and ${\check{X}}$ denote the sequence of the $d_2$ most and the $d_1$ least significant bits of each key symbol, respectively. The reconciliation module performs the following steps: (iii-a) Based on the variance of her binned raw key and the samples $X_A^\text{pe}$ and $X_B^\text{pe}$, Alice determines $d_1$, $d_2$, and the code rate $R$ such that the expected leakage is minimized w.r.t. the entropy in Bob’s symbols, and transmits these parameters to Bob. (iii-b) Then Alice communicates ${\check{X}}_A$ to Bob who reconciles ${\check{X}}_B$ simply by setting $ {\check{X}}_B:={\check{X}}_A $. Hence, the errors which are left in Bob’s key $X_B$ are reduced to the errors in ${\hat{X}}_B$. Non-binary LDPC reconciliation is used to correct ${\hat{X}}_B$ as described in the next step. (iii-c) Both Alice and Bob split their ${\hat{X}}_A$ and ${\hat{X}}_B$ into blocks ${\hat{X}}_A^{(\ell)}$ and ${\hat{X}}_B^{(\ell)}$, $\ell=1,\dots, \frac{N-k}{n}$, each with $n=10^5$ elements of ${\hat{\chi}}$. For this step we identify $ {\hat{\chi}}$ with GF($2^{d_2}$), the Galois field with $2^{d_2}$ elements. For each block ${\hat{X}}_A^{(\ell)}$, Alice uses the parity check matrix $H$ of an LDPC code over GF($2^{d_2}$) and rate $R$ to calculate the syndrome $s^{(\ell)}:=H\cdot {\hat{X}}_A^{(\ell)}$. Alice sends the syndrome $s^{(\ell)}$ to Bob. For all elements $j \in GF(2^{d_2})$ and for all indices $i\in\{1,\dots,n\}$ in the block Bob calculates the conditional probability that $({\hat{X}}_A^{(\ell)})_i=j$, given that Bob has obtained $({\hat{X}}_B^{(\ell)})_i$ and given Alice’s value $({\check{X}}_A^{(\ell)})_i$. Bob uses these probabilities to initialize a non-binary belief propagation decoder. The non-binary belief propagation decoder operates in the probability domain using the multi-dimensional Hadamard transform to speed up the check node operations [@Barnault2003]. Using the syndrome $s^{(\ell)}$ and the conditional probabilities mentioned above, this decoder calculates Bob’s estimate $\tilde{X}_A^{(\ell)}$ of Alice’s block $X_A^{(\ell)}$. We have constructed parity check matrices of non-binary LDPC codes over Galois fields of order 32, 64, 128, and 256 with code rates $R\in\{0.50, 0.51, \dots, 0.95\}$. Each LDPC code has a variable-node degree of two, is check-concentrated, and has a block length of $10^5$ symbols. We used the progressive edge-growth algorithm [@Hu2005] to construct binary codes in a first step. Then each edge has been assigned a random non-zero element of the corresponding Galois field.[@Hu2005] Alice and Bob have access to all non-binary parity check matrices. In our proof-of-principle experiment the error reconciliation step took about $2$h on a single CPU core for the largest data set of $2\times 10^8$ samples. Taking into account the about $30$min to measure the data, real-time error reconciliation could in principle be achieved by splitting the task to e.g. $5$ CPU cores. Alternatively, to speed up the computation LDPC decoder algorithms with reduced complexity could be employed [@Voicila2010]. \(iv) After each block has been corrected, a confirmation step establishes the correctness of the protocol using a family $H$ of (almost) two-universal hash functions with $\text{Prob}_{h\in_R H}(h(x_1)=h(x_2))\le \epsilon_c$ for all $x_1\ne x_2$. For each block Alice chooses a hash function $h$ randomly from $H$ and communicates her choice to Bob. Alice and Bob apply this hash function to their blocks $X_A^{(\ell)}$ and $\tilde{X}_A^{(\ell)}$ and exchange the results. If their results agree the probability that Alice’s and Bob’s blocks are different is bounded from above by $\epsilon_c$. If their results disagree their blocks are definitely different, and they discard them. \(v) Finally, Alice and Bob feed the sequence of all confirmed blocks into the privacy amplification module. Given the accumulated leakage $\ell_{\text{LK}}$ in bits from the previous protocol steps the secure key length is calculated according to Ref  as $$\ell = (N-k) ( \log \frac{1}{c(\delta)} - \log \gamma(d_\text{pe}^0 + \mu) ) - \ell_{\text{LK} } - \log\frac{1}{\epsilon}\ ,$$ where $c(\delta)\approx \delta^2/(2\pi)$ and $\gamma$ is a bound on the correlation between Alice and Bob depending on the previously agreed average distance threshold $d_\text{pe}^0$ and statistical fluctuations $\mu$. Alice chooses a hash function randomly from a two-universal hash family and communicates her choice to Bob. 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--- abstract: 'Quantum networks provide a prominent platform for realizing quantum information processing and quantum communication, with entanglement being a key resource in such applications. Here, we describe the dissipative transport protocol for entangled states, where entanglement stored in the first node of quantum network can be transported with high fidelity to the second node via a 1D chiral waveguide. In particular, we exploit the directional asymmetry in chirally-coupled single-mode ring resonators to transport entangled states. For the fully chiral waveguide, Bell states, multipartite $W$-states and and Dicke states can be transported with fidelity as high as $0.954$, despite the fact that the communication channel is noisy. Our proposal can be utilized for long-distance distribution of multipartite entangled states between the quantum nodes of the open quantum network.' author: - 'Wai-Keong Mok' - Davit Aghamalyan - 'Jia-Bin You' - Tobias Haug - Wenzu Zhang - Ching Eng Png - 'Leong-Chuan Kwek' bibliography: - 'Bibliography.bib' title: 'Long-distance dissipation-assisted transport of entangled states via a chiral waveguide' --- *Introduction.*Quantum networks [@kimble2008quantum; @reiserer2015cavity] are essential for realizing distributed quantum computing and large scale quantum communication, with entanglement being a key resource in such applications. In this context, the main task and at the same time an outstanding challenge is the high fidelity transfer of quantum states over long distances despite having noise and dissipation present in the quantum channel [@nielsen2002quantum]. It is well known that in open quantum systems, dissipation arises as a result of the system coupling with the reservoir [@breuer2002theory], which consequently causes decoherence in the system. In this Letter, we show that the dissipative channel can be used for transporting entangled states. In general, long-distance processes in open quantum systems are challenging because of non-Markovian effects due to the non-negligible time delay between the nodes of a quantum network. Non-Markovianity has been shown to be detrimental to both quantum state transfer and entanglement generation between the nodes [@Gonzalez_Ballestero_2013; @fang2018non]. There have been several theoretical proposals [@cirac1997quantum; @nikolopoulos2014quantum; @PhysRevLett.118.133601; @dlaska2017robust; @stannigel2010optomechanical; @stannigel2011optomechanical; @yao2013topologically; @ramos2016non; @zheng2013persistent; @van2019long] as well as actual experimental realizations [@hofmann2012heralded; @ritter2012elementary; @rosenfeld2017event] for the quantum state transfer(QST) of a single qubit in quantum optical networks, where fast information transfer is achieved with help of photons (“flying qubits”). In all the above proposals, there are few demanding requirements that are hard to be met experimentally: external control pulses that have non-trivial temporal shapes (photonic wave packets are required to be time-symmetric), time-dependent cavity-atom and fiber-atom interaction strengths. Moreover, to the best of our knowledge, there are no existing protocols for long-distance entanglement transfer in the optical frequency domain. On the other hand, spin chains can alleviate the issue of sensitive control of system parameters and realize quantum systems with minimal control (coupling constants are fixed in time), and entanglement transfer has been demonstrated in several theoretical manuscripts [@bose2007quantum; @rafiee2011entanglement; @bayat2010entanglement; @sousa2014pretty; @ji2015quantum; @man2014controllable; @vieira2018almost; @banchi2011long] in Heisenberg-type spin chains. However, these systems can only realize short-distance state transfer, as experimentally one is limited by the number of spins. It is also widely believed that increasing the length of a spin chain will worsen transfer fidelities due to dispersion effects [@banchi2011long]. Quite remarkably, using chiral waveguides, the merits of quantum optical networks (fast information transfer with “flying qubits”) and spin chain networks (minimal control over system parameters) can be combined. In quantum optics, chirality arises, for instance, in atom-waveguide coupled systems when the symmetry of photon emission in the left and right directions is broken [@lodahl2017chiral]. This effect appears as a result of spin-orbit coupling, and has been experimentally demonstrated in photonic waveguides [@sollner2015deterministic]. Chiral systems have been shown to be fruitful for realizing quantum networks [@PhysRevA.91.042116; @ramos2016non; @mahmoodian2016quantum]. In Ref. [@gonzalez2015chiral], it was argued that the maximum achievable concurrence between two atoms is 1.5 times higher as compared to the non-chiral counterparts. Interestingly, systems with perfect chirality realize the paradigm of cascaded systems [@PhysRevLett.70.2273; @gardiner1993driving; @carmichael2007statistical], where two systems are coupled unidirectionally without information backflow. Cascaded systems, even when separated by long distances, can then be described under the Born-Markov approximation with retardation effects accounted for by a simple redefinition of the time and phase of the second node. Here, we exploit the Markovianity provided by cascaded systems as a suitable platform to achieve high-fidelity entanglement transfer, despite the noise being present in the quantum channel. Motivated by Refs. [@gonzalez2015chiral; @PhysRevLett.118.133601], we couple ring cavities with chiral waveguides to obtain unidirectional effective coupling between the cavities. In particular, we benefit from Markovian dynamics thanks to the cascaded systems naturally arising from the high chirality. Each node of our quantum network consists of a $N$-particle atomic ensemble which is coupled to the ring cavity. It is important to highlight that ring cavities introduce greater control over the system compared to the bare atom-fiber coupled case, where the transport fidelity is significantly lowered due to all-to-all long-range interactions between the atoms. By suitably optimizing over the system parameters, we demonstrate the transport of maximally entangled Bell states, Dicke states and $W$-states for up to 20 qubits. For clarity, we remark that the entanglement transport here is not necessarily QST, although the transport of $W$-states and Bell states can be applied to the QST of unknown qubit and qutrit states respectively. Compared to other schemes, our minimal control proposal has various advantages. Firstly, the scheme works in the weak coupling regime with no external driving field required. Also, the optimal transport of entanglement occurs dynamically. This potentially can lead to faster transport compared to steady state schemes [@PhysRevA.91.042116]. Moreover, the entanglement transport is not dependent on the distance between the atoms. *Chiral waveguide-QED system.*The system in consideration comprises two nodes coupled to a 1D waveguide, shown in Fig. . Each node comprises $N$ qubits coupled to a single cavity mode where the transition frequencies of the qubits and resonant frequency of the cavity are $\omega_l^{(j)}$ and $\omega_{cj}$ respectively. The atom-cavity coupling strength is given by $g_l^{(j)}$. The bosonic operators for the cavity mode are $a_j^\dag$ and $a_j$, satisfying the canonical commutation relation $[a_j, a_{j^\prime}^\dag] = \delta_{j j^\prime}$. The waveguide is treated as a common reservoir, with bosonic operators $b_\lambda^\dag (\omega)$ and $b_\lambda (\omega)$ satisfying the commutation relation $[b_\lambda (\omega), b_{\lambda^\prime}^\dag (\omega^\prime)] = \delta_{\lambda \lambda^\prime} \delta(\omega - \omega^\prime)$. The interaction strength between the cavities and the waveguide (at position $x_j$) is characterized by the decay rate $\gamma_{\lambda}$. Here we assume that cavity losses into non-waveguide modes are negligible. The spontaneous decay of the qubits is described by an interaction with independent baths at a decay rate $\Gamma_{jl}$, where the first index denotes the cavity and the second index denotes the qubit. The bath operators $c_l^{(j) \dag} (\omega)$ and $c_l^{(j)} (\omega)$ satisfy the commutation relation $[c_l^{(j)} (\omega), c_{l^\prime}^{(j^\prime)\dag} (\omega^\prime)] = \delta_{j j^\prime} \delta_{l l^\prime} \delta(\omega - \omega^\prime)$. By tracing out the waveguide mode, and applying the Born-Markov approximation, the Lindblad master equation for the system can be found as [@PhysRevA.91.042116] (setting $\hbar = 1$, details in Appendix A) $$\begin{split} \dot{\rho} &= -i [H_{\text{eff}}, \rho] + \gamma_L \mathcal{D} [e^{ikx_1} a_1 + e^{ikx_2} a_2] \rho \\ &+ \gamma_R \mathcal{D} [e^{-ikx_1} a_1 + e^{-ikx_2} a_2] \rho + \sum_{j,l} \Gamma_{jl} \mathcal{D}[\sigma_l^{(j)}] \rho \end{split} \label{eq:master_eqn}$$ with the effective Hamiltonian $$\begin{split} H &= \sum_{j,l} [ \omega_l^{(j)} \sigma_l^{(j)\dag} \sigma_l^{(j)}+ \omega_{cj} {a_j}^\dag a_j + g_l^{(j)} ({a_j}^\dag \sigma_l^{(j)} + \text{H.c.}) ] \\ &- i\frac{\gamma_L}{2} (e^{ikD} a_1^\dag a_2 - \text{H.c.}) - i \frac{\gamma_R}{2} (e^{ikD} a_2^\dag a_1 - \text{H.c.}) \end{split}$$ where $D = |x_2 - x_1|$ is the distance between the nodes. In the following, we will study the transport of entangled qubit states between the nodes mediated by the waveguide. The case of $N=2$ is first presented to illustrate Bell state transport. *Transport of Bell states with chiral couplings.* Here, we exploit the directional asymmetry by using a chiral light-matter interface, with $\gamma_L = 0, \gamma_R \neq 0$ [@PhysRevLett.118.133601]. Using chiral couplings, the setup is essentially a cascaded quantum system [@PhysRevLett.70.2273] where the first node is coupled to the second node unidirectionally without backflow of information. In this case, the setup we consider can be used to study long-distance entanglement transport despite the Born-Markov approximation used, since retardation effects in a cascaded quantum system is accounted for by a simple redefiniton of the time and phase of the second node [@carmichael2007statistical]. For simplicity, we assume that the qubit decay rates are much smaller than the cavity decay rates and can be neglected, and the nodes are identical, i.e. $\omega_{l}^{(j)} = \omega_0$, $\omega_{cj} = \omega_c$, $g_l^{(j)} = g_j$, for all $j,l \in \{1,2\}$. The qubits in the first node are denoted by $1a,1b$ while the qubits in the second node are denoted by $2a,2b$. We first prepare the qubits $1a,1b$ in the Bell state $\ket{\Psi^+} = \frac{1}{\sqrt{2}} (\ket{eg} + \ket{ge})$, and consider resonant conditions $\omega_c = \omega_0$ with cavity coupling strength $g_1 = g_2 = 0.3 \gamma_R$. As shown in Fig. , the concurrence of $1a,1b$ decreases to nearly zero at some time, while concurrence of $2a, 2b$ rises from zero to a maximum of around 0.91. The state fidelity of $2a,2b$ compared to the initial Bell state is around 0.954. This shows that a good entanglement transport can be accomplished. For the case of perfect chirality ($\chi = 1$), due to the cascaded nature of the system, this result is independent of the distance $D$ between the qubits. For the non-chiral case in Fig. where $\gamma_L = \gamma_R$, the maximum concurrence is only around 0.58, even with the optimal distance of $kD = \pi$, where $k$ is the wavenumber of the photon $k = 2\pi / \lambda$ with $\lambda$ the corresponding wavelength. Comparing the fidelity of the qubit state of $2a,2b$ (denoted $\rho_{2}(t)$) with the initial entangled state of $1a,1b$ (denoted $\rho_{1}(0)$) such that $\mathcal{F} = \left(\operatorname{Tr}\sqrt{\sqrt{\rho_{1}(0)} \rho_{2}(t) \sqrt{\rho_{1}(0)}} \right)^2$, Fig. shows that the maximum fidelity transported, $\mathcal{F}_{\text{max}}$, is around 0.951 (green dashed line), a significant improvement over the non-chiral case in Fig. which gives $\mathcal{F}_{\text{max}} \approx 0.78$. Thus, chiral coupling drastically improves the entanglement transport between the nodes. To find the optimal coupling $g_1 = g_2 = g$, we plot the maximum transported concurrence $C_{\text{max}}$ of $2a,2b$ against $g_1$ and $g_2$. As shown in Fig. , the transported concurrence is maximal $(C_{\text{max}} \approx 0.905)$ around $g_1 = g_2 \approx 0.3\gamma_R$. Intuitively, for small couplings, the entanglement does not transport effectively to the cavity, thus the transport is weak. For strong couplings however, the Rabi oscillations between the cavity and the qubits become more significant, which is detrimental to the transport of entanglement via the waveguide. It can also be seen from Fig. that $g_1 = g_2$ is an optimal condition for good entanglement transport. To illustrate the effect of chirality on the transport, we compare the maximum transported concurrence for different chirality. We comment that although non-Markovian effects should in general be taken into account if one considers long distances with imperfect chirality, however, this is not required as long as the entanglement is transported much faster than the timescale for information backflow to occur. Our results indicate that such conditions can be easily achieved for sufficiently long waveguides. Here, chirality is defined as $\chi \equiv (\gamma_R - \gamma_L)/(\gamma_R + \gamma_L)$. Fig. shows the comparison for different chirality. For the fully chiral waveguide ($\chi = 1$), $C_{\text{max}}$ is independent of the inter-nodal distance $D$, as previously mentioned. This is simply due to the cascaded nature of the setup. However, when $\gamma_L \neq 0$, $C_\text{max}$ depends on the distance between the nodes. The peak at $D = 0.5 \lambda$ is a result of the localisation of the photon wavefunction between the nodes [@Gonzalez_Ballestero_2013]. The sensitivity of $C_{\text{max}}$ to fluctuations around this ‘sweet spot’ decreases as $\chi$ gets closer to 1. In general, the entanglement transport worsens with decreasing chirality. Intuitively, this can be due to two factors: (i) leakage of excitation from the first node through the left port via $\gamma_L$, which decreases the probability of the second node being excited; (ii) information backflow from the second node back to the first node, which can be detrimental to the transport process. Thus, using chirality, both problems can be addressed simultaneously, leading to good entanglement transport. Next, we look at the maximum transported fidelity with different initial states of $1a,1b$. To this end, we prepare the qubits in system 1 in the state $$\ket{\psi_1} = \cos \theta \ket{eg} + e^{i\phi} \sin \theta \ket{ge}, \quad \theta \in [0, \pi], \phi \in [0,2 \pi]$$ while the qubits in system 2 are initially in the ground state. The cavities are all in the vacuum state initially. From Fig. , the maximum transported fidelity $(\mathcal{F}_{\text{max}} = 0.951)$ occurs near $\phi =0, \theta = \pi/4$ which corresponds to the Bell state $\ket{\Psi^+}$. The case of $\mathcal{F}_{max} = 0$ occurs near $\phi = \pi, \theta = \pi/4$ which corresponds to the Bell state $\ket{\Psi^-}$. This is because $\ket{\Psi^-}$ is a dark state of the TCM, and thus does not decay with time. However, $\ket{\Psi^-}$ can be easily transported with the same fidelity of $\mathcal{F}_{\text{max}} = 0.951$ by imposing an $\pi$ phase difference between the two cavity-qubit couplings in the same node, i.e. $g_{1}^{(j)} = - g_{2}^{(j)}$. We also consider the initial state $$\ket{\psi_2} = \cos \theta \ket{gg} + e^{i\phi} \sin \theta \ket{ee}, \quad \theta \in [0, 2\pi], \phi \in [-\pi,\pi]$$ with the Bell states $\ket{\Phi^+} = \frac{1}{\sqrt{2}} (\ket{gg} + \ket{ee})$ and $\ket{\Phi^-} = \frac{1}{\sqrt{2}} (\ket{gg} - \ket{ee})$. As shown in Fig. , the maximum transported fidelity is independent of $\phi$. The transported fidelity $\mathcal{F}_{\text{max}} \approx 0.954$ at $\theta = \pi/4, \phi=0$ and $\theta = \pi/4, \phi=\pi$ corresponds to the Bell states $\ket{\Psi^{\pm}}$ respectively. The lowest $\mathcal{F}_{\text{max}} \approx 0.88$ occurs at $\theta = \pi / 2$ which is reasonable since that corresponds to the case of transporting a two-excitation state $\ket{ee}$ to qubits initially prepared in ground state $\ket{gg}$. Overall, we have shown that good transport of entanglement is possible for all the Bell states. *Transport of multipartite entanglement with chiral couplings.*Here, we demonstrate a generalisation of the entanglement transport scheme, by using $N$ qubits per node. Intuitively, it is clear that states with permutation invariance and low excitations can be transported well using this scheme. In Fig. we show the transport of Dicke states denoted by $\ket{^{N}D_{k}}$ which is an equal superposition of $k$ excitations over $N$ qubits. It can be seen that the three-qubit $W$-state $\ket{W_3} = \ket{^{3} D_1}$ is transported with a fidelity of $\mathcal{F}_{\text{max}} \approx 0.954$, while the two-excitation states $\ket{^{3}D_2}$ and $\ket{^{4} D_2}$ are transported with a lower fidelity of 0.905 due to increased leakage of excitation from the second node of the quantum network. $W$-states are extremely useful for quantum information and communication applications as they are more robust states for encoding single qubit states. Moreover, $W$-states have the unique property (contrary to say, GHZ states) that even if one particle is lost, the rest of $N-1$ qubits will remain in the entangled state. In Fig. , the optimal $g_j$ for the transport of $\ket{W_N}$ is plotted. We numerically show that the optimal transport condition for any $\ket{W_N}$ is given by $\sqrt{N} g_{\text{opt}} / \gamma_R \approx 0.43$. In fact, any $\ket{W_M}$ can be mapped onto any $\ket{W_N}$ ($M \neq N$ in general) with the same fidelity of 0.954 as long as this condition is satisfied on each node. To study the effects of detuning on $W$-state transport, we add a random fluctuation $\Delta_a \in [-\delta_a, \delta_a] $ to the qubit frequencies in Fig. and $\Delta_c \in [-\delta_c , \delta_c] $ to the cavity frequencies in Fig. . The result shows that while the scheme is more robust against fluctuations in cavity frequencies, good transport can also be achieved for $\delta_a \leq 0.1$. *Role of imperfections.* The analysis in the previous sections neglected qubit losses by assuming that the decay rate of the cavity is much larger than that of the qubit decay rates. Here, we look at the entanglement transport with qubit losses. Specifically, we prepare the initial state of $1a,2a$ in the Bell state $\ket{\Psi^+}$ and set all qubit decay rates to be equal ($\Gamma_{jl} = \Gamma$) for simplicity. The optimal case is presented for each value of $\Gamma$. In general, for larger $\Gamma$, the required $g_{\text{opt}}$ increases, in order to transport the excitation to the ring cavity before it is dissipated from the qubits. Increasing the qubit decay rate, the fidelity decreases as shown in Fig. . Here, we set the inter-nodal distance to be at the ‘sweet spot’ $kD = \pi$. A comparison between the chiral ($\chi = 1$) and non-chiral ($\chi = 0$) cases shows that as long as the qubit decay rate is within $\Gamma < 0.1 \gamma_R$, the chiral system remains advantageous over the ideal non-chiral case in terms of entanglement transport. As mentioned earlier, the entanglement transport at $kD = \pi$ is also relatively insensitive to small imperfections in chirality. Thus, perfect chirality is not required for the transport scheme to work well. *Conclusion.* In this Letter, we have proposed a protocol for transporting entanglement between the two nodes in open quantum network, where we demonstrated that dissipation can be useful to achieve the task, contrary to the common notion that dissipation creates decoherence. By coupling ring cavities with a chiral 1D waveguide, we demonstrate entanglement transport, with the entangled state stored in the atomic ensembles which are coupled to the ring cavities. Consequently, one node of our quantum network consists of an atomic ensemble which is coupled to the ring cavity, with the communication channel realized by the chiral waveguide. We have found optimal system parameters for the transport of maximally entangled Bell states and for up to 20-qubit $W$-states. As an application of our results, the quantum transport of $W$-states and Bell states can be exploited to achieve QST of unknown qubit and qutrit states respectively. We highlight that our proposal requires minimal control over the system parameters contrary to other proposals which require external pulses with demanding temporal shapes and time-dependent cavity couplings [@cirac1997quantum; @nikolopoulos2014quantum; @PhysRevLett.118.133601; @dlaska2017robust; @stannigel2010optomechanical; @stannigel2011optomechanical; @yao2013topologically; @ramos2016non; @zheng2013persistent; @van2019long]. Moreover, since the entanglement transport is achieved dynamically it is faster compared to its steady state counterparts. Finally, our protocol can easily be applied to long-distance transport by utilising the Markovianity in cascaded systems. This can potentially be significant for the efficient distribution of entanglement within a quantum network. D. A. and W.K. M. would like to acknowledge Marc-Antoine Lemonde for helpful discussions and feedback. The authors thank Jingu Pang for the waveguide diagram. The IHPC A\*STAR Team would like to acknowledge the support from the National Research Foundation Singapore (Grants No. NRF2017NRFNSFC002-015, No. NRF2016-NRF-ANR002, No. NRF-CRP 14-2014-04) and A\*STAR SERC (Grant No. A1685b0005). D. A., L.C. K. and J. Y. acknowledges support from National Research Foundation Singapore (Grant No. 2014NRF-CRP002-042).

--- abstract: 'We perform an asymptotic analysis of general particle systems arising in collective behavior in the limit of large self-propulsion and friction forces. These asymptotics impose a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity variables. The limit models are obtained by averaging with respect to the fast dynamics. We can include all typical effects in the applications: short-range repulsion, long-range attraction, and alignment. For instance, we can rigorously show that the Cucker-Smale model is reduced to the Vicsek model without noise in this asymptotic limit. Finally, a formal expansion based on the reduced dynamics allows us to treat the case of diffusion. This technique follows closely the gyroaverage method used when studying the magnetic confinement of charged particles. The main new mathematical difficulty is to deal with measure solutions in this expansion procedure.' author: - 'Mihai Bostan [^1] , J. A. Carrillo [^2]' date: ' ()' title: 'Asymptotic Fixed-Speed Reduced Dynamics for Kinetic Equations in Swarming' --- #### Keywords: Vlasov-like equations, Measure solutions, Swarming, Cucker-Smale model, Vicsek model, Laplace-Beltrami operator. #### AMS classification: 92D50, 82C40, 92C10. Introduction {#Intro} ============ This paper is devoted to continuum models for the dynamics of systems involving living organisms such as flocks of birds, school of fish, swarms of insects, myxobacteria... The individuals of these groups are able to organize in the absence of a leader, even when starting from disordered configurations [@ParEde99]. Several minimal models describing such self-organizing phenomenon have been derived [@VicCziBenCohSho95; @GreCha04; @CouKraFraLev05]. Most of these models include three basic effects: short-range repulsion, long-range attraction, and reorientation or alignment, in various ways, see [@HW] and particular applications to birds [@HCH09] and fish [@BTTYB; @BEBSVPSS]. We first focus on populations of individuals driven by self-propelling forces and pairwise attractive and repulsive interaction [@LevRapCoh00; @DorChuBerCha06]. We consider self-propelled particles with Rayleigh friction [@ChuHuaDorBer07; @ChuDorMarBerCha07; @CarDorPan09; @UAB25], leading to the Vlasov equation in $d=2,3$ dimensions: $$\label{Equ1} \partial _t { f ^\varepsilon}+ v \cdot \nabla _x { f ^\varepsilon}+ a ^{ \varepsilon}(t,x) \cdot \nabla _v { f ^\varepsilon}+ \frac{1}{{ \varepsilon}} { \mathrm{div}_v}\{{ f ^\varepsilon}{ (\alpha - \beta |v|^2) v}\}= 0,\;\;(t,x,v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d$$ where ${ f ^\varepsilon}= { f ^\varepsilon}(t,x,v) \geq 0$ represents the particle density in the phase space $(x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d$ at any time $t \in {{{\bf}R}}_+$, $a ^{ \varepsilon}$ stands for the acceleration $$a^{ \varepsilon}(t,\cdot) = - \nabla _x U \star \rho ^{ \varepsilon}(t, \cdot ),\;\;\rho ^{ \varepsilon}(t, \cdot ) = { \int _{{{{\bf}R}}^d} \!{ f ^\varepsilon}(t, \cdot, v)\;\mathrm{d}v }\, ,$$ and $U$ is the pairwise interaction potential modelling the repelling and attractive effects. Here, the propulsion and friction forces coefficients $\alpha ^{ \varepsilon}= \frac{\alpha}{{ \varepsilon}}>0$, $\beta ^{ \varepsilon}= \frac{\beta}{{ \varepsilon}} >0$ are scaled in such a way that for ${ \varepsilon}\to 0$ particles will tend to move with asymptotic speed $\sqrt{\tfrac{\alpha}\beta}$. These models have been shown to produce complicated dynamics and patterns such as mills, double mills, flocks and clumps, see [@DorChuBerCha06]. Assuming that all individuals move with constant speed also leads to spatial aggregation, patterns, and collective motion [@CziStaVic97; @EbeErd03]. Another source of models arises from introducing alignment at the modelling stage. A popular choice in the last years to include this effect is the Cucker-Smale reorientation procedure [@CS2]. Each individual in the group adjust their relative velocity by averaging with all the others. This velocity averaging is weighted in such a way that closer individuals in space have more influence than further ones. The continuum kinetic version of them leads to Vlasov-like models of the form in which the acceleration is of the form $$a^{ \varepsilon}(t,\cdot) = - H \star f^{ \varepsilon}(t, \cdot )\, ,$$ where $\star$ stands for the $(x,v)$-convolution, abusing a bit on the notation, with the nonnegative interaction kernel $H:{{{\bf}R}}^{2d}\longrightarrow {{{\bf}R}}^d$. In the original Cucker-Smale work, the interaction is modelled by $H(x,v)=h(x)v$, with the weight function $h$ being a decreasing radial nonnegative function. We refer to the extensive literature in this model for further details [@HT08; @HL08; @CFRT10; @review; @MT11]. In this work, we will consider the Vlasov equation where the acceleration includes the three basic effects discussed above, and then takes the form: $$\label{accel} a^{ \varepsilon}(t,\cdot) = - \nabla _x U \star \rho ^{ \varepsilon}(t, \cdot ) - H \star f^{ \varepsilon}(t, \cdot )\, .$$ We will assume that the interaction potential $U\in C^2_b({{{\bf}R}}^d)$, $U$ bounded continuous with bounded continuous derivatives up to second order, and $H(x,v)=h(x)v$ with $h\in C^1_b({{{\bf}R}}^d)$ and nonnegative. Under these assumptions the model - can be rigorously derived as a mean-field limit [@Neu77; @BraHep77; @Dob79; @CCR10; @BCC11] from the particle systems introduced in [@DorChuBerCha06; @CS2]. We will first study in detail the linear problem, assuming that the acceleration $a = a(t,x)$ is a given global-in-time bounded smooth field. We investigate the regime ${ \varepsilon}\searrow 0$, that is the case when the propulsion and friction forces dominate the potential interaction between particles. At least formally we have $$\label{EquAnsatz} { f ^\varepsilon}= f + { \varepsilon}{ f ^{(1)}}+ { \varepsilon}^2 f ^{(2)} + ...$$ where $$\label{Equ2} { \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} = 0$$ $$\label{Equ3} \partial _t f + { \mathrm{div}_x}(fv) + { \mathrm{div}_v}(f a(t,x)) + { \mathrm{div}_v}\{{ f ^{(1)}}{ (\alpha - \beta |v|^2) v}\} = 0\,,$$ up to first order. Therefore, to characterize the zeroth order term in the expansion we need naturally to work with solutions whose support lies on the sphere of radius $r := \sqrt{\alpha/\beta}$ denoted by $r{ {\bf}{S}}$ with ${ {\bf}{S}}= \{v\in {{{\bf}R}}^d : |v| = 1\}$. In turn, we need to work with measure solutions to which makes natural to set as functional space the set of nonnegative bounded Radon measures on ${{{\bf}R}}^d\times{{{\bf}R}}^d$ denoted by ${\cal M}_b ^+ ({{{\bf}R}}^d\times{{{\bf}R}}^d)$. We will be looking at solutions to which are typically continuous curves in the space ${\cal M}_b ^+ ({{{\bf}R}}^d\times{{{\bf}R}}^d)$ with a suitable notion of continuity to be discussed later on. We will denote by ${ f ^\varepsilon}(t,x,v)\, \mathrm{d}(x,v)$ the integration against the measure solution ${ f ^\varepsilon}(t,x,v)$ of at time $t$. For the sake of clarity, this is done independently of being the measure ${ f ^\varepsilon}(t)$ absolutely continuous with respect to Lebesgue or not, i.e., having a $L^1({{{\bf}R}}^d\times{{{\bf}R}}^d)$ density or not. \[Kernel\] Assume that $(1+|v|^2)F \in {\cal M}_b ^+ ({{{\bf}R}}^d)$. Then $F$ is a solution to if and only if ${ \mathrm{supp\;}}F \subset \{0\} \cup r { {\bf}{S}}$. The condition appears as a constraint, satisfied at any time $t \in {{{\bf}R}}_+$. The time evolution of the dominant term $f$ in the Ansatz will come by eliminating the multiplier ${ f ^{(1)}}$ in , provided that $f$ verifies the constraint . In other words we are allowed to use those test functions $\psi (x,v)$ which remove the contribution of the term ${ \mathrm{div}_v}\{ { f ^{(1)}}{ (\alpha - \beta |v|^2) v}\}$ [*i.e.,*]{} $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { (\alpha - \beta |v|^2) v}\cdot \nabla _v \psi \;{ f ^{(1)}}(t,x,v)\, \mathrm{d}(x,v) } = 0.$$ Therefore we need to investigate the invariants of the field ${ (\alpha - \beta |v|^2) v}\cdot \nabla _v$. The admissible test functions are mainly those depending on $x$ and $v/|v|, v \neq 0$. The characteristic flow $(s,v) \to {\cal V}(s;v)$ associated to $\tfrac1{ \varepsilon}{ (\alpha - \beta |v|^2) v}\cdot \nabla _v$ $$\frac{\mathrm{d}{\cal V}}{\mathrm{d}s} = \frac1{ \varepsilon}{ (\alpha - \beta \left |{\cal V}(s;v)\right |^2) {\cal V}(s;v)},\;\;{\cal V}(0;v) = v$$ will play a crucial role in our study. It will be analyzed in detail in Section \[LimMod\]. Notice that the elements of ${ \{0\} \cup r { {\bf}{S}}}$ are the equilibria of ${ (\alpha - \beta |v|^2) v}\cdot \nabla _v $. It is easily seen that the jacobian of this field $$\partial _v \{ { (\alpha - \beta |v|^2) v}\} = (\alpha - \beta |v|^2 ) I - 2 \beta v \otimes v$$ is negative on $r{ {\bf}{S}}$, saying that $r{ {\bf}{S}}$ are stable equilibria. The point $0$ is unstable, $\partial _v \{ { (\alpha - \beta |v|^2) v}\} |_{v = 0}=\alpha I$. When ${ \varepsilon}\searrow 0$ the solutions $({ f ^\varepsilon})_{ \varepsilon}$ concentrate on ${ {{{\bf}R}}^d \times ( \{0\} \cup r { {\bf}{S}})}$, leading to a limit curve of measures even if $({ f ^\varepsilon})_{ \varepsilon}$ were smooth solutions. We can characterize the limit curve as solution of certain PDE whenever our initial measure does not charge the unstable point $0$. \[MainResult\] Assume that $a \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$, $(1 + |v|^2) { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{(x,v) :|v|\geq r_0>0\}$. Then $({ f ^\varepsilon})_{ \varepsilon}$ converges weakly $\star$ in ${ L^\infty ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}{}$ towards the solution of the problem $$\label{Equ22} \partial _t f + { \mathrm{div}_x}(fv) + { \mathrm{div}_v}\left \{f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \right \} = 0$$ $$\label{Equ23} { \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} = 0$$ with initial data $f(0) = { \left \langle { f ^{\mathrm{in}}}\right \rangle }$ defined by $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) { \left \langle { f ^{\mathrm{in}}}\right \rangle }(x,v)\, \mathrm{d}(x,v)} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi \left (x, r { \frac{v}{|v|}}\right ) { f ^{\mathrm{in}}}(x,v)\, \mathrm{d}(x,v)}\,,$$ for all $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}$. In the rest, we will refer to ${ \left \langle { f ^{\mathrm{in}}}\right \rangle }$ as the projected measure on the sphere of radius $r$ corresponding to ${ f ^{\mathrm{in}}}$. Let us point out that the previous result can be equivalently written in spherical coordinates by saying that $f(t,x,\omega)$ is the measure solution to the evolution equation on $(x,\omega)\in{{{\bf}R}}^d \times r { {\bf}{S}}$ given by $$\partial _t f + { \mathrm{div}_x}(f\omega) + { \mathrm{div}_\omega}\left \{f { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \right \} = 0 \,.$$ These results for the linear problem, when $a(t,x,v)$ is given, can be generalized to the nonlinear counterparts where $a(t,x)$ is given by . The main result of this work is (see Section \[MeaSol\] for the definition of ${ { {\cal P}}_1}$): \[MainResult2\] Assume that $U\in C^2_b({{{\bf}R}}^d)$, $H(x,v)=h(x)v$ with $h\in C^1_b({{{\bf}R}}^d)$ nonnegative, ${ f ^{\mathrm{in}}}\in { {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{(x,v) :|x| \leq L_0, r_0\leq |v| \leq R_0\}$ with $0<r_0<r<R_0<\infty$. Then for all $\delta>0$, the sequence $({ f ^\varepsilon})_{ \varepsilon}$ converges in $C([\delta,\infty);{ {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d)})$ towards the measure solution $f(t,x,\omega)$ on $(x,\omega)\in{{{\bf}R}}^d \times r { {\bf}{S}}$ of the problem $$\label{Equ22n} \partial _t f + { \mathrm{div}_x}(f\omega) - { \mathrm{div}_\omega}\left \{f { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }\left(\nabla_x U\star \rho + H\star f \right) \right \} = 0$$ with initial data $f(0) = { \left \langle { f ^{\mathrm{in}}}\right \rangle }$. Moreover, if the initial data ${ f ^{\mathrm{in}}}$ is already compactly supported on $B_{L_0} \times r { {\bf}{S}}$, then the convergence holds in ${ C ({{{\bf}R}}_+ ; {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d))}$. Let us mention that the evolution problem on ${{{\bf}R}}^d \times r { {\bf}{S}}$ was also proposed in the literature as the continuum version [@DM08] of the Vicsek model [@VicCziBenCohSho95; @CouKraFraLev02] without diffusion for the particular choice $U=0$ and $H(x,v)=h(x) v$ with $h(x)$ some local averaging kernel. The original model in [@VicCziBenCohSho95; @CouKraFraLev02] also includes noise at the particle level and was derived as the mean filed limit of some stochastic particle systems in [@BCC12]. In fact, previous particle systems have also been studied with noise in [@BCC11] for the mean-field limit, in [@HLL09] for studying some properties of the Cucker-Smale model with noise, and in [@DFL10; @FL11] for analyzing the phase transition in the Vicsek model. In the case of noise, getting accurate control on the particle paths of the solutions is a complicated issue and thus, we are not able to show the corresponding rigorous results to Theorems \[MainResult\] and \[MainResult2\]. Nevertheless, we will present a simplified formalism, which allows us to handle more complicated problems to formally get the expected limit equations. This approach was borrowed from the framework of the magnetic confinement, where leading order charged particle densities have to be computed after smoothing out the fluctuations which correspond to the fast motion of particles around the magnetic lines [@BosAsyAna; @BosTraEquSin; @BosGuiCen3D; @BosNeg09]. We apply this method to the following (linear or nonlinear) problem $$\label{Equ31} \partial _t { f ^\varepsilon}+ { \mathrm{div}_x}\{{ f ^\varepsilon}v\} + { \mathrm{div}_v}\{ { f ^\varepsilon}a\} + \frac{1}{{ \varepsilon}} { \mathrm{div}_v}\{ { f ^\varepsilon}{ (\alpha - \beta |v|^2) v}\} = \Delta _v { f ^\varepsilon}$$ with initial data ${ f ^\varepsilon}(0) = { f ^{\mathrm{in}}}$ where the acceleration $a \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$ and ${ f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. By applying the projection operator ${ \left \langle \cdot \right \rangle }$ to , we will show that the limiting equation for the evolution of $f(t,x,\omega)$ on $(x,\omega)\in{{{\bf}R}}^d \times r { {\bf}{S}}$ is given by $$\label{Equ22Diff} \partial _t f + { \mathrm{div}_x}(f\omega) + { \mathrm{div}_\omega}\left \{f { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \right \} = \Delta_\omega f$$ where $\Delta_\omega$ is the Laplace-Beltrami operator on $r { {\bf}{S}}$. Our paper is organized as follows. In Section \[MeaSol\] we investigate the stability of the characteristic flows associated to the perturbed fields $v \cdot \nabla _x + a \cdot \nabla _v + \frac{1}{{ \varepsilon}} { (\alpha - \beta |v|^2) v}\cdot \nabla _v $. The first limit result for the linear problem (cf. Theorem \[MainResult\]) is derived rigorously in Section \[LimMod\]. Section \[NLimMod\] is devoted to the proof of the main Theorem \[MainResult2\]. The new formalism to deal with the treatment of diffusion models is presented in Section \[DiffMod\]. The computations to show that these models correspond to the Vicsek models, written in spherical coordinates, are presented in the Appendix \[A\]. Measure solutions {#MeaSol} ================= Preliminaries on mass transportation metrics and notations {#prelim} ---------------------------------------------------------- We recall some notations and result about mass transportation distances that we will use in the sequel. For more details the reader can refer to [@Vi1; @CT]. We denote by ${ { {\cal P}}_1}({{{\bf}R}}^d)$ the space of probability measures on ${{{\bf}R}}^d$ with finite first moment. We introduce the so-called *Monge-Kantorovich-Rubinstein distance* in ${ { {\cal P}}_1}({{{\bf}R}}^d)$ defined by $$W_1(f,g) = \sup \left \{ \left |\int_{{{{\bf}R}}^d} \varphi(u) (f(u)-g(u))\, \mathrm{d} u \right |, \varphi \in \lip({{{\bf}R}}^d), \lip(\varphi)\leq 1 \right \} $$ where $\lip({{{\bf}R}}^d)$ denotes the set of Lipschitz functions on ${{{\bf}R}}^d$ and $\lip(\varphi)$ the Lipschitz constant of a function $\varphi$. Denoting by $\Lambda$ the set of transference plans between the measures $f$ and $g$, i.e., probability measures in the product space ${{{\bf}R}}^d \times {{{\bf}R}}^d$ with first and second marginals $f$ and $g$ respectively $$f(y) = \int_{{{{\bf}R}}^d} \pi (y,z)\,\mathrm{d}z,\;\;g(z) = \int_{{{{\bf}R}}^d} \pi (y,z)\,\mathrm{d}y$$ then we have $$W_1(f, g) = \inf_{\pi\in\Lambda} \left\{ \int_{{{{\bf}R}}^d \times {{{\bf}R}}^d} \vert y - z \vert \, \pi(y, z)\,\mathrm{d}(y,z) \right\}$$ by Kantorovich duality. ${ { {\cal P}}_1}({{{\bf}R}}^d)$ endowed with this distance is a complete metric space. Its properties are summarized below, see[@Vi1]. \[w2properties\] The following properties of the distance $W_1$ hold: 1. [**Optimal transference plan:**]{} The infimum in the definition of the distance $W_1$ is achieved. Any joint probability measure $\pi_o$ satisfying: $$W_1(f, g) = \int_{{{{\bf}R}}^d \times {{{\bf}R}}^d} \vert y - z \vert \, \mathrm{d}\pi_o(y, z)$$ is called an optimal transference plan and it is generically non unique for the $W_1$-distance. 2. [**Convergence of measures:**]{} Given $\{f_k\}_{k\ge 1}$ and $f$ in ${ { {\cal P}}_1}({{{\bf}R}}^d)$, the following two assertions are equivalent: - $W_1(f_k, f)$ tends to $0$ as $k$ goes to infinity. - $f_k$ tends to $f$ weakly $\star$ as measures as $k$ goes to infinity and $$\sup_{k\ge 1} \int_{\vert v \vert > R} \vert v \vert \, f_k(v) \, \mathrm{d}v \to 0 \, \mbox{ as } \, R \to +\infty.$$ Let us point out that if the sequence of measures is supported on a common compact set, then the convergence in $W_1$-sense is equivalent to standard weak-$\star$ convergence for bounded Radon measures. Finally, let us remark that all the models considered in this paper preserve the total mass. After normalization we can consider only solutions with total mass $1$ and therefore use the Monge-Kantorovich-Rubinstein distance in ${ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d)$. From now on we assume that the initial conditions has total mass $1$. Estimates on Characteristics ---------------------------- In this section we investigate the linear Vlasov problem $$\label{Equ10} \partial _t { f ^\varepsilon}+ { \mathrm{div}_x}\{{ f ^\varepsilon}v\} + { \mathrm{div}_v}\{ { f ^\varepsilon}a\} + \frac{1}{{ \varepsilon}} { \mathrm{div}_v}\{ { f ^\varepsilon}{ (\alpha - \beta |v|^2) v}\} = 0,\;\;(t,x,v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d$$ $$\label{Equ11} { f ^\varepsilon}(0) = { f ^{\mathrm{in}}}$$ where $a \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$ and ${ f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. \[DefMeaSol\] Assume that $a \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$ and ${ f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. We say that ${ f ^\varepsilon}\in { L^\infty ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}{}$ is a measure solution of - if for any test function $\varphi \in { C^1_c ({{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\{\partial _t + v \cdot \nabla _x + a \cdot \nabla _v + \frac{1}{{ \varepsilon}} { (\alpha - \beta |v|^2) v}\cdot & \nabla _v \}\varphi { f ^\varepsilon}(t,x,v)\, \mathrm{d}(x,v)\;\mathrm{d}t} \\ &+ { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \varphi (0,x,v) { f ^{\mathrm{in}}}(x,v) \, \mathrm{d}(x,v) } = 0.\end{aligned}$$ We introduce the characteristics of the field $v\cdot \nabla _x + a \cdot \nabla _v + \frac{1}{{ \varepsilon}} { (\alpha - \beta |v|^2) v}\cdot \nabla _v $ $$\frac{\mathrm{d}{ X ^\varepsilon }}{\mathrm{d}s} = { V ^\varepsilon }(s),\;\;\frac{\mathrm{d}{ V ^\varepsilon }}{\mathrm{d}s} = a(s, { X ^\varepsilon }(s)) + \frac{1}{{ \varepsilon}} { (\alpha - \beta \left |{ V ^\varepsilon }(s)\right |^2) { V ^\varepsilon }(s)}$$ $${ X ^\varepsilon }(s=0) = x,\;\;{ V ^\varepsilon }(s = 0) = v.$$ We will prove that $({ X ^\varepsilon }, { V ^\varepsilon })$ are well defined for any $(s,x,v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d$. Indeed, on any interval $[0,T]$ on which $({ X ^\varepsilon }, { V ^\varepsilon })$ is well defined we get a bound $$\sup _{s \in [0,T]} \{|{ X ^\varepsilon }(s) | + |{ V ^\varepsilon }(s) | \} < +\infty$$ implying that the characteristics are global in positive time. For that we write $$\label{charnew} \frac12\frac{\mathrm{d}|{ V ^\varepsilon }|^2}{\mathrm{d}s} = a(s, { X ^\varepsilon }(s))\cdot { V ^\varepsilon }(s) + \frac{1}{{ \varepsilon}} ( \alpha - \beta |{ V ^\varepsilon }(s) |^2) |{ V ^\varepsilon }(s)|^2.$$ and then, we get the differential inequality $$\frac{\mathrm{d}|{ V ^\varepsilon }|^2}{\mathrm{d}s} \leq 2\|a\|_{{ L^\infty}} |{ V ^\varepsilon }(s)| + \frac{2}{{ \varepsilon}} ( \alpha - \beta |{ V ^\varepsilon }(s) |^2) |{ V ^\varepsilon }(s)|^2$$ for all $s\in [0,T]$, so that $$\sup _{s \in [0,T]} |{ V ^\varepsilon }(s) | < +\infty,\;\;\sup _{s\in [0,T]} |{ X ^\varepsilon }(s) | \leq |x| + T \sup _{s\in [0,T]} |{ V ^\varepsilon }(s) | < +\infty.$$ Once constructed the characteristics, it is easily seen how to obtain a measure solution for the Vlasov problem -. It reduces to push forward the initial measure along the characteristics, see [@CCR10] for instance. For any $t \in {{{\bf}R}}_+$ we denote by ${ f ^\varepsilon}(t)$ the measure given by $$\label{EquDefMea} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) { f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(({ X ^\varepsilon }, { V ^\varepsilon })(t;0,x,v)){ f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}\,,$$ for all $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}$. Then the application $t \to { f ^\varepsilon}(t)$, denoted ${ f ^{\mathrm{in}}}\#({ X ^\varepsilon }, { V ^\varepsilon })(t;0,\cdot,\cdot)$ is the unique measure solution of , , belongs to ${ C ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}$ and satisfies $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}, t \in {{{\bf}R}}_+.$$ The arguments are straightforward and are left to the reader. We only justify that ${ f ^\varepsilon}\in { C ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}$ meaning that for any $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ the application $t \to { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,\,\psi(x,v) { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}}$ is continuous. Choose $\psi\in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. Then, for any $0 \leq t_1 < t_2$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { f ^\varepsilon}(t_2,x,v) &\,{ \mathrm{d}(x,v)}} - { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { f ^\varepsilon}(t_1,x,v)\,{ \mathrm{d}(x,v)}} \\ &= { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \left[\psi (({ X ^\varepsilon }, { V ^\varepsilon })(t_2;t_1, x, v)) - \psi (x,v)\right]{ f ^\varepsilon}(t_1,x,v)\,{ \mathrm{d}(x,v)}}.\end{aligned}$$ Taking into account that $({ X ^\varepsilon }, { V ^\varepsilon })$ are locally bounded (in time, position, velocity) it is easily seen that for any compact set $K \subset {{{\bf}R}}^d \times {{{\bf}R}}^d$ there is a constant $C(K)$ such that $$|{ X ^\varepsilon }(t_2; t_1, x, v) - x| + |{ V ^\varepsilon }(t_2; t_1, x, v) - v| \leq |t_2 - t_1 | C(K),\;\;(x,v) \in K.$$ Our conclusion follows easily using the uniform continuity of $\psi$ and that $\|{ f ^\varepsilon}(t_1) \|_{{\cal M}_b} = \|{ f ^{\mathrm{in}}}\|_{{\cal M}_b}$. Notice also that the equality holds true for any bounded continuous function $\psi$. We intend to study the behavior of $({ f ^\varepsilon})_{ \varepsilon}$ when ${ \varepsilon}$ becomes small. This will require a more detailed analysis of the characteristic flows $({ X ^\varepsilon }, { V ^\varepsilon })$. The behavior of these characteristics depends on the roots of functions like $A + \frac{1}{{ \varepsilon}} (\alpha - \beta \rho ^2 ) \rho$, with $\rho \in {{{\bf}R}}_+$, $A \in {{{\bf}R}}$. \[NegA\] Assume that $A < 0$ and $ 0 < { \varepsilon}< 2\alpha r /(|A| 3 \sqrt{3})$. Then the equation ${ \lambda ^\varepsilon}(\rho) := { \varepsilon}A + (\alpha - \beta \rho ^2 ) \rho = 0$ has two zeros on ${{{\bf}R}}_+$, denoted ${ \rho ^\varepsilon _1}(A), { \rho ^\varepsilon _2}(A)$, satisfying $$0 < { \rho ^\varepsilon _1}< \frac{r}{\sqrt{3}} < { \rho ^\varepsilon _2}< r$$ and $${ \lim _{\varepsilon \searrow 0}}\frac{{ \rho ^\varepsilon _1}}{{ \varepsilon}} = \frac{|A|}{\alpha},\;\;\;\;\;\;{ \lim _{\varepsilon \searrow 0}}\frac{r - { \rho ^\varepsilon _2}}{{ \varepsilon}} = \frac{|A|}{2\alpha}$$ where $r = \sqrt{\alpha/\beta}$. It is easily seen that the function ${ \lambda ^\varepsilon}$ increases on $[0,r/\sqrt{3}]$ and decreases on $[r/\sqrt{3}, +\infty[$ with change of sign on $[0,r/\sqrt{3}]$ and $[r/\sqrt{3}, r]$. We can prove that $({ \rho ^\varepsilon _1})_{ \varepsilon}, ({ \rho ^\varepsilon _2})_{ \varepsilon}$ are monotone with respect to ${ \varepsilon}>0$. Take $0 < { \varepsilon}< { \overline{\varepsilon}}< 2\alpha r /(|A| 3 \sqrt{3})$ and observe that ${ \lambda ^\varepsilon}> \lambda ^{{ \overline{\varepsilon}}}$. In particular we have $$\lambda ^{{ \overline{\varepsilon}}} ({ \rho ^\varepsilon _1}) < { \lambda ^\varepsilon}({ \rho ^\varepsilon _1}) = 0 = \lambda ^{{ \overline{\varepsilon}}} (\rho _1 ^{{ \overline{\varepsilon}}})$$ implying ${ \rho ^\varepsilon _1}< \rho _1 ^{{ \overline{\varepsilon}}}$, since $\lambda ^{{ \overline{\varepsilon}}}$ is strictly increasing on $[0, r/\sqrt{3}]$. Similarly we have $$\lambda ^{{ \overline{\varepsilon}}} ({ \rho ^\varepsilon _2}) < { \lambda ^\varepsilon}({ \rho ^\varepsilon _2}) = 0 < \lambda ^{{ \overline{\varepsilon}}} (\rho _2 ^{{ \overline{\varepsilon}}})$$ and thus ${ \rho ^\varepsilon _2}> \rho _2 ^{{ \overline{\varepsilon}}}$, since $\lambda ^{{ \overline{\varepsilon}}}$ is strictly decreasing on $[r/\sqrt{3}, r]$. Passing to the limit in ${ \lambda ^\varepsilon}(\rho _k ^{ \varepsilon}) = 0, k \in \{1,2\}$ it follows easily that $${ \lim _{\varepsilon \searrow 0}}{ \rho ^\varepsilon _1}= 0,\;\;{ \lim _{\varepsilon \searrow 0}}{ \rho ^\varepsilon _2}= r.$$ Moreover we can write $$\alpha = \frac{\mathrm{d}}{\mathrm{d}\rho }\{(\alpha - \beta \rho ^2 ) \rho \} |_{\rho = 0} = { \lim _{\varepsilon \searrow 0}}\frac{[\alpha - \beta ({ \rho ^\varepsilon _1})^2]{ \rho ^\varepsilon _1}}{{ \rho ^\varepsilon _1}} = - { \lim _{\varepsilon \searrow 0}}\frac{{ \varepsilon}A}{{ \rho ^\varepsilon _1}} \nonumber$$ and $$-2 \alpha = \frac{\mathrm{d}}{\mathrm{d}\rho }\{(\alpha - \beta \rho ^2 ) \rho \} |_{\rho = r} = { \lim _{\varepsilon \searrow 0}}\frac{[\alpha - \beta ({ \rho ^\varepsilon _2})^2]{ \rho ^\varepsilon _2}}{{ \rho ^\varepsilon _2}- r} = - { \lim _{\varepsilon \searrow 0}}\frac{{ \varepsilon}A}{{ \rho ^\varepsilon _2}- r} \nonumber$$ saying that $${ \lim _{\varepsilon \searrow 0}}\frac{{ \rho ^\varepsilon _1}}{{ \varepsilon}} = \frac{|A|}{\alpha},\;\;{ \lim _{\varepsilon \searrow 0}}\frac{r - { \rho ^\varepsilon _2}}{{ \varepsilon}} = \frac{|A|}{2\alpha}.$$ The case $A>0$ can be treated is a similar way and we obtain \[PosA\] Assume that $A > 0$ and $ { \varepsilon}>0$. Then the equation ${ \lambda ^\varepsilon}(\rho) := { \varepsilon}A + (\alpha - \beta \rho ^2 ) \rho = 0$ has one zero on ${{{\bf}R}}_+$, denoted ${ \rho ^\varepsilon _3}(A)$, satisfying $${ \rho ^\varepsilon _3}>r,\;\;{ \lim _{\varepsilon \searrow 0}}\frac{{ \rho ^\varepsilon _3}- r}{{ \varepsilon}} = \frac{|A|}{2\alpha}.$$ Using the sign of the function $\rho \to { \varepsilon}\|a\|_{{ L^\infty}{}} + (\alpha - \beta \rho ^2 ) \rho$ we obtain the following bound for the kinetic energy. \[KinBou\] Assume that $a \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$, $(1 + |v|^2) { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ and let us denote by ${ f ^\varepsilon}$ the unique measure solution of , . Then we have $$\left \|{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,|v|^2 { f ^\varepsilon}(\cdot,x,v)\,{ \mathrm{d}(x,v)}}\right \|_{{ L^\infty}({{{\bf}R}}_+)} \leq { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! [({ \rho ^\varepsilon _3})^2 + |v|^2] { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}.$$ We know that $$\frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }|^2 \leq 2\|a\|_{{ L^\infty}{}} |{ V ^\varepsilon }(t)|+ \frac{2}{{ \varepsilon}} (\alpha - \beta |{ V ^\varepsilon }(t) |^2 ) |{ V ^\varepsilon }(t)|^2=\frac{2}{{ \varepsilon}}|{ V ^\varepsilon }(t)|{ \lambda ^\varepsilon}(|{ V ^\varepsilon }(\overline{t})| ),\;\;t \in {{{\bf}R}}_+.$$ By comparison with the solutions of the autonomous differential equation associated to the righthand side, we easily deduce that $$|{ V ^\varepsilon }(t;0,x,v)| \leq \max \{ |v|, { \rho ^\varepsilon _3}(\|a\|_{{ L^\infty}{}})\}\,,$$ for any $T \in {{{\bf}R}}_+, (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d$. This yields the following bound for the kinetic energy $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! |v|^2{ f ^\varepsilon}(T,x,v)\,{ \mathrm{d}(x,v)}} &= { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! |{ V ^\varepsilon }(T;0,x,v)|^2 { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}} \\ &\leq { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! [({ \rho ^\varepsilon _3})^2 + |v|^2] { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}.\end{aligned}$$ The object of the next result is to establish the stability of ${ V ^\varepsilon }$ around $|v| = r$. We will show that the characteristics starting at points with velocities inside an annulus of length proportional to ${ \varepsilon}$ around the sphere $r{ {\bf}{S}}$ get trapped there for all positive times for small ${ \varepsilon}$. \[RStab\] Assume that ${ \varepsilon}\|a\|_{{ L^\infty}{}} < 2\alpha r /(3\sqrt{3})$ and that ${ \rho ^\varepsilon _2}(-\|a\|_{{ L^\infty}{}}) \leq |v| \leq { \rho ^\varepsilon _3}(\|a\|_{{ L^\infty}{}})$. Then, for any $(t,x) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d$ we have $${ \rho ^\varepsilon _2}(-\|a\|_{{ L^\infty}{}}) \leq |{ V ^\varepsilon }(t;0,x,v)| \leq { \rho ^\varepsilon _3}(\|a\|_{{ L^\infty}{}}).$$ As in previous proof, we know that $$\frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }|^2 \leq \frac{2}{{ \varepsilon}}|{ V ^\varepsilon }(t)|{ \lambda ^\varepsilon}(|{ V ^\varepsilon }(\overline{t})| ),\;\;t \in {{{\bf}R}}_+\,.$$ By comparison with the constant solution ${ \rho ^\varepsilon _3}$ to the autonomous differential equation associated to the righthand side, we get that $\sup _{t \in {{{\bf}R}}_+} |{ V ^\varepsilon }(t;0,x,v)| \leq { \rho ^\varepsilon _3}$. Assume now that there is $T>0$ such that $|{ V ^\varepsilon }(T) | < { \rho ^\varepsilon _2}$ and we are done if we find a contradiction. Since $|{ V ^\varepsilon }(0) |= |v| \geq { \rho ^\varepsilon _2}$, we can assume that $\min _{t \in [0,T]} |{ V ^\varepsilon }(t) | > { \rho ^\varepsilon _1}>0$ by time continuity. Take now ${ \overline{t}}\in [0,T]$ a minimum point of $t \to |{ V ^\varepsilon }(t)|$ on $[0,T]$. Obviously ${ \overline{t}}>0$ since $$|{ V ^\varepsilon }({ \overline{t}}) | \leq |{ V ^\varepsilon }(T)| < { \rho ^\varepsilon _2}\leq |v| = |{ V ^\varepsilon }(0)|.$$ By estimating from below in and using that ${ \overline{t}}$ is a minimum point of $t \to |{ V ^\varepsilon }(t)|>0$ on $[0,T]$, we obtain $$0 \geq \frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }({ \overline{t}})| \geq - \|a\|_{{ L^\infty}{}} + \frac{(\alpha - \beta |{ V ^\varepsilon }({ \overline{t}})|^2)|{ V ^\varepsilon }({ \overline{t}})| }{{ \varepsilon}} =\frac{{ \lambda ^\varepsilon}( |{ V ^\varepsilon }({ \overline{t}})| )}{{ \varepsilon}}.$$ But the function ${ \lambda ^\varepsilon}$ has negative sign on $[0,{ \rho ^\varepsilon _1}] \cup [{ \rho ^\varepsilon _2}, +\infty[$. Since we know that $\min _{t \in [0,T]} |{ V ^\varepsilon }(t)| > { \rho ^\varepsilon _1}$, it remains that $$\min _{t \in [0,T]} |{ V ^\varepsilon }(t)| = |{ V ^\varepsilon }({ \overline{t}})| \geq { \rho ^\varepsilon _2}$$ which contradicts the assumption $|{ V ^\varepsilon }(T)| < { \rho ^\varepsilon _2}$. Let us see now what happens when the initial velocity is outside $[{ \rho ^\varepsilon _2}(-\|a\|_{{ L^\infty}{}}), { \rho ^\varepsilon _3}(\|a\|_{{ L^\infty}{}})]$. In particular we prove that if initially $v \neq 0$, then ${ V ^\varepsilon }(t), t \in {{{\bf}R}}_+$ remains away from $0$. We actually show that the characteristics starting away from zero speed but inside the sphere $r{ {\bf}{S}}$ will increase their speed with respect to its initial value while those starting with a speed outside the sphere $r{ {\bf}{S}}$ will decrease their speed with respect to its initial value, all for sufficiently small ${ \varepsilon}$. \[ZeroStab\] Consider ${ \varepsilon}>0$ such that ${ \varepsilon}\|a\|_{{ L^\infty}{}} < 2\alpha r /(3\sqrt{3})$.\ 1. Assume that ${ \rho ^\varepsilon _1}(- \|a\|_{{ L^\infty}{}}) < |v| < { \rho ^\varepsilon _2}(- \|a\|_{{ L^\infty}{}})$. Then for any $(t,x) \in {{{\bf}R}}_+ ^\star \times {{{\bf}R}}^d$ we have $${ \rho ^\varepsilon _1}(- \|a\|_{{ L^\infty}{}}) < |v| < |{ V ^\varepsilon }(t;0,x,v)|\leq{ \rho ^\varepsilon _3}( \|a\|_{{ L^\infty}{}}).$$ 2. Assume that ${ \rho ^\varepsilon _3}( \|a\|_{{ L^\infty}{}}) < |v|$. Then for any $(t,x) \in {{{\bf}R}}_+ ^\star \times {{{\bf}R}}^d$ we have $${ \rho ^\varepsilon _2}(- \|a\|_{{ L^\infty}{}}) \leq |{ V ^\varepsilon }(t;0,x,v) | < |v|.$$ 1\. Notice that if $|{ V ^\varepsilon }(T;0,x,v)| = { \rho ^\varepsilon _2}$ for some $T>0$, then we deduce by Proposition \[RStab\] that ${ \rho ^\varepsilon _2}\leq |{ V ^\varepsilon }(t) | \leq { \rho ^\varepsilon _3}$ for any $t >T$ and thus $|{ V ^\varepsilon }(t;0,x,v) | \geq { \rho ^\varepsilon _2}> |v|, t \geq T$. It remains to establish our statement for intervals $[0,T]$ such that $|{ V ^\varepsilon }(t) | < { \rho ^\varepsilon _2}$ for any $t \in [0,T]$. We are done if we prove that $t \to |{ V ^\varepsilon }(t)|$ is strictly increasing on $[0,T]$. For any $\tau \in ]0,T]$ let us denote by ${ \overline{t}}$ a maximum point of $t \to |{ V ^\varepsilon }(t)|>0$ on $[0,\tau]$. If ${ \overline{t}}\in [0,\tau[$ we have $\frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }({ \overline{t}})| \leq 0$ and thus $$0 \geq \frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }({ \overline{t}})|\geq - \|a\|_{{ L^\infty}{}} + \frac{(\alpha - \beta |{ V ^\varepsilon }({ \overline{t}})|^2)|{ V ^\varepsilon }({ \overline{t}})| }{{ \varepsilon}} =\frac{{ \lambda ^\varepsilon}( |{ V ^\varepsilon }({ \overline{t}})| )}{{ \varepsilon}}.$$ By construction $|{ V ^\varepsilon }({ \overline{t}})| < { \rho ^\varepsilon _2}$ and moreover, $$|{ V ^\varepsilon }({ \overline{t}})| = \max _{[0,\tau]} |{ V ^\varepsilon }| \geq |v| > { \rho ^\varepsilon _1}\,,$$ and thus, ${ \lambda ^\varepsilon}( |{ V ^\varepsilon }(t)| )>0$ for all $t\in [0,T]$. Consequently, we infer that $t \to |{ V ^\varepsilon }(t)|$ is strictly increasing on $[0,T]$ since $$\frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }(t)|\geq - \|a\|_{{ L^\infty}{}} + \frac{(\alpha - \beta |{ V ^\varepsilon }(t)|^2)|{ V ^\varepsilon }(t)| }{{ \varepsilon}} =\frac{{ \lambda ^\varepsilon}( |{ V ^\varepsilon }(t)| )}{{ \varepsilon}} >0\,.$$ Therefore we have ${ \overline{t}}= \tau$ saying that $|{ V ^\varepsilon }(\tau)| \geq |v|$ for any $\tau \in [0,T]$. 2\. As before, it is sufficient to work on intervals $[0,T]$ such that $|{ V ^\varepsilon }(t) | > { \rho ^\varepsilon _3}(\|a\|_{{ L^\infty}{}})$ for any $t \in [0,T]$. We are done if we prove that $t \to |{ V ^\varepsilon }(t)|$ is strictly decreasing on $[0,T]$. We have for any $t \in [0,T]$ $$\frac{\mathrm{d}}{\mathrm{d}t} |{ V ^\varepsilon }(t)|\leq \|a\|_{{ L^\infty}{}} + \frac{(\alpha - \beta |{ V ^\varepsilon }(t)|^2)|{ V ^\varepsilon }(t)| }{{ \varepsilon}} =\frac{{ \lambda ^\varepsilon}( |{ V ^\varepsilon }(t)| )}{{ \varepsilon}} <0$$ where for the last inequality we have used $|{ V ^\varepsilon }(t) | > { \rho ^\varepsilon _3}, t \in [0,T]$. The limit model {#LimMod} =============== We investigate now the stability of the family $({ f ^\varepsilon})_{ \varepsilon}$ when ${ \varepsilon}$ becomes small. After extraction of a sequence $({ \varepsilon}_k)_k$ converging to $0$ we can assume that $({ f ^{\varepsilon _k}})_k$ converges weakly $\star$ in $L^\infty({{{\bf}R}}_+;{\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))$, meaning that $${ \lim _{k \to +\infty }}{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\varphi (t,x,v) { f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\varphi (t,x,v) f (t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t}$$ for any $\varphi \in { L^1({{{\bf}R}}_+; C^0 _c ({{{\bf}R}}^d \times {{{\bf}R}}^d))}{}$. Using the weak formulation of - with test functions $\eta (t) \varphi (x,v)$, $\eta \in C^1 _c ({{{\bf}R}}_+)$, $\varphi \in C^1 _c ({{{\bf}R}}^d \times {{{\bf}R}}^d)$ one gets $$\begin{aligned} { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\{\eta ^{\;\prime} (t) \varphi + \eta (t) v \cdot \nabla _x \varphi + \eta (t) a \cdot \nabla _v \varphi \}{ f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}&\;\mathrm{d}t}\\ + \frac{1}{{ \varepsilon}_k} { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\eta (t) { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi { f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}&\;\mathrm{d}t} \\ = -{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \eta (0) &\varphi (x,v) { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}.\end{aligned}$$ Multiplying by ${ \varepsilon}_k$ and passing to the limit for $k \to +\infty$ yields $${ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\eta (t) { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi f (t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} = 0$$ and therefore one gets for any $t \in {{{\bf}R}}_+$ and $\varphi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi f (t,x,v)\,{ \mathrm{d}(x,v)}} = 0.$$ Under the hypothesis $(1 + |v|^2) { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we deduce by Proposition \[KinBou\] that $( 1 + |v|^2) f(t) \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ and therefore, applying the $(x,v)$ version of Proposition \[Kernel\] (whose proof is detailed in the sequel), we obtain $${ \mathrm{supp\;}}f(t) \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}}),\;\;t \in {{{\bf}R}}_+.$$ The proof of Proposition \[Kernel\] is based on the resolution of the adjoint problem $$- { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi = \psi (v),\;\;v \in {{{\bf}R}}^d$$ for any smooth righthand side $\psi$ with compact support in $^c({ \{0\} \cup r { {\bf}{S}}})$. (of Proposition \[Kernel\]) It is easily seen that for any $F \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, ${ \mathrm{supp\;}}F \subset { \{0\} \cup r { {\bf}{S}}}$ and any $\varphi \in { C^1_c ({{{\bf}R}}^d)}{}$ we have $${ \int _{{{{\bf}R}}^d} \!{ (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi (v) F(v)\,{ \mathrm{d}v}} = 0$$ saying that ${ \mathrm{div}_v}\{F { (\alpha - \beta |v|^2) v}\} = 0$. Assume now that ${ \mathrm{div}_v}\{F { (\alpha - \beta |v|^2) v}\} = 0$ for some $F \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ and let us prove that ${ \mathrm{supp\;}}F \subset { \{0\} \cup r { {\bf}{S}}}$. We introduce the flow ${\cal V} = {\cal V}(s;v)$ given by $$\label{Equ4} \frac{\mathrm{d}{\cal V}}{\mathrm{d}s} = ( \alpha - \beta |{\cal V} (s;v) |^2 ) {\cal V } (s;v),\;\;{\cal V}(0;v) = v.$$ A direct computation shows that ${ \frac{v}{|v|}}$ are left invariant $${ (\alpha - \beta |v|^2) v}\cdot \nabla _v \left ( { \frac{v}{|v|}}\right ) = (\alpha - \beta |v|^2 ) { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }{ \frac{v}{|v|}}= 0$$ and therefore $${\cal V} (s;v) = |{\cal V}(s;v)| { \frac{v}{|v|}},\;\;v \neq 0.$$ Multiplying by ${\cal V}(s;v) / |{\cal V}(s;v)|$ yields $$\frac{\mathrm{d}}{\mathrm{d}s}|{\cal V}| = ( \alpha - \beta |{\cal V} (s;v) |^2 ) |{\cal V } (s;v)|$$ whose solution is given by $$|{\cal V}(s;v)| = |v| \frac{r e ^{\alpha s}}{\sqrt{|v|^2 ( e ^{2\alpha s} - 1) + r^2}}$$ Finally one gets $${\cal V}(s;v) = \frac{r e ^{\alpha s}}{\sqrt{|v|^2 ( e ^{2\alpha s} - 1) + r^2}}\;v,\;\;s \in ]S(v),+\infty[$$ with $S(v) = - \infty$ if $0 \leq |v| \leq r$ and $S(v) = \frac{1}{2\alpha} \ln \left ( 1 - \frac{r^2}{|v|^2} \right ) < 0$ if $|v| > r$. Notice that the characteristics ${\cal V} (\cdot;v)$ are well defined on ${{{\bf}R}}_+$ for any $v \in {{{\bf}R}}^d$ and we have $$\lim _ {s \to +\infty} {\cal V}(s;v) = r { \frac{v}{|v|}}\;\mbox{ if } v \neq 0,\;\;\lim _ {s \to +\infty} {\cal V}(s;v) =0\;\mbox{ if } v = 0$$ and $$\lim _{s \searrow S(v)} |{\cal V}(s)| = 0\mbox{ if }0 \leq |v| < r,\;\lim _{s \searrow S(v)} |{\cal V}(s)| =r\mbox{ if } |v| = r,\;\lim _{s \searrow S(v)} |{\cal V}(s)| =+\infty\;\mbox{ if } |v| >r.$$ Let us consider a $C^1$ function $\psi = \psi (v)$ with compact support in $^c ({ \{0\} \cup r { {\bf}{S}}})$. We intend to construct a bounded $C^1$ function $\varphi = \varphi (v)$ such that $$- { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi = \psi (v),\;\;v \in {{{\bf}R}}^d.$$ Obviously, if such a function exists, we may assume that $\varphi (0) = 0$. Motivated by the equality $$- \frac{\mathrm{d}}{\mathrm{d}s} \{\varphi ({\cal V}(s;v)) \}= \psi ({\cal V}(s;v)),\;\;0 \leq |v| < r,\;\;- \infty < s \leq 0$$ and since we know that $\lim _{s \to - \infty} {\cal V} (s;v) = 0$ for any $0 \leq |v| < r$, we define $$\label{Equ7} \varphi (v) = - \int _{-\infty} ^ 0 \psi ( {\cal V}(\tau; v))\;\mathrm{d}\tau,\;\;0 \leq |v| < r.$$ Let us check that the function $\varphi$ in is well defined and is $C^1$ in $|v|<r$. The key point is that $\psi $ has compact support in $^c ({ \{0\} \cup r { {\bf}{S}}})$ and therefore there are $0 < r_1 < r_2 < r < r_3 < r_4 < +\infty$ such that $ { \mathrm{supp\;}}\psi \subset \{ v \in {{{\bf}R}}^d \;:\; r_1 \leq |v| \leq r_2 \} \cup \{ v \in {{{\bf}R}}^d \;:\; r_3 \leq |v| \leq r_4\}. $ It is easily seen that $\tau \to |{\cal V} (\tau; v)|$ is strictly increasing for any $0 < |v| < r$. Therefore, for any $|v| \leq r_1$ we have $ |{\cal V} (\tau; v) | \leq |{\cal V} (0; v) | = |v| \leq r_1,\;\;\tau \leq 0 $, implying that $$\varphi (v) = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau; v))\;\mathrm{d}\tau = 0,\;\;0 \leq |v| \leq r_1.$$ For any $v$ with $r_1 < |v| < r_2$ there are $\tau _1 < 0 < \tau _2$ such that $ |{\cal V}(\tau _1; v)| = r_1 < r_2 = |{\cal V}(\tau _2; v)|. $ The time interval between $\tau _1$ and $\tau _2$ comes easily by writing $$\frac{\frac{\mathrm{d}}{\mathrm{d}\tau}|{\cal V}(\tau) |}{(\alpha - \beta |{\cal V}(\tau)|^2)|{\cal V}(\tau) |}= 1$$ implying that $$|\tau _2 | + |\tau _1 | = \tau _2 - \tau _1 = \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }.$$ From the equality $$\varphi (v) = - \int _{-\infty} ^{\tau _1} \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau - \int _{\tau _1} ^0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau = - \int _{\tau _1} ^0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau\,,$$ we deduce that $$\label{Equ8} |\varphi (v) | \leq |\tau _1 | \; \|\psi \|_{C^0} \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0}.$$ Assume now that $r_2 \leq |v| < r$. There is $\tau _2 \geq 0$ such that $v = {\cal V} ( \tau_2 ; r_2 { \frac{v}{|v|}})$ and therefore $$\begin{aligned} \varphi (v) & = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau = - \int _{-\infty} ^ 0 \psi ({\cal V}(\tau + \tau _2;r_2 { \frac{v}{|v|}}))\;\mathrm{d}\tau \\ & = - \int _{-\infty} ^ {-\tau _2} \psi ({\cal V}(\tau + \tau _2 ;r_2 { \frac{v}{|v|}}))\;\mathrm{d}\tau = - \int _{-\infty} ^ {0} \psi ({\cal V}(\tau ;r_2 { \frac{v}{|v|}}))\;\mathrm{d}\tau = \varphi \left ( r_2 { \frac{v}{|v|}}\right).\end{aligned}$$ In particular, the restriction of $\varphi$ on $r_2 \leq |v| < r$ satisfies the same bound as in $$|\varphi (v) | \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0},\;\;r_2 \leq |v| < r.$$ It is easily seen that $\varphi $ is $C^1$ on $0 \leq |v| < r$. For that it is sufficient to consider $r_1 \leq |v| \leq r_2$. Notice that $$\frac{\partial {\cal V}}{\partial v} (\tau; v) = \frac{|{\cal V}(\tau;v)|}{|v|} \left ( I - \frac{{\cal V}(\tau;v) \otimes {\cal V}(\tau;v)}{r^2} ( 1 - e ^ {-2\alpha \tau } ) \right)$$ and therefore the gradient of $\varphi$ remains bounded on $r_1 \leq |v| \leq r_2$ $$\nabla _v \varphi (v) = - \int _{\tau _1} ^ 0 \frac{^ t \partial {\cal V}}{\partial v }(\tau; v) \nabla \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau$$ since on the interval $\tau \in [\tau _1, 0]$ we have $|{\cal V}(\tau;v)| \in [r_1, |v|] \subset [r_1, r_2]$. Taking now as definition for $|v| = r$ $$\varphi (v) = \varphi \left ( r_2 { \frac{v}{|v|}}\right )\,,$$ we obtain a bounded $C^1$ function on $|v| \leq r$ satisfying $$- { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi = \psi (v),\;\;|\varphi (v) | \leq \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2 ) \rho }\; \|\psi \|_{C^0},\;|v|\leq r.$$ We proceed similarly in order to extend the above function for $|v| > r$. We have for any $s>0$ $$- \varphi ({\cal V}(s;v)) + \varphi (v) = \int _0 ^s \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau,\;\;|v|> r.$$ As $\lim _{s \to +\infty} {\cal V}(s;v) = r { \frac{v}{|v|}}$ we must take $$\varphi (v) = \lim _{s \to +\infty}\left \{\varphi ( {\cal V}(s;v)) + \int _0 ^s \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau \right \} = \varphi \left (r{ \frac{v}{|v|}}\right ) + \int _0 ^{+\infty} \psi ({\cal V}(\tau;v))\;\mathrm{d}\tau,\;\;|v| >r.\nonumber$$ Clearly, for any $|v| > r$ the function $\tau \to |{\cal V}(\tau;v)|$ is strictly decreasing. Therefore, for any $r < |v| \leq r_3$ we have $$\varphi (v) = \varphi \left (r{ \frac{v}{|v|}}\right )= \varphi \left (r_2{ \frac{v}{|v|}}\right )$$ since $|{\cal V}(\tau;v)|\leq |v| \leq r_3$ and $\psi ({\cal V}(\tau;v)) = 0$, $\tau \geq 0$. If $r_3 < |v| < r_4$ let us consider $\tau _4 < 0 < \tau _3$ such that $ |{\cal V}(\tau _3;v)| = r_3 < r_4 = |{\cal V}(\tau _4;v)|. $ The time interval between $\tau _4$ and $\tau _3$ is given by $$|\tau _3 | + |\tau _4 | = \tau _3 - \tau _4 = \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } < +\infty\,,$$ and therefore one gets for $r_3 < |v| < r_4$ $$\begin{aligned} |\varphi (v) | &\leq \left | \varphi \left ( r { \frac{v}{|v|}}\right ) \right | + \left |\int _0 ^{\tau _3} \!\!\!\!\psi ({\cal V}(\tau;v))\;\mathrm{d}\tau \right | \nonumber \\ & \leq \left [ \int _{r_1} ^ {r_2} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } + \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } \right ] \|\psi \|_{C^0}.\label{Equ9}\end{aligned}$$ Consider now $|v|\geq r_4$. There is $\tau _4 \geq 0$ such that $r_4 { \frac{v}{|v|}}= {\cal V} (\tau_4; v)$ implying that $$\begin{aligned} \varphi (v) & = \varphi \left ( r { \frac{v}{|v|}}\right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; v)) \;\mathrm{d}\tau = \varphi \left ( r { \frac{v}{|v|}}\right ) + \int _{\tau _4} ^{+\infty} \psi ({\cal V} (\tau; v)) \;\mathrm{d}\tau \\ & = \varphi \left ( r { \frac{v}{|v|}}\right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; {\cal V}(\tau _4;v))) \;\mathrm{d}\tau = \varphi \left ( r { \frac{v}{|v|}}\right ) + \int _0 ^{+\infty} \psi ({\cal V} (\tau; r_4 { \frac{v}{|v|}})) \;\mathrm{d}\tau \\ & = \varphi \left ( r_4 { \frac{v}{|v|}}\right ).\end{aligned}$$ We deduce that the restriction of $\varphi $ on $\{v :|v| \geq r_4\}$ satisfies the same bound as in . Moreover the function $\varphi $ is $C^1$ on $\{v:|v|\geq r\}$, with bounded derivatives. Indeed, it is sufficient to consider only the case $r_3 \leq |v| \leq r_4$, observing that $$\begin{aligned} \nabla _v \varphi (v) = \frac{r_2}{|v|} { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\nabla _v \varphi \left ( r_2 { \frac{v}{|v|}}\right ) + \int _0 ^{\tau _3} \frac{^t \partial {\cal V}}{\partial v }(\tau;v)\nabla \psi ({\cal V}(\tau;v)) \;\mathrm{d}\tau \nonumber\end{aligned}$$ $$|{\cal V} (\tau; v)| \in [r_3, |v| ] \subset [r_3, r_4],\;\tau \in [0,\tau_3],\;\;|\tau _3| + |\tau _4| = \int _{r_4} ^ {r_3} \frac{\mathrm{d}\rho}{(\alpha - \beta \rho ^2) \rho } < +\infty.$$ By construction we have $- { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi = \psi (v)$, $|v| >r$. Consider a $C^1$ decreasing function on ${{{\bf}R}}_+$ such that $\chi |_{[0,1]} = 1, \chi _{[2,+\infty[} = 0$. We know that $${ \int _{{{{\bf}R}}^d} \!{ (\alpha - \beta |v|^2) v}\cdot \nabla _v \left \{ \varphi (v) \chi \left ( \frac{|v|}{R} \right ) \right \}\,F(v)\,{ \mathrm{d}v}} = 0,\;\;R>0\,,$$ saying that $${ \int _{{{{\bf}R}}^d} \!\chi \left ( \frac{|v|}{R} \right ){ (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi \;F(v)\,{ \mathrm{d}v}} + { \int _{{{{\bf}R}}^d} \!(\alpha - \beta |v|^2) \varphi (v) \frac{|v|}{R} \chi ^{\;\prime} \left ( \frac{|v|}{R} \right ) \;F(v)\,{ \mathrm{d}v}} = 0.$$ Since $\varphi$ and $\psi = - { (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi $ are bounded and $F$ has finite mass and kinetic energy, we can pass to the limit for $R \to +\infty$, using the dominated convergence theorem. We obtain for any $C^1$ function $\psi$, with compact support in $^c({ \{0\} \cup r { {\bf}{S}}})$ $${ \int _{{{{\bf}R}}^d} \!\psi (v) F(v)\,{ \mathrm{d}v}} = - { \int _{{{{\bf}R}}^d} \!{ (\alpha - \beta |v|^2) v}\cdot \nabla _v \varphi\, F(v)\,{ \mathrm{d}v}} = 0.$$ Actually the previous equality holds true for any continuous function $\psi$ with compact support in $^c({ \{0\} \cup r { {\bf}{S}}})$, since ${ \int _{{{{\bf}R}}^d} \!F(v)\,{ \mathrm{d}v}} < +\infty$, so that ${ \mathrm{supp\;}}F \subset { \{0\} \cup r { {\bf}{S}}}$. In order to obtain stability for $({ f ^{\varepsilon _k}})_k$ we need to avoid the unstable equilibrium $v = 0$. For that we assume that the initial support is away from zero speed: there is $r_0 >0$ (eventually small, let us say $r_0 < r$) such that $$\label{Equ20} { \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|v| \geq r_0\}.$$ \[UnifSupp\] Under the hypothesis we have for any ${ \varepsilon}>0$ small enough $${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|v| \geq r_0\},\;\;t \in {{{\bf}R}}_+.$$ Take ${ \varepsilon}>0$ such that ${ \varepsilon}\|a\|_{{ L^\infty}{}} < 2\alpha r /(3 \sqrt{3})$ and ${ \rho ^\varepsilon _1}(- \|a\|_{{ L^\infty}{}}) < r_0$. For any continuous function $\psi = \psi (x,v)$ with compact support in ${{{\bf}R}}^d \times \{v\;:\; |v| < r_0\}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}} & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t;0,x,v), { V ^\varepsilon }(t;0,x,v)){ f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}} \\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t;0,x,v), { V ^\varepsilon }(t;0,x,v) ){\bf 1}_{\{|v| \geq r_0 \}}{ f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}}.\end{aligned}$$ But for any $|v| \geq r_0 > { \rho ^\varepsilon _1}$ we know by Proposition \[ZeroStab\] that $|{ V ^\varepsilon }(t;0,x,v)| > |v| \geq r_0$, implying that $\psi ({ X ^\varepsilon }(t), { V ^\varepsilon }(t)) = 0$. Therefore one gets $\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}{\psi(x,v) { f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}} = 0$ saying that ${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v):|v| \geq r_0\}$. We are ready now to establish the model satisfied by the limit measure $f$. The idea is to use the weak formulation of , with test functions which are constant along the flow of ${ (\alpha - \beta |v|^2) v}\cdot \nabla _v$, in order to get rid of the term in $\frac{1}{{ \varepsilon}}$. These functions are those depending on $x$ and ${ \frac{v}{|v|}}$. Surely, the invariants ${ \frac{v}{|v|}}$ have no continuous extensions in $v = 0$, but we will see that we can use it, since our measures ${ f ^\varepsilon}$ vanish around $v = 0$. (of Theorem \[MainResult\]) We already know that $f$ satisfies . Actually, since ${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{(x,v):|v|\geq r_0\}, t \in {{{\bf}R}}_+, { \varepsilon}>0$, we deduce that ${ \mathrm{supp\;}}f(t) \subset \{(x,v):|v| \geq r_0\}$ and finally ${ \mathrm{supp\;}}f(t) \subset {{{\bf}R}}^d \times r { {\bf}{S}}, t \in {{{\bf}R}}_+$. We have to establish and find the initial data. Consider a $C^1$ decreasing function $\chi $ on ${{{\bf}R}}_+$ such that $\chi |_{[0,1]} = 1, \chi _{[2,+\infty[} = 0$. For any $\eta = \eta (t) \in C^1_c ({{{\bf}R}}_+)$, $\varphi = \varphi (x,v) \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we construct the test function $$\theta (t,x,v) = \eta (t) \left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ] \varphi \left ( x, r{ \frac{v}{|v|}}\right ).$$ Notice that $\theta $ is $C^1$ and $\theta = 0$ for $|v| \leq \frac{r_0}{2}$. When applying the weak formulation of - with $\theta$, the term in $\frac{1}{{ \varepsilon}}$ vanishes. Indeed, we can write $$\begin{aligned} \frac{1}{{ \varepsilon}}{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\eta (t) & { (\alpha - \beta |v|^2) v}\cdot \nabla _v \left \{\left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ]\varphi \left ( x, r{ \frac{v}{|v|}}\right ) \right \}{ f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \nonumber \\ & = \frac{1}{{ \varepsilon}} \int _{{{{\bf}R}}_+} \eta (t) \int _{|v|\geq r_0} { (\alpha - \beta |v|^2) v}\cdot \nabla _v \left \{ \varphi \left ( x, r{ \frac{v}{|v|}}\right ) \right \}{ f ^\varepsilon}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t = 0.\nonumber\end{aligned}$$ For the term containing $\partial _t \theta$ we obtain the following limit when $k \to +\infty$ $$\begin{aligned} T_1 ^k := { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\partial _t \theta { f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \to &{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\partial _t \theta f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \\ & = \int _{{{{\bf}R}}_+} \eta ^{\;\prime} (t) \int _{|v|\geq r_0} \varphi \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = \int _{{{{\bf}R}}_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = \int _{{{{\bf}R}}_+} \eta ^{\;\prime} (t) \int _{|v| = r} \varphi \left ( x, v\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\partial _t ( \eta \varphi )f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t}.\end{aligned}$$ Similarly, one gets $$\begin{aligned} T_2 ^k := { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\! v \cdot \nabla _x \theta { f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \to & { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!v \cdot \nabla _x \theta f (t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \\ & = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!v \cdot \nabla _x ( \eta \varphi ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t}.\nonumber\end{aligned}$$ For the term containing $a \cdot \nabla _v \theta$ notice that on the set $|v| \geq r_0$ we have $$a \cdot \nabla _v \theta = \eta (t) a \cdot \nabla _v \left \{ \varphi \left ( x, r{ \frac{v}{|v|}}\right )\right \} = \eta (t) \frac{r}{|v|}a \cdot { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }(\nabla _v \varphi ) \left ( x, r{ \frac{v}{|v|}}\right )$$ and therefore we obtain $$\begin{aligned} T_3 ^k := { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!& a \cdot \nabla _v \theta { f ^{\varepsilon _k}}(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \to { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!a \cdot \nabla _v \theta f (t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t} \nonumber \\ & = \int _{{{{\bf}R}}_+} \eta (t) \int _{|v|\geq r_0} \frac{r}{|v|} { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot (\nabla _v \varphi ) \left ( x, r{ \frac{v}{|v|}}\right ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t \\ & = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\;\;\;{ \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot \nabla _v (\eta \varphi ) f(t,x,v)\,{ \mathrm{d}(x,v)}\;\mathrm{d}t}.\nonumber\end{aligned}$$ For treating the term involving the initial condition, we write $$\begin{aligned} T_4 : = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \theta (0,x,v) { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}} &= { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \eta (0) \varphi \left ( x, r { \frac{v}{|v|}}\right ) { f ^{\mathrm{in}}}(x,v)\,{ \mathrm{d}(x,v)}} \\ &= { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \eta (0) \varphi (x,v) { \left \langle { f ^{\mathrm{in}}}\right \rangle }(x,v)\,{ \mathrm{d}(x,v)}}.\end{aligned}$$ Passing to the limit for $k \to +\infty$ in the weak formulation $T_1 ^ k + T_2 ^ k + T_3 ^ k + T_4 = 0$ yields the problem $$\partial _t f + { \mathrm{div}_x}\{f v \} + { \mathrm{div}_v}\left \{f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \right \} = 0,\;\;f(0) = { \left \langle { f ^{\mathrm{in}}}\right \rangle }$$ as desired. \[ConstraintPropagation\] The constraint is propagated by the evolution equation . This comes by the fact that the flow $(X,V)$ associated to the field $v \cdot \nabla _x + { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot \nabla _v$ leaves invariant ${{{\bf}R}}^d \times r{ {\bf}{S}}$. Indeed, if $(X,V)$ solves $$\frac{\mathrm{d}X}{\mathrm{d}s} = V(s),\;\;\frac{\mathrm{d}V}{\mathrm{d}s} = \left (I - \frac{V(s) \otimes V(s)}{|V(s)|^2} \right ) a(s, X(s))$$ $$X(s;0,x,v) = x,\;\;V(s;0,x,v) = v \neq 0$$ then $$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}s}|V(s)|^2 = \left (I - \frac{V(s) \otimes V(s)}{|V(s)|^2} \right ) a(s, X(s)) \cdot V(s) = 0$$ saying that $|V(s;0,x,v)| = |v|$ for any $(s,x) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d$. In particular, for any continuous function $\psi = \psi (x,v)$ with compact support in $^c ({{{\bf}R}}^d \times r{ {\bf}{S}})$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) f(s,x,v)\,{ \mathrm{d}(x,v)}} & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (X(s;0,x,v), V(s;0,x,v)) { \left \langle { f ^{\mathrm{in}}}\right \rangle }(x,v)\,{ \mathrm{d}(x,v)}} \\ & = \int _{|v| = r} \psi (X(s;0,x,v), V(s;0,x,v)) { \left \langle { f ^{\mathrm{in}}}\right \rangle }(x,v)\,{ \mathrm{d}(x,v)}= 0\end{aligned}$$ since ${ \mathrm{supp\;}}{ \left \langle { f ^{\mathrm{in}}}\right \rangle } \subset {{{\bf}R}}^d \times r{ {\bf}{S}}$. Therefore for any $s \in {{{\bf}R}}_+$ we have ${ \mathrm{supp\;}}f(s) \subset {{{\bf}R}}^d \times r{ {\bf}{S}}$ implying that ${ \mathrm{div}_v}\{f(s){ (\alpha - \beta |v|^2) v}\} = 0, s \in {{{\bf}R}}_+$. \[Uni\] By the uniqueness of the solution for with initial data ${ \left \langle { f ^{\mathrm{in}}}\right \rangle }$, we deduce that all the family $({ f ^\varepsilon})_{ \varepsilon}$ converges weakly $\star$ in ${ L^\infty ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}{}$. The non linear problem {#NLimMod} ====================== Up to now we considered the stability of the linear problems - for a given smooth field $a = a(t,x) \in { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$. We concentrate now on the non linear problem $$\label{Equ41} \partial _t { f ^\varepsilon}+ { \mathrm{div}_x}\{{ f ^\varepsilon}v\} + { \mathrm{div}_v}\{ { f ^\varepsilon}a^{ \varepsilon}\} + \frac{1}{{ \varepsilon}} \{ { f ^\varepsilon}{ (\alpha - \beta |v|^2) v}\}= 0,\;\;(t,x,v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d$$ with $a^{ \varepsilon}= - \nabla _x U \star \rho ^{ \varepsilon}- H \star { f ^\varepsilon}.$ The well posedness of the non linear equation comes by fixed point arguments in suitable spaces of measures, and it has been discussed in [@CCR10; @BCC12] in the measure solution framework. We summarize next the properties of the solutions $({ f ^\varepsilon})_{{ \varepsilon}>0}$. \[ExiUniNonLin\] Assume $h \in C^1_b ({{{\bf}R}}^d), U \in C^2 _b ({{{\bf}R}}^d)$ and $( 1 + |v|^2 ) { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. For all ${ \varepsilon}>0$, there is a unique solution $({ f ^\varepsilon}, a ^{ \varepsilon}) \in C({{{\bf}R}}_+;{ {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}) \times { L^\infty ( {{{\bf}R}}_+ ; W^{1,\infty}({{{\bf}R}}^d ))}{}$ to $$\label{Equ43} \partial _t { f ^\varepsilon}+ { \mathrm{div}_x}\{{ f ^\varepsilon}v \} + { \mathrm{div}_v}\{{ f ^\varepsilon}{ a ^\varepsilon}\} + \frac{1}{{ \varepsilon}}{ \mathrm{div}_v}\{ { f ^\varepsilon}{ (\alpha - \beta |v|^2) v}\} = 0,\;\;(t,x,v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times {{{\bf}R}}^d$$ $$\label{Equ44} { a ^\varepsilon}= - \nabla _x U \star \int _{{{{\bf}R}}^d} { f ^\varepsilon}\;{ \mathrm{d}}v - H \star { f ^\varepsilon},\;\;H(x,v) = h(x)v$$ with initial data ${ f ^\varepsilon}(0) = { f ^{\mathrm{in}}}$, satisfying the uniform bounds $$\sup _{{ \varepsilon}>0, t \in {{{\bf}R}}_+} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! |v|^2 { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}}<+\infty$$ $$\sup _{{ \varepsilon}>0} \|{ a ^\varepsilon}\|_{L^\infty({{{\bf}R}}_+;L^\infty({{{\bf}R}}^d))} = :A <+\infty,\;\;\sup _{{ \varepsilon}>0} \|\nabla _x { a ^\varepsilon}\|_{L^\infty({{{\bf}R}}_+;L^\infty({{{\bf}R}}^d))} = :A_1 <+\infty.$$ Moreover, if the initial condition satisfies $${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|x| \leq L_0, r_0 \leq |v| \leq R_0 \}$$ for some $L_0 >0, 0 < r_0 < r < R_0 < +\infty$, then for any ${ \varepsilon}>0$ small enough we have $${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|x|\leq L_0 + t R_0, r_0 \leq |v| \leq R_0 \},\;\;t \in {{{\bf}R}}_+.$$ Here, we only justify the uniform bounds in ${ \varepsilon}$, the rest is a direct application of the results in [@CCR10; @BCC12]. The divergence form of guarantees the mass conservation $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,{ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \,{ f ^{\mathrm{in}}}(x,v)\;{ \mathrm{d}(x,v)}},\;\;t \in {{{\bf}R}}_+.$$ Notice that the term $- { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}$ balances the momentum $$\int _{{{{\bf}R}}^{2d}}{\!\! v { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}\;{ \mathrm{d}(x,v)}} = \int _{{{{\bf}R}}^{4d}}{\!\! h(x-x^{\prime}) (v ^{\prime}-v) { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}= 0$$ and decreases the kinetic energy $$\begin{aligned} \int _{{{{\bf}R}}^{2d}}{\!\!\!|v|^2 { \mathrm{div}_v}\{ { f ^\varepsilon}H \star { f ^\varepsilon}\}\;{ \mathrm{d}(x,v)}} &= 2\int _{{{{\bf}R}}^{4d}}{{\!\!\!\!h(x-x^{\prime}) (v ^{\prime}-v) \cdot v { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}} \\ & = - \int _{{{{\bf}R}}^{4d}}{{\!\!h(x-x^{\prime}) |v - v ^{\prime}|^2 { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime)}{ \mathrm{d}(x,v)}}.\end{aligned}$$ In particular, as $|v|^2 { f ^{\mathrm{in}}}\in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, then the kinetic energy $\int _{{{{\bf}R}}^{2d}}{|v|^2 { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}}$ remains bounded, uniformly in time $t \in {{{\bf}R}}_+$ and ${ \varepsilon}>0$. Indeed, using the continuity equation one gets $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! v \cdot (\nabla _x U \star \rho ^{ \varepsilon}) { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}} = \frac{1}{2}\frac{{ \mathrm{d}}}{{ \mathrm{d}}t} \int _{{{{\bf}R}}^d}{(U \star \rho ^{ \varepsilon}(t))(x) \rho ^{ \varepsilon}(t,x)\;{ \mathrm{d}}x}$$ and after multiplying by $\frac{|v|^2}{2}$ together with , we obtain $$\begin{aligned} \frac{{ \mathrm{d}}}{{ \mathrm{d}}t} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! &\,\,\left (\frac{|v|^2}{2} +\frac{U \star \rho ^{ \varepsilon}}{2} \right ) { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}} - \frac{1}{{ \varepsilon}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! (\alpha |v|^2 - \beta |v|^4) { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}} \nonumber \\ & = - \frac{1}{2}\int_{{{{\bf}R}}^{4d }}h(x-x^{\prime}) |v - v ^{\prime}|^2 { f ^\varepsilon}(t, x^\prime, v^\prime){ f ^\varepsilon}(t,x,v)\;{ \mathrm{d}}(x^\prime, v^\prime){ \mathrm{d}(x,v)}\leq 0\,.\label{energy}\end{aligned}$$ Consider now $t ^{ \varepsilon}$ a maximum point on $[0,T], T>0$, of the total energy $$W^{ \varepsilon}(t) = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \left (\frac{|v|^2}{2} + \frac{U \star \rho ^{ \varepsilon}}{2} \right ) { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}},\;\;t \in [0,T].$$ If $t^{ \varepsilon}= 0$ then it is easily seen that for any $t \in [0,T]$ $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \frac{|v|^2}{2} { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}} \leq { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \frac{|v|^2}{2} { f ^{\mathrm{in}}}(x,v) \;{ \mathrm{d}(x,v)}} + \|U \| _{{ L^\infty}} \left ( { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { f ^{\mathrm{in}}}\;{ \mathrm{d}(x,v)}}\right ) ^2.$$ If $t ^{ \varepsilon}\in ]0,T]$ then $\frac{{ \mathrm{d}}}{{ \mathrm{d}}t} W ^{ \varepsilon}(t^{ \varepsilon}) \geq 0$ implying from by moment interpolation in $v$ that $$\sup _{{ \varepsilon}>0, T>0} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! (1 + |v|^4) { f ^\varepsilon}(t ^{ \varepsilon},x,v) \;{ \mathrm{d}(x,v)}} <+\infty$$ and thus the inequality $W^{ \varepsilon}(t) \leq W ^{ \varepsilon}(t ^{ \varepsilon}), t \in [0,T]$ yields $$\begin{aligned} \sup _{{ \varepsilon}>0, t \in [0,T]} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \frac{|v|^2}{2} { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}} \leq & \sup _{{ \varepsilon}>0, T>0} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \frac{|v|^2}{2} { f ^\varepsilon}(t ^{ \varepsilon},x,v) \;{ \mathrm{d}(x,v)}} \\ & + \|U \|_{{ L^\infty}} \left ( { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { f ^{\mathrm{in}}}\;{ \mathrm{d}(x,v)}}\right ) ^2 <+\infty.\end{aligned}$$ Therefore the kinetic energy remains bounded on $[0,T]$, uniformly with respect to ${ \varepsilon}>0$, and the bound does not depend on $T>0$. The uniform bounds for ${ a ^\varepsilon}$ come immediately by convolution with $\nabla _x U$ and $H$, thanks to the uniform estimate $$\sup _{{ \varepsilon}>0, t \in {{{\bf}R}}_+} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! |v| { f ^\varepsilon}(t,x,v)} < +\infty.$$ We analyze the support of $({ f ^\varepsilon})_{{ \varepsilon}>0}$. Take ${ \varepsilon}>0$ small enough such that ${ \varepsilon}A < 2 \alpha r /(3\sqrt{3})$ and ${ \rho ^\varepsilon _1}(-A) < r_0,\; { \rho ^\varepsilon _3}(A) < R_0$. By Proposition \[UnifSupp\] we already know that $${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|v | \geq r_0\},\;\;t \in {{{\bf}R}}_+.$$ For any continuous function $\psi = \psi (x,v)$ with compact support in ${{{\bf}R}}^d \times \{ v\in {{{\bf}R}}^d\;:\;|v| > R_0\}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { f ^\varepsilon}(t,x,v) \;{ \mathrm{d}(x,v)}} & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t), { V ^\varepsilon }(t)) { f ^{\mathrm{in}}}(x,v) \;{ \mathrm{d}(x,v)}} \\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi ({ X ^\varepsilon }(t), { V ^\varepsilon }(t) ) { {\bf 1}_{\{r_0 \leq |v| \leq R_0\}}}{ f ^{\mathrm{in}}}(x,v) \;{ \mathrm{d}(x,v)}}.\end{aligned}$$ We distinguish several cases:\ 1. If $r_0 \leq |v| < { \rho ^\varepsilon _2}(-A)$ we deduce by Proposition \[ZeroStab\] that $ |v| < |{ V ^\varepsilon }(t;0,x,v)| \leq { \rho ^\varepsilon _3}(A) < R_0,\;\;t \in {{{\bf}R}}_+, { \varepsilon}>0. $ 2\. If ${ \rho ^\varepsilon _2}(-A) \leq |v| \leq { \rho ^\varepsilon _3}(A)$ we obtain by Proposition \[RStab\] that $ { \rho ^\varepsilon _2}(-A) \leq |{ V ^\varepsilon }(t;0,x,v)| \leq { \rho ^\varepsilon _3}(A) < R_0,\;\;t \in {{{\bf}R}}_+, { \varepsilon}>0. $ 3\. If ${ \rho ^\varepsilon _3}(A) < |v| \leq R_0$ one gets thanks to Proposition \[ZeroStab\] $ { \rho ^\varepsilon _2}(-A) \leq |{ V ^\varepsilon }(t;0,x,v) | < |v| \leq R_0. $ In all cases $({ X ^\varepsilon }, { V ^\varepsilon })(t;0,x,v)$ remains outside the support of $\psi$, implying that $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}} = 0.$$ Thus for any $t \in {{{\bf}R}}_+$ and ${ \varepsilon}>0$ small enough one gets $${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;r_0 \leq |v | \leq R_0\}.$$ Consider $\theta \in C^1 ({{{\bf}R}})$ non decreasing, verifying $\theta (u) = 0$ if $u \leq 0$, $\theta (u) >0$ if $u>0$. Applying the weak formulation of - with the test function $\theta (|x| - L_0 - t R_0)$ yields $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \theta (|x| - L_0 -& t R_0) { f ^\varepsilon}(t,x,v)\;{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \theta (|x| - L_0 ){ f ^{\mathrm{in}}}(x,v)\;{ \mathrm{d}(x,v)}} \\ & + \int _0 ^t { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \theta ^{\prime}(|x| - L_0 - s R_0) \left ( v \cdot \frac{x}{|x|} - R_0\right ){ f ^\varepsilon}(s,x,v)\;{ \mathrm{d}(x,v)}} { \mathrm{d}}s \leq 0\end{aligned}$$ implying that ${ \mathrm{supp\;}}{ f ^\varepsilon}(t) \subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d :|x| \leq L_0 + t R_0\}, t \in {{{\bf}R}}_+$. The uniform bound for the total mass allows us to extract a sequence $({ \varepsilon}_k)_k \subset {{{\bf}R}}_+ ^\star$ convergent to $0$ such that $({ f ^{\varepsilon _k}})_k$ converges weakly $\star$ in ${ L^\infty ({{{\bf}R}}_+ ; {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d))}{}$. The treatment of the non linear term requires a little bit more, that is convergence in $C({{{\bf}R}}_+;{ {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{})$ or at least in $C([\delta, +\infty[;{ {\cal P}_1 ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{})$ for any $\delta >0$. The key argument for establishing that is emphasized by the lemma \[TimeEstimate\] Consider ${ \varepsilon}>0$ small enough.\ 1. For any $(x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d$ with $r_0 \leq |v| < { \rho ^\varepsilon _2}(-A) - { \varepsilon}$, the first time $t^{ \varepsilon}_1 = t ^{ \varepsilon}_1 (x,v)$ such that $|{ V ^\varepsilon }(t^{ \varepsilon}_1;0,x,v) | = { \rho ^\varepsilon _2}(-A) - { \varepsilon}$ satisfies $$t ^{ \varepsilon}_1 \leq \frac{{ \varepsilon}}{2\beta r_0 ^2} \ln \left ( \frac{r - r_0 }{{ \varepsilon}} \right ).$$ 2. For any $(x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d$ with ${ \rho ^\varepsilon _3}(A) + { \varepsilon}< |v| \leq R_0$, the first time $t^{ \varepsilon}_2 = t ^{ \varepsilon}_2 (x,v)$ such that $|{ V ^\varepsilon }(t^{ \varepsilon}_2;0,x,v) | = { \rho ^\varepsilon _3}(A) + { \varepsilon}$ satisfies $$t ^{ \varepsilon}_2 \leq \frac{{ \varepsilon}}{2\beta r ^2} \ln \left ( \frac{R_0 - r}{{ \varepsilon}} \right ).$$ 1\. During the time $[0,t ^{ \varepsilon}_1]$ the velocity modulus $|{ V ^\varepsilon }(t)|$ remains in $[r_0, { \rho ^\varepsilon _2}(-A) - { \varepsilon}] \subset [{ \rho ^\varepsilon _1}(-A), { \rho ^\varepsilon _2}(-A)]$ and we can write for any $t \in [0, t ^{ \varepsilon}_1]$ $$\frac{{ \varepsilon}\frac{{ \mathrm{d}}|{ V ^\varepsilon }|}{{ \mathrm{d}}t }}{- { \varepsilon}A + (\alpha - \beta |{ V ^\varepsilon }(t) |^2 ) \;|{ V ^\varepsilon }(t) |}\geq \frac{\frac{{ \mathrm{d}}|{ V ^\varepsilon }|}{{ \mathrm{d}}t }}{{ a ^\varepsilon}(t, { X ^\varepsilon }(t)) \cdot \frac{{ V ^\varepsilon }(t)}{|{ V ^\varepsilon }(t)|} + \frac{1}{{ \varepsilon}}(\alpha - \beta |{ V ^\varepsilon }(t) |^2 ) \;|{ V ^\varepsilon }(t) |} = 1$$ since $- { \varepsilon}A + (\alpha - \beta u^2)u$ is positive for $u \in [{ \rho ^\varepsilon _1}(-A), { \rho ^\varepsilon _2}(-A)]$. Integrating with respect to $t \in [0, t ^{ \varepsilon}_1]$ yields $$t ^{ \varepsilon}_1 (x,v) \leq { \varepsilon}\int _{|v|} ^{{ \rho ^\varepsilon _2}(-A) - { \varepsilon}} \frac{{ \mathrm{d}}u }{- { \varepsilon}A + (\alpha - \beta u ^2 ) u } \leq { \varepsilon}\int _{r_0} ^{{ \rho ^\varepsilon _2}(-A) - { \varepsilon}} \frac{{ \mathrm{d}}u }{- { \varepsilon}A + (\alpha - \beta u ^2 ) u }.$$ Recall that ${ \rho ^\varepsilon _2}(-A)$ is one of the roots of $u \to - { \varepsilon}A + (\alpha - \beta u ^2 ) u$ and therefore a direct computation lead to $$- { \varepsilon}A + (\alpha - \beta u ^2 ) u = \beta ({ \rho ^\varepsilon _2}- u ) [u ^2 + u { \rho ^\varepsilon _2}+ ({ \rho ^\varepsilon _2}) ^2 - r^2] \geq 2 \beta r_0 ^2 ( { \rho ^\varepsilon _2}- u), \;\;u \in [r_0, { \rho ^\varepsilon _2}],\;{ \varepsilon}\;\mbox{small enough }$$ implying that $$t ^{ \varepsilon}_1 (x,v) \leq \frac{{ \varepsilon}}{2\beta r_0 ^2} \int _{r_0} ^{{ \rho ^\varepsilon _2}- { \varepsilon}} \frac{{ \mathrm{d}}u }{ { \rho ^\varepsilon _2}- u} = \frac{{ \varepsilon}}{2\beta r_0 ^2}\ln \left (\frac{{ \rho ^\varepsilon _2}- r_0}{{ \varepsilon}} \right ) \leq \frac{{ \varepsilon}}{2\beta r_0 ^2}\ln \left (\frac{r - r_0}{{ \varepsilon}} \right ).$$ 2. During the time $[0,t ^{ \varepsilon}_2]$ the velocity modulus $|{ V ^\varepsilon }(t)|$ remains in $[{ \rho ^\varepsilon _3}(A) + { \varepsilon}, R_0] \subset [{ \rho ^\varepsilon _3}(A), +\infty[$ and we can write for any $t \in [0, t ^{ \varepsilon}_2]$ $$\frac{{ \varepsilon}\frac{{ \mathrm{d}}|{ V ^\varepsilon }|}{{ \mathrm{d}}t }}{ { \varepsilon}A + (\alpha - \beta |{ V ^\varepsilon }(t) |^2 ) \;|{ V ^\varepsilon }(t) |}\geq \frac{\frac{{ \mathrm{d}}|{ V ^\varepsilon }|}{{ \mathrm{d}}t }}{{ a ^\varepsilon}(t, { X ^\varepsilon }(t)) \cdot \frac{{ V ^\varepsilon }(t)}{|{ V ^\varepsilon }(t)|} + \frac{1}{{ \varepsilon}}(\alpha - \beta |{ V ^\varepsilon }(t) |^2 ) \;|{ V ^\varepsilon }(t) |} = 1$$ since $ { \varepsilon}A + (\alpha - \beta u^2)u$ is negative for $u \in [{ \rho ^\varepsilon _3}(A), +\infty[$. Integrating with respect to $t \in [0, t ^{ \varepsilon}_2]$ yields $$t ^{ \varepsilon}_2 (x,v) \leq { \varepsilon}\int _{|v|} ^{{ \rho ^\varepsilon _3}(A) + { \varepsilon}} \frac{{ \mathrm{d}}u }{ { \varepsilon}A + (\alpha - \beta u ^2 ) u } \leq { \varepsilon}\int _{R_0} ^{{ \rho ^\varepsilon _3}(A) + { \varepsilon}} \frac{{ \mathrm{d}}u }{ { \varepsilon}A + (\alpha - \beta u ^2 ) u }.$$ By direct computation we obtain $${ \varepsilon}A + (\alpha - \beta u ^2 ) u = - \beta (u - { \rho ^\varepsilon _3}) [u ^2 + u { \rho ^\varepsilon _3}+ ({ \rho ^\varepsilon _3}) ^2 - r^2] \leq -2 \beta r ^2 ( u - { \rho ^\varepsilon _3}), \;\;u \geq { \rho ^\varepsilon _3},\;{ \varepsilon}\;\mbox{small enough }$$ implying that $$t ^{ \varepsilon}_2 (x,v) \leq \frac{{ \varepsilon}}{2\beta r ^2} \int _{{ \rho ^\varepsilon _3}+ { \varepsilon}} ^{R_0} \frac{{ \mathrm{d}}u }{ u - { \rho ^\varepsilon _3}} = \frac{{ \varepsilon}}{2\beta r ^2}\ln \left (\frac{R _0 - { \rho ^\varepsilon _3}}{{ \varepsilon}} \right ) \leq \frac{{ \varepsilon}}{2\beta r ^2}\ln \left (\frac{R_0 - r}{{ \varepsilon}} \right ).$$ We intend to apply Arzela-Ascoli theorem in $C({{{\bf}R}}_+;{ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d))$ in order to extract a convergent sequence $({ f ^{\varepsilon _k}})_k$ with ${ \lim _{k \to +\infty }}{ \varepsilon}_k = 0$. We need to establish the uniform equicontinuity of the family $({ f ^\varepsilon})_{{ \varepsilon}>0}$. The argument below is essentially similar to arguments in [@CCR10]. \[UnifEquiCont\] 1. If the initial data is well prepared [*i.e.,*]{} ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d\;:\;|x|\leq L_0, |v| = r\}$ then there is a constant $C$ (not depending on $t \in {{{\bf}R}}_+, { \varepsilon}>0$) such that $$W_1 ({ f ^\varepsilon}(t), { f ^\varepsilon}(s)) \leq C | t - s|,\;\;t, s \in {{{\bf}R}}_+, { \varepsilon}>0.$$ 2. If ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d\;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$ then there is a constant $C$ (not depending on $t \in {{{\bf}R}}_+, { \varepsilon}>0$) such that for any $\delta >0$ we can find ${ \varepsilon}_\delta$ satisfying $$W_1 ({ f ^\varepsilon}(t), { f ^\varepsilon}(s)) \leq C |t - s|,\;\; t,s \geq \delta,\;\;0 < { \varepsilon}< { \varepsilon}_\delta.$$ 1\. Consider $\varphi = \varphi (x,v)$ a Lipschitz function on ${{{\bf}R}}^d \times {{{\bf}R}}^d$ with $\lip (\varphi ) \leq 1$. For any $t, s \in {{{\bf}R}}_+, { \varepsilon}>0$ we have $$\begin{aligned} \left | { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \!\!\!\varphi ( { f ^\varepsilon}(t) - { f ^\varepsilon}(s)){ \mathrm{d}(x,v)}}\right | &= \left | { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \!\!\!\{ \varphi ({ X ^\varepsilon }(t), { V ^\varepsilon }(t)) - \varphi ({ X ^\varepsilon }(s), { V ^\varepsilon }(s))\}{ f ^{\mathrm{in}}}(x,v){ \mathrm{d}(x,v)}}\right | \\ & \leq { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \{ |{ X ^\varepsilon }(t) - { X ^\varepsilon }(s)| + |{ V ^\varepsilon }(t) - { V ^\varepsilon }(s)|\} { {\bf 1}_{\{|v| = r\}}} { f ^{\mathrm{in}}}{ \mathrm{d}(x,v)}}.\end{aligned}$$ Thanks to Proposition \[RStab\] we have for any $(\tau, x, v) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d \times r { {\bf}{S}}$ $$\frac{{ \rho ^\varepsilon _2}(-A) - r}{{ \varepsilon}} \leq \frac{|{ V ^\varepsilon }(\tau;0,x,v)| - r}{{ \varepsilon}} \leq \frac{{ \rho ^\varepsilon _3}(A) - r}{{ \varepsilon}}$$ and it is easily seen, integrating the system of characteristics between $s$ and $t$, that $$|{ X ^\varepsilon }(t;0,x,v) - { X ^\varepsilon }(s;0,x,v) | = \left | \int _s ^ t { V ^\varepsilon }(\tau;0,x,v) \;\mathrm{d}\tau\right | \leq R_0 |t-s|$$ and $$\begin{aligned} \left | { V ^\varepsilon }(t;0,x,v) - { V ^\varepsilon }(s;0,x,v) \right | & \leq \left | \int _s ^t \left \{ |a^{ \varepsilon}(\tau, { X ^\varepsilon }(\tau))| + \frac{|\alpha - \beta |{ V ^\varepsilon }(\tau) | ^2 | \;|{ V ^\varepsilon }(\tau) |}{{ \varepsilon}} \right \}{ \mathrm{d}}\tau \right |\nonumber \\ & \leq |t -s | \left \{ A + \beta ( r + R_0) R_0 \max \left ( \frac{{ \rho ^\varepsilon _3}(A) - r}{{ \varepsilon}}, \frac{r - { \rho ^\varepsilon _2}(-A) }{{ \varepsilon}} \right ) \right \}.\end{aligned}$$ Our conclusion comes immediately by Propositions \[NegA\], \[PosA\].\ 2. Consider $\delta >0$ and ${ \varepsilon}_\delta $ small enough such that $\frac{{ \varepsilon}}{2\beta r_0 ^2} \ln \left ( \frac{r - r_0}{{ \varepsilon}} \right ) < \delta$, $\frac{{ \varepsilon}}{2\beta r ^2} \ln \left ( \frac{R_0 - r}{{ \varepsilon}} \right ) < \delta$ for $0 < { \varepsilon}< { \varepsilon}_\delta$. For any Lipschitz function $\varphi $ with $\lip (\varphi ) \leq 1$ and any $t, s \geq \delta$ we have $$\left | { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \!\!\!\!\varphi ( { f ^\varepsilon}(t) - { f ^\varepsilon}(s) ) \;{ \mathrm{d}(x,v)}}\right | \leq \!\!{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \{|{ X ^\varepsilon }(t) - { X ^\varepsilon }(s) | + |{ V ^\varepsilon }(t) - { V ^\varepsilon }(s)| \} { {\bf 1}_{\{r_0 \leq |v| \leq R_0\}}} { f ^{\mathrm{in}}}\;{ \mathrm{d}(x,v)}}.$$ For any $(\tau, x) \in {{{\bf}R}}_+ \times {{{\bf}R}}^d$, ${ \rho ^\varepsilon _2}(-A) - { \varepsilon}\leq |v| \leq { \rho ^\varepsilon _3}(A) + { \varepsilon}$ we have by Propositions \[RStab\], \[ZeroStab\] $${ \rho ^\varepsilon _2}(-A) - { \varepsilon}\leq |{ V ^\varepsilon }(\tau;0,x,v) | \leq { \rho ^\varepsilon _3}(A) + { \varepsilon}.$$ The same conclusion holds true for any $\tau \geq \delta$, $x \in {{{\bf}R}}^d$ and $|v| \in [r_0, { \rho ^\varepsilon _2}(-A) - { \varepsilon}[ \cup ]{ \rho ^\varepsilon _3}(A) + { \varepsilon}, R_0]$, thanks to Lemma \[TimeEstimate\], since $\delta > \max \{ t^{ \varepsilon}_1 (x,v), t^{ \varepsilon}_2 (x,v)\}$ (after a time $\delta$, the velocity modulus $|{ V ^\varepsilon }(\tau;0,x,v)|$ is already in the set $\{w\;:\;{ \rho ^\varepsilon _2}(-A) - { \varepsilon}< |w| < { \rho ^\varepsilon _3}(A) + { \varepsilon}\}$). Our statement follows as before, integrating the system of characteristics between $s$ and $t$. Applying Arzela-Ascoli theorem, we deduce that there is a sequence $({ \varepsilon}_k)_k \subset {{{\bf}R}}_+ ^\star$, convergent to $0$ such that $${ \lim _{k \to +\infty }}W_1 ({ f ^{\varepsilon _k}}(t), f(t)) = 0 \mbox{ uniformly for } t \in [0,T],\;\; T>0$$ for some $f \in C({{{\bf}R}}_+;{ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d))$ if ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{(x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d\;:\;|x| \leq L_0,|v| = r\}$ and $${ \lim _{k \to +\infty }}W_1 ({ f ^{\varepsilon _k}}(t), f(t)) = 0 \mbox{ uniformly for } t \in [\delta ,T],\;\; T>\delta >0$$ for some $f \in C({{{\bf}R}}_+ ^\star;{ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d))$ if ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{(x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d\;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$. It is easily seen that if the initial condition is well prepared then there is a constant $C$ cf. Proposition \[UnifEquiCont\] such that $ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \in {{{\bf}R}}_+. $ The same is true for not prepared initial conditions ${ f ^{\mathrm{in}}}$. Take $\delta >0$ and ${ \varepsilon}_\delta$ as in Proposition \[UnifEquiCont\]. For any $0 < { \varepsilon}< { \varepsilon}_\delta$ we have $ W_1 ({ f ^\varepsilon}(t), { f ^\varepsilon}(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ For $k$ large enough we have ${ \varepsilon}_k < { \varepsilon}_\delta$ and therefore $ W_1 ({ f ^{\varepsilon _k}}(t), { f ^{\varepsilon _k}}(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ Passing to the limit as $k$ goes to infinity yields $ W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \geq \delta. $ Since the constant $C$ does not depend on $\delta$ one gets $$W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s >0.$$ In particular we deduce that $f$ has a limit as $t$ goes to $0$ since $( { { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d), W_1)$ is a complete metric space and therefore we can extend $f$ by continuity at $t = 0$. The extended function, still denoted by $f$, belongs to $C({{{\bf}R}}_+;{ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d))$ and satisfies $$W_1 (f(t), f(s)) \leq C |t -s |,\;\;t, s \in {{{\bf}R}}_+.$$ The above convergence allows us to handle the non linear terms. We use the following standard argument [@Dob79; @CCR10]. \[NonLinTerm\] Consider $f,g \in { { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d)$ compactly supported $ { \mathrm{supp\;}}f \cup { \mathrm{supp\;}}g \subset \{ (x,v) \in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|x|\leq L, |v| \leq R\}$, and let us consider $$a_f = - \nabla _x U \star \int _{{{{\bf}R}}^d} f \;{ \mathrm{d}}v - H \star f,\;\;a_g = - \nabla _x U \star \int _{{{{\bf}R}}^d} g \;{ \mathrm{d}}v - H \star g.$$ Then we have $$\|a_f - a_g \|_{L^\infty ( {{{\bf}R}}^3 \times B_R)} \leq \left \{\|\nabla _x ^2 U \|_{{ L^\infty}} + \left (\|h \|^2 _{{ L^\infty}} + 4 R ^2 \|\nabla _x h \| ^2 _{{ L^\infty}} \right ) ^{1/2} \right \}W_1 (f,g)$$ where $B_R$ stands for the closed ball in ${{{\bf}R}}^d$ of center $0$ and radius $R$. Take $\pi $ to be a optimal transportation plan between $f$ and $g$. Then for any $x \in {{{\bf}R}}^d$ we have, using the marginals of $\pi$ $$\begin{aligned} | (\nabla _x U \star f) (x) - (\nabla _x U \star g) (x) | &= \left | { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \nabla _x U ( x - x^\prime) \{ f({ x^{\prime}}, { v^{\prime}}) - g({ x^{\prime}}, { v^{\prime}})\}\;{ \mathrm{d}(x^\prime,v^\prime)}} \right | \\ & = \left | { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! [\nabla _x U (x - { x^{\prime}}) - \nabla _x U (x - { x^{\prime \prime}})]{ \mathrm{d}}\pi ({ x^{\prime}}, { v^{\prime}}, { x^{\prime \prime}}, { v^{\prime \prime}})}} \right | \\ & \leq \|\nabla _x ^2 U \|_{{ L^\infty}{}} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! |{ x^{\prime}}- { x^{\prime \prime}}| \;{ \mathrm{d}}\pi ({ x^{\prime}}, { v^{\prime}}, { x^{\prime \prime}}, { v^{\prime \prime}})}} \\ & \leq \|\nabla _x ^2 U \|_{{ L^\infty}{}} W_1 (f,g).\end{aligned}$$ The estimate for $H \star f - H \star g$ follows similarly observing that on the support of $\pi$, which is included in $\{({ x^{\prime}}, { v^{\prime}}, { x^{\prime \prime}}, { v^{\prime \prime}})\in {{{\bf}R}}^{4d}\;:\; |{ v^{\prime}}|\leq R, |{ v^{\prime \prime}}| \leq R\}$ we have $$\begin{aligned} |h(x- { x^{\prime}}) (v- { v^{\prime}}) - h(x- { x^{\prime \prime}}) & (v- { v^{\prime \prime}})| \\ & \leq |h(x- { x^{\prime}})( { v^{\prime \prime}}- { v^{\prime}})| + |h(x- { x^{\prime}})- h(x- { x^{\prime \prime}})| \;|v - { v^{\prime \prime}}| \\ & \leq \left ( \|h \|^2 _{{ L^\infty}} + 4 R ^2 \|\nabla _x h \| ^2 _{{ L^\infty}} \right ) ^{1/2} \left ( |{ x^{\prime}}- { x^{\prime \prime}}| ^2 + |{ v^{\prime}}- { v^{\prime \prime}}|^2 \right ) ^{1/2}.\end{aligned}$$ We are ready now to prove Theorem \[MainResult2\]. (of Theorem \[MainResult2\]) The arguments are the same as those in the proof of Theorem \[MainResult\] except for the treatment of the non linear terms. We only concentrate on it. Consider $({ f ^{\varepsilon _k}})_k$ with ${ \lim _{k \to +\infty }}{ \varepsilon}_k = 0$ such that ${ \lim _{k \to +\infty }}W_1 ({ f ^{\varepsilon _k}}(t), f(t)) = 0$ uniformly for $t \in [0,T], T>0$ if ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \;:\;|x|\leq L_0, |v| = r\}$ and ${ \lim _{k \to +\infty }}W_1 ({ f ^{\varepsilon _k}}(t), f(t)) = 0$ uniformly for $t \in [\delta,T], T>\delta >0$ if ${ \mathrm{supp\;}}{ f ^{\mathrm{in}}}\subset \{ (x,v) \;:\;|x|\leq L_0, r_0 \leq |v| \leq R_0\}$ for some function $f \in C({{{\bf}R}}_+;{ { {\cal P}}_1}({{{\bf}R}}^d \times {{{\bf}R}}^d))$. Thanks to Proposition \[Kernel\] we deduce (for both prepared or not initial data) that $${ \mathrm{supp\;}}f(t) \subset \{ (x,v)\in {{{\bf}R}}^d \times {{{\bf}R}}^d \;:\;|v| = r\},\;\;t>0.$$ The previous statement holds also true at $t = 0$, by the continuity of $f$. The time evolution for the limit $f$ comes by using the particular test functions $$\theta (t,x,v) = \eta (t) \left [ 1 - \chi \left ( \frac{2|v|}{r_0}\right ) \right ] \varphi \left ( x, r{ \frac{v}{|v|}}\right )$$ with $\eta \in C^1_c ({{{\bf}R}}_+)$, $\varphi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. From now on we consider only the not prepared initial data case (the other case is simpler). We recall the notation ${ a ^\varepsilon}= - \nabla _x U \star \int _{{{{\bf}R}}^d} { f ^\varepsilon}\;{ \mathrm{d}}v - H \star { f ^\varepsilon}$ and we introduce $a = - \nabla _x U \star \int _{{{{\bf}R}}^d} f \;{ \mathrm{d}}v - H \star f$. Since $f$ satisfies the same bounds as $({ f ^\varepsilon})_{ \varepsilon}$, we deduce that $\|a\|_{{ L^\infty}{}} \leq A, \|\nabla _x a\|_{{ L^\infty}{}} \leq A_1$. For any $\delta >0$ we can write $$\begin{aligned} \label{EquBil} &\left |{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\left \{ { a ^{\varepsilon _k}}\cdot \nabla _v \theta \;{ f ^{\varepsilon _k}}- a \cdot \nabla _v \theta \;f\right \}{ \mathrm{d}(x,v)}\!\!\;\mathrm{d}t} \right | \leq \left |\int _0 ^\delta { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { a ^{\varepsilon _k}}\cdot \nabla _v \theta { f ^{\varepsilon _k}}\;{ \mathrm{d}(x,v)}{ \mathrm{d}}t } \right | \nonumber \\ &\qquad + \left |\int _0 ^\delta { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \nabla _v \theta \;f \;{ \mathrm{d}(x,v)}{ \mathrm{d}}t} \right | + \left | \int _\delta ^{+\infty} \!\!{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \left \{ { a ^{\varepsilon _k}}\cdot \nabla _v \theta \;{ f ^{\varepsilon _k}}- a \cdot \nabla _v \theta \;f \right \}\;{ \mathrm{d}(x,v)}{ \mathrm{d}}t} \right | \nonumber \\ \leq &\, 2 A \delta \|\nabla _v \theta \|_{C^0} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { f ^{\mathrm{in}}}\;{ \mathrm{d}(x,v)}} + \left | \int _\delta ^{+\infty}\!\! { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! ({ a ^{\varepsilon _k}}- a) \cdot \nabla _v \theta \;{ {\bf 1}_{\{|v|\leq R_0\}}}{ f ^{\varepsilon _k}}\;{ \mathrm{d}(x,v)}}{ \mathrm{d}}t \right | \nonumber \\ & + \left | \int _\delta ^{+\infty}\!\! { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \nabla _v \theta \;({ f ^{\varepsilon _k}}- f)\;{ \mathrm{d}(x,v)}}{ \mathrm{d}}t \right |.\end{aligned}$$ We keep $\delta >0$ fixed and we pass to the limit when $k$ goes to infinity. Lemma \[NonLinTerm\] implies that the second term in the last right hand side can be estimated as $$\|{ a ^{\varepsilon _k}}- a \|_{{ L^\infty}({{{\bf}R}}^d \times B_{R_0})} = \| a_{{ f ^{\varepsilon _k}}} - a_f \|_{{ L^\infty}({{{\bf}R}}^d \times B_{R_0})} \leq C(R_0) W_1 ({ f ^{\varepsilon _k}}(t), f(t)) \to 0 \;\mbox{ when } k \to +\infty$$ uniformly for $t \in [\delta, T]$, implying, for $T$ large enough $$\left | \int _\delta ^{+\infty}\!\! \int _{|v| \leq R_0}{ ({ a ^{\varepsilon _k}}- a) \cdot \nabla _v \theta { f ^{\varepsilon _k}}\;{ \mathrm{d}(x,v)}}{ \mathrm{d}}t \right | \leq C(R_0)\| \theta \|_{C^1} \int _\delta ^T W_1 ({ f ^{\varepsilon _k}}(t), f(t))\;{ \mathrm{d}}t \to 0$$ when $k$ goes to infinity. For the third term in the right hand side of we use the weak $\star$ convergence ${ \lim _{k \to +\infty }}{ f ^{\varepsilon _k}}(t) = f(t)$ in ${ {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ for any $t\geq \delta$, cf. Proposition \[w2properties\] $${ \lim _{k \to +\infty }}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \nabla _v \theta ({ f ^{\varepsilon _k}}(t)- f(t)) \;{ \mathrm{d}(x,v)}} = 0,\;\;t\geq \delta$$ and we conclude by the Lebesgue dominated convergence theorem $${ \lim _{k \to +\infty }}\int _\delta ^{+\infty} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \nabla _v \theta ({ f ^{\varepsilon _k}}(t,x,v) - f(t,x,v) )\;{ \mathrm{d}(x,v)}}{ \mathrm{d}}t = 0\,.$$ Passing to the limit in when $k$ goes to infinity, we obtain $$\limsup _{k \to +\infty} \left |{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\!\left \{ { a ^{\varepsilon _k}}\cdot \nabla _v \theta { f ^{\varepsilon _k}}- a \cdot \nabla _v \theta f\right \}\;{ \mathrm{d}(x,v)}\!\!\;\mathrm{d}t} \right | \leq 2 A \delta \|\nabla _v \theta \|_{C^0} \,.$$ Sending $\delta$ to $0$ we obtain that $${ \lim _{k \to +\infty }}{ \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\! { a ^{\varepsilon _k}}\cdot \nabla _v \theta \; { f ^{\varepsilon _k}}\;{ \mathrm{d}(x,v)}\!\!\;\mathrm{d}t} = { \int _{{{{\bf}R}}_+} \!\int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\!\!\!\! a \cdot \nabla _v \theta \; f\;{ \mathrm{d}(x,v)}\!\!\;\mathrm{d}t}\,.$$ Diffusion models {#DiffMod} ================ We intend to introduce a formalism which will allow us to investigate in a simpler manner the asymptotic behavior of and . This method comes from gyrokinetic models in plasma physics: when studying the magnetic confinement we are looking for averaged models with respect to the fast motion of particles around the magnetic lines. The analysis relies on the notion of gyro-average operator [@BosTraEquSin], which is a projection onto the space of slow time depending functions. In other words, projecting means smoothing out the fluctuations with respect to the fast time variable, corresponding to the high cyclotronic frequency. This projection appears like a gyro-average operator. Here the arguments are developed at a formal level. We first introduce rigorously the projected measure on the sphere $r{ {\bf}{S}}$ for general measures. Let $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ be a non negative bounded measure on ${{{\bf}R}}^d \times {{{\bf}R}}^d$. We denote by ${ \left \langle f \right \rangle }$ the measure corresponding to the linear application $$\psi \to { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v)\,{ {\bf 1}_{v = 0}} f(x,v)\,{ \mathrm{d}(x,v)}} + { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi\left ( x , r { \frac{v}{|v|}}\right ) { {\bf 1}_{v \neq 0}} f(x,v)\,{ \mathrm{d}(x,v)}}\,,$$ for all $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}$, [*i.e.,*]{} $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi(x,v) { \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}} = \int _{v = 0} \psi(x,v) f(x,v)\,{ \mathrm{d}(x,v)}+ \int _{v \neq 0} \psi \left ( x , r { \frac{v}{|v|}}\right ) f(x,v)\,{ \mathrm{d}(x,v)}\,,$$ for all $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}$. Observe that ${ \left \langle f \right \rangle }$ is a non negative bounded measure, $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \;\;{ \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \;\;f(x,v)\,{ \mathrm{d}(x,v)}},$$ with ${ \mathrm{supp\;}}{ \left \langle f \right \rangle } \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$. We have the following characterization. \[VarChar\] Assume that $f$ is a non negative bounded measure on ${{{\bf}R}}^d \times {{{\bf}R}}^d$. Then ${ \left \langle f \right \rangle }$ is the unique measure $F$ satisfying ${ \mathrm{supp\;}}F \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$, $$\int _{v\neq 0} \psi \left ( x , r { \frac{v}{|v|}}\right )F(x,v)\,{ \mathrm{d}(x,v)}= \int _{v \neq 0}\psi \left ( x , r { \frac{v}{|v|}}\right )f(x,v)\,{ \mathrm{d}(x,v)},\;\;\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$$ and $F = f$ on ${{{\bf}R}}^d \times \{0\}$. The measure ${ \left \langle f \right \rangle }$ defined before satisfies the above characterization. Indeed, ${ \mathrm{supp\;}}{ \left \langle f \right \rangle } \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$. Taking now $\psi (x,v) = \varphi (x) \chi (|v|/\delta)$ with $\varphi \in C^0 _c ({{{\bf}R}}^d)$ and $\delta >0$ one gets $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) { \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}} = &\,\int _{v = 0} \varphi (x) f(x,v)\,{ \mathrm{d}(x,v)}\\ &+ \int _{v \neq 0} \varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) f(x,v)\,{ \mathrm{d}(x,v)}.\end{aligned}$$ Passing to the limit for $\delta \searrow 0$ yields $$\int _{v = 0} \varphi (x) { \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}= \int _{v = 0} \varphi (x) f(x,v)\,{ \mathrm{d}(x,v)},\;\;\varphi \in { C^0_c ({{{\bf}R}}^d)}{}$$ meaning that ${ \left \langle f \right \rangle } = f$ on ${{{\bf}R}}^d \times \{0\}$. Therefore one gets for any $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ $$\begin{aligned} \int_{v \neq 0} \psi \left ( x , r { \frac{v}{|v|}}\right ){ \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}& = \int_{|v| = r} \psi ( x , v ){ \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}\\ & = \int _{v \neq 0} \psi (x,v) { \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}\\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \left \langle f \right \rangle }}(x,v)\,{ \mathrm{d}(x,v)}- \int _{v = 0}\psi { \left \langle f \right \rangle }(x,v)\,{ \mathrm{d}(x,v)}\\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \left \langle f \right \rangle }}(x,v)\,{ \mathrm{d}(x,v)}- \int _{v = 0}\psi f(x,v)\,{ \mathrm{d}(x,v)}\\ & = \int _{v \neq 0} \psi \left ( x , r { \frac{v}{|v|}}\right ) f(x,v)\,{ \mathrm{d}(x,v)}.\end{aligned}$$ Conversely, let us check that the above characterization exactly defines the measure ${ \left \langle f \right \rangle }$. For any $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) F(x,v)\,{ \mathrm{d}(x,v)}} & = \int _{v = 0} \psi F(x,v)\,{ \mathrm{d}(x,v)}+ \int _{v \neq 0} \psi F(x,v)\,{ \mathrm{d}(x,v)}\\ & = \int _{v = 0} \psi (x,v) f(x,v)\,{ \mathrm{d}(x,v)}+ \int _{v \neq 0} \psi \left ( x, r { \frac{v}{|v|}}\right ) F(x,v)\,{ \mathrm{d}(x,v)}\\ & = \int _{v = 0} \psi (x,v) f(x,v)\,{ \mathrm{d}(x,v)}+ \int _{v \neq 0} \psi \left ( x, r { \frac{v}{|v|}}\right )f(x,v)\,{ \mathrm{d}(x,v)}\end{aligned}$$ saying that $F = { \left \langle f \right \rangle }$. By Proposition \[VarChar\] it is clear that ${ \left \langle \cdot \right \rangle }$ leaves invariant the measures with support in ${{{\bf}R}}^d \cup ({ \{0\} \cup r { {\bf}{S}}})$. Consider $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$. We say that ${ \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} \in {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d)$ if and only if there is a constant $C>0$ such that $${ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { (\alpha - \beta |v|^2) v}\cdot \nabla _v \psi f(x,v)\,{ \mathrm{d}(x,v)}} \leq C \|\psi \|_{{ L^\infty}{}},\;\;\psi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}.$$ In this case there is a bounded measure $\mu$ such that $$- { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { (\alpha - \beta |v|^2) v}\cdot \nabla _v \psi f(x,v)\,{ \mathrm{d}(x,v)}} = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi \mu },\;\;\psi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}.$$ By definition we take ${ \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} = \mu$. The main motivation for the construction of the projection ${ \left \langle \cdot \right \rangle }$ is the following result. \[ZeroAve\] For any $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ such that $ { \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\}\in {\cal M}_b ({{{\bf}R}}^d \times {{{\bf}R}}^d)$ we have ${ \left \langle { \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} \right \rangle } = 0$. Let us take ${ \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} = \mu$. We will check that the zero measure $0$ satisfies the characterization of ${ \left \langle \mu \right \rangle }$ in Proposition \[VarChar\]. Clearly ${ \mathrm{supp\;}}0 = \emptyset \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$. For any $\varphi (x) \in { C^0_c ({{{\bf}R}}^d)}{}$ we have $$\begin{aligned} \int _{v = 0} \varphi (x) \mu(x,v)\,{ \mathrm{d}(x,v)}& = { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \varphi (x) \chi \left ( \frac{|v|}{\delta} \right ) \mu(x,v)\,{ \mathrm{d}(x,v)}} \\ & = - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \varphi (x) \chi ^{\;\prime} \left ( \frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta |v|^2) f(x,v)\,{ \mathrm{d}(x,v)}}= 0\end{aligned}$$ by dominated convergence, since $$\left | \chi ^{\;\prime} \left ( \frac{|v|}{\delta} \right )\frac{|v|}{\delta} ( \alpha - \beta |v|^2) \right |\leq \alpha \sup _{u \geq 0} |\chi ^{\;\prime} (u) u | + \beta \delta ^2 \sup _{u \geq 0} |\chi ^{\;\prime} (u) u ^3|.$$ Therefore we deduce that ${ \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} = 0$ on ${{{\bf}R}}^d \times \{0\}$. Consider now $\psi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ and lets us compute $$\begin{aligned} \int _{v \neq 0} \psi \left (x,r { \frac{v}{|v|}}\right ) & \mu(x,v)\,{ \mathrm{d}(x,v)}= { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi \left (x,r { \frac{v}{|v|}}\right ) \left ( 1 - \chi \left ( \frac{|v|}{\delta} \right ) \right ) \mu(x,v)\,{ \mathrm{d}(x,v)}} \\ & = { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi \left (x,r { \frac{v}{|v|}}\right ) \chi ^{\;\prime}\left ( \frac{|v|}{\delta} \right ) \frac{|v|}{\delta} (\alpha - \beta |v|^2) f(x,v)\,{ \mathrm{d}(x,v)}} = 0\end{aligned}$$ since $v \cdot \nabla _v \{ \psi (x, r { \frac{v}{|v|}})\} = 0$. By density, the same conclusion holds true for any $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ and thus ${ \left \langle { \mathrm{div}_v}\{f { (\alpha - \beta |v|^2) v}\} \right \rangle } = 0$. \[SimplerAve\] When $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ does not charge ${{{\bf}R}}^d \times \{0\}$, ${ \left \langle f \right \rangle }$ is given by $${ \mathrm{supp\;}}{ \left \langle f \right \rangle } \subset {{{\bf}R}}^d \times r{ {\bf}{S}},\;\;\int _{v \neq 0} \psi \left ( x, r { \frac{v}{|v|}}\right ) { \left \langle f \right \rangle } = \int _{v \neq 0} \psi \left ( x, r { \frac{v}{|v|}}\right ) f,\;\;\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$$ or equivalently $$\label{Equ34} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \left \langle f \right \rangle } } = \int _{v \neq 0} \psi \left ( x, r { \frac{v}{|v|}}\right ) f,\;\;\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}.$$ Using Proposition \[ZeroAve\] we can obtain, at least formally, the limit model satisfied by $f = { \lim _{\varepsilon \searrow 0}}{ f ^\varepsilon}$. By we know that ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$. The time evolution of $f$ comes by eliminating ${ f ^{(1)}}$ in . For that it is sufficient to project on the subspace of the measures satisfying the constraint , [*i.e.,*]{} to apply ${ \left \langle \cdot \right \rangle }$. $$\label{Equ35} { \left \langle \partial _t f \right \rangle } + { \left \langle { \mathrm{div}_x}\{f v\} \right \rangle } + { \left \langle { \mathrm{div}_v}\{ f a \} \right \rangle } = 0.$$ It is easily seen that ${ \left \langle \partial _t f \right \rangle } = \partial _t { \left \langle f \right \rangle } = \partial _t f $ since ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$ and therefore ${ \left \langle f \right \rangle } = f$. We need to compute the last two terms in . We show that \[TransportAve\] Assume that $a = a(x)$ is a bounded continuous field. Then we have the following equalities $${ \left \langle { \mathrm{div}_x}\{f v \} \right \rangle } = { \mathrm{div}_x}\{f v \}\;\;\mbox{ if } \;{ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times ({ \{0\} \cup r { {\bf}{S}}})$$ $${ \left \langle { \mathrm{div}_v}\{ f a \} \right \rangle } = { \mathrm{div}_v}\left \{ f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \right \} \;\;\mbox{ if } \;{ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times r{ {\bf}{S}}.$$ As a consequence, yields the transport equation obtained rigorously in Theorems [\[MainResult\]]{} and [\[MainResult2\]]{}. For any $\psi \in { C^1_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we have $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi & { \left \langle { \mathrm{div}_x}\{f v \} \right \rangle }} = \int _{v = 0} \psi { \mathrm{div}_x}\{f v \} + \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_x}\{f v \}\\ & = { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \chi \left ( \frac{|v|}{\delta}\right)}{ \mathrm{div}_x}\{f v \}} + { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! { \psi \left ( x, r \frac{v}{|v|}\right )}\left ( 1 - { \chi \left ( \frac{|v|}{\delta}\right)}\right ) { \mathrm{div}_x}\{f v \}} \\ & = - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! v \cdot \nabla _x \psi { \chi \left ( \frac{|v|}{\delta}\right)}f} - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! v \cdot \nabla _x { \psi \left ( x, r \frac{v}{|v|}\right )}\left ( 1 - { \chi \left ( \frac{|v|}{\delta}\right)}\right ) f} \\ & = - \int _{v = 0} v \cdot \nabla _x \psi f - \int _{v \neq 0} v \cdot \nabla _x { \psi \left ( x, r \frac{v}{|v|}\right )}f \\ & = - { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! v \cdot \nabla _x \psi f } = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \mathrm{div}_x}\{f v \}}\end{aligned}$$ saying that ${ \left \langle { \mathrm{div}_x}\{f v \} \right \rangle } = { \mathrm{div}_x}\{f v \}$. Assume now that ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times r{ {\bf}{S}}$. It is easily seen that ${ \mathrm{div}_v}(fa)$ does not charge ${{{\bf}R}}^d \times \{0\}$. Indeed, for any $\psi \in { C^0_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ we have by dominated convergence $$\begin{aligned} \int _{v = 0} \psi { \mathrm{div}_v}(fa) & = { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \chi \left ( \frac{|v|}{\delta}\right)}{ \mathrm{div}_v}(fa)} \\ & = - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \nabla _v \psi { \chi \left ( \frac{|v|}{\delta}\right)}f} - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! a \cdot \frac{v}{|v|} \frac{1}{\delta} { \chi ^{\;\prime}\left ( \frac{|v|}{\delta}\right)}\psi f } = 0.\end{aligned}$$ Therefore we can use $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \left \langle { \mathrm{div}_v}(fa) \right \rangle }} & = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_v}(fa) \\ & = { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \left ( 1 - { \chi \left ( \frac{|v|}{\delta}\right)}\right ) { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_v}(fa) }\\ & = - { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \left ( 1 - { \chi \left ( \frac{|v|}{\delta}\right)}\right ) \frac{r}{|v|} { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot (\nabla _v \psi ) \left ( x, r{ \frac{v}{|v|}}\right ) f } \\ & \quad + { \lim _{\delta \searrow 0}}{ \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \;\;\frac{1}{\delta} { \chi ^{\;\prime}\left ( \frac{|v|}{\delta}\right)}\frac{v}{|v|} \cdot a { \psi \left ( x, r \frac{v}{|v|}\right )}f } \\ & = - \int _{v \neq 0} { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot \nabla _v \psi f = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi \;{ \mathrm{div}_v}\left \{f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \right \}}.\end{aligned}$$ We investigate now the limit when ${ \varepsilon}\searrow 0$ of the diffusion model . We are done if we compute ${ \left \langle \Delta _v f \right \rangle }$ for a non negative bounded measure with support contained in ${{{\bf}R}}^d \times r{ {\bf}{S}}$. As before we can check that $\Delta _v f$ does not charge ${{{\bf}R}}^d \times \{0\}$ and therefore, thanks to , we obtain after some computations $$\label{Equ37} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi { \left \langle \Delta _v f \right \rangle }} = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}\Delta _v f = \int _{v \neq 0} \Delta _v \left \{ { \psi \left ( x, r \frac{v}{|v|}\right )}\right \}f,\;\;\psi \in { C^2_c ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}.$$ \[ZeroHom\] For any function $\varphi \in C^2 ({{{\bf}R}}^d \setminus \{0\})$ and any $r >0$ we have $$\Delta _v \left \{ \varphi \left ( r { \frac{v}{|v|}}\right ) \right \} = \left ( \frac{r}{|v|}\right ) ^2 { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }: \partial ^2 _v \varphi \left ( r { \frac{v}{|v|}}\right ) - 2 \frac{r}{|v|} \frac{v \cdot \nabla _v \varphi \left ( r { \frac{v}{|v|}}\right ) }{|v|^2},\;\;v \neq 0.$$ Combining , Lemma \[ZeroHom\] and the fact that ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times r { {\bf}{S}}$ we obtain $$\begin{aligned} { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) { \left \langle \Delta _v f \right \rangle } } & = \int _{v \neq 0} \left [ { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }: \partial _v ^2 \psi (x,v) - 2 \frac{v \cdot \nabla _v \psi (x,v)}{|v|^2} \right]f \nonumber \\ & = { \int _{{{{\bf}R}}^d \times {{{\bf}R}}^d}\!\!\! \psi (x,v) { \mathrm{div}_v}\left \{ { \mathrm{div}_v}\left [ f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\right ] + 2 f \frac{v}{|v|^2} \right \}}. \nonumber\end{aligned}$$ We deduce the formula $${ \left \langle \Delta _v f \right \rangle } = { \mathrm{div}_v}\left \{ { \mathrm{div}_v}\left [ f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\right ] + 2 f \frac{v}{|v|^2} \right \}$$ for any $f$ satisfying ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times r { {\bf}{S}}$ and the limit of the Vicsek model when ${ \varepsilon}\searrow 0$ becomes $$\label{equnew} \partial _t f + { \mathrm{div}_x}(fv) + { \mathrm{div}_v}\left \{ f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \right \} = { \mathrm{div}_v}\left \{ { \mathrm{div}_v}\left [ f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\right ] + 2 f \frac{v}{|v|^2} \right \}$$ with the initial condition $f(0) = { \left \langle { f ^{\mathrm{in}}}\right \rangle }$, as stated in . Spherical coordinates and the Laplace-Beltrami operator {#A} ======================================================= In this appendix, we show the computations to relate the equations written in original variables $(x,v)$ to the equations in spherical coordinates $(x,\omega)$. Our limit densities have their support contained in ${{{\bf}R}}^d \times r { {\bf}{S}}$ and thus reduce to measures on ${{{\bf}R}}^d \times r{ {\bf}{S}}$. For example, let us consider the measure on ${{{\bf}R}}^d \times r{ {\bf}{S}}$ still denoted by $f$, given by $${ \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega) f(x,\omega)\,\mathrm{d}(x,\omega)} = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{} f(x,v)\,{ \mathrm{d}(x,v)}$$ for any function $\psi \in { C^0 _c ({{{\bf}R}}^d \times r { {\bf}{S}})}{}$. In particular, to any $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$ not charging ${{{\bf}R}}^d \times \{0\}$ it corresponds ${ \left \langle f \right \rangle } \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, with ${ \mathrm{supp\;}}{ \left \langle f \right \rangle } \subset {{{\bf}R}}^d \times r { {\bf}{S}}$, whose characterization is $${ \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega){ \left \langle f \right \rangle }(x,\omega)\,\mathrm{d}(x,\omega)} = \int _{v \neq 0}{ \psi \left ( x, r \frac{v}{|v|}\right )}f(x,v)\,{ \mathrm{d}(x,v)}.$$ We intend to write the previous limit models (in Theorems \[MainResult\], \[MainResult2\], and ) in spherical coordinates. \[SpherCoord\] Assume that $f \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times {{{\bf}R}}^d)}{}$, ${ \mathrm{supp\;}}f \subset {{{\bf}R}}^d \times r { {\bf}{S}}$ and let us denote by $F \in { {\cal M}_b ^+ ({{{\bf}R}}^d \times r { {\bf}{S}})}$ its corresponding measure on ${{{\bf}R}}^d \times r { {\bf}{S}}$. Therefore we have $${ \left \langle { \mathrm{div}_x}(fv) \right \rangle } = { \mathrm{div}_x}(F \omega),\;\;{ \left \langle { \mathrm{div}_v}(fa) \right \rangle } = { \mathrm{div}_\omega}\left \{F { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \right \},\;\;{ \left \langle \Delta _v f \right \rangle } = \Delta _\omega F.$$ Thanks to Proposition \[TransportAve\] we have for any $\psi \in { C^1 _c ({{{\bf}R}}^d \times r { {\bf}{S}})}{}$ $$\begin{aligned} { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega) { \left \langle { \mathrm{div}_x}(fv) \right \rangle }} & = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_x}(fv) = - \int _{v\neq 0} v \cdot \nabla _x { \psi \left ( x, r \frac{v}{|v|}\right )}f\\ & = - \int _{v \neq 0} r { \frac{v}{|v|}}\cdot \nabla _x { \psi \left ( x, r \frac{v}{|v|}\right )}f = - { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \omega \cdot \nabla _x \psi (x, \omega) F}\end{aligned}$$ and thus ${ \left \langle { \mathrm{div}_x}(fv) \right \rangle } = { \mathrm{div}_x}(F \omega)$. Similarly we can write $$\begin{aligned} { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega) { \left \langle { \mathrm{div}_v}(fa) \right \rangle }} & = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \left \langle { \mathrm{div}_v}(fa) \right \rangle }(\mathrm{d}(x,v)) \nonumber \\ & = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \mathrm{div}_v}\left \{f { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a\right \} \nonumber \\ & = - \int _{v \neq 0} \frac{r}{|v|}{ \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\nabla _v{ \psi \left ( x, r \frac{v}{|v|}\right )}f \nonumber \\ & = - \int _{v \neq 0} { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }a \cdot { \left ( I - \frac{v \otimes v}{|v|^2} \right ) }\nabla _v { \psi \left ( x, r \frac{v}{|v|}\right )}f \nonumber \\ & = - { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \cdot { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }\nabla _v \psi (x, \omega) F}\nonumber \\ & = - { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \cdot \nabla _\omega \psi (x, \omega) F}\nonumber \\ & = { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x, \omega) { \mathrm{div}_\omega}\left \{ F{ \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a\right \}}\nonumber\end{aligned}$$ and therefore $${ \left \langle { \mathrm{div}_v}(fa) \right \rangle } = { \mathrm{div}_\omega}\left \{F { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \right \}.$$ Here ${ \mathrm{div}_\omega}$ stands for the divergence along $r{ {\bf}{S}}$ (notice that ${ \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a$ is a tangent field of $r{ {\bf}{S}}$) and $\nabla _\omega = { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }\nabla _v$ is the gradient along $r{ {\bf}{S}}$. For the last assertion we appeal to the following well known result asserting that the Laplace-Beltrami operator coincides with the Laplacian of the degree zero homogeneous extension, see also [@BCC12]. \[LaplaceBeltrami\] Consider $\varphi = \varphi (\omega)$ a $C^2$ function on $r{ {\bf}{S}}$ and we denote by $\Phi = \Phi (v)$ its degree zero homogeneous extension on ${{{\bf}R}}^d \setminus \{0\}$ $$\Phi (v) = \varphi \left ( r { \frac{v}{|v|}}\right ),\;\;v \neq 0.$$ Therefore we have for any $\omega \in r { {\bf}{S}}$ $$\Delta _\omega \varphi (\omega) = \Delta _v \Phi (\omega).$$ Let us come back to the proof of Proposition \[SpherCoord\]. For any $\psi \in C^2_c ({{{\bf}R}}^d \times r { {\bf}{S}})$ we introduce its degree zero homogeneous extension $\Psi (x,v) = { \psi \left ( x, r \frac{v}{|v|}\right )}$. Thanks to Proposition \[LaplaceBeltrami\] we can write $$\begin{aligned} { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x,\omega) { \left \langle \Delta _v f \right \rangle } }& = \int _{v \neq 0} { \psi \left ( x, r \frac{v}{|v|}\right )}{ \left \langle \Delta _v f \right \rangle } \nonumber = \int _{v \neq 0} \Psi (x,v) \Delta _v f = \int _{v \neq 0} \Delta _v \Psi f \\ & = \int _{|v| = r} \Delta _\omega \psi (x,v) f = { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \Delta _\omega \psi (x,\omega) F} = { \int _{{{{\bf}R}}^d \times r { {\bf}{S}}}\!\!\! \psi (x,\omega) \Delta _\omega F}\end{aligned}$$ meaning that ${ \left \langle \Delta _v f \right \rangle } = \Delta _\omega F$. For the sake of completeness, we finally write the equations in spherical coordinates in ${{{\bf}R}}^3$. We introduce the spherical coordinates $\omega = r (\cos \theta \cos \varphi, \cos \theta \sin \varphi, \sin \theta)$ with the angle variables $(\theta, \varphi ) \in ]-\pi/2, \pi/2[ \times [0,2\pi [$, and the orthogonal basis of the tangent space to $r{ {\bf}{S}}$ $$e_\theta = (- \sin \theta \cos \varphi, - \sin \theta \sin \varphi, \cos \theta),\;\;e_\varphi = (- \cos \theta \sin \varphi, \cos \theta \cos \varphi, 0)$$ with $|e_\theta| = 1,\;|e_\varphi| = \cos \theta$. For any smooth function $u$ on $r{ {\bf}{S}}$ we have $$\nabla _\omega u = (\nabla _\omega u \cdot e_\theta) e_\theta + (\nabla _\omega u \cdot e _\varphi ) \frac{e_\varphi}{\cos ^2 \theta} = \frac{1}{r} \partial _\theta u \;e _\theta + \frac{1}{r\cos ^2 \theta} \partial _\varphi u \;e _\varphi$$ and for any smooth tangent field $\xi = \xi _\theta e _\theta + \xi _\varphi e _\varphi $ we have $${ \mathrm{div}_\omega}\xi = \frac{1}{r} \left \{\frac{1}{\cos \theta} \partial _\theta (\xi _\theta \cos \theta) + \partial _\varphi \xi _\varphi \right \}.$$ The coordinates of the tangent field $\xi := F { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a$ are $ \xi _\theta = \xi \cdot e _\theta = F a_\theta,\;\;\xi _\varphi = \frac{\xi \cdot e _\varphi }{\cos ^2 \theta } = F a_\varphi $ and we obtain $${ \left \langle { \mathrm{div}_v}(fa) \right \rangle } = { \mathrm{div}_\omega}\left \{ F { \left ( I - \frac1{r^2}({\omega \otimes \omega}) \right ) }a \right \} = \frac{1}{r} \left \{ \frac{1}{\cos \theta} \partial _\theta (F a_\theta \cos \theta) + \partial _\varphi ( F a_\varphi ) \right \}.$$ The spherical Laplacian is given by $$\begin{aligned} \Delta _\omega F & = { \mathrm{div}_\omega}(\nabla _\omega F) = \frac{1}{r} \left \{\frac{1}{\cos \theta} \frac{\partial}{\partial \theta} \left ( \frac{\cos \theta}{r} \partial _\theta F\right ) + \frac{\partial}{\partial \varphi } \left ( \frac{1}{r \cos ^2 \theta} \partial _\varphi F \right ) \right \}\\ & = \frac{1}{r^2}\left \{ \frac{1}{\cos \theta} \frac{\partial}{\partial \theta} ( \cos \theta \;\partial _\theta F ) + \frac{1}{\cos ^2 \theta} \;\partial ^2 _\varphi F \right \}.\end{aligned}$$ The limit transport equation obtained in for ${{{\bf}R}}^3$ is $$\partial _t F + \omega \cdot \nabla _x F + \frac{1}{r} \left \{\frac{\partial _\theta (F a_\theta \cos \theta )}{\cos \theta} + \partial _\varphi ( F a _\varphi ) \right \} = \frac{1}{r^2} \left \{\frac{1}{\cos \theta} \frac{\partial}{\partial \theta} ( \cos \theta \;\partial _\theta F ) + \frac{1}{\cos ^2 \theta} \;\partial ^2 _\varphi F \right \}.$$ We recall here the proof of Proposition \[LaplaceBeltrami\]. It is a consequence of a more general result. \[MoreGenRes\] Let us consider a function $\varphi = \varphi (v) \in C^2 ({{{\bf}R}}^d)$, $d \geq 2$ which writes in polar coordinates $ \varphi (v) = { \tilde{\varphi}}(\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = { \frac{v}{|v|}}\in { {\bf}{S}}. $ Therefore for any $v \neq 0$ we have $$\Delta _v \varphi (v) = \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho { \tilde{\varphi}}) + \frac{1}{\rho ^2 } \Delta _\sigma { \tilde{\varphi}}(\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = { \frac{v}{|v|}}.$$ Consider a smooth function $\psi = \psi (v) \in C^2$ with compact support in ${{{\bf}R}}^N \setminus \{0\}$, which writes in polar coordinates $ \psi (v) = { \tilde{\psi}}(\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = { \frac{v}{|v|}}\in { {\bf}{S}}. $ We have $$\frac{\partial { \tilde{\varphi}}}{\partial \rho } = \nabla _v \varphi \cdot \sigma,\;\;\nabla _v \varphi = (\nabla _v \varphi \cdot \sigma ) \sigma + (I - \sigma \otimes \sigma) \nabla _v \varphi = \frac{\partial { \tilde{\varphi}}}{\partial \rho }\;\sigma + \nabla _{\omega = \rho \sigma} { \tilde{\varphi}}$$ and $$\frac{\partial { \tilde{\psi}}}{\partial \rho } = \nabla _v \psi \cdot \sigma,\;\;\nabla _v \psi = (\nabla _v \psi \cdot \sigma ) \sigma + (I - \sigma \otimes \sigma) \nabla _v \psi = \frac{\partial { \tilde{\psi}}}{\partial \rho }\;\sigma + \nabla _{\omega = \rho \sigma} { \tilde{\psi}}.$$ Integrating by parts yields $$\begin{aligned} - { \int _{{{{\bf}R}}^d} \!\Delta _v \varphi \;\psi (v) \;\mathrm{d}v} & = { \int _{{{{\bf}R}}^d} \!\nabla _v \varphi \cdot \nabla _v \psi \;\mathrm{d}v} = \int_{{{{\bf}R}}_+} \int _{S^{N-1}} \left \{ \frac{\partial { \tilde{\varphi}}}{\partial \rho} \frac{\partial { \tilde{\psi}}}{\partial \rho} + \frac{1}{\rho ^2} \nabla _\sigma { \tilde{\varphi}}\cdot \nabla _\sigma { \tilde{\psi}}\right \} \;\mathrm{d}\sigma \rho ^{N-1} \;\mathrm{d}\rho \nonumber \\ & = - \int _{S^{N-1}} \int _{{{{\bf}R}}_+} { \tilde{\psi}}\frac{\partial}{\partial \rho } \left ( \rho ^{N-1} \frac{\partial { \tilde{\varphi}}}{\partial \rho} \right ) \;\mathrm{d}\rho\;\mathrm{d}\sigma - \int _{{{{\bf}R}}_+} \frac{\rho ^{N-1}}{\rho ^2} \int _{S^{N-1}} { \tilde{\psi}}\;\Delta _\sigma { \tilde{\varphi}}\;\mathrm{d}\sigma \;\mathrm{d}\rho \nonumber \\ & = - { \int _{{{{\bf}R}}^d} \!\psi (v) \left \{ \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho { \tilde{\varphi}}) + \frac{1}{\rho ^2 } \Delta _\sigma { \tilde{\varphi}}\right \} \;\mathrm{d}v}\end{aligned}$$ and therefore $$\Delta _v \varphi (v) = \frac{1}{\rho ^{N-1}} \frac{\partial}{\partial \rho} ( \rho ^{N-1} \partial _\rho { \tilde{\varphi}}) + \frac{1}{\rho ^2 } \Delta _\sigma { \tilde{\varphi}}(\rho, \sigma),\;\;\rho = |v| >0,\;\;\sigma = { \frac{v}{|v|}}.$$ (of Proposition \[LaplaceBeltrami\]) The degree zero homogeneous extension $\Phi (v) = \varphi \left ( r { \frac{v}{|v|}}\right )$ does not depend on the polar radius $\Phi (v) = \tilde{\Phi} (\sigma) = \varphi (\omega = r \sigma),\;\;\sigma = { \frac{v}{|v|}}.$ Thanks to Proposition \[MoreGenRes\], we deduce $ \Delta _v \Phi = \frac{1}{\rho ^2} \Delta _\sigma \tilde{\Phi} = \frac{r^2}{\rho ^2} \Delta _\omega \varphi . $ Taking $\rho = r $, which means $v = r \sigma = \omega$ we obtain $ \Delta _v \Phi (\omega) = \Delta _\omega \varphi (\omega),\;\;\omega \in r { {\bf}{S}}. $ Acknowledgments {#acknowledgments .unnumbered} --------------- JAC was supported by projects MTM2011-27739-C04-02 and 2009-SGR-345 from Agència de Gestió d’Ajuts Universitaris i de Recerca-Generalitat de Catalunya. 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--- abstract: 'An exact analytical derivation is presented, showing that the Ising model on the Cayley tree exhibits a line of third order phase transition points, between temperatures $T_2=2k_B^{-1}J\ln ({\sqrt 2}+1) $ and $T_{BP}=k_B^{-1}J\ln (3)$, and a line of fourth order phase transitions between $T_{BP}$ and $\infty$, where $k_B$ is the Boltzmann constant, and $J$ is the nearest-neighbor interaction parameter.' author: - 'Borko D. Stoši'' c' - 'Tatijana Stoši'' c' - 'Ivon P. Fittipaldi' title: | Third and Fourth Order Phase Transitions:\ Exact Solution for the Ising Model on the Cayley Tree --- Ever since the pioneering works on phase transitions and critical phenomena, it has been clear that phase transitions of higher order are conceptually possible, but, to the best of our knowledge, up to date there has been no rigorous proof of existence of a single system exhibiting a phase transition of finite order higher then two. One notable attempt in this direction is the work of M" uller-Hartmann and Zittartz [@muller], interpreting the series expansion of the free energy of the Ising model on the Cayley tree in the limit of small field $H\longrightarrow 0^+$, in terms of phase transitions of all even orders (from two to infinity). More precisely, their claim was that the susceptibility of order $2\ell$ (i.e. the 2$\ell$-th derivative of the free energy with respect to field), diverges at temperature $T_{2\ell}=k_B^{-1}J\ln [1+2/(B^{1-1/2\ell}-1)]$ for $\ell=1,2,\dots ,\infty$, where $B$ is the tree branching number (coordination number minus 1). Thus, the zero field susceptibility was found to diverge from $T=0$ to $T_2=k_B^{-1}J\ln [({\sqrt B}+1)/({\sqrt B}-1)]$, representing a line of second order phase transitions (with respect to field), fourth order susceptibility was found to diverge between $T_2$ and $T_4=k_B^{-1}J\ln [(B^{3/4}+1)/(B^{3/4}-1)]$, representing a line of fourth order phase transitions, while the infinite order phase transition was found to occur at the Bethe-Pierls temperature $T_{BP}=k_B^{-1}J\ln \left[(B+1)/(B-1)\right]$. It is shown in this work through an exact analytical calculation of the recursion relations for the derivatives of the free energy, that the exact expressions contain terms previously unnoticed [@muller] in the approximate analysis. Interpratation of these results requires some special consideration of symmetry breaking, the classical approach not being adequate for the system at hand. It is found that the third order derivative diverges between $T_2$ and $T_{BP}$ (all the lower derivatives being finite), representing a line of third order phase transitions, while the fourth order derivative diverges between $T_{BP}$ and $\infty$ (where all lower derivatives are finite), representing a fourth order phase transition line. We consider the nearest-neighbor Ising model with the Hamiltonian $${\cal H}=-J\sum_{\langle nn\rangle}S_iS_j-H\sum_{i}S_i,\label{one}$$ where $J$ is the coupling constant, $H$ is the external magnetic field, $S_i=\pm 1$ is the spin at site $i$, and $\langle nn\rangle$ denotes summation over the nearest-neighbor pairs. For simplicity, hereafter we consider only the tree with $B=2$, while the analysis which follows may be directly generalized for arbitrary tree branching number. Following Eggarter [@egarter], we further consider systems situated on a single $n$-generation branch of a Cayley tree, composed of two $(n-1)$-generation branches connected to a single initial site. Thus, the $n$-generation branch consists of $N_n=2^{n+1}-1$ spins, the $0$-generation branch being a single spin. The exact recursion relations for the partial partition functions of any two consecutive generation branches are easily derived [@egarter] to be $$Z_{n+1}^{\pm}=y^{\pm 1}\left[ x^{\pm 1}Z_{n}^{+} + x^{\mp 1}Z_{n}^{-}\right] ^2,\label{two}$$ where $Z_{n}^{+}$ and $Z_{n}^{-}$ denote the partition functions restricted by fixing the initial spin (connecting the two $n$-generation branches) into the $\{ +\}$ and $\{ -\}$ position, respectively, and where we have used the notation $x\equiv exp(\beta J)$ and $y\equiv exp(\beta H)$, with $\beta=1/k_BT$ denoting the reciprocal of the product of the Boltzmann constant $k_B$ and the temperature $T$. The usual approach of attempting to establish the field dependent expression for the partition function in the thermodynamic limit, then finding its field derivatives, and finally taking the limit $H\longrightarrow 0$ yields only approximate solutions for the zero-field susceptibility [@muller; @heimburg; @matsuda]. In an earlier work [@SSF], current authors have derived an exact expression for the zero-field magnetization and susceptibility, by using the strategy of finding the recursion relations for the [*field derivatives*]{} of the partition function, taking the limit $H\longrightarrow 0$, and only then performing the actual iterations to reach the thermodynamic limit. Here we extend this approach to find higher derivatives of the partition function. Equation (\[two\]) can be formally differentiated with respect to field, to find the recursion relations for the field derivatives of the partition function. Up to the forth derivative we find the following recursion relations $$\begin{aligned} &&{\frac{\partial Z_{n+1}^{\pm}}{\partial h}}= y^{\pm 1} \left ( \pm {\Gamma_n^{0\pm}}^{2}+2 {\Gamma_n^{0\pm}}{\Gamma_n^{1\pm}} \right ), \label{three}\end{aligned}$$ $$\begin{aligned} {\frac{\partial^2 Z_{n+1}^{\pm}}{\partial h^2}}=&& y^{\pm 1} \left [ {\left(\Gamma_n^{0\pm}\right)}^{2} \pm 4{\Gamma_n^{0\pm}}{\Gamma_n^{1\pm}} +2 {\left(\Gamma_n^{1\pm}\right)}^2 +\right.\nonumber\\ && \left.+2 {\Gamma_n^{0\pm}}{\Gamma_n^{2\pm}} \right] , \label{four}\end{aligned}$$ $$\begin{aligned} {\frac{\partial^3 Z_{n+1}^{\pm}}{\partial h^3}}=&& y^{\pm 1} \left [ \pm {\left(\Gamma_n^{0\pm}\right)}^{2} + 6 {\Gamma_n^{0\pm}}{\Gamma_n^{1\pm}} \pm 6 {\left(\Gamma_n^{1\pm}\right)}^2 \pm \right.\nonumber\\ && \left. \pm 6 {\Gamma_n^{0\pm}}{\Gamma_n^{2\pm}} + 6 {\Gamma_n^{1\pm}}{\Gamma_n^{2\pm}} + 2 {\Gamma_n^{0\pm}}{\Gamma_n^{3\pm}} \right ] , \label{five}\end{aligned}$$ $$\begin{aligned} {\frac{\partial^4 Z_{n+1}^{\pm}}{\partial h^4}}=&& y^{\pm 1} \left [ {\left(\Gamma_n^{0\pm}\right)}^{2} \pm 8 {\Gamma_n^{0\pm}}{\Gamma_n^{1\pm}} + 12 {\left(\Gamma_n^{1\pm}\right)}^2 + \right.\nonumber\\ && +12 {\Gamma_n^{0\pm}}{\Gamma_n^{2\pm}} \pm 24 {\Gamma_n^{1\pm}}{\Gamma_n^{2\pm}} \pm 8 {\Gamma_n^{0\pm}}{\Gamma_n^{3\pm}} + \nonumber\\ && \left. +6{\left(\Gamma_n^{2\pm}\right)}^2 + 8 {\Gamma_n^{1\pm}}{\Gamma_n^{3\pm}} + 2 {\Gamma_n^{0\pm}}{\Gamma_n^{4\pm}}\right ] , \label{six}\end{aligned}$$ where we have used notation $h\equiv \beta H$ and $$\begin{aligned} {\Gamma_n^{i\pm}}\equiv {\frac{\partial^i Z_{n}^{+}}{\partial h^i}}{x^{\pm 1}}+ {\frac{\partial^i Z_{n}^{-}}{\partial h^i}}{x^{\mp 1}}.\end{aligned}$$ Starting from a single spin (0-th generation branch), for which we have ${{\partial^k Z_{0}^{\pm}}/{\partial h^k}}=(\pm 1)^k y^{\pm 1}$, it is straightforward to show by mathematical induction, using (\[two\]-\[six\]), that for [*zero field*]{} ($y=1$) the symmetry equations $$\begin{aligned} %&&Z_n^{+}=Z_n^{-},\nonumber\\ %&&{\frac{\partial Z_{n}^{+}}{\partial \beta H}}= %-{\frac{\partial Z_{n}^{-}}{\partial \beta H}},\nonumber\\ %&&{\frac{\partial^2 Z_{n}^{+}}{\partial (\beta H^2)}}= %{\frac{\partial^2 Z_{n}^{-}}{\partial (\beta H)^2}},\nonumber\\ %&&{\frac{\partial^3 Z_{n}^{+}}{\partial (\beta H^3)}}= %-{\frac{\partial^3 Z_{n}^{-}}{\partial (\beta H)^3}},\nonumber\\ %&&{\frac{\partial^4 Z_{n}^{+}}{\partial (\beta H^4)}}= %{\frac{\partial^4 Z_{n}^{-}}{\partial (\beta H)^4}}, \left.{\frac{\partial^k Z_{n}^{+}}{\partial (\beta H)^k}}\right|_{H=0}= \left.(-1)^k{\frac{\partial^k Z_{n}^{-}}{\partial (\beta H)^k}}\right|_{H=0}, \label{seven}\end{aligned}$$ hold for a branch of arbitrary generation n. It then follows from (\[two\]-\[seven\]) that the moments ${\cal S}_n\equiv \frac{1}{Z_n^+}{\frac{\partial Z_{n}^{+}}{\partial h}}$, ${\cal T}_n\equiv \frac{1}{Z_n^+}{\frac{\partial^2 Z_{n}^{+}}{\partial h^2}}$, ${\cal U}_n\equiv \frac{1}{Z_n^+}{\frac{\partial^3 Z_{n}^{+}}{\partial h^3}}$, and ${\cal V}_n\equiv \frac{1}{Z_n^+}{\frac{\partial^4 Z_{n}^{+}}{\partial h^4}}$ satisfy the recursion relations, $$\begin{aligned} &{\cal S}_{n+1}=1+2\;t\;{\cal S}_n , \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;& {\cal S}_0=1,\nonumber\\ &{\cal T}_{n+1}=1+4\;t\;{\cal S}_n+2\;t^2\;{\cal S}_n^2+2{\cal T}_n, \;\;\;\;\;\;\;\;\;\;&{\cal T}_0=1,\nonumber\\ &{\cal U}_{n+1}=1+6\;t\;{\cal S}_n+6\;t^2\;{\cal S}_n^2+6{\cal T}_n +\;\;\;\;\;\;\;\;\;\;&\nonumber\\ &+6\;t\;{\cal S}_n\;{\cal T}_n+2\;t\;{\cal U}_n &{\cal U}_0=1,\nonumber\\ &{\cal V}_{n+1}=1+8\;t\;{\cal S}_n+12\;t^2\;{\cal S}_n^2+12{\cal T}_n \;\;\;\;\;\;\;\;\;\;&\nonumber\\ &+24\;t\;{\cal S}_n\;{\cal T}_n+8\;t\;{\cal U}_n\;\;\;\;\;\;\;\;\;\;&\nonumber\\ &+6\;{\cal T}_n^2+8\;t^2\;{\cal S}_n\;{\cal U}_n +2\;{\cal V}_n\;\;\;\;\;\;\;\;\;\;&{\cal V}_0=1,\nonumber\\ \label{eight}\end{aligned}$$ where $t\equiv \tanh(\beta J)$. Relations (\[eight\]) can be iterated (by summing the geometric series) to yield closed-form expressions for ${\cal S}_n$, ${\cal T}_n$, ${\cal U}_n$ and ${\cal V}_n$ for arbitrary tree generation $n$, and we find $$\begin{aligned} {\cal S}_n= {\frac {{2}^{n+1}{t}^{n+1}-1}{2\,t-1}} \label{nine}\end{aligned}$$ $$\begin{aligned} {\cal T}_n=&& {\frac {4\,{t}^{4}\left (4\,{t}^{2}\right )^{n}} {\left (2\,{t}^{2}-1\right )\left (2\,t-1\right )^{2}}} -{\frac {{2}^{n+1}\left (t+1\right )^{2}}{2\,{t}^{2}-1}} +{\frac {4\,{t}^{2}\left (2\,t\right )^{n}}{\left (2\,t-1\right )^{2}} +{\frac {2\,{t}^{2}-1}{\left (2\,t-1\right )^{2}}}} \label{ten} % 1 %{\frac {-1+2\,{t}^{2}}{\left (-1+2\,t\right )^{2}}} % 2 %4\,{\frac {{t}^{4}\left (4\,{t}^{2}\right )^{n}}{\left (-1+2\,{t}^{2} %\right )\left (-1+2\,t\right )^{2}}} % 3 %-2\,{\frac {{2}^{n}\left (t+1\right )^{2}}{-1+2\,{t}^{2}}} % 4 %4\,{\frac {{t}^{2}\left (2\,t\right )^{n}}{\left (-1+2\,t\right )^{2}}}\end{aligned}$$ $$\begin{aligned} {\cal U}_n=&& +{\frac {24\,{t}^{5}\left (8\,{t}^{3}\right )^{n}}{\left (2\,t+1\right )\left ( 2\,{t}^{2}-1\right )\left (2\,t-1\right )^{4}}} -{\frac {12\,\left (t+1\right )^{2}t\left (4\,t\right )^{n}}{\left (2\,t -1 \right )\left (2\,{t}^{2}-1\right )}} +{\frac {12\,{t}^{3}\left (4\,{t}^{3}+2\,{t}^{2}-t-2\right )\left (4\,{t}^{2} \right )^{n}}{\left (2\,{t}^{2}-1\right )\left (2\,t-1\right )^{4}}}\nonumber\\ &&+{\frac {\left (2\,t\right )^{n+1}\left (24\,n{t}^{4}-30\,n{t}^{2}+6\,n +40\,{t}^{4}-16\,{t}^{3}-60\,{t}^{2}-2\, t+11\right )}{\left (2\,t-1\right )^{4}\left (2\, t+1\right )}} +{\frac {6\,{2}^{n}\left (t+1\right )^{2}}{\left (2\,t-1\right )\left (2\,{t}^ {2}-1\right )}}\nonumber\\ &&-{\frac {8\,{t}^{3}-6\,{t}^{2}-6\,t+5}{\left (2\,t-1\right )^{4}}} \label{eleven} % 1 %12\,{\frac {{t}^{3}\left (4\,{t}^{3}+2\,{t}^{2}-t-2\right )\left (4\,{ %t}^{2}\right )^{n}}{\left (-1+2\,{t}^{2}\right )\left (-1+2\,t\right ) %^{4}}} % 2 %6\,{\frac {{2}^{n}\left (t+1\right )^{2}}{\left (-1+2\,t\right )\left %(-1+2\,{t}^{2}\right )}} % 3 %24\,{\frac {{t}^{5}\left (8\,{t}^{3}\right )^{n}}{\left (2\,t+1\right %)\left (-1+2\,{t}^{2}\right )\left (-1+2\,t\right )^{4}}} % 4 %-12\,{\frac {\left (t+1\right )^{2}t\left (4\,t\right )^{n}}{\left (-1 %+2\,t\right )\left (-1+2\,{t}^{2}\right )}} % 5 %2\,{\frac {\left (2\,t\right )^{n}t\left (40\,{t}^{4}-16\,{t}^{3}-60\, %{t}^{2}-2\,t+11+24\,{t}^{4}n-30\,{t}^{2}n+6\,n\right )}{\left (2\,t+1 %\right )\left (-1+2\,t\right )^{4}}} % 6 %-{\frac {8\,{t}^{3}-6\,{t}^{2}-6\,t+5}{\left (-1+2\,t\right )^{4}}}\end{aligned}$$ $$\begin{aligned} {\cal V}_n=&& {\frac {48\,{t}^{8}\left (12\,{t}^{2}-5\right )\left (16\,{t}^{4}\right )^{n}}{ \left (2\,t+1\right )\left (8\,{t}^{4}-1\right )\left (2\,{t}^{2}-1\right )^{2} \left (2\,t-1\right )^{5}}} +{\frac {12\,\left (t+1\right )^{4}{4}^{n}}{\left (2\,{t}^{2}-1\right )^{2}}} -{\frac {48\,{t}^{4}\left (t+1\right )^{2}\left (8\,{t}^{2}\right )^{n}} {\left (2\,t-1\right )^{2}\left (2\,{t}^{2}-1\right )^{2}}} \nonumber\\ &&+{\frac {96\,{t}^{6}\left (8\,{t}^{4}+16\,{t}^{3}-6\,t-3\right )\left (8\,{t}^{3}\right )^{n}} {\left (2\,{t}^{2}-1\right )\left (2\,t-1\right )^{5} \left (2\,t+1\right )\left (4\,{t}^{3}-1\right )}} -{\frac {48\,\left (t+1\right )^{2}{t}^{2}\left (4\,t\right )^{n}} {\left (2\,{t}^{2}-1\right )\left (2\,t-1\right )^{2}}} \nonumber\\ &&+{\frac {8\,{t}^{4}\left (64\,{t}^{6}+80\,{t}^{5}-104\,{t}^{4}-120\,{t}^{3}+50\,{t }^{2}+40\,t-1\right )\left (4\,{t}^{2}\right )^{n}} {\left (2\,t-1\right )^{5}\left (2\,t+1\right )\left (2\,{t}^{2}-1\right )^{2}}} +{\frac {96\,\left (t^2-1\right ){t}^{4}\left (n+1\right )\left ( 4\,{t}^{2}\right )^{n}}{\left (2\,{t}^{2}-1\right )\left (2\,t-1\right )^{4}}} \nonumber\\ &&+{8\,\frac {\left (8\,{t}^{4}-20\,{t}^{3}-30\,{t}^{2}+2\,t+7\right ){t}^{2} \left (2\,t\right )^{n}}{\left (2\,t-1\right )^{5}\left (2\,t+1\right )}} +{\frac {48\,{t}^{2}\left (t^2-1\right )\left (n+1\right )\left (2 \,t\right )^{n}}{\left (2\,t-1\right )^{4}}} \nonumber\\ &&-{\frac {{2}^{n+2}\left (t+1\right )\left (320\,{t}^{12}-784\,{t}^{10}-384\,{t}^ {9}+176\,{t}^{8}+372\,{t}^{7}+192\,{t}^{6}-56\,{t}^{4}-45\,{t}^{3}-11\,{t}^{2}+6 \,t+4\right )}{\left (2\,{t}^{2}-1\right )^{2}\left (8\,{t}^{4}-1\right )\left ( 2\,t-1\right )^{2}\left (4\,{t}^{3}-1\right )}} \nonumber\\ &&+{\frac {16\,{t}^{5}-24\,{t}^{4}-8\,{t}^{3}+28\,{t}^{2}-6\,t-5} {\left (2\,t-1\right )^{5}}} \label{twelve} % 1 %96\,{\frac {{t}^{6}\left (8\,{t}^{4}+16\,{t}^{3}-6\,t-3\right )\left ( %8\,{t}^{3}\right )^{n}}{\left (2\,t+1\right )\left (-1+2\,{t}^{2} %\right )\left (-1+2\,t\right )^{5}\left (4\,{t}^{3}-1\right )}} % 2 %12\,{\frac {\left (t+1\right )^{4}{4}^{n}}{\left (-1+2\,{t}^{2}\right )^{2}}} % 3 %{\frac {16\,{t}^{5}-24\,{t}^{4}-8\,{t}^{3}+28\,{t}^{2}-6\,t-5}{\left ( %-1+2\,t\right )^{5}}} % 4 %-48\,{\frac {{t}^{2}\left (t+1\right )^{2}\left (4\,t\right )^{n}}{ %\left (-1+2\,t\right )^{2}\left (-1+2\,{t}^{2}\right )}} % 5 %8\,{\frac {{t}^{4}\left (64\,{t}^{6}+80\,{t}^{5}-104\,{t}^{4}-120\,{t} %^{3}+50\,{t}^{2}+40\,t-1\right )\left (4\,{t}^{2}\right )^{n}}{\left ( %-1+2\,t\right )^{5}\left (-1+2\,{t}^{2}\right )^{2}\left (2\,t+1 %\right )}} % 6 %96\,{\frac {\left (t-1\right )\left (t+1\right ){t}^{4}\left (n+1 %\right )\left (4\,{t}^{2}\right )^{n}}{\left (-1+2\,t\right )^{4} %\left (-1+2\,{t}^{2}\right )}} % 7 %8\,{\frac {\left (8\,{t}^{4}-20\,{t}^{3}-30\,{t}^{2}+2\,t+7\right ){t} %^{2}\left (2\,t\right )^{n}}{\left (-1+2\,t\right )^{5}\left (2\,t+1 %\right )}} % 8 %48\,{\frac {\left (t-1\right )\left (t+1\right ){t}^{2}\left (n+1 %\right )\left (2\,t\right )^{n}}{\left (-1+2\,t\right )^{4}}} % 9 %48\,{\frac {{t}^{8}\left (-5+12\,{t}^{2}\right )\left (16\,{t}^{4} %\right )^{n}}{\left (2\,t+1\right )\left (8\,{t}^{4}-1\right )\left (- %1+2\,{t}^{2}\right )^{2}\left (-1+2\,t\right )^{5}}} % 10 %-4\,{\frac {{2}^{n}\left (t+1\right )\left (320\,{t}^{12}-784\,{t}^{10 %}-384\,{t}^{9}+176\,{t}^{8}+372\,{t}^{7}+192\,{t}^{6}-56\,{t}^{4}-45\, %{t}^{3}-11\,{t}^{2}+6\,t+4\right )}{\left (-1+2\,{t}^{2}\right )^{2} %\left (8\,{t}^{4}-1\right )\left (-1+2\,t\right )^{2}\left (4\,{t}^{3} %-1\right )}} % 11 %-48\,{\frac {{t}^{4}\left (t+1\right )^{2}\left (8\,{t}^{2}\right )^{n %}}{\left (-1+2\,t\right )^{2}\left (-1+2\,{t}^{2}\right )^{2}}}\end{aligned}$$ Before proceeding with the analysis of the above exact expressions, a word is due on symmetry breaking. It has been commonly accepted for this model that all odd derivatives of the free energy with respect to field are identically zero in zero field because of symmetry, and only even derivatives have been analyzed [@muller; @morita; @morita2] (in fact, equations similar to (\[ten\]) and (\[twelve\]) for the second and fourth derivatives have been derived in references [@morita; @morita2] by neglecting odd correlations). However, this is true for any lattice in strictly zero field, as every spin configuration has a mirror image (obtained by flipping all the spins) with exactly the same energy, and inverted sign of the configurational magnetization $\sum_{i} S_i$. The usual procedure of breaking the symmetry by retaining an infinitezimal positive field while taking the thermodynamic limit, and only then taking the zero field limit, is not suitable in the present case because the derivatives of the free energy diverge in wide temperature regions (rather than just in a single critical point). Here it seems more appropriate to break the symmetry by applying an infinitely strong local field of infinitezimal range, which can be implemented by restricting a single spin into one of the two possible orientations, while considering all the possible orientations of all the other spins. In a recent work [@SSF3], the present authors have analyzed the effect of restricting a single spin in the $\{+\}$ orientation on magnetization (first derivative of the free energy with respect to field), for the current system. It was found that symmetry is indeed broken by fixing an arbitrary (surface, bulk or center) spin. While without this restriction magnetization is identically zero in strictly zero field for arbitrary system size, it was shown that fixing any spin leads to magnetic ordering of extremely large systems, in a wide temperature range (even if magnetization does go to zero in the thermodynamic limit for all nonzero temperatures). In the rest of this paper, we shall therefore adopt the strategy of breaking the symmetry by fixing a single (central) spin in the $\{+\}$ (upward) orientation while taking the thermodynamic limit, and we shall henceforth use the term $\ell$-th order susceptibility for the quantities $$\chi_n^{(\ell)}\equiv\frac{\beta^{\ell-1}}{N_n}\, \frac{1}{Z_{n}^{+}} \left.{\frac{\partial^\ell Z_{n}^{+}}{\partial (\beta H)^\ell}}\right|_{H=0} \label{thirteen}$$ for finite size systems, and $$\chi^{(\ell)}\equiv\lim_{n\longrightarrow\infty} \chi_n^{\ell} , \label{fourteen}$$ for the thermodynamic limit. The restricted magnetization, $<m>^{+}\equiv\chi^{(1)}$ is found [@SSF; @melin] to be zero for all nonzero temperatures in the thermodynamic limit, (even if it retains nonzero values in a wide temperature range for systems far exceeding in number of particles the observable Universe [@SSF3]), while 2-nd order susceptibility $\chi_n^{(2)}$ diverges below $T_2=2k_B^{-1}J\ln ({\sqrt 2}+1)$. It was also recently shown [@SSF2] that in the thermodynamic limit the divergence of $\chi^{(2)}$ in the vicinity and at $T_2$ is extremely weak, with critical exponent $\gamma=0$, where susceptibility of a finite tree $\chi_n^{(2)}$ diverges proportionally to the three generation level $n$, as $n\longrightarrow \infty$ . In the rest of this paper, we analyze the third and fourth order susceptibility given by $\chi_n^{(3)}\equiv \beta^2{\cal U}_n/N_n$ and $\chi_n^{(4)}\equiv \beta^3{\cal V}_n/N_n$, respectively. Equations (\[ten\]-\[twelve\]) are deceptive in the sense that at first glance they suggest divergence of second, third and fourth order susceptibilities for arbitrary tree generation level $n$, at points $2 t^2=1$ and $2 t=1$ (corresponding to $T_2$ and $T_{BP}$, respectively), while in addition, the fourth order susceptibility seems to diverge (irrespective of $n$) at $4 t^3=1$ and $8 t^4=1$ (the last corresponding to the M" uller-Hartman and Zittartz temperature $T_4$ [@muller]). In fact, the exact expression corresponding to (\[twelve\]), but obtained by neglecting odd correlations [@morita], indeed does diverge at $8 t^4=1$. On the other hand, a more detailed analysis of (\[ten\]-\[twelve\]) shows that the interplay between the individual terms cancels out these aparent singularities at the mentioned temperature points. In particular, upon series expansion of the exact expressions (\[ten\]-\[twelve\]) arround these points (from either side), it is found that there is no divergence for any finite $n$. We give the leading terms of these expansions for large $n$ in Tabs. \[tab1\] and \[tab1b\]. $t\rightarrow 1/2$ $t\rightarrow 1/\sqrt {2}$ ---------------- --------------------------------- -------------------------------------------------------------------------------------------------------------- $\chi_n^{(2)}$ $\left(9/4\right)\ln 3$ $-n\left(\sqrt {2}+3/2\right)\ln \left(\sqrt {2}-1\right)$ $\chi_n^{(3)}$ $n\left(27/8\right)\ln^2 3$ $n\,{2}^{n/2}\left (21/\sqrt {2}+15\right )\left [\ln \left(\sqrt {2}-1\right)\right ]^{2}$ $\chi_n^{(4)}$ $2^n\left(243/16\right)\ln^3 3$ $-{n}^{2}{2}^{n}\left ({\frac {51}{2}}+18\,\sqrt {2}\right )\left [\ln \left(\sqrt {2}-1\right)\right ]^{3}$ : \[tab1\]Leading terms (for large $n$) of the series expansion of susceptibilities $\chi_n^{(2)}$, $\chi_n^{(3)}$ and $\chi_n^{(4)}$, arround points $2t=1$ and $2t^2=1$, demonstrating that expressions (\[ten\]-\[twelve\]) do not diverge at these points for any finite $n$ (see text for details). ------------------------------------------------------------------------------------------------------------------------------------ $\chi_n^{(4)}$ --------------------------------- -------------------------------------------------------------------------------------------------- $t\rightarrow 2^{-\frac{2}{3}}$ ${{2}^{n}}\left [\ln \left({\frac {{2}^{\frac{2}{3}}+1}{{2}^{\frac{2}{3}}-1}}\right)\right ]^{3} \left ({\frac {417}{16}}\,{2}^{\frac{2}{3}}+{\frac {663}{16}}+33\;{2}^{\frac{1}{3}}\right )$ $t\rightarrow 2^{-\frac{3}{4}}$ ${{2}^{n}}\left [\ln \left({\frac {{2}^{\frac{3}{4}}+1}{{2}^{\frac{3}{4}}-1}}\right)\right ]^{3} \left (\frac{81}{8}\,{2}^{\frac{1}{2}}+\frac{33}{4}\,{2}^{\frac{3}{4}}+ {\frac {225}{16}}+12\;{2}^{\frac{1}{4}}\right )$ ------------------------------------------------------------------------------------------------------------------------------------ : \[tab1b\]Leading terms of $\chi_n^{(4)}$ arround $4t^3=1$ and $8t^4=1$. Consequently, divergence of higher order susceptibilities in the thermodynamic limit in different temperature regions is caused only by diverging generation level $n$, and the finite size scaling is accomplished simply by the formula $\ln\chi_n^{\ell}/n$. In Fig. 1 we show the scaled curves of higher order derivatives, obtained using formulas (\[ten\]-\[twelve\]), for several system sizes. It is seen from Fig. 1 that there are three distinct temperature regions, where each temperature represents a point of second, third, or fourth order phase transition. Between zero and $T_2$ the second derivative diverges (the first derivative being finite), and we have a second order phase transition line. Between $T_2$ and $T_{BP}$ the third derivative diverges (first and second derivatives being finite), representing a line of third order phase transitions, while the fourth order derivative diverges between $T_{BP}$ and $\infty$ (where all lower derivatives are finite), representing a fourth order phase transition line. The explicit expressions for the limiting curves, corresponding to the finite size system curves shown in Fig. 1, are obtained by taking the limit $\kappa^\ell\equiv\lim_{n\longrightarrow\infty} {\ln\chi_n^\ell}/{n}$ , and the results obtained using formulas (\[ten\]-\[twelve\]) for different temperature regions are sistematized in Tab. \[tab2\]. ![\[fig1\] Scaled second, third and fourth-order zero-field susceptibilities, calculated using formulas (\[ten\]-\[twelve\]), for several system sizes $n=64, 128, 256$ and $1024$. The dotted vertical lines indicate the points $T_{2}$ and $T_{BP}$, where the transition changes from second to third, and from third to fourth order, respectively. ](chi_n.ps){width="2.7in"} ---------------- --------------------------------------- -------------------------------------------- ------------------------- $0\leq T\leq T_2$ $T_2\leq T\leq T_{BP}$ $T\geq T_{BP}$ $\left( 1\geq t\geq 1/\sqrt 2\right)$ $\left( 1/\sqrt 2 \geq t \geq 1/2 \right)$ $\left(t\leq1/2\right)$ $\kappa^{(2)}$ $\ln\left(2\,t^2\right)$ $0$ $0$ $\kappa^{(3)}$ $\ln\left(4\,t^3\right)$ $\ln\left(2\,t\right)$ $0$ $\kappa^{(4)}$ $\ln\left(8\,t^4\right)$ $\ln 2$ $\ln 2$ ---------------- --------------------------------------- -------------------------------------------- ------------------------- : \[tab2\]Scaled second, third and fourth order susceptibility, in different temperature regions. In conclusion, after decades of continuous interest, the Ising model on the Cayley tree continues to furnish new insights into critical phenomena. Being a highly unphysical system (with its infinite dimension and finite order of ramification), Cayley tree may turn out a unique structure where phase transitions of order higher then two indeed do exist, and the latter may prove to be of only academic interest. On the other hand, it is possible that current findings may turn out relevant for interpretation of experimental data on finite size branching structures, of the real observable physical world. [ E. M" uller-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893.]{} T.P. Eggarter, Phys. Rev. B 9 (1974) 2989. [ J. von Heimburg and H. Thomas, J. Phys. C 7 (1974) 3433.]{} [ H. Matsuda, Prog. Theor. Phys. 51 (1974) 1053.]{} [ T. Sto[š]{}i[ć]{}, B.D. Sto[š]{}i[ć]{} and I.P. Fittipaldi, J. Mag. Mag. Mater. 177-181 (1998) 185.]{} T. Morita and T. Horiguchi, Prog. Theor. Phys. 54 (1975) 982. T. Morita and T. Horiguchi, J. Stat. Phys. 26 (1981) 665. R. M' elin, J.C. Angl' es d’Auriac, P. Chandra\ and B. Douçot, J. Phys. A 29 (1996) 5773. T. Stošić, B.D. Stošić and I.P. Fittipaldi, Physica A, 320 (2003) 443. T. Stošić, B.D. Stošić and I.P. Fittipaldi, cond-mat/0305581, to appear in Physica A (2005).

--- abstract: 'The effect caused by the presence of a number of distinct time scales in a simple stochastic model for the Earth’s atmosphere temperature fluctuations is studied. The model is described by a dissipative dynamics consisting of a set of coupled stochastic evolution equations. The system shows features that resemble recent observations. In contrast to other approaches, like autoregressive models, the fluctuations of the atmosphere’s temperature depend on parameters with clear physical meaning. A reduced version of the model is constructed and its temporal autocovariance function is explicitly written.' author: - | **Arturo Berrones\ Posgrado en Ingenier' ia de Sistemas\ Universidad Autónoma de Nuevo León\ AP 126 - F, Cd. Universitaria\ San Nicolás de los Garza, NL 66450, México.** title: 'Persistence in a Simple Model for the Earth’s Atmosphere Temperature Fluctuations' --- [Keywords: climate models; stochastic processes; long memory effects; long-range correlations; complex systems.]{} Introduction ============ Interaction among processes with several length and time scales is common to a variety of complex systems. For instance, the long-range temporal correlations found in signals from a variety of fields can be associated with an interplay of a number of time scales  [@berrones; @pyragas; @hasselmann; @diezemmann; @berglund]. In particular, it is an extended belief that the persistence observed in the temperature fluctuations of the Earth’s atmosphere is a consequence of its feedback with slower dynamical components in the climate system, like the oceans and Earth’s surface  [@hasselmann]. Persistence at short time scales is related to the everyday life observation where similar weather conditions are likely to be experienced over a given region in a time scale of a few days. The existence of these short-term correlations make weather forecasting possible. The climate’s persistence is also found for larger time scales, however its characterization is a more difficult task  [@kurnaz]. In some recent experiments, the temperature records from different places around the globe have been analyzed. The observations indicate the existence of universal power-laws $C(t) \sim t^{- r}$ describing the correlations of the temperature fluctuations around its mean seasonal value. Although there is some disagreement about the value of the exponent $r$, it has been firmly established that the persistence in temperature fluctuations can indeed be characterized by power-law autocorrelation functions  [@kurnaz]. It has been reported by some authors that for time scales that range from $\sim 1$ to $\sim 25$ years, correlations measured on data from meteorologic stations placed on small islands decay with an exponent $r \sim 0.4$, while for continental stations data are closer to $r \sim 0.7$  [@govindan]. According to other authors  [@fraedrich], the persistence is even more pronounced for the oceanic regions (being roughly characterized by a $1/f$ noise), while in the inner continents $r \sim 1$ (in terms of the power spectrum, a white noise at low frequencies), with a transition region in the coastal zones in which $r \sim 0.7$. The emergence of long-range temporal correlations is a non-trivial feature that can be used to test models of the Earth’s climate  [@govindan]. The understanding of the long-range temporal correlations is fundamental because they characterize the interaction among the different climate components  [@fraedrich]. There is some controversy with respect to the description made by large scale models of the atmospheric temperature variability  [@fraedrich; @vyushin; @comment; @reply; @kurnaz]. The purpose of the present Letter is to introduce a conceptual stochastic model for the fluctuations of the Earth’s atmospheric temperature, or more precisely, its radiated energy (for a survey on conceptual climate models see, for instance, Imkeller and Monahan  [@imkeller]). It will be shown that the model displays qualitative features that closely resemble observations. In contrast with other simple stochastic models of atmospheric temperature fluctuations, like autoregressive models  [@kiraly; @caballero], all the parameters of the presented model have direct physical interpretation. Additionally, as it will be discussed below, the model introduced here also reveals statistical features that are reminiscent of recent observations on the spatial structure of the climate network. Therefore, this work is intended to be a contribution towards the construction of realistic and unexpensive algorithms for Earth’s climate simulation. The model is based on energy balance  [@cushman]. On Earth, as in other planets with a solid crust, the influx of solar radiation is balanced by the outflow from the surface and the atmosphere. In the simplest description of this process, the Earth is treated as a single point. Let us denote by $y$ and $x$ the global averages of the radiation emitted by the atmosphere and by the surface (oceans and land), respectively. A fraction of the Sun’s total radiation is immediately reflected back into space and another is absorbed by the atmosphere. The remainder of the flux is transmitted through the atmosphere and reaches the surface, which in turn absorbs some of the radiation and reflects the rest. The radiation absorbed by the surface is then radiated back as heat. The surface radiates the absorbed energy in the infrared (IR) region of the spectra. It turns out that the atmosphere is not transparent to IR radiation, essentialy due to the presence of the so-called greenhouse gases. Let us denote by $a$ the fraction of IR radiation absorbed by the atmosphere. A fraction $b$ of the total radiation absorbed by the atmosphere is directed towards the surface and the rest is finally lost into space. All these considerations are put together in the well known zero dimensional energy balance model: $$\begin{aligned} \label{zero} y=A+ax, \\ \nonumber x=B+by,\end{aligned}$$ where constants $A$ and $B$ are the net contribution made to $y$ and $x$ by the solar radiation flux, taking into account that some heat is removed by water evaporation from the surface. The constant $a$ is called the infrared (IR) absorption coefficient. All the constants in Eqs. (\[zero\]) are calculated by averaging over a year and over the entire Earth’s surface. Assuming a blackbody process, the average atmosphere’s temperature is given by $y = \nu {\it T}^{4}$, where $\nu $ is the Stefan–Boltzmann constant. In spite of its simplicity, the zero dimensional energy balance model is capable of predicting with very good accuracy the mean Earth’s surface temperature. Another interesting prediction of the zero dimensional energy balance model is the increment of the mean temperature as the coupling parameters $a$ and $b$ grow. In this Letter a spatially extended and time-dependent generalization of model (\[zero\]) is introduced. The Letter is organized as follows: in Sec. \[model\] the model is introduced and the temporal and spatial correlations are discussed numerically. A reduced version of the model is constructed and formally solved in the framework of the Langevin approach. A discussion of the statistical properties of the solution is given. Some conclusions and future directions are discussed in Sec. \[conclusions\]. The Model {#model} ========= Model (\[zero\]) is generalized by the assumption that energy balance is satisfied locally and a transient time is necessary in order to achieve a stationary state. A set of $N$ cells is considered. In each cell, atmosphere interacts with the surface through the local atmosphere’s IR absorption coefficient $a_n$ and the local fraction of heat that the atmosphere returns towards the surface, $b_n$. Each component, atmosphere and surface, has its own intrinsic local response time. The cells of each component interchange radiation via a diffusive process. The model is written as $$\begin{aligned} \label{model} \dot{y}_n = d_1\Delta y_n - \lambda _n [y_n - (A_n + a_n x_n) ] + \varepsilon _n (t), \\ \nonumber \dot{x}_n = d_2\Delta x_n - \gamma _n [x_n - (B_n + b_n y_n)].\end{aligned}$$ In this equation $(\lambda _n)^{-1}$ and $(\gamma _n)^{-1}$ are the local response times of the atmosphere and the surface, respectively. The symbol $\Delta$ is the discrete Laplacian and $d_1$, $d_2$ represent the diffusion coefficients of each component. The term $\varepsilon _n (t)$ is a Gaussian white noise, without correlations between different cells. The meaning of the rest of the terms follow from the zero dimensional energy balance model (\[zero\]). In particular, $y_n$ gives the radiation emitted by the atmosphere in the site $n$ at a given time. The atmosphere is expected to have shorter intrinsic response times than that of the surface. The radiation emitted by the surface at time $t$ in cell $n$ is represented by $x_n$. The noise reflects the more rapid variations, or $weather$. Periodic boundary conditions are taken. The constants $\lambda _n$, $\gamma _n$, $A_n$, $B_n$, $a_n$ and $b_n$ are assumed to be independent variables, such that averages over index (in the limit $N \to \infty$) give the corresponding values of the parameters of the zero dimensional energy balance model, $A$, $B$, $a$, $b$; and the effective inverse response times for the atmosphere and the surface. The first important thing to notice with model (\[model\]) is that it recovers the zero dimensional energy balance model. This can be seen by averaging Eq. (\[model\]) over the cells in the limit $N \to \infty$. The following reduced version of Eq. (\[model\]) is obtained: $$\begin{aligned} \label{motion} \dot{y}=-\lambda [y-(A+ax)]+\varepsilon(t), \\ \nonumber \dot{x}=-\gamma [x-(B+by)].\end{aligned}$$ From the fact that Eq. (\[motion\]) represents an overdamped dynamics in a parabolic potential, the system converges to a stationary state with mean value given by the solution of the zero dimensional energy balance model. This result shows that model (\[model\]) is capable of representing with good approximation the mean behavior of energy in the coupled atmosphere – surface system. Now it will be argued that Eq. (\[model\]) can give realistic descriptions of temperature fluctuations as well. A large variability over the intrinsic characteristic times of the surface around the globe is expected to exist, as a consequence of the different response times present in the geosphere, hydrosphere, cryosphere and biosphere. At first instance, this situation is modeled by treating the $\gamma _n$’s like independent random variables taken from a uniform probability distribution. On the other hand, under the basis of the relative homogeneity of the atmosphere composition, it will be assumed by now that the response time to perturbations of the atmosphere is the same in all cells. In what follows, the value $\lambda_n = \lambda = 1$ is used. Under these assumptions, the time unit is defined as the mean atmospheric response time. The model represents the interaction between a hierarchy of time scales, ranging from minutes to days to geological times. From this point of view, it would be therefore reasonable to assume that $1/ \lambda$ lies in the intermediate scales, roughly in the range from weeks to months. This and other important aspects about the definition of the parameters in the model are intended to be refined by the author in the near future by close cooperation with climate experts. At this point the main goal is limited to explore the capabilities of the model (\[model\]), in order to give qualitatively realistic statistical descriptions of the temperature fluctuations present in the Earth’s atmosphere. Figure \[fig1\](a) is a log–log plot of the power spectrum of the time series of the atmosphere’s temperature in a particular cell. This time series is obtained from the corresponding time series for $y_n (t)$ under the assumption of a blackbody process. The parameters $a_n$, $b_n$, $A_n$ and $B_n$ are left constant and set to their experimental global averages: $a_n = a = 0.96$, $b_n = b = 0.61$, $A_n = A = 179.36$ ${\rm W}/{\rm m}^{2}$ and $B_n = B = 47.82$ ${\rm W}/{\rm m}^{2}$  [@cushman]. The values $\gamma _n$ are drawn from a uniform distribution on the range $(0, 0.2)$. The noise values $\varepsilon _n (t)$ are uniformly distributed over the interval $(-1, 1) W/m^{2}$. The diffusion coeffiecients are taken as $d_1 = d_2 = 1$. System size is $N=50$. The power spectrum is consistent with a power law at low frequencies, with exponent $\sim -0.25$. This exponent of the power spectrum implies a power-law decay of correlations at large times, $C(t) \propto t^{- r}$, with $r \sim 0.75$. An alternative approach is given in Fig. \[fig1\](b). In order to characterize the correlations, the sum of the temperature values on time at a given cell is studied, resulting in: $$\begin{aligned} \label{rw} Y_n(t)=\sum_{\tau = 1}^{t} T_n(\tau),\end{aligned}$$ where $T_n(\tau)$ stands for the atmosphere’s temperature at cell $n$ and time $\tau$. The signal $Y_n(t)$ is then compared to a random walk. In Fig. \[fig1\](b) the standard deviation of $Y_n(t)$ as a function of time is plotted. For large times (roughly greater than $50$ time units), $\sigma \sim t^{0.648}$, which implies a power-law decay of the autocorrelation function at large times, with exponent $\sim 0.7$. This result is consistent with Fig. \[fig1\](a). 0.5cm 0.5cm An analysis of the radiation spectrum has been carried out for the same experimental setup as above. Results indicate that radiation and temperature spectra are basically equivalent, differing at most by a normalization factor. This numerical finding is consistent with recent studies that suggest that a strongly correlated signal preserves its correlation properties after an even polynomial transformation  [@stanley]. 0.5cm The model is also capable of showing scaling behavior for surface temperature. As an example, the power spectrum of surface temperature fluctuations of an induvidual cell is shown in Fig. \[fig1c\]. The system has the same parameter values as before, but $d_2=0$. Notice that with this choice of parameters, the coupling with the atmosphere is essential for the emergence of scaling in the surface temperature. The model displays an interesting spatial structure. In Fig. \[fig2\] a case is considered in which the system size is $N=200$ and the other parameters are the same as in the case presented in Fig. \[fig1\]. The spatial autocorrelation function $C(n)$ is inferred from the power spectrum of the temperature values vector at a fixed time. A least-square fit of the power spectrum to a power-law function indicates that $S(k) \sim k^{-0.22}$, which implies that the spatial autocorrelation function can be roughly characterized by $C(n) \sim n^{-0.78}$ for spatial scales in the range from one to $200$ cells. Further numerical analysis of the same model setup indicates that if larger spatial scales are considered, the power spectrum displays a crossover to white noise at low frequencies. 0.5cm In a more realistic description, the parameters (besides $\gamma _n$) must have some local variability due, for instance, to differences in the Earth’s albedo and in the solar radiation flux over different regions. As an example, a case in which $A_n = A + \epsilon _n$ and $B_n = B + \epsilon ^{'}_{n}$ is discussed in Fig. \[fig3\](a). The $\epsilon$’s are independent random variables uniformly distributed in the range $(-1, 1)$ ${\rm W}/{\rm m}^{2}$. The other parameters are chosen as before. The power spectrum presented in Fig. \[fig3\](a) indicates a crossover between two different scaling regimes. For scales from $50$ to $1000$ cells the correlation function is consistent with $C(n) \sim n^{-0.75}$, while for shorter scales $C(n) \sim n^{-0.2}$. A situation in which the coupling parameters have also local variability is presented in Fig. \[fig3\](b). The values of $b_n$ are set as $b_n = b + \rho _{n} $, where $\rho _{n}$ is uniformly distributed over the interval $(-0.15, 0.15)$. The parameters $a_n$ are taken as $a_n = 0.98$ for all $n$, which implies a stronger mean coupling. The other parameters are the same as in Fig. \[fig3\](a). In Fig. \[fig3\](b) the correlation decay is faster for the short scales and slower for the large scales in comparision to Fig. \[fig3\](a). This effect can be interpreted as an increment of large scale coherence as the mean coupling grows, while the spatial coherence at short scales decreases due to the increment in the local variability. The scaling of the spatial autocorrelation function displayed by the model is reminiscent of recent observations on the spatial structure of the climate network, which indicate that nodes in the climate system conform to a network with the [*small-world*]{} property  [@tsonis]. This property is related to the presence of significant correlations between distant nodes. 0.5cm 0.5cm The numerical findings strongly suggest that model (\[model\]) shows qualitative features that are close to the observations. However, a more precise definition of the parameters is needed. For instance, a different choice of the scale separation between the $\gamma$’s and $\lambda$ in general lead to different properties of the autocorrelations. Another aspect to be refined is concerned with the already mentioned spatial variability of the parameters. This question is closely related to the definition of the size associated to cells. In a realistic model setup, the parameter values come from spatial averages over the region $n$. In the simple $1d+1$ situation discussed here, those would be global averages over a given latitudinal interval. As already mentioned, these and other relevant questions are intended to be investigated by the author in the near future, working in close contact with climate experts. In order to gain insight into model (\[model\]) it is discussed analytically its reduced version given by Eq. (\[motion\]). Without loss of generality, the constants $A$ and $B$ are chosen equal to zero. The term $\varepsilon(t)$ is a Gaussian white noise, defined through the moments $\left < \varepsilon(t) \right > = 0$, $\left < \varepsilon(t) \varepsilon(t^{'}) \right > = D \delta (t-t^{'})$ and with all higher moments equal to zero. The diffusion constant is a parameter that measures the strength of the noise. The function $\delta (t-t^{'})$ is a Dirac’s delta. In the absence of coupling $x$ and $y$ simply converge exponentially to the stationary state $\left< y \right>=x=0$, $\left< y^{2} \right>= \frac{D}{2\lambda}$ with characteristic times $\frac{1}{\lambda}$ and $\frac{1}{\gamma}$. It is assumed that $\lambda > \gamma $ so one of the dynamics is $fast$ and the other is $slow$. Strictly speaking, the system has three time scales, the third one being associated with the noise. However this time scale has an infinite separation with respect to the other two. In the language of control theory, we can view $y$ like an output system with uncertainties (noise) that has feedback with an input whose response time is different from the time scale of $y$. Applying a Laplace transform over time to Eq. (\[motion\]) and assuming for simplicity the initial conditions $x(0)=y(0)=0$, one gets $$\begin{aligned} \label{algebra} s y(s)=-\lambda y(s) + \lambda a x(s) + \varepsilon(s), \\ \nonumber s x(s)=-\gamma x(s) + \gamma b y(s).\end{aligned}$$ Solving Eq. (\[algebra\]) for $y(s)$ and $x(s)$, and by the use of the Faltung theorem, a formal solution for $y(t)$ in terms of the noise is found. This solution can be used to write explicitly the covariance function, that describes the fluctuations around the mean value. [^1] For large times, the covariance function is given by the following expression: $$\begin{aligned} \label{correlation} \left < y(t)y(t+T) \right > = q_1 e^{\mu _1 T} + q_2 e^{\mu _2 T},\end{aligned}$$ where $$\begin{aligned} q_1 = \frac{D}{16K^2}(\lambda - \gamma + 2k) \left (\frac{\lambda - \gamma + 2K}{\lambda + \gamma + 2K} + \frac{\gamma - \lambda + 2K}{\gamma + \lambda} \right ),\end{aligned}$$ $$\begin{aligned} q_2 = \frac{D}{16K^2}(\gamma - \lambda + 2K) \left (\frac{\gamma - \lambda + 2K}{\gamma + \lambda - 2K} + \frac{\lambda - \gamma + 2K}{\gamma + \lambda} \right ),\end{aligned}$$ $$\begin{aligned} \mu _1 = -\frac{\gamma + \lambda}{2} - K,\end{aligned}$$ $$\begin{aligned} \mu _2 = -\frac{\gamma + \lambda}{2} + K,\end{aligned}$$ $$\begin{aligned} K = \sqrt{\left ( \frac{\gamma +\lambda}{2}\right )^{2} - \gamma \lambda (1-ab)}.\end{aligned}$$ In the region of interest of the parameter space, correlations decay monotonically with a characteristic time $$\begin{aligned} \tau = \frac{1}{ \left( \frac{\gamma + \lambda}{2} \right) - \sqrt{\left(\frac{\gamma + \lambda}{2} \right)^{2} - \gamma \lambda (1-ab) } }.\end{aligned}$$ It must be realized that $\tau$ is greater than any of the two intrinsic times. Therefore, because of the feedback there is an emergence of large memory. The characteristic time becomes infinitely large when $\gamma \to 0$ with $\lambda$ finite. This result is consistent with previous works on reduced models of temperature fluctuations  [@fraedrich2]. 0.5cm The reduced model typically shows a region in which none of the two intrinsic time scales is dominant and correlations cannot be adequately fitted by a single exponential. For instance, with $\gamma = 0.2$, $\lambda = 1$, $a = 0.96$, $b = 0.64$ and $D = 1 (\frac{W}{m^{2}})^{2}$, the covariance function is not exponential for time scales approximately an order of magnitude greater than the intrinsic time of the fast variable, as Fig. \[fig4\] shows. Approximation to a power-law or other types of slow decay by a sum of exponentials with different characteristic times has been discussed in several fields  [@anderson; @bouchaud]. In particular, this mechanism has been already proposed in  [@caballero], in order to explain the persistence found in the atmosphere’s temperature record, by fitting the coefficients of a 3d AR(1) type process to data. Conclusions =========== The features shown by the spatially extended stochastic model presented here motivates the construction of realistic and simple algorithms for the prediction of the Earth’s temperature distribution and fluctuations. In this spatially extended situation the parameters vary locally, so there is a number of characteristic times. In order to gain insight on the extended model, a reduced version of it has been constructed and the covariance function explicitly written. One of the future directions of the work is to conduct a more general study of the spatially extended model, in close connection to climate research to have plausible parameter values. 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--- abstract: 'We provide a new representation-independent formulation of Occam’s razor theorem, based on Kolmogorov complexity. This new formulation allows us to: (i) Obtain better sample complexity than both length-based [@blumer1] and VC-based [@blumer] versions of Occam’s razor theorem, in many applications; and (ii) Achieve a sharper reverse of Occam’s razor theorem than that of [@board]. Specifically, we weaken the assumptions made in [@board] and extend the reverse to superpolynomial running times.' author: - 'Ming Li,' - 'John Tromp,' - Paul Vitányi title: 'Sharpening Occam’s Razor' --- Analysis of algorithms ,pac-learning ,Kolmogorov complexity ,Occam’s razor-style theorems Introduction {#introsec} ============ Occam’s razor theorem as formulated by [@blumer; @blumer1] is arguably the substance of efficient pac learning. Roughly speaking, it says that in order to (pac-)learn, it suffices to compress. A partial reverse, showing the necessity of compression, has been proved by Board and Pitt [@board]. Since the theorem is about the relation between effective compression and pac learning, it is natural to assume that a sharper version ensues by couching it in terms of the [*ultimate*]{} limit to effective compression which is the Kolmogorov complexity. We present results in that direction. Despite abundant research generated by its importance, several aspects of Occam’s razor theorem remain unclear. There are basically two versions. The [*VC dimension-based version*]{} of Occam’s razor theorem (Theorem 3.1.1 of [@blumer]) gives the following upper bound on sample complexity: For a hypothesis space $H$ with $VCdim(H)=d$, $1 \leq d < \infty$, $$\label{vc-sample} m(H,\delta , \epsilon ) \leq \frac{4}{\epsilon} (d \log \frac{12}{\epsilon} + \log \frac{2}{\delta} ).$$ The following lower bound was proved by Ehrenfeucht [*et al*]{} [@ehren]. $$\label{vc-lowerbound} m(H,\delta , \epsilon ) > \max (\frac{d-1}{32 \epsilon}, \frac{1}{\epsilon} \ln \frac{1}{\delta} ).$$ The upper bound in (\[vc-sample\]) and the lower bound in (\[vc-lowerbound\]) differ by a factor $\Theta (\log \frac{1}{\epsilon} )$. It was shown in [@haussler] that this factor is, in a sense, unavoidable. When $H$ is finite, one can directly obtain the following bound on sample complexity for a consistent algorithm: $$\label{direct-sample} m(H,\delta , \epsilon ) \leq \frac{1}{\epsilon} \ln \frac{|H|}{\delta}.$$ For a graded boolean space $H_n$, we have the following relationship between the VC dimension $d$ of $H_n$ and the cardinality of $H_n$, $$d \leq \log |H_n | \leq nd.$$ When $\log |H_n|=O(d)$ holds, then the sample complexity upper bound given by (\[direct-sample\]) can be seen to equal $\frac{1}{\epsilon} (O(d)+\ln \frac{1}{\delta})$ which matches the lower bound of (\[vc-lowerbound\]) up to a constant factor, and thus every consistent algorithm achieves optimal sample complexity for such hypothesis spaces. The [*length-based version*]{} of Occam’s razor theorem then gives the following sample complexity $m$ to guaranty that the algorithm pac-learns: For given $\epsilon$ and $\delta$: $$\label{length-sample} m = \max (\frac{2}{\epsilon} \ln \frac{1}{\delta} , (\frac{(2\ln 2)s^{\beta}}{\epsilon} )^{1/(1-\alpha)} ) ,$$ This bound is based on the [*length-based*]{} Occam algorithm [@blumer]: A [*deterministic*]{} algorithm that returns a consistent hypothesis of length at most $m^\alpha s^\beta$, where $\alpha < 1$ and $s$ is the length of the target concept. In summary, the VC dimension based Occam’s razor theorem may be hard to use and it sometimes does not give the best sample complexity. The length-based Occam’s razor is more convenient to use and often gives better sample complexity in the discrete case. However, as we demonstrate below, the fact that the length-based Occam’s razor theorem sometimes gives inferior sample complexity, can be due to the redundant representation format of the concept. We believe Occam’s razor theorem should be “representation-independent”. That is, it should not be dependent on accidents of “representation format”. (See [@manfred] for other representation-independence issues.) In fact, the sample complexities given in (\[vc-sample\]) and (\[vc-lowerbound\]) are indeed representation-independent. However they are not easy to use and do not give optimal sample complexity. Here, we give a Kolmogorov complexity based Occam’s razor theorem. We will demonstrate that our KC-based Occam’s razor theorem is convenient to use (as convenient as the length based version), gives a better sample complexity than the length based version, and is representation-independent. In fact, the length based version can be considered as a specific computable approximation to the KC-based Occam’s razor. As one of the examples, we will demonstrate that the standard trivial learning algorithm for monomials actually often has a [*better sample complexity*]{} than the more sophisticated Haussler’s greedy algorithm [@hauss]. This is contrary to the commen, but mistaken, belief that Haussler’s algorithm is better in all cases (to be sure, Haussler’s method is superior for target monimials of small length). Another issue related to Occam’s razor theorem is the status of the reverse assertion. Although a partial reverse of Occam’s razor theorem has been proved by [@board], it applied only to the case of polynomial running time and sample complexity. They also required a property of closure under exception list. This latter requirement, although quite general, excludes some reasonable concept classes. Our new formulation of Occam’s razor theorem allows us to prove a more general reverse of Occam’s razor theorem, allowing the arbitrary running time and weakening the requirement of exception list of [@board]. [^1] [**Discussion of Result and Technique:**]{} In our approach we obtain better bounds on the sample complexity to learn the representation of a target concept in the given representation system. These bounds, however, are representation-independent and depend only on the Kolmogorov complexity of the target concept. If we don’t care about the representation of the hypothesis (but that is not the case in this paper) then better “iff Occam style” characterizations of polynomial time learnability/predicatability can be given. They rely on Schapire’s result that “weak learnability” equals “strong learnability” in polynomial time [@Sch90] exploited in [@HeWa95]. For a recent survey of the important related “boosting” technique see [@Sch02]. The use of Kolmogorov complexity is to obtain a bound on the size of the hypotheses class for a fixed (but arbitrary) target concept. Obviously, the results described can be obtained using other proof methods—all true provable statements must be provable from the axioms of mathematics by the inference methods of mathematics. The question is whether a particular proof method facilitates and guides the proving effort. The message we want to convey is that thinking in terms of coding and incompressibility suggest improvements to long-standing results. A survey of the use of the Kolmogorov complexity method in combinatorics, computational complexity, and the analysis of algorithms is [@lv] Chapter 6. Occam’s Razor ============= Let us assume the usual definitions, say Anthony and Biggs [@anthony], and notation of [@board]. For Kolmogorov complexity we assume the basics of [@lv]. In the following $\Sigma, \Gamma$ is are finite [*alphabets*]{}: We consider only discrete learning problems in this paper. The set of finite strings over $\Sigma$ is denoted by $\Sigma^*$ and similarly for $\Gamma$. An element of $\Sigma^*$ is an [*example*]{}, and a [*concept*]{} is a set of examples (a language over $\Sigma$). An [*representation*]{} is an element of $\Gamma^*$. A [*representation system*]{} is a tuple $(R,\Gamma , c , \Sigma )$, where $R \subset \Gamma^*$ is the set of representations, and $c:R \rightarrow 2^{\Sigma^*}$ maps representations to concepts, the latter being languages over $\Sigma$. Hence, given $R$ the mapping $c$ determines a [*concept class*]{}. For example, let $\Gamma$ is the alphabet to express Boolean formulas, $\Sigma = \{0,1\}$, and let $R$ be the subset of disjunctive normal form (DNF) formulas. Let $c$ map each element $r \in R$, say a DNF formula over $n$ variables, to $c(r) \subseteq \{0,1\}^n$ such that every example $e \in c(r)$ viewed as truth-value assignment makes $r$ “true”. That is, if $e=e_1 \ldots e_n$ and we assign “true” or “false” to the $i$th variable in $r$ according to whether $e_i$ equals “0” or “1” then $r$ becomes “true”. Each concept in the thus defined concept class is the set of truth assignments that make a particular DNF formula “true”. A [*pac-algorithm*]{} for a representation system ${\bf R} = (R,\Gamma , c , \Sigma )$ is a randomized algorithm $L$ such that, for every $s,n\geq 1,\epsilon>0,\delta>0,r \in R^{\leq s}$, and every probability distribution $D$ on $\Sigma^{\leq n}$, if $L$ is given $s,n,\epsilon,\delta$ as input and has access to an oracle providing examples of $c(r)$ (the concept represented by $r$) according to $D$, then $L$, with probability at least $1-\delta$, outputs a representation $r' \in R$ approximating the target $r$ in the sense that $D(c(r')\Delta c(r)) \leq \epsilon$. Here, $\Delta$ denotes the symmetric set difference. The acronym “pac” coined by Dana Angluin stands for “probably approximately correct” which aptly captures the requirement the output representation must satisfy according to the definition. The question of interest in pac-learning is how many examples (and running time) a learning algorithm has to qualify as a pac-alpgorithm. The [*running time*]{} and and number of examples ([*sample complexity*]{}) of the pac-algorithm are expressed as functions $t(n,s,\epsilon,\delta)$ and $m(n,s,\epsilon,\delta)$. The following definition generalizes the notion of Occam algorithm in [@blumer]: \[def.kcoccam\] An [*Occam-algorithm*]{} for a representation system ${\bf R} = (R,\Gamma , c , \Sigma )$ is a randomized algorithm which for every $s,n\geq 1, \gamma >0$, on input of a sample consisting of $m$ examples of a fixed target $r\in R^{\leq s}$, with probability at least $1-\gamma$ outputs a representation $r' \in R$ consistent with the sample, such that $K(r' \mid r,n,s) < m/f(m,n,s,\gamma)$, with $f(m,n,s,\gamma)$, the compression achieved, being an increasing function of $m$. The [*length-based version*]{} of (possibly randomized) Occam algorithm can be obtained by replacing $K(r' \mid r,n,s)$ by $|r|$ in this definition. The [*running time*]{} of the Occam-algorithm is expressed as a function $t(m,n,s,\gamma)$, where $n$ is the maximum length of the input examples. \[rem.kco\] An Occam algorithm satisfying a given $f$, achieves a lower bound on the number $m$ of examples required in terms of $K(r' \mid r,n,s)$, the Kolmogorov complexity of the outputted representation conditioned on the target representation, rather than the (maximal) length $s$ of $r$ as in the original Occam algorithm [@blumer] and the length-based version above. This improvement enables one to use information drawn from the hidden target for reduction of the Kolmogorov complexity of the output representation, and hence further reduction of the required sample complexity. We need to show that the main properties of an Occam algorithm are preserved under this generalization. Our first theorem is a Kolmogorov complexity based Occam’s Razor. We denote the minimum $m$ such that $f(m,n,s,\gamma) \geq x$ by $f^{-1}(x,n,s,\gamma)$, where we set $f^{-1}(x,n,s,\gamma)=\infty$ if $f(m,n,s,\gamma) < x$ for every $m$. \[KCoccam\] Suppose we have an Occam-algorithm for ${\bf R} = (R,\Gamma , c , \Sigma )$ with compression $f(m,n,s,\gamma)$. Then there is a pac-learning algorithm for [**R**]{} with sample complexity $$m(n,s,\epsilon,\delta) = \max \left\{\frac{2}{\epsilon}\ln \frac{2}{\delta}, f^{-1}(\frac{2\ln 2}{\epsilon},n,s,\delta/2) \right\},$$ and running time $t_{\mbox{pac}}(n,s,\epsilon,\delta) = t_{\mbox{occam}}(m(n,s,\epsilon,\delta),n,s,\delta/2)$. On input of $\epsilon,\delta,s,n$, the learning algorithm will take a sample of length $m=m(n,s,\epsilon,\delta)$ from the oracle, then use the Occam algorithm with $\gamma=\delta/2$ to find a hypothesis (with probability at least $1-\delta/2$) consistent with the sample and with low Kolmogorov complexity. In the proof we abbreviate $f(m,n,s,\gamma)$ to $f(m)$ with the other parameters implicit. Learnability follows in the standard manner from bounding (by the remaining $\delta/2$) the probability that all $m$ examples of the target concept fall outside the, probability $\epsilon$ or greater, symmetric difference with a bad hypothesis. Let $m = m(n,s,\epsilon,\delta)$. Then $m \geq f^{-1} (\frac{2 \ln 2}{\epsilon} ,n,s, \frac{\delta}{2})$ gives $$\epsilon - \frac{\ln 2}{f(m)} \geq \frac{\epsilon}{2},$$ and therefore $m \geq \frac{2}{\epsilon}\ln \frac{2}{\delta}$ gives $$m(\epsilon - \frac{\ln 2}{f(m)} ) \geq \ln \frac{2}{\delta} .$$ This implies (taking the exponent on both sides and using $1-\epsilon<e^{-\epsilon }$) $$2^{m/f(m)}(1-\epsilon)^{m} \leq \delta/2 .$$ The probability that some concept the Occam-algorithm can output has all $m$ examples being bad is at most the number of concepts of complexity less than $m/f(m)$, times $(1-\epsilon)^m$, which by the above is at most $\delta/2$. When the compression is of the form $$f(m,n,s,\gamma) = \frac{m^{1-\alpha}}{p(n,s,\gamma)},$$ one can achieve a sample complexity of $$\max\left\{\frac{2}{\epsilon}\ln \frac{2}{\delta}, \left( \frac{(2 \ln 2)p(n,s,\delta/2)}{\epsilon} \right)^{1/(1-\alpha)}\right\}.$$ In the special case of total compression, where $\alpha=0$, this further reduces to $$\label{total-compression} \frac{2}{\epsilon}\left\{\max(\ln \frac{2}{\delta},(\ln 2) p(n,s,\delta/2))\right\}.$$ For deterministic Occam-algorithms, we can furthermore replace $2/\delta$ and $\delta/2$ in Theorem \[KCoccam\] by $1/\delta$ and $\delta$ respectively. Essentially, our new Kolmogorov complexity condition is a computationally universal generalization of the length condition in the original Occam’s razor theorem of [@blumer1]. Here, in Theorem \[KCoccam\], we consider the shortest description length over all effective representations, given the target representation, rather than in a specific (syntactical) representation system. This allows us to bound the required sample complexity not by a function of the number of hypotheses (returned representations) of length at most the bound on the length of the target representation, but by a similar function of the number of hypotheses that have a certain Kolmogorov complexity conditioned on the target concept, see Remark \[rem.kco\]. Nonetheless, like in the original Occam’s razor Theorem of [@blumer1], we return a representation of a concept approximating the target concept in the given representation system, rather than a representation outside the system like in Boosting approaches. Suppose we have a concept $c$ and a mis-classified example $x$—an [*exception*]{}. Then, the symmetric difference $c \Delta \{x\}$ classifies $x$ correctly: if $x \not\in c$ then $c \Delta \{x\} = c \bigcup \{x\}$, and if $x \in c$ then $c \Delta \{x\} = c \setminus \{x\}$. An [*exception handler*]{} for a representation system ${\bf R} = (R,\Gamma , c , \Sigma )$ is an algorithm which on input of a representation $r\in R$ of length $s$, and an $x \in \Sigma^{\ast}$ of length $n$, outputs a representation $r' \in R$ of the concept $c(r) \Delta \{x\}$, of length at most $e(s,n)$, where $e$ is called the [*exception expansion*]{} function. The running time of the exception-handler is expressed as a function $t(n,s)$ of the representation and exception lengths. If $t(n,s)$ is polynomial in $n,s$, and furthermore $e(s,n)$ is of the form $s+p(n)$ for some polynomial $p$, then we say ${\bf R}$ is [*polynomially closed under exceptions*]{}. \[sex\] Let $L$ be a deterministic pac-algorithm with $m(n,s,\frac{1}{2n},\gamma)$ the sample size, and let $E$ be an exception handler for a representation system ${\bf R}$. Then there is an Occam algorithm for ${\bf R}$ that for $m$ examples achieves compression $f(m,n,s,\gamma)= \frac{1}{2\epsilon n}$. Moreover, $m \geq 2nm(n,s,\frac{1}{2n},\gamma)$ and where $\epsilon$, depending on $m,n,s,\gamma$, is such that $m(n,s,\epsilon,\gamma)=\epsilon m$ holds. The proof is obtained in a fashion similar to [@board]. Suppose we are given a sample of length $m$ and confidence parameter $\gamma$. Assume without loss of generality that the sample contains $m$ different examples. Define a uniform distribution on these examples with $\mu (x) = 1/m$ for each $x$ in the sample. Let $\epsilon$ be as described. The function $m(n,s,\epsilon,\gamma)$ decreases with increasing $\epsilon$, while the function $\epsilon m$ increases with $\epsilon$ so the two necessarily intersect, under the assumption in the theorem, for some $\epsilon_0$, although it may yield an $\epsilon_0 >\frac{1}{2n}$, giving no actual compression. For example, if $m(n,s,\epsilon,\gamma)= (\frac{1}{\epsilon})^{b}$ for some constant $b$, then $\epsilon_0 = m^{-1/(b+1)}$. Apply $L$ with $\delta = \gamma$ and $\epsilon = \epsilon_0$. With probability $1-\gamma$, it produces a concept which is correct with error $\epsilon$, giving up to $\epsilon m$ exceptions. We can just add these one by one using the exception handler. This will expand the concept size, but not the Kolmogorov complexity. The resulting representation can be described by the $\leq \epsilon m$ examples used plus the $\leq \epsilon m$ exceptions found, Since $L$ is deterministic, this uniquely determines the required consistent concept. The compression achieved is $\frac{m}{2\epsilon mn} = \frac{1}{2\epsilon n}$. This is an increasing function of $m$, since increasing the slope of the function $\epsilon m$ moves its intersection with the function $m(n,s,\epsilon,\gamma)$ to the left, that is, to smaller $\epsilon$. Let ${\bf R} = (R,\Gamma , c , \Sigma )$ be a representation system. The concept ${\mbox{MAJ}}(r_1,r_2,r_3)$ is the set $\{x :$ $x$ belongs to at least two out of the three concepts $c(r_1),c(r_2),c(r_3)\}$. A [*majority-of-three algorithm*]{} for ${\bf R}$ is an algorithm which on input of three representation $r_1,r_2,r_3 \in R^{\leq s}$, outputs a representation $r' \in R$ of the concept ${\mbox{MAJ}}(r_1,r_2,r_3)$ of length at most $e(s)$, where $e$ is called the [*majority expansion*]{} function. The running time of the algorithm is expressed as a function $t(s)$ of the maximum representation length. If $t(s)$ and $e(s)$ are polynomial in $s$ then we say ${\bf R}$ is [*polynomially closed under majority-of-three*]{}. \[maj\] Let $L$ be a deterministic pac-algorithm with sample complexity $m(n,s,\epsilon,\delta) \in o(1/\epsilon^2)$, and let $M$ be a majority-of-three algorithm for the representation system ${\bf R}$. Then there is an Occam algorithm for ${\bf R}$ that for $m$ examples has compression $f(m,n,s,\gamma)=m/3nm(n,s,\frac{1}{2\sqrt{m}},\gamma/3)$. Let us be given a sample of length $m$. Take $\delta = \gamma / 3$ and $\epsilon = \frac{1}{2\sqrt{m}}$. [*Stage 1:*]{} Define a uniform distribution on the $m$ examples with $\mu_1 (x) = 1/m$ for each $x$ in the sample. Apply the learning algorithm. It produces (with probability at least $1-\gamma/3$) a hypothesis $r_1$ which has error less than $\epsilon$, giving up to $\epsilon m = \sqrt{m}/2$ exceptions. Denote this set of exceptions by $E_1$. [*Stage 2:*]{} Define a new distribution $\mu_2(x) = \epsilon$ for each $x \in E_1$, and $\mu_2(x) = (1-|E_1|/2\sqrt{m})/(m-|E_1|)$ for each $x \not\in E_1$. Apply the learning algorithm. It produces (with probability at least $1-\gamma/3$) a hypothesis $r_2$ which is correct on all of $E_1$ and with error less than $\epsilon$ on the remaining examples. This gives up to $\epsilon (m-|E_1|) / (1-|E_1|/2\sqrt{m}) < \sqrt{m}$ exceptions. This set, denoted $E_2$, is disjoint from $E_1$. [*Stage 3:*]{} Define a new distribution on the $m$ examples with $\mu(x) = 1/|E_1 \cup E_2| > \epsilon$ for each $x$ in $E_1\cup E_2$, and $\mu(x) = 0$ elsewhere. Apply the learning algorithm. The algorithm produces (with probability at least $1-\gamma/3$) a hypothesis $r_3$ which is correct on all of $E_1$ and $E_2$. In total the number of examples consumed by the pac-algorithm is at most $3m(n,s,\frac{1}{2\sqrt{m}},\gamma/3)$, each requiring $n$ bits to describe. The three representations are combined into one representation by the majority-of-three algorithm $M$. This is necessarily correct on all of the $m$ examples, since the three exception-sets are all disjoint. Furthermore, it can be described in terms of the examples fed to the deterministic pac-algorithm and thus achieves compression $f(m,n,s,\gamma) = m/3nm(n,s,\frac{1}{2\sqrt{m}},\gamma/3)$. This is an increasing function of $m$ given the assumed subquadratic sample complexity. The following corollaries use the fact that if a representation system is learnable, it must have finite VC-dimension and hence, according to (\[vc-sample\]), they are learnable with sample complexity subquadratic in $\frac{1}{\epsilon}$. Let a representation system ${\bf R}$ be closed under either exceptions or majority-of-three, or both. Then ${\bf R}$ is pac-learnable iff there is an Occam algorithm for ${\bf R}$. Let a representation system ${\bf R}$ be polynomially closed under either exceptions or majority-of-three, or both. Then ${\bf R}$ is deterministically polynomially pac-learnable iff there is a polynomial time Occam algorithm for ${\bf R}$. [*Example.*]{} Consider threshold circuits, acyclic circuits whose nodes compute threshold functions of the form $a_1x_1 + a_2x_2 + \cdots +a_nx_n \geq \delta$, $x_i \in \{0,1\}, a_i,\delta \in N$ (note that no expressive power is gained by allowing rational weights and threshold). A simple way of representing circuits over the binary alphabet is to number each node and use [*prefix-free encodings*]{} of these numbers. For instance, encode $i$ as $1^{|\mbox{bin}(i)|}0\mbox{bin}(i)$, the binary representation of $i$ preceded by its length in unary. A complete node encoding then consists of the encoded index, encoded weights, threshold, encoded degree, and encoded indices of the nodes corresponding to its inputs. A complete circuit can be encoded with a node-count followed by a sequence of node-encodings. For this representation, a majority-of-three algorithm is easily constructed that renumbers two of its three input representations, and combines the three by adding a 3-input node computing the majority function $x_1+x_2+x_3 \geq 2$. It is clear that under this representation, the system of threshold circuits are polynomially closed under majority-of-three. On the other hand they are not closed under exceptions, or under the exception lists of [@board]. [*Example.*]{} Let $h_1 , h_2, h_3$ be 3 $k$-DNF formulas. Then ${\mbox{MAJ}}(h_1,h_2,h_3) = (h_1 \wedge h_2) \vee (h_2 \wedge h_3) \vee (h_3 \wedge h_1)$ which can be expanded into a $2k$-DNF formula. This is not good enough for Theorem \[maj\], but it allows us to conclude that pac-learnability of $k$-DNF implies compression of $k$-DNF into $2k$-DNF. Applications ============ Our KC-based Occam’s razor theorem might be [*conveniently*]{} used, providing better sample complexity than the length-based version. In addition to giving better sample complexity, our new KC-based Occam’s razor theorem, Theorem \[KCoccam\], is easy to use, as easy as the length based version, as demonstrated by the following two examples. While it is easy to construct an artificial system with extremely bad representations such that our Theorem \[KCoccam\] gives [*arbitrarily*]{} better sample complexity than the length-based sample complexity given in (\[length-sample\]), we prefer to give natural examples. [**Application 1: Learning a String.**]{} The DNA sequencing process can be modeled as the problem of learning a super-long string in the pac model [@jiang1; @li]. We are interested in learning a target string $t$ of length $s$, say $s=3 \times 10^9$ (length of a human DNA sequence). At each step, we can obtain as an example a substring of this sequence of length $n$, from a random location of $t$ (Sanger’s Procedure). At the time of writing, $n \approx 500$, and sampling is very expensive. Formally, the concepts we are learning are sets of possible length $n$ substrings of a superstring, and these are naturally represented by the superstrings. We assume a minimal target representation (which may not hold in practice). Suppose we obtain a sample of $m$ substrings (all positive examples). In biological labs, a Greedy algorithm which repeatedly merges a pair of substrings with maximum overlap is routinely used. It is conjectured that Greedy produces a common superstring $t'$ of length at most $2s$, where $s$ is the optimal length (NP-hard to find). In [@blum], we have shown that $s \leq |t'| \leq 4s$. Assume that $|t'| \approx 2s$.[^2] Using the length-based Occam’s razor theorem, that is, Theorem \[sex\] with $K(r' \mid r,s,n)$ in Definition \[def.kcoccam\] replaced by $|r'|$, this length of $2s$ would determine the sample complexity, as in (\[total-compression\]), with $p(n,s,\delta/2)= 2 \cdot 2s$ (the extra factor 2 is the 2-logarithm of the size of the alphabet $\{A,C,G,T\}$). Is this the best we can do? It is well-known that the sampling process in DNA sequencing is a very costly and slow process. We improve the sample complexity using our KC-based Occam’s razor theorem. Let $t$ be the target string of length $s$ and $t'$ be the superstring returned by Greedy of length at most $2s$. Then $$K(t' \mid t,s,n ) \leq 2s (2\log s + \log n) / n .$$ We give $t'$ a short description using some information from $t$. Let $S = \{ s_1 , \ldots , s_m \}$ be the set of $m$ examples (substrings of $t$ of length $n$). Align these substrings with the common superstring $t'$, from left to right. Divide them into groups such that each group’s leftmost string overlaps with every string in the group but does not overlap with the leftmost string of the previous group. Thus there are at most $2s/n$ such groups. To specify $t'$, we only need to specify these $2s/n$ groups. After we obtain the superstring for each group, we re-construct $t'$ by optimally merging the superstrings of neighboring groups. To specify each group, we only need to specify the first and the last string of the group and how they are merged. This is because every other string in the group is a substring of the string obtained by properly merging the first and last strings. Specifying the first and the last strings requires $2 \log s$ bits of information to indicate their locations in $t$ and we need another $\log n$ bits to indicate how they are merged. Thus $K(t'\mid t,s,n) \leq 2s (2 \log s + \log n) / n$. This lemma shows that (\[total-compression\]) can also be applied with $p(n,s,\delta/2)= 2\cdot 2s (2 \log s + \log n) / n$, giving a factor $n / (2\log s + \log n)$ improvement in sample-complexity. Note that in (mammal) genome computation practice, we have $n=500$ and $s=3 \times 10^9$. The sample complexity using the Kolmogorov complexity-based Occam’s razor is reduced over the “length based” Occam’s razor by a multiplicative factor of $n / (2\log s + \log n) \approx \frac{500}{2 \times 31 + 9} \approx 7$. [**Application 2: Learning a Monomial.**]{} Consider boolean space of $\{0,1\}^n$. There are two well-known algorithms for learning monomials. One is the standard algorithm. [**Standard Algorithm.**]{} 1. Initially set the concept representation $M := x_1 \overline{x_1} \ldots x_n \overline{x_n}$ (a conjunction of all literals of $n$ variables—which contradicts every example). 2. For each positive example, delete from the current $M$ the literals that contradict the example. 3. Return the resulting monomial $M$. Haussler [@hauss] proposed a more sophisticated algorithm based on set-cover approximation as follows. Let $k$ be the number of variables in the target monomial, and $m$ be the number of examples used. [**Haussler’s Algorithm.**]{} 1. Use only negative examples. For each literal $x$, define $S_x$ to be the set of negative examples such that $x$ falsifies these negative examples. The sets associated with the literals in the target monomial form a set cover of negative examples. 2. Run the approximation algorithm of set cover, this will use at most $k \log m$ sets or, equivalently, literals in our approximating monomial. It is commonly believed that Haussler’s algorithm has better sample complexity than the standard algorithm [^3] We demonstrate that the opposite is sometimes true (in fact for most cases), using our KC-based Occam’s razor theorem, Theorem \[KCoccam\]. Assume that our target monomial $M$ is of length $n - \sqrt{n}$. Then the length-based Occam’s razor theorem gives sample complexity $n/\epsilon$ for both algorithms, by Formula \[total-compression\]. However, $K(M' \mid M)\leq \sqrt{n}\log 3+O(1)$, where $M'$ is the monomial returned by the standard algorithm. This is true since the standard algorithm always produces a monomial $M'$ that contains [*all*]{} literals of the target monomial $M$, and we need at most $\sqrt{n} \log 3 + O(1)$ bits to specify whether other literals are in (positive or negative) or not in $M'$ for the variables that are in $M'$ but not in $M$. Thus our (\[total-compression\]) gives the sample complexity of $O(\sqrt{n}/\epsilon)$. In fact, as long as $|M| > n/\log n$ (which is most likely to be the case if every monomial has equal probability), it makes sense to use the standard algorithm. Conclusions =========== Several new problems are suggested by this work. If we have an algorithm that, given a length-$m$ sample of a concept in Euclidean space, produces a consistent hypothesis that can be described with only $m^\alpha, \alpha<1$ symbols (including a symbol for every real number; we’re using uncountable representation alphabet), then it seems intuitively appealing that this implies some form of learning. However, as noted in [@board], the standard proof of Occam’s Razor does not apply, since we cannot enumerate these representations. The main open question is under what conditions (specifically on the real number computation model) such an implication would nevertheless hold. Can we replace the exception element or majority of 3 requirement by some weaker requirement? Or can we even eliminate such closure requirement and obtain a complete reverse of Occam’s razor theorem? Our current requirements do not even include things like k-DNF and some other reasonable representation systems. Acknowledgements ================ We wish to thank Tao Jiang for many stimulating discussions. [99]{} M. Anthony and N. Biggs, [*Computational Learning Theory*]{}, Cambridge University Press, 1992. A. Blum, T. Jiang, M. Li, J. Tromp, M. Yannakakis, Linear approximation of shortest common superstrings. [*Journal ACM*]{}, 41:4 (1994), 630-647. A. Blumer and A. Ehrenfeucht and D. Haussler and M. Warmuth, Learnability and the Vapnik-Chervonenkis Dimension. [*J. Assoc. Comput. Mach.*]{}, 35(1989), 929-965. A. Blumer and A. Ehrenfeucht and D. Haussler and M. Warmuth, Occam’s Razor. [*Inform. Process. Lett.*]{}, 24(1987), 377-380. R. Board and L. Pitt, On the necessity of Occam Algorithms. 1990 [*STOC*]{}, pp. 54-63. A. Ehrenfeucht, D. Haussler, M. Kearns, L. Valiant. A general lower bound on the number of examples needed for learning. [*Inform. Computation*]{}, 82(1989), 247-261. D. Haussler. Quantifying inductive bias: AI learning algorithms and Valiant’s learning framework. [*Artificial Intelligence*]{}, 36:2(1988), 177-222. D. Haussler, N. Littlestone, and, M. Warmuth. Predicting $\{0,1\}$-functions on randomly drawn points. [*Information and Computation*]{}, 115:2(1994), 248–292. D.P. Helmbold and M.K. Warmuth, On weak learning, [*J. Comput. Syst. Sci.*]{}, 50:3(1995),551-573. T. Jiang and M. Li, DNA sequencing and string learning, [*Math. Syst. Theory*]{}, 29(1996), 387-405. M. Li. Towards a DNA sequencing theory. [*31st IEEE Symp. on Foundations of Comp. Sci.*]{}, 125-134, 1990. M. Li and P. Vitányi. [*An Introduction to Kolmogorov Complexity and Its Applications*]{}. 2nd Edition, Springer-Verlag, 1997. R. E. Schapire. The strength of weak learnability. Machine Learning, 5:2(1990),197–227. R.E. Schapire, The boosting approach to machine learning: An overview. In: [*MSRI Workshop on Nonlinear Estimation and Classification*]{}, 2002. L. G. Valiant. A Theory of the Learnable. [*Comm. ACM*]{}, 27(11), 1134-1142, 1984. M.K. Warmuth. Towards representation independence in PAC-learning. In [*AII-89*]{}, pp. 78-103, 1989. [^1]: A preliminary version was presented at the [*8th Intn’l Computing and Combinatorics Conference (COCOON)*]{}, held in Singapore, August, 2002. [^2]: Although only the $4s$ upper bound was proved in [@blum], which has since been improved, it is widely believed that $2s$ is the true bound. [^3]: In fact, Haussler’s algorithm is specifically aimed at reducing sample complexity for small target monomials, and that it does.

*Introduction and the model*. Breathing solitary pulses (SPs) emerge as fundamental modes in various optical media, including fiber lasers [Orenstein]{}-[@Kutz], systems based on the periodic dispersion management (DM) [@malomed2006soliton]-[@Ganapathy] or nonlinearity management [@Barcelona], and oscillations of spatial solitons in external traps [@Koby; @Serkin]. In those settings, the breathing dynamics is usually induced by the periodic structure of the system. In this work, we demonstrate that robust breathing SP pairs emerge in uniform systems built as linearly coupled guided modes with opposite group velocity dispersions (GVDs). In fact, completely stable breathing SP pairs can be created without nonlinearity. If the nonlinearity is too strong, it actually destroys the pair. We start with a system of two coupled linear Schrödinger equations, which is a particular case of mixed discrete-continuous systems [@LSE]: $$\frac{\partial A_{j}}{\partial z}+v_{gj}^{-1}\frac{\partial A_{j}}{\partial t% }-\frac{i\beta _{j}}{2}\frac{\partial ^{2}A_{j}}{\partial t^{2}}=i\kappa _{0}A_{3-j}, \label{eqn:2}$$ $j=1,2$, with $v_{gj}^{-1}\equiv \frac{k(\omega _{j})}{b_{j}v_{g}(\omega _{j})}$, $\beta _{j}\equiv \frac{k(\omega _{j})}{b_{j}}\frac{\partial ^{2}k(\omega )}{\partial \omega ^{2}}|_{\omega =\omega _{j}}$, where $% v_{g}(\omega _{j})\equiv \left( {\frac{\partial k(\omega )}{\partial \omega }% |_{\omega =\omega _{j}}}\right) ^{-1}\ $and $\frac{\partial ^{2}k(\omega )}{% \partial \omega ^{2}}|_{\omega =\omega _{j}}$ are the group velocity and GVD, respectively. Here $k(\omega )$ is the frequency-dependent wavenumber for the system’s normal modes, carried by eigenfrequencies $\omega _{j}$ and respective propagation constants $b_{j}$ [@yariv2007photonics]. These coupled-mode equations (CPEs) apply to different physical settings. First, they may describe the co-propagation of two modes with the same carrier frequency in a dual-core waveguide with different GVD coefficients in the cores (due to different core materials or waveguide profiles), cf. Refs. [@Kaup; @Boardman]. In this case, the matching between $b_{1}$ and $% b_{2}$ may be supported by an appropriate spatial modulation (e.g., in a grating coupler [@yariv2007photonics]). A more promising possibility is to embed a pair of waveguides into a photonic-crystal-fiber (PCF) matrix [@PCF]. In the latter situation, the three conditions of the equality between the phase and group velocities in the cores, and opposite GVD coefficients, which are assumed below, can be secured using such parameters as the diameter of the waveguides, the PCF pitch, and the carrier wavelength. Alternatively, the same equations govern the co-propagation of two modes in the same waveguide, carried by different frequencies, with the matching between $\omega _{1}$ and $\omega _{2}$ provided by a suitable temporal modulation. Such temporal gratings are the subject of research in linear [@PhysRevE.75.046607; @1063-7869-48-8-R03] and nonlinear optics [@Alon]. Equations (\[eqn:2\]) can be derived for these physical settings from full CPEs which explicitly contain the spatial and temporal modulations as mechanisms for the wavenumber and frequency matching [@brenda1]. As said above, we focus on the symmetric system, with equal group velocities and opposite GVD for the linearly coupled waves: $\beta _{2}=-\beta _{1}\equiv \beta $, $v_{g1}=v_{g2}\equiv v_{g}$. Equations (\[eqn:2\]) are then rescaled by defining $\xi \equiv \left( |\beta |/T_{0}^{2}\right) z$ and $\tau \equiv (t-z/v_{g})/T_{0}$, where $T_{0}$ is a characteristic temporal width of the input pulse: $$\frac{\partial A_{j}}{\partial \xi }-\frac{i}{2}(-1)^{j}\frac{\partial ^{2}A_{j}}{\partial \ \tau ^{2}}=iK_{0}A_{3-j}. \label{eqn:4}$$ The corresponding SP period, normalized coupling coefficient, and coupling length are $Z_{0}=\pi T_{0}^{2}/\left( 2|\beta |\right) $, $K_{0}=\kappa _{0}T_{0}^{2}/|\beta |$ and $L_{c}\equiv \pi /\left( 2K_{0}\right) $, respectively [@agrawal2010applications; @yariv2007photonics]. Equations (\[eqn:4\]) can be derived from the Lagrangian density, with the asterisk standing for complex conjugate: $$\begin{aligned} \mathcal{L}& =\left( i/2\right) (A_{1}^{\ast }A_{1\xi }-A_{1}A_{1\xi }^{\ast }+A_{2}^{\ast }A_{2\xi }-A_{2}A_{2\xi }^{\ast }) \notag \\ & +(i/2)(|A_{1\tau }|^{2}-|A_{2\tau }|^{2})+K_{0}(A_{1}^{\ast }A_{2}+A_{2}^{\ast }A_{1}). \label{eqn:8}\end{aligned}$$Note that the dispersion relation for plane-wave solution to Eq. ( [eqn:4]{}), $A_{j}=A_{j}^{(0)}\exp \left( iQ\xi -i\Omega \tau \right) $ is $$Q^{2}=K_{0}^{2}+\Omega ^{4}/4, \label{gap}$$hence solitary modes may exist in the respective spectral *gap*, $% Q^{2}<K_{0}^{2}$ [@Kaup]. *Analytical and numerical solution for the linear system*. The commonly known fundamental solution of the single linear Schrödinger equation is a spreading Gaussian. In the case of DM with exactly vanishing path-average GVD, $\bar{\beta}=0$, the solution is a breathing Gaussian which does not suffer spreading. The Kerr nonlinearity stabilizes the solution against spreading at $\bar{\beta}\neq 0$, giving rise to DM solitons [@malomed2006soliton]-[@Ganapathy]. In the present system, robust non-spreading breathing SPs are possible without any DM, due to the action of the GVD terms with opposite signs, linked by the linear coupling. An analytical solution can be constructed for broad SPs, with $T_{0}^{2}\gg 1/K_{0}$, using the variational approximation (VA) [@Chu:93; @PhysRevA.41.6287]. A relevant ansatz, describing rapid oscillations between the two modes, is: $$\{A_{1}(\xi ,\tau ),A_{2}(\xi ,\tau )\}=A(\xi ,\tau )\left\{ i\sin (K_{0}\xi ),\cos (K_{0}\xi )\right\} , \label{eqn:6}$$ where $A(\xi ,\tau )$ is a slowly varying complex amplitude \[i.e., these solutions are looked for near edges of the above-mentioned spectral gap, see Eqs. (\[gap\])\]. The ansatz is substituted into Eq. (\[eqn:8\]), keeping only terms with derivatives of the slowly varying amplitude to yield an effective Lagrangian density, $\mathcal{L}_{\mathrm{eff}}=(1/2)\left[ i(A^{\ast }A_{\xi }-AA_{\xi }^{\ast })-|A_{\tau }|^{2}\cos (2K_{0}\xi )% \right] $. It gives rise to Euler-Lagrange equation with *effective* DM corresponding to $\bar{\beta}=0$, although no DM is present in the underlying CPEs (\[eqn:2\]): $$iA_{\xi }+(1/2)A_{\tau \tau }\cos (2K_{0}\xi )=0. \label{eqn:10}$$ Equation (\[eqn:10\]) gives rise to exact Gaussian solutions for non-spreading breathing SPs: $$A(\xi ,\tau )=\frac{A_{0}T_{0}}{\sqrt{2T_{0}^{2}+K_{0}^{-1}\sin (2K_{0}\xi )}% }\exp \left( -\frac{\tau ^{2}}{2T_{0}^{2}+iK_{0}^{-1}\sin (2K_{0}\xi )}% \right) , \label{eqn:11}$$ where the initial temporal width $T_{0}\gg 1/K_{0}$ and $A_{0}$ are arbitrary real constants. Thus, Eqs. (\[eqn:11\]) and (\[eqn:6\]) produce a breathing SP pair, built as two components swinging at coupling frequency $K_{0}$, multiplied by the common amplitude, $A(\xi ,\tau )$, oscillating at the double frequency. In Figs. \[fig1:Fig1\] and \[fig2:Fig2\], these approximate analytical solutions are compared to results produced by the numerical integration of Eqs. (\[eqn:4\]) by means of the split-step Fourier-transform method [@agrawal2010applications; @Li20082811]. It is seen that non-spreading SP solutions, oscillating almost precisely with period $2\pi /K_{0}$, exist in all cases, the broad pulses being perfectly approximated by the analytical solution, as expected, while for narrow ones the approximation is inaccurate. ![The field evolution in a broad SP pair, with $K_{0}=1$, $T_{0}^{2}=20$. (a,b): The numerical solution of Eq. (\[eqn:4\]) for $\mathrm{Re}(A_{2}(\protect\xi ,\protect\tau ))$ and $\mathrm{Im}(A_{1}(\protect\xi ,\protect\tau ))$, respectively. (c,d): Approximate analytical solution given by Eqs. (\[eqn:6\]) and (\[eqn:11\]) for the same components.[]{data-label="fig1:Fig1"}](Fig1.eps){width="\textwidth"} ![The same as in Fig. \[fig1:Fig1\], but for a narrow SP pair, with $T_{0}=1$.[]{data-label="fig2:Fig2"}](Fig2.eps){width="\textwidth"} To analyze the solutions, we define the following correlator between two pairs of functions $\{f_{1}(\xi ,\tau ),f_{2}(\xi ,\tau )\}$ and $% \{g_{1}(\xi ,\tau ),g_{2}(\xi ,\tau )\}$: $$\mathrm{CO}(\{f_{1},f_{2}\},\{g_{1},g_{2}\},\xi )=\frac{|\langle f_{1}|g_{1}\rangle _{\tau }|+|\langle f_{2}|g_{2}\rangle _{\tau }|}{% ||f_{1}||_{\tau }||g_{1}||_{\tau }+||f_{2}||_{\tau }||g_{2}||_{\tau }}, \label{eqn:13}$$ with the inner product, $\langle f(\xi ,\tau )|g(\xi ,\tau )\rangle _{\tau }\equiv \int_{-\infty }^{+\infty }{f^{\ast }(\xi ,\tau )g(\xi ,\tau )d\tau }$, and the corresponding norm, $||f||_{\tau }=\sqrt{\langle f|f\rangle _{\tau }}$. The correlator takes values $0\leq \mathrm{CO}\leq 1$, with $\mathrm{CO}% =1$ and $\mathrm{CO}=0$ corresponding, severally, to perfect correlation and no correlation. Then, we can evaluate the proximity of the SP to the periodic behavior as $\mathrm{CO}_{2\pi /K_{0}}=\mathrm{CO}(\{A_{1}(\xi ,\tau ),A_{2}(\xi ,\tau )\},\{A_{1}(\xi +2\pi /K_{0},\tau ),A_{2}(\xi +2\pi /K_{0},\tau )\})$, and the consistency between approximate analytical and numerical solutions: $\mathrm{CO}_{\mathrm{AN}}=\mathrm{CO}(\{A_{1\mathrm{N}% },A_{2\mathrm{N}}\},\{A_{1\mathrm{A}},A_{2\mathrm{A}}\})$. In particular, as shown in Fig. \[fig3:Fig3\](a), the latter correlator allows one to assess *how long* the approximate solution remains valid. Naturally, the correlations decay with the increase of $\xi $ and decrease of $T_{0}$. The inset shows the value of the propagation distance, $\xi $, at which $\mathrm{% CO}_{\mathrm{AN}}$ drops to $0.95$, as a function $T_{0}^{-2}$, which demonstrates an exponential decrease of the propagation range, in which the analytical solution is valid, with the decrease of the SP’s width. Further, in Fig. \[fig3:Fig3\](b) the $\mathrm{CO}_{2\pi /K_{0}}$ correlator shows how well the SP pair maintains a periodic evolution pattern. Increasing $T_{0}^{-2}$ causes a parabolic decrease in the correlation, while the degree of the deviation from the perfectly periodic behavior is itself periodic in $\xi $, oscillating at the double frequency, $% 2K_{0}$. ![Correlators computed in the linear system. (a) $\mathrm{CO}_{% \mathrm{AN}}(\protect\xi )$ for different temporal widths $T_{0}$ of the SP input. The inset: $\protect\xi (\mathrm{CO}_{\mathrm{AN}}=0.95)$ vs. $% T_{0}^{-2}$, the continuous line showing a fit to an exponential function. (b) $\mathrm{CO}_{2\protect\pi /K_{0}}(\protect\xi )$ for different $T_{0}$. The inset: $\mathrm{CO}_{2\protect\pi /K_{0}}(\protect\xi =0)$ vs. $% T_{0}^{-2}$, the continuous line showing a fit to a parabola.[]{data-label="fig3:Fig3"}](Fig3.eps){width="\textwidth"} *The nonlinear system*. It is straightforward  to add Kerr terms to the CPEs (\[eqn:4\]) [agrawal2010applications,yariv2007photonics,Li20082811]{}:$$\frac{\partial A_{j}}{\partial \xi }-\frac{i}{2}(-1)^{j}\frac{\partial ^{2}A_{j}}{\partial \ \tau ^{2}}=iK_{0}A_{3-j}+i\Gamma |A_{j}|^{2}A_{j}, \label{eqn:14}$$ where $\Gamma $ is the scaled nonlinearity coefficient. Accordingly, the Lagrangian density (\[eqn:8\]) acquires an extra term, $% \left( \Gamma /2\right) |\left( A_{1}|^{4}+|A_{2}|^{4}\right) $, and the VA can be applied to the nonlinear system as well [Chu:93,PhysRevA.41.6287,Li20082811]{}, again assuming $T_{0}\gg 1/K_{0}$. Using the same ansatz (\[eqn:6\]) as above leads to a nonlinear Schrödinger (NLS) equation featuring a combination of DM with $\bar{\beta}=0$ \[cf. Eq. (\[eqn:10\])\] and nonlinearity management: $$iA_{\xi }+(1/2)A_{\tau \tau }\cos (2K_{0}\xi )+(\Gamma /4)\left[ \cos (4K_{0}\xi )+3\right] |A|^{2}A=0. \label{NLSE}$$This equation can readily produce DM solitons solutions, by dint of methods elaborated in the analysis of the DM and nonlinearity-managed systems [malomed2006soliton]{}-[@Barcelona]. Comparing numerical solutions produced by Eq. (\[NLSE\]) with numerical solutions for the SP pairs produced by simulations of the full system ([eqn:14]{}) demonstrate that the strong nonlinearity tends to gradually destroy the solitons. In Fig. \[fig4:Fig4\] we display two representative cases for $T_{0}=20$ and $T_{0}=1$ with $\Gamma =0.1$. Only the evolution of $\mathrm{Re}\{A_{2}\}$ is shown, as it is sufficient to represent the situation. The nonlinearity starts to affect the SP shape at propagation distances exceeding the nonlinearity length, $\sim 1/\Gamma =10$. Similar to the linear system (cf. Figs. \[fig1:Fig1\] and \[fig2:Fig2\]), the VA, i.e., Eq. (\[NLSE\]), is accurate for broad solitons, and inaccurate for narrow ones. The gradual destruction of the SP by the nonlinearity is naturally explained by the fact that, while the fundamental frequency of its internal oscillations falls into the gap \[see Eq. (\[gap\])\], higher-order harmonics, generated by the cubic nonlinearity, couple to the continuous spectrum, initiating decay of the SP. ![The evolution of the $\mathrm{Re}\{A_{2}\}$ field in the nonlinear system with $\Gamma =0.1$ for a broad SP with $T_{0}=20$: (a) a numerical solution of the full system (\[eqn:14\]); (b) simulations of the VA-produced single NLS equation (\[NLSE\]). (c,d): The same as in (a,b), but for a narrow SP with $T_{0}=1$.[]{data-label="fig4:Fig4"}](Fig4.eps){width="\textwidth"} To explore the effect of the nonlinearity in a systematic way, we computed the correlator $\mathrm{CO}_{2\pi /K_{0}}$, using the numerical solutions of the full system (\[eqn:14\]). The results, shown in Fig. \[fig5:Fig5\], make it evident that stronger nonlinearity worsens the stability of the SP pair. It is worthy to note too that narrower pulses, with smaller $T_{0}$, are more robust against the action of the nonlinearity, which is explained by the similarity of the narrow SPs to the DM solitons, as well as to stationary gap soliton, which exist in the same system [@Kaup]. ![Correlator $\mathrm{CO}_{2\protect\pi /K_{0}}$, characterizing the proximity of the SP pair to a periodically oscillating mode, for different strengths of nonlinearity. The top and bottom panels pertain to narrow and broad pulses, with $T_{0}=2.5$ and $T_{0}=20$, respectively. 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--- abstract: 'In this note we investigate the existence of time-periodic solutions to the $p$-Navier-Stokes system in the singular case of $p\in(1,2)$, that describes the flows of an incompressible shear-thinning fluid. In the $3D$ space-periodic setting and for $p\in[\frac53,2)$ we prove the existence of a regular time-periodic solution corresponding to a time periodic force data which is assumed small in a suitable sense. As a particular case we obtain “regular” steady solutions.' author: - Anna Abbatiello and Paolo Maremonti bibliography: - 'periodici-reference.bib' nocite: '[@*]' title: 'Existence of regular time-periodic solution to shear-thinning fluids.' --- Introduction ============ The existence of time periodic solutions for shear-thinning and electro-rheological fluids was firstly studied in [@Lions-per], where a $p$-Navier-Stokes case for $p\geq 3$ constant is considered, and in [@C] for the $p(t,\x)$-Stokes case. The main purpose of this note is to investigate the question for the full $p$-Navier Stokes system. However the difficulties of the problem lead to consider a special case of the problem if compared with the ones studied in [@C; @Lions-per]. Nevertheless, our results are on the wake of the ones obtained in [@BL][^1]. To better introduce the results contained in [@BL] and to show the ours related to shear-thinning fluids, we state the special problem studied in this note. We consider \[system\] \_t + () - §() +=    (0, T)Ø,\ = 0    (0, T)Ø,\ where $\bv=(v_1, v_2, v_3) $ is the vector field of velocity, $\bv_t$ is the time derivative of $\bv$, $\S$ is the deviatoric part of the Cauchy stress tensor constitutively determined by \[S\] §()=||\^[p-2]{}    p(1, 2), where $\cD\bv:=\frac{1}{2}(\n\bv+(\n\bv)^T)$ symmetric part of the spatial gradient of $\bv$, $\pi$ is the pressure scalar field or mean normal stress, finally $\bF=(f_1, f_2, f_3) $ is the prescribed body force. We study the system with in $(0, T)\times\O$ where $T$ is a positive real number and $\O = (0, 1)^3 \subset \R^3$. We endow the problem with space-periodic boundary conditions, i.e. we require \[periodicity\] \_[|\_j]{}=\_[|\_[j+3]{}]{},   \_[|\_j]{}=\_[|\_[j+3]{}]{},\_[|\_j]{}=\_[|\_[j+3]{}]{}, where $\Gamma_j:=\pd\O\cap\{x_j=0\}$ and $\Gamma_{j+3}:=\pd\O\cap\{x_j=1\}$ with $j=1,2,3$. In [@BL], assuming $p\in (\frac95,2)$ and the data force $\bF$ time periodic of period $T>0$, the authors realize the existence of weak solutions which are time periodic with period $T$. They achieve the result of existence by employing a fixed point theorem. For analogous questions related to the Newtonian fluids, this approach is essentially due to Prodi by the Tychonov fixed point theorem and to Prouse by the Brouwer fixed point theorem, see respectively [@PR] and [@Pru]. Brouwer’s fixed point theorem working in the finite dimensional case, it is employed on the Galerkin approximations (that are finite combinations of elements). Then there is the question of the convergence to a limit of the Galerkin approximation. In [@BL] they solve the convergence questions requiring a restriction to the power $p\in(\frac95,2)$. In this note we follow a different point of view. In the sense that we look for solutions corresponding to small data. This allows us to exhibit the results for a wider range of power-law index $p$ and “more regular solutions”. More precisely, assuming in (\[system\]) $p\in \left[\frac53,2\right)$, we are able to obtain the existence of a time-periodic solution to , subject to a small time-periodic force. The proof employs an idea by Serrin exhibited in [@S]. Assuming the existence of solutions defined for all $t>0$ and their regularity, in the case of $3D$-Navier-Stokes boundary value problem subject to a time periodic force, in [@S] Serrin proves that any solution asymptotically tends to a time periodic solution whose time-period is the same of the force. This approach is formal in the sense that the assumptions on the existence of the solutions enjoying the regularity required by Serrin is still an open problem. Nevertheless, as made in [@M] for $\Omega\equiv\mathbb R^3$, for small data and in suitable function spaces Serrin’s approach can be used with success. In this note we follow the same approach of [@M]. Firstly, for $p\in [\frac{5}{3}, 2)$, under the assumption of small data, we furnish the existence of a regular solution (Theorem\[thmA\]), in a sense specified later, to the system with and . Then, under the assumption of periodicity for force data we realize the time periodic solution. It is the case to make precise the claim “small data force”. Actually we assume that $\bF\in L^\infty(0,\infty;L^q(\O))$, that, as in the case of the Navier-Stokes equations, makes the difference since no time-integrability properties are required. The smallness is required for $\operatorname*{ess\,sup}_{t\geq0} \dm \bF(t)\dm_q$. As a particular case we consider $f$ independent of $t$, which allows us to deduce the existence of steady solutions. Our results of existence seem of some interest in connection with the fact they are obtained for the singular case $p\in[\frac53,2)$. We point out that the limit case of $p=\frac53$ and the regularity $\nabla^2\bv\in L^2(0,T;L^\frac4{4-p}(\Omega))$ is achieved by a trick, employed as in [@CGM], connected with the reverse Hölder’s inequality (see Inequality\[ineq\] and Lemma\[lemma6\]). This represents an improvement with respect the results already known. Actually, in [@MA95] for the problem of kind - provided $\bF\in L^{p'}((0,T)\times\Omega)$ suitable small together with the initial data, it is showed the existence of a solution with $\nabla^2\mathbf{v}\in L^2((0, T); W^{2,p}(\Omega)) $ for $p\in(\frac53,2)$. In [@BDR] for the evolution problem they exhibit a solution local in time subject to an integrable force $\bF\in L^\infty(0,T; W^{1,2}(\Omega))\cap W^{1,2}((0,T); L^2(\Omega))$ with $\nabla^2\bv\in L^{\frac{p(5p-6)}{2-p}}((0, T); W^{2,\frac{3p}{p+1}}(\Omega)) $ for $p\in(\frac75,2)$, and for the steady problem $\nabla^2\bv\in W^{2,\frac{3p}{p+1}}(\Omega) $ where it is assumed $p\in (\frac95,2)$ and space-periodic domains as in our case. We restrict our considerations to the tridimensional case just for the simplicity, but the technique also works in the two-dimensional case and the results hold for $p\in(1,2)$. In order to better explain our results we introduce some spaces of functions. For the sequel it is worth to note that vector valued functions are printed in boldface while scalar ones in italic mode and we do not distinguish between space of scalar functions and space of vector-valued functions. As stated before let $\O$ be the cube $(0, 1)^3 $ in $\R^3$ of points $\x=(x_1, x_2,x_3)$ and let $\O_T$ denote $(0, T)\times\O$. For a domain $G$, that can be $\O$ or $\O_T$, for $q\geq 1$ and $m\in \N$ we keep the notation $(L^q(G), \|\cdot\|_{q,G})$ and $(W^{m,q}(G), \|\cdot\|_{m,q,G})$ for the standard Lebesgue and Sobolev spaces and their associated norms. The subscript “[$_{\rm per}$]{}" means that space-periodic functions having zero mean value are considered (i.e. each considered function $f$ satisfies $f(\x+\e_i)=f(\x)$, $i=1, 2, 3$ where $(\e_1, \e_2,\e_3)$ is the canonical basis of $\R^3$ and $\int_\O f=0$). Next let us establish the notation for the spaces of solenoidal space-periodic functions with values in $\R^3$ by setting $$\mathscr{C}_{\rm per}(\O):=\left\{ \bphi\in \mathcal{C}^\infty_{\rm per}(\O, \R^3), \int_\O \bphi(\x)\,d\x=0, \div\bphi=0 \mbox{ in } \O\right\},$$ and $$J^q_{\rm per}(\O):=\overline{\mathscr{C}_{\rm per}(\O)}^{\|\cdot\|_q}, \ \ J^{m,q}_{\rm per}(\O):=\overline{\mathscr{C}_{\rm per}(\O)}^{\|\cdot\|_{m,q}}.$$ Since there is no confusion, in the sequel we omit the subscript $\null_{\rm per}$. In particular when $q=2$ we denote $J(\O):=J^2(\O)$. Finally, if $X$ is a Banach space then $L^q(0, T; X)$ stands for the Bochner space of functions from $(0, T)$ to $X$ and such that their norm in $X$ is $q$-integrable, as well $ {C}(0,T; X)$ is the space of continuous functions from $[0, T]$ to $X$ endowed of the standard norm of the ${\rm sup}$. The subscript “$t$” denotes the differentiation with respect to time, the symbol $\cD\bph$ is the symmetric part of the gradient of a sufficiently smoooth vector function $\bph$, i.e. $\cD\bph := \frac{1}{2}(\n\bph+(\n\bph)^T)$, $(\cdot, \cdot)$ is used for the scalar product in $L^2(\O)$. A field $\bu: (0, +\infty) \times \O \to\R^3$ is said to be [*time-periodic*]{} with period $\mathcal{T}$ if - $\displaystyle\sup_{t\in[0,\mathcal{T}]} \|\bu(t)\|_X<+ \infty$ with $(X, \|\cdot\|_X)$ a Banach space, - $\|\bu(t+\mathcal{T})-\bu(t)\|_X=0 \ \forall t \geq 0.$ Let us give the notion of regular solution to system with $\S$ defined by , endowed of boundary conditions and completed by providing an initial data $\bv_0$. \[def-d2\] For all $T>0$, let $\bF\in L^{\infty}(0,T; L^{\frac{4}{p}}_{\per}(\O))$ and $\bv_0\in J_{\per}^{1,2}(\O)$, a field $\bv:(0,T)\times \O \to\R^3$ is said to be [solution]{} to system with $\eqref{S}$, and initial data $\bv_0$ if - $\nabla^2\bv\in L^2(0,T; L_{\per}^{\frac{4}{4-p}}(\O))$, $\bv\in C(0, T; J_{\per}^{1,2}(\O)),$ $ \bv_t\in L^2(\O_T),$ - $( \bv_t, \bo) + (\bv\cdot\nabla \bv, \bo) + (\S(\cD\bv), \cD\bo) = (\bF, \bo)$ for all $\bo\in J_{\per}^{1,p}(\Omega)$ and a.e. in $(0, T)$, - $\lim_{t\to0^+} \|\bv(t)-\bv_0\|_2=0$. The main theorems are stated below and they are proved in Section \[sectionA\] and \[section1\] respectively. We denote by $\Lambda$ and $K$ two fixed constants such that $$\label{constant-Lambda}\begin{split} & \Lambda<\min\mbox{$\left\{ \left(\frac{p}{8C_S}\right)^{\frac{2}{3-p}}, \left( \frac{3^{\frac{2-p}{2}}C_SC_K}{2}\right)^{-\frac{2}{3-p}}, (C_S^{-1}C_K^{-1}3^{\frac{p-2}2}(p-1))^{\frac{2}{3-p}}\right\}$}\\ & \mbox{\; and \;} C_SK^2 2^{2-p}\leq \frac{p(p-1)}{8 }\Lambda^{p-1}\,, \end{split}$$ where the constants $ C_S, $ depends on $\Omega$ and Sobolev inequalities, while $C_K$ is the constant of the Korn inequality. \[thmA\] Let $\O$ be a three dimensional cube, $p\in \left[\frac{5}{3},2\right)$. If $\bF\in L^\infty(0,+\infty;L^{\frac{4}{p}}(\O)) $ and $\bv_0\in J^{1, 2}(\O)$ with $$\label{smallv0} \|\nabla\bv_0\|_{L^{2}(\O)}^2<\Lambda \mbox{\; and \;} \operatorname*{ess\,sup}_{t\geq0} \|\bF(t)\|_{L^{\frac{4}{p}}(\O)}\leq K \,,$$ then there exists a solution $\bv(t,\bx)$ to system in the sense of Definition \[def-d2\] such that $$\label{gradient2}\sup_{t\in[0,T]} \|\nabla\bv(t)\|_{2}^2\leq \Lambda \ \ \forall T>0.$$ Our result slightly improves the one for $p\in(\frac{5}{3},2)$ and for the operator with some $\mu>0$ in the negative power of the degenerate operator $\S$ in[@MA95](see also [@MNR Theorem 4.51]). Moreover, in connection with problem , Theorem\[thmA\] is the first which proves global existence assuming small force data in $L^\infty(0,\infty;X)$, with $X$ Banach space. For the proof we modify the techniques from [@BDR] and [@M], where the authors consider respectively $p$-Navier-Stokes and Navier-Stokes systems. The regularity of the $p$-Navier-Stokes system is still open and there are many papers dealing with it. It has to be pointed out that the difficulties appear in the $p$-parabolic case already, and recently the $L^\infty(\varepsilon, T; W^{2,q}(\O))$ regularity up to the boundary with $\varepsilon>0$ and $q\geq2$ is given for both domains, bounded or exterior, see [@CGM] and [@CM]. Other relevant results concerning the regularity of the system of type in the steady and evolutionary cases are given in [@Beir], [@BdV], [@CG], [@CG9], [@DR], [@MNR]. \[thm1\] Let $\O$ be a three dimensional cube, $p\in \left[\frac{5}{3},2\right)$ and let $\bF$ be a time-periodic function with period $\mathcal T$ fulfilling the assumptions in Theorem \[thmA\] Then corresponding to $f$ there exists a unique $\bv_0(\bx) \in J^{1, 2}_{\per}(\O)$ such that $$\|\nabla\bv_0\|_{L^{2}(\O)}^2<\Lambda$$ and solution $\bv(t,\x)$ to system , in the sense of Definition\[def-d2\], which is time-periodic with period $\mathcal T$, and that fulfills $$\label{gradient3}\sup_{t\in[0,\mathcal T]} \|\nabla\bv(t)\|^2_{2}\leq \Lambda.$$ It well known the extinction of a $p$-parabolic solutions for $p\in(1,2)$ (e.g. [@DiB], [@CGM]). The same holds for $p$-Stokes problem (see e.g. [@C], [@Anna]) and the following theorem is a special case of the one proved in [@C]: \[extinction\] Assume that data $\bF$ in system verifies the assumptions of Theorem\[thm1\]. Let $\bv(t,x)$ be the time periodic solution. If the data $f$ admits an extinction instant $t_f\in(0, \mathcal T)$, that is $$\|\bF(t)\|_{\frac{4}{p}}=0 \mbox{ a.e. in } [t_f,\mathcal T],$$ and $t_f+\frac{C_S^{3-p}\Lambda^{2-p}}{2-p}<\mathcal T$, then the time periodic solution $\bv$ admits an instant $\bar t$ such that $$\|\bv(t)\|_{L^{2}(\O)}=0 \mbox{ in } [\bar{t}, \mathcal T] \mbox{ with } \bar{t}\leq t_f+\frac{C_S^{3-p}\Lambda^{2-p}}{2-p},$$ where the constant $\Lambda$ is given in . Since our study is only qualitative, it is worth to highlight that Theorem \[extinction\] gives an estimate of the instant $\bar{t}$ and a lower bound for $\mathcal T$ that could not be optimal. \[cor\] For $p\in \left[\frac{5}{3}, 2\right)$, assume that in system $\bF$ be independent of $t$ and satisfying the bound given in Theorem\[thmA\]. Then there exists a steady solution, in the sense of Definition \[def-d2\], $\bv(\bx)$ to system . Approximating system and some auxiliary results =============================================== Let us introduce the following approximating system: \[system-mu\] \_t + () - §\_() +=    (0, T)Ø, = 0    (0, T)Ø,\ where $$\S_\mu(\cD\bv):=(\mu+|\cD\bv|^2)^{\frac{p-2}{2}}\cD\bv \ \ \ \mbox{with} \ \mu\in(0,1) \mbox{ and } p\in(1, 2),$$ endowed of space-periodic boundary conditions. The operator $\S_\mu$ enjoys the following property, that is proved e.g. in [@DER Lemma 6.3], \[property-diff\] (§\_()-§\_(), -)|-|\^2( + ||+||)\^[p-2]{}3\^[2]{}|-|\^2(+||\^2+||\^2)\^2, for $\bA,\bB\in \R^{3\times 3}_{\rm sym}$, where the symbol $\R^{3\times 3 }_{\rm sym}$ stands for the space of symmetric matrix of dimension $3$.\ In what follows we prove some preliminar results for the proof of the main theorems. Since in this paper we are interested in the singular case of the operator $\S$ (being $p\in(1,2)$, cf. ), our strategy is to provide the existence of regular solutions to system , and then consider the passage to the limit as $\mu\to 0$. Let us construct a solution to system by employing the Faedo-Galerkin method. For the reader’s convenience, we recall some basic relations (cfr. [@MNR]). Consider the eigenfunctions $\{\bo_k\}_{k\in\N}$ of the Stokes operator $\mathcal{A}$ and let $\{\lambda_k\}_{k\in \N}$ be the corresponding eigenvalues. Define $X_n=[\bo_1,\dots,\bo_n]$ and let $P^n$ be the projection $P^n: (W^{1,2}(\O), \|\cdot\|_{1,2}) \to (X_n, \|\cdot\|_{1,2})$, defined as $P^n \bu= \sum_{k=1}^n (\bu, \bo_k) \bo_k $, which satisfies $\|P^n\bu\|_{1,2}\leq \|\bu\|_{1,2}$ (see [@MNR]). We seek the Galerkin approximation $$\label{def-galerkin} \bv^n(t, \bx):= \sum_{k=1}^n c^n_k(t)\bo_k(\bx),$$ as solution to the following system of ordinary differential equations \[galerkin\] (\_t\^n, \_k) + (\^n\^n, \_k) + (§\_(\^n), \_k) = (, \_k)   k=1,…, n with initial condition $\bv^n(0)= P^n\bv_0$ where $\bv_0$ is the prescribed initial data. From Carathéodory Theorem follows the local in time existence of $\bv^n(t, \bx)$ in a time interval $[0, t_n)$, $t_n\leq T$. In Lemma \[lemma8\] we prove that $t_n=T$. To this end we provide the following relations for the Galerkin sequence. Multiplying respectively by $c^n_k$, $\lambda_k c^n_k(t)$ and $\dot{c}^n_k(t)$, summing over $k=1,\dots,n$, we get \[galerkin1\]$$\begin{aligned} \frac{1}{2}\frac{d}{dt}\|\bv^n(t)\|_{2,\O}^2+(\S_\mu(\cD\bv^n), \cD\bv^n)& = (\bF(t), \bv^n(t)), \label{energy-d2}\\ \frac{1}{2}\frac{d}{dt}\|\nabla\bv^n(t)\|_{2,\O}^2+ \!\!\iO\! \n\S_\mu(\cD\bv^n)\cdot \n\cD\bv^n dx&=(\bv^n\!\cdot\!\n\bv^n\!, \Delta\bv^n) - (\bF\!,\Delta\bv^n), \label{grad1}\\ \|\bv^n_t(t)\|_2^2+ (\S_\mu(\cD\bv^n), \cD\bv^n_t)&= (\bv^n\!\cdot\!\n\bv^n\!, \bv^n_t) + (\bF\!, \bv^n_t), \label{vt}\end{aligned}$$ where in we employed that the term $(\bv^n\!\cdot\!\nabla \bv^n\!, \bv^n)$ vanishes since $\bv^n$ is solenoidal, while in we exploited the properties of the eigenvalues $\lambda_k$, the equality $\mathcal{A}\bv^n= -\Delta\bv^n$ (that is a well-known technique in the space-periodic case (cfr. [@MNR], [@DR])) and integration by parts. \[lemma8\] Let $p\in \left(1,2\right)$, $\mu>0$ and $\bF\in L^\infty(0,T;L^{\frac{4}{p}}(\O))$. Let $\bv^n(t, \x)$ be the Galerkin approximation defined in , then it holds $$\frac{1}{2}\|\bv^n(T)\|_{2,\O}^2 + \|\cD\bv^n\|_{p,\O_T}^p \leq CT\|\bF\|_{L^\infty(0,T;L^{\frac{4}{p}}(\O))}^{p'}+\frac{1}{2} \|\bv_0\|_{2,\O}^2 + \mu^{\frac{p}{2}}|\O_T| \ \mbox{ for all } \mu>0.$$ From using the inequality $$\ba{c}\displ \int_\O |\cD\bv^n|^p\,dx=\int_{|\cD\bv^n|^2\geq \mu} |\cD\bv^n|^p\,dx + \int_{|\cD\bv^n|^2< \mu} |\cD\bv^n|^p\,dx \VS\leq 2^{\frac{2-p}{2}} \int_\O (\mu+|\cD\bv^n|^2)^{\frac{p-2}{2}}|\cD\bv^n|^2\,dx+ \int_\O \mu^{\frac{p}{2}}\,dx = 2^{\frac{2-p}{2}} (\S_\mu(\cD\bv^n(t)), \cD\bv^n(t))+ \mu^{\frac{p}{2}}|\O|\ea$$ and integrating with respect to the time between $0$ and $T$, it follows \^n(T)\_[2,Ø]{}\^2 + 2\^\^n\_[p,Ø\_T]{}\^p &\_0\^T (t)\_\^n(t)\_ dt + \_0\_[2,Ø]{}\^2+ \^|Ø\_T|\ &C\_0\^T \^n(t)\_[p]{}dt \_[L\^(0,T;L\^(Ø))]{} + \_0\_[2,Ø]{}\^2+ \^|Ø\_T|\ & CT\^\^n\_[p,Ø\_T]{} \_[L\^(0,T;L\^(Ø))]{} + \_0\_[2,Ø]{}\^2+ \^|Ø\_T|\ & \^n\_[p,Ø\_T]{}\^p + C T \_[L\^(0,T;L\^(Ø))]{}\^[p’]{}+ \_0\_[2,Ø]{}\^2+ \^|Ø\_T| where $C$ is a constant due to the Sobolev embedding $W^{1,p}\hookrightarrow L^{\frac{4}{4-p}}(\O)$, Korn’s and Young’s inequalities, that is unessential for our aims. Finally, straightforward computations yield the assertion. \[lemma1\] Let $p\in \left[\frac{5}{3},2\right)$. For every function $\bv$ sufficiently smooth and space-periodic with respect to $\Omega$ it holds \_[3,Ø]{}\^3 C\_[S]{}\^3 \_[2,Ø]{}\^\_[,Ø]{}\^ with $C_{S} $ independent of $\bv$. Since $$\frac{1}{3}= \frac{3p-4}{3p-2}\frac{1}{2}+\frac{2}{3p-2}\frac{8-3p}{12},$$ interpolating we get \[conti-lem\] \_[3,Ø]{} \_[2,Ø]{}\^\_[,Ø]{}\^. Then the Korn inequality, the Sobolev embedding $W^{1, \frac{4}{4-p}}_{\per}(\O)\hookrightarrow L_{\per}^{\frac{12}{8-3p}}(\O)$ and the Poincaré inequality imply $$\|\n\bv\|_{3,\O} \leq C_{S} \|\nabla \bv\|_{2,\O}^{\frac{3p-4}{3p-2}}\|\nabla\cD\bv\|_{\frac{4}{4-p},\O}^{\frac{2}{3p-2}}$$ where $C_{S}$ is a constant depending on $|\O|$ and $p$. Raising to the power $3$ the proof is concluded. We recall the following Hölder’s inequality (see e.g. [@AF Theorem 2.12]). \[ineq\] Let $0<q<1$ and $q'=\frac{q}{q-1}$. If $f\in L^q(\O)$ and $0<\int_\O|g(x)|^{q'}\,dx<\infty$ then $$\int_\O |f(x)g(x)|\geq \left(\int_\O|f(x)|^q\,dx\right)^{\frac{1}{q}}\left(\int_\O|g(x)|^{q'}\,dx\right)^{\frac{1}{q'}}.$$ As in [@CGM], making use of Inequality \[ineq\] we finely and quickly achieve an estimate for the gradient. \[lemma6\] Let $p\in \left[\frac{5}{3},2\right)$ and $\bF\in L^\infty(0,T; L_{\per}^{\frac{4}{p}}(\O))$. Let $\bv^n$ be the Galerkin approximation defined in , then for $t\in(0,T)$ it holds (\_1 + \^n(t)\_2\^[2]{})\^ \^n(t)\_2\^2+\^n(t)\_\^2 C\_S\^3 (\_1 + \^n(t)\_2\^[2]{})\^ \^n(t)\_[2]{}\^\^n(t)\_\^ +(\_1 + \^n(t)\_2\^[2]{})\^[2-p]{}(t)\_\^2 for all $\mu>0$, with $C_S=C_S(p, |\O|)$ and $\mu_1:=\mu|\Omega|$. Since $$\label{gradS}\ba{l}\displ \iO\! \!\n\S_\mu(\cD\bv^n)\!\cdot \!\n\cD\bv^n=\!\! \iO\! (\mu+|\cD\bv^n|^2)^{\frac{p-2}{2}}|\n\cD\bv^n|^2 + (p-2)\!\!\iO \!(\mu+|\cD\bv^n|^2)^{\frac{p-4}{2}}\!(\n\cD\bv^n \cdot\cD\bv^n)^2 \VS \geq (p-1)\! \iO\!(\mu+ |\cD\bv^n|^2)^{\frac{p-2}{2}}|\n\cD\bv^n|^2, \ea$$ and since (see e.g. [@MNR pag. 225] for more details) $$|(\bv^n\!\cdot\!\n\bv^n\!, \Delta\bv^n)|\leq \|\n\bv^n\|_3^3 ,$$ from , we obtain \[grad2\] \^n(t)\_2\^2+(p-1)(+ |\^n|\^2)\^|\^n|\^2dx\^n(t)\_[3]{}\^3+|(,\^n)|. Employing Lemma \[lemma1\] and the Hölder inequality on the right hand side and Inequality \[ineq\] with $\frac{2}{p-2}$ and $\frac{2}{4-p}$ on the left-hand side, it follows \[lemma-grad\] \^n(t)\_2\^2+(p-1) C\_S\^3 \^n(t)\_[2]{}\^\^n(t)\_\^ +(t)\_ \^n(t)\_ where we set $\mu_1:=\mu|\Omega|$, then multiplying both sides by $(\mu_1 + \|\nabla\bv^n(t)\|_2^{2})^{\frac{2-p}{2}}$ and employing the Young inequality in the term $\|\bF(t)\|_{\frac{4}{p}} \|\nabla\cD\bv^n(t)\|_{\frac{4}{4-p}}$ the assertion follows and the proof is concluded. Proof of Theorem \[thmA\] {#sectionA} ========================= Let $\bv^n$ be the Galerkin sequence defined in fulfilling in $[0,T]$ and the initial condition $\bv^n(0)= P^n\bv_0$. #### **Smallness of $\|\nabla\bv^n(t)\|_2$.** Under the assumptions we prove that \[CIN\]v\^n(t)\_2\^2,nt&gt;0. For sake of brevity let us introduce the following notations $$\varphi(t):=\| \nabla\bv^n(t)\|_2^2, \quad D(t):=\|\nabla\cD\bv^n(t)\|_{\frac{4}{4-p}}^2.$$ By virtue of Lemma \[lemma6\], recalling the definition of $\mu_1$, and $_2$ it holds \[phi1\] (\_1 + (t))\^ (t)+D(t) C\_[S]{}\^3 (\_1 + (t))\^ (t)\^D(t)\^ +(\_1 + (t))\^[2-p]{}K\^2 for all $t\in[0,T]$. By means of Sobolev embedding and Poincaré inequality, for $p\in[\frac{5}{3},2)$ it holds $\varphi(t)\leq C_S D(t) $ with $ C_S $ constant depending on $p$ and $\Omega$ and independent of $n$, then from it follows \[phi2\] (\_1 + (t))\^ (t)+D(t) C\_[S]{}\^3 (\_1 + (t))\^ (t)\^[-]{}C\_S\^D(t)\^[+]{} +(\_1 + (t))\^[2-p]{}K\^2 =A(\_1 + (t))\^ (t)\^D(t)+(\_1 + (t))\^[2-p]{}K\^2 where we set $A:=C_{S}^ {\frac{12p-11}{3p-2}}$. Hence, \[phi3\] (\_1 + (t))\^ (t)+( -A(\_1 + (t))\^ (t)\^) D(t)- (\_1 + (t))\^[2-p]{}K\^20. Assuming $\dm \nabla v_0\dm_2^2<\Lambda$ estimate holds. In this connection, by the continuity of $\dm \nabla v^n(t)\dm_2$, let $\overline t$ be the first instant such that $\dm \n v^n(\overline t)\dm_2^2=\Lambda$. Since and we can also assume $\mu_1$ suitable small, for $t=\overline t$ differential inequality furnishes $$\frac{1}{2}(\mu_1 + \varphi(\overline t))^{\frac{2-p}{2}} \frac{d}{dt} \varphi(\overline t)+ \frac{p}{8} D(\overline t)- \frac{1}{p-1}(2\Lambda)^{2-p}K^2< 0\,.$$ We can employ the Sobolev embedding $W_{\per}^{1,\frac{4}{4-p}}(\O)\hookrightarrow L_{\per}^{2}(\O)$ and the Korn inequality on the left hand side and we achieve \[p3\] (\_1 + (t))\^ (t)+ C\_S\^[-1]{}(t)-(2)\^[2-p]{}K\^2&lt; 0. Since via we conclude that $$\frac{d}{dt}\|\nabla\bv^n(\overline{t})\|_2<0\,,$$ which proves that $\overline t$ does not exist. Further, from , by straightforward computations via , it follows \^n(t)\_2\^2+ p8 \^n(t)\_\^2 (\_1 + )\^(t)\_\^2. Then integrating on $[0,T]$, for all $T$, we arrive at \[grad-dv\] \^n(T)\_2\^2+ \_0\^T \^n(t)\_\^2dt&lt;+ uniformly with respect to $n\in\N$ and $\mu>0$. #### **Other uniform estimates.** As a particular consequence of estimate we get for all $T>0$ and for all measurable set $E\subseteq\Omega$ \[S-mu-n\]0TE |§\_(\^n)|\^[p’]{}dxdt0TE |\^n|\^[p]{}dxdt T|E|\^\^p2. Let us investigate the regularity of $\bv^n_t$. From , by the Hölder and the Young inequalities and the Sobolev embedding $W^{2, \frac{4}{4-p}}\hookrightarrow L^\infty$, it follows $$\begin{split} \|\bv^n_t\|_2^2+ (\S_\mu(\cD\bv^n), \cD\bv^n_t) &\leq \|\bv^n\|_{\infty}\|\n\bv^n\|_2 \|\bv^n_t\|_2+\|\bF\|_{\frac{4}{p}} \|\bv^n_t\|_2\\ &\leq \|\bv^n\|_{\infty}^2\|\n\bv^n\|_2^2+\|\bF\|_{\frac{4}{p}}^2+\frac{1}{2}\|\bv^n_t\|_2^2\\ &\leq C_S\Lambda\|\nabla\cD\bv^n\|_{\frac{4}{4-p}}^2+\|\bF\|_{\frac{4}{p}}^2+\frac{1}{2}\|\bv^n_t\|_2^2. \end{split}$$ Hence, we get $$\frac{1}{2}\int_0^T\!\! \|\bv^n_t(t)\|_2^2dt+\frac{1}{p} \int_0^T\!\!\frac{d}{dt} \|(\mu+|\cD\bv^n(t)|^2)^{\frac{1}{2}}\|_p^p dt\leq C_S\Lambda\int_0^T \|\nabla\cD\bv^n(t)\|_{\frac{4}{4-p}}^2dt+\int_0^T\!\!\|\bF(t)\|_{\frac{4}{p}}^2dt,$$ thus by virtue of we conclude $$\label{vt-final}\begin{split} & \|(\mu+|\cD\bv^n(T)|^2)^{\frac{1}{2}}\|_p^p+\frac{1}{2}\int_0^T\!\! \|\bv^n_t(t)\|_2^2dt \\ &\leq C_S\Lambda\!\!\int_0^T\!\! \|\nabla\cD\bv^n(t)\|_{\frac{4}{4-p}}^2dt+\int_0^T\!\!\|\bF(t)\|_{\frac{4}{p}}^2dt+\|(\mu+|\cD\bv^n(0)|^2)^{\frac{1}{2}}\|_p^pdt <+\infty, \end{split}$$ uniformly with respect to $n$ and $\mu$. #### **Limit as $n\to +\infty$.** From the uniform estimates in Lemma \[lemma8\], , , , , the compact embedding $W^{2,\frac{4}{4-p}}(\O)\hookrightarrow\hookrightarrow W^{1,2}(\O)$, that in particular holds in the considered interval for $p$, and the Aubin Lions Lemma, we get the existence of a subsequence, that we do not relabel, such that the following convergences hold \[limit-n\] $$\begin{aligned} &\bv^n \rightharpoonup \bv^\mu &&\mbox{weakly in } \ L^2(0,T;W^{2,\frac{4}{4-p}}(\O)),\\ &\bv^n_t\rightharpoonup \bv_t^\mu &&\mbox{weakly in } \ L^2(0,T;L^2(\O)),\\ &\bv^n \to \bv^\mu &&\mbox{strongly in } \ L^2(0,T;W^{1,2}(\O)).\label{conv-strong} \end{aligned}$$ The lower semicontinuity of the $L^2$-norm with respect to weak convergence together with imply that \[sup-n22\] v\^(t)\_2\^2,t&gt;0.From the facts $\bv^\mu\in L^2(0,T;W^{2,\frac{4}{4-p}}(\O))$, $\bv^\mu_t\in L^2(0,T;L^2(\O))$, we have that \[CG\] \^C(0, T; J\^[1,2]{}(Ø)).Furthermore, as consequence of it follows that for every $\mu> 0$ $$\nabla \bv^n\to \nabla \bv^\mu \ \ \mbox{a.e. in } \O_T ,$$ then the continuity of the operator $\S_\mu$ implies that for all $\mu> 0$ §\_(\^n)§\_(\^)    Ø\_T. Hence, by virtue of , applying Vitali’s convergence theorem, we get \[conv-operatore\] \_[n+]{} \_[Ø\_T]{} §\_(\^n):=\_[Ø\_T]{} §\_(\^):   L\^p(0, T; W\^[1,p]{}(Ø)). From we know that, fixed $n\geq m$, for $\omega\in X_m$ it holds (\_t\^n, ) + (\^n\^n, ) + (§\_(\^n), ) = (, ), then multiplying by $\eta \in C^1(0,T; \R)$ and integrating on the interval $(0,T)$ we have \_0\^T (\_t\^n, ) + (\^n\^n, ) + (§\_(\^n), ) dt= \_0\^T (, )dt. Taking the limit as $n\to +\infty$, the convergences and imply $$\label{nuova} \int_0^T (\bv_t^\mu, \eta \bo) + (\bv^\mu\cdot\nabla \bv^\mu, \eta\bo) + (\S_\mu(\cD\bv^\mu), \eta\cD\bo) \,dt= \int_0^T (\bF, \eta\bo)\,dt$$ for every $\eta \in C^1(0,T; \R)$ and $\omega \in \cup_{m\geq 1} X_m$. Since regularity property in particular implies $\bv^\mu\in C([0,T);L^\frac{6p}{5p-6}(\Omega))$, by means of a density argument it follows \[form-final0\] (\_t\^, ) + (\^\^, ) + (§\_(\^), ) dt= (, ) J\^[1,p]{} (Ø),t\[0,T). #### **Limit as $\mu \to 0$.** For every $\mu>0$ we exhibited $\bv^\mu$ such that is fulfilled, in addition the estimates , , , are uniform with respect to $\mu$. Then employing the lower semicontinuity of the respective norms they still hold. Thus for $\mu\to 0$ and for a suitable subsequence (that we do not relabel), we get \[limit-mu\] $$\begin{aligned} &\bv^\mu \rightharpoonup \bv &&\mbox{weakly in } \ L^2(0,T;W^{2,\frac{4}{4-p}}(\O)),\\ &\bv^\mu_t\rightharpoonup \bv_t &&\mbox{weakly in } \ L^2(0,T;L^2(\O)),\\ &\bv^\mu \to \bv &&\mbox{strongly in } \ L^2(0,T;W^{1,2}(\O)).\label{TV-c} \end{aligned}$$ As a consequence of we achieve $$\n \bv^\mu\to \n\bv \mbox{ and } \bold S_\mu(\mathcal D v^\mu)\to \bold S(\mathcal D v) \mbox{ a.e. in } \Omega_T.$$ Moreover, it is not difficult to realize that holds also in the case of $\bv$: \[CIN-I\](t)\_2\^2,t&gt;0. Since holds for $\bv^\mu$ too, applying Vitali’s convergence theorem, we get the limit property \[conv-operatore2\] \_[0]{} \_[Ø\_T]{} §\_(\^):=\_[Ø\_T]{} §():   L\^p(0, T; W\^[1,p]{}(Ø)). Again the statements $\bv\in L^2(0,T;W^{2,\frac{4}{4-p}}(\O))$, $\bv_t\in L^2(0,T;L^2(\O))$ imply $$\bv \in C(0, T; J^{1,2}(\O)).$$ Finally, taking the limit as $\mu\to 0$ in , employing the convergences and , and by the same arguments used in the passage to the limit as $n\to+\infty$ we establish \[form-final01\] (\_t, ) + (, ) + (§(), ) dt= (, ) J\^[1,p]{} (Ø),t\[0,T). Proof of Theorem \[thm1\] {#section1} ========================= Let $\b_0$ be in $J_{\per}^{1,2}(\O)$ such that $\|\nabla\b_0\|_2^2\leq \Lambda$ with $\Lambda$ the constant determined in . By virtue of Theorem \[thmA\] there exists a solution, denoted by $\b(\bx,t)$, corresponding to $\b_0$ and $\bF$ to system for any $\mu>0$. For $m,k\in \N$ with $m>k$, set $\b'(t, \bx):=\b(t+(m-k)\mathcal T, \bx)$ with $t\geq 0$. The difference $\bu(t, \bx) := \b'(t,\bx)-\b(t,\bx)$ is solution a.e. in $(0, T)\times\O$ to the system \[system-u\]\_t-(§\_(’)-§\_())=-’--(\_[’]{}-\_),=0 where we exploited the periodicity of $\bF$. We multiply by $\bu$ equation $_1$, integrating over $\O$ it holds $$\frac12\frac{d}{dt} \|\bu\|_2^2 + \iO (\S_\mu(\b)-\S_\mu(\b'))\cdot \cD(\b-\b')\,dx\leq |(\bu\cdot\n\bu,\b')|\,.$$ By virtue of inequality and Hölder’s inequality, we get $$\frac12\frac{d}{dt} \|\bu\|_2^2 + 3^{\frac{p-2}2}\iO|\cD\bu|^2(\mu+ |\cD\b|^2+|\cD\b'|^2)^\frac{p-2}2\,dx\leq \|\bu\|_{\frac{12}{8-3p}}\|\n\bu\|_{\frac{4}{4-p}}\|\b'\|_{\frac{6}{3p-4}}\,.$$ Employing the Sobolev embeddings $W^{1, \frac{4}{4-p}}\hookrightarrow L^{\frac{12}{8-3p}}$ and $W^{1,2}\hookrightarrow L^{\frac{6}{3p-4}}$ and the Poincaré inequality and on the right-hand side, we deduce \[holder-d\]12 \_2\^2 +3\^[2]{} ||\^2(+ ||\^2+|’|\^2)\^dx|(,’)| C\_S\^12 \_\^2, moreover using Inequality \[ineq\] on the left-hand side with $\frac{2}{p-2}$ and $\frac{2}{4-p}$ and the Korn inequality, we obtain $$\frac{d}{dt} \|\bu\|_2^2 +\frac{3^{\frac{p-2}2}C_K\|\n\bu\|_{\frac{4}{4-p}}^2}{ (\mu_1 + \|\nabla\b(t)\|_2^2+ \|\nabla\b'(t)\|_2^2)^\frac{2-p}2} \leq C_S \Lambda^\frac12 \|\n\bu\|_{\frac{4}{4-p}}^{2}$$ where $C_K$ is the constant of the Korn inequality, hence $$\frac12\frac{d}{dt} \|\bu\|_2^2 +\Big[\frac{3^{\frac{p-2}2}C_K}{ (\mu_1 + \|\nabla\b(t)\|_2^2+ \|\nabla\b'(t)\|_2^2)^\frac{2-p}2} - C_S\Lambda^\frac12\Big]\|\n\bu\|_{\frac{4}{4-p}}^2 \leq 0.$$ Then, by virtue of and employing the Sobolev embedding $W^{1, \frac{4}{4-p}}(\O)\hookrightarrow L^2(\O)$, it results \_2\^2 + \_2\^20. Integrating the latter between $0$ and $k\mathcal T$, it follows (kT)\_2 e\^[-kT]{} (0)\_2, and by recalling the definition of $\bu$ yields \[cauchy\](mT)-(kT)\_2e\^[-kT]{} ((m-k)T)-\_0\_22 e\^[-kT]{} C\_S\^, where we used the Poincaré inequality and Theorem \[thmA\]. Set $\b_n(x):=\b(n\mathcal T, x)$. From it results that $\b_n(\x)$ is a Cauchy sequence in $J(\O)$ and thus there exists $\bv_0\in J(\O)$ such that $\b_n\to\bv_0$ strongly in $L^2(\O)$. In addition by virtue of Theorem \[thmA\] $\b_n(\x)$ fulfills $$\|\nabla \b_n\|_2^2\leq \Lambda$$ thus it holds $$\nabla \b_n \rightharpoonup \nabla \bv_0 \ \ \mbox{weakly in } \ L^2(\O),$$ and by means of the lower semicontinuity of the $L^2$-norm we achieve $$\label{v0mu} \|\nabla \bv_0\|_2^2\leq \Lambda.$$ Function $\bv_0$ enjoys the assumptions of Theorem \[thmA\], hence there exists $\bv(t, \bx)$ solution corresponding to $\bv_0(\bx)$ and $\bF(t, \bx)$ to system for any $\mu>0$. The solution $\bv(t, \bx)$ is time periodic of period $\mathcal T$. To prove the periodicity of $\bv(t, \bx)$, set $\bv'(t, \bx):= \b(t+n\mathcal T, \bx)$ and consider $\bw(t, \bx):=\bv'(t, \bx) -\bv(t, \bx)$, which is solution of the following system \_t-(§\_(’)-§\_())=-’’+-\_,=0,(, 0)=(nT, )-\_0(). Multiply by $\bw$, integrate over $\O$ and by analogous arguments made for $\bu(t,\bx)$, concerning the system , we deduce \[vudoppio\] (t)\_2 e\^[-t]{} (0)\_2, hence (t+nT)-(t)\_2 e\^[-t]{} (nT)-\_0\_2. Set $t=\mathcal T$. Taking the limit as $n$ goes to infinity and using the strong convergence of $\b_n$ to $\bv_0$ in $L^2(\O)$, it follows \_0-(T)\_2=0. Thus for any $\mu>0$ we exhibited the existence of an initial data $\bv_0^\mu$ and of a corresponding regular time-periodic solution $\bv^\mu$ to with period $\mathcal T$. Following the arguments in the proof of Theorem \[thm1\] we derive uniform estimates that allow the passage to the limit as $\mu\to 0$. The limit $\bv$ is a regular solution to in the sense of Definition \[def-d2\]. It remains to be proved that $\bv$ is time-periodic. In particular $\bv^\mu$ enjoys the following uniform estimate with respect to $\mu$ (see ) $$\|\nabla\bv^\mu(\mathcal T)\|^2_{L^{2}(\O)}\leq \Lambda.$$ Thus from the compact embedding $W^{1,2}(\O)\hookrightarrow\hookrightarrow L^2$, the limit $\bv$ fulfills \[T\] \^(T)(T)    L\^2(Ø). Moreover, by virtue of and compact embedding there exists $\bv_0$ such that the sequence $\bv_0^\mu$ enjoys in particular \[zero\] \^\_0\_0    L\^2(Ø). Therefore the time-periodicity of $\bv^\mu$ and the convergences and imply that $\bv$ is time periodic with period $\mathcal T$. For the uniqueness, consider $\bv$ and $\bv'$ two time-periodic solutions with period $\mathcal T$ corresponding to $\bF$, then define $\bw(t, \bx):=\bv'(t, \bx) -\bv(t, \bx)$ and following the arguments employed in (where the same symbols have different meaning) we get $$\lim_{n\to +\infty} \|\bv(n\mathcal T)-\bv'(n\mathcal T)\|_2=0,$$ but this contradicts the periodicity of the two solutions. Proof of Theorem \[extinction\] =============================== Theorem \[thm1\] ensures the existence of time-periodic solutions $\bv$ to system . Taking $\bv$ as test function in the weak formulation of , then employing Sobolev embedding, the Korn, the Young inequalities and since $\bF(t)\in L^2(\Omega)$ due to the assumption $\bF(t) \in L^{\frac{4}{p}}(\Omega)$ a.e. in $(0,T)$ with $p<2$, we get (t)\_2\^2+(t)\_p\^p(t)\_2(t)\_2C(t)\_2\^[p’]{}+(t)\_p\^p, i.e. \[ext\] (t)\_2\^2+(t)\_p\^pC(t)\_2\^[p’]{}. Since $\|\bF(t)\|_2=0$ in $[t_f, \mathcal T]$, considering between $t_f$ and $t$ after employed Sobolev embedding, it follows (t)\_2\^2+C\_S\^[-1]{}(t)\_2\^p0, then integrating in $[t_f, t]$ (t)\_2\^[2-p]{}-(2-p)C\_S\^[-1]{}(t-t\_f)+\_0\_2\^[2-p]{}. Thus it holds (t)\_2=0     t:  t\_f+tT. Proof of Corollary \[cor\] ========================== By virtue of Theorem \[thm1\] since $\bF$ is independent of the time, for every $\mathcal T>0$, there exists $\bv(t, x)$ as unique time-periodic solution to system with period $\mathcal T$. Let $\bv^{ \mu}$ the approximation constructed as in the proof of Theorem \[thm1\] that converges to $\bv(t, x)$. In what follows we neglect the indexes $\mu$. We set $\bu:=\bv^\mu(t,x)-\bv^\mu(s,x)$. We consider the equation for $\bu$ deduced from . Hence we consider $$\ba{l}\displ \bu_t-\div(\S_\mu(\bv(t))-\S_\mu(\bv(s)))=-\bu\cdot\n\bv(t)-\bv(s)\cdot\n\bu-\n\pi_{\bu},\VS \div \bu=0 \ea$$ where we employ the independence of $t$ of data $\bF$. By virtue of the regularity of $\bv^\mu$, multiplying by $\bu (t,x)$ and integrating by parts, almost for all $t>0$ we get $$\frac{1}{2}\frac{d}{dt}\|\bu(t)\|_2^2+ 3^{\frac{p-2}2}(p-1)\int_\Omega (\mu+|\cD\bv(t)|^2+|\cD\bv(s)|^2)^{\frac{p-2}{2}}|\cD\bu|^2\leq |(\bu\cdot\nabla\bv, \bu)|. $$ By Hölder’s inequality, the Sobolev embedding $W^{1, \frac{4}{4-p}}_{\per}(\Omega)\hookrightarrow L_{\per}^4(\Omega)$, the Korn inequality and on the right-hand side and Inequality \[ineq\] on the left-hand side, we obtain $$\frac{1}{2}\frac{d}{dt}\|\bu\|_2^2+3^{\frac{p-2}2}(p-1)\frac{\|\cD\bu\|_{\frac{4}{4-p}}^2}{(\mu_1+\|\cD\bv(t)\|_2^2+\|\cD\bv(s)\|^2)^{\frac{2-p}{2}}}\leq\|\nabla\bv\|_2 \|\bu\|_4^2\leq C_SC_K\Lambda^{\frac{1}{2}} \|\cD\bu\|_{\frac{4}{4-p}}^2 . $$ Recalling that $\mu_1:=\mu|\Omega|$, then and straightforward computations imply $$\frac{1}{2}\frac{d}{dt}\|\bu\|_2^2+3^{\frac{p-2}2}(p-1)\frac{\|\cD\bu\|_{\frac{4}{4-p}}^2}{(\mu_1+\Lambda)^{\frac{2-p}{2}}}\leq C_SC_K\Lambda^{\frac{1}{2}} \|\cD\bu\|_{\frac{4}{4-p}}^2.$$ Therefore, by virtue of and since we can fix $\mu$ arbitrarily small, we have $3^\frac{p-2}2(p-1)-(\mu_1+\Lambda)^{\frac{2-p}{2}}C_SC_K\Lambda^{\frac{1}{2}}>0$, then the Sobolev embedding $W^{1, \frac{4}{4-p}}_{\per}(\Omega)\hookrightarrow L_{\per}^2(\Omega)$ gives $$\frac{1}{2}\frac{d}{dt}\|\bu\|_2^2+\frac{3^{\frac{p-2}2}(p-1)-C_SC_K(\mu_1+\Lambda)^{\frac{2-p}{2}}\Lambda^{\frac{1}{2}}}{(\mu_1+\Lambda)^{\frac{2-p}{2}}}\|\bu\|_{2}^2\leq 0.$$ Integrating between $h$ and $h+ \mathcal T$, we get $$\|\bu(h+\mathcal T)\|_2 ^2=\|\bv(h+\mathcal T)-\bv(s)\|_2^2=\|\bv(h )-\bv(s)\|_2^2\leq \exp[ -\mathcal T]\|\bv(h)-\bv(s)\|_2^2.$$ which proves that $\bv^\mu(h)=\bv^\mu(s)$. Since it is true for all $h\in[0,\mathcal T] $, then the solution $\bv^\mu$ is a steady solution. The same is true for the limit $\bv$. [^1]: The authors are grate to the editor in chief Prof. G.P. Galdi who submitted to their attention the interesting paper [@BL].

--- abstract: 'The paper presents a numerical study for the finite element method with anisotropic meshes. We compare the accuracy of the numerical solutions on quasi-uniform, isotropic, and anisotropic meshes for a test problem which combines several difficulties of a corner singularity, a peak, a boundary layer, and a wavefront. Numerical experiment clearly shows the advantage of anisotropic mesh adaptation. The conditioning of the resulting linear equation system is addressed as well. In particular, it is shown that the conditioning with adaptive anisotropic meshes is not as bad as generally assumed.' author: - 'Weizhang Huang[^1]' - 'Lennard Kamenski[^2]' - 'Jens Lang[^3]' bibliography: - 'hkl12.bib' title: | Adaptive finite elements\ with anisotropic meshes[^4] --- Introduction {#sec:introduction} ============ Anisotropic mesh adaptation, i.e., adaptation of the size and shape of mesh elements, has been shown to be of significant advantage for problems with distinct anisotropic features. Moreover, the ability to adjust the shape and orientation of mesh elements has proven to be useful for designing numerical schemes with particular features, e.g., satisfying the discrete maximum principle [@LiHua10] or improving the conditioning of the finite element equations [@D'RoDo97; @KaHuXu12]. In this paper we concentrate on obtaining anisotropic meshes for the purpose of minimizing the numerical solution error. Typically, the optimal shape and orientation of mesh elements depend on the Hessian [@D'Azev91; @ForPer03; @Huang06; @Simpso94] or the first derivatives [@Picass03; @Picass06] of the exact solution of the underlying problem. This is the first major difficulty since the exact solution is usually not available. One possibility to solve this difficulty is to try to recover the approximate Hessian from the numerical solution in the course of the computation. In [@FoMiPe04; @ForPer03], mesh adaptation is based on a residual-based error estimator but still requires Hessian recovery for the solution of the dual problem. Unfortunately, Hessian recovery methods do work very well for interpolation problems but they cannot provide a convergent recovery if applied to linear finite element approximations on non-uniform meshes [@AgLiVa10; @Kamens09PhD], although adaptive finite element methods based on Hessian recovery still provide excellent mesh adaptation in practice [@Dolejs98; @FoMiPe04; @ForPer03; @HuaLi10; @LiHua10; @VasLip99]. The fact that the convergence of adaptive algorithms employing Hessian recovery cannot be proven directly (since the recovered Hessian does not converge to the exact Hessian) explains recent interest in anisotropic mesh adaptation based on some kind of a posteriori error estimates which do not depend on the exact solution of the underlying problem [@AgLiVa08; @AgLiVa09; @AgLiVa10; @ApGrJM04; @CaHuRu01; @FrLaRo96; @GeHaHo07; @HuKaLa10]. Moreover, as shown in [@HuKaLa10 Sect. 5.3], using error estimates could be of advantage for problems exhibiting gradient jumps or similar discontinuities along internal interfaces because methods based on recovery of derivatives could result in unnecessarily high mesh density near discontinuities. For our study we employ the anisotropic mesh adaptation algorithm from [@HuKaLa10] which employs a globally defined hierarchical basis error estimate (HBEE) for obtaining the directional information. In contrast to the recovery-based algorithms, this method adapts the mesh in order to directly minimize the a posteriori error estimate and, thus, relies on the accuracy of the error estimator but does not require recovery of derivatives of the exact solution. In this sense, the algorithm is completely a posteriori. Another major concern when using anisotropic meshes is the conditioning of the finite element equations. Generally speaking, an anisotropic mesh is expected to contain elements of large aspect ratio[^5] and there exists a concern that an anisotropic mesh will lead to extremely ill-conditioned linear algebraic systems and this may weaken the accuracy improvements gained through anisotropic mesh adaptation. Fortunately, as it has been recently shown in [@KaHuXu12], the conditioning of the stiffness matrix with anisotropic meshes is not necessarily as bad as generally assumed, especially in $2D$. In Sect. \[subsec:conditioning\], we will see that even if the condition number of the stiffness matrix with an anisotropic mesh is larger than that with an isotropic mesh, the accuracy gained through anisotropic mesh adaptation still clearly outbalances the conditioning issues, at least for the example considered. The outline of this paper is as follows: a brief description of the adaptation algorithm is given in Sect. \[sec:discretization\] which is followed by the numerical experiment in Sect. \[sec:numericalExperiment\]. The concluding remarks are given in Sect. \[sec:summary\]. Discretization and the mesh adaptation algorithm {#sec:discretization} ================================================ We consider a Dirichlet problem for the Poisson equation $$\begin{cases} - \Delta u = f, & \text{in } \Omega \\ u = 0, & \text{on } \partial\Omega \end{cases} \label{eq:bvp}$$ where $\Omega \subset \mathbb{R}^2$ is a connected bounded polygonal domain. For a given triangulation $\mathcal{T}_h$ of $\Omega$ and the associated linear finite element space $V_h \subset H_0^1(\Omega)$, the linear finite element solution $u_h \in V_h$ of is defined by $$\int_{\Omega} \nabla v_h \cdot \nabla u_h ~ dx = \int_{\Omega} f v_h ~ dx, \quad \forall v_h \in V_h. \label{eq:femIntegral}$$ The finite element space $V_h$ and the finite element solution $u_h$ can be written as $$V_h = \text{span} \{ \phi_1, \cdots, \phi_{n_{int}} \} \quad \text{and} \quad u_h = \sum_{j=1}^{n_{int}} u_j \phi_j , \label{eq:uh}$$ where $\phi_j$ is the standard linear basis function associated with the $j$-th vertex and $n_{int}$ is the number of interior vertices of the triangulation. Substituting into and taking $v_h = \phi_i$ for $i = 1, \dots, n_{int}$ results in the linear system $$A u_h = F, \label{eq:linearSystem}$$ where $$A_{ij} = \int_{\Omega} \nabla \phi_j \cdot \nabla \phi_i ~ dx \quad \text{and} \quad F_i = \int_{\Omega} f \phi_i ~ dx.$$ Note that in order to obtain the finite element solution $u_h$ we need to solve the linear algebraic system . Thus, the accuracy of $u_h$ depends also on the conditioning of this system which in turn is affected by the choice of the mesh. As mentioned in the introduction, there is a concern that anisotropic meshes could lead to extremely ill-conditioned linear systems and this may weaken the accuracy gained with anisotropic mesh adaptation. In our numerical experiment in Sect. \[subsec:conditioning\] we will address this issue in detail. In order to construct $\mathcal{T}_h$ (and, thus, the corresponding $V_h$) we employ the $M$-uniform mesh approach which generates an adaptive mesh as a quasi-uniform one in the metric specified by a symmetric and strictly positive definite tensor $M = M(x)$ [@Huang05a]. The algorithm starts with an initial mesh. For every mesh $\mathcal{T}_h^{(i)}$ we compute the finite element solution $u_h^{(i)}$ which is used to compute a new adaptive mesh for the next iteration step. The new mesh is generated as an $M$-uniform mesh with a metric tensor $M_h^{(i)}$ computed from $u_h^{(i)}$. This yields the sequence $$\mathcal{T}_h^{(0)} \rightarrow u_h^{(0)} \rightarrow M_h^{(0)} \rightarrow \mathcal{T}_h^{(1)} \rightarrow u_h^{(1)} \rightarrow M_h^{(1)} \rightarrow \dots$$ The mesh adaptation process is repeated until the mesh is $M$-uniform within a given tolerance (see [@HuKaLa10 Sect. 4.1] for more details). In our computation we use BAMG (*bidimensional anisotropic mesh generator* [@bamg]) to construct anisotropic meshes for a given metric tensor $M$. Typically, the optimal metric tensor $M_h$ depends on the Hessian of the exact solution [@ForPer01; @ForPer03; @Huang05a] which is usually unknown. In this study we follow [@HuKaLa10] and employ the hierarchical basis a posteriori error estimate (HBEE) to obtain the directional information required for the metric tensor $M_h$. The brief idea is as follows (see [@HuKaLa10] for details). If we have an error estimate $z_h$ such that $${\lVert u - u_h \rVert} \leq C {\lVert z_h \rVert}.$$ for a given norm ${\lVert \cdot \rVert}$ and if it further has the property $\Pi_h z_h \equiv 0$ with $\Pi_h$ being the interpolation operator associated with $V_h$ (which is fulfilled by the HBEE), than the finite element approximation error is bounded by the interpolation error of the error estimate, $$\|u - u_h\| \le C {\lVert z_h \rVert} = C {\lVert z_h - \Pi_h z_h \rVert}. \label{eq:zhIE}$$ Hence, up to a constant, the solution error is bounded by the interpolation error of the error estimate and the mesh can be constructed to minimize the interpolation error of $z_h$; the metric tensor $M_h$ does not depend on the Hessian of the exact solution. In this study, we are concerned with the error measured in the $H^1$ semi-norm, which is the energy norm from . Therefore, instead of using the metric tensor developed in [@HuKaLa10] for the error measured in the $L^2$ norm, we construct the metric tensor which minimizes the interpolation error of $z_h$ measured in the $H^1$ semi-norm. In two dimensions the optimal metric tensor is given element-wise by $$M_{K} = \left\|I + \frac{1}{\alpha_h} |H_K(z_{h})|\right\| \cdot \det\left(I + \frac{1}{\alpha_h} |H_K(z_{h})|\right)^{-\frac{1}{4}} \cdot \left[I + \frac{1}{\alpha_h} |H_K(z_{h})|\right],$$ where $H_K(z_h)$ denotes the Hessian of the (quadratic) hierarchical basis error estimate $z_h$ on element $K$ and $\alpha_h$ is a regularization parameter to ensure that $M_K$ is strictly positive definite. $\alpha_h$ can also be seen as an adaptation intensity control: uniform mesh has $\alpha_h = \infty$ and if $\alpha_h \rightarrow 0$ the mesh becomes more adaptive. Usually, $\alpha_h$ is chosen so that about half of the mesh elements are concentrated in regions where $\det(M)$ is large (see [@Huang06] for more details on the choice of $M_K$ and $\alpha_h$). In our computations we employ the globally defined hierarchical basis error estimate since it contains more directional information of the solution than localized versions [@HuKaLa10 Sect. 5.1]. Moreover, it has been shown that local error estimates can be inaccurate on anisotropic meshes [@DoGrPf99]. To avoid the cost of the exact solution of the global error problem, we use only a few sweeps of the symmetric Gauss-Seidel iteration for the resulting linear system until the relative difference of the old and the new error approximations is under a given relative tolerance. This proves to be adequate for the purpose of mesh adaptation and the computational cost is comparable to that of the Hessian recovery: in the tests, the computation of HBEE is about twice slower than Hessian recovery. Although the validity of the classical hierarchical basis error estimate $z_h$ for the anisotropic case is still unclear, theoretical considerations in [@Grosma06PhD Sect. 6.4] and numerical results in [@HuKaLa10] suggest that the hierarchical basis error estimate is a reliable source of information when a mesh is aligned with the solution. Numerical experiment {#sec:numericalExperiment} ==================== For the numerical experiment we consider a problem in [@Mitche10] which combines multiple difficulties. It is a Dirichlet problem of the Poisson equation $$\begin{cases} - \Delta u = f, & \text{in } \Omega\\ u = g, & \text{on } \partial\Omega \end{cases} \label{eq:Poisson}$$ where $\Omega$ is an L-shaped domain $\Omega = (-1, 1) \times (-1, 1) \setminus [0, 1) \times (-1, 0]$. The functions $f$ and $g$ are chosen such that the exact solution $u$ is given by $$\begin{aligned} u(x,y) &= r^{2/3} \sin(2 \theta /3) + \tan^{-1} \left( 200 \left(\sqrt{x^2 + (y + 3/4)^2} - 3/4 \right)\right) \\ & \quad + e^{-1000 \left( (x + \sqrt{5}/4)^2 + (y + 1/4)^2 \right)} + e^{- 100 (y + 1)},\end{aligned}$$ where $r$ and $\theta$ are the polar coordinates. The solution has - a singular gradient at $(0,0)$ due to a reentrant corner of the L-shaped domain $\Omega$, - a circular wavefront with the center in $(0,-3/4)$ and the radius of $3/4$, - a sharp peak at $(-\sqrt{5}/4, -1/4)$, - and a boundary layer along the line $y = -1$. Figure \[fig:temperaturePlot\] shows the surface and the color plot of a numerical solution. ![Surface and color plots of the numerical solution.[]{data-label="fig:temperaturePlot"}](hkl12_mitchell212_mesh "fig:"){height="0.20\textheight"} ![Surface and color plots of the numerical solution.[]{data-label="fig:temperaturePlot"}](hkl12_mitchell212_color "fig:"){height="0.19\textheight"} Accuracy of the numerical solution {#subsec:optimal:mesh:grading} ---------------------------------- First, we compare the accuracy of the numerical solution for Delaunay (quasi-uniform), adaptive isotropic, and adaptive anisotropic meshes. Examples of mesh types are given in Fig. \[fig:meshExamples\]. We observe that both isotropic and anisotropic adaptive meshes (Figs. \[fig:meshExamples:isotropic\] and \[fig:meshExamples:anisotropic\], respectively) have high mesh density in regions with difficulties but the anisotropic mesh (Fig. \[fig:meshExamples:anisotropic\]) is clearly much better aligned with the steep boundary layer and the wavefront. This is the major difference between the isotropic and anisotropic adaptation: the isotropic adaptation can provide proper mesh density whereas the anisotropic adaptation can provide both proper mesh density *and* proper alignment of the mesh with the anisotropic features of the solution. Fig. \[fig:convergence\] shows the error of the numerical solution measured in the energy norm ${||| u - u_h |||}$, which is equal to $H^1(\Omega)$ semi-norm $|u - u_h|_{H^1(\Omega)}$ for the example considered. The convergence plot shows that an anisotropic adaptive mesh requires ca. 200 times fewer elements than a quasi-uniform mesh in order to achieve the same accuracy and ca. 10 times fewer elements than an isotropic adaptive mesh. In other words, the finite element solution with an anisotropic mesh has a 15 times smaller error than an error of the solution on a quasi-uniform mesh with the same number of elements and 3 times smaller than the error achieved by means of an isotropic adaptive mesh. The asymptotic convergence order of the error in the energy norm is the same for all three kinds of meshes: it is $O(N^{-0.5})$.[^6] This is expected since we cannot have a better convergence order for anisotropic mesh adaptation but can expect a much smaller constant when the solution of the problem has anisotropic features. In our test example we gain more than one order of magnitude in comparison to quasi-uniform meshes and about one half of the order in comparison to the isotropic adaptation. Fig. \[fig:convergence\] provides also an interesting insight into the behaviour of anisotropic mesh adaptation. For very coarse meshes ($N < 300$) the resolution is not good enough to capture the anisotropy of the solution, the mesh is isotropic and has the same error as with isotropic mesh adaptation. The interesting part of the plot is between $N \approx 300$ and $N \approx 1000$, where the algorithm starts to catch the anisotropic features and the error drops quickly. When the anisotropic mesh is fine enough to resolve the anisotropy of the solution ($N > 1000$), the error convergence rate reaches the asymptotic state. ![Energy norm of the finite element error vs. number of mesh elements.[]{data-label="fig:convergence"}](hkl12_mitchell212_h1){width="75.00000%"} Condition number of the stiffness matrix {#subsec:conditioning} ---------------------------------------- In this section, we compute the exact condition number (with respect to the $\lVert\cdot\rVert_2$ matrix norm) for the stiffness matrix of the anisotropic finite elements equations and compare it to the conditioning of the finite element equations with isotropic adaptive and quasi-uniform meshes. The analysis in [@KaHuXu12] for the Laplace operator in 2D shows that the condition number can be bounded by a term depending mainly on the number $N$ and the largest aspect ratio of the mesh elements. In our numerical experiment the maximum aspect ratio is up to $3.8$ for quasi-uniform and isotropic meshes and up to $37.9$ for anisotropic meshes. Thus, the rough estimate on the ratio between the condition numbers of the anisotropic and isotropic systems should be about $37.9/3.8 \approx 10.0 $. This is in perfect agreement with our numerical results presented in Fig. \[fig:condition:unscaled\] which show that the condition number of the linear system with anisotropic meshes is about one order of magnitude higher than that with the isotropic meshes. Notice also the sudden jump in the condition number for the anisotropic case in the range $300 \leq N \leq 1000$: the algorithm starts to catch the anisotropic features of the solution and the maximum aspect ratio of the mesh increases quickly as the mesh becomes more and more anisotropic (cp. the corresponding error decrease in Fig. \[fig:convergence\]). Fig. \[fig:condition:unscaled\] also shows that that the asymptotic behaviour of the condition number with anisotropic meshes is at most $O(N \log N)$ which is only slightly larger than $O(N)$ in the quasi-uniform case. The conditioning with isotropic adaptive meshes is also slightly larger than $O(N)$ although still smaller than $O(N \log N)$. Moreover, if a mesh is only locally anisotropic (as in our example), a proper diagonal scaling can reduce the conditioning of the stiffness matrix so that it is comparable with the condition number in the uniform case (see [@KaHuXu12] for more details on diagonal scaling). Fig. \[fig:condition:scaled\] shows that the asymptotic rate of the conditioning of the scaled stiffness matrix is reduced to essentially $O(N)$, which is comparable to that with uniform meshes. Conclusion {#sec:summary} ========== Our numerical experiment shows that for problems with anisotropy the anisotropic mesh adaptation is clearly superior to the isotropic one. In our example, at least a half order of magnitude could be gained in accuracy by switching from the isotropic mesh adaptation to the anisotropic one. The globally defined hierarchical basis error estimate provides good directional information for the anisotropic mesh generation, provided the number of mesh elements is large enough to resolve the anisotropy of the solution. It is worth pointing out that the results in Fig. \[fig:convergence\] present the error of the final numerical solution, i.e., *after* solving the linear system. Thus, even if the condition number of the linear system with anisotropic meshes is larger than that with isotropic meshes, the accuracy gained through the anisotropic discretization for problems with anisotropic features outbalances possible losses due to the numerical accuracy. ### Acknowledgement {#acknowledgement .unnumbered} The authors are thankful to the anonymous referee for the valuable comments. [^1]: Department of Mathematics, the University of Kansas (<[email protected]>) [^2]: Department of Mathematics, the University of Kansas (<[email protected]>) [^3]: Department of Mathematics, Technische Universit[ä]{}t Darmstadt (<[email protected]>) Center of Smart Interfaces, Technische Universit[ä]{}t Darmstadt Graduate School of Computational Engineering, Technische Universit[ä]{}t Darmstadt [^4]: Supported in part by the National Science Foundation (U.S.A.) through grant DMS-1115118 and the German Research Foundation through grants SFB568/3, SPP1276 (MetStroem) and KA 3215/1-1. [^5]: In this paper the aspect ratio of a triangular element is defined as the longest edge divided by the shortest altitude. For example, an equilateral triangle has an aspect ratio of $2 / \sqrt{3} \approx 1.15$. [^6]: Note that $O(N^{-0.5}) = O(h)$ for quasi-uniform meshes in 2D.

--- abstract: 'We consider the problem of correcting the errors incurred from sending quantum information through a noisy quantum environment by using classical information obtained from a measurement on the environment. For discrete time Markovian evolutions, in the case of fixed measurement on the environment, we give criteria for quantum information to be perfectly corrigible and characterize the related feedback. Then we analyze the case when perfect correction is not possible and, in the qubit case, we find optimal feedback maximizing the channel fidelity.' author: - 'M. Gregoratti' - 'R. F. Werner' title: 'On quantum error-correction by classical feedback in discrete time' --- Introduction ============ Error correction is a key problem in quantum information processing. Without it, decoherence would easily destroy all hopes for quantum computation and quantum cryptography. Two important approaches have been devoloped to combat decoherence. The usual theory of quantum error-correction [@NC; @KW] redundantly encodes the original quantum information in a larger quantum system by a unitary operation which maps the initial Hilbert space into the code space, a subspace of the Hilbert space associated to this larger system. After encoding, the larger system is subjected to noise and then a measurement is performed on the system to diagnose the type of error which occurred. Finally, on the basis of the outcome, a restoring operation is performed to return the system to the original state in the code space. Another approach [@MZ; @Plenio; @ADL; @AWM; @Alber], based on feedback control, has been developed for systems which are continuously monitored during their noisy evolution. It employs the result of this continuous measurement to determine the errors occurred and, on this basis, perform corrections in real time to protect states which are known to lie initially within a certain code space. The continuous measurement can be performed introducing an additional interaction to the evolution of the system or preferably by simply observing the environment after its interaction with the system. Up to now there is no general theory for quantum feedback control, but some correction schemes have been considered and have dealt only with Hamiltonian feedback. Taking from feedback control the idea of a correction scheme based on a measurement performed not on the system but on the environment, we have analyzed in an earlier publication [@GW] the different possible behaviours of a channel with respect to the existence of a measurement allowing perfect correction of information, quantum or classical. No code space is introduced and, for every initial state, the aim is to find proper measurement and restoring operations, not necessarily unitary, to recover the initial information or at least optimally restore it. Here we develop the analysis in [@GW] to evolutions composed of many time steps. We consider quantum information carried by a system undergoing a discrete time Markovian evolution (multi-step channel $T=T^{(n)}\circ\cdots\circ T^{(1)}$) and allow correction operations between any two subsequent steps: every time we perform a measurement on the environment and a restoring operation on the system. We take the measurments on the environment as given and we look for the best restoring operations choosen on the basis of all the outcomes observed so far. First we give conditions for quantum information to be perfectly corrigible and characterize the related feedback. It turns out that classical feedback is not useful because perfect correction of quantum information is possible only if a unique correction at the end would suffice. Then we analyze the case when perfect correction is not possible and, in the qubit case, we find optimal feedback maximizing the channel fidelity. Similarly to the case of perfect correction, but more surprisingly, classical feedback is still useless if quantum information carried by one qubit because again a unique restoring operation performed at the end of the evolution guarantees the same performance of correction. Anyway classical feedback is shown to be helpful when the quantum carrier has a larger Hilbert space. Every Section of the paper considers first the case of quantum information sent through a single-step channel, so that it is possible only one correction after this noisy evolution just as in [@GW], and then the case of a multi-step channel, so that classical feedback is possible during the noisy evolution. In Section \[CorrSch\], we set up the framework and the basic correction scheme. In Section \[PerCorr\] we prove the basic criteria for the existence of a correction scheme for a given channel and given measurements on the environment, and we characterize the related feedback. In Section \[OptRec\] we look for optimal correction scheme when it is not be possible to achieve a complete correction of errors and ... The correction scheme {#CorrSch} ===================== We consider quantum information carried by a quantum system with finite dimensional Hilbert space $\HH$. We work in the Schrödinger picture, so the action of the noisy *channel*, which corrupts the information transforming each input density operator $\rho$ on $\HH$ to a different output density operator $T(\rho)$ on $\HH$, is given by a completly positive, trace preserving, linear map $T:\BB(\HH)\to\BB(\HH)$, where $\BB(\HH)$ denotes the space of all linear operators on $\HH$. A channel $T$ is not physically reversible, i.e. there is no channel $R$ such that $R\circ T=\id$, unless $T$ is a unitary channel $T(\rho)=u\,\rho\,u^*$, with $u\in\mathcal{U}(\HH)$, the group of unitary operators on $\HH$. Single-step channel ------------------- Every channel can be described as the result of a unitary coupling to an environment, followed by the discard of the environment after the interaction. Nevertheless, the initial information carried by $\HH$ is then shared with the environment and indeed, if we controlled perfectly the combined system, then we could recover the initial quantum information simply by reversing the global evolution, so restoring perfectly the input state $\rho$. We do not hypothesize this, but we assume that our control on the environment is good enough to have its initial state pure and to be able to perform any measurement on it after the interaction with $\HH$. The measurement of an observable $X$ on the environment decomposes the channel $T$ into an *instrument*, a family of completely positive maps $T_x$ giving the (non-normalized) output states $T_x(\rho)$ of the subensembles of systems selected according to the result “$x$” of this measurement: the probability of observing $x$ is ${\mathbb{P}}(X=x)={\operatorname{tr}}T_x(\rho)$, the normalized output state for the corresponding subensemble is $T_x(\rho)/{\operatorname{tr}}T_x(\rho)$, and the expectation in that subensemble of a self-adjoint $A\in\BB(\HH)$ is ${\operatorname{tr}}(T_x(\rho) A)/{\operatorname{tr}}T_x(\rho)$. Ignoring the result of the measurement one recovers the original channel $$\label{T&Tx} T=\sum_x T_x\;.$$ Of course, the result $x$ of the measurement gives classical information about the environment after the interaction, and hence also about $\HH$ before the interaction. The idea for the correction scheme is to employ this information to select a proper restoring channel: we introduce a family of channels $R^{(x)}:\BB(\HH)\to\BB(\HH)$ where $R^{(x)}$ can depend in an arbitrary way on $x$. After correction, the state of the subensemble for which the measurement has given the result $x$, will be $R^{(x)}(T_x(\rho))$ up to the normalization factor ${\operatorname{tr}}(T_x(\rho))$. The overall corrected channel is built from these conditional operations by ignoring the intermediate information $x$, and is the sum of these contributions: $$\label{Tcorr} \Tcorr=\sum_x R^{(x)}\circ T_x\;.$$ The goodness of the scheme in restoring quantum information depends on how $\Tcorr$ can be brought close to the ideal channel on $\HH$, i.e. $\id$. Whether or not we can find a good correction scheme in principle depends not only on the noisy channel $T$, but also on the set of decompositions obtainable by a measurment on the environment, which usually depends on the particular coupling which induces the noisy evolution $T$. Anyway the assumption of a pure environment overcomes this problem because it guarantees that every decomposition of $T$ into c.p. summands $T_x$ can be realized by a measurement on the environment. Moreover, in order to correct quantum information, the preferable docompositions are the finest ones, those for which no proper refinement is possible. Therefore we shall consider the non refinable decompositions given by the Kraus representations of $T$: $$\label{Tdec} T(\rho) = \sum_x t_x \, \rho \, t_x^* \;, \qquad \mbox{ where } \sum_x t_x^*t_x = \idty \;.$$ In the following we shall assume that a decomposition is given, that it is associated to the measurement of an observable $X$, and for it we shall consider the problem of perfectly correcting quantum information or, if this is not possible, optimally restoring it. Multi-step channel ------------------ We are especially interested in multi-step channels, where the noisy evolution $T$ is given by $n$ Markovian steps: $$\label{divT} T=T^{(n)}\circ\cdots\circ T^{(1)}\;,$$ where $T^{(k)}:\BB(\HH)\to\BB(\HH)$ are $n$ possibly different channels. In this case we assume to be able to perform a measurement at each step, so that every $T^{(k)}$ is decomposed into $$\label{Tkdec} T^{(k)}(\rho) = \sum_{x_k} t^{(k)}_{x_k} \, \rho \, {t^{(k)}_{x_k}}^* \;, \qquad \mbox{ where } \sum_{x_k} {t^{(k)}_{x_k}}^*t^{(k)}_{x_k} = \idty \;,$$ the channel $T$ is decomposed into $$\label{divTdec} T(\rho) = \sum_{x_1,\ldots,x_n} t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^*\;,$$ and the probability of observing $(x_1,\ldots,x_n)$ is ${\mathbb{P}}(X_1=x_1,\ldots,X_n=x_n) = {\operatorname{tr}}t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^*$. We also assume to be able to interfere in the evolution of the system after each step by appling a restoring channel selected according to the whole information gathered so far. For every $k=1,\ldots,n$, we introduce a family of channels $R^{(x_1,\ldots,x_k)}:\BB(\HH)\to\BB(\HH)$, where $R^{(x_1,\ldots,x_k)}$ can depend in an arbitrary way on $(x_1,\ldots,x_k)$ and it is applied after the measurement of $X_k$ so that the overall corrected channel turns out to be $$\label{divTcorr} \Tcorr(\rho)= \sum_{x_1,\ldots,x_n} R^{(x_1,\ldots,x_n)}\Big(t^{(n)}_{x_n} \cdots R^{(x_1)} \big( t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \big) \cdots {t^{(n)}_{x_n}}^* \Big) \;.$$ So the classical information obtained by the measurement process on the environment is fedback to modify the evolution of the system $\HH$. Notice that this general scheme includes the special cases (compare Fig. \[schemes\]): - $R^{(x_1,\ldots,x_k)}= R^{(x_k)}$ for every $k=1,\ldots,n$, i.e. every restoring channel is selected according only to the last observation $x_k$. - $R^{(x_1,\ldots,x_k)}=\id$ for every $k=1,\ldots,n-1$, i.e. the information given by the measurement on the environment is gathered during the evolution, but every correction is deferred after it, In the following we shall assume that a decomposition is fixed, that it is associated to a measurement process, and, given this, we shall consider the problem of perfect or optimal correction of quantum information. Perfect correction of Quantum Information {#PerCorr} ========================================= Given a channel $T$ with a fixed decomposition or , our scheme can perfectly correct quantum information if one can find restoring channels $R$ such that $$\nonumber \Tcorr=\id\;.$$ Single-step channel ------------------- Let us improve the criterion given in [@GW]. \[PCQI\] Let $T:\BB(\HH)\to\BB(\HH)$ be a channel over a finite dimensional Hilbert space $\HH$. Given a Kraus decomposition $T(\rho)=\sum_x t_x\,\rho\,t_x^*$, the following are equivalent - there exists a family of channels $R^{(x)}:\BB(\HH)\to\BB(\HH)$ perfectly restoring quantum information; - $t_x^*t_x=p_x\,\idty$ for all $x$, with $p_x\geq0$, $\sum_x p_x=1$; - $t_x=\sqrt{p_x}\,u_x$ for all $x$, with $u_x\in\mathcal{U}(\HH)$, $p_x\geq0$, $\sum_x p_x=1$; - $T(\rho) = \sum_x p_x \,u_x\,\rho\,u_x^*$ is a convex combination of unitary channels; - ${\mathbb{P}}(X=x)={\operatorname{tr}}t_x\,\rho\,t_x^*$ is independent of the input state $\rho$. When these conditions hold, then the restoring channels in (a) have to be $$\label{PF} R^{(x)}(\rho)=u_x^*\,\rho\,u_x \;,$$ and the probability law of the outcome is $$\label{opl} {\mathbb{P}}(X=x) = p_x \;.$$ \(a) $\Rightarrow$ (b). Let $R^{(x)}$ be channels such that $\sum_x R^{(x)} (t_x\,\rho\,t_x^*) = \rho$ for every state $\rho$. Then for every $x$, by the “No information without perturbation” Theorem, $R^{(x)}(t_x\,\rho\,t_x^*) = p_x \, \rho$, for some $p_x\geq0$, $\sum_x p_x=1$. Therefore ${\operatorname{tr}}t_x^*t_x\,\rho = {\operatorname{tr}}t_x\,\rho\,t_x^* = {\operatorname{tr}}R^{(x)}({\operatorname{tr}}t_x\,\rho\,t_x^*) = p_x$ for all $\rho$, and so $t_x^*t_x=p_x\,\idty$. \(b) $\Leftrightarrow$ (b’) $\Leftrightarrow$ (b”). These conditions are clearly all equivalent because $\dim \HH < \infty$. (b’) $\Rightarrow$ (c). ${\mathbb{P}}(X=x) = {\operatorname{tr}}t_x\,\rho\,t_x^* = {\operatorname{tr}}\sqrt{p_x}\,u_x^*\,\rho\,\sqrt{p_x}\,u_x = p_x$ for every $\rho$, so (b’) implies (c), and equation , too. \(c) $\Rightarrow$ (a). ${\operatorname{tr}}t_x^*t_x\,\rho = {\operatorname{tr}}t_x\,\rho\,t_x^* = {\mathbb{P}}(X=x)$ which does not depend on $\rho$ and so $t_x^*t_x={\mathbb{P}}(X=x)\,\idty$. Then $t_x = \sqrt{{\mathbb{P}}(X=x)}\,u_x$, where $u_x$ is unitary because $\dim \HH<\infty$, and the channels $R^{(x)}(\rho)=u_x^*\,\rho\,u_x$ perfectly restore quantum information. Now we have to show that conditions (a), (b) and (c) imply . Consider a family of channels $R^{(x)}$ restoring quantum information, with $R^{(x)}(\rho)=\sum_j r^{(x)}_j\,\rho\,{r^{(x)}_j}^*$ and $\sum_j {r^{(x)}_j}^*r^{(x)}_j= \idty$. Then $$\nonumber \sum_{x,j} r^{(x)}_j\,t_x\,\rho\,t_x^*\,{r^{(x)}_j}^* = \rho\;,\qquad\forall\;\rho\;,$$ and using again the “No information without perturbation” Theorem we have $$\nonumber r^{(x)}_j\,t_x = \sqrt{p_{x,j}}\,{\mathrm{e}}^{{\mathrm{i}}\theta(x,j)}\,\idty\;, \qquad \forall\;x,j,$$ for some probability law $p_{x,j}$ and some real function $\theta$. Using (b’), we get $$\nonumber r^{(x)}_j = \sqrt{\frac{p_{x,j}}{p_x}}\,{\mathrm{e}}^{{\mathrm{i}}\theta(x,j)}\,u_x^* \propto u_x^* \qquad \forall\;j,$$ i.e., fixed $R^{(x)}$, all the Kraus oprators $r^{(x)}_j$ have to be proportional to the same unitary operator $u_x^*$, and holds. Let us remark that condition (b) gives a simple criterion for the existence of channels $R$ perfectly restoring quantum information based directly on the Kraus operators of $T$. Moreover it clarifies the structure of channels which allow perfect correction with our scheme: the measurement on the environment has to decompose $T$ into a convex combination of unitary channels, so that the overall evolution can be seen as an average of reversible evolutions $\rho\mapsto u_x\,\rho\,u_x^*$, which occur randomly with probability $p_x$; the measurement detects the transformation occurred and $R^{(x)}(\rho') = u_x^*\,\rho'\,u_x$ restores the initial state $\rho$. This means also, by contition (c), that the measurement on the environment is completely uninformative about the system $\HH$: the output $x$ can be interpreted as classical information about “what happened” to the system $\HH$, but it gives no information about the input system because it is independent of its state. And the uninformative character of the observation, if associated to a Kraus decomposition of $T$, is sufficient for perfect correction. Moreover, if perfect correction is possible, the restoring channels $R$ have to be untiary. We shall call *corrigible* every channel $T$ with a decomposition satisfying condition (b). Just to show how useful can be a suitable measurement on the environment, let us remark that our scheme can perfectly correct quantum information sent through a depolarizing channel, which is known to destroy all quantum information and for which no ordinary quantum error correcting code works (it has zero quantum and classical capacity). Indeed, set $N = \dim\HH$, the depolarising channel $T(\rho)= \frac{1}{N} \idty$ admits the Kraus representation $$\nonumber T(\rho) = \sum_{x,y=1}^N t_{x,y} \,\rho\, t_{x,y}^*\;, \qquad t_{x,y} = \frac{1}{N} \sum_{z=1}^N {\mathrm{e}}^{\frac{2\pi{\mathrm{i}}}{N}zy} \, \ketbra{z+x}{z} = \frac{1}{N}\, u_{x,y}\;,$$ with unitary operators $u_{x,y}$, where $\{|z\rangle\}_{z\in\mathbb{Z}_N}$ denotes a basis labeled cyclically so that addition in $|z+x\rangle$ is modulo $N$. Multi-step channel ------------------ Let us consider now perfect correction for multi-step channels. \[divPCQI\] Let $T=T^{(n)}\circ\cdots\circ T^{(1)}$, with every $T^{(k)}:\BB(\HH)\to\BB(\HH)$, be a multi-step channel over a finite dimensional Hilbert space $\HH$. Given the Kraus decompositions $T^{(k)}(\rho)=\sum_{x_k} t^{(k)}_{x_k}\,\rho\,{t^{(k)}_{x_k}}^*$, T.F.A.E. - there exists a family of channels $R^{(x_1)}$, $R^{(x_1,x_2)}$, …, $R^{(x_1, \ldots, x_n)}$ perfectly restoring quantum information; - every channel $T^{(k)}(\rho)=\sum_{x_k} t^{(k)}_{x_k}\,\rho\,{t^{(k)}_{x_k}}^*$ is corrigible,\ i.e. for all $x_k$ we have $t^{(k)}_{x_k}= \sqrt{p^{(k)}_{x_k}}\,u^{(k)}_{x_k}$, with $u^{(k)}_{x_k}\in\mathcal{U}(\HH)$, $p^{(k)}_{x_k}\geq0$, $\sum_{x_k} p^{(k)}_{x_k}=1$; - ${\mathbb{P}}(X_1=x_1,\ldots,X_n=x_n) = {\operatorname{tr}}t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^*$ is independent of the input state $\rho$. When these conditions hold, then for every $k=1, \ldots, n$ the restoring channels in (a) have to be $$\label{divpF} R^{(x_1, \ldots, x_k)}(\rho)=v^{(x_1, \ldots, x_k)}\,\rho\,{v^{(x_1, \ldots, x_k)}}^* \;, \qquad v^{(x_1, \ldots, x_k)} \in \mathcal{U}(\HH)\;,$$ with $$\label{divLpF} v^{(x_1, \ldots, x_n)} = {u^{(1)}_{x_1}}^*\,{v^{(x_1)}}^*\cdots {v^{(x_1, \ldots, x_{n-1})}}^*\,{u^{(n)}_{x_n}}^*\;,$$ and the probability law of the outcome process is always $$\label{divopl} {\operatorname{tr}}R^{(x_1,\ldots,x_n)}\Big(t^{(n)}_{x_n} \cdots R^{(x_1)} \big( t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \big) \cdots {t^{(n)}_{x_n}}^* \Big) = \prod_{k=1}^n p^{(k)}_{x_k}$$ Therefore, as long as we are interested in perfect correction of quantum information, the multi-step structure of the channel and the possibility of applying feedback during the evolution do not help: perfect correction is possible if and only if every step is corrigible and, in this case, it is enough to make a unique correction at the end. Indeed only unitary corrections are allowed, otherwise the original information would be corrupted by the feedback itself, and then the first $n-1$ unitaries $v^{(x_1, \ldots, x_k)}$ can be chosen arbitrarly, also $v^{(x_1, \ldots, x_k)}=\idty$, provided that the whole evolution is reversed by the last one (eq. ). Again, the channel is corrigible if and only if the measured process is uninformative; moreover its probabilistic law, which cannot be modified by the unitary feedback, is that of a sequence of independent random variables. The proof of Proposition \[divPCQI\] goes via a more general result about the composition of $n$ channels when each channel is decomposed according to a measurement and, not only each channel itself, but also its decomposition, may depend on the previous observations. Therefore, denoted by $x_k$ the outcomes of the measurement at the $k^\mathrm{th}$ step, we want to consider the evolution of a quantum system sent through a sequence of channels on $\BB(\HH)$ $$\label{InstrProc} T^{(x_1,\ldots,x_{k-1})}(\rho)= \sum_{x_k} t^{(x_1,\ldots,x_{k-1})}_{x_k}\,\rho\,{t^{(x_1,\ldots,x_{k-1})}_{x_k}}^*\;, \qquad \sum_{x_k}{t^{(x_1,\ldots,x_{k-1})}_{x_k}}^*t^{(x_1,\ldots,x_{k-1})}_{x_k} = \idty\;,$$ where for the first step, $k=1$, we denote by “0” the empty string of prior results, so we write $$T^{(0)}(\rho)=\sum_{x_1} t^{(0)}_{x_1}\,\rho\,{t^{(0)}_{x_1}}^*\;, \qquad \sum_{x_1}{t^{(0)}_{x_1}}^*t^{(0)}_{x_1} = \idty \;.\nonumber$$ Then the total evolution is $$\label{divTdec0} T(\rho) = \sum_{x_1,\ldots,x_n} t^{(x_1,\ldots,x_{n-1})}_{x_n} \cdots t^{(0)}_{x_1}\, \rho \, {t^{(0)}_{x_1}}^* \cdots {t^{(x_1,\ldots,x_{n-1})}_{x_n}}^*\;.$$ and the next Proposition holds. \[divQIrest0\] Given a finite dimensional Hilbert space $\HH$, let $T:\BB(\HH)\to\BB(\HH)$ be a channel resulting from the application of $n$ decomposed channels $T^{(x_1,\ldots,x_{k-1})}$ . If $T = \id$, then - every channel $T^{(x_1,\ldots,x_{k-1})}(\rho)= \sum_{x_k} t^{(x_1,\ldots,x_{k-1})}_{x_k}\,\rho\,{t^{(x_1,\ldots,x_{k-1})}_{x_k}}^*$ is corrigible; - a channel $T^{(x_1,\ldots,x_{k-1})}$ is unitary if later channels and their decomositions do not depend on the value $x_k$ observed at the $k^{\mathrm{th}}$ step. If $T = \id$, then $$\label{f1} t^{(x_1,\ldots,x_{n-1})}_{x_n} \cdots t^{(0)}_{x_1} = \sqrt{p(x_1,\ldots,x_n)} \, {\mathrm{e}}^{{\mathrm{i}}\theta(x_1,\ldots,x_n)} \, \idty \;, \qquad \forall\;x_1,\ldots,x_n,$$ where $p$ is the joint probability law of the outcomes, independent of $\rho$, and $\theta$ is a real function. \(a) Using the normalization property of Kraus operators, we can immediately show that $T^{(0)}$ is corrigible: $$\nonumber {t^{(0)}_{x_1}}^*t^{(0)}_{x_1} = \sum_{x_2,\ldots,x_n} {t^{(0)}_{x_1}}^*\,{t^{(x_1)}_{x_2}}^* \cdots {t^{(x_1,\ldots,x_{n-1})}_{x_n}}^*t^{(x_1,\ldots,x_{n-1})}_{x_n} \cdots t^{(x_1)}_{x_2} \, t^{(0)}_{x_1} = \sum_{x_2,\ldots,x_n} p(x_1,\ldots,x_n) \, \idty = p(x_1) \, \idty\;,$$ where $p(x_1)$ is the probability of observing $x_1$ at the first step. Now, since $t^{(0)}_{x_1} = \sqrt{p(x_1)}\,u^{(0)}_{x_1}$ for some unitary $u^{(0)}_{x_1}$, we can fix $x_1$ and find that $T^{(x_1)}$ is corrigible: $$\nonumber\begin{split} {t^{(x_1)}_{x_2}}^*t^{(x_1)}_{x_2} &= \frac {1}{p(x_1)}\, u^{(0)}_{x_1} \, {t^{(0)}_{x_1}}^*\,{t^{(x_1)}_{x_2}}^*t^{(x_1)}_{x_2}\,t^{(0)}_{x_1}\,{u^{(0)}_{x_1}}^* \\ & = \frac {1}{p(x_1)}\, u^{(0)}_{x_1} \sum_{x_3,\ldots,x_n} {t^{(0)}_{x_1}}^*\,{t^{(x_1)}_{x_2}}^* \cdots {t^{(x_1,\ldots,x_{n-1})}_{x_n}}^*t^{(x_1,\ldots,x_{n-1})}_{x_n} \cdots t^{(x_1)}_{x_2} \, t^{(0)}_{x_1}\,{u^{(0)}_{x_1}}^* \\ &= \frac {1}{p(x_1)}\, \sum_{x_3,\ldots,x_n} p(x_1,\ldots,x_n)\, \idty = p(x_2|x_1)\, \idty\;, \end{split}$$ where $p(x_2|x_1)$ is the probability of observing $x_2$ at the second step conditioned upon the observation of $x_1$ at the first step. Again we have $t^{(x_1)}_{x_2} = \sqrt{p(x_2|x_1)} \, u^{(x_1)}_{x_2}$ for some unitary $u^{(x_1)}_{x_2}$ and, repeating the same argument, we find for every $k$ $$\label{f2} t^{(x_1,\ldots,x_{k-1})}_{x_k} = \sqrt{p(x_k|x_1,\ldots,x_{k-1})}\, u^{(x_1,\ldots,x_{k-1})}_{x_k}\;, \qquad u^{(x_1,\ldots,x_{k-1})}_{x_k}\in\mathcal{U}(\HH)\;,$$ where $p(x_k|x_1,\ldots,x_{k-1})$ is the probability of observing $x_k$ at the $k^{\mathrm{th}}$ step conditioned upon the observation of $(x_1,\ldots,x_{k-1})$ during the previous steps. Equation gives the polar decomposition of $t^{(x_1,\ldots,x_{k-1})}_{x_k}$, which in this case is unique. \(b) Combining equations and we get $$\nonumber t^{(x_1,\ldots,x_{k-1})}_{x_k} = \sqrt{p(x_k|x_1,\ldots,x_{k-1})}\, {\mathrm{e}}^{{\mathrm{i}}\theta(x_1,\ldots,x_n)} \, {u^{(x_1,\ldots,x_k)}_{x_{k+1}}}^*\cdots {u^{(x_1,\ldots,x_{n-1})}_{x_n}}^*\,{u^{(0)}_{x_1}}^*\cdots {u^{(x_1,\ldots,x_{k-2})}_{x_{k-1}}}^*\;,$$ where, because of the uniqueness of the polar decomposition of $t^{(x_1,\ldots,x_{k-1})}_{x_k}$, $$\nonumber {\mathrm{e}}^{{\mathrm{i}}\theta(x_1,\ldots,x_n)} \, {u^{(x_1,\ldots,x_k)}_{x_{k+1}}}^* \cdots {u^{(x_1,\ldots,x_{n-1})}_{x_n}}^* \, {u^{(0)}_{x_1}}^*\cdots {u^{(x_1,\ldots,x_{k-2})}_{x_{k-1}}}^* = u^{(x_1,\ldots,x_{k-1})}_{x_k} \;,$$ which therefore has to be independent of $x_{k+1},\ldots,x_n$. If moreover $(x_1,\ldots,x_{k-1})$ is such that for all $m>k$ the operators $t^{(x_1,\ldots,x_{m-1})}_{x_m}$ do not depend on $x_k$, then the same is true for the unitary operator ${u^{(x_1,\ldots,x_k)}_{x_{k+1}}}^* \cdots {u^{(x_1,\ldots,x_{n-1})}_{x_n}}^* \, {u^{(0)}_{x_1}}^*\cdots {u^{(x_1,\ldots,x_{k-2})}_{x_{k-1}}}^*$ and so the Kraus operators $t^{(x_1,\ldots,x_{k-1})}_{x_k}$ are all proportional to a same unitary operator $u^{(x_1,\ldots,x_{k-1})}$ and $T^{(x_1,\ldots,x_{k-1})}(\rho) = u^{(x_1,\ldots,x_{k-1})}\,\rho\,{u^{(x_1,\ldots,x_{k-1})}}^*$. *Proof of Proposition \[divPCQI\]. *\ (a) $\Rightarrow$ (b). Let $R^{(x_1,\ldots,x_{k-1})}(\rho) = \sum_{j_k} r^{(x_1,\ldots,x_{k-1})}_{j_k} \,\rho\, {r^{(x_1,\ldots,x_{k-1})}_{j_k}}^*$, $\sum_{j_k}{r^{(x_1,\ldots,x_{k-1})}_{j_k}}^*r^{(x_1,\ldots,x_{k-1})}_{j_k} = \idty$. If $\sum_{\substack{x_1,\ldots,x_n\\j_1,\ldots,j_n}} r^{(x_1,\ldots,x_n)}_{j_n}\,t^{(n)}_{x_n}\cdots r^{(x_1)}_{j_1}\,t^{(1)}_{x_1} \,\rho\, {t^{(1)}_{x_1}}^*\,{r^{(x_1)}_{j_1}}^*\cdots{t^{(n)}_{x_n}}^*\,{r^{(x_1,\ldots,x_n)}_{j_n}}^* = \rho$ for every $\rho$, then this decomposition of the ideal channel is associated to a probability measure $p(x_1,j_1,\ldots,x_n,j_n)$ and Proposition \[divQIrest0\] directly implies $t^{(k)}_{x_k} = \sqrt{p(x_k|x_1,j_1,\ldots,x_{k-1},j_{k-1})}\,u^{(x_1,j_1,\ldots,x_{k-1},j_{k-1})}_{x_k}$, where $u^{(x_1,j_1,\ldots,x_{k-1},j_{k-1})}_{x_k}$ is unitary and where $p(x_k|x_1,j_1,\ldots,x_{k-1},j_{k-1})=p^{(k)}_{x_k}$ and $u^{(x_1,j_1,\ldots,x_{k-1},j_{k-1})}_{x_k}= u^{(k)}_{x_k}$ because $t^{(k)}_{x_k}$ does not depend on $(x_1,j_1,\ldots,x_{k-1},j_{k-1})$. \(b) $\Rightarrow$ (c). If $t^{(k)}_{x_k}= \sqrt{p^{(k)}_{x_k}}\,u^{(k)}_{x_k}$, with $u^{(k)}_{x_k}\in\mathcal{U}(\HH)$, $p^{(k)}_{x_k}\geq0$, $\sum_{x_k} p^{(k)}_{x_k}=1$, then for every $\rho$ $$\nonumber {\mathbb{P}}(X_1=x_1,\ldots,X_n=x_n) = {\operatorname{tr}}t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^* = \prod_{k=1}^n p^{(k)}_{x_k}\;.$$ \(c) $\Rightarrow$ (a). ${\operatorname{tr}}{t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^*t^{(n)}_{x_n} \cdots t^{(1)}_{x_1} \,\rho = {\operatorname{tr}}t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^* = {\mathbb{P}}(X_1=x_1,\ldots,X_n=x_n)$ which does not depend on $\rho$ and so ${t^{(1)}_{x_1}}^* \cdots {t^{(n)}_{x_n}}^*t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}= {\mathbb{P}}(X_1=x_1,\ldots,X_n=x_n)\,\idty$. Then, with the same argument used in the proof of property (a) of Proposition \[divQIrest0\], $$\nonumber t^{(k)}_{x_k} = \sqrt{{\mathbb{P}}(X_k=x_k|X_1=x_1,\ldots,X_{k-1}=x_{k-1})} \, u^{(x_1,\ldots,x_{k-1})}_{x_k}\;,$$ where $u^{(x_1,\ldots,x_{k-1})}_{x_k}$ is unitary, and where ${\mathbb{P}}(X_k=x_k|X_1=x_1,\ldots,X_{k-1}=x_{k-1})={\mathbb{P}}(X_k=x_k)$ and $u^{(x_1,\ldots,x_{k-1})}_{x_k}=u^{(k)}_{x_k}$ because $t^{(k)}_{x_k}$ is independ of $(x_1,\ldots,x_{k-1})$. Then the channels $R^{(x_1,\ldots,x_k)}(\rho)={u^{(k)}_{x_k}}^*\,\rho\,u^{(k)}_{x_k}$ perfectly restore quantum information. When (a), (b) and (c) hold, all the restoring channels $R$ in (a) satisfy by Proposition \[divQIrest0\], and of course for every choice of $v^{(x_1, \ldots, x_k)}$, $k=1,\ldots,k-1$, the channel $R^{(x_1,\ldots,x_n)}$ defined by can perfectly restore quantum information. Finally equation directly follows from condition (b) and equation . $\square$ Optimal recovery of Quantum Information {#OptRec} ======================================= When perfect correction of quantum information is not possible, we would like the restoring channels $R$ which bring the corrected channel $\Tcorr$ as close as possible to $\id$, in some sense. We look for channels $R$ which maximize the *channel fidelity* of the corrected channel, ${\mathcal{F}}(\Tcorr)$. For a channel $T:\BB(\HH)\to\BB(\HH)$, $T(\rho)=\sum_x t_x\,\rho\,t_x^*$, $\dim\HH=N$, denoted by $\Omega$ a maximally entangled unit vector in $\HH\otimes\HH$, the channel fidelity $$\nonumber {\mathcal{F}}(T) = \langle\Omega,T\otimes\id(\ketbra{\Omega}{\Omega})\,\Omega\rangle = \frac{1}{N^2} \sum_x |{\operatorname{tr}}t_x|^2$$ measures how well $T$ preserves quantum information, reaching 1 if and only if $T=\id$. Single-step channel ------------------- Let us recall what was found in [@GW] for the single-step case. Given $T$, chosen a Kraus decomposition and the restoring channels $R^{(x)}$, $$\nonumber R^{(x)}(\rho)=\sum_j r^{(x)}_j \,\rho\, {r^{(x)}_j}^*\;, \qquad \sum_j {r^{(x)}_j}^*r^{(x)}_j=\idty\;,$$ we are interested in $$\label{FidVal} {\mathcal{F}}(\Tcorr) = \frac{1}{N^2} \sum_{x,j} |{\operatorname{tr}}r^{(x)}_j\,t_x|^2\;,$$ and we want to maximize it with respect to all possible families $\{R^{(x)}\}$. \[ORQI\] Let $T:\BB(\HH)\to\BB(\HH)$ be a channel over a finite dimensional Hilbert space $\HH$. Fixed a Kraus decomposition $T(\rho) = \sum_x t_x\,\rho\,t_x^*$, for every family of channels $R^{(x)}:\BB(\HH)\to\BB(\HH)$ let $\Tcorr$ be the corresponding overall corrected channel . Then $$\label{FidMaj} {\mathcal{F}}(\Tcorr) \leq \frac{1}{N^2} \sum_x \Big({\operatorname{tr}}|t_x| \Big)^2\;.$$ Moreover equality holds if and only if $$\label{OTcorr} \Tcorr(\rho) = \sum_x |t_x| \,\rho\, |t_x| \;,$$ and it can always be attained choosing $$\label{OF} R^{(x)}(\rho) = u_x^*\,\rho\,u_x\;,$$ where $u_x$ is a unitary operator in the polar decomposition $t_x=u_x\,|t_x|$. Inequality follows from because the Cauchy-Schwarz inequality for the Hilbert-Schmidt inner product gives $$\label{CS} \sum_j |{\operatorname{tr}}r^{(x)}_j\,t_x|^2 = \sum_j\Big|{\operatorname{tr}}\big(r^{(x)}_j\,u_x|t_x|^{1/2} |t_x|^{1/2}\big) \Big|^2 \leq \sum_j {\operatorname{tr}}\big(|t_x|^{1/2}u_x^*{r^{(x)}_j}^*r^{(x)}_j u_x|t_x|^{1/2}\big) \, {\operatorname{tr}}|t_x| = \Big( {\operatorname{tr}}|t_x|\Big)^2 \;.$$ Equality holds in if and only if equality holds in for every $x$, which occurs if and only if $r^{(x)}_j\,u_x|t_x|^{1/2} = \lambda_{xj}|t_x|^{1/2}$, $\lambda_{xj}\in\mathbb{C}$, for every $x$ and $j$. Then $r^{(x)}_j\,t_x = \lambda_{xj}|t_x|$ and holds. The opposite implication is obvious. Finally, if $R^{(x)}$ are chosen according to , then $\Tcorr$ is given by and equality holds in . The structure of the optimal restoring channels obtained by maximizing ${\mathcal{F}}(\Tcorr)$, is just what one could expect. When a measurement on the environment has given the result $x$, we deal with a subensemble of systems undergone the state transformation $\rho\mapsto t_x\,\rho\,t_x^* = u_x |t_x|\,\rho\,|t_x|u_x^*$, which can be seen as a composition of $\rho\mapsto |t_x|\,\rho\,|t_x|$ followed by $\rho\mapsto u_x\,\rho\,u_x^*$. Unless we are in the trivial case $|t_x|\propto\idty$, only the second transformation is physically reversible, and this is just what the channels do. Therefore ${\mathcal{F}}(\Tcorr)$ can be maximized considering only unitary feedback, even if sometimes non unitary feedback could work as well (for example, every time the polar decomposition of $t_x$ is not unique). Multi-step channel ------------------ Given a multi-step channel $T$, decomposed according to , we allow corrections also before the evolution stops and so we want to maximize the channel fidelity of $\Tcorr$ over all possible families $R^{(x_1)}$, $R^{(x_1,x_2)}$, …, $R^{(x_1,\ldots,x_n)}$ in . We are interested in the optimal feedback, of course, and we want to compare the maximum attainable fidelity with: - the maximum fidelity obtainable only with a unique correction at the end, which is, by Proposition \[ORQI\], $$\label{fid'} {\mathcal{F}}(\Tcorr') = \frac{1}{N^2} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big|\Big)^2\;,$$ where $\Tcorr'$ denotes the corrected channel $$\nonumber \Tcorr'(\rho) = \big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big| \,\rho\, \big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big|\;;$$ - the fidelity obtainable applying step by step the feedback suggested by Proposition \[ORQI\], chosen on the basis of the whole avaible information; the corrected evolution associated to the observation of $(x_1,\ldots,x_n)$ would be $$\label{sbsF}\begin{split} \rho &\mapsto t^{(1)}_{x_1}\, \rho \, {t^{(1)}_{x_1}}^* \mapsto |t^{(1)}_{x_1}|\, \rho \, |{t^{(1)}_{x_1}}| \mapsto t^{(2)}_{x_2}\,|t^{(1)}_{x_1}|\, \rho \, |{t^{(1)}_{x_1}}|\,{t^{(2)}_{x_2}}^* \mapsto \big|t^{(2)}_{x_2}\,|t^{(1)}_{x_1}|\big| \,\rho\, \big|t^{(2)}_{x_2}\,|t^{(1)}_{x_1}|\big| \mapsto \cdots \\ &\mapsto \Big|t^{(n)}_{x_n}\cdots\big|t^{(2)}_{x_2}|t^{(1)}_{x_1}|\big|\Big| \,\rho\, \Big|t^{(n)}_{x_n}\cdots\big|t^{(2)}_{x_2}|t^{(1)}_{x_1}|\big|\Big| := \Tcorr''(\rho)\;, \end{split}$$ with fidelity $$\label{fid''} {\mathcal{F}}(\Tcorr'') = \frac{1}{N^2} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\Big|t^{(n)}_{x_n} \cdots \big|t^{(2)}_{x_2} |t^{(1)}_{x_1}| \big| \Big| \Big)^2\;.$$ Quite surprisingly, these two strategies are equivalent, and optimal, in the qubit case ($\dim\HH=2$), but in the general case ($\dim\HH\geq3$) there is no relationship between them, and neither of them gives the optimal correction. ### Qubit multi-step channels \[divQbORQI\] Let $\dim\HH=2$ and $T=T^{(n)}\circ\cdots\circ T^{(1)}$ be a multi-step channel, with every $T^{(k)}:\BB(\HH)\to\BB(\HH)$. Fixed the Kraus decompositions $T^{(k)}(\rho)=\sum_{x_k} t^{(k)}_{x_k}\,\rho\,{t^{(k)}_{x_k}}^*$, for every family of channels $R^{(x_1)}$, $R^{(x_1,x_2)}$, …, $R^{(x_1,\ldots,x_n)}$, let $\Tcorr$ be the corresponding overall corrected channel . Then $$\label{divQbFidMaj} {\mathcal{F}}(\Tcorr) \leq \frac{1}{4} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big| \Big)^2\;.$$ Moreover equality can always be attained choosing $$\label{divQbOF} R^{(x_1, \ldots, x_k)}(\rho)=v^{(x_1, \ldots, x_k)}\,\rho\,{v^{(x_1, \ldots, x_k)}}^*\;, \qquad v^{(x_1, \ldots, x_k)} \in \mathcal{U}(\HH)\;, \qquad \forall\;k=1,\ldots,n\;,$$ with $v^{(x_1, \ldots, x_n)}={w^{(x_1, \ldots, x_n)}}^*$, where $w^{(x_1, \ldots, x_n)}$ is a unitary operator in the polar decomposition $t^{(n)}_{x_n}\,v^{(x_1, \ldots, x_{n-1})} \cdots v^{(x_1)}\,t^{(1)}_{x_1} = w^{(x_1, \ldots, x_n)}\,\big|t^{(n)}_{x_n}\,v^{(x_1, \ldots, x_{n-1})} \cdots v^{(x_1)}\,t^{(1)}_{x_1}\big|$. The key property in a 2-dimensional Hilbert space is that for every operator $A$ $$\nonumber ({\operatorname{tr}}|A|)^2 = {\operatorname{tr}}(|A|^2) + 2\det|A|\;, \qquad \mbox{ where } \det|A|=|\det A| \leq \frac{1}{2} {\operatorname{tr}}(|A|^2)\;.$$ Then, for every family of channels $R^{(x_1,\ldots,x_{k-1})}(\rho) = \sum_{j_k} r^{(x_1,\ldots,x_{k-1})}_{j_k} \,\rho\, {r^{(x_1,\ldots,x_{k-1})}_{j_k}}^*$, $\sum_{j_k}{r^{(x_1,\ldots,x_{k-1})}_{j_k}}^*r^{(x_1,\ldots,x_{k-1})}_{j_k} = \idty$, $$\nonumber\begin{split} {\mathcal{F}}(\Tcorr) &= \frac{1}{4} \sum_{\substack{x_1,\ldots,x_n\\j_1,\ldots,j_n}} \big|{\operatorname{tr}}r^{(x_1,\ldots,x_n)}_{j_n}\,t^{(n)}_{x_n}\cdots r^{(x_1)}_{j_1}\,t^{(1)}_{x_1}\big|^2 \leq \frac{1}{4} \sum_{\substack{x_1,\ldots,x_n\\j_1,\ldots,j_{n-1}}} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n}\,r^{(x_1,\ldots,x_{n-1})}_{j_{n-1}}\cdots r^{(x_1)}_{j_1}\,t^{(1)}_{x_1}\big| \Big)^2 \\ &=\frac{1}{2} + \frac{1}{2} \sum_{\substack{x_1,\ldots,x_n\\j_1,\ldots,j_{n-1}}} \big|\det t^{(n)}_{x_n}\cdots t^{(1)}_{x_1} \big| \cdot \big|\det r^{(x_1,\ldots,x_{n-1})}_{j_{n-1}} \big| \cdot \big|\det r^{(x_1)}_{j_1}\big| \leq \frac{1}{2} + \frac{1}{2} \sum_{x_1,\ldots,x_n} \big|\det t^{(n)}_{x_n}\cdots t^{(1)}_{x_1} \big| \\ &= \frac{1}{4} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big| \Big)^2\;. \end{split}$$ Analogously for every family of unitary channels $R^{(x_1,\ldots,x_k)}(\rho) = v^{(x_1, \ldots, x_k)}\,\rho\,{v^{(x_1, \ldots, x_k)}}^*$, with $v^{(x_1, \ldots, x_n)} = {w^{(x_1,\ldots,x_n)}}^*$, $$\nonumber\begin{split} {\mathcal{F}}(\Tcorr) &= \frac{1}{4} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n}\,v^{(x_1,\ldots,x_{n-1})}\cdots v^{(x_1)}\,t^{(1)}_{x_1}\big|\Big)^2 =\frac{1}{2} + \frac{1}{2} \sum_{x_1,\ldots,x_n} \big|\det t^{(n)}_{x_n}\,v^{(x_1,\ldots,x_{n-1})}\cdots v^{(x_1)}\, t^{(1)}_{x_1} \big| \\ &=\frac{1}{2} + \frac{1}{2} \sum_{x_1,\ldots,x_n} \big|\det t^{(n)}_{x_n}\cdots t^{(1)}_{x_1} \big| = \frac{1}{4} \sum_{x_1,\ldots,x_n} \Big({\operatorname{tr}}\big|t^{(n)}_{x_n} \cdots t^{(1)}_{x_1}\big|\Big)^2 \;. \end{split}$$ ### Higher dimensional two-step channels {#optimult} When $\dim\HH\geq3$ there is no general relationship between the two fidelities and . Consider on $\mathbb{C}^3$ the multi-step channels $T=T^{(2)}\circ T^{(1)}$ and $S=S^{(2)}\circ S^{(1)}$, where $$\begin{gathered} T^{(1)}(\rho) = T^{(2)}(\rho) = \sum_{x=1}^2 t_x \, \rho \, t_x^*\;, \qquad \qquad S^{(1)}(\rho) = S^{(2)}(\rho) = \sum_{x=1}^2 s_x \, \rho \, s_x^*\;, \nonumber\\ |t_1|^2=|s_1|^2=\begin{pmatrix}1/6&0&0\\0&1/3&0\\0&0&1/2\end{pmatrix}\;, \qquad \qquad |t_2|^2=|s_2|^2=\begin{pmatrix}5/6&0&0\\0&2/3&0\\0&0&1/2\end{pmatrix}\;,\nonumber\\ t_1= u\,|t_1|\;, \qquad t_2= |t_2|\;, \qquad\qquad s_1= v\,|s_1|\;, \qquad s_1= u\,|s_1|\;,\nonumber\end{gathered}$$ where $$u=\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix} \;, \qquad\mbox{and} \qquad v= \frac{1}{\sqrt{3}} \begin{pmatrix}{\mathrm{e}}^{2{\mathrm{i}}\pi/3}&{\mathrm{e}}^{-2{\mathrm{i}}\pi/3}&1\\ {\mathrm{e}}^{-2{\mathrm{i}}\pi/3}&{\mathrm{e}}^{2{\mathrm{i}}\pi/3}&1\\1&1&1\end{pmatrix} \;. \nonumber$$ Then since the absolute values of the Kraus operators coincide, the single-step corrected versions of $T$ and $S$ are the same. Moreover, since these absolute values commute, the iterated absolute values factorize, i.e., $$\Big|t^{(n)}_{x_n}\cdots\big|t^{(2)}_{x_2}\,|t^{(1)}_{x_1}|\big|\big| =|t^{(n)}_{x_n}|\cdots|t^{(2)}_{x_2}|\cdot|t^{(1)}_{x_1}|$$ Hence the greedy strategy of correcting for maximal fidelity at every step, using all available information, gives the same result for these channels as the product of the single-step corrected channels. In particular, $S$ and $T$ are equivalent in this respect, and ${\mathcal{F}}(\Tcorr'')={\mathcal{F}}(\Scorr'')$ (see last line in the table below). On the other hand, we can leave the system undisturbed, and only make an optimal correction at the end, again using all available information. Finally, we may optimize fidelity over all correction schemes. Numerically this achieved by an iteration, developed in [@RW], which efficiently determines the maximum of a general positive linear functional on channels. In the case at hand this is the overall channel fidelity as a function of the correcting channels $R$. Since we can correct the unitaries $u$ and $v$ immediately in every step, it is clear that the channels $S$ and $T$ once again give the same corrected fidelities. The results are summarized in the following table: Channel fidelities $T$ $S$ --------------------------------- -------- -------- optimal correction 0.9584 0.9584 final correction only 0.9556 0.9576 $<$ $>$ optimal correction at each step 0.9570 0.9570 Hence, in contrast to the case of perfect corrigibility, it may help to perform corrections on the way, rather than a single correction at the end (column $T$). Perhaps surprisingly, the optimal correction at the intermediate steps requires some foresight, and depends on what is to follow: otherwise the strategy of correcting for highest fidelity using all available information, which is clearly optimal in the last step, would also have to be optimal in the intermediate steps (compare first and last line). Even leaving out the intermediate correction altogether may be better (column $S$). ### Higher dimensional many-step channels We noted that in the previous example the correction scheme for $\Tcorr''$ is equivalent to multiplying the one-step corrected channels. That is, whether this correction step may use all available information or just the information from the last step, leads to the same corrected channel. We showed this by noting that the absolute values of all Kraus operators involved commute. This will be true in any dimension if there are only two Kraus operators, as in the example. Here we just note that this equality is not a general fact. In fact using random channels one quickly finds examples (3 Hilbert space dimensions, 3 Kraus operators, and 4 time steps), in which using all information or just information from the last step give different results. Surprisingly, once again the comparison may be either way: sometimes using only one-step information is [*better*]{} than using all information. Of course, this is only possible because the greedy strategy is not optimal in the first place. Outlook ======= We would like to spend some words about the quantum error-correction in *continuous* time. Our aim would be to generalize this correction scheme for quantum information corrupted by a noisy Markovian evolution assigned by a Quantum Dynamical Semigroup. The present analysis for a multi-step channel $T= \big(T^{(1)}\big)^n$ would provide the intervalwise treatment, but it is not easy to take the limit to continuous time. The well established theory of continuous measurements gives the possible decompositions (unravelings) of the channel according to the observed trajectoies, but a continuous time formulation of general non-Hamiltonian feedback is missing. There is no general description of the evolution conditioned on the observed process if a feedback depending on the whole measurement record is added. And also if the feedback is supposed to depend only on the present observation, the measured current, still there is no general description for non-Hamiltonian feedback. A theory exists only for Hamiltonian feedback simply proportional to the measured currents [@Wiseman]. Anyway a preliminary study of perfect correction in this framework leads to the same conclusions of Proposition \[divPCQI\]: perfect correction is possible if and only if the measured process is uninformative and in this case a unique correction at the end suffices. 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--- author: - | Xiang Zhang\ Harvard University\ `[email protected]`\ Marinka Zitnik\ Harvard University\ `[email protected]`\ title: '[<span style="font-variant:small-caps;">GNNGuard</span>]{}: Defending Graph Neural Networks against Adversarial Attacks' --- Introduction {#sec:intro} ============ Related Work {#sec:related} ============ Background and Problem Formulation {#sec:background} ================================== [<span style="font-variant:small-caps;">GNNGuard</span>]{} {#sec:method} =========================================================== Experiments {#sec:experiments} =========== Conclusion {#sec:conclusion} ========== [10]{} Xiang Yue, Zhen Wang, Jingong Huang, Srinivasan Parthasarathy, Soheil Moosavinasab, Yungui Huang, Simon M Lin, Wen Zhang, Ping Zhang, and Huan Sun. Graph embedding on biomedical networks: methods, applications and evaluations. , 36(4):1241–1251, 2020. 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--- abstract: 'Using the redMaPPer cluster catalogue based on the Sloan Digital Sky Survey (SDSS) photometry, we investigate the importance of major mergers in the stellar mass build-up of brightest cluster galaxies (BCGs) between $0.08 \leq z \leq 0.50$. We use the SDSS spectroscopy, supplemented with spectroscopic observations from the Southern African Large Telescope at higher redshifts, to identify which BCGs and nearby companions are potential major merger candidates. We use the pair fraction as a proxy for the merger fraction in order to determine how much stellar mass growth the BCGs have experienced due to major mergers. We observe a weak trend of the BCG pair fraction increasing with decreasing redshift, suggesting that major mergers may become more important towards the present day. Major mergers are found to contribute, on average, $24 \pm 14 $ $(29 \pm 17)$ per cent towards the stellar mass of a present day BCG since $z=0.32$ (0.45), assuming that half of the companion’s stellar mass is accreted onto the BCG. Furthermore, using our merger results in conjunction with predictions from two recent semi-analytical models along with observational measurements from the literature, we find that major mergers have sufficient stellar material to account for the stellar mass growth of the intracluster light between $z=0.3$ and $z=0$.' author: - | Danièl N. Groenewald$^{1,2}$[^1], Rosalind E. Skelton$^{1}$, David G. Gilbank$^1$ and S. Ilani Loubser$^2$\ $^{1}$South African Astronomical Observatory, Observatory Road, 7925, Cape Town, South Africa\ $^{2}$North-West University, Potchefstroom, 2520, South Africa bibliography: - 'ms.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: 'The close pair fraction of BCGs since $z=0.5$: major mergers dominate recent BCG stellar mass growth' --- \[firstpage\] galaxies: clusters: general – galaxies: clusters: intracluster medium – galaxies: elliptical and lenticular, cD – galaxies: evolution – galaxies: interactions. Introduction {#sec:intro} ============ Brightest cluster galaxies (BCGs) are the most massive and luminous galaxies in the Universe. These galaxies are typically found close to, or at, the centres of their host clusters [@Jones1984; @Jones1999; @Beers1986; @Rhee1991]. In the hierarchical formation scenario BCGs are thought to form through the accretion of smaller galaxies [e.g. @Ostriker1977; @Richstone1983]. These massive galaxies are also known to have properties that are very different from that of other early-type galaxies (ETGs). In comparison to other ETGs of the same mass, BCGs are found to be larger [@Bernardi2007 and references therein] and have extended light profiles [@Matthews1964; @Tonry1987; @Schombert1988; @Gonzalez2000; @Gonzalez2003]. These unique properties have been attributed to their special location at the cluster centres [@Hausman1978]. These properties may also suggest that the formation of BCGs are different from that of other ETGs [e.g. @Burke2000; @Stott2008; @Stott2010]. Due to the privileged positions of BCGs at the centres of their host clusters, these massive galaxies are expected to experience multiple mergers during their lifetime, making them ideal probes with which to study galaxy formation. Observational evidence, for example tidal tails and distorted isophotes, indicates that some BCGs are experiencing mergers [e.g. @Bernardi2007; @Lauer2007; @2007; @McIntosh2008; @Liu2009; @Liu2015; @Rasmussen2010; @Brough2011], but it is not clear to what extent their stellar mass is being built up by mergers and at what rate. There is currently no consensus in the literature regarding the stellar mass evolution of BCGs between $z=1$ and the present day. Observational studies have found that these massive galaxies changed their mass by factors that range between one [equivalent to no growth; e.g. @Aragon-Salamanca1998; @Whiley2008; @Collins2009; @Stott2010; @Oliva-Altamirano2014] and $1.4 \pm 0.2$ [@Lin2013] or $1.8 \pm 0.3$ [@Lidman2012] since $z \lesssim 1$. Predictions from numerical simulations and semi-analytical models (SAMs) suggest that BCGs form in two ‘phases’. They suggest that star formation dominates the BCG’s mass growth at $z\geq2$, while multiple dry mergers of smaller galaxies dominate the mass assembly at $z\leq1$ [e.g. @DeLucia2007; @Naab2009; @Laporte2013]. The simulations, however, differ in the stellar mass growth they predict for BCGs. For example, [@DeLucia2007] find that BCGs have changed their mass by a factor of 4 from $z = 1 - 0$, mainly through minor[^2] mergers. [@Laporte2013] on the other hand predict a BCG mass growth factor closer to 2 over the same redshift range, however this model finds that both major[^3] and minor mergers contribute significantly towards the mass growth of these massive galaxies. The discrepancies between the BCGs’ observed stellar mass growth and that predicted by simulations are quite apparent. This may be because mergers do not contribute significantly to the stellar mass growth of the BCGs if a significant fraction of the merging mass ends up in the cluster’s intracluster light (ICL). The ICL can be described as stars that are not gravitationally bound to any single galaxy, but rather to the cluster potential. Although the origin of the ICL is not known, it is thought that tidal stripping of satellite galaxies and merger events (involving BCGs) contribute towards the stellar mass growth of the ICL. The ICL is often found to be more concentrated around the central galaxy in a cluster [@Mihos2005; @Rudick2011]. This in turn implies that the formation and evolution of BCGs and the ICL are connected to one another. The recent simulation of [@Laporte2013] predicts that 30 per cent of a companion galaxy’s mass will be distributed into the ICL during a merger with the BCG [see also @Conroy2007; @Puchwein2010]. This brings the model’s predicted BCG mass growth over $0 < z < 1$ into better agreement with the observed stellar mass growth estimates of @Lidman2012 [@Lin2013; @Burke2013] in the same redshift range. If we are to believe that some fraction of the merging mass ends up in the ICL, then this presents us with a scenario that the ICL is being built-up through galaxies that are interacting with the BCGs. [@DeMaio2015] however argues against this idea, stating that the ICL is being built-up by the stripping of satellite galaxies with luminosities $>0.2 \, L^{\star}$ [also see @Contini2014]. Consequently, it is clear that the growth and stellar mass build-up of BCGs and the ICL are linked, however it is not clear to what extent merger events contribute towards this. In this paper we investigate the importance of major mergers in the stellar mass build-up of BCGs between $0.08 \leq z \leq 0.50$. We select BCGs from a photometric cluster catalogue, which has been constructed from the Sloan Digital Sky Survey [SDSS; @York2000]. We use spectroscopic information from the SDSS to identify which BCGs and nearby companions are potential major merger candidates. The close pair fraction is used as a proxy for the merger fraction to determine how much stellar mass growth the BCGs will experience due to mergers. This paper is structured as follows. Section \[sec:sample\_descr\] contains details pertaining to the redMaPPer catalogue; in Section \[sec:method\] the methods are described. The results on the pair fraction and merger-inferred stellar mass growth of the BCGs are shown and discussed in Section \[sec:results\_disc\]. Conclusions are drawn in Section \[sec:concl\]. Throughout the paper we assume a flat $\Lambda$CDM cosmology with $\Omega_{M} = 1 - \Omega_{\Lambda} = 0.3$ and $H_{0} = 70$ [kms$^{-1}$Mpc$^{-1}$]{}. Magnitudes are given in the AB system and the [@Chabrier2003] initial mass function is used. Data {#sec:sample_descr} ==== Overview of the redMaPPer cluster catalogue {#sec:redM_over} ------------------------------------------- Throughout this paper we use version 5.2 of the red-sequence Matched-filter Probabilistic Percolation cluster catalogue [redMaPPer[^4]; @Rykoff2014]. This catalogue covers roughly 10 000 $\mathrm{deg^{2}}$ of the sky and consists of more than 25000 galaxy clusters which span a redshift range of $0.08 \leq z \leq 0.55$. The clusters in this catalogue have been optically identified using photometry from the Eighth Data Release [DR8; @Aihara2011] of the SDSS. Where possible, we supplement the catalogue with additional spectroscopic redshifts from the SDSS DR12 [@Alam2015]. A detailed description of the redMaPPer cluster catalogue (and construction) can be found in [@Rykoff2014]. Here, we summarize the most important features. Briefly, the redMaPPer cluster detection algorithm photometrically identifies clusters by searching for overdensities of red-sequence galaxies. This relies on a set of galaxies with spectroscopic redshifts that are used to construct a redshift dependent red-sequence model. The spectroscopic redshifts needed for the calibration of the red-sequence model are retrieved from the SDSS Main Galaxy Sample [MGS; @Strauss2002], Luminous Red Galaxy [LRG; @Eisenstein2001], and Baryon Oscillation Spectroscopic Survey [BOSS; @Ahn2012]. This model is then used to photometrically group red galaxies (with luminosities $\geq 0.2 \, L^{\star}$) at similar redshifts into clusters, assuming a radial filter that corresponds to typical cluster sizes. The algorithm iteratively determines a photometric redshift for each cluster based on the calibrated red-sequence model, as measured from the photometrically identified candidate cluster members. ### Probability of being a cluster member {#sec:clus_mem} The candidate cluster members are each assigned to a redMaPPer cluster based on a membership probability ($P_{\rm{MEM}}$). $P_{\rm{MEM}}$ indicates the probability of a galaxy being a red-sequence galaxy that belongs to a specific cluster. This follows directly from the method used by the cluster detection algorithm to identify clusters, i.e. looking for overdensities of red-sequence galaxies, with the luminosity cut and radial filtering mentioned above. redMaPPer provides catalogues of (candidate) cluster members for each cluster, based on the galaxies’ individual properties, including $P_{\rm{MEM}}$, and it is these catalogues which we use throughout our analysis. In Fig. \[fig:PR\_rms\_cuts\_red\_seq\] we show the colour-magnitude diagrams ($g-r$ vs. $m_\mathrm{i}$) of the redMaPPer clusters out to $z=0.35$ (the redshift cut will be explained in the next section). All candidate cluster members and those with $P_{\rm{MEM}} > 0.9$ are shown in grey and green respectively. In each panel it is apparent that members with large $P_{\rm{MEM}}$ values form a tighter sequence, as expected, since these galaxies are more likely to be red-sequence galaxies. We will return to this idea in Section \[sec:fpair\_results\]. For each cluster in the redMaPPer catalogue, the central galaxy identification algorithm of redMaPPer assigns five galaxies probabilities of being the central galaxy (CG) of the cluster ($P_{\rm{CEN}}$). These galaxies are ranked according to probability and are hereafter referred to as the CG candidates. The central probabilities are defined by using a luminosity filter, photometric redshift filter and a local galaxy density filter (discussed below). The product of these three filters produces the overall centering filter that is used to determine the central probabilities [see equation 67 of @Rykoff2014]. The redshift filter that is used in this equation is slightly broader than the cluster red-sequence filter in order to allow galaxies with slight colour offsets from the red-sequence to be considered as CG candidates, as it is possible for the CGs to have experienced residual amounts of star formation and therefore have bluer photometric colours [$\sim2$ per cent of the CGs in redMaPPer are blue; @Rykoff2014]. The galaxy with the highest $P_{\mathrm{CEN}}$ is not necessarily the brightest galaxy. This is because, apart from the luminosity, the local galaxy density around the CG candidate is also considered. Central galaxies are expected to be found in the highest density central regions of clusters. Consequently, the redMaPPer algorithm gives higher preference to galaxies in denser regions than those in less dense regions. A less luminous galaxy located in a denser environment than the most luminous galaxy may therefore have a higher $P_{\mathrm{CEN}}$. The total magnitudes of the galaxies in the redMaPPer catalogue are given by the $i$-band `cModel_Mag` [$m_{\mathrm{i}}$; @Abazajian2004], while the colours of the galaxies are determined using `modelMag` [@Abazajian2004] in the $u, g, r, i$ and $z$-bands. All magnitudes and colours have been corrected for Galactic extinction using the dust maps of [-@Schlegel1998]. {width="15cm" height="13cm"} ### Richness of the redMaPPer clusters {#sec:richn} The catalogue contains clusters with richnesses[^5] of $\lambda \geq 20 \, S(z)$, where $S(z)$[^6] is a correction factor that is used to account for the survey depth of the SDSS. We show the effect of the survey depth on the measured completeness of the redMaPPer clusters as a function of their halo masses $(M_{h})$ in Fig. \[fig:redM\_completeness\_fig\]. The $M_{h}$ have been derived using the halo mass-richness relation of [@Rykoff2012], assuming richness[^7] is a proxy for $M_{h}$. The various lines in Fig. \[fig:redM\_completeness\_fig\] are constructed from the values in figure 22 of [@Rykoff2014]. The redMaPPer algorithm used five redshift bins to determine the completeness of the catalogue as a function of richness. The lines in this figure join the mean redshift value of each bin. At $z\leq0.35$, the galaxy catalogue is volume limited and the survey depth is brighter than the fiducial luminosity cut of $0.2 \, L^{\star}$ (therefore $S(z)=1$). At these redshifts the cluster catalogue is 50 per cent complete down to clusters with $M_{h} \gtrsim 0.2 \times 10^{15} \, \mathrm{M_{\odot}}$. At $z>0.35$, however, the magnitude limit of the survey causes only the most massive clusters to be detected. This causes the richness detection threshold to increase with redshift. In this work, we restrict our analysis to only consider clusters in the volume limited sample at $z\leq0.35$. We further only consider redMaPPer clusters with halo masses above the 50 per cent $M_{h}$ completeness limit. This is done to maximize the number of clusters in the evolutionary sequence (as discussed in Section \[sec:evol\_seq\_clusters\]).  ### BCGs - Identification and stellar masses {#sec:BCG_selection} In our work we select the first CG candidate (the galaxy with the highest $P_{\rm{CEN}}$) in each cluster as the BCG candidate. The redMaPPer catalogue does not supply any stellar mass $(M_{\star})$ estimates for the galaxies. We determine stellar mass estimates using version 4.1 of the [kcorrect]{} code [@Blanton2007] which uses templates based on the [@Bruzual2003] models. Briefly, spectral energy distribution fitting is used to derive the $M_{\star}$ of the galaxies by fitting their observed `modelMag` magnitudes (in the $u, g, r, i $ and $z$-bands) against a range of spectral templates from [@Bruzual2003] assuming a [@Chabrier2003] initial mass function. The best-fitting stellar template is determined through $\chi^{2}$-minimization, whereafter a mass-to-light ratio is determined from this template and used to convert the luminosity of the galaxy to stellar mass. The SALT sample --------------- We extend the BCG pair fraction analysis out to moderately high redshifts ($0.4 \leq z \leq 0.5$) using the Gaussian Mixture Brightest Cluster Galaxy catalogue [GMBCG[^8]; @Hao2010] with follow-up spectroscopy from the Southern African Large Telescope [SALT; @Buckley2006; @ODonoghue2006]. We require a sample with a high spectroscopic completeness in order to identify pairs that are likely to merge. The spectroscopic completeness of the SDSS drops significantly around $z=0.4$, so we have supplemented the SDSS data with additional redshifts from SALT. The GMBCG cluster catalogue is a photometrically-identified cluster sample similar to redMaPPer and was used in the early stages of this work. When redMaPPer became available, we switched to using this due to several advantages it offered, such as explicit BCG probabilities, candidate member galaxy probabilities, etc. However, spectroscopic follow-up was already well underway for the GMBCG sample. In this work, we treat them as two independent cluster samples, in the same way as comparing our redMaPPer work with those of others in the literature, and will show that our results are compatible. We refer the reader to [@Hao2010] for details on the GMBCG catalogue. Briefly, the catalogue spans a redshift range of $0.1 < z < 0.55$ and contains 55 000 clusters, optically identified from the SDSS DR7 [@Abazajian2009] as overdensities of red-sequence galaxies. The positions of the BCGs and the photometric redshifts of their host clusters are provided in the catalogue. Here we are interested in the 4814 BCGs that are found in the redshift range $0.4 \leq z \leq 0.5$. We use the method outlined in Section \[sec:pair\_selection\] to select the close pairs that are included in the SALT sample. Briefly, all companions within a physical separation distance of 50 kpc and 1.5 magnitudes (in the $i$-band) of the BCGs were retrieved from the DR7 database. The spectroscopic redshifts (if available) of these galaxies were retrieved from the SDSS DR12 [@Alam2015]. We further restricted the SALT sample to only include pairs where either the BCG or companion (in each pair) had a SDSS spectroscopic redshift $(z_{\mathrm{spec}})$. SALT observations were used to determine the remaining galaxy’s $z_{\mathrm{spec}}$ to determine whether the galaxies are potential merger candidates or not. Only 16 of the close pairs satisfied these criteria. Twelve pairs were successfully observed over two semesters spanning 2013 November - 2014 April (proposal ID: 2013-2-RSA-008, PI: Groenewald) and 2014 May - October (proposal ID: 2014-1-RSA\_OTH-009, PI: Groenewald). ### Observations and reductions Using the Robert Stobie Spectrograph [RSS; @Burgh2003; @Kobulnicky2003] on SALT we obtained longslit spectroscopy for the close pairs. A slit with a width of $2\arcsec$ was centred on the BCG in each pair and aligned in such a way that both the BCG and companion were observed in a single observation. This allowed us to determine the relative velocities between these two galaxies using the same wavelength calibration. The RSS observations used the PG900 grating which covers the main optical features that we are interested in over $4500 - 7500$ Å. Each close pair in the SALT sample was observed for a total of 106 minutes, split over two observation blocks (each observation block consisted of $2 \times 20$ minute exposures). Basic data reductions, i.e. gain and cross-talk correction as well as bias subtraction were performed as part of the automated reduction pipeline of SALT [@Crawford2010]. We performed cosmic ray rejection on the science images by using the `LACosmic` package [@2001]. Wavelength calibrations were then performed with standard `IRAF` [@Tody1986; @Tody1993] tasks. We determined the spectroscopic redshift of each galaxy in the SALT sample by fitting the observed SALT spectrum against the ‘Early-type’ SDSS galaxy spectral template (hereafter reference spectrum). The rest wavelengths of the Calcium II $H$ and $K$ absorption lines in the reference spectrum were shifted to match those in the galaxy’s observed SALT spectrum. In Table \[table:SALT\_sample\] we present a summary of the galaxies in the SALT sample along with their SDSS spectroscopic redshifts (if available). For each close pair we indicate the velocity difference as derived from the SALT spectroscopy. The BCG and companion in each pair were observed in the same slit and share the same wavelength calibration. Thus the velocity differences, which are our ultimate goal, are more robust than if we had used a combination of SALT and SDSS redshifts for each pair. -- ----------- ------------ ------------- --------------------- -- BCG 190.404850 $-$0.668280 $0.4587 \pm 0.0001$ Companion 190.406416 $-$0.667039 — BCG 128.317079 0.108010 — Companion 128.316370 0.108480 $0.4748 \pm 0.0001$ BCG 153.259460 0.812003 $0.4063 \pm 0.0001$ Companion 153.259030 0.809660 — BCG 27.261019 $-$0.639814 $0.3584 \pm 0.0001$ Companion 27.261605 $-$0.639844 — BCG 346.949880 0.948270 $0.3686 \pm 0.0001$ Companion 346.949133 0.947189 — BCG 186.754036 0.765942 — Companion 186.753290 0.767780 $0.5153 \pm 0.0002$ BCG 148.316850 1.272180 $0.3648 \pm 0.0001$ Companion 148.316048 1.272079 — BCG 206.032182 1.948250 — Companion 206.031830 1.948720 $0.5432 \pm 0.0001$ BCG 223.098640 0.949810 $0.4830 \pm 0.0001$ Companion 223.099826 0.950909 — BCG 210.260650 0.275680 — Companion 210.258767 0.275656 $0.4757 \pm 0.0001$ BCG 218.763280 3.109750 $0.3829 \pm 0.0001$ Companion 218.763793 3.111125 — BCG 357.091165 0.741055 — Companion 357.088710 0.741300 $0.4113 \pm 0.0001$ -- ----------- ------------ ------------- --------------------- -- Method - The pair fraction and merger-inferred stellar mass growth of the BCGs {#sec:method} ============================================================================== Constructing an evolutionary cluster sequence {#sec:evol_seq_clusters} --------------------------------------------- The stellar masses of BCGs are known to correlate with the halo masses of their host clusters, with more massive halos hosting more massive BCGs [e.g. @Edge1991; @Burke2000; @Brough2008; @Stott2008; @Stott2010; @Stott2012; @Whiley2008; @Collins2009; @Hansen2009; @Lidman2012]. Consequently, it is important to take the halo mass growth of the clusters into account when the stellar mass growth of BCGs is investigated, to ensure that that BCGs at high redshifts are compared to their likely descendants at lower redshifts. This idea has already been explored by other works in the literature [@Lidman2012; @Lin2013; @Oliva-Altamirano2014; @Zhang2016]. In our work we use an approach similar to that implemented by [@Lidman2012] to construct an evolutionary cluster sequence using evolving $M_{h}$ limits. We assume that the BCGs in this evolutionary sequence are progenitors/descendants of one another to derive the merger-inferred mass growth of the BCGs. Briefly, we construct an evolutionary cluster sequence by identifying the low redshift descendants of the redMaPPer clusters in our highest redshift bin $(z=0.35)$. This is done by evolving the $M_{h}$ of these clusters forward in time using the mean mass accretion rates (MMAR) from the [-@Fakhouri2010] model (see their equation 2). Through these MMARs we are able to determine what the corresponding halo masses of these high redshift clusters will be at later times, allowing us to construct an evolving $M_{h}$ limit as a function of redshift. We divide the clusters into four equal sized redshift bins with a width of 0.067. The bin width is chosen to be larger than the typical uncertainty on the photometric redshift of the clusters ($\sim0.02$), which reduces the chances that clusters will be scattered in and out of adjacent redshift bins. Secondly, it is small enough to ensure we have multiple bins, each with a robust number of clusters, with which to study the BCGs’ merger-inferred stellar mass growth. We find a total of 5432 clusters that form part of the evolutionary sequence. These clusters (along with the evolving $M_{h}$ limit) are shown in Fig. \[fig:Mhalo\_match\_no\_upper\_50\]. In Table \[table:f\_pair\] we present a summary of the number of clusters in each redshift bin along with their $M_{h}$ ranges. In Section \[sec:fpair\_results\] we test how the pair fraction of the BCGs is influenced when the halo mass growth of the clusters is not taken into account.  The close pair selection {#sec:pair_selection} ------------------------ We begin by only considering galaxies that are brighter than 21.5 magnitudes in the $i$-band. This is the magnitude where the SDSS is 95 per cent complete for galaxies. This limit was determined by comparing the galaxy number counts in a region on the celestial equator from the SDSS to the Stripe 82 survey [@Adelman-McCarthy2007], which is $\sim2$ mag deeper. From the *redMaPPer catalogue* we construct a close galaxy pair sample of 1336 pairs by searching for all photometric galaxies within a physical separation distance $(r_{\mathrm{sep}})$ of $7 \leq r_{\mathrm{sep}} \leq 50$ kpc from the BCGs. Galaxies within this search radius are referred to as companions. The lower limit is imposed since this is the minimum separation distance down to which the SDSS can resolve individual galaxies over our redshift range. A close pair sample that has only been constructed using photometric information will inevitably suffer from contamination due to line-of-sight projections. Although the contamination in clusters is higher than in the field (including contamination from cluster members themselves), the same techniques can be used to obtain the close pair fraction. In this work we use spectroscopy to correct for this contamination by determining whether the photometrically identified companion galaxies are bound to their host BCGs. In order for close pairs to be considered potential merger candidates, we require the galaxies to have a velocity difference of $\Delta v \leq 300$[kms$^{-1}$]{} [see e.g. @Burbidge1975; @Ellison2013; @Kitzbichler2008]. We have chosen this cut, which is somewhat stricter than many other observational works [e.g. @Lin2004a; @Lin2008; @Lopez-Sanjuan2012; @Robotham2014], in order to use the results of @Kitzbichler2008 to estimate merging timescales for each pair. These field merger timescales can be applied to close pairs in clusters [see section 4.2 of @Lidman2013 for motivation]. We use a compilation of the SDSS spectroscopic galaxy samples of which the BOSS survey supplies the deepest spectroscopy, up to a magnitude limit of 19.9 magnitudes in the $i$-band. We therefore further restrict our close pair sample to only include pairs with galaxies brighter than the BOSS magnitude limit. Our final sample consists of 1016 photometric pairs, of which 320 ($\sim31$ per cent) have spectroscopy. We correct for the spectroscopic incompleteness of our sample when the BCG pair fraction is measured (Section \[sec:fpair\]). To ensure that we are relatively complete for companions to the BOSS magnitude limit, we only consider neighbouring galaxies with a stellar mass ratio within 1:4 of the BCG (see panel b of Fig. \[fig:mag\_lim\_21\_5\]). This maximizes the redshift range over which we are complete. We define all potential merger candidates with stellar mass ratios between 1:1$-$1:4 as major mergers. For illustrative purposes, we show the SDSS cutouts of eight major merger candidates from our close pair sample in Fig. \[fig:SDSS\_cutouts\]. {width="15cm" height="12cm"}   \   \   Defining the pair fraction {#sec:fpair} -------------------------- We ultimately want to determine the fraction of BCGs that are in bound pairs, i.e. BCGs that will merge with their companions by $z=0$ (hereafter loosely referred to as ‘bound companions’). This is defined as follows: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,BC}}}{\mathrm{N_{BCGs}}} \label{eq:bound_merg_frac}$$ where $\mathrm{N_{BCGs}}$ is the total number of BCGs that form part of the evolutionary sequence (as derived in Section \[sec:evol\_seq\_clusters\]) and $\mathrm{N_{BCGs,BC}}$ represents the number of BCGs with bound major merger companions. We can expand Eq. \[eq:bound\_merg\_frac\] to the following: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,C}}}{\mathrm{N_{BCGs}}} \times \frac{\mathrm{N_{BCGs,BC}}}{\mathrm{N_{BCGs, C}}} \label{eq:merg_frac}$$ where the first term is the fraction of BCGs with companions and the second term is the fraction of these companions that are bound (contamination correction). $\mathrm{N_{BCGs,C}}$ is the number of BCGs that have one or more companion(s) as described in Section \[sec:pair\_selection\]. In the case where only photometry is available, the second term (contamination correction) in Eq. \[eq:merg\_frac\] may be set to a constant value [e.g. 0.5 in @Edwards2012]. We obtain a measurement for the BCG pair fraction in two cases. We start with the simple case, where only photometry is used. This is used in studies which do not have spectroscopy available [such as @Edwards2012]. Thereafter we consider the case where there is spectroscopic information. Where spectroscopic information is available for some of the sample, we have more information from which to determine whether companions are bound to the BCGs. Rather than assuming a constant value for the contamination correction in Eq. \[eq:merg\_frac\], we derive a more detailed correction by grouping galaxies into bins of colours, magnitude and separation distance. In the $i$-th bin the pair fraction in redshift bin $j$ is $$(f_{\mathrm{pair}})_{j} \, = \, \frac{\sum_{i} \, C_{i}^{-1} \, \mathrm{N_{BCGs,SC}}_{i}}{\sum_{i} \, \mathrm{N_{BCGs}}_{i}} \label{eq:merg_frac_c_corr}$$ where $\mathrm{N_{BCGs,SC}}$ is the number of BCGs with spectroscopic companions and $C$ is the applied correction for spectroscopic incompleteness (derived in Appendix \[ap:corr\_spec\_comp\]). In our high redshift bin $0.4 \leq z \leq 0.5$, due to the low spectroscopic completeness of the SDSS at these redshifts, we used the SALT sample to determine the contamination correction (second term in Eq. \[eq:merg\_frac\]) i.e: $$f_{\mathrm{pair}} \, = \, \frac{\mathrm{N_{BCGs,C}}}{\mathrm{N_{BCGs}}} \times \frac{\mathrm{N_{BCGs,BC(SALT)}}}{\mathrm{N_{BCGs, C(SALT)}}} \label{eq:salt_bcgs_merger_frac}$$ We have assumed here that the fraction of potential merger pairs in the SALT sample is representative of those found in the photometric pair sample (selected from the GMBCG catalogue). The SALT pairs are not different from those in the photometric pair sample since the same $r_{\mathrm{sep}}$ and $\Delta m_{\mathrm{i}}$ criteria have been used to select pairs in both samples. The only difference between these two samples is the redshift requirement used for the SALT pair selection. Since the BCGs are bright, red galaxies, they are very likely to be observed by the SDSS spectroscopic surveys, so we do not expect a bias against spectroscopy for these galaxies. The number of pairs observed with SALT should therefore be representative of the number of pairs in the photometric sample. The uncertainty on $f_{\mathrm{pair}}$ represents the 68.3 per cent $(1\sigma)$ binomial confidence limit and is calculated using the beta confidence interval as described by [@Cameron2011]. The mass growth of the BCGs {#sec:mass_merger_growth} --------------------------- We now continue on to the main aim of the paper, measuring the merger-inferred stellar mass growth of the BCGs as a function of redshift. In order to do this we first calculate average stellar masses of the BCGs, denoted $\langle M_{\star}\rangle$, in each redshift bin. The uncertainties on $\langle M_{\star}\rangle$ are determined using bootstrap resampling with 1000 realizations. We estimate average merger timescale, denoted $\langle t_{\mathrm{merge}} \rangle$, for each close pair using the results of @Kitzbichler2008. Using the Millennium Simulation [@Springel2005], they find that the average $t_{\mathrm{merge}}$ for galaxies with $r_{\mathrm{sep}} \leq 50$ kpc and $\Delta v \leq 300$ [kms$^{-1}$]{} can be given by the following: $$\langle t_{\mathrm{merge}} \rangle = 2.2\,\mathrm{Gyr} \, \frac{r_{\mathrm{sep}}}{50 \, \mathrm{kpc}} \, \Big( \frac{M_{\star,\mathrm{com}}}{5.5 \times 10^{10} \, \mathrm{M_{\odot}}}\Big)^{-0.3} \, \Big(1 + \frac{z}{8}\Big) \label{eq:t_merge}$$ where $M_{\star,\mathrm{com}}$ is the total stellar mass of the companion at its observed redshift (determined using [kcorrect]{}), $r_{\mathrm{sep}}$ is the physical separation distance (in kpc) and $z$ is the photometric redshift of the cluster. The uncertainties on $\langle t_{\mathrm{merge}} \rangle$ are given by $1\sigma$ standard deviation and are propagated through to the final measured fractional mass growths. Various works have commented that the [@Kitzbichler2008] merger timescales are significantly longer than the estimate from dynamical friction or the orbital period [e.g. @Conroy2007a; @Bertone2009; @Conselice2009a; @Kauffmann2010; @Lotz2011]. The uncertainties on $\langle t_{\mathrm{merge}} \rangle$ in all the redshift bins are large enough to encompass differences with other methods (see Table \[table:redM\_mass\_growth\]). The merger rate, denoted $R_{\mathrm{merge}}$, of the sample, i.e. the number of mergers per BCG per Gyr, is defined as follows: $$R_{\mathrm{merge}} = \frac{f_{\mathrm{pair}}}{\langle t_{\mathrm{merge}} \rangle} \label{eq:R_merge}$$ The uncertainty on $R_{\mathrm{merge}}$ is determined using the standard propagation of errors. Semi-analytical models, for example [@Conroy2007; @Puchwein2010; @Laporte2013] have shown that $30-80$ per cent of the companion’s stellar mass ends up in the ICL during a merger with the BCG. During a merger, we assume that 50 per cent of the companion’s stellar mass is transfered to the BCG [$f_{\mathrm{mass}} = 0.5$; also used by @Liu2009; @Liu2015; @Burke2013; @Lidman2013]. To account for the possible range of $f_{\mathrm{mass}}$ values, we assign an uncertainty of 20 per cent to our assumed $f_{\mathrm{mass}}$ value of 50 per cent (i.e. $f_{\mathrm{mass}} = 0.5 \pm 0.2$), which is propagated through to the measured fractional mass growths. In order to calculate how much mass $(\Delta M)$ major mergers add to a BCG from redshift $z_{i}$ down to $z=0$, we use the following: $$\Delta M \, (z = z_{i} - 0) \, = \, R_{\mathrm{merge}} \times T_{\mathrm{LB}} \times \langle M_{\star} \rangle_{\mathrm{com}} \, (z_{i}) \times f_{\mathrm{mass}} \label{eq:delta_M}$$ where $T_{\mathrm{LB}}$ and $\langle M_{\star} \rangle_{\mathrm{com}}(z_{i})$ respectively gives the lookback time and the average stellar mass of the companions at $z_{i}$. The fractional contribution $(F)$ made by major mergers to the stellar mass of a present day BCG, denoted $\langle M_{\star} \rangle_{\mathrm{BCG}} (z=0)$, is defined as follows: $$F = \frac{\Delta M (z = z_{i} - 0)}{\langle M_{\star} \rangle_{\mathrm{BCG}} (z=0)} \label{eq:mass_merge_growth}$$ with $$\langle M_{\star} \rangle_{\mathrm{BCG}} (z=0) = \langle M_{\star} \rangle_{\mathrm{BCG}} (z_{i}) + \Delta M (z = z_{i} - 0) + \Delta M_{\mathrm{other}} \label{eq:mass_bcg_z0}$$ where $\langle M_{\star} \rangle_{\mathrm{BCG}}(z_{i})$ is the average stellar mass of the BCGs at redshift $z_{i}$ and $\Delta M_{\mathrm{other}}$ accounts for other sources of mass accretion, i.e. star formation and minor mergers, from redshift $z_{i}$ down to $z=0$. We do not take this term into consideration because we are unable to estimate the contribution from minor mergers in this work, as we are incomplete for these systems[^9]. Secondly, we expect the contribution from star formation to be negligible for these low redshift BCGs. It is rare to find BCGs in the local Universe that are actively forming stars [less than 1 per cent; e.g. @Liu2012; @Fraser-McKelvie2014]. These starbursts have been found to contribute only $\sim1-3$ per cent [e.g. @Sarazin1983; @Cardiel1995; @Pipino2009; @Liu2012; @Loubser2016] to the stellar mass of the BCGs. The uncertainty on $F$ is determined using the standard propagation of errors. Results and Discussion {#sec:results_disc} ====================== The redshift evolution of the pair fraction {#sec:fpair_results} ------------------------------------------- ### The redMaPPer pair fraction The redshift evolution of the pair fractions from $0.08 \leq z \leq 0.35$ for BCGs selected from the redMaPPer catalogue are shown in Fig. \[fig:fpair\] and summarized in Table \[table:f\_pair\]. The pair fraction measured within 50 kpc using three different methods to correct for projection effects is shown in the top panel of Fig. \[fig:fpair\], while the bottom panel compares the spectroscopically-corrected results measured within 30 and 50 kpc. A close pair sample constructed using only photometry with no background correction will inevitably suffer from projection effects. The resulting pair fraction can therefore be considered as an upper limit (red symbols and line in the top panel). To correct for this contamination, we have applied a 50 per cent contamination correction [as used by @Edwards2012 blue symbols and line]. We compare this pair fraction to that derived using the available spectroscopy, corrected for incompleteness as described above (grey symbols and line). We find that the spectroscopic $C$-correction reduces the photometric pair fraction to $\sim6$ per cent while the simple assumption of 50 per cent contamination reduces the photometric pair fraction to $\sim10$ per cent. Several studies in the literature, e.g. [@LeF`evre2000; @Kartaltepe2007; @2009; @Lopez-Sanjuan2012; @Keenan2014] parameterised the evolution of the pair fraction with the following power-law function: $$\begin{aligned} f_{\mathrm{pair}} \, (z) = f_{\mathrm{pair}}(z=0) \times (1 + z)^{m} \label{eq:fm_fit}\end{aligned}$$ The power law fits to each of our three measurements are shown by the dashed and dotted lines in the figure. The negative power indices in each case suggest that the individual pair fractions increase slightly with decreasing redshift. The pair fraction derived using the spectroscopic $C$-correction has a steeper redshift evolution than photometric and the 50 per cent contamination-corrected pair fractions. The bottom panel of Fig. \[fig:fpair\] compares the spectroscopic-corrected $f_{\mathrm{pair}}$ as measured within 30 kpc (black symbols and line) to that within 50 kpc. Here again, the negative power indices suggest that the pair fractions are increasing with decreasing redshift. The pair fraction within 50 kpc is a factor of $\sim2$ larger than that measured within 30 kpc. This is close to the expected difference in normalization for different radii found in previous works (e.g. from the two-point correlation function, @Bell2006a). Although the observed evolution of $f_{\mathrm{pair}}$ is slightly steeper within the smaller radius, it is consistent within the uncertainty to that measured within the larger search radius of 50 kpc. Recall that the clusters in redMaPPer have been detected by looking for overdensities of red-sequence galaxies (Section \[sec:redM\_over\]). This may tempt the reader to think that the given cluster members are only red-sequence galaxies. If this is the case then we may miss potential merger candidates that are not located on the red-sequence, causing us to potentially underestimate the pair fraction of the BCGs. In Fig. \[fig:PR\_rms\_cuts\_red\_seq\] we showed that galaxies with high probabilities $(P_{\rm{MEM}} > 0.9)$ form a tighter sequence, as expected, since they are more likely to be red-sequence galaxies. Those with smaller $P_{\rm{MEM}}$ values, on the other hand, are more scattered in colour. This suggests that some galaxies that are not on the red-sequence (i.e. galaxies in the blue cloud) are also included in the catalogue. We investigate this further using a method similar to that of [@Lu2009] to separate red-sequence galaxies from their blue counterparts. We refer the reader to [@Lu2009] for details. Briefly, we obtained the width of the red-sequence by fitting a single Gaussian against the galaxies with high probabilities of being red-sequence cluster members $(P_{\rm{MEM}} > 0.9)$. Galaxies within $2\sigma$ of the red-sequence are considered to be part of the red-sequence while those below this limit are classified as blue galaxies. We find that a small fraction (7 per cent) of the galaxies in redMaPPer are blue. Querying the SDSS DR8 database, we find an additional $\sim 200$ blue galaxies neighbouring our BCGs that are not included in the redMaPPer catalogue. Including these ‘missed’ galaxies into our close pair sample and remeasuring the photometric pair fraction, results in an increase of less than one per cent. It is therefore clear that these ‘missing’ blue galaxies have a negligible influence on the average BCG pair fraction. Several studies in the literature use luminosity rather than stellar mass to identify potential major merger candidates [e.g. @Liu2009; @Liu2015; @Edwards2012; @Burke2013; @Keenan2014; @Burke2015 and others]. We find that the selecting pairs based on their luminosity ratio rather than their mass ratio has no perceivable effect on our individually measured BCG pair fractions. Recall that the pair fractions quoted here have been derived by only considering the BCGs that form part of the evolutionary cluster sequence (i.e. the halo masses of the clusters have been taken into account). We find these pair fractions to be consistent within the uncertainties to that measured in the case where we did not take the host clusters’ $M_{h}$ growth into account. Additionally, we divide the cluster sample into a subsample of low and high mass clusters by using a fixed[^10] $M_{h}$ cut of $\sim 0.3 \times 10^{15} \, M_{\odot}$. The resulting pair fractions of these two subsamples are also consistent within the uncertainties. These results may suggest that the pair fraction of BCGs involved in major mergers does not depend on the halo mass of their host clusters. \  --------------- ---------- ------ ------------------- ------------------- ------------------- ------------------- $0.08 - 0.15$ $> 0.29$ 329 $0.183 \pm 0.025$ $0.091 \pm 0.012$ $0.057 \pm 0.018$ $0.053 \pm 0.015$ $0.15 - 0.21$ $> 0.26$ 789 $0.195 \pm 0.015$ $0.097 \pm 0.008$ $0.071 \pm 0.011$ $0.048 \pm 0.009$ $0.21 - 0.28$ $> 0.24$ 1572 $0.204 \pm 0.012$ $0.102 \pm 0.006$ $0.058 \pm 0.008$ $0.035 \pm 0.006$ $0.28 - 0.35$ $> 0.22$ 2742 $0.181 \pm 0.010$ $0.090 \pm 0.005$ $0.046 \pm 0.007$ $0.019 \pm 0.004$ --------------- ---------- ------ ------------------- ------------------- ------------------- ------------------- ### The SALT pair fraction In order for a close galaxy pair to be considered a potential merger candidate, we require $r_{\mathrm{sep}} \leq 50$ kpc and $\Delta v \leq 300$ [kms$^{-1}$]{}. We find that seven of the 12 close pairs in the SALT sample satisfy this criteria (see Table \[table:SALT\_sample\]). Using Eq. \[eq:salt\_bcgs\_merger\_frac\], we find the measured bound BCG pair fraction to be $0.21^{+0.04}_{-0.05}$ between $0.4 \leq z \leq 0.5$. If instead we consider a separation radius of 30 kpc, we find that three of the pairs in the SALT sample can be considered potential merger candidates. This results in a bound BCG pair fraction of $0.09^{+0.06}_{-0.03}$ (as derived using Eq. \[eq:salt\_bcgs\_merger\_frac\]). ### Literature comparison In Fig. \[fig:fpair\_lit\_comp\_30kpc\_evolB\_SALT\] we show the comparison of our measured pair fraction against values obtained from the literature. We compare against the results of studies conducted in clusters and the field. The majority of the studies discussed hereafter have derived pair fractions within $r_{\mathrm{sep}} \leq 30$ kpc. We have therefore chosen to only show the spectroscopically corrected pair fraction we obtained by considering close pairs within the same physical separation distance (black points).  ### Cluster studies: {#cluster-studies .unnumbered} Both [@McIntosh2008] and [@Liu2009] have studied the BCG pair fraction (over $0.01 \leq z \leq 0.12$) by searching for close pairs within 30 kpc. They further restricted their samples to only consider close pairs where both the BCG and companion show signs of morphological distortions (i.e. diffuse tails and asymmetries in the inner isophotes). These distortions indicate that the galaxies are in the process of merging. By only considering major mergers (luminosity ratios $\leq$ 1:4), [@McIntosh2008] found that 38 of their 845 close pairs were morphologically distorted, giving $f_{\mathrm{pair}} = 0.04 \pm 0.01$, assuming Poisson errors (red point). Similarly, [@Liu2009] found $f_{\mathrm{pair}} = 0.03 \pm 0.01$ (18/515) by assuming Poisson errors (blue point). Only our first redshift bin overlaps with the redshift range used in the above-mentioned studies $(z \leq 0.12)$. We measure a pair fraction of $0.05 \pm 0.01$ at $z\sim 0.11$. The main difference between our study and that of [@McIntosh2008; @Liu2009] is the technique used to select mergers: morphological distortions vs. close pairs. Mergers identified through morphological distortions are in the final stages of merging. Galaxies in these pairs are typically expected to merge within $\sim0.2$ Gyr [e.g. @Patton2002; @Hernandez-Toledo2005; @Lotz2011]. The close pair technique on the other hand selects mergers that are in various stages of the merging process (early to final stages). For major merger pairs with a separation distance of $\sim 50$ kpc, [@Lotz2011] estimate a merger timescale of roughly 0.6 Gyr. For pairs with $r_{\mathrm{sep}} \sim 30$ kpc, $t_{\mathrm{merge}}$ decreases to 0.33 Gyr [@Lotz2011]. The merger timescales of the pairs in our sample are therefore longer (on average) than those in [@McIntosh2008; @Liu2009]. It is therefore not unreasonable that we find a higher pair fraction. We are not able to measure the three dimensional velocities of galaxies so we can only use their line-of-sight velocities from spectroscopy. By imposing a velocity difference (as done in our analysis), one is able to identify pairs of galaxies that are potentially gravitationally bound to one another. False-positive pairs may still be included in the sample, despite the velocity cut, making our pair fraction potentially higher than [@McIntosh2008; @Liu2009]. The velocity cut on the other hand may also exclude real bound pairs from the sample, if they have larger velocity differences than used in this work. However, when we use $\Delta v \leq 500$ [kms$^{-1}$]{} to identify potential merger candidates [also used by @Lin2004a; @Lin2008; @Lopez-Sanjuan2012; @Robotham2014], we find that it only increases $f_{\mathrm{pair}}$ by $\sim 0.03$ per cent. Thus we find that the exclusion of real bound pairs, which do not satisfy our velocity cut, has a negligible effect on the measured pair fraction. In a later study, [@Liu2015] extended their analysis to investigate the role of major mergers in the mass growth of BCGs between $0.3 \leq z \leq 0.6$ (with a median value of $z\sim 0.43$). Using the same methodology as [@Liu2009], they find that 4 of their 29 BCGs have morphological distortions, giving $f_{\mathrm{pair}} = 0.14 \pm 0.07$, assuming Poisson errors (green point). Using data from SALT we measure a bound pair fraction of $0.09 ^{+0.06}_{-0.03}$ at $z=0.45$ within 30 kpc (magenta point). This is lower than that found by [@Liu2015], however the pair fractions are consistent within the uncertainties. ### Field studies: {#field-studies .unnumbered} There are numerous studies over the years that have investigated the close pair fraction of galaxies in the field [e.g. @Carlberg2000; @Patton2000; @Lin2004a; @Lin2008; @Kartaltepe2007; @Hsieh2008; @Bundy2009; @2009; @Jogee2009; @Robaina2010; @Lopez-Sanjuan2012; @Keenan2014; @Robotham2014]. The shaded region in Fig. \[fig:fpair\_lit\_comp\_30kpc\_evolB\_SALT\] represents the range of pair fractions that are measured in the field. We only indicate the pair fractions of two field studies as examples. They were chosen because they defined major mergers in the same way as we do in this paper. Most studies of the pair fraction in the field have been for galaxies of approximately $L^{\star}$ rather than the very massive BCGs we consider here. Nevertheless, it is useful to compare the range of values found in the field to that measured in clusters. This is done to determine how the redshift evolution of the major merger pair fractions in these two environments compare. @Jogee2009 [@Keenan2014; @Robotham2014] investigated the close pair fraction of field galaxies with $M_{\star} \geq (2 - 5) \times 10^{10} \, \mathrm{M_{\odot}}$ over $0.1 \leq z \leq 1.2$. The close pairs were selected to be within $r_{\mathrm{sep}} \leq 30$ kpc and $\Delta v \leq 500$ [kms$^{-1}$]{}, although [@Keenan2014] also imposed a lower $r_{\mathrm{sep}}$ limit of 7 kpc on their close pair sample selection. These studies all considered major mergers, although the major merger definition differed slightly from study to study. [@Jogee2009] used stellar mass ratios of (1:4), while [@Robotham2014] used a mass ratio of (1:3). [@Keenan2014] in turn used luminosity ratios of (1:4) to define major mergers. The results from these studies suggested that the pair fraction experienced a very mild evolution since $z\sim1$. [@Robaina2010] on the other hand measured the pair fraction (within 30 kpc) of field galaxies with $M_{\star} > 5 \times 10^{10} \, \mathrm{M_{\odot}}$ by using the amplitude of the two-point correlation function. Although they do not impose a specific mass ratio criteria, they find that most of their merger sample ($\sim 90$ per cent) consists of major mergers (i.e. mass ratios between 1:1$-$1:4). They measure a very mild evolution in the pair fraction since $z\sim1$. The studies of [@Lin2008; @Lopez-Sanjuan2012] used $10 \lesssim r_{\mathrm{sep}} \lesssim 50$ kpc and $\Delta v \leq 500$ [kms$^{-1}$]{} to identify close galaxy pairs in the field between $0.2 \leq z \leq 0.9$. The galaxies in these studies had $M_{\star} \geq 10^{11} \, \mathrm{M_{\odot}}$. Both these studies only considered major mergers, however [@Lopez-Sanjuan2012] used stellar mass ratios of (1:4) to identify major mergers, while [@Lin2008] used luminosity ratios of (1:4) instead. These studies also separated the red and blue galaxies in their samples to investigate the pair fraction for each galaxy population. [@Lopez-Sanjuan2012] found that the major merger rate for blue galaxies decreased rapidly with decreasing redshift, while this evolution was slower for red galaxies. [@Lin2008] observed the same trend for their blue galaxies, however they found that the red galaxy pair fraction increased with decreasing redshift. The same results were observed by other studies that separated red (or early-type) and blue (or late-type) galaxies [e.g. @Bundy2009; @2009]. More interesting though is that they found that the major merger fraction of early-type galaxies was higher than that of late-type galaxies. [@Robotham2014] found a similar result, i.e. the major merger pair fraction of high mass galaxies was higher than for lower mass galaxies. [@Liu2009] in turn found that the pair fraction (specifically for major mergers) increased with halo mass. Although a direct comparison between our observed $f_{\mathrm{pair}}$ and that measured in the field is beyond the scope of this work, we find that all of the above-mentioned field studies indicate that little to no evolution took place in $f_{\mathrm{pair}}$ since $z \sim 1$. Our results on the other hand suggest that the major merger pair fraction of BCGs (in clusters, by definition) increases with decreasing redshift, a result that was also found by [@Lin2008] for the red galaxies in their field sample. Stellar mass growth due to mergers ---------------------------------- In Table \[table:redM\_mass\_growth\] we present a summary of the fractional contribution made by major mergers to the stellar mass of a present day BCG since a given redshift (assuming an evolving merger rate). Recall that these results have been derived under the assumption that half of the stellar mass of the companion is accreted onto the BCG $(f_{\mathrm{mass}} = 0.5 \pm 0.2)$. We find that major mergers contributed $24 \pm 14$ per cent (on average) to the stellar mass of a present day BCG since $z=0.32$. When we also take the SALT sample into consideration, we find the fractional contribution made by major mergers to the stellar mass of a present day BCG to be $29 \pm 19$ per cent, on average, since $z=0.45$. We note that since the merger rate is very constant with redshift (see Table \[table:redM\_mass\_growth\]), the mass growth results are very similar if we integrate Eq. \[eq:delta\_M\] using a fit to the measured merger rates or assume the rate is constant in each redshift bin, given by the rate at the centre of that bin. [cccl]{} & & &\ & & &\ \ 0.11 & $0.79 \pm 0.53$ & $0.07 \pm 0.06$ & $13 \pm 8$\ 0.18 & $0.69 \pm 0.52$ & $0.10 \pm 0.08$ & $20 \pm 12$\ 0.25 & $0.63 \pm 0.50$ & $0.09 \pm 0.08$ & $24 \pm 15$\ 0.32 & $0.77 \pm 0.55$ & $0.06 \pm 0.05$ & $24 \pm 14$\ \ \ 0.45 & $0.89 \pm 0.42$ & $0.24 \pm 0.12$ & $29 \pm 17$\ ### Literature comparison In Fig. \[fig:fracmassg\_lit\_cumm\_salt\] we show the comparison of our measured BCG mass growth (due to major mergers within $7 \leq r_{\mathrm{sep}} \leq 50$ kpc) to that derived from studies in the literature. We only show the mass growth results (due to major mergers) that have been derived under the assumption that half of the companion’s mass is accreted onto the BCG. The solid and dashed lines respectively indicate the best-linear fit, obtained using least-square fitting, to the observational studies (represented by the filled circles) and simulations (represented by the shaded regions).  Using morphological distortions as an indication of mergers, [@Liu2009] found that major mergers would contribute $15 \pm 10$ per cent on average to the stellar mass of a present day BCG from $z=0.7$ (blue point) at a mean rate of $2.5 \pm 1.7$ per cent per Gyr (assuming $f_{\mathrm{mass}} = 0.5$). [@Liu2015] found that major mergers, on average, contributed $35 \pm 15$ per cent to the stellar mass of a present BCG since $z=0.6$ by assuming $f_{\mathrm{mass}} = 0.5$ (green point). In a separate study, [@Burke2015] used close pairs (identified within 50 kpc) to study the stellar mass build-up of 23 BCGs over $0.18 < z < 0.90$. Their sample included mergers down to stellar mass ratios of 1:20 and therefore considered both major and minor mergers. They found that these mergers increased the BCGs’ stellar masses by a factor of 1.2 since $z = 0.9$. They estimate that $32 \pm 10$ per cent of the total merging mass is locked up in major mergers (with mass ratios of 1:2$-$1:5, cyan point). Extending this analysis out to $0.8 < z < 1.4$, [@Burke2013] found that major mergers (mass ratios of 1:1$-$1:2) contributed $53 \pm 30$ per cent to the stellar mass of present day BCGs since $z\sim1$ (grey point), while the remainder was contributed by minor mergers (mass ratios of $>$1:3$-$1:20). At $z\sim1$ the @DeLucia2007 [@Laporte2013] semi-analytical models (represented by the red and purple dashed lines) predict larger mass growth results for the BCGs than what is observed, with the SAM of [@DeLucia2007] predicting the largest growth. At lower redshifts, however, both these models predict mass growth results that are in agreement with the observations. There may be two sources responsible for the discrepancy at higher redshifts. The first is that the SAMs consider both major and minor mergers. The second source may lie in the manner in which the SAMs take the efficiency of mergers into account, specifically the fraction of the accreted mass that ends up in the ICL. The SAM of [@Laporte2013] predicts a mass growth that is in better agreement with what is observed, perhaps because they include an ICL prescription, with 30 per cent of the companion mass transferred to the ICL in each merger. [@DeLucia2007] does not account for loss to the ICL. This illustrates the need to take the stellar mass growth of the ICL into account when the BCG stellar mass growth is investigated. Contribution of mergers towards the stellar mass build-up of the ICL -------------------------------------------------------------------- Galaxies merging with BCGs have been proposed as a mechanism through which the stellar mass of the ICL grows. Tidal stripping of satellite galaxies are also thought to contribute towards the stellar mass growth of the ICL. This idea is taken into consideration by the more recent simulations that are used to study ICL formation [e.g. @Contini2014; @DeMaio2015]. The simulations can be used study the mass assembly of BCGs and taking the tidal stripping of galaxies into account during the ICL formation may serve to alleviate the discrepancies found between the predicted and observed stellar mass growth of the BCGs. Observations and simulations of the ICL suggest that the stellar mass of the ICL has increased by a factor of $4-5$ since $z=1$ [e.g. @Krick2007; @Murante2007; @Rudick2011; @Burke2012; @Burke2015]. The N-body simulations of [@Murante2007; @Conroy2007; @Puchwein2010] predict that $30-80$ per cent of the BCGs’ merging mass should be distributed into the ICL in order to reproduce the distribution observed in nearby clusters. Combined with the number of mergers that BCGs are expected to experience [@Edwards2012; @Burke2013; @Lidman2013], one is faced with a scenario where the ICL is being built up through galaxies that are interacting with BCGs. If we are to believe the model predictions that much of the mass from companions that merge with BCGs is distributed into the ICL, then this could explain why the stellar masses of the BCGs remain relatively unchanged (or only slightly increase) with decreasing redshift. We investigate whether major mergers provide sufficient stellar material to explain the stellar mass growth of the ICL from $z=0.3$ and $z=0$. We denote the amount of stellar mass added to the ICL due to mergers as $(\Delta M_{\star})_{\mathrm{mergers}}$ and the stellar mass growth of the ICL between two redshift bins by $(\Delta M_{\star})_{\mathrm{ICL}}$. If $(\Delta M_{\star})_{\mathrm{ICL}} \gg (\Delta M_{\star})_{\mathrm{mergers}}$, it implies that mergers do not provide enough stellar mass to account for the growth of the ICL. We investigate this idea by combining values from the literature with our own measurements. We were not able to obtain a stellar mass estimate of the ICL at either $z=0.3$ or $z=0$ from the literature, however [@DeMaio2015] investigated the ICL stellar mass build-up in four clusters at $0.44 \leq z \leq 0.57$ (with a median redshift of $z=0.5$) that span a halo mass range of $0.6-2.6 \times 10^{15} \, \mathrm{M_{\odot}}$. They measured the ICL luminosities of these clusters within a radius of 110 kpc from their centres (see their table 6). Because their $M_{h}$ range overlaps with ours, we take the average of their measured ICL luminosities, denoted $\langle L^{\star} \rangle_{\mathrm{ICL}}(z=0.5)$, as a proxy of the ICL’s average stellar mass at $z=0.5$, denoted $\langle M_{\star} \rangle_{\mathrm{ICL}}(z=0.5)$. We then use the predicted stellar mass growth factor $(f)$ of the ICL from @Contini2014 [see their figure 6][^11] to determine $\langle M_{\star} \rangle_{\mathrm{ICL}}(z=0)$ and $\langle M_{\star} \rangle_{\mathrm{ICL}}(z=0.3)$ as described below. $$\begin{aligned} (\Delta M_{\star})_{\mathrm{ICL}} = \langle M_{\star} \rangle_{\mathrm{ICL}}(z=0) - \langle M_{\star} \rangle_{\mathrm{ICL}}(z=0.3) \label{eq:ICL_mass_growth_1}\end{aligned}$$ with $$\begin{aligned} \begin{split} \langle M_{\star} \rangle_{\mathrm{ICL}}(z=0) & = \frac{\langle M_{\star} \rangle_{\mathrm{ICL}}(z=0.5)}{f(z=0.5-0)} \\ & = \frac{\langle L^{\star} \rangle_{\mathrm{ICL}}(z=0.5) \times (M_{\star})_{L^{\star}}}{f(z=0.5-0)} \label{eq:ICL_mass_0} \end{split}\end{aligned}$$ where $\langle L^{\star} \rangle_{\mathrm{ICL}}(z=0.5)$ denotes the average ICL luminosity of the clusters at $z=0.5$ and has a value of $7.4 \, L^{\star}$ [obtained from @DeMaio2015]. The stellar mass of a $L^{\star}$ galaxy, denoted $(M_{\star})_{L^{\star}}, is \sim 5 \times 10^{10} \, \mathrm{M_{\odot}}$. $$\begin{aligned} \langle M_{\star} \rangle_{\mathrm{ICL}}(z=0.3) = \langle M_{\star} \rangle_{\mathrm{ICL}}(z=0) \, \times \, f(z=0.3-0) \label{eq:ICL_mass_growth_3}\end{aligned}$$ where $f(z=0.5-0) = 0.5$ and $f(z=0.3-0) = 0.75$. These are the predicted stellar mass growth factors of the ICL between $z=0.5-0$ and $z=0.3-0$ from [@Contini2014]. Substituting these values into the relevant equations above, we find $(\Delta M_{\star})_{\mathrm{ICL}} \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$. We measure the amount of stellar mass contained in the close companions of BCGs to be $\sim 4 \times 10^{11} \, \mathrm{M_{\odot}}$, of which half is transferred to the ICL. Compared to the ICL stellar mass estimate given above, we find there is sufficient stellar material in the galaxies that are likely to merge with the BCGs as major mergers to account for the stellar mass growth of the ICL between $0.0 \leq z \leq 0.3$. This may also imply that our assumption of 50 per cent mass transfer to the BCG during mergers is enough to grow the ICL, arguing against a higher mass transfer. Conclusions {#sec:concl} =========== In this paper, we examine the role that mergers play in the stellar mass build-up of BCGs between $0.08 \leq z \leq 0.50$. For this purpose we identify close galaxy pairs within a 50 kpc physical search radius and assume that these close pairs are diagnostics of mergers. The close pair fraction is then used to determine how much stellar mass growth the BCGs have experience over the redshift range of interest due to major mergers. - We observe a weak trend that the spectroscopically corrected BCG pair fraction increases with decreasing redshift, suggesting that major mergers may become more important towards the present day. The evolution of the pair fractions (Eq. \[eq:fm\_fit\]) within 30 and 50 kpc are respectively given by $m = -6.42 \pm 2.58$ and $m = -2.30 \pm 1.27$. - Since $z=0.32$ we find the fractional contribution made by major mergers to the stellar mass of a present day BCG to be $24 \pm 14 $ per cent, on average. - Using data from SALT we extend our study to $z=0.45$. From this redshift, we find the fractional contribution made by major mergers to the stellar mass of a present day BCG to be $29 \pm 17$ per cent, on average. - We also investigate whether mergers provide sufficient stellar material to explain the stellar mass growth of the ICL from $z=0.3$ to $z=0$. Using the predicted stellar mass growth factor from [@Contini2014] we find that the ICL has increased its stellar mass by $(\Delta M_{\star})_{\mathrm{ICL}} \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$ between $z=0.3$ and 0. From our analysis of the amount of mass contained within the close companions of BCGs, we estimate that major mergers distributed about $(\Delta M_{\star})_{\mathrm{mergers}} \sim 2 \times 10^{11} \, \mathrm{M_{\odot}}$ of stellar mass into the ICL between $z=0.3$ to $z=0$. Our findings imply that galaxies that are likely to merge with the BCG as major mergers provide enough stellar material to account for the stellar mass growth of the ICL (at least from $z \leq 0.3$). Acknowledgements {#acknowledgements .unnumbered} ================ We thank the anonymous referee for the many helpful suggestions that improved this paper. The authors gratefully acknowledge the financial support of the National Research Foundation (NRF) towards this project. DNG and RES are funded under the Professional Development Programme by the NRF. DNG would like to thank Risa Wechsler, Danilo Marchesini and Claire Burke for helpful discussions. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is <http://www.sdss3.org/>. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. This work is based in part on observations taken with the Southern African Large Telescope (SALT), proposals 2013-2-RSA-008 and 2014-1-RSA\_OTH-009. We thank the South African Astronomical Observatory for allocating us the time. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Investigating the spectroscopic completeness of the close pair sample {#ap:corr_spec_comp} ===================================================================== Not every galaxy in the close pair sample has a spectroscopic redshift. This may simply be because the galaxies did not meet the selection criteria of the SDSS’s spectroscopic surveys or fiber collision problems caused them not to be targeted for spectroscopy. The minimum separation between the SDSS spectroscopic fibers is $55\arcsec$ and if two objects are closer than this, then only one will be observed [@Zehavi2002; @Blanton2003]. Repeat observations however mean that many galaxies closer than $55\arcsec$ are observed in practice. Our close pair sample is therefore spectroscopically incomplete, which in turn will cause close pairs to be missed because one or more galaxies in these pairs have no spectroscopy. We correct for spectroscopic incompleteness by determining how the completeness of our close pair sample changes as a function of the selection criteria used in the SDSS spectroscopic galaxy samples (summarised in Table \[table:spec\_sample\_selec\]). The angular separation ($r_{\mathrm{ang}}$) between the galaxies in the close pairs also needs to be considered since fiber collisions also influence the spectroscopic completeness of the close pair sample. Through this approach we assign each galaxy in the close pair sample a spectroscopic weight, which is used to determine the pair fraction. ------ ------------------ -------------------------------- ----------- MGS $r_{\mathrm{petro}} < 17.77$ LRG Cut I ($z<0.4$) $r_{\mathrm{petro}} \leq 19.2$ $(g - r)$ Cut II ($z>0.4$) $r_{\mathrm{petro}} \leq 19.5$ $(r - i)$ BOSS LOWZ ($z<0.4$) $r_{\mathrm{cmod}} \leq 19.6$ $(g - r)$ CMASS ($z>0.4$) $i_{\mathrm{cmod}} \leq 19.9$ $(r - i)$ ------ ------------------ -------------------------------- ----------- : Summary of the selection criteria used in the MGS, LRG and BOSS sample to target galaxies for spectroscopy. Column 2 and 3 give the magnitude limit and galaxy colour selection of each survey respectively. Due to the transition of the 4000 Å break from the *g* to *r*-band at $z\sim0.4$, both the LRG and BOSS surveys have to use two different selection criteria to select galaxies for spectroscopic follow-up below and above this redshift.[]{data-label="table:spec_sample_selec"} The correction for spectroscopic incompleteness (hereafter $C$-correction) we apply to the close pair sample is determined as a function of four variables: the $r$-band Petrosian magnitude ($r_{\mathrm{petro}}$), $(g - r)$ and $(r - i)$ colours of the galaxies along with the pairs’ angular separation distances, $r_{\mathrm{ang}}$. The $C$-correction is given as: $$\begin{aligned} C\, = \frac{N_{\mathrm{spec. \,pairs}}(r_{\mathrm{petro}}, \mathit{g - r}, \mathit{r - i}, r_{\mathrm{ang}}) }{N_{\mathrm{phot. \,pairs}}(r_{\mathrm{petro}}, \mathit{g - r}, \mathit{r - i}, r_{\mathrm{ang}})} \label{eq:spec_comp_fac}\end{aligned}$$ where $N_{\mathrm{spec. \,pairs}}$ and $N_{\mathrm{phot. \,pairs}}$ respectively indicates the number of pairs in the spectroscopic and photometric samples. We find that the completeness of the MGS, LRG and BOSS samples are fairly constant with $r_{\mathrm{ang}}$. The $C$-correction of the close pairs is therefore only dependent on three variables, i.e. the $r_{\mathrm{petro}}$ magnitudes, the $(g - r)$ and $(r - i)$ colours. The $C$-correction of the pairs are determined by constructing a three-dimensional cube where each of the three variables represent an axis (see Fig. \[fig:C\_rpetro\_gr\_ri\_all\_Ngals\_MGS\_LRG\_BOSS\]). The galaxies in our close pair sample are divided into $r_{\mathrm{petro}}$ magnitude slices of $\sim1.4$ mag to ensure a smooth $C$-correction transition between the voxels. The $(g - r)$ vs. $(r - i)$ colour diagram of the galaxies in each magnitude slice is then plotted. Thereafter, Eq. \[eq:spec\_comp\_fac\] is used to determine the $C$-correction of each voxel. The spectroscopic completeness of the pair sample is expected to drop around $m_{\mathrm{i}} = 19.9$ mag since this is the magnitude limit of the BOSS survey (see Table \[table:spec\_sample\_selec\]). This roughly corresponds to $m_{\mathrm{r}} = 20.8$ mag (refer to last panel of Fig. \[fig:C\_rpetro\_gr\_ri\_all\_Ngals\_MGS\_LRG\_BOSS\]). For any galaxy with a magnitude fainter than this, we set the $C$-correction equal to zero. As previously mentioned spectroscopy is vital in order to determine which companions are bound to their host BCGs. To ensure that we have a sample with a high spectroscopic completeness with which the BCG pair fraction can be determined, we further restrict our close pair sample to only include galaxies brighter than the BOSS magnitude limit. At this magnitude, we expect to be complete for major mergers (see Fig. \[fig:mag\_lim\_21\_5\]). The pair fraction is then determined using Eq. \[eq:merg\_frac\_c\_corr\]. {width="15cm" height="14cm"} \[lastpage\] [^1]: E-mail: [email protected] [^2]: Typically defined in the literature as mergers with stellar mass ratios of $>$ 1:4 $-$ 1:20 [e.g. @Edwards2012; @Burke2013; @Burke2015]. [^3]: Typically defined as mergers with stellar mass ratios of 1:1 to 1:4 [e.g. @Jogee2009; @Lopez-Sanjuan2012; @Robotham2014]. [^4]: <http://risa.stanford.edu/redmapper/> [^5]: Richness is defined as the number of red-sequence galaxies within a cluster that are brighter than $0.2 \, L^{\star}$. [^6]: Given by equation 23 of [@Rykoff2014]. [^7]: We correct the redMaPPer richness estimates for the survey depth using $\lambda / S(z)$. We use these in the remainder of the paper and refer to them as the corrected richnesses. [^8]: [http://home.fnal.gov/$\tt{\sim}$jghao/gmbcg$\_$sdss$\_$catalog.html](http://home.fnal.gov/~jghao/gmbcg_sdss_catalog.html) [^9]: Between $0.08 \leq z \leq 0.20$, we are complete for mergers with mass ratios down to 1:6 (see Section \[sec:pair\_selection\] and panel c of Fig. \[fig:mag\_lim\_21\_5\]). [^10]: This cut has been varied, however the results remain unchanged. [^11]: They investigated the ICL growth in clusters by updating the [@DeLucia2007] semi-analytical model to include various implementations for the formation of the ICL. Specifically, the ICL is assumed to form from the stellar stripping of satellite galaxies and mergers with the BCGs (assuming that 20 per cent of the companion’s stellar mass ends up in the ICL during a merger event with the BCG).

--- abstract: 'For the quadratic Poincaré gauge theory of gravity (PG) we consider the FLRW cosmologies using an isotropic Bianchi representation. Here the considered cosmologies are for the general case: all the even and odd parity terms of the quadratic PG with their respective scalar and pseudoscalar parameters are allowed with no *a priori* restrictions on their values. With the aid of a manifestly homogeneous and isotropic representation, an effective Lagrangian gives the second order dynamical equations for the gauge potentials. An equivalent set of first order equations for the observables is presented. The generic behavior of physical solutions is discussed and illustrated using numerical simulations.' author: - 'Fei-Hung Ho' - Hsin Chen - 'James M. Nester' - 'Hwei-Jang Yo' title: '[General Poincaré Gauge Theory Cosmology]{}' --- Introduction ============ All the known fundamental physical interactions can be formulated in a common framework: as [*local gauge theories*]{}. This is true especially of gravity; however Einstein’s general relativity (GR), when viewed as a theory for a dynamic spacetime metric, does not so well reveal its affinity with other gauge theories. (For a recent discussion of the fundamental gauge theory nature of gravity, see Ref. [@GR100].) Physically (and geometrically) it is reasonable to consider gravity as a gauge theory of the local Poincaré symmetry of Minkowski spacetime. A formulation of gravity based on local spacetime geometry gauge symmetry, the quadratic Poincaré gauge theory of gravity (PG, a.k.a. PGT) was worked out some time ago [@HHKN; @Hehl80; @HS80; @MieE87; @HHMN95; @GFHF96; @Blag02], and now there is a comprehensive reader with reprints and comments [@BlagoHehl]. The PG is formulated in terms of Riemann-Cartan spacetime geometry, having a metric and a metric compatible connection. Such a connection has, in general, both curvature and torsion. Independent of the metric, the connection has six possible dynamic connection modes; they carry spin-$2^{\pm}$, spin-$1^{\pm}$, spin-$0^{\pm}$. A good dynamic mode should transport positive energy and should not propagate outside the forward null cone. Investigations (especially Refs. [@HS80; @SN80]) of the linearized (even parity) quadratic PG theory found that at most three modes can be simultaneously dynamic; all the acceptable cases were tabulated; many combinations of three modes are satisfactory to linear order. Complementing this, the Hamiltonian analysis revealed the related constraints [@BMNI83]. Then detailed investigations of the Hamiltonian and propagation [@CNY98; @HNZ96; @yo-nester-99; @yo-nester-02] concluded that effects due to nonlinearities in the constraints could be expected to render all of these cases physically unacceptable except for the two “scalar modes”, carrying spin-$0^+$ and spin-$0^-$. In order to further investigate the dynamical possibilities of these PG scalar modes, Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological models were considered. Using a $k=0$ model it was found that the $0^+$ mode naturally couples to the acceleration of the universe and could account for the present day observations [@YN07; @SNY08]; this model was then extended to include the $0^-$ mode [@JCAP09]. There is no known fundamental reason why the gravitational coupling should respect parity. Odd parity terms with pseudoscalar parameters were first introduced into the quadratic PG some time ago [@OPZ], but this innovation was not followed up until more recent times. Now there is renewed interest in all the possible couplings between even and odd parity modes. The odd parity terms with their pseudoscalar coupling parameters were rediscovered and included in the PG [@BHN; @diakonov; @BH]. After systematically developing the general parity PG theory, in BHN [@BHN] the two scalar torsion mode PG Lagrangian was extended to include the appropriate pseudoscalar coupling constants that provide cross parity coupling (often referred to as “parity violating” terms) and the FLRW cosmological models were formulated. The BHN cosmologies were then further explored [@iard10; @HN; @AdP]. The dynamics of the PG BHN model was expected to be clearly revealed in purely time dependent solutions, hence [*homogeneous cosmologies*]{} were investigated. The two dynamical connection modes carry spin 0$^+$ and spin 0$^-$ (referred to as [*scalar*]{} modes or, more specifically, as the [*scalar*]{} and [*pseudoscalar*]{} mode). Consequently in a homogeneous situation they cannot pick out any spatial direction, and thus they have no interaction with spatial anisotropy, so for a study of their dynamics it is most simple and appropriate to look to [*isotropic*]{} models. For the PG BHN model, following the technique used in [@JCAP09], an effective Lagrangian and Hamiltonian as well as a system of first order dynamical equations for Bianchi class A isotropic homogeneous cosmological models was constructed, and some sample evolution was presented which showed the effect of the cross parity coupling. The normal modes were identified, and it was shown analytically how they control the late time asymptotics. A numerical evolution example was presented which shows that the asymptotic late time normal mode evolution is a good approximation [@iard10; @HN; @AdP]. Recently Karananas took on the major task of analyzing the modes in the general quadratic PG including all the odd parity terms in the Lagrangian. At first it seemed that the general PG theory (including all the scalar and pseudoscalar parameters) would allow for all the possible connection modes to have *good* (i.e., “no ghosts, no tachyons”) propagation. However, more careful checking showed that this is not possible after all [@Karananas], nevertheless it does seem that the healthy parameter space of the theory is extended by the inclusion of the odd parity terms. In view of this it is timely to reconsider cosmology for the general quadratic PG, with no *a priori* restrictions on the parameters. We would like to note that the PG theory is a viable alternative to GR. While there are no observations that confirm non-vanishing torsion, there are no observations that exclude torsion, only some mild constraints on the magnitude. For a detailed discussion and a recent assessment of the observational status of torsion the reader may consult Ni [@ni]. Briefly, we just mention that torsion naturally couples to spin, to directly detect it may require highly spin polarized materials. On the other hand it may be that torsion could play a significant role on the cosmological scale. Since the 1970s cosmological models for the Einstein-Cartan (EC) theory and the Poincaré gauge theory of gravity (PG) have been developed. For the EC theory see, e.g., Kopczynśki [@Kopczynski], Trautman [@Trautman], Tafel [@Tafel], and Kuchowicz [@Kuch]. The special interest then was the possibility that models with torsion could avoid singularities, however it was soon noted that torsion is likely to make singularities more severe rather than prevent them, see, e.g., Kerlick [@Kerlick] and [@NesIsen77]. Minkevich *et al*. [@Min8083; @MN95; @MG06; @MGK07] developed torsion cosmology in a series of papers. For the early work on PG cosmology see Goenner and Müller-Hoissen [@GMH]. A more recent report on the status of the subject was given by Puetzfeld [@Peutzfeld], which surveyed virtually all of the papers on the subject up to 2004; as far as we know it only overlooked [@NesIsen77]. For the recent work by our group see [@YN07; @SNY08; @JCAP09; @iard10; @HN; @AdP; @ChenThes]. For other recent works see [@WW09; @Pop10; @ALX10; @AL12; @LSX09; @GLT; @TLG]. The objective of this present work is to begin the analysis of the *general quadratic PG cosmology*. Specifically we consider homogeneous isotropic cosmologies, using an isotropic Bianchi representation to obtain a manifestly homogeneous formulation. This allows us to obtain the dynamical equations via an effective Lagrangian. We find a set of 6 first order equations and discuss the behavior of generic solutions showing some typical numerical evolution. the Poincaré Gauge Theory ========================= Riemann-Cartan geometry ----------------------- The Poincaré gauge theory of gravity (PG) [@HHKN; @Hehl80; @HS80; @MieE87; @HHMN95; @GFHF96; @Blag02; @BlagoHehl] is based on spacetime with a Riemann-Cartan geometry, i.e., a Lorentz signature metric with a metric compatible connection. The two sets of gauge potentials are, respectively, for *translations* the orthonormal coframe and for *Lorentz-rotations* the metric compatible (Lorentz Lie algebra valued) connection one-forms: $$\vartheta^\alpha=e^\alpha{}_i dx^i,\qquad \Gamma^{\alpha\beta}=\Gamma^{[\alpha\beta]}{}_idx^i.$$ The associated field strengths are the *torsion* and *curvature* 2-forms: $$\begin{aligned} T^\alpha&:=&d\vartheta^\alpha+\Gamma^\alpha{}_\beta\wedge\vartheta^\beta=\frac12 T^\alpha{}_{\mu\nu} \vartheta^\mu\wedge \vartheta^\nu,\label{torsion2}\\ R^\alpha{}_\beta&:=&d\Gamma^\alpha{}_\beta+\Gamma^\alpha{}_\gamma\wedge\Gamma^\gamma{}_\beta=\frac12 R^\alpha{}_{\beta\mu\nu}\vartheta^\mu\wedge\vartheta^\nu,\label{curv2}\end{aligned}$$ which satisfy the respective first and second *Bianchi identities*: $$DT^\alpha\equiv R^\alpha{}_\beta\wedge\vartheta^\beta, \qquad DR^\alpha{}_\beta\equiv0.\label{BianchiId}$$ The metric is $g=-\vartheta^0\otimes\vartheta^0+\delta_{ab}\vartheta^a\otimes\vartheta^b$, $a,b=1,2,3$. The coframe along with the Hodge dual gives a convenient dual basis for forms: $\eta^{\alpha\beta\dots}:=*\vartheta^{\alpha\beta\dots}$, where $\vartheta^{\alpha\beta\dots}:=\vartheta^\alpha\wedge\vartheta^\beta\cdots$. The volume 4-form is $\eta:=*1=(4!)^{-1}\eta_{\alpha\beta\mu\nu}\vartheta^{\alpha\beta\mu\nu}\equiv\vartheta^{0123}$, with $\eta_{\alpha\beta\mu\nu}$ being the totally anti-symmetric Levi-Civita tensor. General Lagrangian ------------------ The Lagrangian density for the PG is taken to have a Yang-Mills type form, including up to quadratic terms in the field strengths. It has the general structure $$\begin{aligned} \label{quadraticL} {\cal L}_{\rm PG} \sim \frac{1}{\kappa}\left(\Lambda+ \text{curvature} +\text{torsion}^2\right) + \frac{1}{\varrho}\,\text{curvature}^2\,,\end{aligned}$$ which includes two types of terms with different physical dimensions: $\kappa:=8\pi G/c^4$ is the usual gravitational constant, and $\varrho^{-1}$ has the dimensions of action. $\Lambda$ is the cosmological constant. The associated field equations for the coframe and connection are obtained by varying with respect to these potentials. This gives dynamical equations for the potentials of the quasi-linear second order type with the respective material sources (obtained from some ${\cal L}_\psi={\cal L}(\vartheta^\alpha,\psi,D\psi)$, a minimally coupled matter Lagrangian) being the energy-momentum, $\sim\delta{\cal L}_\psi/\delta\vartheta^\mu$, and the spin density $\sim\delta{\cal L}_\psi/\delta\Gamma^{\alpha\beta}$ 3-forms. The dynamical equations have the general form $$\begin{aligned} \kappa^{-1}(\Lambda+ \hbox{curvature} + D \hbox{ torsion} + \hbox{torsion}^2)+ \varrho^{-1}\hbox{curvature}^2&=& \hbox{energy-momentum},\qquad\\ \kappa^{-1}\hbox{torsion}+\varrho^{-1} D\hbox{ curvature}&=& \hbox{spin}.\end{aligned}$$ From these two equations, with the aid of the Bianchi identities (\[BianchiId\]) (and also via a Noether theorem type argument) one can obtain the conservation of energy-momentum and angular momentum expressions. In complete detail the general quadratic PG Lagrangian 4-form is [@BHN] $$\begin{aligned} \nonumber {\cal L}_{\rm PG} &=& \frac{1}{2\kappa}\Bigl(a_0 R\eta+b_0 X\eta -2\Lambda\eta + \textstyle\sum \limits_{I=1}^{3}a_{I}{}^{(I)}T^\alpha\wedge *{}^{(I)} T_\alpha\Bigr)\nonumber\\ && + \frac{1}{{\kappa}}\left( {\sigma}_{1}{} ^{(1)}T^{\alpha}\wedge\, ^{(1)}T_{\alpha} + {\sigma}_{2}{} ^{(2)}T^{\alpha}\wedge{} ^{(3)}T_{\alpha}\right)\cr & & -\frac{1}{2\varrho} \Bigl(\textstyle\sum\limits_{I=1}^{6}w_{I}{}^{(I)}R^{\alpha\beta}\wedge*{}^{(I)} R_{\alpha\beta}\Bigr)\,\nonumber \\ &&-\frac{1}{2{\varrho}}\left(\ {\mu}_{1}{} ^{(1)}R^{\alpha\beta}\wedge{} ^{(1)}R_{\alpha\beta} + {\mu}_{2}{} ^{(2)}R^{\alpha\beta}\wedge{} ^{(4)}R_{\alpha\beta} \right. \nonumber\\ & & \qquad + \left. {\mu}_{3}{}^{(3)}R^{\alpha\beta}\wedge{} ^{(6)}R_{\alpha\beta} + {\mu}_{4}{} ^{(5)}R^{\alpha\beta}\wedge{} ^{(5)}R_{\alpha\beta} \right).\label{PGgeneralL}\end{aligned}$$ Here $R$ is the scalar curvature and $X$ is the pseudoscalar curvature ($X\equiv -\frac12R_{\alpha\beta\mu\nu}\eta^{\alpha\beta\mu\nu}$). The torsion has been decomposed into three algebraically irreducible pieces: $T^\alpha={}^{(1)}T^\alpha+{}^{(2)}T^\alpha+{}^{(3)}T^\alpha$, which are, respectively, a pure tensor (16 components), the trace (vector), and a totally antisymmetric part (dual to an axial vector). Similarly the curvature 2-form has been decomposed into a sum of 6 algebraically irreducible pieces: $R^\alpha{}_\beta=\sum_{I=1}^6{}^{(I)}R^\alpha{}_\beta$, namely, in numerical order: weyl, pair-commutator, pseudoscalar, ricci-symmetric, ricci-antisymmetric, and scalar. The respective number of components is (10,9,1,9,6,1). In the above Lagrangian the parameters $\Lambda,a_0,a_I,w_I$ (which multiply even parity 4-forms) are scalars, and the parameters $b_0,\sigma_I,\mu_I$ (which multiply odd parity 4-forms) are pseudoscalars. The general theory has 11 scalar plus 7 pseudoscalar parameters. But they are not all physically independent. They are subject to 1 even parity and 2 odd total differentials, leaving effectively 10 scalar + 5 pseudoscalar = 15 “physical” parameters, as we will briefly explain, referring to Refs. [@BHN; @BH] for details. Topological terms ----------------- Not all of the above parameters are physically independent, since there are 3 topological invariants. Without changing the field equations, one can add to the Lagrangian 4-form (\[PGgeneralL\]) any multiple of the (odd parity) *Nieh-Yan identity* [@NY]: $$T^\alpha\wedge T_\alpha-R_{\alpha\beta}\wedge\vartheta^{\alpha\beta} \equiv d(\vartheta^\alpha\wedge T_\alpha).$$ From the irreducible decomposition given in Eqs. (6,7) in [@BH], one can see that this allows one to make parameter changes of the type $$\Delta(b_0,\sigma_1,\sigma_2)=(-f_1,f_1/2,f_1).\label{nieh-yan}$$ Also one can add a multiple of the (even parity) *Euler 4-form* $R^{\alpha\beta}\wedge R^{\gamma\delta}\eta_{\alpha\beta\gamma\delta}$. Because of the 2nd Bianchi identity (\[BianchiId\]b), this makes no contributions to the field equations. From the irreducible decomposition of Eqs. (14b,16,18) in [@BH], it can be seen that this induces parameter changes of the type $$\Delta(w_1,w_2,w_3,w_4,w_5,w_6)=(-f_3,f_3,-f_3,f_3,-f_3,-f_3).\label{euler}$$ Furthermore once can add a multiple of the (odd parity) *Pontryagin* 4-form $R^\alpha{}_\beta\wedge R^\beta{}_\alpha$. Again, thanks to the 2nd Bianchi identity, this has no effect on the field equations. For the irreducible decomposition of Eqs. (14a,16,20) in [@BH], one sees that this induces parameter changes of the type $$\Delta(\mu_1,\mu_2,\mu_3,\mu_4)=(-f_2,-2f_2,-2f_2,-f_2).\label{pointryagin}$$ The actual physical equations will only depend on combinations of the 18 parameters that are invariant under such transformations. Minimal versions ---------------- The topological identities could be used to eliminate some parameters and thus simplify computations. Baekler and Hehl used the Nieh-Yan, Euler, and Pontryagin forms to respectively eliminate the parameters $\sigma_1,\ w_1,\ \mu_1$ and arrive at their Lagrangian (see Eq. (56) in [@BH]); it has no explicit weyl curvature terms, ${}^{(1)}R^\alpha{}_\beta$ (this is the largest curvature part, having 10 components). One of the many alternatives is to instead use the Euler and Pontryagin forms to respectively eliminate the parameters $w_2,\ \mu_2$; then the resulting Lagrangian would not contain any terms involving “paircom” ${}^{(2)}R^\alpha{}_\beta$ (a less familiar part of the curvature with 9 components). The decision is simply a practical question, not a fundamental issue: one could select the most convenient choice for a particular application, although this complicates the comparison of results obtained via different choices. The computations presented here are completely general: all the PG parameters are allowed to have any values, no special restrictions or choices have been imposed. Generic PG cosmology kinematics =============================== Following [@HN; @AdP; @ChenThes], for homogeneous, isotropic Bianchi type I and IX (respectively equivalent to FLRW $k=0$ and $k=+1$) cosmological models we take the isotropic orthonormal coframe to have the form $$\vartheta^0:=dt,\qquad \vartheta^a:=a\sigma^a.$$ Here $a=a(t)$ is the scale factor and $\sigma^a$ depends on the (not needed here) spatial coordinates in such a way that $$d\sigma^a=\zeta\epsilon^a{}_{bc}\sigma^b \wedge\sigma^c$$ (here $\epsilon_{abc}\equiv\epsilon_{[abc]}$ is the 3D Levi-Civita symbol), where $\zeta=0$ for Bianchi I and $\zeta=1$ for Bianchi IX, thus $\zeta^2=k$, the FLRW spatial Riemannian curvature parameter. (We remark that our *derivation* here of the dynamical cosmological equations applies only to the cases $k=0$ and 1. The $k=-1$ case does not admit a manifestly isotropic representation and needs to be treated separately [@ChenThes; @MT72]. In fact, the dynamical equations obtained here are actually valid also for $k=-1$; the details justifying this will be presented elsewhere.) As a consequence of isotropy, the only non-vanishing connection one-form coefficients are necessarily of the form $$\Gamma^a{}_0=\psi(t)\, \sigma^a,\qquad \Gamma^a{}_b=\chi(t)\epsilon^a{}_{bc}\, \sigma^c.$$ From the definition (\[curv2\]), $R^\alpha{}_\beta:=d\Gamma^\alpha{}_\beta+\Gamma^\alpha{}_\gamma\wedge\Gamma^\gamma{}_\beta$, all the independent nonvanishing curvature 2-form components are found to be $$\begin{aligned} R^{ab}&=&\dot \chi dt\wedge \epsilon^{ab}{}_{c}\sigma^c+[\psi^2-(\chi-\zeta)^2+\zeta^2]\sigma^a\wedge \sigma^b,\\ R^a{}_0&=&\dot \psi dt\wedge \sigma^a -\psi(\chi-\zeta) \epsilon^a{}_{bc}\sigma^b\wedge\sigma^c .\end{aligned}$$ Consequently, the scalar and pseudoscalar curvatures are, respectively, $$\begin{aligned} R&=&6[a^{-1}\dot \psi+a^{-2}(\psi^2-[\chi-\zeta]^2+\zeta^2)], \label{R} \\ X&=&6[a^{-1}\dot \chi+2a^{-2}\psi(\chi-\zeta)].\label{X}\end{aligned}$$ Regarding the other irreducible parts of the curvature, for FLRW cosmogies we know that parts 1 and 5 (Weyl and antisymmetric Ricci) identically vanish [@BHN]. For part 4 (traceless Ricci symmetric) we find $$^{(4)}R^{0b}{}_{0c}=\frac1{12} \tilde R\delta^b_c,\qquad ^{(4)}R^{ab}{}_{cd}=-\frac1{12} \tilde R\delta^{ab}_{cd},$$ $$\tilde R:=6[a^{-1}\dot\psi-a^{-2}(\psi^2-[\chi-\zeta]^2+\zeta^2)].$$ For part 2 (paircom) we find $$\begin{aligned} ^{(2)}R_{ab0c}&=&-{}^{(2)}R_{0cab}=\frac1{12} \tilde X\epsilon_{abc},\\ \tilde X&:=&6[a^{-1}\dot\chi-2a^{-2}\psi(\chi-\zeta)]. $$ Via the torsion expressions that will be given next, these results are equivalent to Eqs. (147) and (149) in BHN [@BHN]. Because of isotropy, the only nonvanishing torsion tensor components are of the form $$T^a{}_{0b}=u(t)\delta^a_b, \qquad T^a{}_{bc}=- 2x(t)\epsilon^a{}_{bc},$$ where $u$ and $x$ are referred to as the [*scalar*]{} and [*pseudoscalar*]{} torsion, respectively. They occur in the vector and axi-vector parts of the torsion. The tensor part, ${}^{(1)}T^\alpha$, vanishes for isotropic cosmologies, so it will play no role in our considerations. From the torsion 2-form definition (\[torsion2\]), $T^\mu:=D\vartheta^\mu:=d\vartheta^\mu+\Gamma^\mu{}_\nu\wedge\vartheta^\nu$, the relations between these torsion components and the gauge variables are found to be $$u=a^{-1}(\dot a-\psi), \qquad x= a^{-1}(\chi-\zeta). \label{fchi}$$ In terms of the *Hubble function*, $H:=a^{-1}\dot a$, we have $$a^{-1}\psi=H-u,\label{Hu}$$ a relation which clearly shows that [*kinematically*]{} the scalar torsion $u$ [*directly couples*]{} to the rate of [*expansion*]{} of the universe. From the symmetry assumptions of our model, the source material energy-momentum tensor is [*necessarily*]{} of the fluid form. Thus it can be described by an effective energy density $\rho$, pressure $p$, and flow vector along the cosmological time axis. Here we assume that the source spin density is negligible; this is a quite reasonable physical assumption except for the very early universe. Quadratic curvature terms ------------------------- For isotropic cosmology, since the curvature pieces 1 and 5 vanish, the quadratic curvature terms in the general Lagrangian (\[PGgeneralL\]) involving these pieces (i.e., those with the parameters $w_1,\ w_5,\ \mu_1,\ \mu_4$) are not relevant. The BHN model investigations [@BHN; @HN; @AdP] considered only those quadratic curvature terms involving the scalar and pseudoscalar curvature, $R$ and $X$. The quadratic terms involving these pieces (with the parameters $w_3,\ w_6,\ \mu_3$) were the only ones included in the BHN Lagrangian. In comparison with the BHN model considered in those earlier studies the general quadratic PG (gqPG) cosmological Lagrangian 4-form gains three new quadratic curvature terms: $$-\frac{1}{2\varrho}\left[w_2 {}^{(2)}R_{\alpha\beta} \wedge * {}^{(2)}R^{\alpha\beta} +w_4 {}^{(4)}R_{\alpha\beta} \wedge * {}^{(4)}R^{\alpha\beta} +\mu_2 {}^{(2)}R_{\alpha\beta} \wedge {}^{(4)}R^{\alpha\beta} \right].$$ For homogeneous-isotropic cosmology the respective values are $$\begin{aligned} {}^{(2)}R_{\alpha\beta} \wedge * {}^{(2)}R^{\alpha\beta}&=&-\frac1{12}\tilde X^2\eta,\\ {}^{(4)}R_{\alpha\beta} \wedge * {}^{(4)}R^{\alpha\beta}&=&\frac1{12}\tilde R^2\eta,\\ {}^{(2)}R_{\alpha\beta} \wedge {}^{(4)}R^{\alpha\beta}&=&-\frac1{12}\tilde X\tilde R\eta.\end{aligned}$$ Here $\eta=a^3 dt\wedge \sigma^1\wedge\sigma^2\wedge\sigma^3$ is the proper 4-volume 4-form. Topological terms in cosmology ------------------------------ Let us see in detail how certain combinations of terms add up to a total time derivative for these cosmologies. For the even parity Euler quadratic curvature terms, the pattern from (\[euler\]) is of the form $|{}^{(2)}R|^2-|{}^{(3)}R|^2+|{}^{(4)}R|^2-|{}^{(6)}R|^2$: $$\begin{aligned} \frac{a^3}{12}\{-\tilde X^2+X^2+\tilde R^2-R^2\}&\equiv&3a^3\big\{- [a^{-1}\dot\chi-2a^{-2}\psi(\chi-\zeta)]^2\nonumber\\ &&\quad+[a^{-1}\dot \chi+2a^{-2}\psi(\chi-\zeta)]^2\nonumber\\ &&\quad+[a^{-1}\dot\psi-a^{-2}(\psi^2-\chi^2+2\zeta\chi)]^2\nonumber\\ &&\quad-[a^{-1}\dot \psi+a^{-2}(\psi^2-\chi^2+2\zeta\chi)]^2\big\}\nonumber\\ &\equiv&12[2\dot\chi\psi(\chi-\zeta)-\dot\psi(\psi^2-\chi^2+2\zeta\chi)],\end{aligned}$$ the quadratic parts cancel and the cross terms add up to an expression which is a total time derivative of $12\psi\chi^2-4\psi^3-24\zeta\psi\chi$. For the odd parity Pontryagin form the pattern is $\langle{}^{(2)}R\times {}^{(4)}R\rangle+\langle{}^{(3)}R\times {}^{(6)}R\rangle$: $$\begin{aligned} \frac{a^3}{12}\{-\tilde X\tilde R+RX\}&\equiv&3a^3\big\{[-a^{-1}\dot\chi+2a^{-2}\psi(\chi-\zeta)] [a^{-1}\dot\psi-a^{-2}(\psi^2-\chi^2+2\zeta\chi)]\nonumber\\ &&\quad+[a^{-1}\dot \psi+a^{-2}(\psi^2-\chi^2+2\zeta\chi)] [a^{-1}\dot\chi+2a^{-2}\psi(\chi-\zeta)]\big\}\nonumber\\ &\equiv&6[\dot\chi(\psi^2-\chi^2+2\zeta\chi)+2\psi\dot\psi(\chi-\zeta)],\end{aligned}$$ which again is a total time derivative. Effective Lagrangian ==================== For homogenous cosmologies for our models (which are in Bianchi class A), as in GR [@MT72], one can obtain an effective Lagrangian by integrating over space. Our [*effective Lagrangian*]{} for the gqPG $L_{\mathrm{eff}}=L_\mathrm{PG}+L_{\mathrm{int}}$ includes the [**interaction**]{} Lagrangian: $L_{\mathrm{int}}= pa^3$, where $p=p(t)$ is the source fluid pressure. The gravitational Lagrangian for the gqPG cosmology is $$\begin{aligned} L_\mathrm{PG} &=& \frac{1}{2\kappa}(a_0R+b_0X-2\Lambda)a^3+\frac3{2\kappa}(-a_2u^2+4a_3x^2-4\sigma_2ux)a^3\nonumber\\ &&-\frac1{24\varrho}[-w_2\tilde X^2-w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde R+\mu_3RX]a^3.\quad \label{generalL}\end{aligned}$$ (Remark: The sign of the $\sigma_2$ term in the gravitational Lagrangian here differs from that used in our earlier works [@HN; @AdP]. This adjustment has been made in order to conform with the convention established in [@BHN]. Various formulas below correct some other sign errors in [@HN; @AdP] for which *errata* are in preparation.) Although our model makes sense for all values of the parameters, there are certain inequalities that identify what can be regarded as the *physical* range. For *least action* one should have *positive* “kinetic energy”, hence the terms with quadratic time derivatives should have positive coefficients, which yields the requirements $$a_2<0,\quad w_2+w_3>0, \quad w_4+w_6<0, \quad 4 (w_2+w_3)(w_4+w_6)+(\mu_3-\mu_2)^2<0.$$ In the following, for simplicity, we often take units such that $\kappa=1=\varrho$. In the final results these factors can be easily restored by multiplying $\{a_0,a_2,a_3,b_0,\Lambda,\sigma_2\}$ by $\kappa^{-1}$ and $\{w_2,w_3,w_4,w_6,\mu_2,\mu_3\}$ by $\varrho^{-1}$. Useful combinations ------------------- In the general effective Lagrangian (\[generalL\]), by using the fact that $\tilde R^2,\ \tilde X^2,\ \tilde X\tilde R$ differ from $R^2,\ X^2,\ XR$ only by having opposite sign cross terms, the quadratic curvature terms can be re-expressed as follows: $$\begin{aligned} &&-\frac1{24}[-w_2\tilde X^2-w_3X^2+w_4\tilde R^2+w_6R^2-\mu_2\tilde X\tilde R+\mu_3RX]a^3\nonumber\\ &&\qquad\equiv-\frac1{24}[-w_{3+2}X^2+w_{6+4}R^2+\mu_{3-2}RX]a^3 \nonumber\\ &&\qquad\qquad-\frac1{24}[-w_2(\tilde X^2-X^2)+w_4(\tilde R^2-R^2)-\mu_2(\tilde X\tilde R-XR) ]a^3\nonumber\\ &&\qquad\equiv -\frac1{24}[-w_{2+3}X^2+w_{4+6}R^2+\mu_{3-2}RX]a^3\nonumber\\ &&\qquad\qquad -12w_2\dot\chi\psi(\chi-\zeta) +6{w_4}\dot\psi[\psi^2-(\chi-\zeta)^2+\zeta^2]\nonumber\\ &&\qquad\qquad -3\mu_2[2\psi(\chi-\zeta)\dot\psi+(\psi^2-\chi^2+2\chi\zeta)\dot\chi]\nonumber\\ &&\qquad \equiv-\frac1{24}[-w_{2+3}X^2+w_{4+6}R^2+\mu_{3-2}RX]a^3\nonumber\\ &&\qquad\qquad +3w_{4-2}[-\dot\psi(\chi-\zeta)^2+2\dot\chi\psi(\chi-\zeta)] +\hbox{total time derivative}.\label{quadcurv}\end{aligned}$$ Here, in order to have a more compact presentation, we introduced certain convenient parameter combinations: $$w_{2+3}:=w_2+w_3, \quad w_{4-2}:=w_4-w_2,\quad w_{4+6}:=w_4+w_6, \quad \mu_{3-2}:=\mu_3-\mu_2,$$ which are invariant under the allowed topological transformations (\[euler\]), (\[pointryagin\]) and hence are *physically meaningful*. Energy function --------------- The gqPG energy function can be readily obtained from the earlier work [@HN] Eq. (20). This is because the gravitational Lagrangian has the usual form of a sum of terms homogeneous in “velocities” (i.e., time derivatives): $L_\mathrm{PG}=L_0+L_1+L_2$, consequently, from Euler’s theorem, the associated [*energy function*]{} is of the familiar “kinetic-minus-potential” form: $L_2-L_0$. As Eq. (\[quadcurv\]) shows in detail, the “new” terms that we consider with parameters $w_2,w_4,\mu_4$ are—to order 2 and 0—*identical* to those from $w_3,w_6,\mu_3$, respectively, so those parts *contribute in exactly the same way*, while the order 1 terms do not effect the energy function. Hence we have $$\begin{aligned} {\cal E}_\mathrm{PG}&:=&\frac{\partial L_\mathrm{PG}}{\partial \dot \psi}\dot\psi+\frac{\partial L_\mathrm{PG}}{\partial \dot\chi}\dot\chi+\frac{\partial L_\mathrm{PG}}{\partial \dot a}\dot a-L_\mathrm{PG}=L_2-L_0\nonumber\\ &=&a^3\biggl\{\Lambda-3a_0(H^2+ka^{-2})+\frac32 \tilde a_2(u^2-2Hu) -6\tilde a_3x^2-6\tilde\sigma_2x(H-u)\nonumber\\ &&\qquad-\frac{w_{4+6}}{24}\left[R^2-12R\left\{(H-u)^2-x^2+ka^{-2}\right\}\right] \nonumber\\ &&\qquad +\frac{w_{2+3}}{24}\left[X^2-24Xx(H-u)\right]\nonumber\\ &&\qquad-\frac{\mu_{3-2}}{24}\left[RX-6X\left\{(H-u)^2-x^2+ka^{-2}\right\}-12Rx(H-u)\right]\biggr\},\label{energyfunction}\end{aligned}$$ where we introduced the convenient modified parameters $$\tilde a_2:=a_2-2a_0,\quad \tilde a_3:=a_3-\frac12 a_0,\quad \tilde \sigma_2:=\sigma_2+b_0.$$ The energy value (\[energyfunction\]) has the form $-a^3\rho$ where, from a comparison with Eq. (167) in BHN [@BHN], it can be seen that $\rho$ is the value of the material [*energy density*]{}. Since $L_{\rm PG}$ is time independent, the energy function (\[energyfunction\]) satisfies an [*energy conservation*]{} relation: $$\dot{\cal E}_{\rm PG} =-\frac{\delta L_{\rm PG}}{\delta \psi} {\dot \psi}-\frac{\delta L_{\rm PG}}{\delta \chi} {\dot \chi}-\frac{\delta L_{\rm PG}}{\delta a} {\dot a}=\frac{\delta L_{\rm int}}{\delta a}{\dot a}=3pa^2{\dot a},$$ which, with the interpretation of $\rho$ mentioned above, is just the perfect fluid work-energy relation: $$-\frac{d(\rho a^3)}{dt}=p\frac{da^3}{dt}.$$ Dynamical equations ------------------- With the aid of the formulas for the torsion and curvature scalars in terms of the gauge variables (\[R\]), (\[X\]), (\[fchi\]), the Euler-Lagrange equations for the gqPG cosmology can be obtained. The $\psi$ equation is $$\begin{aligned} \frac{d}{dt}\frac{\partial L_\mathrm{PG}}{\partial \dot \psi}&=&\frac{d}{dt}\left(\frac{a^{2}}{2}\left[6a_0-w_{4+6}R-\frac{\mu_{3-2}}2X\right] -3w_{4-2}(\chi-\zeta)^2\right)\nonumber\\ &=&\frac{\partial L_\mathrm{PG}}{\partial \psi}\nonumber\\ &=&3(a_2u+2\sigma_2x)a^2\nonumber\\ && +\left[6a_0-w_{4+6}R-\frac{\mu_{3-2}}2X \right]a\psi \nonumber \\ &&+\left\{\left[6b_0-\frac{\mu_{3-2}}2 R+w_{3+2}X\right]a+6w_{4-2}\dot\chi\right\}(\chi-\zeta).\label{Epsi}\end{aligned}$$ Reexpressed in terms of observables $(R,\ X, u,\ x, H)$ the $\psi$ equation becomes $$\begin{aligned} -\frac{w_{4+6}}2\dot R-\frac{\mu_{3-2}}4 \dot X&=&-\left[-3 \tilde a_2-w_{4+6}R-\frac{\mu_{3-2}}2 X\right]u\nonumber\\ && +\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]x\nonumber\\ &&+w_{4-2}[2X-24(H-u)x]x. \label{dotRdotX} $$ The last line includes qualitatively new types of terms when compared to the corresponding result in the earlier investigations [@HN; @AdP] based on the BHN model [@BHN]. The $\chi$ equation is $$\begin{aligned} \frac{d}{dt}\frac{\partial L_\mathrm{PG}}{\partial \dot \chi}&=&\frac{d}{dt}\left\{\frac{a^{2}}{2}\left[6b_0-\frac{\mu_{3-2}}2R+w_{3+2}X \right]+6w_{4-2}\psi(\chi-\zeta)\right\}\nonumber\\ &=&\frac{\partial L_\mathrm{PG}}{\partial \chi}\nonumber\\ &=&6(2a_3x-\sigma_2u)a^2\nonumber\\ && -\left\{\left[6a_0-w_{4+6}R -\frac{\mu_{3-2}}2 X\right]a+6w_{4-2}\dot\psi\right\}(\chi-\zeta) \nonumber \\ & &+\left\{\left[6b_0-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]a+6w_{4-2}\dot\chi\right\}\psi.\label{Echi}\end{aligned}$$ Reexpressed in terms of observables this becomes $$\begin{aligned} -\frac{\mu_{3-2}}4 \dot R+\frac{w_{2+3}}2\dot X&=&-\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]u\nonumber\\ &&+\left[12\tilde a_3+w_{4+6}R+\frac{\mu_{3-2}}2 X\right]x\nonumber\\ &&-w_{4-2}(2R-12[(H-u)^2-x^2+k a^{-2}])x. \label{dotXdotR} $$ Again the last line shows novel interaction terms compared to those seen in the earlier BHN model investigations. For the $a$ equation, however, the result has exactly the same form as that found in the earlier BHN model investigations [@HN; @AdP] since, by (\[quadcurv\]), the $w_{4-2}$ terms are independent of the scale factor $a$, and so make no contribution: $$\begin{aligned} \frac{d}{dt}\frac{\partial L_\mathrm{PG}}{\partial \dot a}&=&\frac{d}{dt}\left(-a^{2}3[a_2u+2\sigma_2x]\right)=\frac{\partial L_\mathrm{PG}}{\partial a} +\frac{\partial L_\mathrm{int}}{\partial a}\nonumber\\ &=&3a^{-1}L-\left(\frac{a_0}2-\frac{w_{4+6}}{12}R-\frac{\mu_{3-2}}{24}X\right)[a^2R+6(\psi^2-[\chi-\zeta]^2+\zeta^2)]\nonumber\\ &&-\left(\frac{b_0}2+\frac{w_{2+3}}{12}X-\frac{\mu_{3-2}}{24}R\right)[a^2X+12\psi(\chi-\zeta)]\nonumber\\ &&+3a^2(a_2u+2\sigma_2x)u-6a^2[2a_3x-\sigma_2u]x+3pa^2.\end{aligned}$$ Reexpressed in terms of observables this takes the form $$\begin{aligned} -3(a_2\dot u+2\sigma_2 \dot x) &=& a_0R+b_0X-3\Lambda+3p+6H[a_2u+2\sigma_2x]\nonumber\\ &&+\frac3{2}(-a_2u^2+4a_3x^2-4\sigma_2ux)\nonumber\\ &&+\frac{a_0}2\left[-6(H-u)^2+6x^2-6ka^{-2}\right] -6b_0x(H-u)\nonumber \\ &&-\frac{w_{4+6}}{24}\{R^2-12R[(H-u)^2-x^2+ka^{-2}]\}\nonumber\\ &&-\frac{\mu_{3-2}}{24}\{RX-12R(H-u)x-6X[(H-u)^2-x^2+ka^{-2}]\}\nonumber\\ &&+\frac{w_{2+3}}{24}\{X^2-24X(H-u)x\}.\label{Ea} $$ Two further useful relations can be obtained by taking the time derivatives of the torsion (\[fchi\]) and using the curvature definitions (\[R\],\[X\]): $$\begin{aligned} \qquad\dot x&=&\frac{X}6-Hx-2x(H-u),\label{xdot}\\ \dot H -\dot u&=&\frac{R}6-H(H-u)-[(H-u)^2-x^2+ka^{-2}]\label{Hudot}.\end{aligned}$$ Rearranging (\[Ea\]) using (\[xdot\]) yields $$\begin{aligned} -3a_2\dot u &=& a_0\{R-3[(H-u)^2-x^2+ka^{-2}]\}-3\Lambda+3p\nonumber\\ &&+\tilde\sigma_2[X-6(H-u)x] +\frac3{2}(-a_2u^2+4a_3x^2)+6a_2Hu\nonumber\\ &&-\frac1{24}w_{4+6}\{R^2-12R((H-u)^2-x^2 +ka^{-2})\}\nonumber\\ &&-\frac1{24}\mu_{3-2}\{RX-12R(H-u)x-6X[(H-u)^2-x^2+ka^{-2}]\}\nonumber\\ &&+\frac1{24}w_{2+3}[X^2-24X(H-u)x]. \label{udot3p}\end{aligned}$$ Using the expression for the material energy density gives a more compact alternative: $$\begin{aligned} 3a_2\dot u &=& -a_0R-\tilde\sigma_2X+\rho-3p+4\Lambda\nonumber\\ &&+3a_2(u^2-3Hu)-12a_3x^2. \label{udotrho3p} $$ Alternative derivations ----------------------- The system of dynamical equations can also be obtained from simpler Lagrangians that can be found using the topological identities (\[euler\],\[pointryagin\]). These alternatives serve as good checks of the calculation. One “minimal” Lagrangian has the previously studied BHN form plus the $w_4$ term. This additional quadratic term can be rewritten as follows: $$\begin{aligned} a^3\tilde R^2&\equiv& a^3(\tilde R^2-R^2)+a^3R^2\nonumber\\ & \equiv& -4\cdot6^2\dot\psi[\psi^2-(\chi-\zeta)^2+\zeta^2]+a^3 R^2\nonumber\\ &\equiv& a^3R^2 +4\cdot 6^2 \dot\psi(\chi-\zeta)^2 + \hbox{ total time derivative}.\end{aligned}$$ This leads to “extra” terms in the $\psi$ and $\chi$ equations which are equivalent to those found above. Furthermore, one could instead use the Euler identity to eliminate the $w_4$ term. This leads to another “minimal” version with the effective Lagrangian being BHN plus the $w_2$ term. The additional quadratic curvature term can be written as $$\begin{aligned} a^3\tilde X^2&\equiv& a^3X^2+a^3(\tilde X^2-X^2)\nonumber\\ &\equiv& a^3X^2 -8\cdot6^2 \dot\chi\psi(\chi-\zeta).\end{aligned}$$ This leads to the same “extra” terms in the $\psi$ and $\chi$ equations as found above. First order equations ===================== The above equations (\[Epsi\]), (\[Echi\]), and (\[Ea\]) are 3 [*second order*]{} equations for the *gauge potentials* $\psi,\chi,a$. However they can be used to construct a set of 6 [*first order*]{} equations for the *observable* quantities: namely (\[dotRdotX\]), (\[dotXdotR\]), (\[xdot\]), (\[Hudot\]), and (\[udot3p\]) or (\[udotrho3p\]) along with the Hubble relation. For convenience we repeat them together here: $$\begin{aligned} -\frac{w_{4+6}}2\dot R-\frac{\mu_{3-2}}4 \dot X&=&-\left[-3 \tilde a_2-w_{4+6}R-\frac{\mu_{3-2}}2 X\right]u\nonumber\\ && +\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]x\nonumber\\ &&+w_{4-2}[2X-24(H-u)x]x, \label{dotRdotX'} \\ -\frac{\mu_{3-2}}4 \dot R+\frac{w_{2+3}}2\dot X&=&-\left[6\tilde\sigma_2-\frac{\mu_{3-2}}2 R+w_{2+3}X\right]u\nonumber\\ &&+\left[12\tilde a_3+w_{4+6}R+\frac{\mu_{3-2}}2 X\right]x\nonumber\\ &&-w_{4-2}(2R-12[(H-u)^2-x^2+k a^{-2}])x, \label{dotXdotR'} $$ $$\begin{aligned} \dot a&=&aH, \label{hubble'}\\ \qquad\dot x&=&\frac{X}6-3Hx+2xu,\label{xdot'}\\ \dot H -\dot u&=&\frac{R}6-2H^2+3Hu-u^2+x^2-ka^{-2},\label{Hudot'} \\ a_2\dot u &=&\frac13( -a_0R-\tilde\sigma_2X+\rho-3p+4\Lambda)\nonumber\\ &&+a_2(u^2-3Hu)-4a_3x^2. \label{udotrho3p'}\end{aligned}$$ The associated material energy density is $$\begin{aligned} \rho &=&-\Lambda+3a_0[(H-u)^2-x^2+ka^{-2}]\nonumber\\&&-\frac32 a_2(u^2-2Hu) +6 a_3x^2+6\tilde\sigma_2x(H-u)\nonumber\\ &&\qquad+\frac{w_{4+6}}{24}\left[R^2-12R\left\{(H-u)^2-x^2+ka^{-2}\right\}\right] \nonumber\\ &&\qquad+\frac{\mu_{3-2}}{24}\left[RX-6X\left\{(H-u)^2-x^2+ka^{-2}\right\}-12Rx(H-u)\right]\nonumber\\ &&\qquad -\frac{w_{2+3}}{24}\left[X^2-24Xx(H-u)\right]\label{density}.\end{aligned}$$ The two dynamical equations (\[dotRdotX’\],\[dotXdotR’\]) can be resolved for $\dot R$ and $\dot X$ by inverting the symmetric matrix $$\frac14\left( \begin{array}{cc} -2w_{4+6} & -\mu_{3-2} \\ -\mu_{3-2} & 2w_{2+3} \\ \end{array} \right). \label{kineticT}$$ The qualitatively “new” terms appear only in the second lines of these 2 relations. These “new” terms exclude a special class of solutions with certain constant values for the scalar curvatures $R$ and $X$. When $w_{4-2}$ vanishes we are back at the BHN case investigated earlier [@BHN; @HN; @AdP]. That model, with one further parameter restriction, admits a certain class of constant curvature solutions [@HN15]. It should be noted that we have written our equations in a form in which they are valid for all ranges of the parameters, including both the “physical” and “unphysical” ranges, as well as degenerate cases. Generically there are 3 degrees of freedom, but in certain degenerate cases there are less, such as when $a_2$ vanishes or when the matrix (\[kineticT\]) has less than maximal rank. Then some of the above equations change their character. They may become constraint equations rather than dynamical equations. A particularly interesting subclass is the generalized Einstein-Cartan system, wherein all the quadratic curvature parameters vanish; this includes GR as a special case. An analysis of that subclass is given in Ref. [@HN15]. From here on we confine our attention the *generic* case, with no special degeneracies. Linearized equations and late time normal modes {#linana} =============================================== The fluid pressure $p$, the cosmological constant $\Lambda$, and the spatial curvature parameter $k$ play essentially the same roles here in the gqPG as they play in the GR FLRW cosmologies. Hence, in order to better see the novel features of gqPG cosmology, we take them to vanish in our further considerations. Following the procedure used in [@JCAP09; @AdP], [*for vanishing $\Lambda$ and $k$*]{}, by dropping higher than linear order terms in $H$, $u$, $x$, $R$, $X$ in the dynamical equations, our model can be linearized. (It should be noted that under these assumptions $\rho$ and hence $p$ are non-linear.) This leads to the first order [*linearized*]{} versions of the dynamical equations. Using this procedure for the BHN model one zero frequency and two dynamical normal modes were identified. Furthermore, it was argued that the dynamical variables would have a fall off of $a^{-3/2}$. Hence the dynamical variables rescaled by $a^{3/2}$ would at late time just approach the linearized modes. This behavior was confirmed by numerical calculations [@AdP]. A comparison of the gqPG equations considered here with those of the BHN model considered in [@AdP] shows that the new terms that appear in the general system, those with the coefficient $w_{4-2}$, are all *non-linear*. Hence the *linearization* of the gqPG is essentially identical with that of the BHN model, and thus the gqPG cosmology has essentially the same linearized normal modes and frequencies as were found earlier for the BHN model. Consequently the “late time” normal mode analysis for the $\Lambda=0=k$ case (as discussed in [@AdP]) applies to general quadratic PG cosmology virtually unchanged. That the gqPG really does show essentially these same qualities has been demonstrated in our numerical analysis described in the next section. We note that the zero frequency mode for the gqPG is $$z:= a_0(H-u)+\frac{{a}_2}{2}u+\tilde{\sigma}_2x,\label{zero}$$ (from Eqs. (\[dotRdotX’\])–(\[udotrho3p’\]) it can be verified that this is indeed constant to linear order) while the details of the normal modes and frequencies are, as mentioned, essentially identical with those presented in [@AdP]. numerical ========= \[fig1\] ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- For the BHN model the effects of the PG pseudoscalar parameters $\mu_{3},\ \tilde\sigma_2$ that algebraically cross couple the spin $0^+$ and $0^-$ modes had been investigated earlier [@HN]. The linearized equations and the associated normal modes and their frequencies were found [@AdP]. It was noted that (with vanishing $p,\ \Lambda,\ k$) from the conserved energy condition $a^3\rho=$const in an expanding universe one could infer that the observables $H,\ u,\ x,\ R,\ X$ rescaled by $a^{3/2}$ would at late time satisfy the linearized equations. This was supported by numerical evolution that indeed showed the expected damping and the various observables as superpositions of two normal modes and one zero frequency mode. Likewise for the gqPG FLRW cosmology, the parameters $\tilde\sigma_2,\ \mu_{3-2}$ will play the same kind of role, and again (with vanishing $p,\ \Lambda,\ k$) from the conserved energy condition $a^3\rho=$const one can infer that the observables rescaled by $a^{3/2}$ should at late time behave like the linearized equation solutions. To verify this expectation and to see the effects of the new parameter $w_{4-2}$ we follow the numerical analysis technique used previously [@SNY08; @JCAP09; @HN; @AdP]. Based on certain sets of well-tested appropriate parameters, typical effects of the new parameter $w_{4-2}$ can be seen from the parameter choices $$\begin{aligned} &&a_0=1; \quad a_{2}=-0.85;\quad a_{3}=0.35;\quad w_{4+6}=-1.3; \quad w_{2+3}=0.6; \nonumber\\ &&\tilde\sigma_2=0;\quad \mu_{3-2}=0; \quad w_{4-2}=1,\ 0.5,\ 0,\ -0.5,\ -1.\end{aligned}$$ We have chosen the cross parity pseudoscalar coupling parameters (whose effects were explored in [@HN]) to vanish here in order to more clearly see the effects of the “new” cross parity coupling $w_{4-2}$. The initial values we used are $$\begin{aligned} &&a(t_0)=50,\quad H(t_0)=1,\quad u(t_0)=0.335,\quad x(t_0)=0.3, \nonumber\\ &&R(t_0)=2.18, \quad X(t_0)=2.0,\end{aligned}$$ where $t_0=1$, the present time of universe. Fig. 1 displays typical evolutions for the observables $a,\ H,\ u,\ x,\ R,\ X,\ \rho,\ \ddot a$, showing how the behavior depends on the “new” parameter $w_{4-2}$. For the same parameter values and initial conditions the rescaled variables $a^{3/2}H$, $a^{1/2}\textrm{d}^2a/\textrm{d}t^2$, $a^{3/2}R$, $a^{3/2}X$, $a^{3/2}u$, $a^{3/2}x$, $a^{3/2}z$, $a^3\rho$ are plotted in Fig. 2. From Fig. 1 one can see the typical damped oscillation behavior for $H,\ u,\ x,\ R,\ X$ that is obtained with generic parameter values in the physical range. Note that the Hubble parameter $H$ is decreasing on average, but in repeating cycles is sometimes increasing. This is reflected in the acceleration $\ddot a$. On average the expansion is slowing down but has repeating cycles of acceleration. From Fig 2 one can readily see that the amplitude for the oscillating observables rescaled by $a^{3/2}$ indeed is tending to a constant at late time. The nearly constant value of $a^3\rho$ is a good indication of the numerical stability accuracy. The plot of $a^{3/2}z$ is a good indicator that, for this set of parameters and initial values, before $t=10$ the linearized approximation is not so good but it is not bad after $t=20$. We note a rather dramatic change in these waveforms when $w_{4-2}$ is changed from 1 to 0.5 to 0, but a relatively smoother change in the range 0 to $-0.5$ to $-1$. As expected, for rescaled $H,\ \ddot a,\ R,\ X,\ u,\ x$ the waveforms are approximately harmonic, with nearly uniform amplitude and wavelength. This is especially the case for $w_{4-2}=0$ (which is equivalent to the BHN model) except at the very earliest times. For nonvanishing values of the parameter $w_{4-2}$ these observables are not very harmonic at early times, such as $t<10$. But at later times, like $t>15$, they are quite harmonic. Thus, as expected, the parameter $w_{4-2}$ can have a large effect at early time, but it does not have much effect at later time. Whatever wave pattern it produced is at late time preserved. These features are just what we expected from an examination of the $w_{4-2}$ terms in the dynamical equations. From (\[dotRdotX\]) and (\[dotXdotR\]) it is clear that the $w_{4-2}$ terms are nonlinear, and in an expanding universe they will at late time have no effect. The $w_{4-2}$ terms couple the two dynamical modes of even and odd parity in a quite different fashion than that associated with the parameters $\mu_{3-2},\ \tilde\sigma_2$. The latter are *pseudoscalar* parameters that produce a linear coupling of the modes. But $w_{4-2}$ is a *scalar* parameter controlling a non-linear coupling of a kind of magnetic type, as revealed by the $w_{4-2}$ terms in Eqs. (\[Epsi\]) and (\[Echi\]) which have the form of a product of a gauge variable with the time derivative of a gauge variable. \[fig2\] -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} {width="50.00000%"}{width="50.00000%"} -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- Discussion ========== Here we have considered FLRW cosmological models for the general quadratic PG, including all the possible even and odd parity terms with their respective scalar and pseudoscalar parameters. For the general PG all six connection modes are in principle dynamic—however the cosmological symmetries essentially only allow *scalar* dynamic connection modes. Generically the general quadratic PG cosmology has, in addition to the usual metric scale factor, just two dynamical connection modes with spin-$0$: $0^+$ and $0^{-}$, effectively a scalar and a pseudoscalar. In addition to the two of GR ($\Lambda, a_0$), the gqPG FLRW cosmology has 8 more physical parameters ($ a_2,\ a_3,\ \tilde \sigma_2,\ w_{3-2},\ w_{4+6},\ \mu_{3-2},\ w_{4-2}$). We used a manifestly homogeneous and isotropic Bianchi representation to develop an effective Lagrangian, an energy expression, and the 2nd order dynamical equations for the 3 gauge potentials. An equivalent first order system of 6 equations for the observables was obtained. Certain inequalities were noted that identify a preferred physical range of parameters. The late time dynamical behavior for generic solutions within the physical parameter range was discussed and illustrated using numerical evolution. It turns out that the gqPG has effectively just one more free parameter ($w_{4-2}$) beyond what the earlier works [@BHN; @HN; @AdP] considered; typical dynamical effects of this parameter were discussed and illustrated via numerical evolution. The two dynamic connection modes do not directly couple to any known matter. Hence they could be relatively large and unnoticed. The $0^+$ scalar mode couples directly to the Hubble expansion rate. Torsion could play a role in the observable universe. It might help in the understanding of some of the present puzzles associated with inflation, dark matter, and dark energy. These topics should be explored within the general quadratic PG cosmology presented here. 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--- abstract: 'Quantum communication is rapidly gaining popularity due to its high security and technological maturity. However, most implementations are limited to just two communicating parties (users). Quantum communication networks aim to connect a multitude of users. Here we present a fully connected quantum communication network on a city wide scale without active switching or trusted nodes. We demonstrate simultaneous and secure connections between all 28 pairings of 8 users. Our novel network topology is easily scalable to many users, allows traffic management features and minimises the infrastructure as well as the user hardware needed.' author: - Siddarth Koduru Joshi - Djeylan Aktas - Sören Wengerowsky - Martin Lončarić - Sebastian Philipp Neumann - Bo Liu - Thomas Scheidl - Željko Samec - Laurent Kling - '[Alex Qiu]{}' - Mario Stipčević - 'John G. Rarity' - Rupert Ursin title: ' A trusted-node-free eight-user metropolitan quantum communication network ' --- [^1] Quantum communication networks present a revolutionary step in the field of quantum communication [@scarani2009security; @pirandola]. Despite real world demonstrations of Quantum Key Distribution (QKD) [@wengerowsky2018field; @hipp2016demonstration; @wang2015experimental; @patel2012coexistence; @qiu2014chinalink], the difficulty of scaling the standard two-user QKD protocols to many users has prevented the large scale adoption of quantum communication. Thus far, quantum networks relied upon one or more problematic features: trusted nodes [@Peev2009; @Sasaki2011; @Stucki2011; @Xu2009; @FieldQKD_Wang14] that are a potential security risk, active switching  [@Toliver2003; @Chen:10-chinese-access-network; @Elliott2005; @Chang2016] which restricts both functionality and connectivity and most recently wavelength multiplexing [@Soeren2018] with limited scalability. The ultimate goal of quantum communication research is to enable widespread connectivity, much like the current internet, with security based on the laws of physics rather than computational difficulty. To achieve this, a quantum network must be scalable, allow users with dissimilar hardware, be compatible with traffic management techniques, must not limit permitted network topologies and as far as possible avoid potential security risks like trusted nodes. So far, all demonstrated QKD networks fall in three broad categories. First, trusted node networks [@Peev2009; @Sasaki2011; @Stucki2011; @Xu2009] where some or all nodes in a network are assumed to be safe [from eavesdropping]{}. In most practical networks, it is rare to be able to trust every connected user. Furthermore, such networks tend to utilise multiple copies of both the sender and receiver hardware at each node, thereby increasing the cost prohibitively. Second, actively switched or “access networks” where only certain pairs of users are allowed to exchange a key at a time [@Herbauts13]. Similarly point to multipoint networks are useful in niche applications and have been shown using passive beamsplitters [@Townsend1997; @Choi2011; @Frohlich2013], active switches [@Toliver2003; @Chen:10-chinese-access-network; @Elliott2005; @Chang2016], and frequency multiplexing [@Aktas2016; @Chang2016; @Zhu15; @lim2008]. Lastly, fully connected quantum networks which can be based on high-dimensional/multi-partite entanglement to share entanglement resources between several users [@Torma1999; @pivoluska2017layered]. However, the extreme complexity of changing the dimensionality of the state produced by the source makes this approach unscalable. Fortunately, fully connected networks (i.e. where every user is connected to every other user directly) can be achieved using multiplexing and bipartite entanglement [@Soeren2018]. Nevertheless, the scheme in Ref [@Soeren2018] requires $\bigO(n^2)$ wavelength channels for $n$ users, which prevents the technology from being scaled to more than a few users. Here we present a city-wide quantum communication network, with 8 users, that forms a fully connected graph/network (where each user exchanges a secure key with every other user simultaneously) while requiring only 8 wavelength channel pairs, minimal user hardware (i.e., 2 detectors and a polarisation analysis module), and no trusted nodes. The quadratic improvement in resources (the number of channels) used is due to passively multiplexing using both wavelength filters and beamsplitters (BS). Further, to the best of our knowledge, we have demonstrated the largest quantum network without trusted nodes to date. Similar to Ref [@Soeren2018], just one source of polarisation entangled photon pairs is shared passively between all users and requires neither trust in the service provider nor adaptations to add or remove users. We performed a full QKD experiment on a city wide scale in deployed fibres with a mixture of Superconducting Nanowire Single Photon Detectors (SNSPDs) and a Single Photon Avalanche Diode (SPAD). Further, we demonstrate a new topology with $\bigO(n)$ scaling of all resources consumed using only 16 wavelength and 2 BS channels to distribute 8 entangled states, among the 28 links, between 8 users in a fully connected graph using only one fibre and polarisation analysis module per user. {width="80.00000%"} By using a combination of standard telecom Dense Wavelength Division Multiplexers (DWDM) with 100GHz channel spacing and BS multiplexing using in-fibre beamsplitters, we were able to distribute bi-partite entangled states between all users from just one source of polarisation entangled photon pairs. The network architecture requires only 16 wavelength channels to fully interconnect 8 users as opposed to the 56 channels that would be necessary following our earlier scheme [@Soeren2018]. The network architecture is best understood when divided into different layers of abstraction as shown in Fig. \[fig:layers\]. The bottom “physical layer” represents the actual infrastructure that supports the network and comprises of a central Quantum Network Service Provider (QNSP) and the user hardware connected to the QNSP via distribution fibres. In the physical layer, the network topology requires only one fibre between each user and the service provider, while in the logical/connection layer the topology naturally forms a fully connected graph between all 28 unique pairs formed by 8 users (see Fig \[fig:layers\]). Every user is equipped with a polarisation analysis module that implements a passive basis choice using two single photon detectors each as shown in Fig \[fig:source+wdms\]. Users can demultiplex the incident photons such that each detector receives fewer wavelength channels to improve their signal-to-noise ratio and therefore the key rate. We generate secure keys between all 28 links formed by pairs of users. Four of these links can be chosen to have *premium* connections with increased key rates, which when combined with active switching can provide traffic management on the network. Lastly, we demonstrate that our network is capable of supporting a mixture of both SNSPD and SPAD based user platforms. {width="95.00000%"} The details of the QNSP are shown to the left of Fig. \[fig:source+wdms\]. It consists of both the source of polarisation entangled photon pairs and the Multiplexing Unit (MU) comprising WDMs and beamsplitters. All multiplexing is performed in a single MU, co-located with the source in our implementation. To take advantage of existing fibre infrastructure, channels for many users can be sent along fewer fibres and multiple MUs, at various locations closer to clusters of users, can be used to create this quantum network. The user hardware – a Polarization Analysis Module (PAM) and two single photon detectors – is shown in Fig. \[fig:source+wdms\] inset b). Photons incident on a PAM are directed by a beamsplitter along either the short path where they are measured in the horizontal/vertical polarisation (HV) basis, or along the long path and through a half-wave plate such that they are measured in the diagonal/anti-diagonal polarisation (DA) basis. The overall result is that the physical layer constitutes a relatively simple hub and spoke topology, while in the logical layer, every pair of users always share an entangled photon pair. We conceptually divide the 8 users of our network referred to as Alice (A), Bob (B), Chloe (C), Dave (D), Feng (F), Gopi (G), Heidi (H) and Ivan (I), into two sub-nets of 4 users (see Fig \[fig:layers\]). A sub-net uses wavelength multiplexing to form a fully connected network among its members – A,B,C,D. Beamsplitters are then used in each of the wavelength channels to duplicate the first sub-group creating a second set of 4 interconnected users – F,G,H,I. Thus entanglement is shared between every pair of users except AF, BG, CH and DI as the above splitting also gives rise to connections between the two sets. Two additional wavelength pairs are then distributed between these pairs of users to create the fully connected network of 8 users with only 16 wavelength channels. Each pair of users performs a standard BBM92 [@BBM92] protocol where the photons shared with all other users are treated as background noise. A narrow coincidence window, optimised in post processing and typically about 130ps, ensures that this noise only contributes minimally to the Quantum Bit Error Rate (QBER). ![**Temporal cross correlation histograms**: Each pair of users identify photon pairs by their arrival time using $g^{(2)}$ histograms. The data shown here was collected over one hour for user Gopi (G) during the laboratory demonstration of the network. Users G and A share two sets of correlated wavelengths to enable higher key rates therefore they share two sets of $g^{(2)}$ peaks (GA$_1$ and GA$_2$). Information as to which detector(s) clicked was obscured by all users. Extended Data Figure \[fig:Histgb\] shows the histograms between each pair of detectors for the users G and B however, this more detailed graph contains information about the measurement outcome and cannot be used to generate a key. []{data-label="fig:g2"}](Fig03_Histo.pdf){width="45.00000%"} All multiplexing and demultiplexing in the experiment is performed with standard telecommunications equipment. The experimental setup (shown in Fig. \[fig:source+wdms\]) uses a broadband source of polarisation entangled photon pairs at telecommunications wavelengths similar to that described in Ref [@Soeren2018]. Comparable sources have also been reported in Refs. [@lim2008; @Herbauts13; @Aktas2016; @Zhu:19; @Chen:18] where a $\sim 775\,$nm pump downconverts in a Type 0 MgO:PPLN crystal to produce signal and idler photons with a Full Width at Half Maximum (FWHM) bandwidth of $\sim$[60]{}nm centred at [1550.217 nm]{} (roughly corresponding to the International Telecommunication Union’s (ITU) 100GHz DWDM grid at channel 34 (Ch34) centred at 1550.12nm). Due to energy conservation during the down conversion process, only frequencies equidistant from half the pump frequency can support entangled pairs. Thus, wavelengths corresponding to the channel pairs {Ch33,Ch35}, {Ch32,Ch36}, {Ch31,Ch37}, and so on are entangled with each other. The wide signal and idler spectrum [was]{} demultiplexed into 8 wavelength pairs as above and each user is given a combination of wavelengths according to the table in Fig \[fig:layers\]. Thus each user receives 4 [wavelength]{} multiplexed channels simultaneously via one single mode fibre and 32 different entangled states are shared between the 28 different pairs of users. {width="90.00000%"} The experiment was performed in two stages. In the first stage the QNSP, MU, the 8 users each connected to the QNSP/MU with a single fibre $\sim$10m in length, and the 16 detectors were situated in a single laboratory in the Nano Science and Quantum Information (NSQI) building in Bristol. To demonstrate the stability of our network we recorded data for [18.4]{}hours as shown in Fig \[fig:stability\]. To be able to account for finite key effects with a security parameter of [$10^{-5}$]{}, we computed the private key once every 10 minutes and the figure shows the average secure key generation rate per second in each 10min period for each of the 28 links (see section \[sec:methods\]). The total secure key obtained is shown in extended data Table \[tab:short\]. Users A through H used superconducting nanowire detectors from Photon Spot while Ivan used a combination of one SNSPD and one InGaAs Single Photon Avalanche photoDiode (SPAD). We note that the use of heterogeneous detectors did not significantly impact the key generation rates. Alice Bob Chloe Dave Feng Gopi Heidi Ivan ------- ------- ------- ------- ------- ------- ------- ------- ------- Alice 31621 9102 6081 15529 39054 7159 979 Bob 23798 24208 83645 42062 8893 10400 Chloe 16480 14931 18735 9495 13155 Dave 42 17170 12544 3999 Feng 19925 10470 593 Gopi 10448 3953 Heidi 1700 : **Total secure key (bits) over long distance links for the city wide metropolitan network demonstration.** We connected 4 locations/users across the city of Bristol as shown in Fig \[fig:source+wdms\] via deployed fibre in a loop back configuration. Two other users were sent signals through fibre spools and the remaining two were connected via short (10m) fibres. The distances of all 28 links are given in Extended Data Table \[tab:length\]. Considering finite-size-effects, we measured for $\sim27$ minutes to obtain the final secure key shown. Here, we set the failure probability of phase error estimation to $10^{-5}$. Extended Data Figure \[fig:Keystability\] shows the overall stability of the key for 7hours. \[tab:long\] In the second stage the connection between the user and the MU was replaced for 6 users by long distance links. Furthermore, the SPAD was exchanged between Ivan and Gopi. Alice was connected via a 12.6km spool with a loss of 13.3dB, Chloe was connected to a loop-back from a laboratory in the first floor of the physics building of the University of Bristol with a total distance of 463m and 1.36dB loss, Dave used a 4.34km spool (15.7dB), Feng looped back from the basement of the Merchant Ventureres Building (MVB) through 1.625km of deployed fibre (and a loss of 2.04dB), Heidi utilised a loop-back connection from the ground floor of Queen’s Building with a loss of 1.68dB and a total distance of 1.624km, Ivan was connected to another loop-back from the server room of One Cathedral Square in the city centre totalling 3.10km (2.57dB). Bob and Gopi continued to be connected via short fibres. Thus, the 28 links varied in the effective separation of users from 16.6km to $\sim$10m. This shows the versatility of the network architecture as both a local area network and a city wide metropolitan area network. Tab. \[tab:long\] shows the secure key rate over these long distances in deployed fibre and in fibre spools. The QBER, and hence the secure key rate, in our proof of principle experiment was limited by two main experimental imperfections: Firstly, a more careful fibre neutralisation procedure using the manual Fibre Polarisation Controllers would significantly improve the QBER. Secondly, the alignment of the HWP and extinction ratio of the Polarising BeamSplitter (PBS) used in each user’s PAM could be further optimised (see supplementary information Sec \[key\]). Further optimisation of the secure key rate is possible by adjusting the pump power in the source thus, changing the pair generation rate (see Extended Data Figure \[fig:pow\]). We cannot adjust the pair generation rate in each pair of wavelength channels separately. Thus the optimum pump power is strongly effected by the different types/alignment of user hardware (like detectors and PAMs) in the network. An overview of the operational parameters of the detectors used are found in Sec \[sec:methods\]. Nevertheless, the measured QBER proved to be stable in an 18hour laboratory test (see Extended Data Figure \[fig:qberstability\]) and resulted in a stable and positive overall secure key rate in a 7 hour metropolitan quantum communication network demonstration (see Extended Data Figure \[fig:Keystability\]). Using a low-cost design for the polarisation analysis module at each node, any pair of communicating users (say Alice and Bob) obtain three peaks in their temporal cross-correlation histogram $g^{(2)}$ between each detector of A and B for each correlated pair of wavelengths they share (see Fig. \[fig:g2\] and Extended Data Figure \[fig:Histgb\]). The central peak corresponds to all measurements where A and B chose the same measurement basis, while the side peaks correspond to A and B choosing different measurement bases[^2]. We note that each user only shares the time of a detection event and not which detector clicked as required by the protocol. Since the two detectors used by a user can have different delays or jitters, every user must characterise their setup and modify the time tags before they are shared. We have successfully realised a complete entanglement based quantum communication network with improved scaling, traffic management, and long distance links via deployed fibre throughout the city. We have shown the effectiveness of a new and improved network architecture. Our fully connected scheme can be modified at the software or hardware level to create any desired sub-graph. Further, by multiplexing states intended for several users into a single fibre and demultiplexing them later on, our architecture can easily support any desired complex network (see Supplementary information section \[use\]). [As the number of users increases, the QNSP can choose to use additional wavelength channels, which (up to a limit based on noise counts as discussed in the supplementary information) minimally impacts the key rates of all existing users. This detrimental impact can be completely negated by users selectively detecting only the desired wavelengths. Alternatively, to increase the number of users, the QNSP can use additional beamsplitters which would reduce key rates but drastically increase the number of users on the network with the fewest additional wavelength channels. However, this would irrecoverably affect the key rates of all users. In our network, the physical topology grows linearly with each additional user requiring only one additional PAM and fibre. Photons intended for multiple users can also be multiplexed through the same fibre to optimise cost/convenience of distribution. We note that the network is also capable of producing all possible sub-graphs, adding or removing users and changing the allocation of premium connections without altering the source of entanglement.]{} Detectors are a significant resource for individual users therefore, we significantly lower the financial cost per user by sending several channels onto the same detector at the expense of a slight increase in QBER. However, this can be mitigated by demultiplexing the signal on each detector to multiple detectors. The all passive trusted-node-free implementation could allow us to use active switching to incorporate additional functionality such as traffic and bandwidth management, Software Defined Networking, etc. The [$\sim$ 17km, or more,]{} range of the network, as we have demonstrated, is more than enough to create Local and Metropolitan Area Networks interconnecting end users throughout a city or building. This range can be further extended by repeaters, reduced detector jitter, wavelength demultiplexing to several detectors and wavelength selective switching to the same detector. The number of users that can be connected to a given network is limited by available resource, losses, and marginal increase in error rate with each simultaneous connection established (with a given detector). The error rate is theoretically limited by the probability that uncorrelated photon detection events can accidentally occur in any given coincidence time window. In terms of resource, the network scales linearly with respect to user hardware and [number of deployed fibres.]{} On the service provider’s side, the number of wavelength channels can be increased drastically by using closer-spaced narrower-band WDMs and by generating broader band downconversion. For example, periodically poled fibres are a very promising method of the latter [@Chen:18]. Using polarisation preserving methods of multiplexing, such as an on chip design, would eliminate the need for most Fibre Polarisation Controllers (FPCs). We were able to demonstrate this by connecting several extra kilometres of fibre to many users and maximising the key rate using only the FPCs on each user’s module instead of the FPCs of the service provider. Further, simulations show that the network topology can be extended to 32 (49) users divided into 2 (7) subnets while maintaining a reasonable secure key rate (see Extended Data Figure \[fig:numnodes\] in supplementary information). The long term goal of a full fledged quantum internet requires quantum communication networks that support other forms of quantum information processing or other quantum technologies. The vast majority of such applications rely on the distribution of entanglement between several nodes making the current architecture an ideal candidate for further study. Acknowledgements {#acknowledgements .unnumbered} ---------------- The research leading to this work has received funding from the Engineering and Physical Science Research Council (EPSRC) Quantum Communications Hub EP/M013472/1 and equipment procured by the QuPIC project EP/N015126/1. We also acknowledge the Ministry of Science and Education (MSE) of Croatia, contract No. KK.01.1.1.01.0001. We acknowledge financial support from the Austrian Research Promotion Agency (FFG) project ASAP12-85 and project SatNetQ 854022. We gratefully acknowledge assistance from Ashley Bligh and Ian Stubbs of the University of Bristol networks team as well as Graham Marshall of QET labs, University of Bristol for provisioning parts of the fibre network used in this research. We thank Vitomir Stanišić of RBI-ZFK Workshop for his help designing and making the mechanical components for the user PAMs. We also thank Roland Blach of IQOQI for help making the FPCs and Abdul Waris Ziarkash for helpful discussions. Author Contributions {#author-contributions .unnumbered} -------------------- The Source was built by SN and SW, the nodes by ML, ZS, SKJ and SW and the remaining setup by DA and SKJ. The network topology/architecture was conceived by SKJ with help from SW and DA. Security analysis was performed by BL and TS. [Test, monitoring and secure key analysis]{} software was written by BL. The Project was supervised by RU, JR and MS who coordinated the 3 teams while the whole project was lead by SKJ. Simulations were performed by AQ and SKJ with help from SN and SW. LK provided logistic support and helped with the detection of single photons. ML, and ZS produced the low-cost optics and optomechanics. The experiment was performed by SKJ, DA, SW, SN, ML, LK, ZS, TS, LB. The paper was written by SKJ with the help of DA, SW and BL. All authors discussed the results and commented on the manuscript. Methods {#sec:methods} ======= Quantum Network Service Provider -------------------------------- The network consists of the Quantum Network Service Provider (QNSP), distribution fibres and users. The QNSP is comprised of a photon pair source set up to prepare bipartite polarisation entangled states and a Multiplexing Unit (MU). The source is pumped by diagonally polarised light $|D\rangle$, from a CW pump laser emitting at which passes through a dichroic mirror and a polarising BeamSplitter (PBS) that defines the input and output of a Sagnac loop (see Fig. \[fig:source+wdms\]) [@Kim2005; @lim2008; @fedrizzi2007wavelength]. The horizontally (vertically) polarised pump light thus propagates anti-clockwise (clockwise) inside the loop. A half-wave plate (HWP) after the transmission port of the PBS rotates the polarisation by to vertical. This allows for injecting light in both directions of a [5]{}cm-long Magnesium Oxide doped periodically poled Lithium Niobate (MgO:ppLN) bulk crystal with a poling period of 19.2$\mu$m in which vertically polarised pump photons are converted to vertically polarised signal and idler photon pairs through type-0 spontaneous parametric down-conversion, *i.e.* $|V\rangle \rightarrow |V_{\rm{\textit{s}}}\rangle|V_{\rm{\textit{i}}}\rangle$. The photon pair contribution in the clockwise direction is rotated by the HWP and therefore becomes $|H_s\rangle|H_i\rangle$. Both contributions are then coherently combined at the PBS and isolated from the pump light thanks to the dichroic mirror, thus creating a maximally polarisation-entangled state with two different wavelengths, $\lambda_1$ and $\lambda_2$, located symmetrically about the central wavelength at : $$\label{eq:state} |\Phi\rangle =\frac{1}{\sqrt{2}}( |V_{\lambda_1}V_{\lambda_2}\rangle + |H_{\lambda_1}H_{\lambda_2}\rangle)\,.$$ The MU consists of off-the-shelf dense wavelength division multiplexer filters (DWDM). In addition, the MU also has a set of 50:50 fibre beamsplitters. These components were spliced together to distribute photon pairs from the source to each of the 8 distribution fibres as shown in Figs \[fig:layers\] and \[fig:source+wdms\]. The spatial mode containing the signal and idler photons from the source was coupled into one single mode fibre and spectrally split by a thin-film DWDM (with a channel spacing and FWHM of ) into 32 channels as defined by the International Telecommunication Union (ITU) in G.694.1. We selected 16 of these channels symmetrically with respect to the degeneracy wavelength of , which corresponds to ITU channel 34 (see Fig. \[fig:source+wdms\] inset a) ). On the red side of the spectrum we used ITU frequency channels 26-33 and channels 35-42 on the blue side. Due to the well defined pump wavelength of the CW laser and energy conservation during down-conversion, we obtained polarisation entanglement between pairs of channels (26 & 42, 27 & 41, 28 & 40, and so on). Each of the 16 wavelength channels is then split by a beamsplitter, such that in total 32 possible pairs of correlated photons are available. Using further add-drop multiplexers before and after the beamsplitters, four channels were combined into each single-mode fibre to every user. Since the partner photons for each channel can be found in two other fibres, each of the users now holds eight polarisation-entangled connections to other users. This means, each user is connected to all the other users, featuring one doubled connection. Fibre Polarisation Controllers (FPCs) were used to ensure that the reference frame of polarisation in the source is (nearly) identical to that of the polarisation analysis module (PAM). It was not necessary to compensate all channels independently. Similar wavelengths were compensated together. At the end each user received 4 channels (see Figs \[fig:layers\] and \[fig:source+wdms\]) via a single distribution fibre and used a polarisation analysis module to measure in the HV or DA polarisation basis. The distribution fibres were all single mode for 1550nm but of varying lengths and specifications. Several of the fibres were deployed across the university and the city of Bristol. We conducted two experimental runs, the first with short distribution fibres and the second with varying link lengths as shown in Extended Data Table \[tab:length\]. Users ----- Each user in the network is equipped with the polarisation analysis module (PAM) and two single photon detectors. The PAM enables passive switching between photon polarisation measurements in two orthonormal bases (either HV or DA). A BeamSplitter (BS) at the PAM’s input randomly directs incoming photons either through the short optical path to the PBS and measurement in HV basis, or through the long optical path with an achromatic half-wave plate providing a $45\degree$ polarisation rotation and the same PBS for measurement in DA basis. The difference between the long and short free-space paths corresponds to 3.7ns of time delay between, resulting in polarisation analysis in different time bins[@Zhu:19]. We designed the PAM to be completely passive, compact, portable, simple and cheap to mass-produce and align, but still robust and stable. At each PAM’s input and outputs to two detectors, SMF28e single mode fibres are connected to collimators/couplers with custom produced 15.7 mm effective focal length (at 1550 nm) SF11 glass singlet lenses (AR coated for 1500-1600 nm), with x-y, tip/tilt and focus adjustment. Cube BS and PBS are mounted on kinematic platforms for rotation and tip/tilt adjustment. The achromatic HWP is mounted in a manual precision rotation mount. The long optical path was realised using unprotected gold mirrors on tip/tilt kinematic mounts. Commercially available BS, PBS, HWP and unprotected gold mirrors were used (Thorlabs BS012, PBS104, AHWP05M-1600 and PFSQ10-03-M03). BS characteristics are $T=(56 \pm 8)\%$, $R = (44 \pm 8)\%$ depending on input polarisation and orientation angle. The PBS in use have extinction ratios $T_{p}:T_{s} > 1000:1$, $R_{s}:R_{p}$ roughly between 20:1 to 100:1, transmission efficiency $T_{p} > 90\%$ and reflection efficiency $R_{s} > 99.5\%$. Achromatic HWP retardance accuracy is $<\lambda/300$. Complete production of lenses and optomechanics as well as assembly was done at the Ruer Bošković Institute, in the optical and mechanical workshops of the Division of Physical Chemistry. The PAM outputs are fibre coupled and launched into 2 Single photon detectors. We used 15 superconducting nanowire single–photon detectors (SNSPDs) from Photon Spot with detection efficiencies ranging from $\sim$70 to 90%, a jitter of between $\sim$80 to 60ps (including the measurement device) and dark counts of $\sim$1 kHz and 1 InGaAs avalanche single photon avalanche detector (SPAD) from ID Quantique, model ID230, which has 20% efficiency, $\sim$100 ps jitter and dark counts of $\sim$0.05 kHz. After detection the optical signal is converted into an electronic pulse and read out at a 18-channel time tagging module (Swabian instrument model Time Tagger Ultra). Using a laptop we performed an on-the-fly computation of coincidences, basis reconciliation and secure key rate estimation for all 28 QKD links. Secure key generation --------------------- Due to the design of PAMs and the Continuous Wave (CW) pump of the source, we only know whether the measurement basis choice (of a pair of users) was matched or not. Here the information of which detector clicked directly encodes the measurement outcome without revealing the measurement basis choice. Thus, the information, in which basis a detected photon was measured, is not available. Suppose the time delay between Alice’s HA detector to Bob’s HA detector is longer than the delay between Alice’s VD detector and Bob’s VD detector, then by looking at the $g^{(2)}$ histogram, Eve can identify two different delays, which means that Eve can guess what the measurement outcome was and thus what the key bit could be. Thus, all users must [merge the time tags of all detectors into a single data set without which-channel or which-detector information. They then exchange this data]{} to synchronise the time offset between different correlated detector pairs. Then, all users exchange their merged time-tags with each other via the authenticated public channel(s). After the time-tags are shared among all users, they calculate a temporal cross-correlation histogram ($g^{(2)}$) to find the coincidences. As the channel information is kept private, users can assign “0” or “1” to the measurement outcomes where they both detected a photon within the coincidence window and happened to measure in the same basis. Eve could learn the position of matched pulses, but doesn’t know the value of the matched key bit due to lack of channel information. After obtaining the sifted keys, all users perform the error correction and privacy amplification procedures to extract the final secure key. For each pair of users, the final secure key length $n_f$ can be calculated by $$\label{equ:finalkeylength} n_f \geq n_s\left[1-H_2\left(e_p^U\right) - f\left(e_b\right)H_2\left(e_b\right)\right]\,,$$ where $n_s$ is the sifted key length, $e_b$ is the measured quantum bit error rate, $e_p^U$ is the estimated upper-bound of phase error rate, $H_2\left(x\right)$ is the binary Shannon entropy and $f\left(x\right)$ is the error correction efficiency. Since we could not divide our sifted key into two individual bases ($Z$ and $X$), here we analyse the phase error rate in the mixed basis case. Given failure probability $\xi_{ph}$, the upper-bound of phase error rate can be estimated by $$\label{equ:phaseErrRate} e_p^U= \alpha e_b + \left(1+\alpha\right)\sqrt{\frac{\ln4 -\ln {\xi_{ph}}}{n_s}}\,,$$ where $\alpha \geq 1$, and the phase error probability is $\alpha$ times bit error probability. In our experiment, we used passive measurement modules with 50:50 beamsplitters to perform unbiased measurement basis choices, which results in $\alpha=1$ [@Fung2010]. We believe that similar arguments can be made to show security even in the case of bias in the measurement basis choice. From the experimental data, we can infer this bias under the assumption that the fibre-coupling for both detectors of the PAM is equal. Assume that the basis choice bias of Alice and Bob is $p_A$ and $p_B$, then the coincidence counts of the left histogram peak between a pair of users (as seen in the histograms of Extended Data Figure \[fig:Histgb\]) is related to $p_A(1-p_B)$, the coincidence counts of the middle peak are related to $p_Ap_B + (1-p_A)(1-p_B)$, the coincidence count of the right peak is related to $(1-p_A)p_B$. Therefore, when the basis choice is biased, one could first measure $p_A$ and $p_B$ by monitoring the coincidence counts in the left, middle and right peaks. Then, one can estimate $\alpha$ using these values of $p_A$ and $p_B$. Data availability ----------------- All data from this publication is stored for at least 10 years on the University of Bristol’s Research Data Storage Facility (RDSF). The processed data for the findings in this paper is available publicly from the RDSF, the raw data consisting of timetag files is too large to host publicly and available from the corresponding author on request. Supplementary Information {#appendix .unnumbered} ========================= Scalability {#scale} ----------- We have demonstrated an eight-user quantum communication network enabled by a single source and wavelength/BS multiplexing. The number of wavelength channels available depends on the available WDM technology and the bandwidth of the polarisation entangled photon pair source. The current source has a bandwidth of $\sim$60nm ($\sim$7492GHz) and the WDM channels used have a $\sim$0.8nm (100GHz) bandwidth. This limits us to a maximum of $\sim$75 wavelength channels. However, broader bandwidth sources (such as supercontinuum based sources of polarisation entanglement) and or closer spaced WDM channels (such as the upcoming 10 GHz DWDM standard) would allow at least a few hundred channels. Thus the real limit to the maximum number of users in such a quantum network is the QBER, specifically the contribution from “accidentals”. Given a particular photon flux incident on the detectors of two users, there is a probability that two uncorrelated detection events happen to occur within the chosen coincidence window. Such coincidences are called accidentals. In our experiment, a user opens a different coincidence window for every incident wavelength channel as needed because of the uncompensated propagation times of different wavelengths via the fibres and DWDM channels. Thus increasing the number of wavelengths a single user receives increases the accidental rate and the QBER, effectively reducing the secure key rate. The QNSP can correct for the propagation delays between different wavelength and BS, when all users are connected by fixed lengths of fibre, dramatically increasing the secure key rates because each user will then need to consider only a single coincidence window. Similarly the users have two strategies to increase their secure key rates: use multiple detectors for each measurement outcome, each of which detects fewer wavelength channels; or selectively choose which wavelengths are incident on their detectors based on the connection(s) desired with select other users. In the absence of such methods, minimising the number of wavelength channels is the best way to increase secure key rates. By using $k$-fold beamsplitters, we can create $\frac{n}{k}$ subnets between $n$ users. Each subnet forms a fully connected network with WDMs only using $\frac{n}{k}\left(\frac{n}{k} -1 \right)$ wavelength channels. Additional wavelength channels are needed to interconnect the different subgroups. This can be done in two ways: First, we treat each subnet effectively as a single user in a $k$ user network and create a fully connected network with $k(k-1)$ additional wavelength channels. This implies using $\frac{n}{k}$-fold beamsplitters to distribute each of these extra $k(k-1)$ channels to all $\frac{n}{k}$ users in a subnet. When considering the link between just two users, beamsplitters can be viewed as losses. Thus mixing $k$-fold and $\frac{n}{k}$-fold beamsplitters can result in significantly different coincidence rates between different sets of channels/users. Consequently, the optimal pump power (i.e. pair rate emitted by the source) will be different for various channels. With a single source, we cannot optimise this independently which in turn leads to sub optimal key rates. Thus in the second method, we can impose the constraint that all wavelength channels are split using only $k$-fold beamsplitters. Here, each subnet requires $\frac{n}{k^2}$ wavelengths and the network requires a total of $\frac{n}{k}\left(\frac{n}{k} -1 \right) + \frac{n}{k^2}k(k-1)$ wavelength channels. We note that the above formulae are valid only when $n$, $k$, $\frac{n}{k}$ and $\frac{n}{k^2}$ are all integers. To create networks with any integer number of users, it is possible to create a larger network that satisfies the above conditions and not connect all users as and how required. With a fixed pump power of the entangled photon source, the net effect of BS multiplexing a correlated wavelength pair is akin to “time sharing” the key. When considering just two users, it can be thought of as additional loss. On the other hand, wavelength multiplexing introduces a new source of key. Both techniques adversely affect the signal to noise ratio however, the additional noise due to more wavelength channels can be avoided by filtering. Thus a good topology makes use of a significant amount of wavelength multiplexing supplemented by BS multiplexing. Extended Data Figure \[fig:numnodes\] shows the secure key rate for different numbers of users as a function of the transmission loss. For convenience, we have focused on the use of 2-fold beamsplitters (solid lines) and networks where $n = k^2$, using $k$-fold beamsplitters and $k$ subnets (dashed lines). Maintaining a reasonable secure key rate, within the constraints of the current experiment, is possible for both 32 users in 2 subnets and even 49 users in 7 subnets. This demonstrates how our network architecture can be used for very large and complex Local Area Quantum Networks. Adapting the network for practical use-cases {#use} -------------------------------------------- In our proof of principle demonstration, the source of polarisation entangled photon pairs was adjacent to the Multiplexing Unit (MU) that distributed these pairs among the users. The users were in turn each connected to the MU by single long fibres. In a real world scenario, the MU need not be adjacent to the source nor must the MU be a single unit. Photons from the source can be split and combined in any physical location or in several locations to take advantage of available fibre infrastructure. In our experiment the source was separated from the MU by a 8m long fibre and each component in the MU was separated by between 2 and 7.5m of optical fibre. Further, our network topology allows the flexibility needed for a wide variety of use-cases. Longer distance, or higher speed links can co-exist on the same network; these can be implemented by using multiple detectors on certain nodes where each detector measures some of the incident wavelengths. Access networks are often preferred for low bandwidth use-cases, however users sacrifice anonymity when they request the Quantum Network Service Provider (QNSP) to establish a connection with another chosen user. Here, our topology is capable of supporting an anonymous access network controlled by each user instead of the QNSP. Optimising the key rate {#key} ----------------------- It is possible to significantly improve the key rate beyond what is shown in the main paper in several ways. First, increasing the pump power increases the number of photon pairs. Given the detector jitter, losses and QBER there exists an optimum pump power at which the secure key rate (in bits per second) is maximised. Extended Data Figure \[fig:pow\] shows the key rate of all 28 links in the network measured at 9 different pump powers. Note that when the photon flux is excessive, a secure key cannot always be generated. This is because of the increased contribution of uncorrelated singles to the QBER via accidental counts. Reducing the detector jitter is thus the best way to further increase the key rates. Using a single source for the entire network limits our control over the individual pair production rates for each correlated wavelength pair. Thus using different types of detectors strongly influences the optimal pump power. In addition to different detectors, the alignment of individual PAMs of each user and the FPCs contribute to the overall network performance. Second, using a pulsed pump as discussed in the supplementary information of Ref [@Soeren2018] would help reduce the QBER and significantly increase the key rates. Third, in our experiment, we utilised several manual Fibre Polarisation Controllers (FPCs) which were needed to maintain the polarisation entangled state at each stage of the multiplexing, demultiplexing, beamsplitting and distribution. For expediency and to demonstrate the success of our network topology, we considered it sufficient when each of the FPCs were aligned with $>$ 97% visibility. A better fibre neutralisation would have resulted in improved key rates. Lastly, in an attempt to keep the costs of each user’s Polarisation Analysis Module (PAM) to a minimum, we used readily available sub-par components which we estimate contribute to the overall QBER by up to an additional 1%. Alice Bob Chloe Dave Feng Gopi Heidi Ivan ------- ------- ------ ------- ------- ------- ------- ------- ------- Alice 8.67 9.54 8.48 11.22 14.08 8.14 6.15 Bob 8.04 7.33 13.28 5.26 6.62 5.21 Chloe 12.27 11.10 6.14 7.12 12.10 Dave 9.17 5.25 17.01 9.95 Feng 7.27 9.59 3.65 Gopi 5.25 3.26 Heidi 5.49 : **Total secure key (Mega bits) for the laboratory demonstration** as measured continuously over 18.45 hours after accounting for all finite key size effects. \[tab:short\] Alice Bob Chloe Dave Feng Gopi Heidi Ivan ------- ------- ------- ------- ------- ------- ------- ------- ------- Alice 12642 13095 16971 14257 12642 14256 15735 Bob 473 4350 1636 20 1634 3113 Chloe 4803 2089 473 2087 3566 Dave 5965 4350 5963 7442 Feng 1636 3250 4728 Gopi 1635 3113 Heidi 4727 : **Length of each link in meters** in the metropolitan network shown in meters. Bob and Gopi were users separated by 10m of fibre each from the QNSP. Alice and Dave were connected to $\sim$12.6km and $\sim$4.3km spools of fibre. The remaining users were connected via loopback to various locations across the city of Bristol as shown in Fig \[fig:source+wdms\]. Each link was characterised by an OTDR and the measurements shown are the link distances in fibre between each pair of users. \[tab:length\] {width="90.00000%"} {width="90.00000%"} {width="95.00000%"} {width="95.00000%"} {width="65.00000%"} [^1]: Correspondence and requests for materials should be addressed to Siddarth Koduru Joshi [^2]: Since we do not explicitly note down the basis choice, we must assume that the BS ensures that we measure in both necessary measurement bases. In general it is sufficient to assume that the BS has a bound on its splitting ratio. [(See Supplementary information for more details on the security)]{}

--- abstract: 'The modern theory of orbital magnetization (OM) was developed by using Wannier function method, which has a formalism similar with the Berry phase. In this manuscript, we perform a numerical study on the fate of the OM under disorder, by using this method on the Haldane model in two dimensions, which can be tuned between a normal insulator or a Chern insulator at half filling. The effects of increasing disorder on OM for both cases are simulated. Energy renormalization shifts are observed in the weak disorder regime and topologically trivial case, which was predicted by a self-consistent $T$-matrix approximation. Besides this, two other phenomena can be seen. One is the localization trend of the band orbital magnetization. The other is the remarkable contribution from topological chiral states arising from nonzero Chern number or large value of integrated Berry curvature. If the fermi energy is fixed at the gap center of the clean system, there is an enhancement of $|M|$ at the intermediate disorder, for both cases of normal and Chern insulators, which can be attributed to the disorder induced topological metal state before localization.' address: - '$^1$ SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China' - '$^2$ College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China' - '$^3$ School of Physics and Electronic Engineering, Guangzhou University, 510006 Guangzhou, China' - '$^4$ School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China' - '$^5$ Microelectronic Instrument and Equipment Research Center, Institute of Microelectronics of Chinese Academy of Sciences, Beijing 100029, China' - '$^6$ School of Microelectronics, University of Chinese Academy of Sciences, Beijing 100049, China' - '$^7$ Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China' author: - 'Si-Si Wang$^{1,2}$, Yan-Yang Zhang$^3$, Ji-Huan Guan$^{1,4}$, Yan Yu$^{1,4}$, Yang Xia$^{5,6}$ and Shu-Shen Li$^{1,2,7}$' title: Numerical Study of Disorder on the Orbital Magnetization in Two Dimensions --- Introduction ============ The quantum theory of orbital magnetization (OM) was brought to the forefront by the end of the 20th century. This theory was first derived by using linear-response methods which was only used to calculate the OM changes instead of the OM itself[@Mauri1996; @Mauri1996B; @Mauri2002; @Sebastiani2002; @Sebastiani2001]. After 2005, a systematic quantum mechanical method was proposed to calculate the OM itself of crystalline insulators in the Wannier representation[@Marzari1997; @Thonhauser2005; @Resta2005], which is consistent with semi-classical derivations[@Xiao2005; @CFang2009]. It was further generalized to metals and Chern insulators[@Ceresoli2006; @Marrazzo2016; @Bianco]. These developments lead to modern theory of OM in solids[@Thonhauser2011; @Aryasetiawan2019]. This modern theory of OM is expressed as a momentum space Brillouin-zone integral function, with a formalism similar to that of Berry phase[@CFang2009; @Thonhauser2011; @Berry1984]. Therefore, the OM manifests itself in an peculiar way in topological materials. A typical example is the large and energy dependent OM in the bulk gap as a result of chiral edge states associated with nonzero Chern number[@Ceresoli2006; @Bianco]. Experimentally, the OM can be measured by the Compton scattering of photons[@ComptonScattering; @DeterminationOM0], x-ray absorption or x-ray magnetic circular dichroism spectroscopy[@DeterminationOM; @DeterminationOM2]. For example, in the coexistence of spin and orbit magnetic moments, the orbital magnetic moment can be obtained from the total one by deducting the spin counterpart measured from the magnetic Compton scattering in terms of an applied magnetic field[@DeterminationOM0]. Several materials possessing remarkable bulk orbital magnetic moment have been recently proposed[@Nikolaev2014; @CorrelatedElectron; @RoleOfBerryPhase; @Liu2019; @WeylSemimetal] or observed[@DeterminationOM0; @DeterminationOM; @DeterminationOM2; @DeterminationOM3]. It was also found that the OM plays an important role in the process of magnetization switching operation[@MagnetizationSwitching]. Most researches of OM so far have been focused on clean crystals. It is well known that strong disorder will eventually induce localization[@Anderson1958; @Hatsugai1999; @YY2012; @YY2013; @Localization19B]. Disorder also leads to rich phenomena in topological materials even in two dimensions, for example, the topological Anderson insulator[@YY2012; @TAI; @TAI2018] and the topological metal[@TopologicalMetal2; @TopologicalMetal]. The essential physics underlying these quantum transports is the “translational” motion of electrons from one terminal to another through the sample. The OM, on the other hand, is related to the angular momentum of the circular motion, which can be further separated into the itinerant circular, the local circular and the topological (boundary circular) contributions[@Thonhauser2005; @Ceresoli2006; @Bianco]. The effects of disorder on the OM and its process towards localization are still open questions. Based on a self-consistent $T$-matrix approximation, it was concluded that the effect of weak disorder in two dimensions is simply an energy renormalization, i.e., a shift of orbital magnetization function along the energy axis[@GBZhu2012]. In this manuscript, we perform numerical studies of the OM $M$ in disordered two-dimensional (2D) systems, based on the Haldane model[@Haldane1988; @Haldane2004] with a tunable Chern number $C$. Starting from a clean system with Chern number $C=0$ or $1$, the development of fermi energy dependent $M$ with increasing disorder is investigated. Meanwhile, the development of the intrinsic anomalous Hall conductance (AHC) $\sigma_{xy}^{\mathrm{int}}$ is also presented as a reference indication for disorder induced topological transition. Based on these numerical results, we demonstrate that, although the self-consistent $T$-matrix approximation can capture some features of the band OM in the weak disorder regime, it cannot predict the localization trend, and the contribution associated with chiral states which are important when $\sigma_{xy}^{\mathrm{int}}\gg 0$. If the fermi energy $\mu$ is fixed at the gap center of the clean system, we find there is always a peak of $|M|$ at the intermediate disorder, in both cases of $C=0,1$, before the final localization at strong disorder. This may correspond to a disorder induced metal state with non-quantized $\sigma_{xy}^{\mathrm{int}}$[@TopologicalMetal2; @TopologicalMetal]. This manuscript is organized as follows. In sections II and III, the details of the model and calculation methods are introduced. Then, the results for topologically trivial ($C=0$) and nontrivial ($C=1$) phases are represented in Sections IV and V, respectively. The Model ========== In order to incorporate the effects from topology, we adopt the Haldane model which can be tuned between a Chern insulator (quantum anomalous Hall effect) with $C=1$ and a normal insulator with $C=0$[@Haldane1988]. This is a tight binding model defined on a 2D honeycomb lattice, with a real nearest-neighbor hopping $t_1$ (set to be 1 as the energy unit), complex next nearest-neighbor hoppings $t_2e^{\pm{i\phi}}$ and staggered potentials $\pm{\Delta}$, where $\pm$ corresponds to A and B sublattices respectively, as illustrated in Fig. \[structure\]. ![ The honeycomb lattice of the Haldane model. Solid (open) points indicate the A (B) sublattice sites with onsite energy $+\Delta$ ($-\Delta$). $t_1$ represents the hopping amplitude of nearest-neighbor sites with different sublattice. $t_2e^{i\phi}$ represents the complex hopping amplitude of next nearest-neighbor sites with the same sublattice. Arrows on the dashed lines denote the positive sign of the phase $\phi$. []{data-label="structure"}](Fig01.eps){width="70.00000%"} Let $\bm{a_1},\bm{a_2},\bm{a_3}$ be the unit vectors from a site on sublattice A to its three nearest-neighbor sites on sublattice B, and $\bm{b_1}=\bm{a_2}-\bm{a_3},\bm{b_2}=\bm{a_3}-\bm{a_1}, \bm{b_3}=\bm{a_1}-\bm{a_2}$ be the next nearest-neighbor vectors from this site to its three nearest-neighbor sites on sublattice A. Now the Hamiltonian of Haldane model can be expressed in $k$-space as a $2\times 2$ matrix $$\label{Hk} \eqalign{ H(\bm{k})=2t_2\cos{\phi}\big(\sum_i\cos(\bm{k}\cdot\bm{b_i})\big)I+\\ t_1\big(\sum_i\big[\cos(\bm{k}\cdot\bm{a_i})\sigma^1+ \sin(\bm{k}\cdot\bm{a_i})\sigma^2\big]\big)+\\ \big[\Delta-2t_2\sin{\phi}\big(\sum_i\sin(\bm{k}\cdot\bm{b_i})\big)\big]\sigma^3 } $$ where $\sigma^i$ are Pauli matrices acting on the space of sublattice. Nonzero $\phi$ breaks time-reversal symmetry and this can give rise to nonzero OM and Chern number[@Xiao2010; @Thouless1982; @Nakai2016]. This model’s topological phase depends on the value of the parameter ratio $|\frac{\Delta}{t_2}|$. It is topologically nontrivial with Chern number $|C|=1$ when $|\frac{\Delta}{t_2}|<3\sqrt{3}\sin{\phi}$. The real space Hamiltonian can be obtained by an inverse Fourier transformation of Eq. (\[Hk\]). In Fig. \[disper\], we present the band structures of a ribbon with zigzag edges for two typical parameter settings, corresponding to $C=0$ in Panel (a) and $C=1$ in Panel (b), respectively. In both cases, the bulk gap region is around the energy range $(0,2)$. Red curves in Fig. \[disper\](b) are topological edge states traversing the bulk gap arising from $C=1$ of the valence band. Due to the topological origin, the existence of this pair of edge states is therefore robust for any kind of edge cut from the bulk crystal. Details of Haldane model¡¯s edge states associated with different edges are available in Ref. [@HaldaneEdges]. In the following, we will investigate the OM $M$ and its constituent parts \[Eq. (\[magnet4\])\] for these two typical cases. ![ The band structure of the Haldane model in the quasi-one-dimensional ribbon geometry with width $N_y=100$ without aesny disorder. The two panels correspond to different onsite energy $\Delta$: (a) $\Delta=4.0$, (b) $\Delta=1.6$. The other model parameters are identical in both panels: $t_1=2.0, t_2=2/3, \phi=0.7\pi$. []{data-label="disper"}](Fig02.eps){width="90.00000%"} The METHODS ============ In a 2D crystalline solid, the quantum mechanical description of electronic OM, the orbital magnetic moment per unit volume (area in 2D discussed here), can be formulated in $k$-space as[@Thonhauser2005; @Ceresoli2006; @Thonhauser2011; @Liu2019] $$\label{magnet} \eqalign{ M(\mu)&=M_{\mathrm{LC}}(\mu)+M_{\mathrm{IC}}(\mu)-2\mu N_{\mathrm{BC}}(\mu), \\ M_{\mathrm{LC}}&=\frac{e}{2 c} \mathrm{Im}\sum_n\int_{\varepsilon_{nk}\leqslant\mu} \frac{\mathrm{d}\bm{k}}{(2\pi)^2}\,\langle\partial_{\bm{k}}u_{n\bm{k}}| \times H_{\bm{k}} |\partial_{\bm{k}}u_{n\bm{k}}\rangle \\ M_{\mathrm{IC}}&=\frac{e}{2 c}\mathrm{Im}\sum_n\int_{\varepsilon_{nk}\leqslant\mu} \frac{\mathrm{d}\bm{k}}{(2\pi)^2}\,\varepsilon_{n\bm{k}}\langle\partial_{\bm{k}}u_{n\bm{k}}| \times|\partial_{\bm{k}}u_{n\bm{k}}\rangle \\ N_{\mathrm{BC}}&=\frac{e}{2 c}\mathrm{Im}\sum_n \int_{\varepsilon_{nk}\leqslant\mu}\frac{\mathrm{d}\bm{k}}{(2\pi)^2}\, \langle\partial_{\bm{k}}u_{n\bm{k}}| \times|\partial_{\bm{k}}u_{n\bm{k}}\rangle\\ &=\frac{e}{4\pi c}\mathcal{C}, } $$ where three terms correspond to the local circulation (LC), the itinerant circulation (IC) and the Berry curvature (BC) contributions, respectively. Here $|u_{n\bm{k}}\rangle$ is the cell-periodic Bloch function, $c$ is the vacuum speed of light, and $\varepsilon_{nk}$ is the Bloch eigenvalue, so that $H_{\bm{k}}|u_{n\bm{k}}\rangle=\varepsilon_{n\bm{k}}|u_{n\bm{k}}\rangle$. All summations in Eq. (\[magnet\]) are over occupied bands $n$ up to the fermi energy $\mu$. In the third term, $N_{\mathrm{BC}}$ is proportional to the (intrinsic) AHC $\sigma_{xy}^{\mathrm{int}}$ as [@RoleOfBerryPhase; @Xiao2010] $$\label{EqHallConductivity} \sigma_{xy}^{\mathrm{int}}=\frac{e^2}{h}\mathcal{C}=\frac{2ec}{\hbar} N_{\mathrm{BC}}.$$ Here, the dimensionless number $\mathcal{C}$ is quantized as the Chern number $C$ (the topological invariant) when $\mu$ is in the gap, but may not be quantized if $\mu$ is not in the gap. When $\mathcal{C}\neq 0$, this term corresponds to the contribution from the magnetic moment of chiral edge states[@Ceresoli2006]. In numerical calculations, the derivatives $|\partial_{\alpha}u_{n\bm{k}}\rangle \equiv |\partial_{k_{\alpha}}u_{n\bm{k}}\rangle$ in Eq. (\[magnet\]) have to be evaluated on a mesh of discretized Brillouin zone. However, this cannot be done by a simple finite-deference, since the gauges of wavefunctions $| u_{n\bm{k}}\rangle$ on neighboring grid points cannot be fixed. Instead, we use the discretized covariant derivative [@Marzari1997; @Ceresoli2006; @Souza2004] $$\label{EqCovariantDerivative} |\tilde{\partial}_\alpha u_{n\bm{k}}\rangle,$$ which involves linear combinations of occupied states under $\mu$, and a local gauge fixing around a definite grid point. See Appendix A of Ref. [@Ceresoli2006] for details. This definition guarantees itself to be numerically gauge invariant. Let us define[@Ceresoli2006; @Souza2008] $$\label{fgh2} \eqalign{ \tilde{f}_{\bm{k},i}&=\frac{1}{v}\epsilon_{ijl}q_{i}\sum_n\mathrm{Im} \langle\tilde{\partial}_{j} u_{nk}|\tilde{\partial}_{l}u_{nk}\rangle\\ \tilde{g}_{\bm{k},i}&=\frac{1}{v}\epsilon_{ijl}q_{i}\sum_n\mathrm{Im} \langle\tilde{\partial}_{j} u_{nk}|H_{\bm{k}}| \tilde{\partial}_{l}u_{nk}\rangle\\ \tilde{h}_{\bm{k},i}&=\frac{1}{v}\epsilon_{ijl}q_{i}\sum_{nm} \varepsilon_{nm\bm{k}}\mathrm{Im} \langle\tilde{\partial}_{j} u_{mk}| \tilde{\partial}_{l} u_{nk}\rangle\\ } $$ where $q_{i}$ represents the primitive reciprocal vectors of the discretized $\bm{k}$ mesh in the $i$-th direction, and $v$ denotes the volume of the unit cell in the mesh. Now Eq. (\[magnet\]) can be transformed into another form as [@Ceresoli2006; @Souza2008]: $$\begin{aligned} \label{magnet2} \eqalign{ &M_{\mathrm{LC}}=\frac{-1}{2c(2\pi)^2}\int_{\mathrm{BZ}}\mathrm{d}\bm{k} \tilde{g}_{\bm{k}}, \\ &M_{\mathrm{IC}}=\frac{-1}{2c(2\pi)^2}\int_{\mathrm{BZ}}\mathrm{d}\bm{k} \tilde{h}_{\bm{k}}, \\ &N_{\mathrm{BC}}=\frac{1}{2c(2\pi)^2}\int_{\mathrm{BZ}}\mathrm{d}\bm{k} \tilde{f}_{\bm{k}}, } $$ where $\tilde{f}_{\bm{k}}\equiv \tilde{f}_{\bm{k},z}$ (similarly for $\tilde{g}$ and $\tilde{h}$) is the only nonzero component for a 2D crystal. Besides the numerical computability of $|\tilde{\partial}_\alpha u_{n\bm{k}}\rangle$, another merit of Eqs. (\[fgh2\]) and (\[magnet2\]) is that now the components $M_{\mathrm{LC}}$ and $M_{\mathrm{IC}}$ are separately gauge invariant, even in the multi-band case[@Ceresoli2006]. Gauge invariant quantities like those defined in (\[magnet2\]) should be potentially observable[@Xiao2010; @Scott1962; @Hugu1971]. As stated in Eq. (\[EqHallConductivity\]), $N_{\mathrm{BC}}$ is proportional to the AHC $\sigma_{xy}^{\mathrm{int}}$[@Xiao2010; @QAHRMP]. As for the other two terms, we adopt an alternate combination defined in Ref. [@Souza2008] as $$\label{magnet4} \eqalign{ M(\mu)&=M_{\mathrm{SR}}^{(\mathrm{I})}(\mu)+\Delta M_0 (\mu)-2{\mu}N_{\mathrm{BC}}(\mu),\\ M_{\mathrm{SR}}^{(\mathrm{I})}&=M_{\mathrm{LC}}-M_{\mathrm{IC}},\\ \Delta{M_0}&=2 M_{\mathrm{IC}}. } $$ Here, $M_{\mathrm{SR}}^{(\mathrm{I})}$ is found to be proportional to the differential absorption of right and left circular polarized light verified by the $\textit{f}$-sum rule[@PM1998] which corresponds to the self-rotation of the WFs, namely, the intraorbital part of OM $M$. On the other hand, $\Delta M_0$ is the interorbital part which corresponds to a boundary current circulation[@Xiao2006], The AHC $\sigma_{xy}^{\mathrm{int}}$ and OM $M$ are results of an integration over energy up to $\mu$. It is also insightful to investigate their densities with respect to energy[@Yang2011; @Qiao2016]. Specifically, we define $$\begin{aligned} \label{cdensity} \rho_{\sigma}(\mu)=\lim\limits_{\triangle{E}\rightarrow 0}\frac{\sigma_{xy}^{\mathrm{int}}(\mu+\triangle{E})-\sigma_{xy}^{\mathrm{int}}(\mu)}{\triangle{E}} $$ and $$\begin{aligned} \label{mdensity} \rho_{M}(\mu)=\lim\limits_{\triangle{E}\rightarrow 0}\frac{M(\mu+\triangle{E})-M(\mu)}{\triangle{E}}. $$ In the following, we will focus on how $M$ and its constituent components defined in Eq. (\[magnet4\]) change with respect to disorder strength. The disorder is simulated as a random potential $U(i)$ on each site $i$, where $U(i)$ is uniformly distributed within the interval $(-\frac{W}{2},\frac{W}{2})$ , with $W$ the disorder strength. To be compatible with the $k$ space formalism introduced above[@Thonhauser2005; @Bianco; @Thonhauser2011], we use a disordered sample with size $L_x \times L_y$ as a supercell (repeating unit) of a superlattice, so that the translation symmetry can be restored on a larger scale, and the results are free from artificial effects from existing edges. This is equivalent to twisted boundary conditions on both directions[@Hatsugai1999; @YY2012; @Xiao2010; @Sheng1997; @Essin2007; @Song2016]. Due to the presence of disorder fluctuation, each data point in the following will be an average over disorder ensembles, with typically 1000 random configurations. Results and Discussion ====================== Topologically Trivial Phase --------------------------- Let us start from a topologically trivial phase as shown in Fig. \[disper\] (a), which is a normal insulator at half filling with Chern number $C=0$. In Fig. \[phi=0.7C=0\] (a), we plot the AHC $\sigma_{xy}^{\mathrm{int}}$ as a function of fermi energy $\mu$, for different disorder strengths $W$. At the weakest disorder ($W=3$, black line), the energy interval $\mu\in(0,2)$ with $\sigma_{xy}^{\mathrm{int}}=0$ corresponds to the energy gap. With increasing disorder, this gap shrinks to zero at around $W=6$. After that, $\sigma_{xy}^{\mathrm{int}}$ grows remarkably to a considerable but non-quantized value $\sim 0.35\frac{e^2}{h}$ at $W\sim 10$, before the localization ($\sigma_{xy}^{\mathrm{int}}\sim0$) at strong disorder $W=15$. Such an emergence of nonzero $\sigma_{xy}^{\mathrm{int}}$ under increasing disorder is not rare in systems without particle-hole symmetry, and can be attributed to disorder induced band inversion [@YY2012; @YY2013; @TopologicalMetal; @StatTI]. The non-quantization of $\sigma_{xy}^{\mathrm{int}}$ suggests that the system is in the metallic state, similar to that before the appearance of topological Anderson insulator[@YY2013; @TopologicalMetal2; @TopologicalMetal]. However, since a stable energy gap or mobility gap cannot be formed before another disorder induced band inversion into a trivial Anderson insulator at strong disorder[@Hatsugai1999; @YY2012; @YY2013], this system cannot develop into a topological Anderson insulator with a quantized topological invariant (Chern number here). The development of OM $M(\mu)$ under disorder is the main focus of this work. The numerical results are presented in Fig. \[phi=0.7C=0\] (b). The first observation is that $M$ has opposite signs for the valance and conduction bands respectively. At weak disorder $W=3$, they are separated by the gap with constant $M$. In a previous work based on the self-consistent $T$-matrix approximation for weak disorder, it has been predicted that the effect of weak disorder on $M(\mu)$ is just an energy renormalization, i.e., a global shift of $M(\mu)$ profile along the $\mu$ axis[@GBZhu2012]. With the increasing of disorder strength $W$ shown in Fig. \[phi=0.7C=0\] (b), as predicted, the $M(\mu)$ profiles associated with the valance band (with $M>0$) and the conduction band (with $M<0$) do shift along the energy axis, but in opposite directions respectively (illustrated by blue arrows). An important feature that has not been captured in the self-consistent $T$-matrix approximation is the reduction of the magnitude $|M|$ with increasing $W$ for most of the band ranges, which corresponds to the localization tendency of the orbital motion. The opposite directional shifts of $M(E)$ for conduction and valence bands at weak disorder can be attributed to the energy renormalization from disorder. Consider a generic two-band model $$H_2=h_x\sigma_x+h_y\sigma_y+h_z\sigma_z.\label{EqTwoBand}$$ The effect of non-magnetic disorder to the this model can be calculated within the first Born approximation as a self energy $\Sigma$. Its real part is[@Beenakker] $$\label{EqReSigma} \mathrm{Re}\Sigma \propto W^2 \mathrm{Re} \int \frac{\epsilon - h_x\sigma_x - h_y\sigma_y-h_z\sigma_z }{\epsilon^2-h_x^2 - h_y^2 - h_z^2 } d^3h,$$ where $\epsilon$ is the eigen-energy, and it is $\epsilon^+>0$ ($\epsilon^-<0$) for the conduction (valance) band. Expressed in terms of $\sigma_i$, $\mathrm{Re}\Sigma$ is also a $2\times2$ matrix. Its diagonal elements $$\label{EqReSigmaDiag} \propto W^2 \int \frac{\epsilon^{\pm} }{(\epsilon^{\pm})^2-h_x^2 - h_y^2 - h_z^2 } d^3h,$$ will contribute to the energy renormalization, i.e., band shifts, with opposite signs for conduction and valence bands respectively. This approaching of two topologically trivial bands at weak disorder can also be understood in a simpler context as a second perturbation[@YY2012], which plays an important role in forming the topological Anderson insulator. ![ The fermi energy dependences for the topologically trivial case corresponding to Fig. \[disper\] (a). The AHC $\sigma_{xy}^{\mathrm{int}}$ \[Panel (a)\] and OM $M$ \[Panel (b)\] as functions of fermi energy $\mu$, for different disorder strength $W$. Panels (c) and (d) are their energy densities, respectively. Each data point is an average over 1000 disorder configurations with the supercell size $L_x=L_y=L=6$. The blue arrows indicate the shift direction of the $M(\mu)$ profiles, as predicted in [@GBZhu2012]. The dashed lines in (b) and (d) indicate the location of gap center in the clean limit.[]{data-label="phi=0.7C=0"}](Fig03.eps){width="90.00000%"} The energy densities associated with $\sigma_{xy}^{\mathrm{int}}$ and $M$ are presented in Fig. \[phi=0.7C=0\] (c) and (d). Despite strong fluctuations, some important information can still be drawn. For the AHC density in Fig. \[phi=0.7C=0\] (c), the sharp peaks ($M>0$) and valleys ($M<0$) at $W\sim 10$ correspond to the creation of topological charges (Chern numbers) with opposite signs soon after the disorder induced band inversion[@Hatsugai1999; @YY2012; @Xiao2010]. This picture confirms again the origin of the appearance of $\sigma_{xy}^{\mathrm{int}}$ peaks around $W\sim 10$ in Fig. \[phi=0.7C=0\] (a). As another result, the chiral edge states associated with these nonzero topological charges give rise to remarkable contributions to $M$. This is reflected by the valleys with largest $|\rho_{M}|$ in Fig. \[phi=0.7C=0\](d), which also appear at $W\sim 10$. After an overall view of $M(\mu)$, now we concentrate on the OM of an insulator[@Thonhauser2005; @Bianco], by fixing the fermi energy $\mu$ in the bulk gap center. In Fig. \[fgh-C=0\] (a), we plot $M$ (black solid line) and $\sigma^{\mathrm{int}}_{xy}$ (red dashed line) as functions of disorder strength $W$, at the fermi energy $\mu=1$ \[indicated as the dashed line in Fig. \[phi=0.7C=0\] (c) and (d)\], which is near the gap center in the clean limit. The AHC $\sigma^{\mathrm{int}}_{xy}$ is identically zero until the band closing at $W\sim 6$. Notice that $M$ is changing during this process. This is not surprising since Chern number (here 0) is a topological invariant of the band while OM is not, and any distortion of the band (e.g., from disorder) may influence the value of OM even when $\mu$ is in the gap. With the appearance of nonzero $\sigma^{\mathrm{int}}_{xy}$, $|M|$ starts to increase more quickly. The magnitudes of both quantities arrive at the maximum value $|M|_{\mathrm{max}}$ and $\sigma^{\mathrm{int}}_{xy,\mathrm{max}}$ at the intermediate disorder around $W\sim 10$. From the size dependence of $|M|_{\mathrm{max}}$ in the inset of Fig. \[fgh-C=0\] (a), it can be seen that this nonzero magnetization is expected to persist to the thermodynamic limit. This enhancement of orbital magnetic moment at intermediate disorder reflects the emergence of the metal state from another respect. Similarly, magnetic impurities was also found to induce remarkable OM in a Rashba electron gas[@MagneticImpurity]. After this peak at disorder $W\sim 10$, all electronic motions go towards a final localization $M,\sigma^{\mathrm{int}}_{xy}\sim 0$ in the strong disorder limit. In order to obtain more intuitions, we scrutinize the behaviors of three constituent components of $M$: $M_{\mathrm{SR}}^{(\mathrm{I})}$, $\Delta{M_0}$ and $-2{\mu}N_{\mathrm{BC}}$ defined in Eq. (\[magnet4\]). In Fig. \[fgh-C=0\] (b), they are also presented as functions of disorder strength $W$, at the fermi energy $\mu=1$ near the gap center of the clean limit. Due to the vanishing of Chern number, the contribution from $-2{\mu}N_{\mathrm{BC}}$ term (blue line with triangular dots) is small, and is actually zero before the band touching at $W\sim 7$. However, the components $M_{\mathrm{SR}}^{(\mathrm{I})}$ (green line with square dots) and $\Delta{M_0}$ (red line with circle dots) are almost one order of magnitude larger than $M$ itself (black line), but with opposite signs. This cancelation makes the magnitude of total magnetization $M$ rather small. This feature is consistent with previous analytical results[@CFang2009; @GBZhu2012]. ![ Developments of the normal insulator under disorder $W$, by fixing the fermi energy at $\mu=1$ \[indicated as dashed line in Fig. \[phi=0.7C=0\] (b) and (d)\]. (a) The OM $M$ (black line) and anomalous Hall conductance $\sigma_{xy}^{\mathrm{int}}$ (red dashed line). The inset is the peak value $M_{(\mathrm{max})}$ versus the linear size $L$ of supercells. (b) The OM $M$ (black line) and its constituent components (line with symbols) defined in Eq. (\[magnet4\]).[]{data-label="fgh-C=0"}](Fig04.eps){width="60.00000%"} Topologically Non-Trivial Phase ------------------------------- Now we turn to the case of a Chern insulator at half filling, with the band structure as presented in Fig. \[disper\] (b). The development of AHC $\sigma_{xy}^{\mathrm{int}}$ are plotted in Fig. \[Figphi=0.7C=1\] (c). The $C=1$ plateau around $\mu\in(0,2)$ can be clearly seen in the weak disorder regime, reflecting the robust edge states in the bulk gap. With increasing disorder, the width of this plateau shrinks and finally results in a collapse after $W>6$. The associated AHC densities $\rho_{\sigma}(\mu)$ for different disorder strengths are presented in Fig. \[Figphi=0.7C=1\] (c). At the weakest disorder ($W=3$, black line), $\rho_{\sigma}(E)$ consists of one peak and one valley separated by a horizontal line with $\rho_{\sigma}(E)=0$, which correspond to the valance band with positive Chern number, the conduction band with negative Chern number, and the bulk gap, respectively. With increasing disorder, the gap shrinks and nonzero Chern numbers annihilate after the band touching. This is a well known process of Anderson localization for a Chern insulator at strong disorder, which also corresponds to a disorder induced band inversion[@Hatsugai1999; @YY2012; @Song2016; @SS2019]. The development of $M(\mu)$ under increasing disorder is presented in Fig. \[Figphi=0.7C=1\](b). At the weakest disorder $W=3$ (black line), similar to the topologically trivial case, $M$ also possesses opposite signs in the valence and conduction bands respectively. Now in the gap region, $\mu\in(0,2)$, $M(\mu)$ is linearly decreasing instead of constant as in Fig. \[phi=0.7C=0\] (b). This originates from the chiral edge states in the bulk gap, so that[@Ceresoli2006] $$\label{EqMEdgeStates} \frac{dM}{d\mu}=-\frac{C}{2\pi c}.$$ This is a direct consequence from the last term of Eq. (\[magnet4\]), which is the only energy-dependent contribution when the fermi energy is in the bulk gap. From its energy density $\rho_M$ \[black line in Fig. \[phi=0.7C=0\] (d)\], we can see that dominating contributions are indeed from the bulk gap around $(0,2)$ and nearby band edges. In other words, in the topologically nontrivial case, Berry curvature related chiral states play an important role in the orbital magnetization. With increasing disorder, this linear region of $M$ shrinks gradually, due to the narrowing of the bulk gap. Meanwhile, the magnitudes of band orbital magnetization $M$ decrease almost monotonically in most of the *band* region, as a result of the localization tendency. Now, due to the strong modulation of chiral edge states pinned around the gap region, these $M(\mu)$ profiles associated with both bands do not exhibit prominent global shifts along the energy axis. This is different from the previous case with $C=0$ shown in Fig. \[phi=0.7C=0\] (b), and also different from that predicted from the self-consistent $T$-matrix approximation[@GBZhu2012]. Therefore, the development of most band orbital magnetization $M$ under disorder in Fig. \[Figphi=0.7C=1\](b) looks simpler than that in Fig. \[phi=0.7C=0\]: just a monotonic decreasing of magnitudes towards localization $|M|=0$ in strong disorder limit. ![ Similar to Fig. \[phi=0.7C=0\], but corresponding to the topologically nontrivial case shown in Fig. \[disper\] (b).[]{data-label="Figphi=0.7C=1"}](Fig05.eps){width="90.00000%"} Fig. \[fgh-C=1\] focuses on the developments of the Chern *insulator*, i.e., by fixing fermi energy $\mu=1$ fixed at the gap center of the clean limit. Fig. \[fgh-C=1\] (a) is the OM $M$ (black solid line) and AHC $\sigma^{\mathrm{int}}_{xy}$ (red dashed line) under increasing disorder. Different from most of the *band* OM with monotonic dependence on $W$, now there is a peak of $|M|$ at an intermediate disorder $W\sim 6$, just when the AHC plateau starts to collapse. The size dependence of this peak value \[inset of Fig. \[fgh-C=1\] (a)\] slows down after $L>10$, so we believe the peak value of $M$ will also approach a stable one in the thermodynamic limit $L\rightarrow \infty$. Similar to the previous normal insulator case, this peak is closely related to the emergence of a disorder induced metal state with non-quantized Hall conductance[@YY2013; @TopologicalMetal2; @TopologicalMetal]. Analogous remarkable disorder enhancement of orbital magnetic moment around the collapse of the AHC plateau was also found in bilayer quantum anomalous Hall systems, where it manifests itself as a peak of orbital magnetoelectric coupling[@SS2019]. We conjecture that such disorder enhancement of orbital magnetic motion[@YY2013; @TopologicalMetal2; @TopologicalMetal; @SS2019] is an indication of a “topological metal”[@TopologicalMetal2; @TopologicalMetal] before localization. This also offers an option to finding materials with remarkable OM. The developments of corresponding three components is shown in Fig. \[fgh-C=1\] (b). The most noticeable difference from the trivial state in Fig. \[fgh-C=0\] (b), is the remarkable contribution from the Berry curvature term $-2\mu N_{\mathrm{BC}}$ (blue triangles) due to nonzero Chern insulator. ![ Similar to Fig. \[fgh-C=0\], but corresponding to the topologically nontrivial case shown in Fig. \[disper\] (b). []{data-label="fgh-C=1"}](Fig06.eps){width="60.00000%"} Summary ======= In summary, the OM $M$ in two dimension under disorder is studied, based on the two-band Haldane model whose Chern number can be conveniently tuned. Starting from a normal insulator, disorder will bring two bands together, and induce a “topological metal” with nonzero AHC $\sigma_{xy}^{\mathrm{int}}$ in the band touching region. This metallic state corresponds to a disorder induced peak of OM. On the other hand, the OM profiles associated with both bands are shifting along the energy axis, consistent with previous analytical predictions. Besides, our numerical simulations show that there is always a magnitude reduction accompanying with the shifts, reflecting the localization tendency of orbital motions. Starting from a Chern insulator with a fixed fermi energy in the gap of clean limit, there also appears an $|M|$ peak with increasing disorder, almost simultaneous with the collapse of the quantized Chern number. As for the band OM, it is greatly influenced by the contribution from chiral edge states pinned at the bulk gap, and is therefore deviated from the energy renormalization picture derived from the self-consistent T-matrix approximation. We thank Prof. G. B. Zhu (Heze University) for beneficial discussions. This work was supported by National Natural Science Foundation of China under Grants No. 11774336 and No. 61427901. 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--- abstract: 'We have measured the transmission spectrum of the extremely inflated hot Jupiter WASP-52b using simultaneous photometric observations in SDSS $u''$, $g''$ and a filter centred on the sodium doublet (NaI) with the ULTRACAM instrument mounted on the 4.2m William Herschel Telescope. We find that Rayleigh scattering is not the dominant source of opacity within the planetary atmosphere and find a transmission spectrum more consistent with wavelength-independent opacity such as from clouds. We detect an in-transit anomaly that we attribute to the presence of stellar activity and find that this feature can be more simply modelled as a bright region on the stellar surface akin to Solar faculae rather than spots. A spot model requires a significantly larger planet/star radius ratio than that found in previous studies. Our results highlight the precision that can be achieved by ground-based photometry with errors in the scaled planetary radii of less than one atmospheric scale height, comparable to HST observations.' author: - | J. Kirk$^1$[^1], P. J. Wheatley$^1$[^2], T. Louden$^1$, S. P. Littlefair$^2$, C. M. Copperwheat$^3$, D. J. Armstrong$^{1,4}$, T. R. Marsh$^1$ and V. S. Dhillon$^{2,5}$\ $^1$Department of Physics, University of Warwick, Coventry, CV4 7AL, UK\ $^2$Department of Physics and Astronomy, University of Sheffield, Sheffield, S3 7RH, UK\ $^3$Astrophysics Research Institute, Liverpool John Moores University, Liverpool, L3 5RF, UK\ $^4$ARC, School of Mathematics and Physics, Queen’s University Belfast, University Road, Belfast BT7 1NN, UK\ $^5$Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38205, La Laguna, Spain bibliography: - 'Report\_ref.bib' title: 'Transmission spectroscopy of the inflated exoplanet WASP-52b, and evidence for a bright region on the stellar surface' --- stars: planetary systems – stars: individual: WASP-52 – stars: starspots Introduction ============ Transmission spectroscopy using HST/STIS was used to detect the first spectral feature of any exoplanet, that of the narrow core of atomic sodium originating from the upper atmosphere of HD209458b [@Charbonneau2002]. Broad-band opacity sources were later detected in this planet’s atmosphere in the form of the pressure broadened wings of NaI [@Sing2008_spectra] and a blueward slope in the transmission spectrum interpreted as Rayleigh scattering by H$_2$ [@Etangs2008_HD209]. This feature was also seen in HST data for HD189733b (@Pont2008; @Sing2011), however the absence of the broad wings of NaI suggested the presence of silicate condensates in the upper atmosphere of the planet [@Etangs2008_HD189]. Recent studies of hot Jupiters with HST have revealed a diverse array of atmospheres from clear to cloudy (e.g. @Sing2016). These studies have included the detections of Rayleigh scattering by high altitude hazes also in WASP-12b [@Sing2013], additional unknown optical absorbers in HAT-P-1b (@Nikolov2014), a cloud deck masking the majority of the NaI absorption and near infrared water features in WASP-31b [@Sing2015] and a clear atmosphere consistent with solar and sub-solar metallicities in WASP-39b (@Sing2016; @Fischer2016). Ground-based measurements of hot Jupiter transmission signals have also found success, beginning with the detection of the narrow component of NaI in HD189733b [@Redfield2008] and HD209458b (@Snellen2008; @Langland-Shula2009). More recent results have included further detections of sodium (e.g. @Wood2011; @Zhou2012), potassium (e.g. @Sing2011_XO2b; @Wilson2015), blueward scattering slopes (e.g. @Jordan2013; @Stevenson2014), and atmospheres dominated by clouds or hazes (e.g. @Gibson2013_wasp29; @Mallonn2016). Although there has been success with ground-based spectroscopic observations they are more susceptible to systematics arising from, for example, differential slit losses (e.g. @Sing2012). A simpler approach, which avoids this problem, is to use photometers to perform simultaneous broadband multi-wavelength measurements. These studies have included evidence for large blueward slopes (e.g. @Southworth2012; @Southworth2015; @Mancini2016), enhanced absorption around the alkali metal lines (e.g. @Mancini2013; @Bento2014) and transmission spectra most consistent with clouds (e.g. @Mallonn2015_hat12). By observing the planetary radius at carefully selected wavelengths we can probe for the existence of Rayleigh scattering and the broad absorption wings of the NaI doublet that are expected to be present in the cloud free atmospheres of hot Jupiters [@SeagerSasselov]. Transmission spectroscopy is often focussed on inflated hot Jupiters; planets with very low density due to their large radii and relatively low masses. As a result of their low densities, low surface gravities and high temperatures these planets have large atmospheric scale heights, $H$, given by $$\label{eq:scaleheight} H = \frac{kT}{g\mu}$$ where $k$ is Boltzmann’s constant, $T$ is the temperature of the planet, $g$ is the acceleration due to gravity and $\mu$ is the mean molecular mass. The outermost $\sim$ 5 scale heights may account for up to 10% of the cross sectional area of a hot Jupiter, where different atmospheric species can affect the observed transmission spectrum [@Brown2001]. The relative size of the planet’s scale height to the stellar disc governs the amplitude of the transmission signal. WASP-52b [@Hebrard2013] is an extremely inflated hot Jupiter, with a mass of 0.46M$\mathrm{_{J}}$ and radius of 1.27R$\mathrm{_{J}}$ giving it a mean density of 0.299g/cm$^3$. It orbits its 0.87M$_{\odot}$ K2V host star with a period of 1.75 days. Due to the combination of the inflated planetary radius with the small radius of the host (0.79R$_{\odot}$), it shows a deep transit in the WASP photometry (2.7%). From the table of system parameters (Table \[tab:wasp52\]; @Hebrard2013), and assuming a Jupiter mean molecular mass of 2.3 times the mass of a proton, the scale height of WASP-52b is calculated to be 731km. This makes WASP-52b an exceptional target for transmission spectroscopy as the difference in the transit depth corresponding to one atmospheric scale height is $4.4 \times 10^{-4}$, at least three times stronger than that of HD189733b. In this paper we present multi-wavelength observations of WASP-52b taken using the high speed multi-band photometer ULTRACAM [@Dhillon2007]. Observations ============ WASP-52b was observed on the night of the 7th September 2012 using the ULTRACAM [@Dhillon2007] instrument on the 4.2m William Herschel Telescope (WHT), La Palma. ULTRACAM is a high speed triple beam CCD photometer. Incoming light is split into three bandpasses, using two dichroics, and re-imaged onto three CCDs at a resolution of 0.3" per pixel, with a field of view of 5’. ULTRACAM is particularly useful for the ground-based application of transmission spectroscopy as it simultaneously takes measurements at three different wavelengths, enabling the transit depth to be measured as a function of wavelength (e.g. @Copperwheat2013; @Bento2014). The use of frame transfer CCDs allow for high frame rates (up to 300Hz) with little dead time (24ms), which is useful to avoid saturation when observing bright stars and enables many more sky flats to be taken. The observations were made using SDSS $u'$ ($\lambda_{central}$ = 3557Å, FWHM = 599Å) and $g'$ ($\lambda_{central}$ = 4825Å, FWHM = 1379Å) filters and a filter centred on the NaI doublet ($\lambda_{central}$ = 5912Å, FWHM = 312Å). These filters were selected to probe for Rayleigh scattering by observing the difference in transit depth between the $u'$ and $g'$ bands and to search for the broad wings of the sodium doublet with the NaI filter. The observations were performed with moderate telescope defocussing ($\sim 3$ arcsec) in windowed mode with exposure times of 0.76 seconds in the red and green channels and a cadence of 0.79 seconds. Due to the reduced photon count in the blue channel multiples of 10 frames were averaged on-chip before readout, leading to a 7.9 second exposure time in this channel. The moon was at 54% illumination on the night of our observations and we analysed the data with an airmass varying from 1.41 $\rightarrow$ 1.06 $\rightarrow$ 2.00. All the data were reduced using the ULTRACAM data reduction pipeline[^3] with bias subtraction and flatfielding performed in the standard way. Aperture photometry was performed for all frames using a fixed aperture. Initially, many reductions were performed with a variety of aperture sizes and the signal-to-noise ratio was calculated for each using the ratio of aperture counts to aperture errors. The optimal aperture radius was found to be 18 pixels with a sky annulus of inner radius 23 pixels and outer radius 27 pixels. Some of the observations of WASP-52 were taken through cloud, seen as dips in transmission in the raw light curves, which had to be removed before analysis. These were well defined, discrete events, with good quality data taken between the clouds. The cloudy data were removed using an iterative process. An array of running medians was calculated over a sliding box of 600 data points in the red and green arms and 60 points in the blue. This array of running medians was then subtracted from the raw data to flatten it. The median absolute deviation (MAD) of this subtracted array was calculated and sigma clipping performed to remove those data points lying at $\geq 6\sigma$ from the MAD. This process was then repeated with a smaller sliding box of 400 data points in the red and green and 40 in the blue with a final sigma cut at 5$\sigma$ from the MAD. The sigma clipping was performed on both the target and comparison star independently and only those frames that passed the sigma cut for both were kept, resulting in the removal of 20% of the data. After the sigma clipping and binning of the red and green channels to the cadence of the blue channel, there were 2494 data points in each of the three light curves at a cadence of 7.9 seconds giving us excellent sampling even after the cleaning of the data. Differential photometry was subsequently performed to remove the worst effects of telluric extinction using a comparison star with similar magnitude and colour to WASP-52 (BD+08 5023, 23:14:12.026 +08:50:56.28). This star has a V magnitude of 10.59 and B-V colour of 0.86, whilst WASP-52 has a V magnitude of 12.22 and B-V colour of 0.82. The comparison star was checked and found to be photometrically stable. Combinations of fainter stars in the field were tested as comparison stars but led to more scatter in the differential light curve than division by the single, bright comparison. Data Analysis ============= Light curve fitting with analytic model --------------------------------------- We initially fitted the differential light curves with analytic limb-darkened transit light curves [@MandelAgol] using a Markov Chain Monte Carlo (MCMC) algorithm, implemented through the <span style="font-variant:small-caps;">emcee</span> [@emcee] Python package. A quadratic limb darkening law was used and fit simultaneously with a long time-scale trend so as not to bias the derived radii. In the red and green channels, this trend was fit with a second order polynomial whilst in the blue it was fit as a function of airmass since it was clearly related to extinction. In order to fit the airmass term in the blue channel, an extinction coefficient that varied quadratically in time was used, which replicated the trend well. An extinction coefficient that varied linearly in time was also tested but could not fit the sharp downturn in the blue light curve at the end of the night (Fig. \[fig:mandel\_agol\]). The scaled semi major axis, $a/R_{*}$, inclination of the orbit and the time of mid transit, $T_{0}$, were tied across the three light curves when fitting. The parameters that were fit individually in each of the channels were the ratio of planet to star radius $R_{P}/R_{*}$, the second limb darkening coefficient $u2$, and the parameters defining the long time scale trend. The first limb darkening coefficient, $u1$, was held fixed in the fitting as there is a degeneracy between the limb darkening parameters which can affect the light curve solution [@Southworth2008]. The limb darkening coefficients and priors were chosen from the tables of [@Claret2011]. Uniform priors were adopted for all the model parameters, with the MCMC walkers started at the values from [@Hebrard2013]. The resulting fits of the analytic limb-darkened transit light curves are shown in Fig. \[fig:mandel\_agol\]. The strongest residual across the whole light curve in all three wavelengths is seen during transit and is consistent across the three bands. This residual is akin to the planet occulting areas of stellar activity. We considered the possibility that this anomaly could have been associated with the use of incorrect limb darkening coefficients but no choice of coefficients could replicate this feature. Fitting of star spot model {#sec:spot_model} -------------------------- The presence of the in-transit anomaly after the fitting of limb-darkened analytic light curves motivated the use of spot models. Star spot occultations have been seen in the transit light curves of several planets, including HD189733b (@Pont2007; @Sing2011), TrES-1b (@Rabus2009; @Dittmann2009), CoRoT-2b (@Wolter2009; @Huber2009; @Silva-Valio2010), HAT-P-11b (@Sanchis-Ojeda2011_hat11; @Deming2011; @Beky2014_hat11), WASP-4b (@Sanchis-Ojeda2011_wasp4; @Hoyer2013), WASP-19b (@Mancini2013; @Reed2013; @Huitson2013; @Mandell2013; @Sedaghati2015), HATS-2b [@Mohler2013], Kepler-63b [@Sanchis-Ojeda2013], Qatar-2b [@Mancini2014], and HAT-P-36b [@Mancini2015]. If a planet occults a spot (a region cooler than the surrounding photosphere) it blocks less of the stellar flux than compared with its transit across the hotter pristine stellar disc. This results in a bump during transit and therefore a smaller derived planetary radius. Star spot activity is not unexpected for WASP-52 since [@Hebrard2013] found modulations in its light curve and chromospheric emission peaks in the CaII H+K lines. Using these modulations they calculated the rotation period of WASP-52 to be 16.4 days.  Spot crossing events have been modelled with a variety of techniques, such as <span style="font-variant:small-caps;">prism</span> [@Reed2013], <span style="font-variant:small-caps;">soap-t</span> [@Oshagh2013] and <span style="font-variant:small-caps;">spotrod</span> [@Beky2014]. For this analysis we used <span style="font-variant:small-caps;">spotrod</span> to generate the spot affected light curves and wrote an MCMC wrapper around these generated light curves to fit them simultaneously across the bands and with the same long time-scale trends as before. We chose to use <span style="font-variant:small-caps;">spotrod</span> due to the speed of its integration, which uses polar coordinates in the projection plane. The integration with respect to the polar coordinate is done analytically so that only the integration with respect to the radial coordinate needs to be performed numerically. To calculate the projection of the planet on the stellar surface, <span style="font-variant:small-caps;">spotrod</span> calculates the arrays of planar orbital elements $\xi$ and $\eta$, using the formalism of [@Pal2009], and assumes the same limb darkening law for the spot as for the star. The system parameters were again fit across the three light curves simultaneously (although this time fitting for impact parameter, $b$, rather than the inclination, as required by <span style="font-variant:small-caps;">spotrod</span>) and with the addition of the parameters defining the spots. The fitting of one spot was tested but was unable to fit both bumps on either side of the transit mid-point, therefore two spots were used in further analysis. The parameters defining each spot were the longitude, latitude, radius ratio of spot to star, and ratio of the spot flux to stellar flux (with 1 being a spot with the same flux as the pristine photosphere and 0 being a spot with zero flux). We held $u1$ fixed as before but now also put Gaussian priors on $u2$ with means equal to the values from [@Claret2011] and standard deviations from the propagated errors in the effective temperature and surface gravity of the host star. This prior was necessary as the limb darkening and spot models can play off each other in trying to fit the transit shape. The MCMC was initiated with the system parameters equal to those in [@Hebrard2013] and was run for 10000 steps in burn in and another 10000 steps in the production run. There were 31 fitted parameters with 124 walkers. The parameters governing a spot’s characteristics are correlated with one another. The correlation between spot contrast and size has been shown previously by e.g. [@Pont2007], [@Wolter2009], [@Reed2013] and [@Beky2014]. We initiated the MCMC starting positions with spots at various, randomly selected, latitudes on the stellar surface so as not to bias the fits and test for convergence. After the burn in phase the error bars in the data points were rescaled to give a reduced $\chi^2$ of unity. After the first MCMC chain, a second MCMC was run but this time with the parameters that were tied across channels fixed to the results from the first run ($a/R_{*}$, $b$, $T_{0}$ and the spot sizes and positions). Correlations with these parameters cause $R_{P}/R_{*}$ to move up and down together across the three wavelengths, contributing to the uncertainty in the absolute planetary radius in each of the bands. Since we are concerned with the shape of the transmission spectrum, we are interested in the relative radii between the bands and not the absolute planetary radius, thus motivating the second run of the MCMC with fixed system parameters. We present the best fitting spot model in Fig. \[fig:spots\], after the second MCMC run, with the results in Table \[tab:fit\_results\] and transmission spectrum in Fig. \[fig:trans\_spec\] (blue squares). With the sizes and contrasts calculated from <span style="font-variant:small-caps;">spotrod</span>, we were able to create a schematic of the stellar surface (Fig. \[fig:stellar\_surface\], left panel) and consider what filling factor would reproduce the derived contrasts (section \[sec:spot\_properties\] and Fig. \[fig:stellar\_surface\], right panel). This figure displays the large regions of stellar activity along the transit chord. The second spot crossing event comprises of a smaller region of higher contrast (0.2 in the $g'$ band, Table \[tab:fit\_results\]). The error in the contrast of this dark spot also allowed for a larger, less dark spot.    Fitting of bright region model {#sec:facula_model} ------------------------------ As an alternative interpretation, the in-transit anomaly was also modelled as a bright feature analogous to Solar faculae. Solar faculae are bright regions on the solar photosphere which display limb brightening behaviour (e.g. @Unruh1999). They are often co-spatial with sunspots but not perfectly so (e.g. @Haywood2016). The effects of occultations of bright regions in exoplanet transits have been discussed by [@Oshagh2014] and could lead to an observable anomaly in transit data. There has not yet been any conclusive evidence of a facula occultation in a transit light curve although [@Mohler2013] found evidence for a hot spot in GROND photometry of HATS-2b. They detected a bright feature in the Sloan-$g$ band, which covered the CaII lines, that was consistent with a chromospheric plage occultation. <span style="font-variant:small-caps;">spotrod</span> was also used to model the facula scenario but instead of two individual spots with flux ratios $<1$, a single feature was modelled with a flux ratio of $>1$. This fitting method produced the fits seen in Fig. \[fig:plages\] with the results in Table \[tab:fit\_results\]. This model was able to reproduce the in-transit anomaly with an equally good fit as the two spot model but with six fewer parameters. The transmission spectrum resulting from the facula model is also shown in Fig. \[fig:trans\_spec\] (red triangles). In contrast to the flat transmission spectrum resulting from the fitting of spots, the fitting of a facula led to a slope in the planetary radius increasing towards the red (Fig. \[fig:trans\_spec\]).  [l c c c c ]{} Parameter & Symbol & $u'$ & $g'$ & NaI\ *2 spot model* & & & &\ Time of mid-transit (MJD)$^a$ & $T_0$ & — & 56178.12742 $ \pm 0.00005$ & —\ Scaled semi major axis$^a$ & $a/R_{*}$ & — & 6.99 $ \pm 0.04$ & —\ Impact parameter$^a$ & $b$ & — & 0.656 $^{+0.006}_{-0.007}$ & —\ Radius ratio & $R_{P}/R_{*}$ & 0.1759 $^{+0.0005}_{-0.0004}$ & 0.1751 $ \pm 0.0004$ & 0.1757 $ \pm 0.0008$\ Limb darkening coeff.$^b$ & $u_1$ & 0.643 & 0.529 & 0.381\ Limb darkening coeff. & $u_2$ & 0.169 $ \pm 0.008$ & 0.233 $ \pm 0.008$ & 0.290 $^{+0.009}_{-0.010}$\ Spot 1 x position$^a$ & $x_{1}$ & — & -0.19 $ \pm 0.03$ & —\ Spot 1 y position$^a$ & $y_{1}$ & — & 0.91 $^{+0.03}_{-0.04}$ & —\ Spot 1 radius ratio$^a$ & $r_{1}/R_{*}$ & — & 0.36 $^{+0.04}_{-0.03}$ & —\ Spot 1 contrast & $\rho_{1}$ & 0.61 $ \pm 0.02$ & 0.66 $ \pm 0.02$ & 0.63 $ \pm 0.04$\ Spot 2 x position$^a$ & $x_{2}$ & — & 0.38 $^{+0.02}_{-0.007}$ & —\ Spot 2 y position$^a$ & $y_{2}$ & — & 0.69 $^{+0.02}_{-0.03}$ & —\ Spot 2 radius ratio$^a$ & $r_{2}/R_{*}$ & — & 0.07 $^{+0.08}_{-0.006}$ & —\ Spot 2 contrast & $\rho_{2}$ & 0.17 $\pm 0.07$ & 0.19 $\pm 0.07$ & 0.3 $\pm 0.1$\ *Facula model* & & & &\ Time of mid-transit (MJD)$^a$ & $T_0$ & — & 56178.12740 $ \pm 0.00004$ & —\ Scaled semi major axis$^a$ & $a/R_{*}$ & — & 7.23 $ \pm 0.05$ & —\ Impact parameter$^a$ & $b$ & — & 0.593 $^{+0.008}_{-0.009}$ & —\ Radius ratio & $R_{P}/R_{*}$ & 0.1632 $\pm 0.0003$ & $0.1641 \pm 0.0003 $& 0.1657 $\pm 0.0006$\ Limb darkening coeff.$^b$ & $u_1$ & 0.643 & 0.529 & 0.381\ Limb darkening coeff. & $u_2$ & 0.161 $\pm 0.009$ & 0.239 $ \pm 0.009$ & $0.29 \pm 0.01$\ Facula x position$^a$ & $x_{1}$ & — & 0.123 $^{+0.007}_{-0.008}$ & —\ Facula y position$^a$ & $y_{1}$ & — & 0.62 $ \pm 0.03$ & —\ Facula radius ratio$^a$ & $r/R_{*}$ & — & 0.20 $^{+0.01}_{-0.02}$ & —\ Facula contrast & $\rho$ & 1.119 $\pm 0.007$ & 1.089 $\pm 0.008$ & $1.08 \pm 0.01$\ \ \ \ \ \ \ Discussion ========== Spots or faculae? {#sec:spots_vs_faculae} ----------------- It is difficult to distinguish between the spot and facula models of sections \[sec:spot\_model\] and \[sec:facula\_model\] using the quality of the fits alone. Application of the Bayesian Information Criterion (BIC; given by $BIC = \chi^2 + k\ln N$ with $k$ free parameters and $N$ data points) favours the facula model because of the similar $\chi^2$ and 6 fewer free parameters, with a value of $> 10$ lower than that of the spot model. However, our prior knowledge of transit light curves tends to favour the spot model because spots are more commonly detected (e.g. e.g. @Pont2007; @Sing2011; @Mancini2013; @Reed2013) than faculae or bright regions (e.g. @Mohler2013). A second approach to distinguishing between these models is to compare our fitted planet radius with that from independent studies. It can be seen in the model light curves of Fig. \[fig:spots\] and \[fig:plages\] that our spot and facula models imply different underlying depths of transit. In the spot model the anomaly is treated as two bumps (bottom panel of Fig. \[fig:spots\]), and so the underlying transit is deeper than in the facula model, where the anomaly is treated as a single dip (Fig. \[fig:plages\]). This difference can be seen in our fitted planet/star radius ratios (Table \[tab:fit\_results\]) where the spot model implies a planet radius that is 15 sigma larger than the value from the facula model. The planet/star radius ratio has been previously measured by [@Hebrard2013] using 6 individual transits, and found to be $0.1646 \pm 0.0012$. This is consistent with our value from the facula model (within 1 sigma), and inconsistent with our value from the spot model ($>6$ sigma discrepancy). While this paper was under review, another study of WASP-52b was published on arXiv [@Mancini_wasp52] including a further 8 transits. Spots were clearly detected in 5 of these light curves and the system parameters measured accounted for these spots. Again, we find that the planet/star radius ratio is consistent with a facula model (within 1 sigma) and inconsistent with our spot model (12 sigma discrepancy). We also consider the possibility that unocculted spots could be the cause of the discrepant $R_P/R*$ values we derive from the spot model but find this would require a total dimming of the star of 12%, which we consider unlikely. This is discussed in more detail in section \[sec:unocculted\]. We conclude that the comparison of our planet/star radius ratios with independent studies strongly favours the presence of a facula in our observed transit. Nevertheless, in sections \[sec:spot\_properties\] and \[sec:facula\_properties\] we discuss the implications of both models. ### Spot properties {#sec:spot_properties} The best fitting spot model indicates two large regions of stellar activity (Fig. \[fig:stellar\_surface\], left panel) which are consistent with the majority of the in-transit data being activity affected (Fig. \[fig:spots\]). Although spot modelling codes, including <span style="font-variant:small-caps;">spotrod</span>, fit spots as circular areas on the stellar surface, in reality these may be areas of several small spots which may be arranged in more complex configurations on the stellar surface. If the features in our light curves really were single spots with temperatures consistent with those expected for K stars, we would expect to see large amplitude bumps and a strong colour dependence. When considering the spots resulting from our modelling however, we believe that the spots are more accurately interpreted as active regions, with spots and pristine photosphere contained within, instead of two large individual spots (Fig. \[fig:stellar\_surface\]). When considering these complexes of smaller star spots, it is useful to consider the area ratio of spots to uncontaminated photosphere within each of these regions, which we define here as the filling factor, $f$. This quantity can be related to the spot’s contrast, $\rho$, through $$\label{eq:filling_factor} \rho = \frac{f F_{\bullet}(\lambda) + (1-f)F_{*}(\lambda)}{F_{*}(\lambda)}$$ where $F_{\bullet}$ is the flux of the spot and $F_{*}$ is the flux of the star. [@Berdyugina2005] plotted the observed temperature differences of spots for several different stellar effective temperatures. For K stars, the spot temperature differences ($\Delta$T) lie in the region of 1250 – 1500K. Due to the degeneracy between spot size and contrast the spot contrasts have large error bars in Fig. \[fig:stellar\_surface\] (right panel). However, by fixing the spot sizes and positions at the best fitting values from the first MCMC run, the second run gave the relative uncertainty in the spot contrasts across the bands rather than the absolute uncertainty, as with the planetary radii. When considering the relative spot contrasts (Fig. \[fig:stellar\_surface\], right panel, solid error bars) it can be seen that the contrasts and colour dependences of these active regions are consistent with a $\Delta$T of 1500K, given a single filling factor $f$. For the smaller spot (Fig. \[fig:stellar\_surface\], right panel, red error bars) we infer a higher filling factor (approximately 0.85), and for the larger spot (Fig. \[fig:stellar\_surface\], right panel, blue error bars) a lower filling factor (approximately 0.4). The larger spot can be understood as a large active region on the stellar surface with a relatively low density of smaller dark spots. The smaller spot however has a high density of smaller dark spots leading to the greater contrast. ### Facula properties {#sec:facula_properties} As discussed in section \[sec:spots\_vs\_faculae\] the occultation of a bright region, analogous to a Solar facula, is our favoured interpretation for these data. Unlike spots, the flux contrasts of faculae increase at the stellar limb as they display limb brightening behaviour (e.g. @Unruh1999) and so the assumption that the modelled spots follow the same limb darkening as the star does not hold in this case. Therefore Eqn. \[eq:filling\_factor\] cannot be used to calculate the temperature of the facula so we report only the flux contrasts of the facula in Table \[tab:fit\_results\] which indicate the hot spot is $\sim 10\%$ brighter in all three bands. Solar faculae are also able to display such contrasts however, at the Solar limb with a high viewing angle (e.g. @Unruh1999; @Ahern2000). At the viewing angle of our facula, we would only expect to see a $\sim3\%$ contrast if this were a Solar facula (e.g. @Ahern2000). However, since WASP-52 is a more active and later type star it is possible it could display higher contrast faculae. Measuring the residual red noise -------------------------------- In order to measure the systematic red noise in the residuals, we observed how the fractional root mean square (RMS) of the residuals with respect to the flux varied with binning of the data. Fig. \[fig:rms\_vs\_bins\] shows the RMS for a number of different bin sizes. The line overlaid corresponds to pure, Gaussian, white noise with a gradient equal to $1/\sqrt{N}$ where $N$ is the number of points per bin. The factor, $\beta$, quantifies how the actual binning gradient deviates from pure white noise [@Winn2008]. The $\beta$ factors found here show that the residuals do deviate from pure white noise but with amplitudes of only 0.5 mmag. ![The residual RMS scatter once the spot model has been subtracted for NaI (top panel), $g'$ (middle panel) and $u'$ (bottom panel). The error bars on the data points were rescaled to give a $\chi^{2}_{\nu}$ = 1. $\beta$ quantifies how the measured noise deviates from pure Gaussian white noise (solid black lines).[]{data-label="fig:rms_vs_bins"}](rms_vs_bins.pdf) Unocculted spots {#sec:unocculted} ---------------- Just as the effects of occulted spots or bright regions must be taken into account when calculating the planetary radius, so must the effects of those regions which are not seen in the transit. As the photometric modulation of Sun-like stars with high magnetic activity is dominated by dark spots (e.g. @Shapiro2014), we only consider the effect of unocculted spots on our derived transit depths. Unocculted spots cause the observed stellar flux to decrease, increasing the transit depth as compared to a spot free surface. This effect needs to be taken into account with a wavelength dependent depth correction. Following the formalism of [@Sing2011], the variation in transit depth, $\Delta d/d$, can be related to the fractional decrease of the stellar flux due to unocculted spots at a reference wavelength $\lambda_0$, $\Delta f(\lambda_{0},t)$, the flux of the pristine stellar disc, $F_{\lambda}^{T_{star}}$, and the flux of the spot, $F_{\lambda}^{T_{spot}}$, through $$\frac{\Delta d}{d} = \Delta f(\lambda_{0},t) \left(1 - \frac{F_{\lambda}^{T_{spot}}}{F_{\lambda}^{T_{star}}}\right)/\left(1 - \frac{F_{\lambda_0}^{T_{spot}}}{F_{\lambda_0}^{T_{star}}}\right)$$ leading to a variation in the ratio of planet radius to stellar radius of $$\Delta(R_{p}/R_{star}) \approx \frac{1}{2}\frac{\Delta d}{d}(R_p/R_{star})$$ We used stellar atmosphere models (ATLAS9, @Kurucz1993) to generate synthetic spectra of the star and a spot with the maximum temperature contrast of 1500K cooler than the surrounding photosphere [@Berdyugina2005]. If we rearrange the above equations for $\Delta f(\lambda_{0},t)$, we can estimate the percentage of total dimming required to bring our derived $R_P/R_*$ values into agreement with those of [@Hebrard2013]. We find that the total dimming must be $>12\%$ at a reference wavelength of 6000Å, considerably higher than the 1% amplitude measured by [@Hebrard2013]. As a result of this, we do not believe that unocculted spots are the cause of the discrepancy in the $R_P/R_*$ values resulting from the spot model. To calculate what effect unocculted spots have on the shape of our transmission spectra, we used a total dimming of 1% at a reference wavelength of 6000Åto calculate $\Delta(R_{P}/R_{*})$ in each of our 3 light curves. The difference in $\Delta(R_{P}/R_{*})$ between the red and the blue light curves is 10 times smaller than the 1 sigma error bars and so unocculted spots do not affect the shape of our transmission spectrum (Fig. \[fig:trans\_spec\]) or the conclusions we draw from it. Transmission spectrum {#sec:trans_spec} --------------------- Figure \[fig:trans\_spec\] displays the derived transmission spectrum for WASP-52b. On this plot are the results from modelling the in-transit anomaly both as spots and as a single facula. The errors in the transmission spectra are those after fitting with the wavelength independent parameters held fixed at the best fitting values from the first MCMC run ($b$, $a/R_{*}$, $T_0$, spot/facula sizes and positions). This resulted in errors of less than one planetary atmospheric scale height. In the presence of Rayleigh scattering, the expected slope of the planetary radius as a function of wavelength is given by [@Etangs2008_HD189]: $$\frac{dR_{p}}{d\ln\lambda} = \frac{k}{\mu g} \alpha T$$ where $\mu$ is the mean molecular mass of an atmospheric particle taken to be 2.3 times the mass of a proton, $k$ is the Boltzmann constant, $g$ is the planet’s surface gravity, $\alpha = -4$ as expected for Rayleigh scattering, and $T$ we take as the equilibrium temperature. Using our derived values for the planet’s surface gravity (Table \[tab:wasp52\]) we are able to rule out Rayleigh scattering in this atmosphere with $> 3\sigma$ confidence (Fig. \[fig:trans\_spec\]). The spot model gives a $\chi^2$ of 14.52 for Rayleigh scattering and 2.00 for a flat line, each with 2 degrees of freedom. Therefore the spot model strongly favours a flat transmission spectrum. For the case of the facula model, the transmission spectrum shows an increase in planetary radius towards the red which is not well fit by either a flat line or Rayleigh scattering. The implied increased radius in the red could indicate the detection of the broad wings of NaI. An absence of Rayleigh scattering has been seen in a handful of planets to date, such as HAT-P-32b [@Gibson2013_hat32], WASP-29b [@Gibson2013_wasp29] and HAT-P-19b [@Mallonn2015]. The absence of Rayleigh scattering and any broad spectral features has been attributed to clouds in the upper atmosphere of the planet obscuring any such features. It could be that clouds are masking the spectral features of WASP-52b also.  Updated system parameters ------------------------- As outlined in section \[sec:spots\_vs\_faculae\], we compared our results to those of [@Hebrard2013] (Table \[tab:wasp52\]). We note that [@Baluev2015] also studied WASP-52b in looking for transit timing variations but the results of the parameters that they fit for, $R_{P}/R_{*}$ and $b$, are within the 1$\sigma$ errors of [@Hebrard2013]. All the parameters in Table \[tab:wasp52\] marked with $^b$ were derived from the $g'$ band transit light curve alone, following the prescription of [@Seager2003]. The planet surface gravity was calculated using the equation from [@Southworth2007] which can be derived from the transit light curve but with the addition of the stellar reflex velocity for which we use the value from [@Hebrard2013]. The parameters that we have held fixed in our fitting (semi major axis and period) and those which we do not derive are left blank in Table \[tab:wasp52\]. The parameters marked with $^c$ were derived assuming a stellar radius equal to that of [@Hebrard2013]. As discussed in section \[sec:spots\_vs\_faculae\], the differing implied transit depths of the spot and facula models result in significantly different system parameters (Table \[tab:wasp52\]). When modelled as two spots, we find a transit depth $>3\sigma$ larger than that of [@Hebrard2013], whereas the facula model results in a transit depth consistent with that of [@Hebrard2013]. [|c|c|c|c|c|c|]{} Parameter (units) & Symbol & This work (Spots) & This work (Facula) & [@Hebrard2013]\ Orbital period (days) & $P$ & — & — & $ 1.7497798 \pm 0.0000012 $\ Semi major axis (AU) & $a$ & — & — & $ 0.0272 \pm 0.0003 $\ Orbital eccentricity & $e$ & — & — & 0 (fixed)\ Planetary equilibrium temperature (K) & $T_{\rm P}$ & — & — & $ 1315 \pm 35 $\ Transit epoch (HJD-2450000.0) (days)$^a$ & $T_0$ & $6178.62742 \pm 0.00005$ & $6178.62740 \pm 0.00004$ & $ 5793.68143 \pm 0.00009 $\ Scaled stellar radius $^a$ & $R_*/a$ & $0.1431 \pm 0.0009$ & $0.1383 \pm 0.0008 $ & $ 0.1355 \pm 0.0020 $\ Impact parameter ($R_*$) $^a$ & $b$ & 0.656 $^{+0.006}_{-0.007}$ & $ 0.593 ^{+0.008}_{-0.009}$ & $ 0.60 \pm 0.02 $\ Orbital inclination ($^\circ$) $^a$ & $i_p$ & $84.62 \pm 0.07$ & $85.30 \pm 0.08 $ & $ 85.35 \pm 0.20 $\ Transit duration (days)$^b$ & $t_T$ & $ 0.0777 \pm 0.0006$ & $0.0772 \pm 0.0006$ & $ 0.0754 \pm 0.0005 $\ Planet/star area ratio$^b$ & $(R_{\rm p}/R_*)^2$ & $0.0306 \pm 0.0005$ & $0.0270 \pm 0.0002 $ & $ 0.0271 \pm 0.0004 $\ Stellar density ($\rho_\odot$)$^b$ & $\rho_*$ & $1.50 \pm 0.03$ & $ 1.66 \pm 0.03 $ & $ 1.76 \pm 0.08 $\ Stellar radius (R$_\odot$) & $R_*$ & — & — & $ 0.79 \pm 0.02 $\ Planet radius (R$_{\rm J}$)$^c$ & $R_{\rm p}$ & $ 1.34 \pm 0.04$ & $ 1.26 \pm 0.03 $ & $ 1.27 \pm 0.03 $\ Stellar reflex velocity (km s$^{-1}$) & $K_1$ & — & — & $ 0.0843 \pm 0.0030 $\ Planet surface gravity (cgs) & $\log g_{\rm p}$ & $ 2.75\pm0.02 $ & $2.83 \pm 0.02$ & $ 2.81 \pm 0.03 $\ Planet density ($\rho_{\rm J}$)$^c$ & $\rho_{\rm p}$ & $ 0.383 \pm 0.024 $ & $0.204\pm0.009$ & $ 0.22 \pm 0.02 $\ Planet mass (M$_{\rm J}$)$^c$ & $M_{\rm p}$ & $ 0.38\pm0.02$ & $ 0.41 \pm 0.03 $ & $ 0.46 \pm 0.02 $\ \ \ \ \ Conclusions =========== In this paper we have measured the optical transmission spectrum of the highly inflated hot Jupiter WASP-52b. Our multi-wavelength light curves, observed with WHT/ULTRACAM, have demonstrated that 0.5 mmag precision can be achieved with high quality ground-based photometry. This leads to errors in our planetary radii of less than one planetary atmospheric scale height, comparable to recent HST observations (e.g. @Nikolov2014; @Sing2015; @Fischer2016). Our transit light curves revealed the presence of regions of stellar activity whose in-transit anomalies were modelled using <span style="font-variant:small-caps;">spotrod</span> [@Beky2014]. Although modelled as discrete events, it is likely that the planet in fact transits across latitudes of high stellar activity which may be near continuous across the transit chord. We find this activity can be most simply modelled as a bright region akin to Solar faculae, which results in system parameters consistent with independent studies. The light curves can also be fit with two regions of dark spots but this requires a planet/star radius ratio inconsistent with these studies. As a result, the occultation of a bright region on the host star is the favoured interpretation of the feature seen in the light curves. We find that Rayleigh scattering is not the dominant source of opacity within the planetary atmosphere. When modelling the in-transit anomalies as spots, we derive a transmission spectrum consistent with wavelength-independent absorption by clouds and find no evidence for any broad sodium absorption although we cannot rule out the presence of the narrow line core. For our favoured facula model, we find an increasing planetary radius towards the red optical which could be interpreted as evidence for the broad wings of NaI. This work highlights the need for high precision photometry at multiple wavelengths simultaneously to detect and study the effects of stellar activity, in the form of spots and bright regions, on the derived transit parameters. We thank the anonymous referee for their helpful comments which improved the quality of the paper. J.K. and T.L. are supported by Science and Technology Facilities Council (STFC) studentships. P.W. and T.R.M. are supported by an STFC consolidated grant (ST/L00073). D.J.A. acknowledges funding from the European Union Seventh Framework programme (FP7/2007- 2013) under grant agreement No. 313014 (ETAEARTH). V.S.D, S.P.L and ULTRACAM are supported by an STFC consolidated grant (ST/M001350). [^1]: [email protected] [^2]: [email protected] [^3]: http://deneb.astro.warwick.ac.uk/phsaap/software/ultracam/\ html/index.html

--- author: - 'Wen-Du Li' - 'and Wu-Sheng Dai' title: '[A duality of fields]{}\' --- [GBK]{}[song]{} [!!]{} [ !a! @toks= @toks= ]{} [ !b! @toks= @toks= ]{} [abstract[It is shown that there exists a duality among fields. If a field is dual to another field, the solution of the field can be obtained from the dual field by the duality transformation. We give a general result on the dual fields. Different fields may have different numbers of dual fields, e.g., the free field and the $\phi^{4}$-field are self-dual, the $\phi^{n}$-field has one dual field, a field with an $n$-term polynomial potential has $n+1$ dual fields, and a field with a nonpolynomial potential may have infinite number of dual fields. All fields which are dual to each other form a duality family. This implies that the field can be classified in the sense of duality, or, the duality family defines a duality class. Based on the duality relation, we can construct a high-efficiency approach for seeking the solution of field equations: solving one field in the duality family, all solutions of other fields in the family are obtained immediately by the duality transformation. As examples, we consider some $\phi^{n}$-fields, general polynomial-potential fields, and the sine-Gordon field. ]{}]{} Introduction ============ In this paper, we show that there exists a duality between fields. Furthermore, we show that the duality can serve as a method of solving field equations. Consider a scalar field with the potential $V\left( \phi\right) $. The Lagrangian is$$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-V\left( \phi\right)$$ and the field equation is$$\square\phi+m^{2}\phi+\frac{\partial V\left( \phi\right) }{\partial\phi}=0. \label{fieldeq}$$ It will be shown that the field with the potential $V\left( \phi\right) $ may have dual fields determined by some other potentials. A field may have different numbers of dual fields. A $\phi^{n}$-field has one dual field, a field with an $n$-term polynomial potential has $n+1$ dual fields, a field with a nonpolynomial potential may have infinite number of dual fields, and there are also self-dual fields. Dual fields form a duality family; all fields in the family are dual to each other. This allows us to classify fields based on the duality: a duality family forms a duality class. The duality can serve as a high-efficiency approach for seeking the solution of field equations. The duality relation relates a field and its dual field, so the duality relation can be used to find the solution of a field equation from the solution of the field equation of its dual field. In a duality family, if one field in the family is solved, then the solution of all other fields in the family can be obtained immediately by the duality transformation. Concretely, we construct the dual field of the $\phi^{n}$-field, the general-polynomial-field ($V\left( \phi\right) $ is a general polynomial of $\phi$), and the sine-Gordon field. In physics, the duality plays an important role. The duality bridges two different physical systems and reveals inherent relations in physics. Various dualities are found in many physical problems. The $S$-duality (strong–weak duality) relates a strongly coupled theory to an equivalent theory with a small coupling constant, such as the duality between two perturbed quantum many-body systems [@zarei2017strong], the $(2+1)$-dimensional duality of free Dirac or Majorana fermions and strongly-interacting bosonic Chern-Simons-matter theories [@chen2018strong], the electric-magnetic duality [@bae2001electric; @seiberg1995electric; @hatsuda1999electric], the duality of higher spin free massless gauge fields [@boulanger2003note], the duality in the couple of gauge field to gravity [@igarashi1998electric], and in string theory [@duff1994four; @font1990strong]. The AdS/CFT duality has been found in many physical areas [@maldacena1997large; @witten1998anti; @witten1998anti2; @aharony2000large; @d2004supersymmetric]. The fluid/gravity duality bridges fluid systems which is described by the Navier-Stokes equation and spacetime which is described by the Einstein equation [@bredberg2012navier; @hubeny2011fluid; @compere2011holographic; @hao2015flat; @quiros2000dual; @bhattacharyya2008nonlinear; @bhattacharyya2009forced; @bhattacharyya2008conformal; @ashok2014forced; @bhattacharyya2009incompressible; @compere2012relativistic; @pinzani2015towards; @wu2013fluid]. The gravoelectric duality is another duality of spacetime [@dadhich2002most; @nouri1999spacetime; @dadhich2000electromagnetic; @dadhich1998duality; @dadhich2000gravoelectric]. In string theory there is an T-duality [@alvarez1995introduction; @giveon1994target]. In section \[phin\], we consider the duality of the $\phi^{n}$-field. In section \[polynomial\], we consider the duality of the general polynomial potential. In section \[sine-Gordon\], we consider the duality of the sine-Gordon field. In section \[general\], we discuss an approach of the construction of dual fields. In section [@generalpotential], we give a general result on dual fields. In section \[example\], we provide some examples. The conclusion is given in section \[conclusion\]. Moreover, we add an appendix on the solution of field equations. The $\phi^{n}$-field \[phin\] ============================= In this section, we consider the duality of the $\phi^{n}$-field. The duality ----------- *Two scalar fields* $\phi\left( x^{\mu}\right) $ and $\varphi\left( y^{\mu}\right) $* with the potentials*$$\begin{aligned} V\left( \phi\right) & =\lambda\phi^{a},\label{Vphi}\\ U\left( \varphi\right) & =\eta\varphi^{A} \label{Uphiw}$$ *are dual to each other if*$$\frac{2}{2-a}=\frac{2-A}{2}. \label{power}$$ *The dual fields are related by the following duality relations:*$$\begin{aligned} \phi & \rightarrow\varphi^{\frac{a}{2-a}},\label{phitans}\\ x^{\mu} & \rightarrow\frac{2}{2-a}y^{\mu},\text{ \ }\mu=0,1,\ldots \label{xtans}$$ *and*$$\begin{aligned} \lambda & \rightarrow-\mathcal{G},\label{xie}\\ G & \rightarrow-\eta, \label{Eeta}$$ *where* $$\begin{aligned} G & =\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{a},\label{Ephi}\\ \mathcal{G} & =\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi +\frac{1}{2}m^{2}\varphi^{2}+\eta\varphi^{A} \label{Evarphi}$$ *are two Lorentz scalars corresponding to* $\phi$ *and* $\varphi$*, respectively.* The duality relation can be verified directly. The field equation of the potential (\[Vphi\]) is $$\square\phi+m^{2}\phi+a\lambda\phi^{a-1}=0. \label{phieqa}$$ The duality transformations (\[phitans\]) and (\[xtans\]) give$$\square\phi\rightarrow\frac{2-a}{2}\varphi^{\frac{a}{2-a}}\square\varphi +\frac{a}{2}\varphi^{\frac{2}{2-a}-2}\partial_{\mu}\varphi\partial^{\mu }\varphi. \label{2dtansx}$$ Substituting the transformation (\[2dtansx\]) into the field equation (\[phieqa\]) gives$$\frac{2-a}{2}\varphi^{\frac{a}{2-a}}\square\varphi+\frac{a}{2}\varphi ^{\frac{2}{2-a}-2}\partial_{\mu}\varphi\partial^{\mu}\phi+m^{2}\varphi ^{\frac{2}{2-a}}+a\lambda\varphi^{\frac{2\left( a-1\right) }{2-a}}=0. \label{equatans1}$$ The coupling constant $\lambda$ in the potential $V\left( \phi\right) $, by the duality transformation (\[xie\]), is replaced by:$$\lambda\rightarrow-\left( \frac{1}{2}\partial_{\mu}\varphi\partial^{\mu }\varphi+\frac{1}{2}m^{2}\varphi^{2}+\eta\varphi^{A}\right) . \label{lbdG}$$ Using the duality relation (\[power\]), we arrive at the field equation of $\varphi$,$$\square\varphi+m^{2}\varphi+A\eta\varphi^{A-1}=0. \label{varphieqa}$$ Solving field equations by means of the duality ----------------------------------------------- The duality relation bridges various fields and can serve as a method of solving field equations. Once a field equation is solved, its dual field is also solved. It can be checked that the field equation with the potential (\[phieqa\]) has an implicit solution (Appendix \[appendix\]):$$\beta_{\mu}x^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left( \frac{1}{2}m^{2}\phi^{2}+\lambda\phi^{a}-G\right) }}d\phi=0, \label{solphia}$$ where $\beta_{\mu}$ is a constant. Substituting the duality transformations (\[phitans\]) and (\[xtans\]) into the solution (\[solphia\]) gives$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\varphi^{2}+\left( -G\varphi^{-\frac{2a}{2-a}}\right) -\left( -\lambda\right) \right] }}d\varphi=0. \label{solivarphiA1}$$ This is just a solution of the field equation with $U\left( \varphi\right) =-G\varphi^{-\frac{2a}{2-a}}$. By the duality relations (\[power\]), (\[xie\]), (\[Eeta\]), and Eq. (\[Evarphi\]), we can see that this is the solution of the field equation (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left( \frac{1}{2}m^{2}\varphi^{2}+\eta\varphi^{A}-\mathcal{G}\right) }}d\varphi=0.$$ The general polynomial potential \[polynomial\] =============================================== In this section, we consider the duality between the field with general polynomial potentials. The general polynomial is a superposition power series containing arbitrary real-number powers. The duality ----------- *Two scalar fields* $\phi\left( x^{\mu}\right) $ and $\varphi\left( y^{\mu}\right) $* of the potentials*$$\begin{aligned} V\left( \phi\right) & =\lambda\phi^{a}+\sum_{n}\kappa_{n}\phi^{b_{n}},\label{GPPV}\\ U\left( \varphi\right) & =\eta\varphi^{A}+\sum_{n}\sigma_{n}\varphi ^{B_{n}} \label{GPPU}$$ *are dual to each other if*$$\begin{aligned} \frac{2-A}{2} & =\frac{2}{2-a},\label{aandA}\\ 2-b_{n} & =\frac{2}{2-A}\left( 2-B_{n}\right) \text{ or }\frac{2}{2-a}\left( 2-b_{n}\right) =2-B_{n}. \label{bandB}$$ *The dual fields are related by the following duality relations:*$$\begin{aligned} \phi & \rightarrow\varphi^{\frac{2}{2-a}},\label{phitanspol}\\ x^{\mu} & \rightarrow\frac{2}{2-a}y^{\mu},\text{ \ }\mu=0,1,\ldots \label{xtanspol}$$ *and*$$\begin{aligned} \lambda & \rightarrow-\mathcal{G},\label{lamG}\\ G & \rightarrow-\eta,\label{Geta}\\ \kappa_{n} & \rightarrow\sigma_{n}, \label{kapsig}$$ *where*$$\begin{aligned} G & =\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi^{2}+V\left( \phi\right) ,\label{Gphin}\\ \mathcal{G} & =\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi +\frac{1}{2}m^{2}\varphi^{2}+U\left( \varphi\right) \label{Ghua}$$ *are two Lorentz scalars corresponding to* $\phi$ *and* $\varphi$*, respectively.* The field equation with the potential (\[GPPV\]) is$$\square\phi+m^{2}\phi+a\lambda\phi^{a-1}+\sum_{n}b_{n}\kappa_{n}\phi^{b_{n}-1}=0. \label{phieqan}$$ The duality transformations (\[phitanspol\]) and (\[xtanspol\]) give$$\square\phi\rightarrow\frac{2-a}{2}\varphi^{\frac{a}{2-a}}\square\varphi +\frac{a}{2}\varphi^{\frac{2}{2-a}-2}\partial_{\mu}\varphi\partial^{\mu }\varphi. \label{2dtansxn}$$ Substituting the transformation into the field equation (\[phieqan\]) gives an equation of $\varphi$:$$\frac{2-a}{2}\varphi^{\frac{a}{2-a}}\square\varphi+\frac{a}{2}\varphi ^{\frac{2}{2-a}-2}\partial_{\mu}\varphi\partial^{\mu}\phi+m^{2}\varphi ^{\frac{2}{2-a}}+a\lambda\varphi^{\frac{2\left( a-1\right) }{2-a}}+\sum _{n}b_{n}\kappa_{n}\varphi^{\frac{2\left( b_{n}-1\right) }{2-a}}=0. \label{equatansn1}$$ The transformation of the coupling constant is given by the duality transformation (\[lamG\]):$$\lambda\rightarrow-\left( \frac{1}{2}\partial_{\mu}\varphi\partial^{\mu }\varphi+\frac{1}{2}m^{2}\varphi^{2}+\eta\varphi^{A}+\sum_{n}\sigma_{n}\varphi^{B_{n}}\right) .$$ Then we arrive at the field equation of $U\left( \varphi\right) =\eta \varphi^{A}+\sum_{n}\sigma_{n}\phi^{B_{n}}$ with the duality relations (\[aandA\]), (\[bandB\]), and (\[kapsig\]): $$\square\varphi+m^{2}\varphi+A\eta\varphi^{A-1}+\sum_{n}B_{n}\sigma_{n}\varphi^{B_{n}-1}=0.$$ It can be seen from the potential (\[GPPV\]) that in a general polynomial potential $V\left( \phi\right) =\lambda\phi^{a}+\sum_{n}\kappa_{n}\phi^{b_{n}}$, every term in the potential can be chosen as the first term in $V\left( \phi\right) $ to play the role of $\lambda\phi^{a}$. Different choices give different dual fields. For an $n$-term polynomial potential, there are $n$ choices. Therefore, all potentials who are dual to each other form a duality family with $n+1$ members. Solving field equations by means of the duality ----------------------------------------------- Similarly, the field equation (\[phieqan\]) has an implicit solution:$$\beta_{\mu}x^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\phi^{2}+\left( \lambda\phi^{a}+\sum_{n}\kappa_{n}\phi^{b_{n}}\right) -G\right] }}d\phi=0. \label{solpol}$$ Substituting the duality transformations (\[phitanspol\]) and (\[xtanspol\]) into the solution (\[solpol\]) and using the duality relations (\[aandA\]) and (\[bandB\]), we arrive at$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\varphi^{2}+\left( -G\varphi^{-\frac{2a}{2-a}}+\sum_{n}\kappa _{n}\varphi^{\frac{2b_{n}-2a}{2-a}}\right) -\left( -\lambda\right) \right] }}d\varphi=0. \label{solpol2}$$ This is just a solution of the field equation with $U\left( \varphi\right) =-G\varphi^{-\frac{2a}{2-a}}+\sum_{n}\kappa_{n}\varphi^{\frac{2b_{n}-2a}{2-a}}$. By the duality relations (\[aandA\]), (\[bandB\]), (\[lamG\]), (\[Geta\]), (\[kapsig\]), and Eq. (\[Ghua\]), we can see that this is the solution of the field equation with the potential (\[varphieqa\]):$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\varphi^{2}+U\left( \varphi\right) -\mathcal{G}\right] }}d\varphi=0.$$ Obviously, for an $n$-term polynomial potential, once a potential in the duality family is solved, the other $n$ potentials are immediately solved. The sine-Gordon equation \[sine-Gordon\] ======================================== The sine-Gordon potential is not a polynomial potential. We first expand the sine-Gordon potential as a power series. Each term in the expansion is a power function, which has a duality discussed in section \[phin\]. Calculating the duality of each term of the series and then summing up the series give the dual potential of the sine-Gordon potential. The duality ----------- The sine-Gordon equation is [@Rajaraman1987solitons]$$\square\phi+\frac{m^{3}}{\sqrt{\lambda}}\sin\left( \frac{\sqrt{\lambda}}{m}\phi\right) =0. \label{SinGeq}$$ First rewrite the sine-Gordon Lagrangian, $\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{m^{4}}{\lambda}\left[ 1-\cos\left( \frac{\sqrt{\lambda}}{m}\phi\right) \right] $, as$$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-V\left( \phi\right)$$ with $$V\left( \phi\right) =\frac{m^{4}}{\lambda}\left[ 1-\cos\left( \frac {\sqrt{\lambda}}{m}\phi\right) \right] -\frac{1}{2}m^{2}\phi^{2}. \label{Vphimphi2}$$ Expanding $V\left( \phi\right) $ gives$$V\left( \phi\right) =-\frac{\lambda}{24}\phi^{4}-\sum_{n=3}^{\infty}\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}\phi^{2n}. \label{Veff}$$ The leading term in Eq. (\[Veff\]) is the $\phi^{4}$ term. First regarding the expansion (\[Veff\]) as a polynomial (though it is indeed a series), we can obtain the dual potential by the duality transformation of the general polynomial potential, Eqs. (\[aandA\]), (\[bandB\]), (\[lamG\]), (\[Geta\]), and (\[kapsig\]):$$\begin{aligned} A & =4,\text{ \ }B_{n}=4-2n,\text{ \ }n=3,4,5,\ldots,\nonumber\\ \eta & =-G,\text{ \ }\sigma_{n}=-\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}. \label{sineGT}$$ The duality transformations here, by Eqs. (\[phitanspol\]) and (\[xtanspol\]), are $$\begin{aligned} \phi & \rightarrow\varphi^{-1},\label{SinKphiphi}\\ x^{\mu} & \rightarrow-y^{\mu}. \label{SinKxy}$$ The potential of the dual field can be obtained by substituting the duality relation (\[sineGT\]) into Eq. (\[GPPU\]) and then summing up the series:$$\begin{aligned} U\left( \varphi\right) & =-G\varphi^{4}-\sum_{n=3}^{\infty}\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}\varphi^{4-2n}\nonumber\\ & =-G\varphi^{4}+\frac{m^{4}}{\lambda}\varphi^{4}\left( 1+\cos\frac {\sqrt{\lambda}}{m\varphi}\right) -\frac{1}{2}m^{2}\varphi^{2}. \label{SinGUphi}$$ The field equation of the dual field then reads$$\square\phi-4G\varphi^{3}+\frac{4m^{4}}{\lambda}\varphi^{3}\left( 1-\cos \frac{\sqrt{\lambda}}{m\varphi}\right) -\frac{m^{3}}{\sqrt{\lambda}}\varphi^{3}\sin\frac{\sqrt{\lambda}}{m\varphi}=0.$$ Different solution of $\phi$ gives different $G$ and different $G$ gives different dual potentials. The expansion (\[Veff\]) has infinite terms, so the duality family has infinite members. ### The $1+n$-dimensional solution The sine-Gordon equation (\[SinGeq\]) has a $1+n$-dimensional solution [@Rajaraman1987solitons] $$\phi=\frac{4m}{\sqrt{\lambda}}\operatorname{arccot}\left( i\cot\left( \frac{\beta_{\mu}x^{\mu}}{2\sqrt{-\beta^{2}}}m\right) \right) . \label{sgslo}$$ Substituting the solution (\[sgslo\]) into Eq. (\[Gphin\]) gives $$G=\frac{2m^{4}}{\lambda}.$$ Then the potential of the dual field, by Eq. (\[SinGUphi\]), reads $$U\left( \varphi\right) =-\frac{m^{4}}{\lambda}\varphi^{4}\left( 1+\cos \frac{\sqrt{\lambda}}{m\varphi}\right) -\frac{1}{2}m^{2}\varphi^{2}. \label{Uvarphi}$$ The field equation of the dual field is then$$\square\varphi+\frac{4m^{4}}{\lambda}\varphi^{3}\left( 1+\cos\frac {\sqrt{\lambda}}{m\varphi}\right) +\frac{m^{3}}{\sqrt{\lambda}}\varphi ^{2}\sin\frac{\sqrt{\lambda}}{m\varphi}=0.$$ The solution of the dual field can be obtained by substituting the duality transformations (\[phitanspol\]) and (\[xtanspol\]) into the solution (\[sgslo\]):$$\varphi=\left[ \frac{4m}{\sqrt{\lambda}}\operatorname{arccot}\left( i\cot\left( \frac{-\beta_{\mu}y^{\mu}}{2\sqrt{-\beta^{2}}}m\right) \right) \right] ^{-1}.$$ ### The $1+1$-dimensional solution The sine-Gordon equation (\[SinGeq\]) also has a $1+1$-dimensional solution [@ablowitz1973method; @ablowitz1991solitons],$$\phi=\frac{4m}{\sqrt{\lambda}}\arctan\left( \frac{\cos\left( \frac{\sqrt{2}}{2}mt\right) }{\cosh\left( \frac{\sqrt{2}}{2}mx\right) }\right) . \label{sgslo1}$$ Substituting the solution (\[sgslo\]) into (\[Gphin\]) gives$$G=\frac{8m^{4}}{\lambda}\left[ \cos\left( \sqrt{2}mt\right) +\cosh\left( \sqrt{2}mx\right) +2\right] ^{-1}.$$ The potential of the dual field, by Eq. (\[SinGUphi\]), reads$$U\left( \varphi\right) =-\frac{8m^{4}}{\lambda}\left[ \cos\left( \sqrt {2}mt\right) +\cosh\left( \sqrt{2}mx\right) +2\right] ^{-1}\varphi ^{4}+\frac{m^{4}}{\lambda}\varphi^{4}\left( 1-\cos\left( \frac{\sqrt {\lambda}}{m\varphi}\right) \right) -\frac{1}{2}m^{2}\varphi^{2}. \label{Uvarphibian}$$ The field equation of the dual field is$$\square\varphi-\frac{32m^{4}}{\lambda}\left[ \cos\left( \sqrt{2}mt\right) +\cosh\left( \sqrt{2}mx\right) +2\right] ^{-1}\varphi^{3}+\frac{4m^{4}}{\lambda}\varphi^{3}\left( 1-\cos\frac{\sqrt{\lambda}}{m\varphi}\right) -\frac{m^{3}}{\sqrt{\lambda}}\varphi^{3}\sin\frac{\sqrt{\lambda}}{m\varphi}=0. \label{Uvarphibianeq}$$ The solution of the dual field can be obtained by substituting the duality transformations (\[SinKphiphi\]) and (\[SinKxy\]) into the solution (\[sgslo1\]):$$\varphi=\left[ \frac{4m}{\sqrt{\lambda}}\arctan\left( \frac{\cos\left( \frac{\sqrt{2}}{2}mt\right) }{\cosh\left( \frac{\sqrt{2}}{2}mx\right) }\right) \right] ^{-1}.$$ This is the solution of the field equation (\[Uvarphibianeq\]). The duality family of the sine-Gordon field ------------------------------------------- In the above, by regarding the expansion of the sine-Gordon potential as a polynomial potential, we obtain the dual potential of the sine-Gordon potential. In section \[polynomial\], we write a polynomial potential in the form of $V\left( \phi\right) =\lambda\phi ^{a}+\sum_{n}\kappa_{n}\phi^{b_{n}}$, in which the first term $\lambda\phi ^{a}$ is arbitrarily chosen from the polynomial and the other terms in the polynomial are incorporated into the sum. However, every term in $V\left( \phi\right) $ can be chosen as the first term and different choices lead to different dual potentials. In the following, we give a general discussion on the dual potential of the sine-Gordon potential by choosing different terms in the expansion as the first term. Rewriting the expansion of $V\left( \phi\right) $, Eq. (\[Vphimphi2\]), as $$V\left( \phi\right) =-\frac{\left( -1\right) ^{\chi}}{\left( 2\chi\right) !}\frac{\lambda^{\chi-1}}{m^{2\left( \chi-2\right) }}\phi^{2\chi}-\sum_{n=2\left( n\neq\chi\right) }^{\infty}\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}\phi^{2n},\ \chi=2,3,4\cdots. \label{Veff1}$$ It can seen that the expansion (\[Vphimphi2\]) is just the special case of Eq. (\[Veff1\]) with $\chi=2$. Similarly, regarding the expansion (\[Veff1\]) as a “polynomial”, we obtain the dual potential by the duality transformation of the general polynomial potential, Eqs. (\[aandA\]), (\[bandB\]), (\[lamG\]), (\[Geta\]), and (\[kapsig\]):$$\begin{aligned} A & =\frac{2\chi}{\chi-1},\text{ \ }B_{n}=\frac{2\left( n-\chi\right) }{1-\chi},\text{ \ }\nonumber\\ \eta & =-G,\text{ \ }\sigma_{n}=-\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }},\text{\ }n=2,3,4,\ldots,\text{ and }n\neq\chi. \label{sigkappa}$$ The duality transformations (\[phitanspol\]) and (\[xtanspol\]) then become$$\begin{aligned} \phi & \rightarrow\varphi^{\frac{1}{1-\chi}},\label{phiphi}\\ x^{\mu} & \rightarrow\frac{1}{1-\chi}y^{\mu}. \label{xtoy}$$ Substituting the duality relation into Eq. (\[GPPU\]) and summing up the series give the dual potential of $V\left( \phi\right) $ $$\begin{aligned} U\left( \varphi\right) & =-G\varphi^{\frac{2\chi}{\chi-1}}-\sum _{n=2^{\prime}}^{\infty}\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}\varphi^{\frac{2\left( n-\chi\right) }{1-\chi}}\\ & =-G\varphi^{\frac{2\chi}{\chi-1}}-\frac{m^{4}}{\lambda}\varphi^{\frac {2\chi}{\chi-1}}\left[ \cos\left( \frac{\sqrt{\lambda}}{m}\varphi^{\frac {1}{1-\chi}}\right) -1\right] -\frac{1}{2}m^{2}\varphi^{2}.\end{aligned}$$ The field equation is then$$\square\varphi-\frac{2\chi}{\chi-1}G\varphi^{\frac{\chi+1}{\chi-1}}-\frac{2\chi}{\chi-1}\frac{m^{4}}{\lambda}\varphi^{\frac{\chi+1}{\chi-1}}\left[ \cos\left( \frac{\sqrt{\lambda}}{m}\varphi^{\frac{1}{1-\chi}}\right) -1\right] +\frac{1}{1-\chi}\frac{m^{3}}{\sqrt{\lambda}}\varphi^{\frac{\chi}{\chi-1}}\sin\left( \frac{\sqrt{\lambda}}{m}\varphi^{\frac{1}{1-\chi}}\right) =0.$$ Substituting the solution (\[sgslo\]) into Eq. (\[Gphin\]) gives$$G=\frac{2m^{4}}{\lambda}.$$ Then we arrive at$$\begin{aligned} U\left( \varphi\right) & =-\frac{2m^{4}}{\lambda}\varphi^{\frac{2\chi }{\chi-1}}-\sum_{n=2^{\prime}}^{\infty}\frac{\left( -1\right) ^{n}}{\left( 2n\right) !}\frac{\lambda^{n-1}}{m^{2\left( n-2\right) }}\varphi ^{\frac{2\left( n-\chi\right) }{1-\chi}}\nonumber\\ & =-\frac{m^{4}}{\lambda}\varphi^{\frac{2\chi}{\chi-1}}\left( 1+\cos \frac{\sqrt{\lambda}}{m}\varphi^{\frac{1}{1-\chi}}\right) -\frac{1}{2}m^{2}\varphi^{2}-\left( -\frac{\lambda}{m^{4}}\right) ^{\chi-1}\frac{m^{2\chi}}{\left( 2\chi\right) !}.\end{aligned}$$ The field equation of the dual field then reads $$\square\varphi-\frac{2\chi}{\chi-1}\frac{m^{4}}{\lambda}\varphi^{\frac{\chi +1}{\chi-1}}\left[ \cos\left( \frac{\sqrt{\lambda}}{m}\varphi^{\frac {1}{1-\chi}}\right) +1\right] +\frac{1}{1-\chi}\frac{m^{3}}{\sqrt{\lambda}}\varphi^{\frac{\chi}{\chi-1}}\sin\left( \frac{\sqrt{\lambda}}{m}\varphi^{\frac{1}{1-\chi}}\right) =0.$$ Substituting the duality relations (\[phiphi\]) and (\[xtoy\]) into the solution (\[sgslo\]) gives the solution of the dual potential:$$\varphi=\left[ \frac{4m}{\sqrt{\lambda}}\operatorname{arccot}\left( i\cot\left( -\frac{1}{1-\chi}\frac{\beta_{\mu}y^{\mu}}{2\sqrt{-\beta^{2}}}m\right) \right) \right] ^{1-\chi}.$$ Fields with general potentials \[generalpotential\] =================================================== For nonpolynomial potentials, like that in the case of the sine-Gordon fields, we can first expand the potential as a series, then construct the dual field with the help of the duality relation of the general polynomial potential given in section \[polynomial\], and sum up the series. Expand the potential $V\left( \phi\right) $ as$$V\left( \phi\right) =\lambda\phi^{a}+\sum_{n}\kappa_{n}\phi^{b_{n}}. \label{Vphiseries}$$ Using the duality relations (\[aandA\]), (\[bandB\]), (\[lamG\]), (\[Geta\]), and (\[kapsig\]), we obtain the series of the dual potential,$$\begin{aligned} U\left( \varphi\right) & =-G\varphi^{\frac{2a}{a-2}}+\sum_{n}\kappa _{n}\varphi^{\frac{2a}{a-2}-\frac{2b_{n}}{a-2}}\nonumber\\ & =-G\left( \varphi^{\frac{2}{2-a}}\right) ^{-a}+\left( \varphi^{\frac {2}{2-a}}\right) ^{-a}\sum_{n}\kappa_{n}\left( \varphi^{\frac{2}{2-a}}\right) ^{b_{n}}. \label{Uvarphiseries1}$$ By Eq. (\[Vphiseries\]) we have $\sum_{n}\kappa_{n}\phi^{b_{n}}=V\left( \phi\right) -\lambda\phi^{a}$, so the series in Eq. (\[Uvarphiseries1\]) can be summed up as $$\begin{aligned} U\left( \varphi\right) & =-G\left( \varphi^{\frac{2}{2-a}}\right) ^{-a}+\left( \varphi^{\frac{2}{2-a}}\right) ^{-a}\left[ V\left( \varphi^{\frac{2}{2-a}}\right) -\lambda\left( \varphi^{\frac{2}{2-a}}\right) ^{a}\right] \nonumber\\ & =-G\varphi^{\frac{2a}{a-2}}+\varphi^{\frac{2a}{a-2}}V\left( \varphi ^{\frac{2}{2-a}}\right) -\lambda.\end{aligned}$$ This is a general result on dual fields. Constructing the dual fields from the solution \[general\] =========================================================== In this section, we discuss an approach of constructing the dual field from the solution. In the implicit solution of the field equation (\[fieldeq\]),$$\beta_{\mu}x^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\phi^{2}+V\left( \phi\right) -G\right] }}d\phi=0, \label{solution}$$ by writing the duality transformation in the following general form, $$\begin{aligned} \phi & \rightarrow f\left( \varphi\right) ,\\ x^{\mu} & \rightarrow g^{\mu}\left( y\right) ,\end{aligned}$$ the solution (\[solution\]) becomes$$\beta_{\mu}g^{\mu}\left( y\right) +\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}f^{2}\left( \varphi\right) +V\left( f\left( \varphi\right) \right) -G\right] }}f^{\prime}\left( \varphi\right) d\varphi=0. \label{dualtr1}$$ In order that Eq. (\[dualtr1\]) is still a solution of a field equation, a simple choice is$$g^{\mu}\left( y\right) =\chi y^{\mu} \label{gyxy}$$ with $\chi$ a constant. Then Eq. (\[dualtr1\]) with the duality transformation (\[gyxy\]) becomes$$\beta_{\mu}y^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left\{ \frac{1}{2}m^{2}\chi2\frac{f^{2}\left( \varphi\right) }{\left[ f^{\prime}\left( \varphi\right) \right] ^{2}}+\frac{\chi^{2}}{\left[ f^{\prime}\left( \varphi\right) \right] ^{2}}V\left[ f\left( \varphi\right) \right] -G\frac{\chi^{2}}{\left[ f^{\prime}\left( \varphi\right) \right] ^{2}}\right\} }}d\varphi=0. \label{dualtr2}$$ Requiring that Eq. (\[dualtr2\]) is still a solution of the field equation and comparing Eqs. (\[solphia\]) and (\[dualtr2\]), we can choose (1) one of the three terms as the mass term which should be in proportion to $\varphi^{2}$, (2) one of the other terms as the constant term, and (3) the remaining term as the potential term $U\left( \varphi\right) $. Different choices give different dual fields. The duality of polynomial fields: examples \[example\] ====================================================== In this section, we give some examples of the method of solving the field equation by the duality transformation. The self-duality: free fields and $\phi^{4}$-fields --------------------------------------------------- The self-duality means that the dual field of a field is the field itself. The free field and the $\phi^{4}$-field are self-dual. This can be seen directly from the duality relation (\[power\]): if $a=0$, then $A=0$; if $a=$ $4$, then $A=4$. Take the $\phi^{4}$-field as an example. The field equation of the $\phi^{4}$-field is $$\square\phi+m^{2}\phi+4\lambda\phi^{3}=0, \label{p4}$$ which has a soliton solution [@polyanin2016handbook], $$\phi=\frac{im}{2\sqrt{\lambda}}\tanh\left( \alpha t+\beta x_{1}+\gamma x_{2}-\frac{x_{3}}{2}\sqrt{4\alpha^{2}-4\beta^{2}-4\gamma^{2}-2m^{2}}+\delta\right) . \label{p4sol}$$ For the $\phi^{4}$-field potential, by Eqs. (\[phitans\]) and (\[xtans\]), the duality transformations are$$\begin{aligned} \phi & \rightarrow\varphi^{-1},\label{phi4phitrans}\\ x^{\mu} & \rightarrow-y^{\mu}. \label{xtoyphi4}$$ The field equation (\[p4\]) under the duality transformation becomes$$-\varphi^{-2}\square\varphi+2\varphi^{-3}\partial_{\mu}\varphi\partial^{\mu }\phi+m^{2}\varphi^{-1}+4\lambda\varphi^{-3}=0. \label{P41}$$ The coupling constant $\lambda$, by the duality relation (\[xie\]) and Eq. (\[Evarphi\]), should be replaced by$$\lambda\rightarrow-\frac{1}{2}\partial_{\mu}\varphi\partial^{\mu}\varphi -\frac{1}{2}m^{2}\varphi^{2}-\eta\varphi^{4}. \label{kappap4}$$ Substituting the replacement (\[kappap4\]) into the field equation (\[P41\]) gives $$\square\varphi+m^{2}\varphi+4\eta\varphi^{3}=0,$$ which is the field equation with $U\left( \varphi\right) =\eta\varphi^{4}$. The duality transformation of the solution is straightforward. Substituting the solution (\[p4sol\]) into Eq. (\[Ephi\]) gives$$G=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{m^{2}}{2}\phi ^{2}+\lambda\varphi^{4}=-\frac{m^{4}}{16\lambda}.$$ Substituting the transformations (\[phi4phitrans\]) and (\[xtoyphi4\]) into the solution (\[p4sol\]) and using Eq. (\[Eeta\]) give the solution of $U\left( \varphi\right) =\frac{m^{4}}{16\lambda}\varphi^{4}$:$$\varphi=-\frac{2\sqrt{\lambda}}{im}\coth\left( \alpha\tau+\beta y_{1}+\gamma y_{2}-\frac{y_{3}}{2}\sqrt{4\alpha^{2}-4\beta^{2}-4\gamma^{2}-2m^{2}}-\delta\right) . \label{varphisol1}$$ Putting $\frac{m^{4}}{16\lambda}=\eta$, we obtain the solution of $U\left( \varphi\right) =\eta\varphi^{4}$:$$\begin{aligned} \varphi & =\frac{im}{2\sqrt{\eta}}\coth\left( \alpha\tau+\beta y_{1}+\gamma y_{2}-\frac{y_{3}}{2}\sqrt{4\alpha^{2}-4\beta^{2}-4\gamma^{2}-2m^{2}}-\delta\right) \nonumber\\ & =\frac{im}{2\sqrt{\eta}}\tanh\left( \alpha\tau+\beta y_{1}+\gamma y_{2}-\frac{y_{3}}{2}\sqrt{4\alpha^{2}-4\beta^{2}-4\gamma^{2}-2m^{2}}+\delta^{\prime}\right)\end{aligned}$$ with the constant $\delta^{\prime}=-\delta+i\frac{\pi}{2}$. The $\phi^{1}$-field and the $\phi^{-2}$-field ---------------------------------------------- The dual field of the $\phi^{1}$-field, by the duality relation (\[power\]), is the $\phi^{-2}$-field, i.e., the fields $$\begin{aligned} V\left( \phi\right) & =\lambda\phi,\label{Vphi1}\\ U\left( \varphi\right) & =\eta\varphi^{-2} \label{Uphiwm2}$$ are dual to each other. The duality transformations, by Eqs. (\[phitans\]) , (\[xtans\]), and (\[Eeta\]), are$$\begin{aligned} \phi & \rightarrow\varphi^{2},\label{Tphiphiw}\\ x^{\mu} & \rightarrow2y^{\mu},\label{Txy}\\ \eta & \rightarrow-G. \label{etamG}$$ For the field $\phi$ with the potential (\[Vphi1\]),$$G=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi ^{2}+\lambda\phi. \label{hphi1}$$ ### The $1+3$-dimensional solution The field equation of $\phi$, Eq. (\[Vphi1\]), has a solution$$\phi=\exp\left( \alpha t+\beta x_{1}+\gamma x_{2}+i\sqrt{-m^{2}-\alpha ^{2}+\beta^{2}+\gamma^{2}}x_{3}\right) -\frac{\lambda}{m^{2}}. \label{phi4dsol}$$ Substituting the solution (\[phi4dsol\]) into Eq. (\[hphi1\]) gives $$G=-\frac{\lambda^{2}}{2m^{2}}. \label{G1m2}$$ Substituting the duality transformations (\[Tphiphiw\]) and (\[Txy\]) into the solution (\[phi4dsol\]) and using the duality relation (\[etamG\]), we arrive at the solution of $U\left( \varphi\right) =\frac{\lambda^{2}}{2m^{2}}\varphi^{-2}$: $$\varphi=\left[ \exp\left( 2\alpha\tau+2\beta y_{1}+2\gamma y_{2}+2i\sqrt{-m^{2}-\alpha^{2}+\beta^{2}+\gamma^{2}}y_{3}\right) -\frac{\lambda }{m^{2}}\right] ^{1/2}. \label{varphi4dsol}$$ This is just the solution of the dual potential $U\left( \varphi\right) =\eta\varphi^{-2}$ with $\frac{\lambda^{2}}{2m^{2}}=\eta$: $$\varphi=\left[ \exp\left( 2\alpha\tau+2\beta y_{1}+2\gamma y_{2}+2i\sqrt{-m^{2}-\alpha^{2}+\beta^{2}+\gamma^{2}}y_{3}\right) -\frac {\sqrt{2\eta}}{m}\right] ^{1/2}.$$ ### The $1+1$-dimensional solution The field equation of $\phi$, Eq. (\[Vphi1\]), has a $1+1$-dimensional solution,$$\phi=e^{\alpha t}\sinh\left( \sqrt{\alpha^{2}+m^{2}}x\right) -\frac{\lambda }{m^{2}}. \label{phi4dsol2}$$ Substituting the solution (\[phi4dsol2\]) into Eq. (\[hphi1\]) gives$$G=-\frac{\lambda^{2}}{2m^{2}}-\frac{\alpha^{2}+m^{2}}{2}e^{2\alpha t}.$$ Substituting the duality transformations (\[Tphiphiw\]) and (\[Txy\]) into the solution (\[phi4dsol2\]) and using the duality relation (\[etamG\]), we arrive at the solution of $U\left( \varphi\right) =\left( \frac {\lambda^{2}}{2m^{2}}+\frac{\alpha^{2}+m^{2}}{2}e^{2\alpha t}\right) \varphi^{-2}$: $$\varphi=\left[ e^{2\alpha\tau}\sinh\left( 2\sqrt{\alpha^{2}+m^{2}}y\right) -\frac{\lambda}{m^{2}}\right] ^{1/2}.$$ The $\phi^{3}$-field and the $\phi^{6}$-field --------------------------------------------- The dual field of the $\phi^{3}$-field, by the duality relation (\[power\]), is the $\phi^{6}$-field, i.e., the fields $$\begin{aligned} V\left( \phi\right) & =\lambda\phi^{3},\label{Vphi3}\\ U\left( \varphi\right) & =\eta\phi^{6} \label{Uphi6}$$ are dual to each other. The duality transformations by Eqs. (\[phitans\]) , (\[xtans\]), and (\[Eeta\]) are$$\begin{aligned} \phi & \rightarrow\varphi^{-2},\label{Tphiphiw36}\\ x^{\mu} & \rightarrow-2y^{\mu}. \label{Txy36}$$ and$$\eta\rightarrow-G. \label{etamG36}$$ The field equation of $\phi$, Eq. (\[Vphi3\]), has a solution$$\phi=-\frac{m^{2}}{6\lambda}\left[ 3\tanh^{2}\left( \alpha x_{1}+\beta x_{2}+\gamma x_{3}+\frac{\sqrt{4\alpha^{2}+4\beta^{2}+4\gamma^{2}+m^{2}}}{2}t\right) -1\right] . \label{phi34dsol}$$ For the field $\phi$ with the potential (\[Vphi3\]),$$G=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi ^{2}+\lambda\phi^{3}=\frac{m^{6}}{54\lambda^{2}}. \label{hphi3}$$ Substituting the duality transformations (\[Tphiphiw36\]) and (\[Txy36\]) into the solution (\[phi34dsol\]) and using the duality relation (\[etamG36\]), we obtain the solution of $U\left( \varphi\right) =-\frac{m^{6}}{54\lambda^{2}}\phi^{6}$: $$\varphi=\left\{ -\frac{m^{2}}{6\lambda}\left[ 3\tanh^{2}\left( 2\alpha y_{1}+2\beta y_{2}+2\gamma y_{3}+\sqrt{4\alpha^{2}+4\beta^{2}+4\gamma ^{2}+m^{2}}\tau\right) -1\right] \right\} ^{-1/2}. \label{varphi4dsol36}$$ We then arrive at the solution of $U\left( \varphi\right) =\eta\phi^{6}$ with $-\frac{m^{6}}{54\lambda^{2}}=\eta$:$$\varphi=\left\{ \frac{i\sqrt{6\eta}}{2m}\left[ 3\tanh^{2}\left( 2\alpha y_{1}+2\beta y_{2}+2\gamma y_{3}+\sqrt{4\alpha^{2}+4\beta^{2}+4\gamma ^{2}+m^{2}}\tau\right) -1\right] \right\} ^{-1/2}.$$ $V\left( \phi\right) =\lambda\phi^{3}+\kappa\phi$ and its duality ------------------------------------------------------------------- The massless scalar field with the polynomial potential $$V\left( \phi\right) =\lambda\phi^{3}+\kappa\phi\label{Vphi31}$$ has two dual potentials. Taking $a=3$ in the duality relations (\[aandA\]) and (\[bandB\]) gives $A=6$ and $B_{n}=4$, i.e., the dual potential is $$U\left( \varphi\right) =\eta\varphi^{6}+\sigma\varphi^{4}. \label{Uvarphi64}$$ Taking $a=1$ in the duality relations (\[aandA\]) and (\[bandB\]) gives $A=-2$ and $B_{n}=4$, i.e., the dual potential is$$U\left( \varphi\right) =\eta\varphi^{-2}+\sigma\varphi^{4}. \label{Uvarphi-24}$$ Obviously, the fields with the potentials (\[Uvarphi64\]) and (\[Uvarphi-24\]) are also dual. The massless field equation of the potential (\[Vphi31\]) has a solution $$\phi=i\sqrt{\frac{\kappa}{\lambda}}\left[ \frac{2}{\sqrt{3}}-\sqrt{3}\tanh^{2}\left( \alpha t-\frac{x}{2}\sqrt{4\alpha^{2}-2i\left( 3\kappa\lambda\right) ^{1/2}}\right) \right] . \label{sol31}$$ For the field $\phi$ with the potential (\[Vphi31\]),$$G=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi ^{2}+\lambda\phi^{3}=\frac{2i\kappa^{3/2}}{3\sqrt{3\lambda}}.$$ From the solution of the potential (\[Vphi31\]), we can obtain the solution of its dual potentials, Eqs. (\[Uvarphi64\]) and (\[Uvarphi-24\]), by means of the duality relation. ### $U\left( \varphi\right) =\eta\varphi^{6}+\sigma\varphi^{4}$ The duality transformations given by Eqs. (\[phitanspol\]), (\[xtanspol\]), (\[Geta\]), and (\[kapsig\]) are $$\begin{aligned} \phi & \rightarrow\varphi^{-2},\nonumber\\ x^{\mu} & \rightarrow-2y^{\mu},\nonumber\\ \eta & \rightarrow-G,\nonumber\\ \sigma & \rightarrow\kappa.\end{aligned}$$ Substituting the duality relation into the solution (\[sol31\]) gives the solution of $U\left( \varphi\right) =-\frac{2i\kappa^{3/2}}{3\sqrt{3\lambda }}\varphi^{6}+\kappa\varphi^{4}$: $$\varphi=i\sqrt{\frac{\kappa}{\lambda}}\left[ \frac{2}{\sqrt{3}}-\sqrt{3}\tanh^{2}\left( -2\alpha\tau+y\sqrt{4\alpha^{2}-2i\left( 3\kappa \lambda\right) ^{1/2}}\right) \right] ^{-1/2}. \label{sol641}$$ This is the solution of the dual potential $U\left( \varphi\right) =\eta\varphi^{6}+\sigma\varphi^{4}$ when $-\frac{2i\kappa^{2}}{3\sqrt {3\kappa\lambda}}=\eta$ and $\kappa=\sigma$: $$\varphi=\left( \frac{\sigma}{3\eta}\right) ^{1/2}\left[ 1-\frac{3}{2}\tanh^{2}\left( -2\alpha\tau+2y\sqrt{\alpha^{2}+\frac{\sigma^{2}}{3\eta}}\right) \right] ^{-1/2}.$$ ### $U\left( \varphi\right) =\eta\varphi^{-2}+\sigma\varphi^{4}$ Similarly, the solution of the potential (\[Uvarphi-24\]) can be achieved by the duality transformations (\[phitanspol\]), (\[xtanspol\]), (\[Geta\]), and (\[kapsig\]):$$\begin{aligned} \phi & \rightarrow\varphi^{2},\nonumber\\ x^{\mu} & \rightarrow2y^{\mu},\nonumber\\ \eta & \rightarrow-G,\nonumber\\ \sigma & \rightarrow\lambda.\end{aligned}$$ Substituting into the solution (\[sol31\]) gives the solution of $U\left( \varphi\right) =-\frac{2i\kappa^{2}}{3\sqrt{3\kappa\lambda}}\varphi ^{-2}+\lambda\varphi^{4}$,$$\varphi=\left[ 2i\sqrt{\frac{\kappa}{3\lambda}}-i\sqrt{\frac{3\kappa}{\lambda}}\tanh^{2}\left( 2\alpha\tau-y\sqrt{4\alpha^{2}-2i\left( 3\kappa\lambda\right) ^{1/2}}\right) \right] ^{1/2}. \label{sol-24}$$ This is the solution of the dual potential $U\left( \varphi\right) =\eta\varphi^{-2}+\sigma\varphi^{4}$ when $-\frac{2i\kappa^{2}}{3\sqrt {3\kappa\lambda}}=\eta$ and $\sigma=\lambda$:$$\varphi=\left( \frac{\eta}{2\sigma}\right) ^{1/6}\left[ 3\tanh^{2}\left( 2\alpha\tau-y\sqrt{4\alpha^{2}+6\left( \frac{\sigma^{2}\eta}{2}\right) ^{1/3}}\right) -2\right] ^{1/2}.$$ $V\left( \phi\right) =\lambda\phi^{n}+\kappa_{1}\phi ^{2n-2}+\kappa_{2}$ and its duality ------------------------------------------------------ The massive scalar field with the polynomial potential$$V\left( \phi\right) =\lambda\phi^{n}+\kappa_{1}\phi^{2n-2}+\kappa_{2},\text{ \ \ }n\geq3, \label{Vphin}$$ has the following dual potentials. By the duality relations (\[aandA\]) and (\[bandB\]), $a=n$ corresponds to $A=\frac{2n}{n-2}$, $B_{1}=-2$ and $B_{2}=\frac{2n}{n-2}$, i.e., the dual potential is$$U\left( \varphi\right) =\eta\varphi^{\frac{2n}{n-2}}+\sigma_{1}\varphi ^{-2}+\sigma_{2}\varphi^{\frac{2n}{n-2}}; \label{Uphin1}$$ $a=2n-2$ corresponds to $A=\frac{2n-2}{n-2}$, $B_{1}=1$, and $B_{2}=\frac{2n-2}{n-2}$, i.e., the dual potential is$$U\left( \varphi\right) =\eta\varphi^{\frac{2n-2}{n-2}}+\sigma_{1}\varphi+\sigma_{2}\varphi^{\frac{2n-2}{n-2}}. \label{Uphin2}$$ The fields with the potentials (\[Uphin1\]) and (\[Uphin2\]) are also dual. The field equation with the potential (\[Vphi31\]) has a solution [@polyanin2016handbook]$$\phi=\left[ \frac{\sqrt{\lambda^{2}-2\kappa_{1}m^{2}}}{m^{2}}\cos\left( m\left( 2-n\right) \left( t\cosh\alpha+x\sinh\alpha\right) +\beta\right) -\frac{\lambda}{m^{2}}\right] ^{\frac{1}{2-n}}. \label{soln2n-2}$$ For the field $\phi$ with the potential (\[Vphin\]),$$G=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}m^{2}\phi ^{2}+\lambda\phi^{3}=\kappa_{2}.$$ From the solution of the potential (\[soln2n-2\]), we can obtain the solution of its dual potentials (\[Uphin1\]) and (\[Uphin2\]) by means of the duality relation. ### $U\left( \varphi\right) =\kappa_{1}\varphi^{-2}$ The duality transformations given by Eqs. (\[phitanspol\]), (\[xtanspol\]), (\[Geta\]), and (\[kapsig\]) are$$\begin{aligned} \phi & \rightarrow\varphi^{\frac{2}{2-n}},\nonumber\\ x^{\mu} & \rightarrow\frac{2}{2-n}y^{\mu},\nonumber\\ \eta & \rightarrow-G,\nonumber\\ \sigma_{1} & \rightarrow\kappa_{1},\nonumber\\ \sigma_{2} & \rightarrow\kappa_{2}.\end{aligned}$$ Substituting into the solution (\[soln2n-2\]) gives the solution of $U\left( \varphi\right) =-\kappa_{2}\varphi^{\frac{2n}{n-2}}+\kappa _{1}\varphi^{-2}+\kappa_{2}\varphi^{\frac{2n}{n-2}}=\kappa_{1}\varphi^{-2}$: $$\varphi=\left[ \frac{\sqrt{\lambda^{2}-2\kappa_{1}m^{2}}}{m^{2}}\cos\left( 2m\left( \tau\cosh\alpha+y\sinh\alpha\right) +\beta\right) -\frac{\lambda }{m^{2}}\right] ^{1/2}. \label{sol22-n}$$ ### $U\left( \varphi\right) =\lambda\varphi$ Similarly, the solution of the potential (\[Uphin2\]) can be achieved by the duality transformation given by Eqs. (\[phitanspol\]), (\[xtanspol\]), (\[Geta\]), and (\[kapsig\]): $$\begin{aligned} \phi & \rightarrow\varphi^{\frac{1}{2-n}},\nonumber\\ x^{\mu} & \rightarrow\frac{1}{2-n}y^{\mu},\nonumber\\ \eta & \rightarrow-G,\nonumber\\ \sigma_{1} & \rightarrow\lambda,\nonumber\\ \sigma_{2} & \rightarrow\kappa_{2}.\end{aligned}$$ Substituting into the solution (\[soln2n-2\]) gives the solution of $U\left( \varphi\right) =-\kappa_{2}\varphi^{\frac{2n-2}{n-2}}+\lambda\varphi+\kappa_{2}\varphi^{\frac{2n-2}{n-2}}=\lambda\varphi$:$$\varphi=\frac{\sqrt{\lambda^{2}-2\kappa_{1}m^{2}}}{m^{2}}\cos\left( m\left( \tau\cosh\alpha+y\sinh\alpha\right) +\beta\right) -\frac{\lambda}{m^{2}}. \label{sol12-n}$$ Conclusion \[conclusion\] ========================= In this paper, we reveal a duality between fields. The dual fields are related by the duality relation. It is shown that fields who are dual to each other form a duality family. For polynomial potentials, the duality family consists of finite number of dual fields, for nonpolynomial potentials, e.g., the sine-Gordon field, the duality family consists of infinite number of dual fields. Some fields, e.g., the free field and the $\phi^{4}$-field, are self-duality. The existence of the duality family inspires us to classify fields based on the duality. A duality family is a duality class. In a duality class, the different of fields is only a duality transformation. The duality of fields provides a high-efficiency approach for solving field equations. Once one field equation is solved, all of its dual fields are solved by the duality relation. That is, in a duality family, we only need to solve one of its member. As examples, we solve the $\phi^{-2}$-field from the solution of its dual field, $\phi^{1}$-field, solve the field equation of the dual field of the sine-Gordon field from the sine-Gordon field, etc. In further work, we will consider the quantum theory of dual fields. For example, the relation of the Feynman rule of dual fields. Especially, in quantum field theory we will consider the duality in the heat kernel method [@vassilevich2003heat] and in the scattering spectrum method [@graham2009spectral; @pang2012relation; @li2015heat]. In these methods we can calculate the one-loop effective action and the vacuum energy [@dai2009number; @dai2010approach]. We may observe the relation of the one-loop effective action and the vacuum energy of dual fields. Moreover, we will consider the duality of spinor fields and vector fields. A solution of the scalar field equation \[appendix\] ==================================================== In this appendix we show the field equation$$\square\phi+m^{2}\phi+\frac{\partial V\left( \phi\right) }{\partial\phi}=0, \label{motequ}$$ has an implicit solution$$\beta_{\mu}x^{\mu}+\int\frac{\sqrt{-\beta^{2}}}{\sqrt{2\left[ \frac{1}{2}m^{2}\phi^{2}+V\left( \phi\right) -G\right] }}d\phi\equiv F_{1}\left( x^{\mu},\phi\right) =0. \label{gensol}$$ From the solution (\[gensol\]) we obtain $$\partial_{\mu}\phi=-\frac{\frac{\partial F_{1}\left( x^{\mu},\phi\right) }{\partial x^{\mu}}}{\frac{\partial F_{1}\left( x^{\mu},\phi\right) }{\partial\phi}}=-\frac{\beta_{\mu}\sqrt{2\left[ \frac{1}{2}m^{2}\phi ^{2}+V\left( \phi\right) -G\right] }}{\sqrt{-\beta^{2}}}. \label{rdphi1}$$ Furthermore, by Eq. (\[rdphi1\]), letting$$\partial_{\mu}\phi+\frac{\beta_{\mu}\sqrt{2\left[ \frac{1}{2}m^{2}\phi ^{2}+V\left( \phi\right) -G\right] }}{\sqrt{-\beta^{2}}}\equiv F_{2}\left( x^{\mu},\phi,\partial_{\mu}\phi\right) =0,$$ we arrive at $$\begin{aligned} \square\phi & =\partial^{\mu}\partial_{\mu}\phi=\frac{\partial}{\partial x_{\mu}}\partial_{\mu}\phi\nonumber\\ & =-\frac{\frac{\partial F_{2}\left( x^{\mu},\phi,\partial_{\mu}\phi\right) }{\partial x_{\mu}}}{\frac{\partial F_{2}\left( x^{\mu},\phi,\partial_{\mu }\phi\right) }{\partial_{\mu}\phi}}-\frac{\frac{\partial F_{2}\left( x^{\mu },\phi,\partial_{\mu}\phi\right) }{\partial\phi}}{\frac{\partial F_{2}\left( x^{\mu},\phi,\partial_{\mu}\phi\right) }{\partial_{\mu}\phi}}\frac {\partial\phi}{\partial x_{\mu}}\nonumber\\ & =-\frac{\partial F_{2}\left( x^{\mu},\phi,\partial_{\mu}\phi\right) }{\partial\phi}\partial^{\mu}\phi\nonumber\\ & =-\left( m^{2}\phi+\frac{\partial V\left( \phi\right) }{\partial\phi }\right) .\end{aligned}$$ This is the field equation (\[motequ\]). 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--- abstract: 'The one clean qubit model (or the DQC1 model) is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, the model can solve several problems (such as calculating Jones polynomials) whose classical efficient solutions are not known. Furthermore, it was shown that if the output probability distribution of the one clean qubit model can be classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level. Is it possible to improve the multiplicative error hardness result to a constant total variation distance error one like other sub-universal quantum computing models such as the IQP model, the Boson Sampling model, and the Fourier Sampling model? In this paper, we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Interestingly, the anti-concentration lemma can be easily shown by using the special property of the one clean qubit model that each output probability is so small that no concentration occurs.' author: - Tomoyuki Morimae title: Hardness of classically sampling one clean qubit model with constant total variation distance error --- Introduction ============ The one clean qubit model (or the DQC1 model) first introduced by Knill and Laflamme [@KnillLaflamme] is a restricted model of quantum computing where only a single input qubit is pure and all other input qubits are maximally mixed. In spite of the severe restriction, surprisingly, the model can solve several problems whose efficient classical solutions are not known, such as the spectral density estimation [@KnillLaflamme], testing integrability [@integrability], calculations of the fidelity decay [@fidelity_decay], and approximations of the Jones polynomial, HOMFLY polynomial, and Turaev-Viro invariant [@ShorJordan08; @Passante09; @JordanWocjan09; @JordanAlagic]. Furthermore, it was recently shown that if the output probability distribution of the one clean qubit model is classically efficiently sampled with a constant multiplicative error, then the polynomial hierarchy collapses to the second level [@MFF; @FKMNTT]. (Here, we say that a probability distribution $\{p_z\}_z$ is sampled by a machine $M$ with a multiplicative error $\epsilon\ge0$ if $$\begin{aligned} |p_z-q_z|\le \epsilon p_z\end{aligned}$$ is satisfied for all $z$, where $\{q_z\}_z$ is the output probability distribution of $M$.) Since a collapse of the polynomial hierarchy is not believed to happen in computer science, the result suggests the impossibility of classically simulating the one clean qubit model. Similar hardness results for constant multiplicative error sampling were also shown for other sub-universal quantum computing models, such as the IQP model [@BJS] and the Boson Sampling model [@AA]. The requirement of constant multiplicative error sampling is, however, strong, and sampling with a constant total variation distance error (or the L1-norm error) is considered as more appropriate. (Here, we say that a probability distribution $\{p_z\}_z$ is sampled by a machine $M$ with a total variation distance error $\epsilon\ge0$ if $$\begin{aligned} \sum_z|p_z-q_z|\le\epsilon\end{aligned}$$ is satisfied, where $\{q_z\}_z$ is the output probability distribution of $M$.) In fact, the hardness results with constant total variation distance errors were shown for the IQP model [@BMS], the Boson Sampling model [@AA], and the Fourier Sampling model [@FU] (assuming some conjectures). Is it possible to show a similar constant-total-variation-distance-error hardness result for the one clean qubit model? In this paper, we show that it is indeed possible if we accept a modified version of the average case hardness conjecture. Our proof is similar to those of Refs. [@AA; @BMS; @FU], but there is one interesting difference which is specific to the one clean qubit model: the anti-concentration lemma can be easily shown. For the Boson Sampling model and the Fourier Sampling model, the anti-concentration lemma is a conjecture [@AA; @FU]. For the IQP model, it is shown with some calculations by using a special structure of IQP circuits [@BMS]. For the present case, as we will see later, the anti-concentration lemma is easily shown by using a special property of the one clean qubit model that each probability is so small that no concentration occurs. Average case hardness conjecture ================================ As in the cases of other sub-universal quantum computing models, such as the IQP model [@BMS], the Boson Sampling model [@AA], and the Fourier Sampling model [@FU], we need a conjecture so-called “average case hardness conjecture", which claims that the $\#$P-hardness for the worst case can be lifted to an average case. To show our result, which is a hardness of efficient classical sampling of the one clean qubit model with a constant total variation distance error, we need the following conjecture. [**Conjecture**]{}: For each $n$, there exists a discrete set ${\mathcal U}_{n+1}$ of uniformly-generated polynomial-time $(n+1)$-qubit unitary operators such that calculating $$\begin{aligned} f(z,U)\equiv \langle z|U(|0\rangle\langle0|\otimes I^{\otimes n})U^\dagger|z\rangle\end{aligned}$$ with a multiplicative error less than 1/2 for more than 1/6 fraction of $(z,U)\in\{0,1\}^{n+1}\times{\mathcal U}_{n+1}$ is $\#$P-hard. Here, $I\equiv|0\rangle\langle0|+|1\rangle\langle1|$ is the two-dimensional identity operator, and $|z\rangle$ is the computational-basis state corresponding to the bit string $z$. (For example, if $z=010$, $|z\rangle=|010\rangle=|0\rangle\otimes|1\rangle\otimes|0\rangle$.) Unfortunately, we do not know whether the conjecture is true or not, but we can show that it is true at least for the worst case. Here, we give two proofs for the worst-case $\#$P-hardness. First proof ----------- Let us consider a unitary operator $U$ such that $$\begin{aligned} U^\dagger&=&\Big[I\otimes|0\rangle\langle0|^{\otimes n} +X\otimes(I^{\otimes n}-|0\rangle\langle0|^{\otimes n})\Big] (I\otimes C),\end{aligned}$$ where $C$ is an $n$-qubit IQP circuit. Then, $$\begin{aligned} f(0^{n+1},U)&=&|\langle0^n|C|0^n\rangle|^2.\end{aligned}$$ For certain IQP circuits $C$, $\langle0^n|C|0^n\rangle$ is related to the partition function, $Z$, of the Ising model [@BMS; @FujiiMorimae] and the gap function, $gap(f)$, of a degree-3 polynomial $f$ over ${\mathbb F}_2$ [@BMS]: $$\begin{aligned} \langle0^n|C|0^n\rangle&=&\frac{Z}{2^n},\\ \langle0^n|C|0^n\rangle&=&\frac{gap(f)}{2^n}.\end{aligned}$$ It is known that calculating $|Z|^2$ and $gap(f)^2$ with constant multiplicative errors is $\#$P-hard [@BMS; @FujiiMorimae]. Hence calculating $|\langle0^n|C|0^n\rangle|^2=f(0^{n+1},U)$ with constant multiplicative errors is $\#$P-hard for certain unitary operators $U$, which shows the correctness of the conjecture for the worst case. Second proof ------------ We define two unitary operators $U_1$ and $U_2$ as $$\begin{aligned} U_1^\dagger&=& \Big[ (I\otimes|0\rangle\langle0|+X\otimes|1\rangle\langle1|) \otimes I^{\otimes n-1}\Big] (I\otimes V),\\ U_2^\dagger&=& \Big[ \Big( I\otimes|0\rangle\langle0|^{\otimes 2} +X\otimes(I^{\otimes 2}-|0\rangle\langle0|^{\otimes 2}) \Big) \otimes I^{\otimes n-2}\Big](I\otimes V),\end{aligned}$$ where $V$ is an $n$-qubit unitary operator. Then, we obtain $$\begin{aligned} f(0^{n+1},U_1)&=&\langle0^n|V^\dagger(|0\rangle\langle0| \otimes I^{\otimes n-1})V|0^n\rangle,\\ f(0^{n+1},U_2)&=&\langle0^n|V^\dagger(|0\rangle\langle0|^{\otimes 2} \otimes I^{\otimes n-2})V|0^n\rangle.\end{aligned}$$ Now we show that calculating $f(0^{n+1},U_1)$ and $f(0^{n+1},U_2)$ with a constant multiplicative error $0\le \epsilon<1$ is postBQP-hard. Since ${\rm postBQP}={\rm PP}$ [@postBQP] and ${\rm P}^{\rm PP}={\rm P}^{\#{\rm P}}$, it means that the calculation is $\#$P-hard. Proof is as follows. Let us assume that there exists an algorithm that calculates $a$ and $b$ such that $$\begin{aligned} |f(0^{n+1},U_1)-a|&\le& \epsilon f(0^{n+1},U_1),\\ |f(0^{n+1},U_2)-b|&\le& \epsilon f(0^{n+1},U_2).\end{aligned}$$ Let $L$ be a language in postBQP. Then, for any polynomial $r$, there exists a uniform family $\{V_x\}_x$ of polynomial-time quantum circuits such that - If $x\in L$ then $P_{V_x}(o=0|p=0)\ge1-2^{-r}$. - If $x\notin L$ then $P_{V_x}(o=0|p=0)\le2^{-r}$. Here, $$\begin{aligned} P_{V_x}(o=0|p=0)=\frac{P_{V_x}(o=0,p=0)}{P_{V_x}(p=0)},\end{aligned}$$ $P_{V_x}(o=0,p=0)$ is the probability that $V_x$ outputs $(o,p)=(0,0)$, and $P_{V_x}(p=0)$ is the probability that $V_x$ outputs $p=0$. Let us construct $U_1$ and $U_2$ by using $V_x$. Then, for any polynomial $r$, if $x\in L$, $$\begin{aligned} \frac{b}{a} &\ge& \frac{1-\epsilon}{1+\epsilon} \frac{f(0^{n+1},U_2)} {f(0^{n+1},U_1)}\\ &\ge& \frac{1-\epsilon}{1+\epsilon} (1-2^{-r}),\end{aligned}$$ and if $x\notin L$, $$\begin{aligned} \frac{b}{a} &\le& \frac{1+\epsilon}{1-\epsilon} \frac{f(0^{n+1},U_2)} {f(0^{n+1},U_1)}\\ &\le& \frac{1+\epsilon}{1-\epsilon} 2^{-r}.\end{aligned}$$ Therefore, if we can calculate $a$ and $b$, we can solve $L$. 0 ![ (a) Circuit $U_1$. (b) Circuit $U_2$. []{data-label="figure"}](figure.eps){width="40.00000%"} Main result =========== If we accept the conjecture, we can show the following theorem, which is the main result of the present paper. [**Theorem**]{}: If there exists a probabilistic polynomial-time classical algorithm that outputs $z\in\{0,1\}^{n+1}$ with probability $q_z(U)$ such that $$\begin{aligned} \sum_{z\in\{0,1\}^{n+1}}|p_z(U)-q_z(U)|\le\epsilon\end{aligned}$$ for any $U\in\cup_n{\mathcal U}_{n+1}$, then the polynomial hierarchy collapses to the third level. Here, $\epsilon=\frac{1}{36}$ and $$\begin{aligned} p_z(U)\equiv\langle z|U\Big(|0\rangle\langle0|\otimes \frac{I^{\otimes n}}{2^n}\Big) U^\dagger|z\rangle.\end{aligned}$$ The theorem says that if the output probability distribution $p_z(U)$ of the one clean qubit model can be classically efficiently sampled with the total variation distance error $\epsilon$, then the polynomial hierarchy collapses to the third level. [**Proof**]{}: Now let us give a proof of the theorem. Our proof is similar to those of Refs. [@AA; @BMS; @FU] except that the anti-concentration lemma can be easily shown. Let $\delta>0$ be a parameter specified later. From Markov’s inequality, $$\begin{aligned} \mbox{Pr}_{z,U} \Big[|p_z(U)-q_z(U)|\ge \frac{\epsilon}{2^{n+1}\delta}\Big] &\le&\frac{2^{n+1}\delta}{\epsilon} \frac{1}{2^{n+1}|{\mathcal U}_{n+1}|} \sum_{U,z}|p_z(U)-q_z(U)|\\ &\le&\delta.\end{aligned}$$ From Stockmeyer’s Counting Theorem [@Stockmeyer], there exists an ${\rm FBPP}^{\rm NP}$ algorithm that outputs $\tilde{q}_z(U)$ such that $$\begin{aligned} |\tilde{q}_z(U)-q_z(U)|\le\frac{q_z(U)}{poly}.\end{aligned}$$ Therefore, $$\begin{aligned} |\tilde{q}_z(U)-p_z(U)|&\le&|\tilde{q}_z(U)-q_z(U)|+|q_z(U)-p_z(U)|\\ &\le&\frac{q_z(U)}{poly}+|q_z(U)-p_z(U)|\\ &=&\frac{p_z(U)+q_z(U)-p_z(U)}{poly} +|q_z(U)-p_z(U)|\\ &\le&\frac{p_z(U)+|q_z(U)-p_z(U)|}{poly} +|q_z(U)-p_z(U)|\\ &=&\frac{p_z(U)}{poly}+|q_z(U)-p_z(U)|\Big(1+\frac{1}{poly}\Big)\\ &<&\frac{p_z(U)}{poly} +\frac{\epsilon}{2^{n+1}\delta}\Big(1+\frac{1}{poly}\Big)\end{aligned}$$ with more than $1-\delta$ fraction of $(z,U)$. Let $S\subseteq\{0,1\}^{n+1}\times {\mathcal U}_{n+1}$ be the set of $(z,U)$ such that $$\begin{aligned} \frac{\epsilon}{2^{n+1}\delta}\le\frac{p_z(U)}{3}.\end{aligned}$$ Since $$\begin{aligned} p_z(U)&=&\langle z|U\Big(|0\rangle\langle0|\otimes\frac{I^{\otimes n}}{2^n} \Big)U^\dagger|z\rangle\\ &=&\frac{1}{2^n} \langle z|U(|0\rangle\langle0|\otimes I^{\otimes n} )U^\dagger|z\rangle\\ &\le&\frac{1}{2^n}\times1=\frac{1}{2^n},\end{aligned}$$ for all $(z,U)$, and $$\begin{aligned} \sum_{z\in\{0,1\}^{n+1}}p_z(U)=1\end{aligned}$$ for all $U$, we obtain $$\begin{aligned} 1&=&\frac{1}{|{\mathcal U}_{n+1}|} \sum_{U,z}p_z(U)\\ &=&\frac{1}{|{\mathcal U}_{n+1}|} \sum_{(z,U)\in S}p_z(U) +\frac{1}{|{\mathcal U}_{n+1}|}\sum_{(z,U)\notin S}p_z(U)\\ &<&\frac{1}{2^n|{\mathcal U}_{n+1}|}|S| +\frac{2^{n+1}|{\mathcal U}_{n+1}|-|S|}{|{\mathcal U}_{n+1}|} \frac{3\epsilon}{2^{n+1}\delta},\end{aligned}$$ which means $$\begin{aligned} \frac{|S|}{2^{n+1}|{\mathcal U}_{n+1}|}&>& \frac{1-\frac{3\epsilon}{\delta}} {2-\frac{3\epsilon}{\delta}}.\end{aligned}$$ Therefore, $$\begin{aligned} |\tilde{q}_z(U)-p_z(U)| &<&\frac{p_z(U)}{poly}+\frac{p_z(U)}{3}\Big(1+\frac{1}{poly}\Big)\\ &=&p_z(U)\Big(\frac{1}{3}+\frac{1}{poly}\Big)\end{aligned}$$ for more than $$\begin{aligned} F\equiv1-\delta-\frac{1}{2-\frac{3\epsilon}{\delta}}\end{aligned}$$ fraction of $(z,U)$. For example, if we take $\delta=6\epsilon$, $$\begin{aligned} F=\frac{1}{6}.\end{aligned}$$ Note that $$\begin{aligned} p_z(U)=\frac{f(z,U)}{2^n}.\end{aligned}$$ Therefore, the above result means that there exists an ${\rm FBPP}^{\rm NP}$ algorithm that outputs $\tilde{q}_z(U)$ such that $$\begin{aligned} |\tilde{q}_z(U)2^n-f(z,U)| &<&f(z,U)\Big(\frac{1}{3}+\frac{1}{poly}\Big) <\frac{1}{2}f(z,U)\end{aligned}$$ for more than 1/6 fraction of $(z,U)$. If our average-case hardness conjecture is true, it means the collapse of the polynomial hierarchy to the third level. TM is supported by JST ACT-I No.JPMJPR16UP, the Grant-in-Aid for Scientific Research on Innovative Areas No.15H00850 of MEXT Japan, and the JSPS Grant-in-Aid for Young Scientists (B) No.26730003 and No.17K12637. [99]{} E. Knill, and R. Laflamme, Power of one bit of quantum information. Phys. Rev. Lett. [**81**]{}, 5672 (1998). D. Poulin, R. Laflamme, G. J. Milburn, and J. P. 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--- abstract: 'Algorithm configuration methods optimize the performance of a parameterized heuristic algorithm on a given distribution of problem instances. Recent work introduced an algorithm configuration procedure (“Structured Procrastination”) that provably achieves near optimal performance with high probability and with nearly minimal runtime in the worst case. It also offers an *anytime* property: it keeps tightening its optimality guarantees the longer it is run. Unfortunately, Structured Procrastination is not *adaptive* to characteristics of the parameterized algorithm: it treats every input like the worst case. Follow-up work (“LeapsAndBounds”) achieves adaptivity but trades away the anytime property. This paper introduces a new algorithm, “Structured Procrastination with Confidence”, that preserves the near-optimality and anytime properties of Structured Procrastination while adding adaptivity. In particular, the new algorithm will perform dramatically faster in settings where many algorithm configurations perform poorly. We show empirically both that such settings arise frequently in practice and that the anytime property is useful for finding good configurations quickly.' author: - | Robert Kleinberg\ Department of Computer Science\ Cornell University\ `[email protected]` Kevin Leyton-Brown\ Department of Computer Science\ University of British Columbia\ `[email protected]` Brendan Lucier\ Microsoft Research\ `[email protected]` Devon Graham\ Department of Computer Science\ University of British Columbia\ `[email protected]` bibliography: - 'main.bib' title: 'Procrastinating with Confidence: Near-Optimal, Anytime, Adaptive Algorithm Configuration' --- Introduction ============ Algorithm configuration is the task of searching a space of *configurations* of a given algorithm (typically represented as joint assignments to a set of algorithm parameters) in order to find a single configuration that optimizes a performance objective on a given distribution of inputs. In this paper, we focus exclusively on the objective of minimizing average runtime. Considerable progress has recently been made on solving this problem in practice via general-purpose, heuristic techniques such as ParamILS [@hutter-aaai07a; @hutter-jair09a], GGA [@ansotegui-cp09a; @ansotegui-ijcai15a], irace [@birattari-gecco02a; @lopez-ibanez-tech11a] and SMAC [@hutter-bayesopt11; @hutter-lion11a]. Notably, in the context of this paper, all these methods are *adaptive*: they surpass their worst-case performance when presented with “easier” search problems. Recently, algorithm configuration has also begun to attract theoretical analysis. While there is a large body of less-closely related work that we survey in Section \[sec:related\], the first nontrivial worst-case performance guarantees for general algorithm configuration with an average runtime minimization objective were achieved by a recently introduced algorithm called *Structured Procrastination (SP)* [@ijcai17]. This work considered a worst-case setting in which an adversary causes every deterministic choice to play out as poorly as possible, but where observations of random variables are unbiased samples. It is straightforward to argue that, in this setting, any fixed, deterministic heuristic for searching the space of configurations can be extremely unhelpful. The work therefore focuses on obtaining candidate configurations via random sampling (rather than, e.g., following gradients or taking the advice of a response surface model). Besides its use of heuristics, SMAC also devotes half its runtime to random sampling. Any method based on random sampling will eventually encounter the optimal configuration; the crucial question is the amount of time that this will take. The key result of @ijcai17 is that SP is guaranteed to find a near-optimal configuration with high probability, with worst-case running time that nearly matches a lower bound on what is possible and that asymptotically dominates that of existing alternatives such as SMAC. Unfortunately, there is a fly in the ointment: SP turns out to be impractical in many cases, taking an extremely long time to run even on inputs that existing methods find easy. At the root, the issue is that SP treats every instance like the worst case, in which it is necessary to achieve a fine-grained understanding of every configuration’s runtime in order to distinguish between them. For example, if every configuration is very similar but most are not quite ${\varepsilon}$-optimal, subtle performance differences must be identified. SP thus runs every configuration enough times that with high probability the configuration’s runtime can accurately be estimated to within a $1+{\varepsilon}$ factor. <span style="font-variant:small-caps;">LeapsAndBounds</span> and <span style="font-variant:small-caps;">CapsAndRuns</span> {#sec:beyond} -------------------------------------------------------------------------------------------------------------------------- @weisz2018leapsandbounds introduced a new algorithm, <span style="font-variant:small-caps;">LeapsAndBounds (LB)</span>, that improves upon Structured Procrastination in several ways. First, LB improves upon SP’s worst-case performance, matching its information-theoretic lower bound on running time by eliminating a log factor. Second, LB does not require the user to specify a runtime cap that they would never be willing to exceed on any run, replacing this term in the analysis with the runtime of the optimal configuration, which is typically much smaller. Third, and most relevant to our work here, LB includes an adaptive mechanism, which takes advantage of the fact that when a configuration exhibits low variance across instances, its performance can be estimated accurately with a smaller number of samples. However, the easiest algorithm configuration problems are probably those in which a few configurations are much faster on average than all other configurations. (Empirically, many algorithm configuration instances exhibit just such non-worst-case behaviour; see our empirical investigation in the Supplementary Materials.) In such cases, it is clearly unnecessary to obtain high-precision estimates of each bad configuration’s runtime; instead, we only need to separate these configurations’ runtimes from that of the best alternative. LB offers no explicit mechanism for doing this. LB also has a key disadvantage when compared to SP: it is not anytime, but instead must be given fixed values of ${\varepsilon}$ and $\delta$. Because LB is adaptive, there is no way for a user to anticipate the amount of time that will be required to prove $({\varepsilon},\delta)$-optimality, forcing a tradeoff between the risks of wasting available compute resources and of having to terminate LB before it returns an answer. <span style="font-variant:small-caps;">CapsAndRuns (CR)</span> is a refinement of LB that was developed concurrently with the current paper; it has not been formally published, but was presented at an ICML 2018 workshop [@weisz2018capsandruns]. CR maintains all of the benefits of LB, and furthermore introduces a second adaptive mechanism that does exploit variation in configurations’ mean runtimes. Like LB, it is not anytime. Our Contributions ----------------- Our main contribution is a refined version of SP that maintains the anytime property while aiming to observe only as many samples as necessary to separate the runtime of each configuration from that of the best alternative. We call it “Structured Procrastination with Confidence” (SPC). SPC differs from SP in that it maintains a novel form of lower confidence bound as an indicator of the quality of a particular configuration, while SP simply uses that configuration’s sample mean. The consequence is that SPC spends much less time running poorly performing configurations, as other configurations quickly appear better and receive more attention. We initialize each lower bound with a trivial value: each configuration’s runtime is bounded below by the fastest possible runtime, $\kappa_0$. SPC then repeatedly evaluates the configuration that has the most promising lower bound.[^1] We perform these runs by “capping” (censoring) runs at progressively doubling multiples of $\kappa_0$. If a run does not complete, SPC “procrastinates”, deferring it until it has exhausted all runs with shorter captimes. Eventually, SPC observes enough completed runs of some configuration to obtain a nontrivial upper bound on its runtime. At this point, it is able to start drawing high-probability conclusions that other configurations are worse. Our paper is focused on a theoretical analysis of SPC. We show that it identifies an approximately optimal configuration using running time that is nearly the best possible in the worst case; however, so does SP. The key difference, and the subject of our main theorem, is that SPC also exhibits near-minimal runtime beyond the worst case, in the following sense. Define an $({\varepsilon},\delta)$-suboptimal configuration to be one whose average runtime exceeds that of the optimal configuration by a factor of more than $1+{\varepsilon}$, even when the suboptimal configuration’s runs are capped so that a $\delta$ fraction of them fail to finish within the time limit. A straightforward information-theoretic argument shows that in order to verify that a configuration is $({\varepsilon},\delta)$-suboptimal it is sufficient—and may also be necessary, in the worst case—to run it for $O({\varepsilon}^{-2} \cdot \delta^{-1} \cdot {\text{OPT}})$ time. The running time of SPC matches (up to logarithmic factors) the running time of a hypothetical “optimality verification procedure” that knows the identity of the optimal configuration, and for each suboptimal configuration $i$ knows a pair $({\varepsilon}_i,\delta_i)$ such that $i$ is $({\varepsilon}_i,\delta_i)$-suboptimal and the product ${\varepsilon}_i^{-2} \cdot \delta_i^{-1}$ is as small as possible. SPC is anytime in the sense that it first identifies an $({\varepsilon},\delta)$-optimal configuration for large values of ${\varepsilon}$ and $\delta$ and then continues to refine these values as long as it is allowed to run. This is helpful for users who have difficulty setting these parameters up front, as already discussed. SPC’s strategy for progressing iteratively through smaller and smaller values of ${\varepsilon}$ and $\delta$ also has another advantage: it is actually faster than starting with the “final” values of ${\varepsilon}$ and $\delta$ and applying them to each configuration. This is because extremely weak configurations can be dismissed cheaply based on large $({\varepsilon}, \delta)$ values, instead of taking more samples to estimate their runtimes more finely. Other Related Work {#sec:related} ------------------ There is a large body of related work in the multi-armed bandits literature, which does not attack quite the same problem but does similarly leverage the “optimism in the face of uncertainty” paradigm and many tools of analysis [@lai1985asymptotically; @auer2002finite; @bubeck2012regret]. We do not survey this work in detail as we have little to add to the extensive discussion by @ijcai17, but we briefly identify some dominant threads in that work. Perhaps the greatest contact between the communities has occurred in the sphere of hyperparameter optimization [@bergstra2011algorithms; @thornton2013auto; @li2016hyperband] and in the literature on bandits with correlated arms that scale to large experimental design settings [@kleinberg2006anytime; @kleinberg2008multi; @chaudhuri2009parameter; @bubeck2011x; @srinivas2012information; @cesa2012combinatorial; @munos2014bandits; @shahriari2016taking]. In most of this literature, all arms have the same, fixed cost; others [@guha2007approximation; @tran2012knapsack; @badanidiyuru2013bandits] consider a model where costs are variable but always paid in full. (Conversely, in algorithm configuration we can stop runs that exceed a captime, yielding a potentially censored sample at bounded cost.) Some influential departures from this paradigm include @kandasamy2016multi, @ganchev2010censored, and most notably @li2016hyperband; reasons why these methods are nevertheless inappropriate for use in the algorithm configuration setting are discussed at length by @ijcai17. Recent work has examined the learning-theoretic foundations of algorithm configuration, inspired in part by an influential paper of @gupta2017pac that framed algorithm configuration and algorithm selection in terms of learning theory. This vein of work has not aimed at a general-purpose algorithm configuration procedure, as we do here, but has rather sought sample-efficient, special-purpose algorithms for particular classes of problems, including combinatorial partitioning problems (clustering, max-cut, etc) [@balcan2017learning], branching strategies in tree search [@balcan2018learning], and various algorithm selection problems [@Vitercik2018]. Nevertheless, this vein of work takes a perspective similar to our own and demonstrates that algorithm configuration has moved decisively from being solely the province of heuristic methods to being a topic for rigorous theoretical study. Model ===== We define an algorithm configuration problem by the 4-tuple $(N, \Gamma, R, \kappa_0)$, where these elements are defined as follows. $N$ is a family of (potentially randomized) algorithms, which we call *configurations* to suggest that a single piece of code instantiates each algorithm under a different parameter setting. We do not assume that different configurations exhibit any sort of performance correlations, and can so capture the case of $n$ distinct algorithms by imagining a “master algorithm” with a single, $n$-valued categorical parameter. Parameters are allowed to take continuous values: $|N|$ can be uncountable. We typically use $i$ to index configurations. $\Gamma$ is a probability distribution over input instances. When the instance distribution is given implicitly by a finite benchmark set, let $\Gamma$ be the uniform distribution over this set. We typically use $j$ to index (input instance, random seed) pairs, to which we will hereafter refer simply as instances. $R(i,j)$ is the execution time when configuration $i \in N$ is run on input instance $j$. Given some value of $\theta > 0$, we define $R(i,j,\theta) = \min\{R(i,j), \theta\}$, the runtime capped at $\theta$. $\kappa_0 > 0$ is a constant such that $R(i,j) \geq \kappa_0$ for all configurations $i$ and inputs $j$. For any timeout threshold $\theta$, let $R_\theta(i) = {\mathrm{E}}_{j \sim \Gamma}[R(i,j,\theta)]$ denote the average $\theta$-capped running time of configuration $i$, over input distribution $\Gamma$. Fixing some running time $\bar{\kappa} = 2^{\beta} \kappa_0$ that we will never be willing to exceed, the quantity $R_{\bar{\kappa}}(i)$ corresponds to the expected running time of configuration $i$ and will be denoted simply by $R(i)$. We will write $OPT = \min_i R(i)$. Given $\epsilon > 0$, a goal is to find $i^* \in N$ such that $R(i^*) \leq (1+\epsilon) OPT$. We also consider a relaxed objective, where the running time of $i^*$ is [*capped*]{} at some threshold value $\theta$ for some small fraction of (instance, seed) pairs $\delta$. \[def:eps-delta-opt\] A configuration $i^*$ is *$(\epsilon,\delta)$-optimal* if there exists some threshold $\theta$ such that $R_{\theta}(i^*) \leq (1+\epsilon) OPT$, and $\Pr_{j \sim \Gamma} \big( R(i^*,j) > \theta \big) \leq \delta$. Otherwise, we say $i^*$ is *$(\epsilon, \delta)$-suboptimal*. [^1]: While both SPC and CR use confidence bounds to guide search, they take different approaches. Rather than rejecting configurations whose lower bounds get too large, SPC focuses on configurations with small lower bounds. By allocating a greater proportion of total runtime to such promising configurations we both improve the bounds for configurations about which we are more uncertain and allot more resources to configurations with relatively low mean runtimes about which we are more confident.

--- abstract: 'The Chern-Simons forms for $\mathbb{R}$-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such $\mathbb{R}$-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for $\mathbb{R}$-linear connections of Lie algebroids.' author: - Bogdan Balcerzak title: | Chern-Simons forms for $\mathbb{R}$-linear connections on\ Lie algebroids --- [^1][^2] Introduction ============ We observe that non-linear objects (forms, connections, mappings between modules of cross-sections of vector bundles, which are non–linear over a ring of smooth functions) have increasing meaning in problems of differential geometry. S. Evens, J. H. Lu and A. Weinstein considered especial non–linear connections of Lie algebroids called connections up to homotopy (see [@Evens-Lu-Weinstein]). Crainic and Fernandes [Crainic-up to homotopy]{}, [@Crainic-Fernandes-jets] introduce the Chern character for non–linear connections. They discuss non–linear forms on Lie algebroids with values in a super vector bundle as antisymmetric, multilinear maps over $\mathbb{R}$ (not necessarily multilinear over the ring of smooth functions), which have a local property. Every non–linear connection $\nabla $ establishes on non–linear forms the covariant derivative operator. If $\nabla $ is flat, the Chern character vanishes and the induced covariant derivative operator is the exterior derivative, and in classically way defines the cohomology space. Crainic and Fernandes introduced secondary characteristic classes for connections up to homotopy [@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets]. We stay the question whenever these ideas refer to $\mathbb{R}$-linear forms and $\mathbb{R}$-linear connections – meaning as objects for which it is not supposed a local property. In the paper, using the generalized Stokes formula for $\mathbb{R}$-linear connections on Lie algebroids, we prove the Chern-Simons transgression formula without assumption locality for $\mathbb{R}$-linear connections. This is a helpful starting point to define characteristic classes for $\mathbb{R}$-linear connections on Lie algebroids. Some Crainic and Fernandes ideas we use to extend notions of Chern character and exotic (secondary) characteristic classes to $\mathbb{R}$-linear objects. Moreover, we found some explicit formulae for $\mathbb{R}$-linear Chern-Simons forms. In particular, we gain an direct formula of exotic (secondary) characteristic classes for an $\mathbb{R}$-linear connection as some trace $\mathbb{R}$-linear forms on a Lie algebroid. A *Lie algebroid* is a trip $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $, in which $A$ is a real vector bundle over a manifold $M$, $\rho _{A}:A\rightarrow TM$ (called an *anchor*) is a homomorphism of vector bundles, $\left( \Gamma \left( A\right) ,[\![\bullet ,\bullet ]\!]_{A}\right) $ is an $\mathbb{R}$-Lie algebra and the Leibniz identity$$\lbrack \![a,f\cdot b]\!]_{A}=f\cdot \lbrack \![a,b]\!]_{A}+\rho _{A}\left( a\right) \left( f\right) \cdot b\ \ \ \ \ \text{for all\ \ \ \ }a,b\in \Gamma \left( A\right) ,\ f\in \mathscr{C}^{\infty }\left( M\right)$$holds. Since the representation $\varrho :\mathscr{C}^{\infty }\left( M\right) \rightarrow \limfunc{End}\nolimits_{\mathscr{C}^{\infty }\left( M\right) }\left( \Gamma \left( A\right) \right) $, $\varrho \left( \nu \right) \left( a\right) =\nu \cdot a$, $\nu \in \mathscr{C}^{\infty }\left( M\right) $,$\ a\in \Gamma \left( A\right) $, is faithful ([@Herz], see also [@B-K-W-Primary]), the anchor induces a homomorphism of Lie algebras $\limfunc{Sec}\rho _{A}:\Gamma \left( A\right) \rightarrow \mathscr{X}\left( M\right) $, $a\mapsto \rho _{A}\circ a$. If $\rho _{A}$ is a constant rank (i.e. $\func{Im}\rho _{A}$ is a constant dimensional and completely integrable distribution), we say that $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $ is *regular*. A tangent bundle $TM$ to a manifold $M$ with the identity as an anchor and the bracket of vector fields is an elementary example of a Lie algebroid. For more about Lie algebroids and their properties we refer for example to [@Mackenzie], [Higgins-Mackenzie]{}, [@Kubarski-Lyon], [@Fernandes], [B-K-W-Primary]{}, [@Crainic-Fernandes-jets]. There are Lie functors from many geometric categories to the category of Lie algebroids (see a long list eg in [@Mackenzie], [@Kubarski-Lyon]). Especially meaning in the paper have algebroids of vector bundles. We recall that the module $\mathscr{CDO}\left( E\right) $ of sections of the Lie algebroid $\limfunc{A}\left( E\right) $ of a vector bundle $E$ is the space of all covariant differential operators in $E$, i.e. $\mathbb{R}$-linear operators $\ell :\Gamma \left( E\right) \rightarrow \Gamma \left( E\right) $ such that there exists exactly one $\widetilde{\ell }\in \mathscr{X}\left( M\right) $ with $\ell \left( f\zeta \right) =f\ell \left( \zeta \right) +\widetilde{\ell }\left( f\right) \zeta $ for all $f\in \mathscr{C}^{\infty }\left( M\right) $ and $\zeta \in \Gamma \left( E\right) $; see for example [@Teleman], [@Mackenzie], [@Kubarski-Lyon]. Let $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $ and $\left( B,\rho _{B},[\![\bullet ,\bullet ]\!]_{B}\right) $ be Lie algebroids over the same manifold $M$. A homomorphism $\nabla :A\rightarrow B$ of vector bundles is called an $A$-*connection* in $B$ if $\rho _{B}\circ \nabla =\rho _{A}$ (see [@B-K-W-Primary]). If an $A$-connection $\nabla $ in $B$ is a homomorphism of Lie algebroids ($\nabla $ preserves the Lie brackets) we say that $\nabla $ is *flat*. The notion of an $A$-connection in $B$ generalizes the known notions of connections (for example usual and partial covariant derivatives in vector bundles, a connection in principal bundles, a connection in extensions of Lie algebroids). In the case where $A=TM$ and $B=\limfunc{A}\left( E\right) $ is an algebroid of a vector bundle $E$, $TM$-connections in $\limfunc{A}\left( E\right) $ are one–to–one with covariant derivatives in $E$. For an arbitrary Lie algebroid $A$ and $B=\limfunc{A}\left( E\right) $ we have $A$-connections of $E$ considered in [@Mackenzie], [@Fernandes], [@Crainic-Fernandes-jets]. In case $B=\limfunc{A}\left( P\right) $ is a Lie algebroid of a principal bundle $P$, we get $A$-connections in $P$. In Poisson geometry an especially rule have connections acting from a Lie algebroid $T^{\ast }M$ associated to a given Poisson structure. In these examples a connection $\nabla $ considered as a mapping on modules of cross-sections is linear over $\mathscr{C}^{\infty }\left( M\right) $. By an $\mathbb{R}$*-linear connection *of $A$* *in $B$* *we called an $\mathbb{R}$-linear operator $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $* *such that $$\limfunc{Sec}\rho _{B}\circ \nabla =\limfunc{Sec}\rho _{A}\text{.}$$An $\mathbb{R}$-linear connection of $A$ in the Lie algebroid $A\left( E\right) $ is called the $\mathbb{R}$*-linear connection of* $A$*on the vector bundle* $E$. We call the map$$R^{\nabla }:\Gamma \left( A\right) \times \Gamma \left( A\right) \rightarrow \Gamma \left( B\right) ,\ \ R^{\nabla }\left( \alpha ,\beta \right) =[\![\nabla _{\alpha },\nabla _{\beta }]\!]_{B}-\nabla _{\lbrack \![\alpha ,\beta ]\!]_{A}}$$a *curvature* of $\nabla $. We see that $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $ is flat if $R^{\nabla }=0$. For every Lie algebroid $A$, the adjoint connection $\limfunc{ad}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( A\right) $, $\limfunc{ad}\left( a\right) =[\![a,\bullet ]\!]_{A}$ is an $\mathbb{R}$-linear connection of $A$ on $A$. The notion of an $\mathbb{R} $-linear connection includes so-called non-linear connections and connections up to homotopy on super-vector bundles ([@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets], [@Evens-Lu-Weinstein]); such connections have a local property. Let $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $, $\left( B,\rho _{B},[\![\bullet ,\bullet ]\!]_{B}\right) $ be Lie algebroids over a manifold $M$. An $\mathbb{R}$-multilinear, antisymmetric map $$\omega :\underset{n}{\underbrace{\Gamma \left( A\right) \times \cdots \times \Gamma \left( A\right) }}\longrightarrow \Gamma \left( B\right)$$is called an $\mathbb{R}$*-linear* $n$-*form* on $A$ with values in $B$. The space of all such $\mathbb{R}$-linear $n$-forms will be denoted by $\mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) $, and the space of $\mathbb{R}$-linear forms on $A$ with values in $B$ by$$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) =\bigoplus\limits_{k\geq 0}\mathcal{A}lt_{\mathbb{R}}^{k}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) ,$$where $\mathcal{A}lt_{\mathbb{R}}^{0}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) =\Gamma \left( B\right) $.[ ]{}Observe that if $\nabla :A\rightarrow B$ is an arbitrary $\mathbb{R}$-linear connection, then the curvature $R^{\nabla }$ is an element of $\mathcal{A}lt_{\mathbb{R}}^{2}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) $. We define the covariant differential operator$$d_{\mathbb{R}}^{\nabla }:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet +1}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right)$$for $\mathbb{R}$-linear forms on $A$ with values in $B$ by the classical formula$$\begin{gathered} \left( d_{\mathbb{R}}^{\nabla }\eta \right) \left( a_{1},\ldots ,a_{n+1}\right) =\dsum\limits_{i=1}^{n+1}\left( -1\right) ^{i+1}\nabla _{a_{i}}\left( \eta \left( a_{1},\ldots \hat{\imath}\ldots ,a_{n+1}\right) \right) \\ +\dsum\limits_{i<j}\left( -1\right) ^{i+j}\eta \left( \lbrack \![a_{i},a_{j}]\!]_{A},a_{1},\ldots \hat{\imath}\ldots \hat{\jmath}\ldots ,a_{n+1}\right) .\end{gathered}$$$d_{\mathbb{R}}^{\nabla }$ is an antiderivation in $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( E\right) \right) $ with respect to the product of $\mathbb{R}$-linear forms. A flat $\mathbb{R}$-linear connection $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $ induces, denoted by $H_{\nabla ,\mathbb{R}}^{\bullet }\left( A;B\right) $, the *Lie algebroid* $\mathbb{R}$*-cohomology space with coefficients in* $B$ as the cohomology space of the complex $\left( \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) ,d_{\mathbb{R}}^{\nabla }\right) $. The differential operator $d_{\mathbb{R}}^{\limfunc{Sec}\rho _{A}}$ induced by the anchor, i.e. by the flat[ ]{}$A$-connection in $TM$, will be denoted by $d_{A,\mathbb{R}}$. Since modules $\Gamma \left( M\times \mathbb{R}\right) $ and $\mathscr{C}^{\infty }\left( M\right) $ are isomorphic, it follows that $d_{A,\mathbb{R}} $ is an extension of the exterior derivative from the space $\Omega ^{\bullet }\left( A\right) $ of ($\mathscr{C}^{\infty }\left( M\right) $-linear) differential forms on $A$ to $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $. Let us recall that the cohomology space of the complex $\left( \Omega ^{\bullet }\left( A\right) ,d_{A}\right) $ where $\Omega ^{\bullet }\left( A\right) $$\mathscr{C}^{\infty }\left( M\right) $-linear $A$, $d_{A}=\left. d_{\mathbb{R}}^{\limfunc{Sec}\rho _{A}}\right\vert \Omega ^{\bullet }\left( A\right) :\Omega ^{\bullet }\left( A\right) \rightarrow \Omega ^{\bullet +1}\left( A\right) $, is called the *cohomology of Lie algebroid* and is denoted by $H^{\bullet }\left( A\right) $. Let $E$ be a vector bundle over $M$. Observe that $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is a left module over the algebra $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ with the standard multiplication of forms. Moreover, $$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( \func{End}E\right) \cong \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right)$$as $\mathscr{C}^{\infty }\left( M\right) $-modules by the isomorphism defined in such a way that$$\omega \otimes \phi \longmapsto \omega \wedge \phi .$$ In the paper, we define the Chern-Simons forms for $\mathbb{R}$-linear connections on Lie algebroids. The generalized Chern-Simons formula is derived as a consequence of Stokes’ formula for $\mathbb{R}$-linear forms. The notion of the Chern classes of a vector bundle as cohomology classes of some $\mathbb{R}$-linear Chern-Simons forms is proposed. We show that such classes for a given $\mathbb{R}$-linear connection[ ]{}do not depend on the choice of the connection. In the paper, we discuss the wider then in [@Crainic-Fernandes-jets] for linear connections set of obstructions to the existence of a flat connection of a given Lie algebroid. Using ideas form papers Crainic and Fernandes, we introduce the secondary characteristic classes for arbitrary $\mathbb{R}$-linear connections of Lie algebroids in vector bundles. If an $\mathbb{R}$-linear $A$-connection $\nabla $ on a vector bundle $E$ is metrizable with respect to any metric $h$ in $E$ (i.e. $\nabla h=0$), the defined secondary characteristic classes vanishes. Therefore, secondary characteristic classes of $\nabla $ are obstructions to the existence of an invariant metric with respect to $\nabla $. In [@Crainic-Fernandes-jets] were considered connections up to homotopy (some non-linear connections with a local property). Here we examine all $\mathbb{R}$-linear connections. At the end of the last section we derive some comments on the Chern-Simons forms for $\mathbb{R}$-linear connections (in particular for Lie algebroids over odd dimensional manifolds). The Chern-Simons transgression forms on Lie algebroids and the Chern Character ============================================================================== Let $\left( A,\rho _{A},[\![\cdot ,\cdot ]\!]_{A}\right) $ be a Lie algebroid on a manifold $M$, $E$ a vector bundle over $M$, $k$ a natural number and $\func{pr}_{2}:\mathbb{R}^{k}\times M\rightarrow M$ a projection on the second factor. Consider an $\mathbb{R}$-linear connection $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$. The standard fibrewise trace $\func{Tr}:\Gamma \left( \func{End}E\right) \rightarrow \mathscr{C}^{\infty }\left( M\right) $ on $\func{End}\left( E\right) $ induces a trace $$\func{Tr}_{\ast }:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$such that $\func{Tr}_{\ast }\left( \omega \right) \left( a_{1},\ldots ,a_{n}\right) =\func{Tr}\left( \left( \omega \right) \left( a_{1},\ldots ,a_{n}\right) \right) $. Set (for $p\geq 1$)$$\func{ch}_{p}\left( \nabla \right) =\func{Tr}_{\ast }\left( R^{\nabla }\right) ^{p}\in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$where $\left( R^{\nabla }\right) ^{p}\in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is, for $a_{1},...,a_{2p}\in \Gamma \left( A\right) $, given by$$\left( R^{\nabla }\right) ^{p}\left( a_{1},...,a_{2p}\right) =\frac{1}{2^{p}}\sum\nolimits_{\tau \in S_{2p}}\func{sgn}\tau \cdot R_{a_{\tau \left( 1\right) },a_{\tau \left( 2\right) }}^{\nabla }\circ \cdots \circ R_{a_{\tau \left( 2p-1\right) },a_{\tau \left( 2p\right) }}^{\nabla }.$$The $2p$-form $\func{ch}_{p}\left( \nabla \right) $ is called the *Chern character form* associated to $\nabla $. \[comm\_Tr\_and\_diff\]$d_{A,\mathbb{R}}\circ \func{Tr}_{\ast }=\func{Tr}_{\ast }\circ d_{\mathbb{R}}^{\overline{\nabla }}$ where $\overline{\nabla }:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( \limfunc{End}E\right) $,* *$\overline{\nabla }_{a}=\left[ \nabla _{a},\bullet \right] $. First, recall that the space $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is isomorphic to$$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( \func{End}E\right) .$$Let $\eta \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $, $\varphi \in \Gamma \left( \func{End}E\right) $. Then $\func{Tr}_{\ast }\left( \eta \otimes \varphi \right) =\eta \cdot \func{Tr}\varphi $. It is a simple matter to see that $d_{A,\mathbb{R}}\left( \func{Tr}\varphi \right) =\func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) $. Therefore$$\begin{aligned} d_{A,\mathbb{R}}\func{Tr}_{\ast }\left( \eta \otimes \varphi \right) &=&d_{A,\mathbb{R}}\eta \cdot \func{Tr}\varphi +\left( -1\right) ^{n}\eta \wedge d_{A,\mathbb{R}}\left( \func{Tr}\varphi \right) \\ &=&d_{A,\mathbb{R}}\eta \cdot \func{Tr}\varphi +\left( -1\right) ^{n}\eta \wedge \func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) \\ &=&\func{Tr}_{\ast }\left( d_{A,\mathbb{R}}\eta \otimes \varphi +\left( -1\right) ^{n}\eta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) \\ &=&\func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\left( \eta \otimes \varphi \right) \right) .\end{aligned}$$ $\mathscr{C}^{\infty }\left( \mathbb{R}\times M\right) $-modules $\Gamma \left( \func{pr}_{2}^{\ast }A\right) $ and $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) $ are isomorphic (see [Higgins-Mackenzie]{}) and this way the module of cross-sections of the inverse image$$\func{pr}_{2}^{\;\wedge }\hspace{-0.1cm}\left( A\right) =\left\{ \left( \gamma ,w\right) \in T\left( \mathbb{R}^{k}\times M\right) \times A:\left( \func{pr}_{2}\right) _{\ast }\gamma =\rho _{A}\left( w\right) \right\} \cong T\mathbb{R}^{k}\times A$$of $A$ by $\func{pr}_{2}$ is a $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) $-submodule of $$\mathscr{X}\left( \mathbb{R}^{k}\times M\right) \times \left( \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) \right)$$($T\mathbb{R}^{k}\times A$ is the Cartesian product of Lie algebroids $T\mathbb{R}^{k}$ and $A$, see [@Kubarski-invariant]). We denote cross-sections $0\times a$, $\frac{\partial }{\partial t^{j}}\times 0$ of the vector bundle $T\mathbb{R}^{k}\times A$ briefly by $a$ and $\frac{\partial }{\partial t^{j}}$, respectively. Let $$\Delta ^{k}=\left\{ \left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k};\;\;\;\forall i\;\;t_{i}\geq 0\,,\;\;\sum\nolimits_{i=1}^{k}t_{i}\leq 1\right\}$$be the *standard* $k$*-simplex* in $\mathbb{R}^{k}$. Additionally we set the *standard* $0$*-simplex* as $\Delta ^{0}=\left\{ 0\right\} $. Define$$\dint\nolimits_{\Delta ^{k}}:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet -k}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) ,$$$$\left( \int\nolimits_{\Delta ^{k}}\omega \right) \left( a_{1},...,a_{n-k}\right) =\int\nolimits_{\Delta ^{k}}\omega \left( \frac{\partial }{\partial t^{1}},...,\frac{\partial }{\partial t^{k}},a_{1},...,a_{n-k}\right) _{|\left( t_{1},...,t_{k},\bullet \right) }dt_{1}...dt_{k},$$$$\left( \int\nolimits_{\Delta ^{0}}\omega \right) \left( a_{1},...,a_{n}\right) =\iota _{0}^{\ast }\left( \omega \left( 0\times a_{1},...,0\times a_{n}\right) \right) ,\ \ \ \int\nolimits_{\Delta ^{0}}f=\iota _{0}^{\ast }f$$for all $n\geq 1$, $1\leq k\leq n$, $\omega \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $, $f\in \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) $ and where $\iota _{0}:M\rightarrow \Delta ^{0}\times M$ is an inclusion defined by $\iota _{0}\left( x\right) =\left( 0,x\right) $. In view of the factorization property in $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) $, we conclude that for $\nabla $ there exists exactly one $\mathbb{R}$-linear connection $$\widetilde{\nabla }:\Gamma \left( T\mathbb{R}^{k}\times A\right) \longrightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right)$$of $T\mathbb{R}^{k}\times A$ on $\func{pr}_{2}^{\;\ast }E$* *such that$$\left( \widetilde{\nabla }_{\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) }\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) =\nabla _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right)$$for all $\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) \in \mathscr{X}\left( \mathbb{R}^{k}\times M\right) \times \left( \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) \right) $, $\nu \in \Gamma \left( E\right) $, $t=\left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k} $. In particular, $\left( \widetilde{\nabla }_{\left( 0\times \left( \rho _{A}\circ a\right) ,1\otimes a\right) }\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) =\nabla _{a}\left( \nu \right) $, $a\in \Gamma \left( A\right) $.$\;$The connection$\;\widetilde{\nabla }$ is called the *lifting* of $\nabla $ to $T\mathbb{R}^{k}\times A$. Let $\nabla ^{0},\,\nabla ^{1},\ldots ,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$be $\mathbb{R}$-linear connections of a Lie algebroid $A$ on a vector bundle $E$ and $\widetilde{\nabla }^{0},\,\widetilde{\nabla }^{1},\ldots ,\,\widetilde{\nabla }^{k}:\Gamma \left( T\mathbb{R}^{k}\times A\right) \rightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right) $ be their liftings to $T\mathbb{R}^{k}\times A$. Then there exists an $\mathbb{R}$-linear connection$$\nabla ^{\func{aff}_{k}}:\Gamma \left( T\mathbb{R}^{k}\times A\right) \longrightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right) ,$$called the *affine combination of connections* $\nabla ^{0},\nabla ^{1},\ldots ,\nabla ^{k}$, given by $$\begin{aligned} &&\left( \nabla _{\,\,\,\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) }^{\func{aff}_{k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) \\ &=&\left( 1-\sum\nolimits_{i=1}^{k}t_{i}\right) \cdot \left( \nabla ^{0}\right) _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right) +\sum\nolimits_{i=1}^{k}t_{i}\cdot \left( \nabla ^{i}\right) _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right) .\end{aligned}$$For all $0<k\leq 2p$ we define an $\mathbb{R}$-linear form$$\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) =\int\nolimits_{\Delta ^{k}}\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) \in \mathcal{A}lt_{\mathbb{R}}^{2p-k}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$called the *Chern-Simons form* for $\left( \nabla ^{0},...,\nabla ^{k}\right) $ and additionally we put $cs_{p}\left( \nabla ^{0}\right) =\func{ch}_{p}\left( \nabla ^{0}\right) $. We have the following (useful) Stokes’ formula for $\mathbb{R}$-linear forms on $A$ (see [@Balcerzak-Stokes]) being a generalization of the one for tangent bundles given by R. Bott [@Bott]. For every natural number* *$k$,$$\int\nolimits_{\Delta ^{k}}\circ \,d_{T\mathbb{R}^{k}\times A,\mathbb{R}}+\left( -1\right) ^{k+1}d_{A,\mathbb{R}}\circ \int\nolimits_{\Delta ^{k}}=\dsum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\int\nolimits_{\Delta ^{k-1}}\circ \,\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }, \label{Stokes}$$where $\sigma _{j}^{k}:\mathbb{R}^{k}\rightarrow \mathbb{R}^{k+1}$ for $0\leq j\leq k+1$ are functions defined by $\sigma _{0}^{0}\left( 0\right) =1 $, $\sigma _{1}^{0}\left( 0\right) =0$, and for $t=\left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k}$ by$$\begin{aligned} \sigma _{0}^{k}\left( t\right) &=&\left( 1-\dsum\nolimits_{i=1}^{k}t_{i},t_{1},...,t_{k}\right) , \\ \sigma _{j}^{k}\left( t\right) &=&\left( t_{1},...,t_{j-1},0,t_{j},...,t_{k}\right) ,\;\;1\leq j\leq k+1,\end{aligned}$$and where $\left( \left( \int\nolimits_{\Delta ^{k-1}}\circ \,\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }\right) \omega \right) \left( a_{1},...,a_{n-k+1}\right) $ is, by definition, equal to$$\int\nolimits_{\Delta ^{k-1}}\omega \left( d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{1}}\right) ,...,d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{k-1}}\right) ,a_{1},...,a_{n-k+1}\right) _{|\left( t_{1},...,t_{k-1},\bullet \right) }\hspace{-0.3cm}dt_{1}...dt_{k-1}$$and $$\left( \left( \int\nolimits_{\Delta ^{0}}\circ \,\left( d\sigma _{j}^{0}\times \func{id}_{A}\right) ^{\ast }\right) \omega \right) \left( a_{1},...,a_{n}\right) =\left( \sigma _{j}^{0}\times \func{id}_{M}\circ \iota _{0}\right) ^{\ast }\left( \omega \left( a_{1},...,a_{n}\right) \right)$$if $k\geq 2$, $\omega \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \right) $, $a_{i}\in \Gamma \left( A\right) $, $j\in \left\{ 0,1\right\} $. The following lemma will be useful below in the proof of the Chern-Simons formula for $\mathbb{R}$-linear connections of Lie algebroids. \[lemma\_abc\]Let $a,\,b\in \Gamma \left( A\right) $, $\nu \in \Gamma \left( E\right) $, $t\in \mathbb{R}^{k-1}$, $0\leq j\leq k,$ $1\leq s\leq k, $ $1\leq z\leq k-1$. Denote here the affine combination $\nabla ^{\func{aff}_{k}}$ of$\ \nabla ^{0},\ldots ,\nabla ^{k}$ by $\nabla ^{0,...,k}$. Then - $\left( R_{a,b}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) =\left( R_{a,b}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) ,$ - $\left( R_{\frac{\partial }{\partial \,\tilde{t}\,^{s}},a}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) $ is equal to $\left( R_{\frac{\partial }{\partial \,t^{s}},a}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $ if $1\leq s<j,$ and $\left( R_{\frac{\partial }{\partial \,t^{s-1}},a}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $ if $j\leq s\leq k$, and where $\tilde{t}\,^{i}$ are coordinates of the identity map of $\mathbb{R}^{k}$, - $\left( R_{d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{z}}\right) ,a}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) =\left( R_{\frac{\partial }{\partial t^{z}},a}^{\nabla ^{1...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $. Just calculations. *(The Chern-Simons formula for Lie algebroids and* $\mathbb{R}$*-linear connections)* Let $\left( A,\rho _{A},[\![\cdot ,\cdot ]\!]\right) $ be a Lie algebroid on a manifold $M$, $E$ a vector bundle over $M$, $k\in \mathbb{N}$, $\nabla ^{0},\,...,\,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$ $\mathbb{R}$-linear connections of $A$ on $E$. Then$$\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) =\sum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\,\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...\widehat{\nabla ^{j}}...,\nabla ^{k}\right) \label{ChernSimonsformula}$$for all integer numbers $p$ such that $0<k\leq 2p$ and $d_{A,\mathbb{R}}\left( cs_{p}\left( \nabla ^{0}\right) \right) =0$. From Lemma \[comm\_Tr\_and\_diff\] and the Bianchi identity ($d_{\mathbb{R}}^{\overline{\nabla ^{j}}}\left( R^{\nabla ^{j}}\right) =0$) we deduce that forms $\func{ch}_{p}\left( \nabla ^{0}\right) $ and $\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) $ are closed. Since these forms are closed, applying the Stokes formula (\[Stokes\]) we conclude that $$\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) =\sum\limits_{j=0}^{k}\left( -1\right) ^{j}\int\nolimits_{\Delta ^{k-1}}\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) .$$Let $a_{0},$...,$a_{2p-k}\in \Gamma \left( A\right) $. From the above$$\begin{gathered} \left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) \left( a_{0},...,a_{2p-k}\right) \\ =\dsum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\left( \int\nolimits_{\Delta ^{k-1}}\left( d\sigma _{j}^{k-1}\times \func{id}_{M}\right) ^{\ast }\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) \right) \left( a_{0},...,a_{2p-k}\right) .\end{gathered}$$From the definition of $\left( R^{\nabla ^{\func{aff}_{k}}}\right) ^{p}$ and fact that $R_{\frac{\partial }{\partial \widetilde{t}^{i}},\frac{\partial }{\partial \widetilde{t}^{j}}}^{\nabla ^{\func{aff}_{k}}}=0$ (where $\left( \tilde{t}\,^{1},...,\tilde{t}\,^{k}\right) $ is the identity map on the manifold $\mathbb{R}^{k}$) we observe that the possible non-zero terms in the above sum are the form$$R_{d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{s}}\right) ,a}^{\nabla ^{\func{aff}_{k}}}\circ \cdots \circ R_{b,c}^{\nabla ^{\func{aff}_{k}}}\circ \cdots \circ R_{d,e}^{\nabla ^{\func{aff}_{k}}},\ \ \text{\ }a,b,c,d,e\in \Gamma \left( A\right) .$$Lemma \[lemma\_abc\] now yields that $\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) \left( a_{0},...,a_{2p-k}\right) $ is equal to$$\begin{gathered} \sum_{j=0}^{k}\left( -1\right) ^{j}\hspace{-0.1cm}\int\nolimits_{\Delta ^{k-1}}\hspace{-0.3cm}\func{ch}_{p}\left( \nabla ^{0,...\widehat{j}...,k}\right) \hspace{-0.1cm}\left. \left( \frac{\partial }{\partial t^{1}},...,\frac{\partial }{\partial t^{k-1}},a_{0\,},...,a_{2p-k}\right) \right\vert _{\left( t_{1},...,t_{k-1},\bullet \right) }\hspace{-0.3cm}dt_{1}\ldots dt_{k-1} \\ =\left( \sum_{j=0}^{k}\left( -1\right) ^{j}\,\limfunc{cs}\nolimits_{p}\left( \nabla _{0},...\widehat{\nabla ^{j}}...,\nabla _{k}\right) \right) \left( a_{0\,},...,a_{2p-k}\right) .\end{gathered}$$ *If* $\nabla ^{0},\nabla ^{1},\ldots ,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$*are* $\mathscr{C}^{\infty }\left( M\right) $*-linear connections, then* $\nabla ^{\func{aff}_{k}}$* is a* $\mathscr{C}^{\infty }\left( M\right) $*-linear connection. In this case, we obtain a formula due to property of Chern-Simons transgressions in [Crainic-Fernandes-jets]{} by M. Crainic and R. L. Fernandes.* *Let* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ *be an* $\mathbb{R}$*-linear connection of* $A$* on a vector bundle* $E$*.* *The Chern character forms* $\func{ch}_{p}\left( \nabla \right) \in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ *are closed and their cohomology classes*$$\func{ch}_{p}\left( A,E\right) =\left[ \func{ch}_{p}\left( \nabla \right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p}\left( A;M\times \mathbb{R}\right) \emph{,}$$*do not depend on the choice of the connection* $\nabla $*. Indeed, let* $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ *be* $\mathbb{R}$*-linear connections of* $A$* on* $E$*. According to (\[ChernSimonsformula\]), we have*$$\begin{aligned} d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},\nabla ^{1}\right) \right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla ^{1}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{0}\right) \\ &=&\func{ch}_{p}\left( \nabla ^{1}\right) -\func{ch}_{p}\left( \nabla ^{0}\right) .\end{aligned}$$ In this way we have correctly defined the Chern character$$\func{ch}\left( A,E\right) \in H_{\rho _{A},\mathbb{R}}\left( A;M\times \mathbb{R}\right) .$$ *([@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets]) In the particular case we can obtain the Chern character for a non-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $* of a Lie algebroid* $A$* on a vector bundle* $E$*, i.e. a local* $\mathbb{R}$*-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $*. In the space* $\Omega _{nl}\left( A\right) $*of non-linear differential forms on* $A$* (local* $\mathbb{R}$*-linear forms on* $A$*) we have the differential operator* $d_{nl}=d_{A,\mathbb{R}}|\Omega _{nl}^{\bullet }\left( A\right) :$* *$\Omega _{nl}^{\bullet }\left( A\right) \rightarrow \Omega _{nl}^{\bullet +1}\left( A\right) $. Secondary characteristic classes for $\mathbb{R}$-linear connections and some the Chern-Simons forms for a pair of connections ============================================================================================================================== Let $E$ be a vector bundle over $M$ with a metric $h$ and $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ be an $\mathbb{R}$-linear connection of a Lie algebroid $A$ on $E$. We define an $\mathbb{R}$-linear connection $\nabla ^{h}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$ such that$$\left( \rho _{A}\circ a\right) \left( h\left( s,t\right) \right) =h\left( \nabla _{a}s,t\right) +h\left( s,\nabla _{a}^{h}t\right) ,\ \ \ a\in \Gamma \left( A\right) ,\ s,t\in \Gamma \left( E\right) .$$We can observe that$$R_{a,b}^{\nabla ^{h}}=-\left( R_{a,b}^{\nabla }\right) ^{\ast },\ \ \ \ \ \ a,b\in \Gamma \left( A\right) ,$$where $\left( R_{a,b}^{\nabla }\right) ^{\ast }$ is the adjoint map to $R_{a,b}^{\nabla }$ with respect to $h$. Therefore we obtain the following lemma. \[Lemma\_1\_about\_nabla\_h\]If $\nabla _{0}$, $\nabla _{1}$ are $\mathbb{R}$-linear connections of $A$ on $E$, then - $\limfunc{cs}\nolimits_{p}\left( \nabla _{0}^{h}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla _{0}\right) $, - $\limfunc{cs}\nolimits_{p}\left( \nabla _{0}^{h},\nabla _{1}^{h}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla _{0},\nabla _{1}\right) .$ From the Chern-Simons formula (\[ChernSimonsformula\]) and Lemma [Lemma\_1\_about\_nabla\_h]{} (a) we deduce that $$\begin{aligned} d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla \right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{h}\right) \\ &=&\limfunc{cs}\nolimits_{p}\left( \nabla \right) -\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla \right) \\ &=&0,\end{aligned}$$because $\nabla $ is flat. In particular, we see that $\nabla ^{h}$ is also flat. The cohomology class $\left[ \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p-1}\left( A\right) $ do not depend on the choice of metric $h$. Let $h_{1}$, $h_{2}$ be two metrics on $E$ and let $\nabla ^{M}$ be any $TM$-connection on $E$. Thus $\nabla _{o}=\nabla ^{M}\circ \rho _{A}$ is an $A$-connection on $E$ (i.e. a linear connection). The Chern-Simons formula ([ChernSimonsformula]{}) yields$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{j}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{j}}\right) \label{e1}$$and$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{j}}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{j}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{j}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}\right) \label{e2}$$for $j\in \left\{ 1,2\right\} $. Lemma \[Lemma\_1\_about\_nabla\_h\] implies $\limfunc{cs}\nolimits_{p}\left( \nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}\right) $. From this, (\[e1\]) and (\[e2\]) we get$$\begin{aligned} & \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}}\right) \\ & =d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}},\nabla _{o}^{h_{2}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}},\nabla _{o}^{h_{1}}\right) \right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{2}}\right) \\ & =d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}},\nabla _{o}^{h_{2}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}},\nabla _{o}^{h_{1}}\right) \right) +d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{1}}\right) \\ & -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{2}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{2}}\right) .\end{aligned}$$Because of $\nabla _{o}$ is a linear connection, Proposition 1 from [Crainic-Fernandes-jets]{} yields $\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{1}}\right) $$-\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{2}}\right) $ is an exact form. In this way cohomology classes of $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}}\right) $ and $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}}\right) $ are both equal. *We call* $$\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =\left[ \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p-1}\left( A\right) ,\ \ \ p\in \left\{ 1,\ldots ,\limfunc{rank}E\right\} ,$$the secondary characteristic classes *of an* $\mathbb{R}$*-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $. If there exists in $E$ an invariant metric $h$ with respect to $\nabla $, then $\nabla ^{h}=\nabla $. Then classes $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) $ are equal to zero. Hence these classes are obstructions to the existence of an invariant metric with respect to $\nabla $. We obtain the following theorem analogous to Proposition 2 in [Crainic-Fernandes-jets]{}. Let $\nabla $, $\nabla _{m}$ be $\mathbb{R}$-linear connections of $A$ on $E$ and $\nabla _{m}$ be additionally metric. - If $p$ is even, then $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =0$. - If $p$ is odd, then $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) $ is a closed form and$$\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =\left[ 2\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) \right] .$$ Let $\nabla _{m}$ be metric connection with respect to a metric $h$. On account of the Chern-Simons formula (\[ChernSimonsformula\]), we have$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) .$$Now Lemma \[Lemma\_1\_about\_nabla\_h\] leads to $\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) $, because $\nabla _{m}^{h}=\nabla _{m}$. It follows that$$\begin{aligned} \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) \\ &=&\left( 1+\left( -1\right) ^{p+1}\right) \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) ,\end{aligned}$$which completes the proof. For two $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$ we define an $\mathbb{R}$-linear $1$-form$$\lambda =\nabla ^{1}-\nabla ^{0}\in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) .$$Let us observe that $$R^{\nabla ^{1}}=R^{\nabla ^{0}}+d^{\overline{\nabla }^{0}}\lambda +\left[ \lambda ,\lambda \right] , \label{curvature_and_pair}$$where $d^{\overline{\nabla }^{0}}$is the covariant derivative in $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ determined by $\overline{\nabla }^{0}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( \limfunc{End}E\right) $, $\overline{\nabla }_{a}^{0}=\left[ \nabla _{a}^{0},\bullet \right] $ for all $a\in \Gamma \left( A\right) $, and $\left[ \lambda ,\lambda \right] \in \mathcal{A}lt_{\mathbb{R}}^{2}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is given by $\left[ \lambda ,\lambda \right] \left( a,b\right) =\left[ \lambda \left( a\right) ,\lambda \left( b\right) \right] $ for all $a,b\in \Gamma \left( A\right) $. *[@Balcerzak]* For two $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ the following properties hold: $$(R^{\nabla ^{\limfunc{aff}_{1}}})_{\frac{\partial }{\partial t},a}\left( \nu \circ \limfunc{pr}\nolimits_{2}\right) _{|\left( t,\bullet \right) }=\lambda \left( a\right) \left( \nu \right) , \label{properity_1}$$$$(R^{\nabla ^{\limfunc{aff}_{1}}})_{a,b}\left( \nu \circ \limfunc{pr}\nolimits_{2}\right) _{|\left( t,\bullet \right) }=\left( 1-t\right) \cdot R_{a,b}^{\nabla ^{0}}\left( \nu \right) +t\cdot R_{a,b}^{\nabla ^{1}}\left( \nu \right) +\left( t^{2}-t\right) \cdot \left[ \lambda ,\lambda \right] _{\left( a,b\right) }\left( \nu \right) \label{properity_2}$$for all $a,b\in \Gamma \left( A\right) $, $\nu \in \Gamma \left( E\right) $, $t\in \mathbb{R}$. #### **The Chern-Simons forms of the first and the second rank**   Let $\theta \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $, $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ be an $\mathbb{R}$-linear connection of $A$ on $E$. Therefore $\nabla +\theta $ is also an $\mathbb{R}$-linear $A$-connection on $E$, and $\limfunc{cs}\nolimits_{1}\left( \nabla ,\nabla +\theta \right) \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ is given by $\limfunc{cs}\nolimits_{1}\left( \nabla ,\nabla +\theta \right) \left( a\right) =\limfunc{tr}\left( \theta \left( a\right) \right) $, $a\in \Gamma \left( A\right) $. Moreover, we conclude from (\[properity\_1\]), (\[properity\_2\]) and ([curvature\_and\_pair]{}) that$$\limfunc{tr}\left( R^{\nabla ^{\limfunc{aff}_{1}}}\right) ^{2}\left( \frac{\partial }{\partial t},\bullet \right) _{|\left( t,\bullet \right) }=2\limfunc{tr}\left( \theta \wedge R^{\nabla ^{0}}+t\cdot \theta \wedge d_{\mathbb{R}}^{\overline{\nabla ^{0}}}\theta +t^{2}\cdot \theta \wedge \theta \wedge \theta \right)$$for all $a_{1},a_{2},a_{3}\in \Gamma \left( A\right) $, $t\in \mathbb{R}$, hence$$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =\limfunc{tr}\left( 2\theta \wedge R^{\nabla }+\theta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\theta +\frac{2}{3}\theta \wedge \theta \wedge \theta \right) .$$If $\nabla $ and $\nabla +\theta $ are both flat, then $d_{\mathbb{R}}^{\overline{\nabla }}\theta =-\theta \wedge \theta $, which then yields $$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =-~\frac{1}{3}\limfunc{tr}\left( \theta \wedge \theta \wedge \theta \right) .$$ For every manifold $M$ of an odd dimension $2m-1$, $\limfunc{cs}\nolimits_{m}\left( \nabla ,\nabla +\theta \right) $ is closed. In the case where $M$ is a $3$-dimensional manifold, $\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) $ is closed and is given by the above formula; if additionally $\nabla $ is flat, we see that$$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =\limfunc{tr}\left( \theta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\theta +\frac{2}{3}\theta \wedge \theta \wedge \theta \right) . \label{formulaWZ}$$(\[formulaWZ\]) is a generalization of the known formula for tangent bundles of smooth, compact, oriented, three dimensional manifolds and standard connections (see for example [@Zhang]) to arbitrary rank three vector bundles and $\mathbb{R}$-linear connections. Moreover, we add (see [@Balcerzak]) that if both $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of a Lie algebroid $A$ on a vector bundle $E$ are flat, then the Chern–Simons $\mathbb{R}$-linear form $\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},\nabla ^{1}\right) $ is equal to $\left( -1\right) ^{p+1}\,\frac{\,p!\left( p-1\right) !\,}{\left( 2p-1\right) !}\func{Tr}_{\ast }\left( \lambda ^{2p-1}\right) $. In particular, for any flat $\mathbb{R}$-linear connection $\nabla $ of $A$ on $E$, $\nabla ^{h}$ is also flat and we conclude that the class $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) $ is represented by the form $$\left( -1\right) ^{p+1}\,\frac{\,p!\left( p-1\right) !\,}{\left( 2p-1\right) !}\func{Tr}_{\ast }\left( \omega ^{2p-1}\right) ,$$where $\omega =\nabla ^{h}-\nabla \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $.   [99]{} <span style="font-variant:small-caps;">B. Balcerzak, J. Kubarski and W. Walas</span>,** ***Primary characteristic homomorphism of pairs of Lie algebroids and Mackenzie algebroid*, Banach Center Publ. **54** (2001)**,** 135–173. <span style="font-variant:small-caps;">B. Balcerzak</span>, *Modular classes of Lie algebroids homomorphisms as some the Chern-Simons forms*, Univ. Iagel. Acta Math. **47** (2009), 11–28. <span style="font-variant:small-caps;">B. Balcerzak</span>, *The Generalized Stokes theorem for* $\mathbb{R}$*-linear forms on Lie algebroids*, Accepted to Journal of Applied Analysis; available as preprint , 2011. <span style="font-variant:small-caps;">R. Bott</span>, *Lectures on characteristic classes and foliations,* Springer Lecture Notes in Math. 279, Springer, Berlin, 1972. <span style="font-variant:small-caps;">M. Crainic</span>, *Connections up to homotopy and characteristic classes*, Preprint arXiv, (2000). <span style="font-variant:small-caps;">M. Crainic and R. L. Fernandes</span>, *Secondary Characteristic Classes of Lie Algebroids*, In: Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys. 662, pp. 157–176, Springer, Berlin, 2005. <span style="font-variant:small-caps;">S. Evens, J. H. Lu and A. Weinstein</span>, *Transverse measures, the modular class and a cohomology pairing for Lie algebroids*, Q. J. Math. **50 **(1999), 417–436. <span style="font-variant:small-caps;">R. L. Fernandes</span>, *Lie algebroids, holonomy and characteristic classes*, Adv. in Math. **170** (2002), 119–179. <span style="font-variant:small-caps;">J.-C. Herz</span>, Pseudo-algèbres de Lie, *C. R. Math. Acad. Sci. Paris* **263** (1953), I, 1935–1937, and II, 2289–2291. <span style="font-variant:small-caps;">Ph. J. Higgins and K. C. H. Mackenzie</span>, *Algebraic constructions in the category of Lie algebroids*, J. Algebra **129** (1990), 194–230. <span style="font-variant:small-caps;">J. Kubarski</span>, *The Chern-Weil homomorphism of regular Lie algebroids*, Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon, 1991, 1–69. <span style="font-variant:small-caps;">J. Kubarski</span>, Invariant cohomology of regular Lie algebroids, in: *Analysis and Geometry in Foliated Manifolds* (Proceedings of the VII International Colloquium on Differential Geometry, Santiago de Compostella, Spain, 26–30 July 1994), pp. 137–151, World Sci. Publ., Singapore–New Yersey–London–Hong Kong, 1995. <span style="font-variant:small-caps;">K. C. H. Mackenzie</span>, *General Theory of Lie Groupoids and Lie Algebroids*, London Math. Soc. Lecture Note Ser. 213, Cambridge Univ. Press, 2005. <span style="font-variant:small-caps;">N. Teleman</span>, *A characteristic ring of a Lie algebra extension*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl., vol. **52** (1972),[** **]{}498–506 and 708–711. <span style="font-variant:small-caps;">W. Zhang</span>, *Lectures on Chern-Weil Theory and Witten Deformations*, Nankai Tracts Math., vol 4, World Sci. Publ., New Yersey–London–Singapore–Hong Kong, 2001.   [Bogdan Balcerzak]{} [Institute of Mathematics]{} [Technical University of Łódź]{} [Wólczańska 215]{} [90-924 Łódź, Poland]{} [*E-mail address*: [email protected]]{} [^1]: 2010 *Mathematics Subject Classification*: Primary 53C05; Secondary 58H05, 17B56. [^2]: *Key words and phrases*: Lie algebroid, connections, Lie algebroid cohomology, Chern-Simons forms.

--- abstract: 'Let $G(q)$ be a Chevalley group over a finite field ${{\mathbb{F}}}_q$. By Lusztig’s and Shoji’s work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach.' address: | IAZ - Lehrstuhl für Algebra\ Universität Stuttgart\ Pfaffenwaldring 57\ D–70569 Stuttgart\ Germany author: - Meinolf Geck date: 'November 11, 2017' title: On the values of unipotent characters in bad characteristic --- Introduction {#sec0} ============ Let $G(q)$ be a group of Lie type over a finite field with $q$ elements. This paper is concerned with the problem of computing the values of the irreducible characters of $G(q)$. The work of Lusztig [@L1], [@Lintr] has led to a general program for solving this problem. In this framework, one has to establish certain identities of class functions on $G(q)$ of the form $R_x=\xi \chi_A$, where $R_x$ denotes an “almost character” (that is, an explicitly known linear combination of irreducible characters) and $\chi_A$ denotes the characteristic function of a suitable “character sheaf” on the underlying algebraic group $G$; furthermore, $\xi$ is a scalar of absolute value $1$. This program has been successfully carried out in many cases, see, e.g., Bonnafé [@Bo3], Lusztig [@L3], [@L7], Shoji [@Sclass], [@S7] and Waldspurger [@wald], but not in complete generality. In this paper, we will assume that $G(q)$ is of split type and only consider the above problem as far as unipotent characters of $G(q)$ are concerned, as defined by Deligne and Lusztig [@DeLu]. By Shoji’s work [@S2], [@S3], we know that the desired identities $R_x=\xi \chi_A$ as above hold, but the scalars $\xi$ are not yet determined in all cases. And even in those cases where they are known, this often required elaborate computations. In such cases, the scalars then turn out to behave rather uniformly as $q$ varies (see, e.g., Shoji [@Sclass], [@S7]). The main theoretical result of this paper, Proposition \[p1\], provides a partial, [*a priori*]{} explanation for this phenomenon; the proof is merely an elaboration of ideas which are already contained in Lusztig’s and Shoji’s papers. The fact that we can formulate this result without any assumptions on $q$ essentially relies on Lusztig [@L10], where the “cleanness” of cuspidal character sheaves is established in complete generality and, consequently, the principal results of [@L2a]–[@L2e] (e.g. [@L2e Theorems 23.1 and 25.2]) hold unconditionally. The main observation of this paper is that the statement of Proposition \[p1\] can also be exploited in a different way, as follows. For a given type of group, we consider the base case where $q=p$ is a prime. For a specific value of $p$, we can use ad hoc methods and/or computer algebra systems like [GAP]{} [@gap] to perform all kinds of computations within the fixed finite group $G(p)$. If we succeed in this way to determine the scalars $\xi$ for $G(p)$, then Proposition \[p1\] tells us that the analogous result will hold for any power of $p$. This is particularly relevant for “bad” primes $p=2,3,5$ which, typically, are known to cause additional complications and require separate arguments. We illustrate this procedure with a number of examples. In particular, we determine the scalars $\zeta$ in two cases, where the character table of $G({{\mathbb{F}}}_p)$ is explicitly known; namely, $F_4$, $E_6$ and $p=2$. For type $F_4$, our results complete earlier results of Marcelo–Shinoda [@MaSh]. See also [@GeHe] for the discussion of further examples, where the complete character table of $G({{\mathbb{F}}}_p)$ is not known. We assume some familiarity with the character theory of finite groups of Lie type; see, e.g., [@C2], [@first]. The basic reference for the theory of character sheaves are Lusztig’s papers [@L2a]–[@L2e]. In Section \[sec1\], we review the classification of unipotent characters of $G(q)$ and the analogous classification of the unipotent character sheaves on $G$. These two classifications are known to be the same for $G(q)$ of split type (a fact which has only recently found a conceptual explanation; see Lusztig [@L11]). In Section \[sec2\], we can then formulate in precise terms the problem of equating class functions $R_x=\xi\chi_A$ as above, and establish Proposition \[p1\]. Finally, Sections \[sec3\], \[secF4\], \[secE6\] contain a number of examples where we show how the scalars $\xi$ can be determined using standard functions and algorithms in [GAP]{}. [**Notation.**]{} \[abs11\] Let $\ell$ be a prime such that $\ell\nmid q$. If $\Gamma$ is a finite group, we denote by $\mbox{CF} (\Gamma)$ the vector space of $\overline{{{\mathbb{Q}}}}_\ell$-valued functions on $G^F$ which are constant on the conjugacy classes of $\Gamma$. (We take $\overline{{{\mathbb{Q}}}}_\ell$ instead of ${{\mathbb{C}}}$ since, in the framework of [@DeLu], [@Lintr], class functions on $\Gamma=G(q)$ are constructed whose values are cyclotomic numbers in $\overline{{{\mathbb{Q}}}}_\ell$.) Given $f,f'\in \mbox{CF} (\Gamma)$, we denote by $\langle f,f' \rangle=|\Gamma|^{-1} \sum_{g \in \Gamma} f(g)\overline{f'(g)}$ the standard scalar product of $f,f'$ where the bar denotes a field automorphism which maps roots of unity to their inverses. Let ${{\operatorname{Irr}}}(\Gamma)$ be the set of irreducible characters of $\Gamma$ over $\overline{{{\mathbb{Q}}}}_\ell$; these form an orthonormal basis of $\mbox{CF}(\Gamma)$ with respect to the above scalar product. Unipotent character sheaves and almost characters {#sec1} ================================================= Let $p$ be a prime and $k=\overline{{{\mathbb{F}}}}_p$ be an algebraic closure of the field with $p$ elements. Let $G$ be a connected reductive algebraic group over $k$ and assume that $G$ is defined over the finite subfield ${{\mathbb{F}}}_q\subseteq k$, where $q=p^f$ for some $f{\geqslant}1$. Let $F\colon G\rightarrow G$ be the corresponding Frobenius map. Let $B\subseteq G$ be an $F$-stable Borel subgroup and $T\subseteq B$ be an $F$-stable maximal torus. Let $W=N_G(T)/T$ be the corresponding Weyl group. We assume that $F$ acts trivially on $W$ and that $F(t)=t^q$ for all $t\in T$. Then the group of rational points $G^F=G({{\mathbb{F}}}_q)$ is a finite group of Lie type of “split type”. \[r11\] For each $w\in W$, let $R_w$ be the virtual character of $G^F$ defined by Deligne–Lusztig [@DeLu §1]. Let ${{\mathfrak{U}}}(G^F)$ be the set of unipotent characters of $G^F$, that is, $${{\mathfrak{U}}}(G^F)=\{\rho\in{{\operatorname{Irr}}}(G^F)\mid \langle \rho,R_w\rangle\neq 0\mbox{ for some $w\in W$}\}.$$ Now [@L1 Main Theorem 4.23] provides a classification of ${{\mathfrak{U}}}(G^F)$ in terms of - a parameter set $X(W)$ and a pairing $\{\;,\;\} \colon X(W)\times X(W)\rightarrow \overline{{{\mathbb{Q}}}}_\ell$ (which only depend on $W$), - an embedding ${{\operatorname{Irr}}}(W) \hookrightarrow X(W)$, $\epsilon\mapsto x_\epsilon$. (See [@L1 4.21] for precise definitions; recall that we assume that $F$ acts trivially on $W$). Indeed, there is a bijection $${{\mathfrak{U}}}(G^F)\leftrightarrow X(W),\qquad \rho\leftrightarrow x_\rho,$$ such that for any $\rho\in{{\mathfrak{U}}}(G^F)$ and any $\epsilon\in{{\operatorname{Irr}}}(W)$, we have $$\langle \rho,R_\epsilon\rangle=\delta_\rho\{x_\rho,x_\epsilon\}\qquad \mbox{where} \qquad R_\epsilon:=\frac{1}{|W|} \sum_{w\in W} \epsilon(w)R_w.$$ Here, for $\rho\in{{\mathfrak{U}}}(G^F)$, we define a sign $\delta_\rho=\pm 1$ by the condition that $\delta_\rho D_G(\rho)\in{{\mathfrak{U}}}(G^F)$, where $D_G$ denotes the duality operator on the character ring of $G$; see [@L1 6.8]. Note that [@L1 6.20] identifies $\delta_\rho$ with the sign $\Delta(x_\rho)$ appearing in the formulation of [@L1 4.23]. \[r11a\] Assume that $G$ is simple modulo its centre. The bijection ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ in \[r11\] is not uniquely determined by the properties stated above. We shall make a definite choice according to [@L1 12.6] and the tables in the appendix of [@L1]. In particular, this means the following. Let us fix a square root of $q$ in $\overline{{{\mathbb{Q}}}}_\ell$. Then it is well-known that the irreducible characters of $G^F$ which occur in the character of the permutation representation of $G^F$ on the cosets of $B^F$ are naturally parametrised by the irreducible characters of $W$; see, e.g., [@L1 8.7]. If $\epsilon\in{{\operatorname{Irr}}}(W)$, we denote the corresponding irreducible character of $G^F$ by $\rho_\epsilon$; clearly, $\rho_\epsilon \in {{\mathfrak{U}}}(G^F)$. Hence, under a bijection ${{\mathfrak{U}}}(G^F) \leftrightarrow X(W)$ as in \[r11\], the character $\rho_\epsilon$ will correspond to an element $x_{\rho_\epsilon}$. By [@L1 Prop. 12.6], we automatically have $x_\epsilon=x_{\rho_\epsilon}$ except when $\epsilon(1)= 512$ and $G$ is of type $E_7$, or when $\epsilon(1)=4096$ and $G$ is of type $E_8$. In these exceptional cases, ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ can still be chosen such that $x_\epsilon=x_{\rho_\epsilon}$; see the tables for $E_7$, $E_8$ in the appendix of [@L1]. In order to obtain a full uniqueness statement, one has to take into account Harish-Chandra series and further invariants of the characters in ${{\mathfrak{U}}}(G^F)$, namely, the “eigenvalues of Frobenius” as determined in [@L1 11.2]; see [@DiMi1 Prop. 6.4], [@L10a §3], [@first §4]. \[r12\] The “Fourier matrix” $\Upsilon:= \bigl(\{x,y\} \bigr)_{x,y\in X(W)}$ is hermitian, and $\Upsilon^2$ is the identity matrix (see [@L1 4.14]). For each $x\in X(W)$, the corresponding unipotent “almost character” $R_x$ is defined by $$R_x:=\sum_{\rho\in{{\mathfrak{U}}}(G^F)} \delta_{\rho} \{x_\rho,x\}\rho;\qquad \mbox{see \cite[4.24.1]{L1}}.$$ Note that $R_{x_\epsilon}=R_\epsilon$ for $\epsilon\in{{\operatorname{Irr}}}(W)$. For any $x,y \in X(W)$ we have $$\langle R_x,R_y\rangle=\left\{\begin{array}{cl} 1 & \qquad \mbox{if $x=y$}, \\ 0 & \qquad \mbox{otherwise}.\end{array}\right.$$ Since $\Upsilon^2$ is the identity matrix, we obtain $$\rho=\delta_{\rho}\sum_{x\in X(W)} \{x,x_\rho\}R_x \qquad \mbox{for any $\rho\in{{\mathfrak{U}}}(G^F)$}.$$ Thus, the problem of computing the values of $\rho\in{{\mathfrak{U}}}(G^F)$ is equivalent to the analogous problem for the unipotent almost characters $R_x$, $x\in X(W)$. \[r13\] Let $\hat{G}$ be the set of character sheaves on $G$ (up to isomorphism). These are certain simple perverse sheaves in the bounded derived category $\mathcal{D}G$ of constructible $\overline{{{\mathbb{Q}}}}_\ell$-sheaves on $G$ (in the sense of Beilinson, Bernstein, Deligne [@bbd]), which are equivariant for the action of $G$ on itself by conjugation. For $w\in W$ let $K_w^{\mathcal{L}_0}\in\mathcal{D}G$ be defined as in [@L2a 2.4], where $\mathcal{L}_0=\overline{{{\mathbb{Q}}}}_\ell$ is the constant local system on the maximal torus $T$. Let $\hat{G}^{\text{un}}$ be the set of unipotent character sheaves, that is, those $A\in \hat{G}$ which are constituents of a perverse cohomology sheaf ${^p\!H}^i (K_w^{\mathcal{L}_0})$ for some $w\in W$ and some $i\in{{\mathbb{Z}}}$ (see [@L2a Def. 2.10]). Let $\epsilon \in{{\operatorname{Irr}}}(W)$. In analogy to the above definition of $R_\epsilon$, we formally define $$K_\epsilon^{\mathcal{L}_0}:=\frac{1}{|W|} \sum_{w\in W} \epsilon(w) \sum_{i\in{{\mathbb{Z}}}} (-1)^{i+\dim G}\, {^p\!H}^i(K_w^{\mathcal{L}_0});$$ see [@L2c 14.10.3]. (We write $K_\epsilon^{\mathcal{L}_0}$ in order to avoid confusion with $R_\epsilon$ in \[r11\].) As in [@L2c 14.10.4], we also denote by $(A:K_\epsilon^{\mathcal{L}_0})$ the multiplicity of $A\in\hat{G}^{\text{un}}$ in $K_\epsilon^{\mathcal{L}_0}$ (in the appropriate Grothendieck group). \[r14\] Now [@L2e Theorem 23.1] (see also the comments in [@L10 3.10]) provides a classification of $\hat{G}^{\text{un}}$ in terms of similar ingredients as in \[r11\]. Indeed, let the parameter set $X(W)$, the pairing $\{\;,\;\} \colon X(W)\times X(W)\rightarrow \overline{{{\mathbb{Q}}}}_\ell$ and the embedding ${{\operatorname{Irr}}}(W) \hookrightarrow X(W)$ be as above. Then there is a bijection $$\hat{G}^{\text{un}}\leftrightarrow X(W),\qquad A\leftrightarrow x_A,$$ such that $(A:K_\epsilon^{\mathcal{L}_0}) =\hat{\varepsilon}_{A}\{x_A, x_\epsilon\}$ for any $A\in \hat{G}^{\text{un}}$ and $\epsilon\in{{\operatorname{Irr}}}(W)$. Here, we set $$\hat{\varepsilon}_K:=(-1)^{\dim G-\dim\text{supp}(K)}\qquad \mbox{for any $K\in\mathcal{D}G$},$$ where $\mbox{supp}(K)$ is the Zariski closure of the set $\{g\in G\mid {{\mathcal{H}}}_g^i(K)\neq \{0\} \mbox{ for some $i$}\}$. (Cf.  [@L2c 15.11].) Here, ${{\mathcal{H}}}_g^i(K)$ are the stalks at $g\in G$ of the cohomology sheaves of $K$, for any $i\in {{\mathbb{Z}}}$. Assume that $G$ is simple modulo its centre. Then, again, the bijection $\hat{G}^{\text{un}}\leftrightarrow X(W)$ is not uniquely determined by the above properties. But one obtains a full uniqueness statement by an analogous scheme as in Remark \[r11a\]; see [@L10a §3]. \[r15\] Consider any object $A\in \mathcal{D}G$ and suppose that its inverse image $F^*A$ under the Frobenius map is isomorphic to $A$ in $\mathcal{D}G$. Let $\phi \colon F^*A \stackrel{\sim}{\rightarrow} A$ be an isomorphism. Then $\phi$ induces a linear map $\phi_{i,g}\colon {{\mathcal{H}}}_g^i(A)\rightarrow {{\mathcal{H}}}_g^i(A)$ for each $i$ and $g\in G^F$. This gives rise to a class function $\chi_{A,\phi}\in \mbox{CF}(G^F)$, called “characteristic function” of $A$, defined by $$\chi_{A,\phi}(g)=\sum_i (-1)^i \mbox{Trace}(\phi_{i,g},{{\mathcal{H}}}_g^i(A))\qquad \mbox{for $g \in G^F$},$$ see [@L2b 8.4]. Note that $\phi$ is unique up to a non-zero scalar; hence, $\chi_{A,\phi}$ is unique up to a non-zero scalar. Now assume that $A\in\hat{G}$. Then one can choose an isomorphism $\phi_A \colon F^*A \stackrel{\sim}{\rightarrow} A$ such that the values of $\chi_{A,\phi_A}$ are cyclotomic integers and $\langle \chi_{A,\phi_A}, \chi_{A,\phi_A}\rangle=1$; see [@L2e 25.6, 25.7] (and also the comments in [@L10 3.10]). The precise conditions which guarantee these properties are formulated in [@L2c 13.8], [@L2e 25.1]; note that these conditions specify $\phi_A$ up to multiplication by a root of unity. In the following, we will tacitly assume that $\phi_A$ has been chosen in this way whenever $A\cong F^*A$. \[r16\] Assume that $Z(G)$ is connected and that $G/Z(G)$ is simple; also recall that $F$ is assumed to act trivially on $W$. Let $A\in\hat{G}^{\operatorname{un}}$ and $x\in X(W)$ be such that $x=x_A$. Then $F^*A\cong A$ and $R_x$ is equal to $\chi_{A,\phi_A}$, up to a non-zero scalar multiple. As already mentioned, Shoji’s results also apply to non-split groups and to non-unipotent characters. In [@S2], [@S3] it is assumed, however, that $p$ is “almost good”, that is, the following conditions hold. If $G$ is type $A_n$, $B_n$, $C_n$ or $D_n$, no condition. If $G$ is of type $E_6$, then $p\neq 2$. If $G$ is of type $G_2$, $F_4$ or $E_7$, then $p\neq 2,3$. If $G$ is of type $E_8$, then $p\neq 2,3,5$ (see [@L2e 23.0.1]). Since Lusztig [@L10] has established the “cleanness” of cuspidal character sheaves in full generality, the methods in [@S2], [@S3] which were used to prove Theorem \[r16\] for $G$ of exceptional type and almost good $p$ can now also be applied for any $p$. Consequently, as Shoji pointed out to the author, Theorem \[r16\] holds as stated above, without any condition on $p$. \[s1\] In the setting of Theorem \[r16\], let $A\in \hat{G}^{\operatorname{un}}$ and $x\in X(W)$ be such that $x=x_A$. Recall that $\phi_A\colon F^*A \stackrel{\sim}{\rightarrow} A$ is assumed to be chosen as in \[r15\]. Then we define $0\neq \zeta_A\in\overline{{{\mathbb{Q}}}}_\ell$ by the condition that $R_x=(-1)^{\dim G}\hat{\varepsilon}_A\zeta_A\chi_{A, \phi_A}$. \[r17\] In the setting of \[r11a\], let $\epsilon\in {{\operatorname{Irr}}}(W)$ and consider the corresponding character $\rho_\epsilon$. We have $\delta_{\rho_\epsilon}=1$ and $x_\epsilon=x_{\rho_\epsilon}$. So, using the formula in \[r12\], we obtain: $$\begin{aligned} \rho_\epsilon&=\sum_{x\in X(W)} \{x, x_{\rho_\epsilon}\}R_x= \sum_{x\in X(W)} \{x, x_\epsilon\}R_x\\&=(-1)^{\dim G}\sum_{A\in \hat{G}^{\text{un}}} \hat{\varepsilon}_A\zeta_A\{x_A, x_\epsilon\} \chi_{A,\phi_A}\qquad \mbox{(see Def.~\ref{s1})}\\&=(-1)^{\dim G} \sum_{A\in \hat{G}^{\text{un}}} \zeta_A(A:K_\epsilon^{\mathcal{L}_0}) \chi_{A,\phi_A} \qquad \mbox{(see \ref{r14})}.\end{aligned}$$ Such an expression for $\rho_\epsilon$ as a linear combination of characteristic functions first appeared in [@L2c 14.14]; it is actually an important ingredient in the proof of Theorem \[r16\]. On the other hand, the argument in [@L2c 14.14] relies on an alternative interpretation of the coefficients $\zeta_A$, which we will consider in more detail in the following section. Note that, for a given $A\in\hat{G}^{\text{un}}$, there always exists some $\epsilon\in{{\operatorname{Irr}}}(W)$ such that $(A:K_\epsilon^{\mathcal{L}_0})\neq 0$; see [@L2c 14.12]. Furthermore, by [@L2e 25.2] (and the comments in [@L10 3.10]), the functions $\{\chi_{A,\phi_A}\mid A\in \hat{G}^{\text{un}}\}$ are linearly independent. It follows that the coefficients $\zeta_A$ are uniquely determined by the above system of equations, where $\epsilon$ runs over ${{\operatorname{Irr}}}(W)$. The scalars $\zeta_A$ {#sec2} ===================== We keep the basic assumptions of the previous section; we also assume that $Z(G)$ is connected and $G/Z(G)$ is simple. We fix a square root of $q$ in $\overline{{{\mathbb{Q}}}}_\ell$. For any $A\in\hat{G}^{\text{un}}$, we know by Theorem \[r16\] that $F^*A\cong A$; we assume that an isomorphism $\phi_A\colon F^*A \stackrel{\sim}{\rightarrow} A$ has been chosen as in \[r15\]. Our aim is to get hold of the coefficients $\zeta_A$ in Definition \[s1\]. \[r18\] For the further discussion, it will be convenient to change the notation and label everything by elements of $X(W)$. Thus, via the bijection ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ in \[r11\] (arranged as in Remark \[r11a\]), we can write $${{\mathfrak{U}}}(G^F)=\{\rho_x \mid x \in X(W)\} \qquad \mbox{where}\qquad \rho_\epsilon=\rho_{x_\epsilon} \mbox{ for $\epsilon \in {{\operatorname{Irr}}}(W)$}.$$ For $x \in X(W)$, we write $\delta_x:=\delta_{\rho_x}$. Then $R_y=\sum_{x \in X(W)}\delta_x \{x,y\}\rho_x$ for all $y \in X(W)$. Next, via the bijection $\hat{G}^{\text{un}}\leftrightarrow X(W)$ in \[r14\], we can write $$\hat{G}^{\text{un}}=\{A_x \mid x \in X(W)\}.$$ For $x \in X(W)$, we denote an isomorphism $F^*A_x \cong A_x$ as in \[r15\] by $\phi_x$ and the corresponding characteristic function simply by $\chi_x$. Then the relation in Definition \[s1\] is rephrased as $$R_x=(-1)^{\dim G} \hat{\varepsilon}_x \zeta_x\chi_x \qquad \mbox{where} \qquad \hat{\varepsilon}_x:=\hat{\varepsilon}_{A_x} \mbox{ and } \zeta_x:= \zeta_{A_x}.$$ For $\epsilon\in{{\operatorname{Irr}}}(W)$, the identity in \[r17\] now reads: $$\rho_{\epsilon}=(-1)^{\dim G} \sum_{x\in X(W)} \zeta_x(A_x: K_\epsilon^{\mathcal{L}_0}) \chi_x.$$ \[r21\] Let us fix an integer $m{\geqslant}1$. Then $G$ is also defined over ${{\mathbb{F}}}_{q^m}$ and $F^m\colon G\rightarrow G$ is the corresponding Frobenius map. Clearly, $F^m$ acts trivially on $W$ and we have $F^m(t)= t^{q^m}$ for all $t\in T$. So the whole discussion in Section \[sec1\] can be applied to $F^m$ instead of $F$. As in \[r18\], we write $${{\mathfrak{U}}}(G^{F^m})=\{\rho_x^{(m)} \mid x \in X(W)\}.$$ Again, the unipotent characters of $G^{F^m}$ which occur in the character of the permutation representation of $G^{F^m}$ on the cosets of $B^{F^m}$ are naturally parametrised by ${{\operatorname{Irr}}}(W)$. If $\epsilon\in{{\operatorname{Irr}}}(W)$, we denote the corresponding character of $G^{F^m}$ by $\rho_\epsilon^{(m)}$. As in Remark \[r11a\], the labelling of ${{\mathfrak{U}}}(G^{F^m})$ is arranged such that $$\rho_\epsilon^{(m)}=\rho_{x_\epsilon}^{(m)} \qquad \mbox{for $\epsilon \in {{\operatorname{Irr}}}(W)$}.$$ For $y \in X(W)$, the corresponding unipotent almost character of $G^{F^m}$ is given by $$R_y^{(m)}=\sum_{x \in X(W)} \delta_x\{x,y\}\rho_x^{(m)}.$$ Note that $\delta_{\rho_x^{(m)}}=\delta_{\rho_x}=\delta_x$ by [@L1 4.23, 6.20]. \[r22\] Let $x\in X(W)$. Then $\phi_x\colon F^*A_x \stackrel{\sim}{\rightarrow} A_x$ naturally induces isomorphisms $$F^*(\phi_x)\colon (F^*)^2A_x\stackrel{\sim}{\rightarrow} F^*A_x,\quad (F^*)^2(\phi_x)\colon (F^*)^3A_x\stackrel{\sim}{\rightarrow} (F^*)^2A_x, \quad \ldots,$$ which give rise to an isomorphism $$\tilde{\phi}_x^{(m)}:=\phi_x\circ F^*(\phi_x)\circ \ldots \circ (F^*)^{m-1}(\phi_x) \colon (F^*)^mA_x\stackrel{\sim}{\rightarrow} A_x.$$ We also have a canonical isomorphism $(F^*)^mA_x\cong (F^m)^* A_x$ which, finally, induces an isomorphism $$\phi_x^{(m)} \colon (F^m)^*A_x\stackrel{\sim}{\rightarrow} A_x\qquad \mbox{(see \cite[1.1]{S2})}.$$ The latter isomorphism again satisfies the conditions in \[r15\]; we denote the corresponding characteristic function by $\chi_x^{(m)}\colon G^{F^m}\rightarrow \overline{{{\mathbb{Q}}}}_\ell$. Note that, if $g$ is an element in $G^F$ (and not just in $G^{F^m}$), then $$\chi_x^{(m)}(g)=\sum_i (-1)^i\mbox{Trace}\bigl((\phi_x)_{i,g}^m, {{\mathcal{H}}}_g^i(A_x)\bigr).$$ (See again [@S2 1.1].) As in Definition \[s1\], we define $0\neq \zeta_x^{(m)}\in \overline{{{\mathbb{Q}}}}_\ell$ by the condition that $$R_x^{(m)}=(-1)^{\dim G}\hat{\varepsilon}_x\zeta_x^{(m)}\chi_x^{(m)}.$$ (Thus, if $m=1$, then $\zeta_x^{(1)}=\zeta_x$.) We can now state the main result of this section. \[p1\] In the setting of \[r21\], \[r22\], we have $\zeta_x^{(m)}=\zeta_x^m$ for all $x\in X(W)$. A result of this kind is implicitly contained in Lusztig [@L2c §14] and Shoji [@S2 §2 and 5.19]; the proof will be given in \[r25\]. First, we need some preparations. \[r23\] We recall some constructions from [@L2c §12, §13]. For any $w\in W$, we assume chosen once and for all a representative $\dot{w}\in N_G(T)$. There is a corresponding complex $\bar{K}_{\dot{w}}^{\mathcal{L}_0}\in \mathcal{D}G$ as defined in [@L2c 12.1]. Then $\hat{G}^{\text{un}}$ can also be characterized as the set of isomorphism classes of simple perverse sheaves on $G$ which occur as constituents of a perverse cohomology sheaf ${^p\!H}^i( \bar{K}_{\dot{w}}^{\mathcal{L_0}})$ for some $w\in W$ and some $i\in{{\mathbb{Z}}}$. Furthermore, for each $i$, there is a natural isomorphism $$\varphi_{i,\dot{w}}\colon F^*({^p\!H}^i(\bar{K}_{\dot{w}}^{\mathcal{L}_0})) \stackrel{\sim}{\rightarrow} {^p\!H}^i(\bar{K}_{\dot{w}}^{\mathcal{L}_0}); \qquad \mbox{see \cite[12.2, 13.8]{L2c}}.$$ The advantage of using $\bar{K}_{\dot{w}}^{\mathcal{L}_0}$ instead of $K_w^{\mathcal{L}_0}$ (see \[r13\]) is that $\bar{K}_{\dot{w}}^{\mathcal{L}_0}$ is semisimple (see [@L2c 12.8]). Let us now fix $i$, $w$ and denote $K:={^p\!H}^i (\bar{K}_{\dot{w}}^{\mathcal{L}_0})$. We also set $\varphi:=\varphi_{i, \dot{w}} \colon F^*K\stackrel{\sim}{\rightarrow}K$ and recall that, for each $x\in X(W)$, we are given $\phi_x\colon F^*A_x \stackrel{\sim}{\rightarrow} A_x$. Now, following Lusztig [@L2c 13.8.2], there is a canonical isomorphism $$K\cong \bigoplus_{x\in X(W)} (A_x\otimes V_x)$$ where $V_x$ are finite-dimensional vector spaces over $\overline{{{\mathbb{Q}}}}_\ell$ endowed with linear maps $\psi_x\colon V_x\rightarrow V_x$ such that, under the above direct sum decomposition, the map $\phi_x\otimes \psi_x$ corresponds to the given $\varphi\colon F^*K\stackrel{\sim}{\rightarrow} K$. More precisely, $V_x$ and $\psi_x$ are as follows (cf. [@L2b 10.4] and [@L3 3.5]). We have $V_x=\mbox{Hom}(A_x,K)$ and $$\psi_x(v)=\varphi\circ F^*(v)\circ \phi_x^{-1} \qquad\mbox{for $v\in V_x$},$$ where $F^*(v)\in \mbox{Hom}(F^*A_x,F^*K)$ is the map induced by $v\colon A_x\rightarrow K$. \[r24\] In the setting of \[r23\], all eigenvalues of $\psi_x\colon V_x\rightarrow V_x$ are equal to $\zeta_x q^{(i-\dim G)/2}$. More precisely, Lusztig first shows in [@L2c 13.10] that there is a constant $0\neq \xi_x\in\overline{{{\mathbb{Q}}}}_\ell$ (which only depends on $\phi_x$, the choice of a square root of $q$ and the choice of the representatives $\dot{w}$) such that, for any $i$ and $w$, all eigenvalues of $\psi_x\colon V_x\rightarrow V_x$ are equal to $\xi_x q^{(i-\dim G)/2}$. (The “cleanness” assumption in [@L2c 13.10] holds in general by [@L10].) It is then shown in [@L2c 14.14] that $$\rho_\epsilon=(-1)^{\dim G} \sum_{x\in X(W)} \xi_x(A_x: K_\epsilon^{\mathcal{L}_0}) \chi_x \qquad \mbox{for all $\epsilon \in{{\operatorname{Irr}}}(W)$}.$$ (There are also coefficients $\nu(A_x)$ in the formula in [@L2c 14.14] but these are all equal to $1$ in our situation.) Finally, a comparison with the formula in \[r18\] implies that $\xi_x=\zeta_x$ for all $x\in X(W)$. \[r25\] [*Proof of Proposition \[p1\].*]{} Let $x\in X(W)$. We also fix $i,w$ and place ourselves in the setting of \[r23\]. The whole discussion there can be repeated with $F$ replaced by $F^m$. As in \[r22\], we have isomorphisms $$\tilde{\phi}_x^{(m)}\colon (F^*)^mA_x\stackrel{\sim}{\rightarrow} A_x \qquad \mbox{and}\qquad {\phi}_x^{(m)}\colon (F^m)^*A_x \stackrel{\sim}{\rightarrow} A_x.$$ Analogously, $\varphi\colon F^*K\stackrel{\sim}{\rightarrow} K$ induces isomorphisms $$\tilde{\varphi}^{(m)}\colon (F^*)^mK\stackrel{\sim}{\rightarrow} K\qquad \mbox{and}\qquad \varphi^{(m)}\colon (F^m)^*K\stackrel{\sim}{\rightarrow} K.$$ Let us consider again the canonical isomorphism $$K\cong \bigoplus_{x\in X(W)} (A_x\otimes V_x)\qquad\mbox{where} \qquad V_x=\mbox{Hom}(A_x,K).$$ Here, as before, each $V_x$ is endowed with a linear map $\psi_x^{(m)}\colon V_x \rightarrow V_x$ such that $$\psi_x^{(m)}(v)=\varphi^{(m)} \circ (F^m)^*(v)\circ (\phi_x^{(m)})^{-1} \qquad \mbox{for all $v\in V_x$}.$$ By Theorem \[r24\], the scalar $\zeta_x^{(m)}$ is determined by the eigenvalues of $\psi_x^{(m)}$. Now, a simple induction on $m$ shows that $$\psi_x^m(v)=\tilde{\varphi}^{(m)} \circ (F^*)^m(v)\circ (\tilde{\phi}_x^{(m)})^{-1} \qquad \mbox{for all $v\in V_x$}.$$ Next, we use again that we have canonical isomorphisms $(F^*)^mA_x\cong (F^m)^*A_x$ and $(F^*)^mK\cong (F^m)^*K$. Under these isomorphisms, the above map $(F^*)^m(v)$ corresponds to the map $(F^m)^*(v)\colon (F^m)^*A_x \rightarrow (F^m)^*K$. Hence, we also have $$\psi_x^m(v)=\varphi^{(m)} \circ (F^m)^*(v)\circ (\phi_x^{(m)})^{-1} \qquad \mbox{for all $v\in V_x$}.$$ Thus, we have $\psi_x^{(m)}=\psi_x^m$ for all $x\in X(W)$ and so the eigenvalues of $\psi_x^{(m)}$ are obtained by raising the eigenvalues of $\psi_x$ to the $m$-th power. It remains to use Theorem \[r24\]. This completes the proof of Proposition \[p1\]. \[exp1\] We can, and will assume that $G$ is defined and split over the prime field ${{\mathbb{F}}}_p$ of $k$. Let $F_0\colon G\rightarrow G$ be the corresponding Frobenius map. If $q=p^f$ where $f{\geqslant}1$, then $F= F_0^f$. Hence, Proposition \[p1\] means that it will be sufficient to determine the scalars $\zeta_x$ $(x\in X(W)$) for the group $G^{F_0}= G({{\mathbb{F}}}_p)$. For specific values of $p$ (e.g., bad primes $p=2,3,5$), we may then use ad hoc information which is available, for example, in the Cambridge [ATLAS]{} [@atl], or via computer algebra methods (using [GAP]{} [@gap], [CHEVIE]{} [@jmich]). This is the basis for the discussion of the examples below. Cuspidal character sheaves and small rank examples {#sec3} ================================================== We keep the notation of the previous section; in particular, we label all objects by the parameter set $X(W)$ as in \[r18\]. By [@L3 3.5], the computation of the scalar $\zeta_x$ can be reduced to the case where $A_x$ is a cuspidal character sheaf (in the sense of [@L2a Def. 3.10]). So let us look in more detail at this case. \[exp2\] Assume that $Z(G)=\{1\}$. Let $x\in X(W)$ be such that $A_x\in\hat{G}^{\text{un}}$ is cuspidal. Then there exists an $F$-stable conjugacy class $C$ of $G$ and an irreducible, $G$-equivariant $\overline{{{\mathbb{Q}}}}_\ell$-local system ${{\mathcal{E}}}$ on $C$ such that $F^*{{\mathcal{E}}}\cong {{\mathcal{E}}}$ and $A=\mbox{IC}(\overline{C},{{\mathcal{E}}})[\dim C]$; see [@L2a 3.12]. In particular, $\mbox{supp}(A_x)=\overline{C}$ and so $\hat{\varepsilon}_{A_x}= (-1)^{\dim G-\dim C}$. Let us fix $g_1\in C^F$ and set $A(g_1):=C_G(g_1)/ C_G^\circ(g_1)$. Then $F$ induces an automorphism $\gamma\colon A(g_1) \rightarrow A(g_1)$. We further assume that: - the local system ${{\mathcal{E}}}$ is one-dimensional and, hence, corresponds to a $\gamma$-invariant linear character $\lambda\colon A(g_1) \rightarrow \overline{{{\mathbb{Q}}}}_\ell^\times$ (via [@Ldisc4 19.7]). (This assumption will be satisfied in all examples that we consider.) We form the semidirect product $\tilde{A}(g_1)=A(g_1)\rtimes \langle \gamma\rangle$ such that, inside $\tilde{A}(g_1)$, we have the identity $\gamma(a)=\gamma a\gamma^{-1}$ for all $a\in A(g_1)$. By ($*$), we can canonically extend $\lambda$ to a linear character $$\tilde{\lambda}\colon \tilde{A}(g_1) \rightarrow \overline{{{\mathbb{Q}}}}_\ell^\times,\qquad a\gamma\mapsto \lambda(a).$$ For each $a\in A(g_1)$ we have a corresponding element $g_a\in C^F$, well-defined up to conjugation within $G^F$. (We have $g_a=hg_1h^{-1}$ where $h\in G$ is such that $h^{-1}F(h)\in C_G(g_1)$ has image $a\in A(g_1)$.) We define a class function $\chi_{g_1,\lambda} \colon G^F \rightarrow \overline{{{\mathbb{Q}}}}_\ell$ by $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl} q^{(\dim G-\dim C)/2} \lambda(a) & \quad\mbox{if $g=g_a$ for some $a\in A(g_1)$},\\ 0 & \quad \mbox{if $g\not\in C^F$}.\end{array}\right.$$ Now, we can choose an isomorphism $F^*{{\mathcal{E}}}\stackrel{\sim}{\rightarrow}{{\mathcal{E}}}$ such that the induced map on the stalk ${{\mathcal{E}}}_{g_1}$ is scalar multiplication by $q^{(\dim G-\dim C)/2}$. Then this isomorphism canonically induces an isomorphism $\phi_x\colon F^*A_x \stackrel{\sim}{\rightarrow} A_x$ which satisfies the requirements in \[r15\], and we have $\chi_x=\chi_{A_x, \phi_x}=\chi_{g_1,\lambda}$. (This follows from the fact that $A_x$ is “clean” [@L10], using the construction in [@Ldisc4 19.7].) With this choice of $\phi_x$, we also have for all $m{\geqslant}1$: $$\chi_x^{(m)}(g_a^{(m)})=\left\{\begin{array}{cl} q^{m(\dim G-\dim C)/2} \lambda(a) & \quad\mbox{if $g=g_a^{(m)}$ for some $a\in A(g_1)$},\\ 0 & \quad \mbox{if $g\not\in C^{F^m}$}.\end{array}\right.$$ (Here, $g_a^{(m)}=hg_1h^{-1}$ where now $h\in G$ is such that $h^{-1}F^m(h) \in C_G(g_1)$ has image $a\in A(g_1)$; see again [@Ldisc4 19.7].) The identity in Definition \[s1\] now reads: $$R_x=\sum_{y\in X(W)} \delta_y\{y,x\}\rho_y=(-1)^{\dim C} \zeta_x\chi_{g_1,\lambda}.$$ \[rem1\] Let $G$ be of (split) classical type. Then Shoji has shown that we always have $\zeta_x=1$ for cuspidal $A_x\in \hat{G}^{\text{un}}$; see [@Sclass Prop. 6.7] for $p\neq 2$, and [@S7 Theorem 6.2] for $p=2$. Note that this involves, in each case, the choice of a particular representative in the conjugacy class supporting $A_x$. Since classical groups of low rank appear as Levi subgroups in groups of exceptional type, it will be useful to work out explicitly the relevant identities $R_x=(-1)^{\dim C} \zeta_x\chi_{(g_1,\lambda)}$ for $G$ of type $C_2$, $D_4$ and $p=2$. This also provides a good illustration for: (a) the role of the choice of a class representative as above and (b) the strategy that we will employ when dealing with groups of exceptional type. \[sp4\] Let $G=\mbox{Sp}_4(k)$ be the $4$-dimensional symplectic group. Then $G=\langle x_\alpha(t)\mid \alpha\in \Phi, t\in k \rangle$ with root system $\Phi=\{\pm a,\pm b,\pm (a+b),\pm (2a+b)\}$. The Weyl group $W=\langle s_a,s_b\rangle$ is dihedral of order $8$ and we have $${{\operatorname{Irr}}}(W)=\{1_W,\text{sgn},\text{sgn}_a,\text{sgn}_b,r\}$$ where $1_W$ is the trivial character, $\text{sgn}$ is the sign character, $r$ has degree $2$, and $\text{sgn}_a,\text{sgn}_b$ are linear characters such that $\text{sgn}_a(s_a)=1$, $\text{sgn}_a(s_b)=-1$, $\text{sgn}_b (s_a)=-1$, $\text{sgn}_b(s_b)=1$. By [@C2 p. 468], we have $$X(W)=\{x_1, x_{\text{sgn}}, x_{\text{sgn}_a}, x_{\text{sgn}_b}, x_{r}, x_0\}$$ where $\rho_{1_W}(1)=1$, $\rho_{\text{sgn}}(1)=q^4$, $\rho_{\text{sgn}_a} (1)=\rho_{\text{sgn}_b}(1)=\frac{1}{2}q(q^2+1)$, $\rho_{r}(1)=\frac{1}{2} q(q+1)^2$ and $\rho_{x_0}(1)=\frac{1}{2}q(q-1)^2$. By the explicit description of the Fourier matrices in [@C2 p. 471], we find that $R_{x_0}=\frac{1}{2}(\rho_{r}-\rho_{\text{sgn}_a}-\rho_{\text{sgn}_b}+ \rho_{x_0})$. If $q$ is odd, then the identification of $R_{x_0}$ with a characteristic function of a cuspidal character sheaf is explained in the appendix of Srinivasan [@Sr94]. Now assume that $q=2^f$ where $f{\geqslant}1$. Then, by [@L2d 22.2], there is a unique cuspidal character sheaf $A_0$ on $G$, and it is contained in $\hat{G}^{\text{un}}$. By the explicit description in [@LuSp1 2.7], we have $A_0=A_{x_0}$ and $A_0=\mbox{IC}(\overline{C}, {{\mathcal{E}}})[\dim C]$ where $C$ is the class of regular unipotent elements and ${{\mathcal{E}}}\not\cong \overline{{{\mathbb{Q}}}}_\ell$; we have $\dim G=10$ and $\dim C=8$. Let us fix $$g_1=x_a(1)x_b(1) \in C^F.$$ One checks that $g_1$ has order $4$ and that $g_1$ is conjugate in $G^F$ to $g_1^{-1}=x_b(1)x_a(1)$. Furthermore, $A(g_1)\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}}$ is abelian and $F$ acts trivially on $A(g_1)$. Let $\lambda$ be the non-trivial character of $A(g_1)$. Then, as in Example \[exp2\], we obtain: $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl} q &\quad\mbox{if $g=g_1$},\\ -q & \quad \mbox{if $g=g_1'$},\\ 0 & \quad \mbox{if $g\not\in C^F$}, \end{array}\right.$$ where $g_1'\in C^F$ corresponds to the non-trivial element of $A(g_1)$. We now have $R_{x_0}=\zeta_{x_0}\chi_{g_1,\lambda}$. In order to determine $\zeta_{x_0}$ it is sufficient, by Remark \[exp1\], to consider the case where $q=2$. But $\mbox{Sp}_4({{\mathbb{F}}}_2)$ is isomorphic to the symmetric group $\mathfrak{S}_6$; an explicit isomorphism is described in [@HupI 9.21]. One checks that, under this isomorphism, $g_1=x_a(1) x_b(1)$ corresponds to an element of cycle type $(4,2)$ in $\mathfrak{S}_6$. We also need to identify $\rho_{r}$, $\rho_{\text{sgn}_a}$, $\rho_{\text{sgn}_b}$, $\rho_{x_0}$ in the character table of $\mathfrak{S}_6$. Now, $B({{\mathbb{F}}}_2)$ is a Sylow $2$-subgroup of $\mbox{Sp}_4 ({{\mathbb{F}}}_2)$. Working out the character of the permutation representation on the cosets of this subgroup, one can identify the $5$ characters which are of the form $\rho_\epsilon$ for some $\epsilon\in {{\operatorname{Irr}}}(W)$. Looking also at character degrees, we can then identify $\rho_{\text{sgn}}$ and the sum $\rho_{\text{sgn}_a}+\rho_{\text{sgn}_b}$; finally, $\rho_{x_0}$ corresponds to the sign character of $\mathfrak{S}_6$. By inspection of the table of $\mathfrak{S}_6$, we find that $R_{x_0}(g_1)=2$ (for $q=2$) and, hence, $\zeta_{x_0}=1$ for any $q=2^f$ (using Proposition  \[p1\]). — Of course, this could also be deduced from the explicit knowledge of the “generic” character table of $G^F=\mbox{Sp}_4({{\mathbb{F}}}_q)$ for any $q=2^f$ (see Enomoto [@En]). But the point is that, once $\zeta_{x_0}$ is known in advance, the task of computing such a generic character table is considerably simplified! \[so8\] Let $G=\mbox{SO}_8(k)$ be the $8$-dimensional special orthogonal group. The Weyl group $W$ has $13$ irreducible characters, which are labelled by certain pairs of partitions. By [@C2 p. 471], we have $$X(W)=\{x_\epsilon\mid \epsilon\in{{\operatorname{Irr}}}(W)\} \cup\{x_0\}.$$ By the explicit description of the Fourier matrices in [@C2 p. 472], we find that $R_{x_0}=\frac{1}{2}(\rho_{(21,1)}-\rho_{(22,\varnothing)}- \rho_{(2,11)}+ \rho_{x_0})$; here, $(1,21)$, $(\varnothing,22)$, $(11,2)$ indicate irreducible characters $\epsilon\in{{\operatorname{Irr}}}(W)$ (as in [@C2 p. 449]). If $q$ is odd, then the values of $R_{x_0}$ are explicitly computed in [@GP1 Prop. 4.5] (and this provides an identification of $R_{x_0}$ with a characteristic function of a cuspidal character sheaf). Now assume that $q=2^f$ where $f{\geqslant}1$. Then, by [@L2d 22.3], there is a unique cuspidal character sheaf $A_0$ on $G$, and it is contained in $\hat{G}^{\text{un}}$. Again, by the explicit description in [@LuSp1 3.3], we have $A_0=A_{x_0}=\mbox{IC}(\overline{C},{{\mathcal{E}}}) [\dim C]$ where $C$ is the class of regular unipotent elements and ${{\mathcal{E}}}\not\cong \overline{{{\mathbb{Q}}}}_\ell$; we have $\dim G=28$ and $\dim C=24$. Let us fix $$g_1=x_a(1)x_b(1)x_c(1)x_d(1) \in C^F$$ where $\{a,b,c,d\}$ is a set of simple roots in the root system of type $D_4$. One checks that $g_1$ has order $8$ and that, if $a',b',c',d'$ is any permutation of $a,b,c,d$, then $g_1$ is conjugate in $G^F$ to $x_{a'}(1) x_{b'}(1)x_{c'}(1)x_{d'}(1)$. Furthermore, $A(g_1)\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}}$ is abelian and $F$ acts trivially on $A(g_1)$. Let $\lambda$ be the non-trivial character of $A(g_1)$. As above, we obtain: $$\chi_{g_1,\lambda}(g)=\left\{\begin{array}{cl} q^2 &\quad\mbox{if $g=g_1$},\\ -q^2 & \quad \mbox{if $g=g_1'$},\\ 0 & \quad \mbox{if $g\not\in C^F$}, \end{array}\right.$$ where $g_1'\in C^F$ corresponds to the non-trivial element of $A(g_1)$. Now we have $R_{x_0}=\zeta_{x_0}\chi_{g_1,\lambda}$. In order to show that $\zeta_{x_0}=1$, we can use the known character table of $\mbox{SO}_8^+({{\mathbb{F}}}_2)$; see the [ATLAS]{} [@atl p. 85]. In fact, using an explicit realization in terms of orthogonal $8\times 8$-matrices, one can create $\mbox{SO}_8^+({{\mathbb{F}}}_2)$ as a matrix group in [GAP]{} and simply re-calculate that table using the [CharacterTable]{} function. The advantage of this re-calculation is that [GAP]{} also computes a list of representatives of the conjugacy classes of $\mbox{SO}_8^+({{\mathbb{F}}}_2)$. So one can identify the class to which $g_1$ belongs. Arguing as in the previous example, one can identify the characters $\rho_{x_0}$, $\rho_{(21,1)}$ and the sum $\rho_{(22,\varnothing)}+\rho_{(2,11)}$ in the table of $\mbox{SO}_8^+({{\mathbb{F}}}_2)$. (We omit the details.) In this way, one finds that $R_{x_0} (g_1)=4$ (for $q=2$), as required. \[rsplit\] Assume that $Z(G)=\{1\}$, as above. Let $x\in X(W)$ be such that $A_x$ is cuspidal and let $C$ be the $F$-stable conjugacy class of $G$ such that $\mbox{supp}(A_x)=\overline{C}$. The above examples highlight the importance of singling out a specific representative $g_1\in C^F$ in order to determine a characteristic function $\chi_x$ of $A_x$ and the scalar $\zeta_x$. This problem is, of course, not a new one. If $C$ is a unipotent class and $p$ is a good prime for $G$, there is a notion of “split” elements in $C^F$ which solves this problem in almost all cases; see Shoji’s survey [@S1 §5]. Despite of much further work (e.g., Shoji [@S6]), the question of finding general conditions which single out a distinguished representative $g_1 \in C^F$ appears to be open. In the above examples (and those below), we are able to choose a representative $g_1\in C^F$ according to the following principles: - $g_1$ belongs to $C^{F_0}=C({{\mathbb{F}}}_p)$ (cf. Remark \[exp1\]) and is conjugate in $G({{\mathbb{F}}}_p)$ to all powers $g_1^n$ where $n\in {{\mathbb{Z}}}$ is coprime to the order of $g_1$. - The unipotent part of $g_1$ has a “short” expression in terms of the Chevalley generators $x_\alpha(t)$ of $G$ where $\alpha$ is a root and $t\in{{\mathbb{F}}}_p$. These principles also work in the further examples discussed in [@GeHe]. Type $F_4$ in characteristic $2$ {#secF4} ================================ Throughout this section, we assume that $G$ is simple of type $F_4$ and $p=2$. We have $G=\langle x_\alpha(t)\mid \alpha\in\Phi,t\in k\rangle$ where $\Phi$ is the root system of $G$ with respect to $T$. Let $\{\alpha_1,\alpha_2, \alpha_3,\alpha_4\}$ be the set of simple roots with respect to $B$, where we choose the notation such that $\alpha_1,\alpha_2$ are long roots, $\alpha_3, \alpha_4$ are short roots, and $\alpha_2,\alpha_3$ are not orthogonal. (This coincides with the conventions in Shinoda [@Shi], where the conjugacy classes of $G$ are determined.) Also note that, since $p=2$, we do not have to worry about the precise choice of a Chevalley basis in the underlying Lie algebra. Our aim is to determine the exact relation between characteristic functions of cuspidal unipotent character sheaves and almost characters in this case, using the approach illustrated in the previous section. For this purpose, we essentially rely on the knowledge of the character table of $F_4({{\mathbb{F}}}_2)$; see the [ATLAS]{} [@atl p. 167]. (The correctness of this table has been verified independently in [@veri], and it is available in the library of [GAP]{} [@gap].) We shall also rely on a number of explicit computational results obtained through general algorithms and functions concerning matrix groups and character tables in [GAP]{}. \[r51\] There are seven cuspidal characters sheaves $A_1,\ldots,A_7$ on $G$, described in detail by Shoji [@S2 §7] (based on earlier work of Lusztig and Spaltenstein). Firstly, they are all contained in $\hat{G}^{\text{un}}$; see [@S2 7.6]. Secondly, as in \[exp2\], each $A_j$ corresponds to a pair $(g_1,\lambda)$ where $g_1\in G^F$ and $\lambda\in {{\operatorname{Irr}}}(A(g_1))$ is a non-trivial linear character of $A(g_1)$; see [@S2 §7.2]. Here, in all seven cases, $F$ acts trivially on $A(g_1)$. The correspondences $A_j \leftrightarrow (g_1,\lambda)$ are given as follows. - $A_j\leftrightarrow (u,\lambda_j)$ ($j=1,2$) where $u\in G^F$ is regular unipotent, $\dim C_G(u)=4$; furthermore, $A(u)\cong {{\mathbb{Z}}}/4{{\mathbb{Z}}}$ is generated by the image $\bar{u}\in A(u)$, and $\lambda_j$ are the linear characters such that $\lambda_1(\bar{u})=i$, $\lambda_2(\bar{u})=-i$, where $i=\sqrt{-1}$ is a fixed fourth root of unity. - $A_3\leftrightarrow (u,\lambda)$ where $u\in G^F$ is unipotent such that $\dim C_G(u)=6$ and $A(u)\cong {{\mathbb{Z}}}/2{{\mathbb{Z}}}$; furthermore, $\lambda$ is the non-trivial character of $A(u)$. - $A_4\leftrightarrow (u,\lambda)$ where $u\in G^F$ is unipotent such that $\dim C_G(u)=8$ and $A(u)$ is isomorphic to the dihedral group $D_8$; furthermore, $\lambda$ corresponds to the sign character of $D_8$. - $A_5\leftrightarrow (u,\lambda)$ where $u\in G^F$ is unipotent such that $\dim C_G(u)=12$ and $A(u)$ is isomorphic to the symmetric group $\mathfrak{S}_3$; furthermore, $\lambda$ corresponds to the sign character of $\mathfrak{S}_3$. - $A_j\leftrightarrow (su,\lambda_j)$ ($j=6,7)$ where $s\in G^F$ is semisimple, with $C_G(s)$ isogenous to $\mbox{SL}_3(k)\times \mbox{SL}_3 (k)$, and $u \in C_G(s)^F$ is regular unipotent; we have $\dim C_G(su)=4$. Furthermore, $A(su)\cong {{\mathbb{Z}}}/3{{\mathbb{Z}}}$ is generated by the image $\overline{su} \in A(su)$, and $\lambda_j$ are linear characters such that $\lambda_6 (\overline{su})=\theta$, $\lambda_7(\overline{su})=\theta^2$, where $\theta$ is a fixed third root of unity. (In each case (a)–(d), the conditions on $\dim C_G(u)$ and $A(u)$ uniquely determine $g_1=u$ up to $G$-conjugacy; see Shinoda [@Shi §2]. Furthermore, the class of $s$ in (e) is also uniquely determined such that $C_G(s)$ has type $A_2\times A_2$; see [@Shi §3].) $\begin{array}{clcclc} \hline \text{In \ref{r51}} & \text{representative $g_1$}&A(g_1)&|C_G(g_1)^F| & \text{\sf GAP}\\ \hline \text{(a)}&u_{31}=x_{1000}(1)x_{0100}(1)x_{0010}(1)x_{0001}(1) & {{\mathbb{Z}}}/4{{\mathbb{Z}}}& 4q^4 & {\tt 16a} (76)\\ \text{(b)}&u_{29}=x_{0122}(1)x_{1000}(1)x_{0100}(1)x_{0010}(1) & {{\mathbb{Z}}}/2{{\mathbb{Z}}}& 2q^6 & {\tt 8j} (47)\\ \text{(c)}&u_{24}=x_{1100}(1)x_{0120}(1)x_{0001}(1)x_{0011}(1) & D_8 & 8q^8 & {\tt 8a} (38)\\ \text{(d)}&u_{17}=x_{1110}(1)x_{1220}(1)x_{0011}(1)x_{0122}(1) & \mathfrak{S}_3 & 6q^{12} & {\tt 4l} (20)\\ \text{(e)} & su_{17}=u_{17}s \;\text{ with $s\in G^F$ of order $3$} & {{\mathbb{Z}}}/3{{\mathbb{Z}}}& 3q^4 & {\tt 12o} (68)\\ \hline \multicolumn{5}{c}{\text{(Notation for $u_i$ as in Shinoda \cite[\S 2]{Shi})}}\end{array}$ \[r52\] Let $A_j$ be one of the cuspidal character sheaves in \[r51\]. In Table \[supp2\], we fix a specific element $g_1\in G^F$ for the corresponding pair $(g_1,\lambda)$. The characteristic function $\chi_j:=\chi_{A_j,\phi_{A_j}}$ in \[exp2\] will then be defined with respect to this choice of $g_1$. In the table, we use the following notation. If $\alpha$ is a positive root, written as $\alpha=\sum_{1{\leqslant}i{\leqslant}4} n_i\alpha_i$, then we denote the corresponding root element $x_\alpha(t)$ by $x_{n_1n_2n_3n_4}(t)$. Some further comments about the entries of Table \[supp2\]. \(1) In the cases (a)–(d), $g_1$ is unipotent; let $C$ be the conjugacy class of $g_1$ in $G$. Then we choose $g_1$ to be that representative of $C^F$ in Shinoda’s list [@Shi §2], which has an expression as a product of root elements $x_\alpha(t)$ whose description does not involve the elements $\eta, \zeta\in k$ defined in [@Shi 2.2]. All the coefficients in the expression for $g_1$ are equal to $1$, and so $g_1\in F_4({{\mathbb{F}}}_2)$. \(2) Let $g_1\in F_4({{\mathbb{F}}}_2)$ be as in (1). Then we will need to know to which class of the [GAP]{} character table of $F_4({{\mathbb{F}}}_2)$ this element belongs. This problem is easy for some cases, e.g., $g_1=u_{17}$ since there is a unique class in the [GAP]{} table with centraliser order $6\cdot 2^{12}= 24576$. On the other hand, if $g_1=u_{31}$, then the classes with centraliser order $4\cdot 2^4=64$ are quite difficult to distinguish in the [GAP]{} table, especially the two classes [16a]{}, [16b]{}. In such cases, we use the following (computational) argument. Using the explicit $52$-dimensional matrix realization of $F_4({{\mathbb{F}}}_2)$ from Lusztig [@L12 2.3] (see also [@mylie 4.10]), we create $F_4({{\mathbb{F}}}_2)$ as a matrix group in [GAP]{}. Then we consider the subgroup $$P:=\langle x_{\pm 1000}(1), x_{\pm 0100}(1),x_{\pm 0010}(1), x_{0001}(1) \rangle\subseteq F_4({{\mathbb{F}}}_2).$$ The [GAP]{} function [CharacterTable]{} computes the character table of $P$ (by a general algorithm, without using any specific properties of $P$), together with a list of representatives of the conjugacy classes. There are $214$ conjugacy classes of $P$ and, again by standard algorithms, [GAP]{} can find out to which of these $214$ classes any given element of $P$ belongs. Now the function [PossibleClassFusions]{} determines all possible fusions of the conjugacy classes of $P$ into the [GAP]{} character table of $F_4({{\mathbb{F}}}_2)$. (Here, a “possible class fusion” is a map which assigns to each conjugacy class of $P$ one of the $95$ class labels [1a]{}, [2a]{}, $\ldots$, [30b]{} of the [GAP]{} table of $F_4({{\mathbb{F}}}_2)$, such that certain conditions are satisfied which should hold if the map is a true matching of the classes of $P$ with those of the [GAP]{} table; see the help menu of [PossibleClassFusions]{} for further details.) As might be expected (since there are non-trivial table automorphisms of the character table of $F_4({{\mathbb{F}}}_2)$), the class fusion is not unique; in fact, it turns out that there are $16$ possible fusion maps. But, if $g_1 \in \{u_{17}, u_{24}, u_{29}, u_{31}\}$, then $g_1$ is mapped to the same class label of $F_4({{\mathbb{F}}}_2)$, under each of the $16$ possibilities. Thus, the fusion of $g_1$ is uniquely determined, and this is the entry in the last column of Table \[supp2\]. \(3) Let again $g_1\in F_4({{\mathbb{F}}}_2)$ be as in (1). Having identified the class of $g_1$ in the [GAP]{} table, we can simply check by inspection that $\chi(g_1) \in{{\mathbb{Z}}}$ for all $\chi\in{{\operatorname{Irr}}}(F_4({{\mathbb{F}}}_2))$. Hence, $g_1$ is conjugate within $F_4({{\mathbb{F}}}_2)$ to each power $g_1^n$ where $n$ is a positive integer coprime to the order of $g_1$; in particular, $g_1$, $g_1^{-1}$ are conjugate in $F_4({{\mathbb{F}}}_2)$. \(4) Now let $g_1=su$ be as in case (e). Let $C$ be the $G$-conjugacy class of $su$. Since $s$ has order $3$ and $u$ has order $4$, the element $g_1=su$ has order $12$; furthermore, $|C_G(g_1)^F|=3q^4$. All this also works for the base case where $q=2$ and so we can assume that $g_1 \in C({{\mathbb{F}}}_2)\subseteq F_4({{\mathbb{F}}}_2)$. Then $C({{\mathbb{F}}}_2)$ splits into $3$ classes in $F_4({{\mathbb{F}}}_2)$; all elements in these three classes have order $12$ and centraliser order $3\cdot 2^4=48$. By inspection of the [GAP]{} table, we see that these three classes must be [12o]{}, [12p]{}, [12q]{}; here, [12o]{} has the property that all character values on this class are integers. We now choose $g_1$ to be in [12o]{}; then $g_1$ is conjugate in $F_4({{\mathbb{F}}}_2)$ to all powers $g_1^n$ where $n$ is a positive integer coprime to the order of $g_1$. In particular, $g_1$ is conjugate in $G^F$ (even in $F_4({{\mathbb{F}}}_2)$) to $g_1^5=s^{-1}u$ and, consequently, $s,s^{-1}$ are conjugate in $C_G(u)^F$. Now the [GAP]{} table also shows that $g_1^3=u^3=u^{-1}$ belongs to [4l]{}. We have seen in (2), (3) that both $u_{17}$ and $u_{17}^{-1}$ belong to [4l]{}. Hence, we can assume that $u=u_{17}$. It then also follows from the above results (and Sylow’s Theorem) that, if $t\in C_G(u)^F$ is any element of order $3$, then $tu$ is conjugate to $su$ in $G^F$. $\begin{array}{crc} \hline \rho_x & \rho_x(1) & \text{{\sf GAP}} \\\hline F_4^{II}[1]& 1326 & X.5\\ F_4^{I}[1]& 21658 & X.7 \\ F_4[-1]& 63700 & X.13 \\ \phi_{1,12}''& 99450 & X.14 \\ \phi_{1,12}'& 99450 & X.15 \\ \{F_4[i], F_4[-i]\}& 142884 & \{X.16,X.17\} \\ \{F_4[\theta],F_4[\theta^2]\}& 183600 & \{X.20,X.21\} \\ B_2:(\varnothing.2)& 216580 & X.22 \\ B_2:(11.\varnothing)& 216580 & X.23 \\ \hline \multicolumn{3}{l}{\text{(Notation for $\rho_x$ as in Carter \cite[p.~479]{C2})}} \end{array} \qquad\qquad \begin{array}{crc} \hline \rho_x & \rho_x(1) & \text{{\sf GAP}}\\\hline \phi_{6,6}''& 249900 & X.24 \\ B_2:(1.1)& 269892 & X.25 \\ \phi_{4,8}& 322218 & X.27 \\ \phi_{4,7}''& 358020 & X.30 \\ \phi_{4,7}'& 358020 & X.31 \\ \phi_{6,6}'& 519792 & X.32 \\ \phi_{9,6}''& 541450 & X.33 \\ \phi_{9,6}'& 541450 & X.34 \\ \phi_{12,4}& 584766 & X.37 \\ \phi_{16,5}& 947700 & X.44 \\ \hline \end{array}$ \[r53\] Let us write ${{\mathfrak{U}}}(G^F)=\{\rho_x\mid x\in X(W)\}$ and $\hat{G}^{\text{un}}=\{A_x \mid x \in X(W)\}$ as in \[r18\]. Here, the parameter set $X(W)$ has $37$ elements. It is partitioned into $11$ “families”, which correspond to the special characters of $W$, as defined in [@L1 4.1, 4.2]. The corresponding Fourier matrix has a block diagonal shape, with one diagonal block for each family. The $7$ elements of $X(W)$ which label cuspidal character sheaves are all contained in one family, which we denote by ${{\mathcal{F}}}_0$ and which consists of $21$ elements in total. The elements of ${{\mathcal{F}}}_0$ are given by the $21$ pairs $(u,\sigma)$ where $u\in{{\mathfrak{S}}}_4$ (up to ${{\mathfrak{S}}}_4$-conjugacy) and $\sigma\in{{\operatorname{Irr}}}(C_{{{\mathfrak{S}}}_4}(u))$. The diagonal block of the Fourier matrix corresponding to ${{\mathcal{F}}}_0$ is then described in terms of ${{\mathfrak{S}}}_4$ and the pairs $(u,\sigma)$; see [@C2 §13.6] for further details. The identification of the $21$ unipotent characters $\{\rho_x\mid x\in{{\mathcal{F}}}_0\}$ with characters in the [GAP]{} table of $F_4({{\mathbb{F}}}_2)$ is given in Table \[uni2\]. (This information is available through the [GAP]{} function [DeligneLusztigNames]{}; note that this depends on fixing a labelling of the long and short roots.) The results of Shoji [@S2 Theorem 7.5] show that the cuspidal character sheaves $A_j$ in \[r51\] are labelled as follows by pairs in ${{\mathcal{F}}}_0$: $$\begin{gathered} A_1=A_{(g_4,i)}, \qquad A_2=A_{(g_4,-i)}, \qquad A_3=A_{(g_2,\epsilon)},\\ \{A_4,A_5\}=\{A_{(1,\lambda^3)},A_{(g_2', \varepsilon)}\},\qquad A_6=A_{(g_3,\theta)},\qquad A_7=A_{(g_3,\theta^2)}.\end{gathered}$$ Here, we follow the notation in [@C2 p. 455] for pairs in ${{\mathcal{F}}}_0$. (Note that, a priori, the statement of Theorem \[r16\] alone does not tell us anything about the supporting set of a cuspidal character sheaf $A_x$!) Let us consider in detail $A_6,A_7$. (The remaining cases have been dealt with already by Marcelo–Shinoda [@MaSh]; see Remark \[mash\] below.) Using the $21\times 21$-times Fourier matrix printed in [@C2 p. 456] and the labelling of the unipotent characters $\{\rho_x \mid x \in {{\mathcal{F}}}_0\}$ in terms of pairs $(u,\sigma)$ as above (see [@C2 p. 479]), we obtain the following formulae for the unipotent almost characters labelled by $(g_3,\theta)$ and $(g_3,\theta^2)$: $$\begin{aligned} R_{(g_3,\theta)} & =\textstyle\frac{1}{3}\bigl(\rho_{(12,4)}+F_4^{II}[1]- \rho_{(6,6)'}-\rho_{(6,6)''}+ 2F_4[\theta]-F_4[\theta^2]\bigr),\\ R_{(g_3,\theta^2)} & =\textstyle\frac{1}{3}\bigl(\rho_{(12,4)}+F_4^{II}[1]- \rho_{(6,6)'}-\rho_{(6,6)''} -F_4[\theta]+2F_4[\theta^2]\bigr).\end{aligned}$$ By \[r52\] and Theorem \[r16\], we have $R_{(g_3,\theta)}= \zeta_{(g_3,\theta)}\chi_6$ and $R_{(g_3,\theta^2)}=\zeta_{(g_3,\theta^2)} \chi_7$. \[p2\] With notation as in \[r51\]–\[r53\], we have $\zeta_{(g_3,\theta)}=\zeta_{(g_3,\theta^2)}=1$; hence, $R_{(g_3,\theta)}= \chi_6$ and $R_{(g_3,\theta^2)}=\chi_7$. Let $C$ be the conjugacy class of $g_1=su_{17}$ in $G$ (cf.  Table \[supp2\]). Recall from \[r52\] that the values of $\chi_6$ and $\chi_7$ on $C^F$ are given as follows: $$\chi_6(g)= \left\{\begin{array}{cl} q^4 & \mbox{ (if $g=g_1$)}\\ q^4 \theta& \mbox{ (if $g=g_{\bar{g}_1}$)}\\ q^4\theta^2 & \mbox{ (if $g= g_{\bar{g}_1^2}$)}\end{array}\right. \quad \mbox{and}\qquad \chi_7(g)= \left\{\begin{array}{cl} q^4 & \mbox{ (if $g=g_1$)}\\ q^4 \theta^2& \mbox{ (if $g=g_{\bar{g}_1}$)}\\ q^4\theta & \mbox{ (if $g=g_{\bar{g}_1^2}$)} \end{array}\right.$$ In order to prove that $\zeta_{(g_3,\theta)}=\zeta_{(g_3,\theta^2)}=1$ it is sufficient, by Proposition \[p1\], to consider the base case where $q=2$. So, in the [GAP]{} table of $F_4({{\mathbb{F}}}_2)$, we form the above linear combinations of the irreducible characters, giving $R_{(g_3,\theta)}$ and $R_{(g_3,\theta^2)}$ for the group $F_4({{\mathbb{F}}}_2)$. We find that these linear combinations have value $16$ on the class [12o]{}, which contains $g_1=su_{17}\in C^F$. (Note that this does not depend on how we match $F_4[\theta]$, $F_4[\theta^2]$ with [X.20]{}, [X.21]{} in the [GAP]{} table, since [X.20]{} and [X.21]{} are algebraically conjugate and all values on [12o]{} are integers.) Comparing with the above formulae we see that $\zeta_{(g_3,\theta)}= \zeta_{(g^3,\theta^2)}=1$, as desired. \[mash\] The remaining $A_1,\ldots,A_5$ have already been dealt with by Marcelo–Shinoda [@MaSh Theorem 4.1], by completely different (and, in our opinion, computationally somewhat more complicated) methods. In particular, they showed that $$A_4=A_{(g_3',\varepsilon)}, \quad A_5=A_{(1,\lambda^3)} \quad \mbox{and} \quad \zeta_{A_1}= \zeta_{A_2}= \zeta_{A_3}= \zeta_{A_4}= \zeta_{A_5}=1.$$ In our setting, this can also be shown by exactly the same kind of argument as in the proof of Proposition \[p2\]. Type $E_6$ in characteristic $2$ {#secE6} ================================ Throughout this section, we assume that $G$ is simple of adjoint type $E_6$ and $p=2$. We have $G=\langle x_\alpha(t)\mid \alpha\in\Phi,t\in k\rangle$ where $\Phi$ is the root system of $G$ with respect to $T$. Let $\{\alpha_1,\alpha_2, \alpha_3,\alpha_4,\alpha_5,\alpha_6\}$ be the set of simple roots with respect to $B$, where the labelling is chosen as in the following diagram: (270,53) ( 10,25)[$E_6$]{} (132, 3)[$\alpha_2$]{} ( 61,43)[$\alpha_1$]{} ( 91,43)[$\alpha_3$]{} (121,43)[$\alpha_4$]{} (151,43)[$\alpha_5$]{} (181,43)[$\alpha_6$]{} (125, 5) ( 65,35) ( 95,35) (125,35) (155,35) (185,35) (125,35)[(0,-1)[30]{}]{} ( 65,35)[(1,0)[30]{}]{} ( 95,35)[(1,0)[30]{}]{} (125,35)[(1,0)[30]{}]{} (155,35)[(1,0)[30]{}]{} Again, since $p=2$, we do not have to worry about the precise choice of a Chevalley basis in the underlying Lie algebra. Further note that $G({{\mathbb{F}}}_2)$ is a simple group; it will just be denoted by $E_6({{\mathbb{F}}}_2)$. (For $q$ an even power of $2$, the group $G({{\mathbb{F}}}_q)$ has a normal subgroup of index $3$). Now we essentially rely on the knowledge of the character table of $E_6({{\mathbb{F}}}_2)$, which has been determined by B. Fischer and is contained in the [GAP]{} library [@bre]. (We have $|{{\operatorname{Irr}}}(E_6({{\mathbb{F}}}_2))|=180$.) Although the group is bigger, the further discussion will actually be simpler than that for type $F_4$ in the previous section, since there are fewer cuspidal unipotent character sheaves. \[r61\] By Lusztig [@L2d 20.3] and Shoji [@S3 4.6], there are two cuspidal character sheaves $A_1,A_2\in \hat{G}^{\text{un}}$. Both $A_1$, $A_2$ have the same support, namely, the closure of the $G$-conjugacy class $C$ of an element $g_1=su\in G^F$ such that $s\in G^F$ is semisimple, with $C_G(s)^\circ$ isogneous to $\mbox{SL}_3(k)\times \mbox{SL}_3(k)\times\mbox{SL}_3(k)$, and $u \in C_G^\circ(s)^F$ is regular unipotent. Furthermore, $\dim C_G(g_1)=6$ and $A(g_1)\cong C_3\times C_3'$ where $C_3, C_3'$ are cyclic of order $3$ and $C_3$ is generated by the image $\overline{g}_1\in A(g_1)$. Finally, $A_j\leftrightarrow (g_1,\lambda_j)$ where $\lambda_j$ are linear characters which are trivial on $C_3'$ and such that $\lambda_1(\overline{g}_1)= \theta$, $\lambda_2(\overline{g}_1)=\theta^2$ (where, again, $\theta$ is a fixed third root of unity). Once we fix a specific element $g_1\in C^F$, we obtain well-defined characteristic functions $$\chi_j:=\chi_{g_1,\lambda_j}=\chi_{A_j,\phi_{A_j}} \qquad \mbox{for $j=1,2$ (cf.\ \ref{exp2})}.$$ $\begin{array}{crc} \hline \rho_x & \rho_x(1) & \text{{\sf GAP}} \\\hline \{E_6[\theta],E_6[\theta^2]\}& 45532800& \{X.14,X.15\} \\ D_4,r &120645056 & X.18 \\ (20,10) & 184660800 & X.20 \\ (10,9) & 192047232 & X.22\\ (60,8) & 800196800 & X.29\\ (80,7) & 864212544 & X.30\\ (90,8) & 902358912 & X.31\\ \hline \multicolumn{3}{l}{\text{(Notation for $\rho_x$ as in Carter \cite[p.~480]{C2})}} \end{array}$ \[r62\] Let us write ${{\mathfrak{U}}}(G^F)=\{\rho_x\mid x\in X(W)\}$ and $\hat{G}^{\text{un}}=\{A_x \mid x \in X(W)\}$ as in \[r18\]. Here, the parameter set $X(W)$ has $30$ elements, and it is partitioned into $17$ families. The $2$ elements of $X(W)$ which label cuspidal character sheaves are all contained in one family, which we denote by ${{\mathcal{F}}}_0$ and which consists of $8$ elements in total. The elements of ${{\mathcal{F}}}_0$ are given by the $8$ pairs $(u,\sigma)$ where $u\in{{\mathfrak{S}}}_3$ (up to ${{\mathfrak{S}}}_3$-conjugacy) and $\sigma\in{{\operatorname{Irr}}}(C_{{{\mathfrak{S}}}_3}(u))$. The diagonal block of the Fourier matrix corresponding to ${{\mathcal{F}}}_0$ is then described in terms of ${{\mathfrak{S}}}_3$ and the pairs $(u,\sigma)$; see [@C2 §13.6] for further details. The identification of the $8$ unipotent characters $\{\rho_x\mid x\in{{\mathcal{F}}}_0\}$ with characters in the [GAP]{} table of $E_6({{\mathbb{F}}}_2)$ is given in Table \[uni2a\]. (These characters are uniquely determined by their degrees, except for the two cuspidal unipotent characters $E_6[\theta]$, $E_6[\theta^2]$ which have the same degree.) The results of Shoji [@S3 4.6, 5.2] show that the cuspidal character sheaves $A_j$ in \[r61\] are labelled as follows by pairs in ${{\mathcal{F}}}_0$: $$A_1=A_{(g_3,\theta)}\qquad \mbox{and} \qquad A_2=A_{(g_3,\theta^2)},$$ where we follow again the notation in [@C2 p. 455] for pairs in ${{\mathcal{F}}}_0$. Using the $8\times 8$-times Fourier matrix printed in [@C2 p. 457] and the labelling of the unipotent characters $\{\rho_x \mid x \in {{\mathcal{F}}}_0\}$ in terms of pairs $(u,\sigma)$ as above (see [@C2 p. 480]), we obtain the unipotent almost characters labelled by $(g_3,\theta)$ and $(g_3,\theta^2)$: $$\begin{aligned} R_{(g_3,\theta)} & =\textstyle\frac{1}{3}\bigl(\rho_{(80,7)}+\rho_{(20,10)} -\rho_{(10,9)}-\rho_{(90,8)}+2E_6[\theta]-E_6[\theta^2]\bigr),\\ R_{(g_3,\theta^2)} & =\textstyle\frac{1}{3}\bigl(\rho_{(80,7)}+\rho_{(20,10)} -\rho_{(10,9)}-\rho_{(90,8)}-E_6[\theta]+2E_6[\theta^2]\bigr).\end{aligned}$$ \[r62a\] Let $C$ be the $F$-stable conjugacy class of $G$ such that $\mbox{supp}(A_j)=\overline{C}$ for $j=1,2$ (see \[r61\]). We now fix a specific representative $g_1\in C^F$, as follows. Using the [GAP]{} character table of $E_6({{\mathbb{F}}}_2)$ and the information in Table \[uni2a\], we find (for $q=2$) that $\{R_{(g_3,\theta)}, R_{(g_3, \theta^2)}\}=\{f_1,f_2\}$ where $f_1,f_2 \colon E_6({{\mathbb{F}}}_2)\rightarrow \overline{{{\mathbb{Q}}}}_\ell$ are class functions with non-zero values as follows: $$\begin{array}{cccc} \hline {\sf GAP:} & {\tt 12n} \,\text{(no.~60)} & {\tt 12o} \,\text{(no.~61)} & {\tt 12p} \, \text{(no.~62)} \\ \hline f_1 & 8 & 8\theta & 8\theta^2\\f_2 & 8 & 8\theta^2 & 8\theta\\\hline\end{array}$$ Since $\chi_j$ only has non-zero values on elements in $C^F$, we conclude that $C$ must contain the elements of $E_6({{\mathbb{F}}}_2)$ which belong to the three classes [12n]{}, [12o]{}, [12p]{} in the [GAP]{} table; these three classes have centraliser order $192=3\cdot 2^6$. By inspection of the [GAP]{} table, all character values on [12n]{} are integers, while the character values on [12o]{} are complex conjugates of those on [12p]{}. Thus, we choose $g_1\in C({{\mathbb{F}}}_2)\subseteq C^F$ to be from the class [12n]{} in $E_6({{\mathbb{F}}}_2)$. Then $g_1$ is conjugate in $E_6({{\mathbb{F}}}_2)$ to all powers $g_1^n$ where $n$ is a positive integer coprime to the order of $g_1$. Once $g_1$ is fixed, we obtain characteristic functions $\chi_1,\chi_2$ as in \[r61\]. Then Theorem \[r16\] shows that $R_{(g_3,\theta)}= \zeta_{(g_3,\theta)} \chi_1$ and $R_{(g_3,\theta^2)}=\zeta_{(g_3,\theta^2)} \chi_2$. \[r62b\] Let $g_1\in C({{\mathbb{F}}}_2)$ be as in \[r62a\]. We write $g_1=su=us$ where $s\in E_6({{\mathbb{F}}}_2)$ has order $3$ and $u\in E_6({{\mathbb{F}}}_2)$ is unipotent. As in \[r52\](4) one sees that all such elements $s$ of order $3$ are conjugate in $E_6({{\mathbb{F}}}_2)$. We now identify the unipotent part $u$ of $g_1$, where we use the results and the notation of Mizuno [@Miz]. Let $$u_{15}:=x_{\alpha_2}(1)x_{\alpha_3}(1)x_{\alpha_4}(1)x_{\alpha_2+\alpha_4+ \alpha_5}(1)\in E_6({{\mathbb{F}}}_2),\tag{a}$$ as in [@Miz Lemma 4.3]; in particular, we have $$A(u_{15})\cong \mathfrak{S}_3 \qquad \mbox{and} \qquad |C_G(u_{15})^F|= 6q^{18}(q-1)^2.\tag{b}$$ We claim that the unipotent part $u$ of $g_1$ is conjugate in $E_6({{\mathbb{F}}}_2)$ to $u_{15}$. To see this, we follow a similar approach as in \[r52\]. Using the explicit $78$-dimensional matrix realization of $E_6({{\mathbb{F}}}_2)$ from Lusztig [@L12 2.3] (see also [@mylie 4.10]), we create $E_6({{\mathbb{F}}}_2)$ as a matrix group in [GAP]{}. We consider the subgroup $$L:=\langle x_{\pm \alpha_2}(1),x_{\pm \alpha_3}(1),x_{\pm \alpha_4}(1), x_{\pm \alpha_5}(1) \rangle \subseteq E_6({{\mathbb{F}}}_2);$$ note that $u_{15}\in L$. The group $L$ is simple of order $174182400$. (In fact, $L \cong \mbox{SO}_8^+({{\mathbb{F}}}_2)$.) As in Example \[so8\], the [GAP]{} function [CharacterTable]{} computes the character table of $L$ (by a general algorithm), together with a list of representatives of the conjugacy classes; there are $53$ conjugacy classes of $L$. One notices that there is a unique class of $L$ in which the elements have centraliser order $64$, and one checks using [GAP]{} that our element $u_{15}$ belongs to this class. Now, as in \[r52\], the function [PossibleClassFusions]{} determines all possible fusions of the conjugacy classes of $L$ into the [GAP]{} table of $E_6({{\mathbb{F}}}_2)$. It turns out that there are $2$ possible fusion maps. But under each of these two possibilities, the class containing our element $u_{15}$ is mapped to the class [4k]{} (no. 18) in the [GAP]{} table of $E_6({{\mathbb{F}}}_2)$. Thus, $u_{15}$ belongs to the class [4k]{} in [GAP]{}. According to the [GAP]{} table, the element $g_1^3=u^{-1}$ also belongs to the class [4k]{}; furthermore, all character values on [4k]{} are integers. Hence, $u$, $u^{-1}$, $u_{15}$ are all conjugate in $E_6({{\mathbb{F}}}_2)$, as claimed. To summarize the above discussion, we can assume that the chosen representative $g_1\in C^F$ in \[r62a\] is as follows. We have $g_1= us=su\in E_6({{\mathbb{F}}}_2)$ where $u=u_{15}$ (see \[r62b\](a)) and $s\in E_6({{\mathbb{F}}}_2)$ is any element of order $3$ such that $su=us$. \[r63\] Let $g_1=su_{15}\in E_6({{\mathbb{F}}}_2)\subseteq G^F$ be as above. For $j=1,2$, define $\chi_j=\chi_{(g_1,\lambda_j)} \colon G^F \rightarrow \overline{{{\mathbb{Q}}}}_\ell$ as in Example \[exp2\], with $\lambda_j$ as in \[r61\]. Then $\zeta_{(g_3, \theta)}=\zeta_{(g_3,\theta^2)}=1$; hence, $R_{(g_3,\theta)}=\chi_1$ and $R_{(g_3,\theta^2)}=\chi_2$. Recall from \[r62\] that $R_{(g_3,\theta)}=\zeta_{(g_3, \theta)}\chi_1$ and $R_{(g_3,\theta^2)}=\zeta_{(g_3,\theta^2)} \chi_2$. Now it will be sufficient to show that the scalars $\zeta_{(g_3,\theta)}$ and $\zeta_{(g_3,\theta^2)}$ are equal to $1$ for the special case where $q=2$; see Proposition \[p1\]. But this has been observed in \[r62a\] above; note that, for $q=2$, we have $\{f_1,f_2\}=\{\chi_1,\chi_2\}$ and $g_1$ belongs to the class [12n]{} in the [GAP]{} table. Regardless of how we match the functions in these two pairs, the values on $g_1\in {\tt 12n}$ are equal to $8$ in both cases. Hence, we must have $\zeta_{(g_3, \theta)}=\zeta_{(g_3,\theta^2)}=1$. \[fine6\] We briefly sketch an alternative proof of Proposition \[r63\], following the line of argument in Lusztig [@L3]. Let us fix $g_1=su_{15}=u_{15}s$ as above. Arguing analogously to \[r62\], we can express the unipotent characters $\{\rho_x\mid x\in {{\mathcal{F}}}_0\}$ in terms of the almost characters $\{R_x\mid x\in {{\mathcal{F}}}_0\}$. In particular, we obtain: $$E_6[\theta] =\textstyle\frac{1}{3}\bigl(R_{(80,7)}+R_{(20,10)} -R_{(10,9)}-R_{(90,8)}+2R_{(g_3,\theta)}-R_{(g_3,\theta^2)}\bigr).$$ Now evaluate on $g_1$. The values of the almost characters $R_\epsilon$ ($\epsilon\in{{\operatorname{Irr}}}(W)$) occurring in the above expression can be computed using the character formula in [@C2 7.2.8] (which expresses the values of the Deligne–Lusztig virtual characters $R_w$ in terms of the Green functions of $C_G(s)^\circ$) and the fact that Green functions take value $1$ on regular unipotent elements (see [@DeLu 9.16]). Furthermore, since all character values on $g_1$ are integers, we must have $\zeta_{(g_3,\theta)}= \zeta_{(g_3, \theta^2)}=\pm 1$, and so $R_{(g_3,\theta)}(g_1)= R_{(g_3,\theta^2)}(g_1)=\pm q^3$. This yields the condition that $$E_6[\theta](g_1)=\textstyle\frac{1}{3}\bigl(\pm q^3\,+\text{ known value}\bigr)\in {{\mathbb{Z}}},$$ which uniquely determines the sign of $\zeta_{(g_3,\theta)}$. Note that this argument does not require the knowledge of the complete character table of $E_6({{\mathbb{F}}}_2)$, but we still need to know some information about the representative $g_1$, e.g., the fact that it is conjugate to all powers $g_1^n$ where $n$ is coprime to the order to $g_1$. (A similar argument will actually work for $E_6({{\mathbb{F}}}_q)$, where $q$ is a power of any prime $\neq 3$, once the existence of a representative $g_1$ as above is guaranteed; for $q$ a power of $3$, a different argument is needed. See [@GeHe] for further details.) [**Acknowledgements.**]{} We are indebted to George Lusztig and Toshiaki Shoji for useful comments concerning the proof of Proposition \[p1\]. 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