unveiling the fundamental principles behind complex behavior in matter is a problem at the frontiers of condensed matter physics and embraces cases like the high - temperature superconductors , heavy fermions , and low - dimensional electron liquids .generically , describing the structure and behavior of matter involves studying systems of interacting quantum constituents ( bosons , fermions , spins , gauge fields ) whose fields satisfy the basic laws of quantum mechanics .nevertheless , the plethora of complex phenomena exhibited by nature exceeds our ability to explain them , in part , because _ the whole is not necessarily the sum of its parts _ and thus typical perturbation - like - theory arguments or standard mathematical techniques are not appropriate to disentangle its mysteries . in this paperwe present a unifying algebraic framework for interacting extended quantum systems that enables one to study complex phenomena characterized by the coexistence and competition of various states of matter.  the emergence of such complex phenomena may be the result of very simple , undiscovered , principles that conspire against any straightforward explanation .nonetheless , we expect that two pillars of modern science , symmetry and topology , are key guiding principles behind those fundamental laws of emergence .group theory and geometry have been fundamental to the physics of the twentieth century and we count on them to continue playing such a role . indeed , the notion of symmetry and its breakings have shaped our current conception of nature .understanding the idea of invariance and its corresponding conservation laws turns out to be as fundamental as determining the causes that prevent such harmony and leads to more complex behavior .other kinds of order , not described by broken symmetries , are possible . for example , quantum orders of topological nature distinguish the internal structure of the state without breaking any _ local _ symmetry , and its study is beyond the scope of the present paper .another example is provided by the notion of _ emergent _ symmetries where , as the name indicates , the number of symmetries of the state increases as the temperature is lowered , contrary to the broken symmetry case .these concepts are at the heart of the field of quantum phase transitions that studies the changes that may occur in the macroscopic properties of matter at zero temperature ( i.e. , ) due to changes in the parameters characterizing the system .  in this regard ,the development of exact algebraic methods is one of the most elegant and promising tools toward the complete understanding of quantum phases of matter and their corresponding phase transitions .typically , there are no distinct length or time scales which separates the different competing orders and often these systems are near quantum criticality which makes their study extremely complicated , if not impossible , by the traditional techniques , such as standard mean - field or perturbation theories .the reason which prevents the effective use of these theories is precisely the key for the successful application of the algebraic methods , namely , the absence of a small parameter for the various complex quantum orderings . in other words , one can not systematically apply renormalization group ideas and easily integrate out irrelevant degrees of freedom . on the other hand ,those competing orders are frequently related by symmetry principles characterizing the critical behavior caused by the competing interactions , thus increasing the symmetry group of the effective low - energy physics .  in the present manuscriptwe will be concerned with quantum lattice systems .a quantum lattice is a composite system identified with where is the total number of lattice sites , and associated to each lattice site ( or mode ) is a hilbert space of finite dimension describing the _ local _ modes .the total hilbert space , i.e. , state space of the lattice , is the tensor product of the local modes state spaces , , in the case of distinguishable subsystems .( notice that may support inequivalent tensor product decompositions . )indistinguishability places an additional constraint on the space of admissible states which manifests itself in their symmetry properties ( e.g. , antisymmetric states in the case of fermions ) and , consequently , the physical hilbert space is a subspace of . a pure state of the system ( at ) is simply a vector in , and an observable is a selfadjoint operator .the dynamical evolution of the system is determined by its hamiltonian .the topology of the lattice , determined by the connectivity of the graph induced by the interactions in , is an important element in establishing complexity .unless specified , we will always consider regular lattices of coordination . in the case of quantum continuous systems we can still use the present formalism after discretizing , for example, the space coordinates . going beyond this approachis deferred for a possible later publication .every given physical system is represented by a _ language _ which consists of a set of operators that acts irreducibly on the hilbert space .for example , if our system consists of a collection of interacting electrons a natural language is a set of spin 1/2 fermion field operators . defining a language amounts to establishing the state space used to describe the physical system , while a hamiltonian written in terms of that language comes to setting up the quantum dynamics expected to depict the phenomenon of interest .can we connect the different ( spin - particle - gauge ) languages of nature within a single algebraic framework ?the answer to this question is _ yes _ and the key outcome is a set of _ dictionaries _ relating the languages representing the different physical systems .more precisely , by dictionary we mean an isomorphic ( one - to - one ) mapping connecting two languages .the existence of dictionaries provides not only a tool to explore complexity but leads naturally to the fundamental concept of _ universality _ , or equivalence , meaning that different physical systems display the same behavior .the concepts of language and dictionary are introduced in section [ sec2 ] .previous to our work there were two seemingly unrelated examples of these types of mappings : the jordan - wigner ( 1928 ) and matsubara - matsuda ( 1956 ) transformations .we have not only generalized these ( ) transformations to any spin , spatial dimension and particle statistics but have proved a fundamental theorem that permits the connection of the generators of the different languages which can be used to describe a given physical problem .section [ sec2 ] starts defining the concept of a _bosonic language_. then we prove our fundamental theorem and establish the necessary conditions for application of a given bosonic language to describe the physical system under consideration .the main result of this section is a proof of the existence of mappings between the generators of any pair of languages which belong to the same class , i.e. , that can be used to describe the same physical system .the necessary and sufficient condition for any pair of languages to be connected is that the dimension of their local hilbert spaces be the same .in addition we prove that for each class of languages , there is at least one that is realized by the generators of a lie algebra .to complete this picture , we introduce the notion of _ hierarchical language _ and prove that there is at least one hierarchical language in each class . those languages are , in general , the most convenient ones to characterize the different quantum phases of the system under consideration .  in section [ sec3 ] ,we extend these conclusions to non - bosonic systems by introducing operators which transmute the modes statistics .these operators have a local and a non - local component .the non - local part is a trivial generalization of the transformation introduced by jordan and wigner to map spins =1/2 into spinless fermions .the local component transmutes the statistics associated to the interchange of particles which are sharing the same lattice site . adding the transmutation of statistics to the fundamental theorem completes the characterization of each class of equivalent languages . in other words ,the classes of bosonic languages defined in section [ sec2 ] are expanded in section [ sec3 ] to include fermionic , anyonic , or hybrid ( para- ) languages .the most natural choice of local hilbert space is the one spanned by a single site ( or mode ) basis of dimension .the possibility to decompose the total hilbert space of the problem into subspaces or subsystems which are not necessarily single sites opens up the possibility to generate other dictionaries with unforeseen applications .( remember that may support different tensor product decompositions . )section [ sec3n ] expands on this concept and show the simple case of a dictionary that uses as local hilbert space a bond state space of dimension ( ) , and which is mapped onto a site hilbert space of the same dimension .section [ sec4 ] is devoted to show explicit connections between equivalent languages .the main purpose of this section is to illustrate through examples the application of the dictionaries developed in the previous two sections . out of the many possible transformations predicted by our fundamental theorem and the transmutation of statistics , we selected a few of them which are useful for the applications described in the later sections . however , it is important to note that the procedure described to find the mapping connecting two languages in the same class is very straightforward .in addition , we show that the fractional exclusion statistics algebras emerge naturally from the present formalism .  given a model hamiltonian operator representing a quantum system, we can use our dictionaries to _ translate _ it into another equivalent hamiltonian written in a different language .in other words , we can write down the same hamiltonian operator in many distinct ways according to the different languages that are included in the corresponding class . since some of these languages are naturally associated to specific physical entities ( like spins , particles , gauge fields , etc . ) , the corresponding translation provides a rigorous connection between seemingly unrelated physical systems .this is the main subject of section [ sec5 ] , where we show examples of completely different quantum lattice systems described by the same model .another interesting aspect of these mappings is the potential to unveil hidden symmetries of the hamiltonian , possibly leading to exact or quasi - exact solutions .sometimes particular languages allow us to recognize invariant subspaces of our hamiltonian .when the action of our hamiltonian is restricted to an invariant subspace , the corresponding operator can always be written in a language which is more _ elementary _ than the original one ( i.e. , it realizes a reduced number of degrees of freedom ) . in some cases ,this procedure enables one to recognize the hidden symmetries which lead to the quasi - exact solution of the model considered ( i.e. , the model is integrable within the invariant subspace ) .the quasi - exact solution of the one - dimensional - model belongs to this class of problems .it is important to remark that this quasi - exact solution leads to the exact determination of the quantum phase diagram and the charge excitations of the - model .a new notion beyond landau s concept of broken symmetry is the complementary idea of _ emergent symmetry _ ,i.e. , the fact that new symmetries not realized in the hamiltonian describing the system can emerge at low energies .there are many instances in which the high - symmetry , low - energy , effective theory can be derived in an exact way , i.e. , without appealing to approximate schemes like the renormalization group .for instance , the exact effective theory ( ground state plus any charge excitations ) of the one - dimensional - model is the =1/2 model .it is well known that the model has an infinite number of symmetries which make it exactly solvable by the bethe ansatz .another example is provided by the family of spin hamiltonians for which the ground state is a product of spin singlets .this family includes the majumdar - ghosh model and many other generalizations . these and other simple examples introduced in section [ sec5 - 6 ] illustrate the fundamental concept of emergent symmetry which , as we will show , can be used as a guiding principle to find new states of matter . in a sense that will become clear later, this concept not only formalizes but also provides a systematics to the principle of adiabatic continuity that p.w .anderson has advocated as one of the two most important principles of condensed matter physics ( together with the concept of broken symmetry ) .states like the fermi liquid or the band insulator which do not correspond to any broken symmetry can be characterized by their corresponding emergent symmetries . in this wayone can establish a formal connection between different phases .for instance , the emergent symmetry of the band insulator is analog to the emergent symmetry of the singlet dimer magnetic states .in most cases , the notion of emergent symmetry is approximate ; however , still provides a guiding principle to identify the relevant degrees of freedom , and the nature of the ground state and low - energy excitations .coexistence and competition of different quantum orderings associated to broken continuous symmetries is one of the main subjects of section [ sec6 ] .there we show the fundamental role played by the hierarchical languages for the classification of the possible order parameters and the calculation of the quantum phase diagram of a given model .we note that the hierarchical languages are the most natural ones to provide a unified description of the order parameters characterizing each phase . to illustrate the procedure we consider the example of the heisenberg hamiltonians written in different languages .we then take the local order parameter and reduce its components according to the different subgroups of that can be used to generate an equivalent language ( for instance ) . as an illustration ,we show that the local order parameter for an =1 -spin hamiltonian can be either the usual magnetization or a spin - nematic order parameter . to conclude this section ,we show how to obtain the quantum phase diagram of the bilinear - biquadratic heisenberg model ( for spatial dimensions ) just by writing the hamiltonian in the hierarchical language .it is important to remark that this zero temperature phase diagram was only known for semi - classical spins .the fact that there is no calculation involved in the elaboration of this quantum phase diagram permits the reader to appreciate the power and potential applications of this algebraic framework .the formal connection between lattice models in condensed matter physics and lattice gauge theories of high - energy physics is described in section [ sec7 ] . therewe take advantage of the existing quantum link models and the connection between spins and other degrees of freedom , such as gauge fields , which emerges from our algebraic approach .in addition to the fascinating field of quantum phase transitions and statistical mechanics , our algebraic approach can be applied to the field of quantum information .this new paradigm of information processing takes advantage of the fundamental laws of quantum mechanics to perform operations which can be done at least as efficiently ( i.e. , with polynomial complexity ) as with classical devices . the device that performs the manipulation of informationis named quantum computer and the basic unit of information is the _ qubit _( i.e. , a two - level system ) .the quantum computer ( a quantum many - body system in disguise ) consists of an isolated set of quantum degrees of freedom which can be controlled and manipulated at the quantum level to perform the required operations .the different languages associated with the description of these quantum degrees of freedom are possible programming languages and indeed may realize different models of quantum computation .the development of dictionaries connecting these languages is relevant not only to improve the efficiency of a quantum computer , but also to model and simulate different physical phenomena .the applications of our algebraic framework to the fields of quantum information and quantum computing are discussed in section [ sec8 ] .there we show how each physical realization of a quantum computer has a class of languages associated to the realization of the quantum operations . for instance , if we consider the typical case of interacting =1/2 spins the most natural language is the one generated by the pauli matrices .hence we can use the dictionary provided by the jordan - wigner ( jw ) transformation to simulate a fermionic system . in this regard ,the purpose of this paper is to generalize this idea to other possible realizations of quantum computers .we also discuss the potential applications of this algebraic framework to the description of recent experiments on bosonic systems in optical lattices .finally , in section [ sec9 ] we summarize the main concepts introduced in this paper .the notions of hilbert space and linear maps or operators are central to the formulation of quantum mechanics .pure states ( wave functions ) are elements of a hilbert space and physical observables are associated to hermitian ( self - adjoint ) operators which act on that space , and whose eigenvalues are selected by a measuring apparatus .the role of linear operators in quantum mechanics is not restricted to the representation of physical observables .non - hermitian operators are very often used .that is the case , for instance , of the creation and annihilation operators in second quantization .moreover , the unitary operator that describes the time evolution of the system is in general non - hermitian .these observations simply tell us that linear operators in general ( hermitian and non - hermitian ) play a more general role in quantum mechanics since they provide the mathematical _ language _ which is required to describe the dynamical behavior of our quantum system .what do we mean by the term mathematical _ language _ ? in order to provide a rigorous answer for this questionit is important to first determine what is the mathematical structure associated to the set of quantum operators .quantum operators form a complex linear ( vector ) space under the sum and the multiplication by a scalar over the field of complex numbers . if we augment this vector space with a bilinear operation ( product between two operators ) , the set form an _algebra_. quantum mechanics also requires this operation being non - commutative and associative .this additional structure makes the set of quantum operators form an _associative algebra_. in principle , any operator in this algebra can play a role in the description of our quantum system .however , one can always select a subset ( which forms a basis ) in such a way that any quantum operator can be expressed as a function of the selected ones . for instance, we can use the pauli matrices  to express any quantum operator associated with a spin .the particular subset of operators that we choose to express any quantum operator is the mathematical language that we will use for the quantum description of our system .the elements of this subset will be called _ generators _ of the language .what are the conditions a given set of quantum operators must satisfy to become a language ? how many different languages can be used to describe a quantum system ?what is the connection between the different languages ?what is the most appropriate language to describe a particular system ?what is the relation between language and symmetry generators ? how can languages help us understand the phenomenon of universality ?( it turns out that the notions of language and universality are closely related whenever a common language can be used to describe seemingly unrelated physical phenomena . ) a great part of this paper is dedicated to answer these questions .in particular , in this section we introduce the notion of _ bosonic language _ , we establish a formal connection or _dictionary _ between the different bosonic languages associated with a given bosonic system , and we also establish a formal relation between bosonic languages and _ lie algebras_.  one of the fundamental steps toward a deeper understanding of classical systems was the recognition that dynamical variables which define the phase space are generators of continuous transformations .the set of continuous transformations forms a _ group _ and the infinitesimal generators provide a basis for a lie algebra which is related to the group .a group is a non - empty set which is closed under an associative product , it contains an identity , , and all of its elements are invertible .when the transformations in the group leave the equations of motion invariant the set becomes a symmetry group for the considered system .a real ( complex ) lie algebra is a linear space over the field of real ( complex ) numbers which is closed under a non - associative lie product  ] ) belongs to the algebra of endomorphisms for the vector space over the field of complex numbers , , and these elements are linearly independent . * the elements of a monoid of linear transformations under the associative product in the algebra which acts irreducibly on in the sense that the only subspaces stabilized by are and ( is the null vector ) . * if and are elements of different subsets and , then .the elements of the sets are called generators of the bosonic language . combining the associative product and the additive operation in the algebra of endomorphisms of , , we can define the non - associative lie product  ] . then, the last condition can be reformulated by establishing that the commutator of elements in different subsets is zero }=0 \ , \;\ ; \mbox { if}\;\ ; { \bf i}\neq { \bf j}. \label{bosl}\ ] ] it is also important to notice that the set is not necessarily closed under the regular product ( composition ) or the lie product ( commutator ) . if the set is closed under the lie product , the elements of generate a lie algebra .we will denote the lie algebra by .in addition , since each generator is represented by an endomorphism of there is a particular representation of associated to the bosonic language .the second condition for a bosonic language implies that is irreducible .the third condition implies that the global lie algebra associated to the complete set of generators is the direct sum of local algebras , .therefore , if the set is closed under the lie product , we can represent the bosonic language by the conjunction of the lie algebra , and the irreducible representation : .the dimension of is equal to the dimension of the local hilbert space : .the algebra of endomorphisms for the vector space is the enveloping algebra of .the following fundamental theorem shows that two languages are _ equivalent _ if they have in common the dimension of their local hilbert space , and is finite .in other words , there is always a dictionary connecting both languages , and a given physical phenomena can be described with either one .a corollary of the theorem is that given a particular language one can always find an equivalent one which can be expressed as , i.e. , the operators are generators of a lie algebra in the irreducible representation .in general there are various complex lie algebras which can be used to build equivalent languages .the only condition is that they have to admit an irreducible representation of dimension . clearly , the number of lie algebras satisfying this condition increases with .the demonstration of the fundamental theorem of this section is a direct consequence of the classical theorem of burnside , which plays an important role in the theory of rings .given the above definitions we can enunciate the theorem in the following way :  _ burnside s theorem_. let be a monoid of linear transformations in a finite dimensional vector space over an algebraically closed field , that acts irreducibly on in the sense that the only subspaces stabilized by are and . then contains a basis for end over .end is an abbreviated notation for the ring of endomorphisms of over the field . in quantum mechanical systems, the field corresponds to the complex numbers .the demonstration of the burnside s theorem can be found , for instance , in ref .. an immediate consequence of this theorem is the following one which is the basis for connecting different bosonic languages :  _ fundamental theorem _( `` on the equivalence of languages '' ). given two bosonic languages having the same finite dimension of their local hilbert spaces , , the generators of one of them can be written as a polynomial function of the generators of the other language and vice versa .  _proof _ : the proof is a trivial application of burnside s theorem . we need to notice first that since the dimension is the same for both languages , the spaces are also the same ( vector space of dimension over the field of complex numbers ) .let us consider the monoid of transformations generated by multiplying the generators of the first language until the enlarged set becomes closed under the product .the second condition in the definition of a bosonic language states that acts irreducibly on .since the dimension of is finite , burnside s theorem guarantees that admits a basis for end .therefore , any endomorphism in can be written as a linear combination of endomorphisms in .in particular , the generators of the second language can be written in this way because they belong to end . sinceeach element of is a product of generators of the first language , this concludes the demonstration .this theorem establishes an isomorphism between the algebras of endomorphisms associated to each of the two languages .motivated by this observation we can introduce the notion of classes of equivalent bosonic languages .we will say that two bosonic languages belong to the same class if they have the same dimension of their local hilbert spaces .the fundamental theorem establishes the existence of dictionaries connecting languages within the same class . as a consequence, we can use any bosonic language in the class to describe a given physical phenomena . the natural question which emerges from this resultis : what is the most appropriate language in a given class for describing our particular problem ?there is no generic answer to this question .nonetheless , the following two corollaries give an important hint for problems which are invariant under particular transformations , because they relate the notion of language to the generators of symmetry groups ._ corollary i _ : in each class of bosonic languages there is at least one which is the conjunction of a lie algebra and an irreducible representation ( ) , i.e. , the generators of the bosonic language are generators of the lie algebra in the representation ._ proof _ : first , we need to notice that each class is characterized by the dimension of the local hilbert space .let us consider the group .the lie algebra associated to this group is .the generators of are , =i \delta_{{{\bf i}}{{\bf j } } } \epsilon_{\mu \nu \lambda } s_{{\bf i}}^\lambda \ , \;\;\ ; \mu,\nu,\lambda = x , y , z\ ] ] ( and is the totally antisymmetric levi - civita symbol ) , and there is one irreducible representation of dimension for each possible value of  .therefore , the set in the representation fulfills the three requirements for a bosonic language and operates in a local hilbert space of dimension . since is the set of generators of the lie algebra , making and , we have proved the corollary i.  the proof of corollary i shows that any bosonic problem with a local hilbert space of dimension can be described with -spins of magnitude ( representation ) .the matsubara - matsuda ( mm ) transformation is the simplest application of this corollary to .( indeed , one can construct generalized mm transformations for any  . )we introduce now another definition which is motivated by the next corollary .a given bosonic language will be called _ hierarchical _ if any local physical operator can be written as a linear combination of the generators of the language , i.e. , where , and it is the conjunction of a lie algebra and an irreducible representation .  _ corollary ii _ : in each class of bosonic languages there is one which is hierarchical and its generators are the identity and the generators of in the fundamental representation .  _ proof _ : for each class , with dimension of the local hilbert space , we consider the group with .the generators of the lie algebra associated to this group , , are the identity plus the generators of . since the fundamental representation of has dimension , the conjunction of and this representation is one possible language for the class considered . since the dimension of is , which is the dimension of the vector space end , then the generators of also form a basis for end .the first consequence of corollary ii is that the generators of any language can be expressed as a linear combination of generators of a hierarchical language in the same class .again , the most trivial example is given by the class of bosonic languages containing the spin =1/2 lattice .the generators of any language ( like hard - core bosons or any two level system ) in the same class can be written as a linear combination of the identity and the pauli matrices .we will see later that corollary ii is the key to get a hierarchical classification of the possible broken symmetries of a given physical hamiltonian .we consider now two additional examples that illustrate in detail the contents of the fundamental theorem and the subsequent corollaries .the first example corresponds to hard - core bosons with different flavors .since they are hard core only single occupancy is allowed , i.e. , the eigenvalues of are either or ( , and is the number operator for the flavor at the site ) .to distinguish between site and flavor indices we will adopt the following convention : sites are denoted by latin indices while the different flavors ( or orbitals within the same site ) are labeled by greek indices .the minimal set of operators that we can use to generate a bosonic language which is appropriate for hard - core bosons is : with .it can be shown that this set satisfies the three requirements for a bosonic language .the dimension of the local hilbert space for these endomorphisms is .then by varying the total number of flavors we can generate all possible values of .since each class of bosonic languages is characterized by the value of , these hard - core bosons provide an example of a bosonic language in each class .it is clear that the set is not closed under the lie product .therefore , we can not associate a lie algebra to this minimal bosonic language . however , if we increase the number of generators in the set by including bilinear forms of the type , then the new set , with , becomes closed under the lie product } = { [ } \bar{b}^\dagger_{{{\bf i}}\alpha } , \bar{b}^\dagger_{{{\bf j}}\beta } { ] } = 0  \ , \\ { [ } \bar{b}^{\;}_{{{\bf i}}\beta } , \bar{b}^\dagger_{{{\bf j}}\alpha } { ] } = \delta_{{{\bf i}}{{\bf j } } }  ( \delta_{\alpha \beta}- { \bar n}_{{\bf i}}\delta_{\alpha \beta}-\bar{b}^\dagger_{{{\bf i}}\alpha } \bar{b}^{\;}_{{{\bf i}}\beta } ) \ , \\ { [ } \bar{b}^\dagger_{{{\bf i}}\alpha}\bar{b}^{\;}_{{{\bf i}}\beta } ,  \bar{b}^\dagger_{{{\bf j}}\gamma } { ] } = \delta_{{{\bf i}}{{\bf j } } } \delta_{\beta \gamma } \bar{b}^\dagger_{{{\bf i}}\alpha }  \ .\end{cases } \label{conm}\end{aligned}\ ] ] this means that the extended set is now a set of generators for a lie algebra in a particular representation . from the commutation relations ( eq .( [ conm ] ) ) we can conclude that is the direct sum of an algebra , generated by the identity , and an ( ) algebra generated by : .the representation is the fundamental representation of ( ) .therefore , the new language is a hierarchical one . in section [ sec4a ] we give a more detailed description of this particular language .here we only want to emphasize the practical consequences of the fundamental theorem and its corollaries .the first non - trivial observation is that for each system of interacting multiflavored hard - core bosons , there is an equivalent system of interacting -spins in the fundamental representation ( with the minimal non - zero magnitude ) .for we recover the well - known mm transformation .we can see now that this is the generalization to of the mm transformation . with this example we can envision the broad set of applications derived from the fundamental theorem .another consequence of the theorem is that any physical theory for a bosonic system can be formulated in terms of multiflavored hard - core bosons if the dimension of the local hilbert space is finite .the usefulness of this formulation will depend on the particular system as it is illustrated in the next sections .since the second language is hierarchical we can write down any local physical operator ( endomorphism in ) as a linear combination of its generators. therefore , each of these generators will appear in the hamiltonian under consideration with a power not larger than one .in addition , if the hamiltonian has a global symmetry generated by a direct sum of local transformations : , the symmetry will become explicit by writing the hamiltonian in terms of a hierarchical language .the most basic example is the case of mm hard - core bosons ( ) in a lattice , described by a hamiltonian with a kinetic energy term and a nearest - neighbor density - density interaction .the expression for the hamiltonian in terms of the first language defined by the set is where , refers to nearest - neighbors in an otherwise regular -dimensional lattice . since and are not generators of a lie algebra , the eventual global symmetries of remain hidden in this particular language .however , if we _ translate _ to the second -spin language using the dictionary provided by matsubara and matsuda  we can immediately unveil the hidden symmetries of .the well - known expression for in terms of the generators ( i.e. , the equivalent spin hamiltonian ) is the magnetic couplings , and , are related to the original parameters , and , by the relations : and .it is clear from the last version of ( eq .( [ version2 ] ) ) that the original model has a global invariance if , i.e. , it is in the isotropic heisenberg point .the existence of this -symmetric point has a very important consequence for the phase diagram of the bosonic model of eq .( [ version1 ] ) : if there is a charge density wave ( cdw ) instability at that point , the invariance implies that _ there is also _ a bose - einstein condensation and vice versa .the order parameters of both phases are different components of a unique order parameter in the spin language , i.e. , the staggered magnetization of the antiferromagnetic phase ( ) .the -component of the staggered magnetization is mapped onto the cdw order parameter for the bosonic gas , while the transverse component is equivalent to the order parameter for the bose - einstein condensation .only one of these two phases , which are coexisting at the invariant point , is stable when we depart from the symmetric point in any of both directions in parameter space ( bose - einstein condensation if and cdw if ) . in this very simple examplewe can see the advantages of using a hierarchical language ( in this case ) . in the first place, we can immediately recognize the high symmetry points .secondly , we can describe an eventual broken symmetry state at those points in terms of a unified order parameter .if we were to use a non - hierarchical language to describe the same problem , we would find coexistence of more than one phase at the high symmetry points .the order parameters of each of those phases are different components of the unified order parameter that we simply found with the hierarchical language .these ideas are developed in more detail in section [ sec6 ] .here the aim is to give a flavor of the potential applications of our fundamental theorem .  using the concept of transmutation of statistics , to be introduced in the next section, we will extend the notion of classes of bosonic languages to more general classes containing non - bosonic languages ( i.e. , languages for which eq . ( [ bosl ] ) is replaced by a different algebraic condition ) .the most well - known examples of non - bosonic languages are the fermionic ones for which the commutator of eq .( [ bosl ] ) is replaced by an anticommutator .we will see that there are non - local transformations which map non - bosonic languages into bosonic ones . in this way, the simultaneous application of these transformations and the fundamental theorem provides the natural path toward a unification of the different languages used to describe quantum systems .in the previous section we have shown that two different bosonic languages can be connected ( i.e. , there is a dictionary connecting the two languages ) if they have the same dimension of their local hilbert spaces .however , we know that the bosonic languages do not exhaust all the possible languages of quantum mechanics . the best known example is the fermionic language , whose creation and annihilation operators obey a closed set of _ anticommutation _ relations. there are many other examples of non - bosonic languages that we address below .is it possible to connect these non - bosonic languages to the bosonic ones introduced in the previous section ?a positive answer to this question is given in this section by introducing operators which transform commutators into anticommutators and vice versa .these operators have a local ( related to the generalized pauli exclusion principle ( see section [ sec4d ] ) ) and a non - local ( related to the exchange statistics ) components .the local component is derived at the beginning of this section , while the non - local one is introduced in the second part .we will see that an abelian _ anyonic _ statistics can also be achieved by introducing a continuous parameter in these transformations .( non - abelian statistics is beyond the scope of the present manuscript . )the transmutation of statistics together with the fundamental theorem establish the framework needed to classify the classes of equivalent languages .this means that to describe a given physical problem one can use the original language or any other belonging to the same class .  before proceeding, we would like to mention that it seems as if there are two unrelated notions of anyonic statistics in the literature .one tied to quantum mechanics in first quantization in the coordinate representation , and another derived within the framework of quantum field theory . in both cases the original motivation to introduce such particles was basically as an inherent possibility in the kinematics of ( 2 + 1)-dimensional quantum mechanics and clearly the concepts , if correctly implemented , should be equivalent but are not indeed . in this manuscript ,our anyon notion is consistent with the one developed in quantum field theory where the exclusion properties are preserved under _statistical transmutation_. in this way , fermions can be kinematically transformed into hard - core bosons but not into canonical ones .in addition to bosons , the other type of fundamental particles found in nature are the fermions .we have seen that the notion of bosonic languages is closely related to the concept of lie algebras , since for each class of bosonic languages there is at least one language whose elements are generators of a lie algebra .however , the same can not be done for fermions .this can be easily seen by noticing that the third condition for a bosonic language ( see eq . ( [ bosl ] ) ) is not valid in general for fermions .in addition , the main consequence of the third condition for a bosonic language is that the global lie algebra is the direct sum of the local ones associated to each subset .the generalization of these concepts to _ fermionic languages _ can be done by introducing the notion of lie superalgebras ( see for instance ref .the fermionic languages are associated to lie superalgebras in the same way the bosonic languages are associated to lie algebras .therefore , to give a general definition of a fermionic language we should first introduce and explain the notions of grassman algebras , which are associative superalgebras , and lie superalgebras .since this is beyond the scope of the present paper , we will only consider the fermionic language generated by the canonical creation and annihilation operators and other languages obtained by imposing particular local constraints on the canonical fermions .these generators , together with the identity , generate the heisenberg lie superalgebra . in analogy to the bosonic languages ( see eq .( [ bosl ] ) ) , the lie product ( in the superalgebra ) of two elements acting in different sites ( different subsets , ) is zero .thus , instead of having a direct sum of local lie algebras like in the bosonic case , we have a direct sum of local lie superalgebras . in the case of canonical fermionsthe local lie superalgebras are heisenberg superalgebras .we will start by considering multiflavor fermions (  ] is the _ local transmutator _ , and we are assuming a particular ordering for the flavor index . from the expression for is clear that it is straightforward to verify that the -operators satisfy local canonical anticommutation relations ( eq . ( [ canoc ] )when ) .  in this way we have established a mapping between fermions and bosons which are operating locally ( on a given orbital or mode ) .in other words , we have related the subset of local generators of a bosonic language to the subset of local generators of canonical fermions ( ) .so far , we have only transmuted the commutation relations between generators which belong to the same site or subset .for commutation relations of two generators of different sites we need to introduce a non - local operator .jordan and wigner were the first to introduce such an operator in connection to their one - dimensional ( or ) transformation between spins =1/2 and spinless fermions .the so - called kink operator that they introduced is \ , \label{1d}\ ] ] where is the number operator for spinless fermions at the site .it is clear that for multiflavor canonical fermions we only need to replace by , where denotes the flavor .therefore , eq . ( [ rotloc ] )must be replaced by where for a lattice . even though this is a non - local operator, it does not introduce long - range interactions if the model has only short - range terms .in other words , a given hamiltonian can be written in terms of bosons , fermions or anyons , and the interactions remain short range for all the cases .this is a special characteristic of one dimension .the only consequence of changing the statistics of the particles is a change of the short - range interactions in the original basis .therefore , the concept of particle statistics in one dimension becomes irrelevant since any physical system can be described with a bosonic language without changing the short - range character of the interactions .the next step is the generalization of to higher dimensions .this has been done by fradkin and y.r .wang , who considered the generalization of the traditional jw transformation for the two - dimensional ( ) case , and huerta - zanelli and s. wang , who did the same for higher dimensions .we will see that , in the same way we did for the case , these generalizations can be extended to transform canonical fermions into bosons and vice versa .the generalization given by fradkin for the case is \ ,  \label{2d}\ ] ] here , is the angle between the spatial vector and a fixed direction on the lattice , and is defined to be zero ( see fig .[ fig0 ] ) .again for the case of multiflavor canonical fermions we just need to replace by .we comment that the kink ( or string ) operators constitute a particular case of eq .( [ 2d ] ) with when and equals zero otherwise .   in 2 .the dotted line represents a fixed direction on the lattice.,width=279 ]  a general procedure for finding the string operators in dimension has been given by wang in the way we describe below .following the and cases , the expression proposed for the operator is \ , \ ] ] where is a function to be determined by imposing the transmutation of statistics .it can be shown that this is equivalent to the antisymmetric condition &=&-\exp[i\omega({\bf j},{\bf l } ) ] , \\mbox{if } { { \bf l}}\ne { { \bf j}}\nonumber \\\omega({\bf l},{\bf l})&= & 0 . \label{condition}\end{aligned}\ ] ]the simplest solution for a lattice is where is the step or heaviside function .this is the solution found by jordan and wigner ( see eq . ( [ 1d ] ) ) .the solution introduced in ref . for the 2 case corresponds to however , it has been pointed out by wang that this is not the only possible solution since ,\ ] ] with and , also satisfies eqs .( [ condition ] ) .the advantage of this solution is that its generalization to higher dimensions is straightforward ( , and  ] appears in between the product of the kinetic energy - like terms .the field operator is defined by where represents the lattice gradient (  ] . so unless  ] .it is this second aspect , not related to the exclusion properties , that decides whether the particles are bosons , fermions , or anyons .now we will consider another example of transmutation from bosons to anyons . in this case, we will take the multiflavored hard - core bosons defined by the commutation relations eq .( [ bef ] ) .the particles obtained after the transmutation of the hard - core bosons will be called _type i hard - core anyons _ with ,  \nonumber \\k^{\theta}_{\bf j}&=&\exp[i \frac{\theta}{\pi } \sum_{\substack{{\bf l } } } \omega({\bf l},{\bf j } )  \\tilde{n}_{\bf l } ] \ .\end{aligned}\ ] ] like in the previous example , the local commutation relations are preserved ( , ) =  [ \tilde{a}^\dagger_{{{\bf j}}\alpha},\tilde{a}^\dagger_{{{\bf j}}\alpha}]=0   \ , \\ { [ } \tilde{a}^{\;}_{{{\bf j}}\alpha},\tilde{a}^\dagger_{{{\bf j}}\alpha}{]}=1- 2\tilde{n}_{{{\bf j}}\alpha } \ .\end{cases}\end{aligned}\ ] ] in this particular case , since there is a hard - core condition , the operators also satisfy the following local anticommutation relations thus , the local anticommutation relations are also preserved under statistical transmutation .clearly , eqs .( [ lcf ] ) are the local anticommutation relations for canonical fermions .this is not surprising since the multiflavored hard - core bosons defined by eq .( [ bef ] ) can be transmuted into canonical fermions ( see eq .( [ trans ] ) ) . for the commutation relations involving operators with different indices, we have to define an index ordering like in the previous example . for \theta =  [ \tilde{a}^\dagger_{{{\bf i}}\alpha},\tilde{a}^\dagger_{{{\bf j}}\beta } ] _\theta=0   \ , \\ { [ } \tilde{a}^{\dagger}_{{{\bf j}}\beta } , \tilde{a}^{\;}_{{{\bf i}}\alpha}{]}_\theta=0 \ .\end{cases } \label{hca}\end{aligned}\ ] ] corresponds to bosons which are hard - core in each flavor ( see eq .( [ bef ] ) ) , and to canonical fermions , i.e. , and . for any statistical angle one can put up to a single particle per mode and flavor ( i.e , the particles are _ hard - core _ in each flavor and , therefore , satisfy the pauli exclusion principle ) .it is evident in general that the transmutation of the statistics does not change the local commutation relations and therefore the exclusion rule . in a later sectionwe will show how to generalize the exclusion principle to accommodate up to particles per single quantum state .in particular , we will see that the para - particles , which obey this generalized exclusion principle , can be obtained from particles which are partial transmutations of canonical bosons ( to obtain the parabosons ) or fermions ( to obtain the parafermions ) .the last example of anyonic particles corresponds to the transmutation of the hard - core bosons defined by the commutation relations eq .( [ conm ] ). this concept will be useful for the generalized jw transformations that we introduce in the next section ( these particles will be called _ type ii hard - core anyons _ or simply _ jw particles _ ) with ,  \nonumber \\ k^{\theta}_{\bfj}&=&\exp[i \frac{\theta}{\pi } \sum_{\substack{{\bf l } } } \omega({\bf l},{\bf j } )  \\bar{n}_{\bf l } ] \ .\end{aligned}\ ] ] since the local commutation relations are preserved , we have ( , ) =  [ \bar{a}^\dagger_{{{\bf j}}\alpha},\bar{a}^\dagger_{{{\bf j}}\alpha}]=0   \ , \\ { [ } \bar{a}^{\;}_{{{\bf j}}\alpha},\bar{a}^\dagger_{{{\bf j}}\alpha}{]}=1- \bar{n}_{{{\bf j}}\alpha } - \bar{n}_{{{\bf j } } } \ .\end{cases } \label{canoa1}\end{aligned}\ ] ] again , we need to define an index ordering for the deformed commutation relations involving operators with different indices . for \theta =  [ \bar{a}^\dagger_{{{\bf i}}\alpha},\bar{a}^\dagger_{{{\bf j}}\beta } ] _\theta=0   \ , \\ { [ } \bar{a}^{\dagger}_{{{\bf j}}\beta } , \bar{a}^{\;}_{{{\bf i}}\alpha}{]}_\theta=0 \ .\end{cases } \label{hca2}\end{aligned}\ ] ]  notice that , formally , in all cases we could have considered different angles instead of a single one in the expressions for the local , ( e.g. , ) , and non - local , ( e.g. , ) , transmutators , although we do not see how physically relevant this general situation could be .it is important to stress at this point when the exchange statistics property of the particles ( i.e. , the property attached to the non - local part of the transmutator ) becomes relevant in the description of a particular physical system .it is clear that whenever the system hamiltonian does not permute particles on different sites and , then becomes a symmetry and the hamiltonian is invariant under particle exchange statistics .this is the case in many 1 problems ( e.g. , the model of eq .( [ version1 ] ) or the hubbard model ) and in special 2 problems such as the gauge magnet that we expand on in section [ sec7a ] .figure [ fig01 ] summarizes the fundamental notions of languages and dictionaries connecting them , concepts developed in sections [ sec2 ] and [ sec3 ] .we have seen in the previous section that the only condition to establish an isomorphism between two different languages is that the dimension of their local hilbert spaces be equal ( see fig .[ fig01 ] ) .notice , however , that the choice of local hilbert space depends upon the particular decomposition of the global hilbert space .when we work on a lattice , the most natural decomposition is the one where is the hilbert space associated to each site ( or mode ) .this is the decomposition that we have adopted in our previous examples .another possible decomposition could be the one for which corresponds to the hilbert space of a bond ( i.e. , two sites instead of one ) .in general , there is no restriction in the way one can partition .= 1/2 ladder and a fermionic linear chain . in this case ,bonds ( rungs ) are mapped onto sites.,width=325 ]the purpose of this section is to illustrate with examples the algebraic framework developed in the previous three sections . in the first partwe describe different and spin - particle mappings which are a direct consequence of our fundamental theorem . in these cases we are connecting bosonic languages and , therefore , it is not necessary to transmute the statistics .in contrast , in the second part we describe the generalization of the jw transformation where the transmutation of the statistics plays a fundamental role . in this case , we connect the different representations of -spins to constrained fermions .we devote the third part of this section to show how the fractional exclusion statistics algebras emerge from the present formalism in a natural way .we close the section showing that the notion of parastatistics introduced by green is associated to partial transmutations of the canonical bosonic and fermionic languages . in section [ sec7 ] we connect the gauge - field to spin languages through the quantum link model relation . in this wayone envisions formal relations between all the different languages used to describe the properties of quantum systems .in the course of this section we will see that many well - known transformations are particular examples of applications of both the fundamental theorem and the transmutation of statistics .the connection between the schwinger - wigner bosons and the generators of in different irreducible representations results from a simple application of the fundamental theorem . we will consider a system of schwinger - wigner bosons with different flavors (  ] of bosonic operators is mapped onto the corresponding commutator of fermionic operators }=\delta_{{{\bf i}}{{\bf j } } } \delta_{\beta \gamma } \bar{c}^\dagger_{{{\bf i}}\alpha } \ .\ ] ] finally , to complete this correspondence between commutation and anticommutation relations we have to consider the effect of the transmutation on the commutators  ] ) where represents a basis which span the lie algebra isomorphic to (  ] invariant .the elementary parallel transporters are associated with an ordered pair of points in the lattice ( links ) .for instance , the link ( ) corresponds to the straight path from lattice site to .recently , wiese and co - workers have further elaborated work started by horn in 1981 on a new way of non - perturbative regularization of field theories which they named d - theory . in d - theory the euclidean action in the standard wilsonian formulation of a -dimensional lattice field theory is replaced by a hamilton operator  with quantum link operators on a -dimensional lattice , which constitute generators of an algebra acting on a hilbert space ( e.g , a quantum link model can be realized with quantum links that satisfy an algebra ) . with the local generators of gauge transformations ( with ) and the theory is defined through the quantum partition function , where the trace is taken on the hilbert space .  as emphasized in ref . , d - theory is not a new set of field theories but another lattice regularization and quantization of the corresponding classical models .the main important and attractive feature is the use of discrete quantized variables and finite hilbert spaces .however , it turns out to be necessary to formulate the theory with an additional euclidean dimension ( i.e. , in ) . in this regard dimensional reductionis a fundamental component to relate quantum link models to ordinary gauge theories .the dimensional reduction hypothesis relies on the existence of a massless phase in , assumption that must be verified on a case - by - case basis .the existence of a quantum link model connecting lattice gauge theories to spin ( or other algebraic ) theories opens the possibility of formal connections between gauge theories of high - energy physics and strongly correlated problems of condensed matter .indeed , since there is a connection between the `` gauge world '' and the `` spin world '' and we exhausted the connection between the `` spin world '' and the `` particle world '' the consequence is that one can find isomorphisms between the `` gauge world '' and the `` particle world . ''in this way , for instance , one may look for the exact equivalent of confinement ( in gauge theories ) in the particle ( condensed - matter ) language .even though some of these ideas were speculated in the literature there was no formal rigorous relation established : _ the connection is our fundamental theorem_.   for pedagogical purposes , it is convenient to illustrate the connection between gauge and condensed matter theories by considering the simplest gauge theory , usually called _ gauge magnet_. gauge magnets are gauge - invariant generalizations of the heisenberg models .the corresponding classical model has a parallel transporter associated to each link and the action  ] is imposed . in this waywe see how the formal connections between condensed matter and lattice gauge theories can be exploited to predict new physical behaviors in one or the other field .= 1/2 gauge magnet in the particle language .quantum fluctuations ( qf ) correspond to the horizontal motion of vertical pairs in the active ( black ) plaquettes.,width=309 ]a new challenge in information theory and computer science has recently emerged as the result of applying the fundamental laws of quantum mechanics and using the quantum effects to advantage .this new set of ideas comprise what is known as `` theory of quantum computation and quantum information '' and has as a major objective to process information in a way that exceeds the capabilities of classical information . the device that performs the manipulation of informationis named quantum computer and the standard unit of information is the _ qubit _( i.e. , a two - level system ) .the close relationship between information processing and the physical phenomena leading to it is perhaps the most remarkable aspect of this new paradigm .since information can be represented in many different physical forms , and easily converted from one form to another without changing its meaning , quantum information represents a new abstract archetype for information processing independent of the precise implementation of the quantum computer , only requiring at least one physical representation to be useful .a key fundamental concept in information theory is the realization that a model of computation is intimately connected to a physical system through a closed operator algebra .in other words , each physical system is associated to a certain language ( e.g. , spin =1/2 ) and thus to an algebra realizing it ( e.g. , pauli algebra ) , and that particular algebra may become a possible model of computation .an immediate consequence is that an arbitrary physical system can be simulated by another physical system ( e.g. , a quantum computer ) whenever there exists an isomorphic mapping between the different operator algebras representing the systems .a very simple example is provided in refs . and , where it is shown how to simulate a 1 impurity anderson model using a quantum computer based on an array of spins =1/2 .another resource of quantum information that can help us understand complex phenomena in strongly correlated matter is the notion and measures of _entanglement_.  it is very clear the power that our fundamental theorem has by providing the formal connections ( isomorphisms ) between the different languages of nature .therefore , the implications for quantum information and computation are rather obvious , namely that one can identify quantum resources and define convenient models of computation , or imitate an arbitrary quantum phenomena with a given quantum computer given the appropriate dictionaries to translate nature s language to the _ machine language_. in this way , one can recognize three complementary areas where the concepts of language and dictionaries developed in the present manuscript are of particular relevance , areas we will expand on in the following .a model of computation consists of an algebra of operators , a set of controllable hamiltonians and measurable observables , and an initial state of the physical system .the set of controllable hamiltonians must be universal in the sense that any unitary operation can be performed with such a set .the _ standard model _ of computation is a particular example which uses the pauli operator algebra and the qubit as basic unit .another possible model is the fermion model which is isomorphically related to the standard one through the jw mapping and , thus , is equivalent .the main point is that the choice of model depends upon the nature of the available physical resources and their quantum control .for example , in a liquid nmr quantum computer the nuclear spins of the molecules ( =1/2 ) are the units which can be controlled and the standard model is the appropriate one .  the control of quantum mechanical systems is hampered by quantum noise and decoherence and , therefore , identifying static quantum resources for information processing is a great challenge . given a physical information processing device there will be a language which will the most natural one for the elementary static resources of the device .for instance , if we could control atoms at the quantum level it would be natural to consider a hard - core boson model of computation .the importance of our fundamental theorem is that once one identifies the best quantum resources it allows us to build the model of computation accordingly .physical phenomena can be simulated or _ imitated _ by a quantum network with the help of a quantum computer .imitation is realized through a quantum algorithm which consists of a quantum network with a means to repeat blocks of instructions .a quantum network is defined by a sequence of universal gates ( unitary operations ) , applied to the system for the purpose of information processing , and measurements in a fixed temporal order .the measurement operation is mostly needed to classically access information about the state of the system .every matrix which represents a reversible operation on quantum states can be expressed as a product of the one and two - qubit gates , and the minimum set needed to represent any such matrices is called a universal set of gates .  when trying to simulate a problem using quantum information processing , an important issue is to determine how many physical resources are needed for the solution .the main resources are _ quantum space _ , the number of qubits needed , and _ quantum time _ , the number of quantum gates required .the accounting of algorithmic resources forms the foundations of quantum complexity theory .one of the objectives in quantum information theory is to accomplish imitation efficiently , i.e , with polynomial complexity , and the hope is that quantum imitation is more efficient ( i.e. , needs less resources ) than classical imitation .there are examples that support such hope ( e.g. , fermion simulations with polynomially bounded statistical errors ) , although there is no general proof that indicates the superiority of quantum over classical imitations , regarding efficiency .indeed , there is , so far , no efficient quantum algorithm that can determine the ground state ( or , in general , the spectrum ) of a given hermitian operator , despite occasional claims .it is known that the ability to resolve this question leads to efficient algorithms for np - complete problems like the traveling salesman conundrum .a very important observation , in connection with the notion of efficiency , is a corollary of our fundamental theorem : given two languages , the generators of one of them can be written as a polynomial function , with _polynomial complexity in the number of modes or resources _ , of the generators of the other and vice versa .this result implies that the important algorithmic step of _ translation _ from the language of the system to be imitated to the _ machine language _ does not change the complexity of the quantum space and time .certainly , a general purpose quantum computer is not the only device that allows simulation of physical phenomena in nature ( with its many languages ) .imitation can also be achieved in a conceptually different manner using a quantum simulator .the main distinction is the lack of universality of the latter .an example of a quantum simulator is an optical lattice which is specifically designed to imitate a given physical hamiltonian and where there is limited quantum control .the possibility of control and tunability of the interactions of the elementary constituents offers the potential to design new states of matter .this is of particular relevance in strongly correlated matter where these quantum simulators furnish the benchmark to test theories and approximations .again , the importance of the languages and dictionaries developed in this manuscript is clear and concrete .entanglement , a word that schrdinger coined to distinguish quantum from classical mechanics , is that _ bizarre _ feature of composite quantum systems that led to so much controversy in the past ( like quantum non - locality in the einstein - podolsky - rosen gedanken experiment ) . essentially , _entanglement _ is a quantum property whereby a pure state of a composite quantum system may cease to be determined by the states of its constituent subsystems .entangled pure states are those that have mixed subsystem states .an independent concept is the notion of _ separability _ which refers to the property of a quantum state of a composite system of being able to be written as a direct product of states of its component subsystems .the theory of entanglement , and its generalizations , is currently under development . in ref . entanglement is viewed as an _ observer - dependent _ concept , whose properties are determined by the expectations of a distinguished subspace of observables of the system of interest ( experimental access ) , without reference to a preferred subsystem decomposition .the standard notion of entanglement is recovered when these means are limited to local observables acting on subsystems .a tremendous effort is put in trying to understand the properties of entanglement that can be used as a resource in quantum information .  in a sense , that must be defined more precisely , the notion of entanglement is a relative of the notion of quantum correlations and a very relevant question is whether one can construct useful measures of entanglement to understand the emergence of complex phenomena in strongly correlated matter .a quantum phase transition involves a qualitative change in the correlations of the ground state of the system as a result of tuning parameters of its hamiltonian . in some casesan order parameter is associated to the transition , in others a topological order .it is intuitively expected that this change is also associated to a change in the nature of entanglement .therefore , quantifying and classifying entanglement is very important to characterize a quantum phase transition : can measures of entanglement distinguish between broken and non - broken symmetry phase transitions ?we have introduced an algebraic framework for interacting quantum systems to study complex phenomena characterized by the coexistence and competition of various ordered states . we argued that symmetry , and topology , are key guiding principles behind such complex emergent behavior .emphasis has been made in developing a systematic mathematical structure that allows one to attack these problems within a single unifying approach .the core result of the paper , from which all other results follow , is the proof of a fundamental theorem that permits to connect the various operator languages used in the description of the properties of physical systems .this theorem together with the notion of transmutation of statistics provide the tools necessary to unifying the quantum description of matter . to formalize this unification we needed to rigorously define the concepts of language and dictionary : to model a particular physical phenomena we commonly identify the main degrees of freedom of the problem and associate to them certain operators .one can furnish to the resulting set of operators ( that we call language ) with an algebraic structure and ask whether two different languages have something in common .the fundamental theorem tells us that two languages can be connected whenever the dimension of their local hilbert spaces are equal . we expanded the notion of local hilbert space to embrace different hilbert space decompositions ( we saw , for instance , how to map the hilbert space of a bond to a site ) .the resulting one - to - one language mappings we named dictionaries ( a traditional example of which is the jordan - wigner mapping ) . in the course of the presentation we showed , through example , many different dictionaries relating diverse operator languages . in this way we defined universality of behavior as an equivalence relation between seemingly different physical phenomena which share exactly the same underlying mathematical structure as a result of one - to - one language mappings ( for example , the spin nematic order and bose - einstein condensation of flavored hard - core bosons ) . out of the many languages one can use to describe a given physical problemthere is a class , we named hierarchical language , which has the advantage that any local operator can be expressed as a linear combination of its generators . in this way, hierarchical languages provide the tools necessary to classify order parameters .a new , formally developed , notion is the idea of _ emergent symmetry _, i.e. , the fact that new symmetries not realized in the hamiltonian describing the system can emerge at low energies . from a broader perspective ,an emergent symmetry is a _ non - local _ transformation which commutes with the system hamiltonian and becomes _ local _ when restricted to a low - energy subspace of the hilbert space . in some instances like the quasi - exactly solvable problems , to which the 1 - model belongs , or the family of spin hamiltonians for which the ground state is a product of spin singlets ( e.g. , the majumdar - ghosh model ), the low - energy effective theory can be derived in an exact way and the emergent symmetry is exact . in other cases ,the emergent symmetry is approximate , however , it still provides a guiding principle to identify the nature of the ground state and its low - energy excitations . in the same way that rigidityis related to a broken symmetry phenomenon , localization ( in real or any other space ) is connected to the emergence of a gauge symmetry .indeed , the metal - insulator transition may be interpreted as a discontinous change in the group of emergent symmetries characterizing the two phases .there are several reasons for our algebraic framework to constitute a powerful method to study complex phenomena in interacting quantum systems .most importantly : to connect seemingly unrelated physical phenomena ( e.g. , high- or heavy fermions and quantum spin theories ) ; identify the general symmetry principles behind complex phase diagrams ; unveil hidden symmetries ( and associated order parameters ) to explore new states of matter ; obtain exact solutions of relevant physical models that display complex ordering at certain points in hamiltonian space ; and find new approximations which do not favor any of the competing interactions .the power of the present approach is reflected in the unlimited number of potential applications where they could be used , and ranges from condensed matter and statistical mechanics to lattice gauge theories and quantum information and computation .we hope and envision this formalism will find future applications in contexts different than the ones we have developed here .commutators ( anticommutators ) will be represented by square ( curly ) brackets , i.e. , =ab - ba ] .we may also use \pm = ab \pm ba ] ) is defined on . for the sake of clarity and with no loss of generality , we will always assume that the single - particle basis is orthonormal , i.e. , , but , of course , this condition can be relaxed .for a -dimensional lattice with sites , the operator is defined in terms of a kronecker product as where is the unit matrix , , and is a spin- operator .thus admits a matrix representation of dimension .the theory of zhang constitutes a particular example of a unified ( high - symmetry ) order parameter used to describe the coexistence of more than one phase ( in this case superconductivity and antiferromagnetism ) .however , contrary to the philosophy of this paper , this description relies on the assumption of an symmetry of the low - energy effective theory , assumption whose validity is unclear . according to zhang , this high - symmetry effective theory can be obtained by applying the renormalization group .     if the system has long - range interactions the dimension of the lattice becomes irrelevant , since any -dimensional system can be described in a lower dimensional lattice at the cost of including long - range interactions .g. ortiz and c.d .batista in _ recent progress in many - body theories _ , eds .raymond f bishop , tobias brandes , klaus a gernoth , niels r walet , and yang xian ( world scientific , singapore , 2002 ) , p. 425 ; int .j. of modern phys .b 17 , 5413 ( 2003 ) .suppose that we want to establish a particular ordering for two - flavored particles ( ) in a finite 2 lattice . a possible ordering from to an ordered set of integersi could be where for .s. kamefuchi and y. takahashi , nucl . phys .* 36 * , 177 ( 1962 ) ; c. ryan and e.c.g .sudarshan , _ ibid _ * 47 * , 207 ( 1963 ) ; k. kademova , nucl . phys .b * 15 * , 350 ( 1970 ) ; a.j . bracken and h.s .green , nuovo cim .* 9 * , 349 ( 1972 ) .                other representations with different particle statistics are possible; for example , we could have used a _hard - core _fermion representation .in such a case , the resulting hamiltonian would be similar in form to eq .( [ hamiltjb ] ) with a kinetic energy term modified to include a non - local gauge field .after friedrich august kekul , a german chemist , who solely on intuition proposed the known resonant structure of benzene in 1872 , before quantum mechanics was born .it was linus pauling who formally developed the resonance theory in 1931 .

Abstract: we present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter . we start by showing how to connect different ( spin - particle - gauge ) _ languages _ by means of exact mappings ( isomorphisms ) that we name _ dictionaries _ and prove a fundamental theorem establishing when two arbitrary languages can be connected . these mappings serve to unravel symmetries which are hidden in one representation but become manifest in another . in addition , we establish a formal link between seemingly unrelated physical phenomena by changing the language of our model description . this link leads to the idea of _ universality _ or equivalence . moreover , we introduce the novel concept of _ emergent symmetry _ as another symmetry guiding principle . by introducing the notion of _ hierarchical languages _ , we determine the quantum phase diagram of lattice models ( previously unsolved ) and unveil hidden order parameters to explore new states of matter . hierarchical languages also constitute an essential tool to provide a unified description of phases which compete and coexist . overall , our framework provides a simple and systematic methodology to predict and discover new kinds of orders . another aspect exploited by the present formalism is the relation between condensed matter and lattice gauge theories through quantum link models . we conclude discussing applications of these dictionaries to the area of quantum information and computation with emphasis in building new models of computation and quantum programming languages .


one of the most prevalent methods for testing the fatigue properties of a material is to construct a constant amplitude strain - life curve . in this method , a sampleis strained cyclically between two strain levels until failure .this test is performed on a number of identical samples at different strain magnitudes , and the results are plotted as a strain - life curve .this `` '' curve can be fitted to the basquin - manson - coffin , ( bmc ) equation , where is half of the strain range , is the elastic strain , is the plastic strain , is the fatigue strength coefficient , is the elastic modulus , is the fatigue strength exponent , is the fatigue ductility coefficient , is the fatigue ductility exponent , and is the number of reversals until failure . the low cycle fatigue regime is considered a product of plastic deformation , and the high cycle portion is related primarily to elastic deformation .this approach gives a reasonably accurate prediction of sample life at constant amplitude cyclic strains .however , parts in service are rarely subjected to idealized constant amplitude cyclic strains and instead undergo variable amplitude loading . the palmgren - miner ( pm )rule is a widely used approach for predicting part lifetime under variable amplitude loading .this method hypothesizes that the damage caused by each stress state in a variable amplitude load history is a function of the number of times that the particular cyclic stress state occurs and the number of cycles it would take for the sample to fail from a constant amplitude history at that stress state .  mathematically the palmgren - miner rule is written , where the sum is taken over all stress states with denoting a particular state , is the number of cycles at the stress state , is the number of cycles to failure if the sample is cycled under constant amplitude loading at the stress state , and is a constant .based on the assumptions made by the model , should be ; experimentally it is found to range between and .this variability is evidence of the failure of the palmgren - miner rule to accurately predict fatigue lifetimes .although it is well known that the palmgren - miner rule is inaccurate , its conceptual simplicity and the minimal amount of data necessary for implementation makes it a popular method for estimating fatigue life .improving upon the palmgren - miner rule has been a major focus of researchers studying variable amplitude fatigue .a good review of cumulative damage and life prediction theories through the end of the last century is presented by fatemi and yang . . in recent years modelshave been developed based on a variety of techniques that couple theories of fracture mechanics and empirical observations . other methods take into account the residual stresses caused by the plasticity of the material at the crack tip , and crack tip closing phenomena . although these modeling methods are more accurate than the palmgren - miner rule , they require substantially more experimental data to fit the necessary parameters .  in this papera new method , free of fitting parameters , is demonstrated for estimating strain life under variable amplitude loading .it is unique from other models in that the only data used for input is the constant amplitude strain - life curve and the cyclic stress - strain response . using this model it is possible to accurately predict the variable amplitude strain life of specimens using a relatively small amount of experimental data that can easily be generated . in the section following this introduction the analytical details of the modelare presented .next the experimental procedure and model implementation are explained . in the results and discussion section the measured and predicted strain lifetimesare presented and compared .the theoretical lifetimes predicted from this model are compared to lifetime predictions presented in the literature . in the final sectionthe paper is succinctly summarized .in this model the total damage , , is the sum of the damage of all reversals , , ranging from to , in this definition is the normalized damage caused by the reversal and failure occurs when .the damage caused by each reversal is determined using the well - known constant amplitude strain - life relation . using this approach ,the strain history is incorporated as damage accumulation . the damage accrued in each stepis calculated using a relatively simple algorithm and constant amplitude strain - life data .following examples from ref . , the damage during fatigue is assumed to be due to a single critical crack propagating across the width of the specimen . a good description of the crack growth per reversal , for a constant amplitude strain , is given by the hyperbolic sine function .for a crack size , , the rate of crack tip advance , after reversals , can be written where is the total number of reversals to failure for the given strain amplitude , and is a scaling factor , which will be discussed in more detail later . eq .( [ dadn ] ) is a natural expression of damage per reversal , and has the same functional form as has been used to describe changes in crack growth rates previously . it accurately reflects the phenomena in that during the initial stages of damage the rate of crack propagation is low and as damage is accrued the rate of crack growth increases .other expressions for the rate of crack tip advance , which are phenomenologically similar , are presented elsewhere and will not be discussed here .  using eq .( [ dadn ] ) for damage , the normalized damage due to the reversal is expressed , where is the number of reversals required to achieve the accumulated damage , , were the damage due to a constant amplitude strain range , .the number of reversals to failure at this constant strain range is .the denominator normalizes the damage per reversal such that at failure the total damage is 1 .the current state of damage , , is known and is expressed as the sum of incremental damage , , from eq .( [ di ] ) , where is the number of damaging reversals to cause the damage at the constant amplitude strain .the summations in eqs .( [ di ] ) and ( [ dt ] ) can be approximated as integrals , which allows eq .( [ dt ] ) to be solved to find   the general approach for calculating a specimen life under variable amplitude loading goes as follows .begin by determining the initial total damage .this is typically near zero , if the part begins pristine and the strain - life curve is well defined at all strains of interest .a case for non - zero starting damage will be discussed later in section [ modelimplementation ] .the strain range for the first tensile reversal is , and is calculated using eq .( [ di ] ) and the bmc relation , eq .( [ bmc ] ) , which is fit to constant amplitude strain life data .the value of is added to .the next tensile strain is .the lifetime at this strain range along with , are used to calculate from eq .( [ nt ] ) , which is used in eq .( [ di ] ) to calculate , which is again added to .this process continues until , at which point failure is predicted .an algorithm for implementing this model is demonstrated in section [ modelimplementation ] .published experimental data are used to validate the model . pereira et al . tested p355nl1 steel and compared their experimental results to an effective strain damage model based on the work of duquesnay . colin and fatemi published experimental data for 304l stainless steel and 7075 t6 aluminum .  the load histories used in the calculations were recreated from the descriptions detailed by pereira et al . , and colin and fatemi . the p355nl1 specimens were subjected to a variety of load blocks , including a high to low scheme , a low to high scheme , a low to high to low scheme , and random loading , examples of which are shown in figs .[ fig1 ] , [ fig2 ] , [ fig3 ] , and [ fig4 ] .the 304l stainless steel fatigue - life results were prepared for periodic , fully - reversed overloads , shown in fig .[ fig5 ] , and random loading .the 7075 t6 aluminum samples were subjected to random loading . unlike the shaped loading blocks , the random load history was not reproduced from the literature , but instead a strain history file was created using a random number generator , filtering the random number stream to ensure that each iteration reversed the strain . a representative sample of the random reversal data is shown in fig . [ fig4 ]the model described in section [ model ] above was implemented using the algorithm detailed in this section and applied to the strain history data discussed in section [ experiment ] .to begin , the first two strains from the strain history were converted to stresses using the ramberg - osgood stress - strain relationship , where and were fit from the cyclic stress - strain behavior .the morrow mean stress correction , where is the mean stress , was used to calculate .the value of was used with to calculate from eq .( [ nt ] ) .finally , the damage from this reversal , , was determined from eq .( [ di ] ) .the damage was added to the total damage , .if the total damage was greater than , then the specimen was deemed to have failed due to this reversal , otherwise the procedure was continued using the new value of and the next strain taken from the strain history .this was repeated until failure occurs , when .the constant , in eq . ( [ dadn ] ) , appropriately scales the incremental damage . for this implementationit was selected to be , where was from the bmc equation and was taken as . in this way , scales with the applied strains and the incremental damage , , has the correct functional relationship to the strain amplitude .  the integrated curve , which expresses total damage as a function of the number of cycles , has a general shape that is well known . careful inspection of experimental data of damage as a function of number of cycles reveals that a smooth well fitting curve does not always intercept the damage axis at . in many experiments, damage has been observed to accumulate rapidly to around 5 to 10% early in the specimen s life , and then slow to the crack propagation model that is well known . to account for the rapid damage accumulation that occurs during the initial cycling , the starting damage was assigned to be for all of the variable amplitude lifetime prediction data presented here .a comparison of the published experimental data and the predictions of this work can be seen in figs .[ fig6][fig10 ] . in figs . [ fig6 ] and [ fig7 ] the experimental data for p355nl1 steel , from pereira et al . , is shown for a maximum strain of 1.05% and 2.10% for the loading blocks discussed in section [ experiment ] above . the experimental results are compared to those predicted from the present model in addition to the model of pereira and duquesnay .the current model is in good agreement with both the experimental and theoretical results from peirera et al .the experimental results from colin and fatemi for 304l stainless steel are shown in figs .[ fig8 ] and [ fig9 ] for the loading blocks discussed above at various strain ranges . the results for 7075 t6 aluminum are shown in fig .[ fig10 ] .the accuracy of the present model is compared to the predictions of the palmgren - miner rule both with and without the smith - watson - topper correction .the current model is again in good agreement with the experimental results .it is worth noting that the model presented here agrees with the experimental data both when the palmgren - miner results are non - conservative by an order of 2 or 3 and when they agree well with experiment .this is evidence that the results form the current work are more profound than a simple lifetime reduction from the palmgren - miner rule .the strength of the present model is due to the natural inclusion of the strain history when determining the inflicted damage caused by a strain reversal .both the effect of the immediately preceding strain and the effect of the relative age of a specimen are included . to calculate a strain and the immediately preceding strainmust be known to determine the applied strain range and the mean stress correction .the damage inflicted by a particular strain reversal depends not only on the amplitude , but also on the total state of damage at the instant of the reversal .take the example of a periodic , fully - reversed , overload .if one considers the integrated curve for a constant amplitude cyclic strain , applying a fully reversed overload would advance the position of the subsequent constant amplitude reversals on this curve substantially . from the shape of the curve, it is apparent that an overload early in the specimen s life that increments the damage will have a more substantial impact on the specimen s total life , compared to an overload applied later .the more recent models , such as those of , include residual strains and crack - tip plasticity or crack tip closure phenomena .these models account for the cumulative damage through localized plasticity near the crack tip , or a change in the effective stresses due to crack tip closure .they are more accurate than the palmgren - miner model and other simple models because they account for the ordering of the applied strains and localized damage near the crack tip ; however , they require substantial experimental support . by comparison ,the present model only includes materials information from the cyclic stress - strain curve and the constant amplitude strain life curve .it should be noted that for some analysis involving extreme load cases , a more sophisticated model than the one presented here may be necessary . in particular , cases where residual plasticity will play a dominant role in crack growth rates through strain hardening and residual stresses will likely require a model that explicitly deals with plasticity near the crack tip .one of the greatest engineering challenges of the last years is that of predicting the strain - life relationships of mechanical components undergoing variable amplitude loading . in spite of extensive studies ,no conclusive model has been determined .although many useful models have been developed , many require cumbersome amounts of experimental data . herewe report a new method , free of fitting parameters , for making an accurate variable amplitude strain - life prediction using basic constant amplitude fatigue data .the method is validated using data from experiments performed on p355nl1 steel , 304l stainless steel , and 7075 t6 aluminum . the present model fits the experimental data well for a variety of load spectra and materials and the algorithm is simple to implement .the authors gratefully acknowledge funding from john deere & company .22 natexlab#1#1[2]#2 , , , , . , , , ., , . , , ( ) ., , , ( ) . , , , , ( ) ., , , ( ) . , , , , , , ( ) . , , , ( ) ., , , , ( ) . , , , , , , ( ) . , , , , ( ) ., , , , , , ( ) . , , , ( ) ., , , , ( ) . , , , ( ) ., , , ( ) ., , , , , ., , , , , ( ) . , , , ( ) .

Abstract: a new phenomenological technique for using constant amplitude loading data to predict fatigue life from a variable amplitude strain history is presented . a critical feature of this reversal - by - reversal model is that the damage accumulation is inherently non - linear . the damage for a reversal in the variable amplitude loading history is predicted by approximating that the accumulated damage comes from a constant amplitude loading that has the strain range of the particular variable amplitude reversal . a key feature of this approach is that overloads at the beginning of the strain history have a more substantial impact on the total lifetime than overloads applied toward the end of the cycle life . this technique effectively incorporates the strain history in the damage prediction and has the advantage over other methods in that there are no fitting parameters that require substantial experimental data . the model presented here is validated using experimental variable amplitude fatigue data for three different metals .   fatigue , variable amplitude , model , damage accumulation , prediction , random loading